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SAGE

Creating a viable free open source alternative to
Maple, Mathematica, and Matlab

William Stein, Associate Professor, University of Washington

Poll

How many of you use mathematics software such as Mathematica, Maple, Matlab, Magma PARI, etc.?

History

I started Sage at Harvard in January 2005.
Existing math software not good enough (free or commercial).
Sage-1.0 released February 2006 at Sage Days 1 (San Diego).
Sage Days Workshops 1, 2, ..., 11, at UCLA, UW, Cambridge, Bristol, Austin, France, San Diego, Seattle, etc.
Sage won first prize in Trophees du Libre (November 2007)
Funding from UW, NSF, DoD, Microsoft, Google, Sun, private donors, etc.

Microsoft's Interest in Sage

Support research in cryptography, combinatorics, graph theory, linear algebra, signal processing, etc.

Improve the viability of using Windows for higher performance computing (HPC); Sage is mathematical software that could run on a large cluster with no license fee issues.

Sage is 100% Free

Active user community; 964 members of the sage-support mailing list.

Free webapp -- sagenb.org -- has about 3000 users (and there are other servers at universities around the world..)

sagemath.org

sagenb.org

Advantages of Sage

Python -- general purpose language; huge user base compared to Matlab, Mathematica, Maple and Magma
Cython -- blazingly fast compiled code
Integrated web servers -- (twisted, etc.)
General purpose programming language (not some ad hoc math language)
Free and Open source

Drawbacks of Sage

Maple, Mathematica, and Matlab all have excellent native Microsoft Windows support; Sage doesn't yet.
Sage is new and rapidly growing
Can't be bought -- Sage can't be incorporated into any closed source products

So What is Sage?

Well over 300,000 lines of new Python/Cython code
A Distribution of mathematical software (over 60 third-party packages); builds from source without dependency (over 5 million lines of code)
Nearly 150 developers: developer map
Exact and numerical linear algebra, optimization (numpy, scipy, R, and gsl all included)
Group theory, number theory, combinatorics, graph theory
Symbolic calculus
Coding theory, cryptography and cryptanalysis
2d and 3d plotting
Statistics (Sage includes R)
Overall range of functionality rivals that of Maple, Matlab, and Mathematica

Reference Manual (over 3000 pages, all new)

PART 2 -- Examples

The first example from the Python tutorial:

the_world_is_flat = 1
if the_world_is_flat:
   print "Be careful not to fall off!"

Be careful not to fall off!

Symbolic expressions:

x, y = var('x,y')
type(x)

<class 'sage.calculus.calculus.SymbolicVariable'>

a = 1 + sqrt(2) + pi + 2/3 + x^y
a

x^y + pi + sqrt(2) + 5/3

show(expand(a^3))

<html><div class="math">{x}^{{3 y}}  + {{3 \pi} {x}^{{2 y}} } + {{3 \sqrt{ 2 }} {x}^{{2 y}} } + {5 {x}^{{2 y}} } + {{3 {\pi}^{2} } {x}^{y} } + {{{6 \sqrt{ 2 }} \pi} {x}^{y} } + {{10 \pi} {x}^{y} } + {{10 \sqrt{ 2 }} {x}^{y} } + \frac{{43 {x}^{y} }}{3} + {\pi}^{3}  + {{3 \sqrt{ 2 }} {\pi}^{2} } + {5 {\pi}^{2} } + {{10 \sqrt{ 2 }} \pi} + \frac{{43 \pi}}{3} + \frac{{31 \sqrt{ 2

3} + \frac{395}{27} }}}

Solve equations

var('a,b,x')
show(solve(x^3 + a*x^2 + b == 0, x)[0])

<html><div class="math">x  =  {\left( \frac{{-\sqrt{ 3 } i}}{2} - \frac{1}{2} \right) {\left( \frac{\sqrt{ {b \left( {27 b} + {4 {a}^{3} } \right)} }}{{6 \sqrt{ 3

- \frac{{27 b} + {2 {a}^{3} }}{54} \right)}^{\frac{1}{3}} } + \frac{{\left( \frac{{\sqrt{ 3 } i}}{2} - \frac{1}{2} \right) {a}^{2} }}{{9 {\left( \frac{\sqrt{ {b \left( {27 b} + {4 {a}^{3} } \right)} }}{{6 \sqrt{ 3 }}} - \frac{{27 b} + {2 {a}^{3} }}{54} \right)}^{\frac{1}{3}} }} - \frac{a}{3} }}}

Example: A Huge Integer Determinant

a = random_matrix(ZZ,200,x=-2^127,y=2^127)
time d = a.determinant(); d

2416497820361370388123383955577577680723194259147094503536041541766672185383857614838374290624633492056793610736341426051504453095450214000475427591337149255091690791667690160169983484118651032962983758756214409647615725340948969113773979039363120287051188777437815000911540924059899685201457361771138762543141823558489142384037722952577860809629325881917066027420667363408753295640031399805643322536667324120549268506951783272715680032274950089815645394539639869852012451346690257636397456699016366359469719556276320998724322776826629537434956023026947755858894338669400506836939977907617680233942085676212238994758698441065159782885105905749605914804070935581547459158330386266992785592045616359018138111038892809036287714263465898041267723002338900927589680348503641507296769939411081538390876749291824550411255729123800520415314833099050344967085775718682807363828612203750773050146601742251195892218864983959063242213161558748629428566924435167168331002469780564522560891888203665428763674235951785782376412630227224081048260734997047670178752284673560294196218464506582142144901179152377329515089155496858623890798108433350281344322212839915592607946785457703425249733558972474402196830642355057560564137811357646221394591721190298641319463495530670448304243116467258460321862224460000202349343841713446647481406865596711676419895802211753653728309682519669281567424648028579462362425064215095811694336743031392030525904637902392032037720648623999156036746954408947557041757263219722477302060817248012573380453784066486141510632678783712945489481510363291224579959353441372680247191800986052074624207924680525827630635864626827619301075312000210211273833474173373637536998971829060992173795132468662266899229906868711654047265114268106972833788295757900000486640257316447645804775094511634495073709386911686013832952016177002857442684211142397763514475288714011381693504876249755051022002389557628426544432511893270568864481581118028336689533500838591845874788817111806726777368334663759540768904122304137554506462940282803152529784695365946558925736068483485357833490016245554039618965905783330160163202604530388461788529046447438472604671033312017138030460745698797218042638232625328983707139109172824650228065132287052151098919018418634427420122977641949960842295182881673740552528823381698038338131783638412994633357855298843466554179209538273152110003687969802942520179777568929580144665057674198527075287678630608832646049837271613640664981999274695356826658984285307866038418668960100627284732309950879586909726237698304912065319961022002165296025438054068704036096121479441105522394126105101348346525479236380329689332959814271061522050892179170770274791629001667012806843181596899817517480512217033777735466606027739604673656391401215311835873699225186977310845414096112882777270017646167046614773997568021479496271934254039814649543365504895668506500539869115596580997874797244645583100760131499594532060100363674870964130401812962490970955808278080213844206477636255917164806795654539339915199592891595438524629484802715804846049136125724957098536383656461603463733359365786258507522063290562270313557335881594624220035092980392860499947540823668581111914246689548746051312347325778512123468637203760949038416353903658325724942077373382133642192931605175408645786873621161673862071442298541289705523622359240353874158367459833180146408008127092995108663404501971123354889752946215340682671715359806840501475253795596617097545325742769841927505786423133937174167038413823423397753343196907537135601341052239056045787763453805157585808174750930070565665223937072595022396349281683894183623339887113051321093025968606970721006195025203978568543583706080453120216130966234686699269607688314360112247900835443559981885255290629736484790072605960376285571749103395369114765501475684587542920458554062170911376144537705415749145142652038114205133595227870124188214066153387638585440204597500106628273734146227441609943129610039491637503442724812833652769893313303779499200365625655073870271705896679221391498908366816954705733974541805350593401179622188798473027969545878884744610620695712655639567427862111922423847471025405586861712369650285037828101555507647096685846807593729035075803153884193371503770109009079472349017005627512905575705211084226620756974861482985691245034038054089166580796510590002763146180099282899157215792863523959518335093145395168179152969080796150631176422701034870853098668644856825801588273989065155568601936746061669545923428479085784833115339987165212916847640531650072646649736661574002969843972942495876825390837069969196161449598530776115816730801455610959801569433144986048277631160519391653832521129867595398707705927006428830540894916062458174575543427892704191936729398051339656109320471249448466004375972089131079706231101451947013703288559696275434039897966795132699788372405942953839893534588875065406911472391030482503918520719601927363428369592850792668565724931301636311935017438215132177539163105701040595512626717963093902825060972688637223404965618372490442876585205819165016026133996458455054271612749515364666671495882541830055707889501436816407256799887499654264309426200135691904060967018146272512670288026532367905675550819350825068659168830066417415979004674579443219469620092196709133569184914744242419024851720063670217337537901061478524544000054069549349716616484444074184445619237786055808166503717824452180634865572290112966679701704286713999051689171812385094956366617130929466316379508789023993413312071660885154381762424702205071921756683123684298720906989929550729920116447337549542373553040620798241198454639703241212264818011358891043265435079587335645492919050816567789456944706940960831404040107979266930031873477260267616868404845292413402969576654421705630159929046524390297563542934586257865782993698482342133590222476363307621803838797267051812108240771549769574825393604642798014914794133051973844637186767079721408591539970503925667297876891090450960470158229625633345785426782060784810751132977651981254938369741788101292916789025419458374846405919899716849809426113075555694112653330046878168047148333874320803658842513143351385653695575770455937317390806081634170237352789283722528576365366099857442509091134016179415418984242522135169710661887509472774338447608147237545387372155562636449511168833873336503667026232418810705197663360098199951531725469431094912977343279286891093228218713217969551833834032925867568488200632738191578730184099371255332753548950394665819888113346696815632663461585604168696457762195112136727019745214909682958417785432159169093096814543805626051262523315445449522287546374394986040391102809331950461023909732484197978935550330130184892135873484419603075631396039592602407679211207856561944653155680988268314693525867363056052996184741388168042348687722625028755103969464287207425227434065398521247342776326598306291159629940914499625611530714326956447496803944361099383204693376579538369937294345993317364414363039297789569833314823638061878695653281880728632952164197258942116826148743765135774231392902360341430108973954179398258945860744688488289901084348192952650326743175016858904586873772769517328336342044396590959614369809353964792753594626253777483305951323461799892870151462149385580366994189693989450456566714902155579161597641738018257413140202255271240401308577417384459734585356588854881124417870988841012437860991767067440246535532103088536787702941812981282806387780013069805659891079057412232332919950237402403852222119890131026153099074345040493442028670543382020199339256135436577767741869579963650884852401073925330682272081718599902992060729979079421910482529914325110938211545852939625026908837653126306556939364916617662129831385946029797255406441397217177665687516350688717342277202155966239885493194189805080116881240951336784893301858783247
Time: CPU 3.33 s, Wall: 5.93 s

We can also copy this matrix over to Maple and compute the same determinant there...

maple.with_package('LinearAlgebra')
B = maple(a)
t = maple.cputime()
time c = B.Determinant()    # new fast Determinant in maple (don't use "det"!)
maple.cputime(t)

22.253

c == d

True

We can copy to Magma to. This ability to copy objects around is unique to Sage.

B = magma(a)
t = magma.cputime()
time e = B.Determinant()
magma.cputime(t)

Time: CPU 0.00 s, Wall: 11.36 s
10.69

Example: A Symbolic Expression

x = var('x')

f = sin(3*x)*x+log(x) + 1/(x+1)^2
show(f)

{x \sin \left( {3 x} \right)} + \log \left( x \right) + \frac{1}{{\left( x + 1 \right)}^{2} }

Plotting functions has similar syntax to Mathematica:

plot(f,(0.01,2))

Sage can also has 2d plotting that is almost identical to MATLAB:

from pylab import *
t = arange(0.0, 2.0, 0.01)
s = sin(2*pi*t)
P = plot(t, s)
xlabel('time (s)'); ylabel('voltage (mV)')
title('About as simple as it gets, folks')
grid(True)
savefig('sage.png')
reset()  # make things back like they were before doing from pylab import *

_fast_float_ yields super-fast evaluation of Sage symbolic expressions -- e.g., here it is 10 times faster than native Python!

show(f)

{x \sin \left( {3 x} \right)} + \log \left( x \right) + \frac{1}{{\left( x + 1 \right)}^{2} }

g = f._fast_float_(x)
timeit('g(4.5r)')

625 loops, best of 3: 515 ns per loop

%python   
# %python, so no preparsing so uses pure python
import math
def g(x): return math.sin(3*x)*x + log(x) + 1/(1+x)**2

timeit('g(4.5r)')

625 loops, best of 3: 6.66 µs per loop

Example: Compare Answers with Maple

var('x')
f = sin(3*x)*x+log(x) + 1/(x+1)^2
show(integrate(f))

\frac{\sin \left( {3 x} \right) - {{3 x} \cos \left( {3 x} \right)}}{9} + {x \log \left( x \right)} - \frac{1}{x + 1} - x

The command maple(...) fires up Maple (if you have it!), and creates a reference to a live object:

m = maple(f); m

sin(3*x)*x+ln(x)+1/(x+1)^2

type(m)

<class 'sage.interfaces.maple.MapleElement'>

m.parent()

Maple

m.parent().pid()

24038

os.system('ps ax |grep 24038')

s007  Ss+    0:00.01 /bin/sh /Users/wstein/bin/maple -t
s015  S+     0:00.00 sh -c ps ax |grep 24038
s015  R+     0:00.00 grep 24038
0

Use Maple objects via a Pythonic notation:

show(m.integrate('x'))

1/9\,\sin \left( 3\,x \right) -1/3\,\cos \left( 3\,x \right) x+\ln \left( x \right) x-x- \left( x+1 \right) ^{-1}

mathematica(f).Integrate(x)

-x - (1 + x)^(-1) - (x*Cos[3*x])/3 + x*Log[x] + Sin[3*x]/9

Example: Interactive Image Compression

This illustrates pylab (matplotlib + numpy), Sage plotting, html output, and @interact.

# first just play
import pylab
A = pylab.imread(DATA + 'seattle.png')
graphics_array([matrix_plot(A), matrix_plot(A[0:,0:,2])]).show(figsize=[10,4])

import pylab
A_image = pylab.mean(pylab.imread(DATA + 'seattle.png'), 2)
@interact
def svd_image(i=(20,(1..100)), display_axes=True):
    u,s,v = pylab.linalg.svd(A_image)
    A     = sum(s[j]*pylab.outer(u[0:,j], v[j,0:]) for j in range(i))
    g = graphics_array([matrix_plot(A),matrix_plot(A_image)])
    show(g, axes=display_axes, figsize=(8,3))
    html('<h2>Compressed using %s eigenvalues</h2>'%i)

Example: 3d Plots

var('x,y')
plot3d(sin(x*y^2) -cos(x), (x,-2,2), (y,-2,2))

sphere((0,0,0)) + sphere((0,1,0), color='red', opacity=0.5) + icosahedron((1,1,0), color='orange')

L = []

@interact
def random_list(number_of_points=(10..50), dots=True):
    n = normalvariate
    global L
    if len(L) != number_of_points:
        L = [(n(0,1), n(0,1), n(0,1)) for i in range(number_of_points)]
    G = list_plot3d(L,interpolation_type='nn', texture=Color('#ff7500'),num_points=120)
    if dots: G += point3d(L)
    G.show()

3d plotting (using jmol) is fast even though it does not use Java3d or OpenGL or require any special signed code or drivers.

# Yoda! -- over 50,000 triangles.
from scipy import io
X = io.loadmat(DATA + 'yodapose.mat')
from sage.plot.plot3d.index_face_set import IndexFaceSet
V = X['V']; F3=X['F3']-1; F4=X['F4']-1
Y = IndexFaceSet(F3,V,color='green') + IndexFaceSet(F4,V,color='green')
Y = Y.rotateX(-1)
Y.show(aspect_ratio=[1,1,1], frame=False, figsize=4)
html('"Use the source, Luke..."')

"Use the source, Luke..."

Real World Example: Super Fast Code (using Cython)

to	sage-support 
date	Sat, Jan 31, 2009 at 11:15 AM
Hi,

I received first a MemoryError, and later on Sage reported:

uitkomst1=[]
uitkomst2=[]
eind=int((10^9+2)/(2*sqrt(3)))
print eind
for y in srange(1,eind):
 test1=is_square(3*y^2+1,True)
 test2=is_square(48*y^2+1,True)
 if test1[0] and test1[1]%3==2: uitkomst1.append((y,(2*test1[1]-1)/3))
 if test2[0] and test2[1]%3==1: uitkomst2.append((y,(2*test2[1]+1)/3))
print uitkomst1
een=sum([3*x-1 for (y,x) in uitkomst1 if 3*x-1<10^9])
print uitkomst2
twee=sum([3*x+1 for (y,x) in uitkomst2 if 3*x+1<10^9])
print een+twee

If you replace 10^9 with 10^6, the above listing works properly.

Maybe I made a mistake?

Rolandb

I rewrote Roland's code slightly so it wouldn't waste memory by constructing big lists... but the result was slow.

def f_python(n):
    uitkomst1=[]
    uitkomst2=[]
    eind=int((n+2)/(2*sqrt(3)))
    print eind
    for y in (1..eind):
        test1=is_square(3*y^2+1,True)
        test2=is_square(48*y^2+1,True)
        if test1[0] and test1[1]%3==2: uitkomst1.append((y,(2*test1[1]-1)/3))
        if test2[0] and test2[1]%3==1: uitkomst2.append((y,(2*test2[1]+1)/3))
    print uitkomst1
    een=sum(3*x-1 for (y,x) in uitkomst1 if 3*x-1<10^9)
    print uitkomst2
    twee=sum(3*x+1 for (y,x) in uitkomst2 if 3*x+1<10^9)
    print een+twee

time f_python(10^5)

28868
[(1, 1), (15, 17), (209, 241), (2911, 3361)]
[(1, 5), (14, 65), (195, 901), (2716, 12545)]
51408
Time: CPU 0.74 s, Wall: 0.86 s

time f_python(10^6)

288675
[(1, 1), (15, 17), (209, 241), (2911, 3361), (40545, 46817)]
[(1, 5), (14, 65), (195, 901), (2716, 12545), (37829, 174725)]
716034
Time: CPU 7.14 s, Wall: 7.65 s

While waiting to see if f_python(10^6) would finish, I decided to try the Cython compiler. I declared a few data types, put %cython at the top of the cell, and wham, it got over 200 times faster.

%cython
from sage.all import is_square
cdef extern from "math.h":
    long double sqrtl(long double)
    
def f(n):
    uitkomst1=[]
    uitkomst2=[]
    cdef long long eind=int((n+2)/(2*sqrt(3)))
    cdef long long y, t
    print eind
    for y in range(1,eind):   
        t = <long long>sqrtl(<long long> (3*y*y + 1))
        if t * t == 3*y*y + 1:
            uitkomst1.append((y, (2*t-1)/3))
        t = <long long>sqrtl(<long long> (48*y*y + 1))
        if t * t == 48*y*y + 1:
            uitkomst2.append((y, (2*t+1)/3))
    print uitkomst1
    een=sum([3*x-1 for (y,x) in uitkomst1 if 3*x-1<10^9])
    print uitkomst2
    twee=sum([3*x+1 for (y,x) in uitkomst2 if 3*x+1<10^9])
    print een+twee

time f(10^5)

28868
[(1L, 1L), (4L, 4L), (15L, 17L), (56L, 64L), (209L, 241L), (780L, 900L), (2911L, 3361L), (10864L, 12544L)]
[(1L, 5L), (14L, 65L), (195L, 901L), (2716L, 12545L)]
2
Time: CPU 0.00 s, Wall: 0.00 s

time f(10^6)

288675
[(1L, 1L), (4L, 4L), (15L, 17L), (56L, 64L), (209L, 241L), (780L, 900L), (2911L, 3361L), (10864L, 12544L), (40545L, 46817L), (151316L, 174724L)]
[(1L, 5L), (14L, 65L), (195L, 901L), (2716L, 12545L), (37829L, 174725L)]
2
Time: CPU 0.03 s, Wall: 0.03 s

time f(10^9)

288675135
[(1L, 1L), (4L, 4L), (15L, 17L), (56L, 64L), (209L, 241L), (780L, 900L), (2911L, 3361L), (10864L, 12544L), (40545L, 46817L), (151316L, 174724L), (564719L, 652081L), (2107560L, 2433600L), (7865521L, 9082321L), (29354524L, 33895684L), (109552575L, 126500417L)]
[(1L, 5L), (14L, 65L), (195L, 901L), (2716L, 12545L), (37829L, 174725L), (526890L, 2433601L), (7338631L, 33895685L), (102213944L, 472105985L)]
2
Time: CPU 25.60 s, Wall: 26.50 s

7.14/0.03

238.000000000000

This is not a contrived example. This is a real world example that came up a this weekend. For C-style computations, Sage (via Cython) is as fast as C.

Example: Do some number theory

Plotting an elliptic curve over a finite field

p = 389
E = EllipticCurve(GF(p).random_element())
E.plot()

and finding its group structure...

E.abelian_group()

(Multiplicative Abelian Group isomorphic to C180 x C2, ((262 : 190 : 1), (228 : 0 : 1)))

Creating another random elliptic curve over a bigger finite field and compute its cardinality in seconds:

p = next_prime(10^60)
E = EllipticCurve(GF(p).random_element())
show(E)

y^2 = x^3 + 425356051223648747023108902736172620897278673931104973974164x + 920733503445411966314417165378315096439996780491982901461664

time E.cardinality()

1000000000000000000000000000000753054185359723724596110573310
Time: CPU 0.00 s, Wall: 5.66 s

Some weight 3 modular forms on $\Gamma_1(12)$ :

show(ModularForms(Gamma1(12),3,prec=20).basis())

\begin{array}{l}[q - 6q^{4} - 3q^{5} + 6q^{6} + 5q^{7} + 12q^{8} - 9q^{9} - 6q^{11} - 6q^{12} + 14q^{13} - 24q^{14} + 3q^{15} + 15q^{17} - 31q^{19} + O(q^{20}),\\ q^{2} - 4q^{4} - q^{5} + 3q^{6} + 5q^{7} + 4q^{8} - 6q^{9} - 2q^{10} - 6q^{11} + 12q^{13} - 12q^{14} + 3q^{15} + 8q^{16} + 5q^{17} - 3q^{18} - 31q^{19} + O(q^{20}),\\ q^{3} - 2q^{4} - q^{5} + 2q^{6} + q^{7} + 4q^{8} - 6q^{9} - 2q^{11} - 2q^{12} + 12q^{13} - 8q^{14} + q^{15} + 5q^{17} - 19q^{19} + O(q^{20}),\\ 1 + 528q^{10} - 960q^{11} + 572q^{12} - 1176q^{14} + 3200q^{15} - 420q^{16} - 4224q^{17} - 216q^{18} + 2496q^{19} + O(q^{20}),\\ q + \frac{32429}{72}q^{10} - \frac{7945}{9}q^{11} + \frac{96131}{216}q^{12} + 170q^{13} - \frac{70211}{72}q^{14} + \frac{67282}{27}q^{15} - \frac{23447}{72}q^{16} - \frac{30008}{9}q^{17} - \frac{637}{4}q^{18} + \frac{17615}{9}q^{19} + O(q^{20}),\\ q^{2} + \frac{2458}{9}q^{10} - \frac{4960}{9}q^{11} + \frac{8722}{27}q^{12} - \frac{5626}{9}q^{14} + \frac{49600}{27}q^{15} - \frac{2305}{9}q^{16} - \frac{21824}{9}q^{17} - 133q^{18} + \frac{12896}{9}q^{19} + O(q^{20}),\\ q^{3} + \frac{373}{3}q^{10} - \frac{760}{3}q^{11} + \frac{1435}{9}q^{12} - \frac{931}{3}q^{14} + \frac{7834}{9}q^{15} - \frac{355}{3}q^{16} - \frac{3344}{3}q^{17} - 70q^{18} + \frac{1976}{3}q^{19} + O(q^{20}),\\ q^{4} + \frac{88}{3}q^{10} - \frac{160}{3}q^{11} + \frac{331}{9}q^{12} - \frac{196}{3}q^{14} + \frac{1600}{9}q^{15} - \frac{31}{3}q^{16} - \frac{704}{3}q^{17} - 12q^{18} + \frac{416}{3}q^{19} + O(q^{20}),\\ q^{5} - \frac{1027}{72}q^{10} + \frac{350}{9}q^{11} - \frac{5005}{216}q^{12} + \frac{2989}{72}q^{14} - \frac{3032}{27}q^{15} + \frac{1225}{72}q^{16} + \frac{1450}{9}q^{17} + \frac{35}{4}q^{18} - \frac{793}{9}q^{19} + O(q^{20}),\\ q^{6} - \frac{70}{3}q^{10} + \frac{160}{3}q^{11} - \frac{286}{9}q^{12} + \frac{196}{3}q^{14} - \frac{1600}{9}q^{15} + \frac{79}{3}q^{16} + \frac{704}{3}q^{17} + 22q^{18} - \frac{416}{3}q^{19} + O(q^{20}),\\ q^{7} - \frac{1315}{72}q^{10} + \frac{350}{9}q^{11} - \frac{5005}{216}q^{12} + \frac{3277}{72}q^{14} - \frac{3275}{27}q^{15} + \frac{1225}{72}q^{16} + \frac{1441}{9}q^{17} + \frac{35}{4}q^{18} - \frac{784}{9}q^{19} + O(q^{20}),\\ q^{8} - \frac{88}{9}q^{10} + \frac{160}{9}q^{11} - \frac{250}{27}q^{12} + \frac{196}{9}q^{14} - \frac{1600}{27}q^{15} + \frac{70}{9}q^{16} + \frac{704}{9}q^{17} + 4q^{18} - \frac{416}{9}q^{19} + O(q^{20}),\\ q^{9} - \frac{31}{8}q^{10} + 5q^{11} - \frac{49}{24}q^{12} + \frac{49}{8}q^{14} - \frac{50}{3}q^{15} + \frac{13}{8}q^{16} + 22q^{17} + \frac{7}{4}q^{18} - 13q^{19} + O(q^{20})]\end{array}