4810.sagews

print "Modular form space dimensions"
print "N = row, k = col"
print
print "\t",

lok = 1
hik = 16
Nbound = 23

for k in range(lok, hik):
    print "#" + repr(k), "\t",
print

for N in range(1, Nbound):
    print "#" + repr(N), "\t",
    for k in range(lok, hik):
        print ModularForms(Gamma0(N), k).dimension(), "\t",
    print

Modular form space dimensions N = row, k = col #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13 #14 #15 #1 0 0 0 1 0 1 0 1 0 1 0 2 0 1 0 #2 0 1 0 2 0 2 0 3 0 3 0 4 0 4 0 #3 0 1 0 2 0 3 0 3 0 4 0 5 0 5 0 #4 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 #5 0 1 0 3 0 3 0 5 0 5 0 7 0 7 0 #6 0 3 0 5 0 7 0 9 0 11 0 13 0 15 0 #7 0 1 0 3 0 5 0 5 0 7 0 9 0 9 0 #8 0 3 0 5 0 7 0 9 0 11 0 13 0 15 0 #9 0 3 0 5 0 7 0 9 0 11 0 13 0 15 0 #10 0 3 0 7 0 9 0 13 0 15 0 19 0 21 0 #11 0 2 0 4 0 6 0 8 0 10 0 12 0 14 0 #12 0 5 0 9 0 13 0 17 0 21 0 25 0 29 0 #13 0 1 0 5 0 7 0 9 0 11 0 15 0 15 0 #14 0 4 0 8 0 12 0 16 0 20 0 24 0 28 0 #15 0 4 0 8 0 12 0 16 0 20 0 24 0 28 0 #16 0 5 0 9 0 13 0 17 0 21 0 25 0 29 0 #17 0 2 0 6 0 8 0 12 0 14 0 18 0 20 0 #18 0 7 0 13 0 19 0 25 0 31 0 37 0 43 0 #19 0 2 0 6 0 10 0 12 0 16 0 20 0 22 0 #20 0 6 0 12 0 18 0 24 0 30 0 36 0 42 0 #21 0 4 0 10 0 16 0 20 0 26 0 32 0 36 0 #22 0 5 0 11 0 17 0 23 0 29 0 35 0 41 0

print "Cusp form space dimensions"
print "N = row, k = col"
print
print "\t",

kbound = 16
Nbound = 23

for k in range(1, kbound):
    print "#" + repr(k), "\t",
print

for N in range(1, Nbound):
    print "#" + repr(N), "\t",
    for k in range(1, kbound):
        print CuspForms(Gamma0(N), k).dimension(), "\t",
    print

Cusp form space dimensions N = row, k = col #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13 #14 #15 #1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 #2 0 0 0 0 0 0 0 1 0 1 0 2 0 2 0 #3 0 0 0 0 0 1 0 1 0 2 0 3 0 3 0 #4 0 0 0 0 0 1 0 2 0 3 0 4 0 5 0 #5 0 0 0 1 0 1 0 3 0 3 0 5 0 5 0 #6 0 0 0 1 0 3 0 5 0 7 0 9 0 11 0 #7 0 0 0 1 0 3 0 3 0 5 0 7 0 7 0 #8 0 0 0 1 0 3 0 5 0 7 0 9 0 11 0 #9 0 0 0 1 0 3 0 5 0 7 0 9 0 11 0 #10 0 0 0 3 0 5 0 9 0 11 0 15 0 17 0 #11 0 1 0 2 0 4 0 6 0 8 0 10 0 12 0 #12 0 0 0 3 0 7 0 11 0 15 0 19 0 23 0 #13 0 0 0 3 0 5 0 7 0 9 0 13 0 13 0 #14 0 1 0 4 0 8 0 12 0 16 0 20 0 24 0 #15 0 1 0 4 0 8 0 12 0 16 0 20 0 24 0 #16 0 0 0 3 0 7 0 11 0 15 0 19 0 23 0 #17 0 1 0 4 0 6 0 10 0 12 0 16 0 18 0 #18 0 0 0 5 0 11 0 17 0 23 0 29 0 35 0 #19 0 1 0 4 0 8 0 10 0 14 0 18 0 20 0 #20 0 1 0 6 0 12 0 18 0 24 0 30 0 36 0 #21 0 1 0 6 0 12 0 16 0 22 0 28 0 32 0 #22 0 2 0 7 0 13 0 19 0 25 0 31 0 37 0

S2_11 = CuspForms(Gamma1(11), 3)
tlist = S2_11.basis()
t = tlist[0]

N = 200
q = t.q_expansion(N)
prev = 0

for i in range(10):
    current = randint(1, N)
    print "current =", current
    while current < N and prev != current:
        prev = current
        current = abs(q[current])
        print current

current = 58 680 current = 89 4910 current = 3 0 0 current = 183 394 current = 26 330 current = 77 3382 current = 147 583 current = 118 1691 current = 76 575 current = 28 1120

#    Helper method to obtain basis object for cusp space (see below)
def getBasis(N, gammaType):
    if gammaType == 0:
        S = CuspForms(Gamma0(N), 2)
    elif gammaType == 1:
        S = CuspForms(Gamma1(N), 2)
    else:
        S = CuspForms(Gamma(N), 2)
    return S.basis()

#    Amicable pair search among weight 2 normalized basis cusp forms for fixed level.
#    Returns two-dim list: rows correspond to basis forms of cusp space; cols to amicable pairs for each form.

def pairSearch(N, gammaType, depth):
    #    N = level
    #    gammaType = Gamma0 if 0, Gamma1 if 1, Gamma otherwise
    #    depth = how deep to look into coeffs (go up to (depth - 1)th index)
    
    basis = getBasis(N, gammaType)    
    list = []
    for t in range(len(basis)):
        list.append([])                # append a new sublist for each basis form
        q = basis[t].q_expansion(depth)
        for i in range(1, depth):
            for j in range(i, depth):
                if abs(q[i]) == j and abs(q[j]) == i:
                    list[t].append((i, j))
    return list
    
#    Search for fixed coeff value among weight 2 normalized basis cusp forms for fixed level.
#    Returns two-dim list: rows correspond to basis forms of cusp space; cols to indices of coeffs equal to the input value.

def valueSearch(N, gammaType, value, depth):
    #    N = level
    #    gammaType = Gamma0 if 0, Gamma1 if 1, Gamma otherwise
    #    depth = how deep to look into coeffs (go up to (depth - 1)th index)
    
    basis = getBasis(N, gammaType)
    list = []
    for t in range(len(basis)):
        list.append([])                # append a new sublist for each basis form
        q = basis[t].q_expansion(depth)
        for i in range(1, depth):
            if abs(q[i]) == value:
                list[t].append(i)
    return list

depth = 1000
for N in range(11, 20):
    print pairSearch(N, 0, depth)

[[(1, 1), (2, 2)]] [] [] [[(1, 1)]] [[(1, 1)]] [] [[(1, 1)]] [] [[(1, 1)]]

E = EllipticCurve([0, -1, 0, 0, 1/4])
E.has_cm()

False

print pairSearch(11, 0, 20000)

[[(1, 1), (2, 2)]]

for N in range(11, 40):
    print N, "\t",
    list = valueSearch(N, 0, 1, 1000)
    for t in range(len(list)):
        print len(list[t]),
    print

11 23 12 13 14 38 15 57 16 17 16 18 19 15 20 8 21 64 22 23 19 23 26 23 24 11 25 26 53 51 27 6 28 7 31 29 58 49 30 39 47 41 31 33 31 32 2 33 25 78 15 34 5 14 40 35 37 69 33 36 1 37 13 36 38 32 13 47 51 39 84 28 82

depth = 4000

S = CuspForms(Gamma0(11), 2)
basis = S.basis()
q = basis[0].q_expansion(depth)

list0 = []
list1 = []
list2 = []
count0 = 0
count1 = 0
count2 = 0
for i in range(1, depth):
    absCoeff = abs(q[i])
    if absCoeff == 0:
        count0 += 1
        list0.append((i, count0))
    elif absCoeff == 1:
        count1 += 1
        list1.append((i, count1))
    elif absCoeff == 2:
        count2 += 1
        list2.append((i, count2))
points0 = point2d(list0)
points1 = point2d(list1, rgbcolor = (1, 0, 0))
points2 = point2d(list2, rgbcolor = (0, 1, 0))
show(points0 + points1 + points2)

depth = 4000
M = 7.0

list = []
count = []
for value in range(0, M):
    list.append([])
    count.append(0)

S = CuspForms(Gamma0(11), 2)
basis = S.basis()
q = basis[0].q_expansion(depth)

for i in range(1, depth):
    absCoeff = abs(q[i])
    if absCoeff < M:
        count[absCoeff] += 1
        list[absCoeff].append((i, count[absCoeff]))

points = []        
for value in range(0, M):
    points.append(point2d(list[value], rgbcolor = (value / M, 0, 0)))
show(points[0] + points[1] + points[2] + points[3] + points[4] + points[5] + points[6])