Basic commands
hello world
Programming
57 is composite
[2, 3]
[3, 8]
[5, 24]
[7, 48]
[11, 120]
[13, 168]
[17, 288]
[19, 360]
[23, 528]
Mathematical objects
(, Integer Ring)
(, Rational Field)
(, Complex Field with 200 bits of precision)
(, Finite Field in g of size 3^4)
Rational function field in x over Rational Field
Modular forms
'Modular Group SL(2,Z)'
([ 0 -1]
[ 1 0], [1 1]
[0 1])
Modular Forms space of dimension 1 for Modular Group SL(2,Z) of weight 4 over Rational Field
1
[
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)
]
1/240 + q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6)
126
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for Modular Group SL(2,Z) of weight 12 over Rational Field
[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + 401856*q^14 + 1217160*q^15 + 987136*q^16 - 6905934*q^17 + 2727432*q^18 + 10661420*q^19 + O(q^20)
]
[
1 + 16320/3617*q + 534790080/3617*q^2 + 234174178560/3617*q^3 + 17524001357760/3617*q^4 + 498046875016320/3617*q^5 + O(q^6)
]
(1 + 24*q^2 + 24*q^4 + O(q^6), q + 4*q^3 + 6*q^5 + O(q^6))
<class 'sage.modular.modform.ambient_g1.ModularFormsAmbient_g1_Q_with_category.element_class'>
Modular Forms space of dimension 2 for Congruence Subgroup Gamma1(4) of weight 2 over Rational Field
1 + 24*q^2 + 24*q^4 + 96*q^6 + 24*q^8 + 144*q^10 + 96*q^12 + 192*q^14 + 24*q^16 + 312*q^18 + O(q^20)
Power Series Ring in q over Integer Ring
1 + 8*q + 24*q^2 + 32*q^3 + 24*q^4 + 48*q^5 + 96*q^6 + 64*q^7 + 24*q^8 + 104*q^9 + 144*q^10 + 96*q^11 + 96*q^12 + 112*q^13 + 192*q^14 + 192*q^15 + 24*q^16 + 144*q^17 + 312*q^18 + 160*q^19 + 144*q^20 + 256*q^21 + 288*q^22 + 192*q^23 + 96*q^24 + O(q^25)
True
Some pictures of fundamental domains
(warning: not of the same type as in the lectures)
[, , , ]
Hecke operators
Modular Forms space of dimension 2 for Modular Group SL(2,Z) of weight 12 over Rational Field
Modular Forms space of dimension 7 for Congruence Subgroup Gamma1(7) of weight 3 over Rational Field
Old and new subspaces
(, )
(, )
Modular Forms subspace of dimension 6 of Modular Forms space of dimension 24 for Congruence Subgroup Gamma1(14) of weight 4 over Rational Field
Newforms
[]
[, ]
Error in lines 1-1
Traceback (most recent call last):
File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1234, in execute
flags=compile_flags), namespace, locals)
File "", line 1, in <module>
File "/ext/sage/sage-8.9_1804/local/lib/python2.7/site-packages/sage/modular/modform/constructor.py", line 467, in Newforms
return CuspForms(group, weight, base_ring).newforms(names)
File "/ext/sage/sage-8.9_1804/local/lib/python2.7/site-packages/sage/modular/modform/space.py", line 1642, in newforms
raise ValueError("Please specify a name to be used when generating names for generators of Hecke eigenvalue fields corresponding to the newforms.")
ValueError: Please specify a name to be used when generating names for generators of Hecke eigenvalue fields corresponding to the newforms.
[, , , ]
[, , , ]
(, )
[, , , ]
-functions
L-series associated to the cusp form q - q^2 - 2*q^3 + q^4 + O(q^6)
[]
[, , , ]
(, )
(, )