ModularSymbols(11)
Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field |
ModularSymbols(Gamma1(13),3)
Modular Symbols space of dimension 28 for Gamma_1(13) of weight 3 with sign 0 and over Rational Field Modular Symbols space of dimension 28 for Gamma_1(13) of weight 3 with sign 0 and over Rational Field |
G.<a> = DirichletGroup(13) ModularSymbols(a^2,2)
Modular Symbols space of dimension 4 and level 13, weight 2, character [zeta6], sign 0, over Cyclotomic Field of order 6 and degree 2 Modular Symbols space of dimension 4 and level 13, weight 2, character [zeta6], sign 0, over Cyclotomic Field of order 6 and degree 2 |
M = ModularSymbols(Gamma0(33)); M
Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field |
M.eisenstein_subspace().basis()
((1,0) + (11,3), (3,5) + (3,10) + (3,17) + (3,20) + 3*(11,1) + 3*(11,2) + 4*(11,3), (3,7) + 1/2*(3,10) + 1/2*(3,20) - 1/2*(11,1) - 3/2*(11,2) - 3*(11,3)) ((1,0) + (11,3), (3,5) + (3,10) + (3,17) + (3,20) + 3*(11,1) + 3*(11,2) + 4*(11,3), (3,7) + 1/2*(3,10) + 1/2*(3,20) - 1/2*(11,1) - 3/2*(11,2) - 3*(11,3)) |
M.cuspidal_subspace().basis()
((3,5) - (11,2) + (11,3), (3,7) - (11,2) + (11,3), (3,10) - (11,2) + (11,3), (3,17) - (11,2) + (11,3), (3,20) - (11,2) + (11,3), (11,1) - (11,2)) ((3,5) - (11,2) + (11,3), (3,7) - (11,2) + (11,3), (3,10) - (11,2) + (11,3), (3,17) - (11,2) + (11,3), (3,20) - (11,2) + (11,3), (11,1) - (11,2)) |
M.new_subspace().basis()
((1,0) + 2/5*(3,7) + 2/5*(3,17) + 4/5*(11,1) + 2/5*(11,2) - 1/5*(11,3), (3,5) - (3,17), (3,10) + (3,20) - (11,1) - (11,2) + 2*(11,3)) ((1,0) + 2/5*(3,7) + 2/5*(3,17) + 4/5*(11,1) + 2/5*(11,2) - 1/5*(11,3), (3,5) - (3,17), (3,10) + (3,20) - (11,1) - (11,2) + 2*(11,3)) |
O = M.old_subspace(); O
Modular Symbols subspace of dimension 7 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field Modular Symbols subspace of dimension 7 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field |
O.basis()
((1,0) + (11,3), (3,5) - 3*(11,2) - 3*(11,3), (3,7) - (11,2) - (11,3), (3,10) - (11,2) - 3*(11,3), (3,17) + 3*(11,2) + 3*(11,3), (3,20) + (11,2) + (11,3), (11,1) + (11,2) + 2*(11,3)) ((1,0) + (11,3), (3,5) - 3*(11,2) - 3*(11,3), (3,7) - (11,2) - (11,3), (3,10) - (11,2) - 3*(11,3), (3,17) + 3*(11,2) + 3*(11,3), (3,20) + (11,2) + (11,3), (11,1) + (11,2) + 2*(11,3)) |
t3 = O.hecke_operator(3); t3
Hecke operator T_3 on Modular Symbols subspace of dimension 7 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field Hecke operator T_3 on Modular Symbols subspace of dimension 7 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field |
t3.matrix()
[ 1 0 0 0 0 0 0] [ 12 -1 2 7 0 4 0] [ 4 1 2 1 1 1 0] [ 8 0 4 3 1 3 -1] [-12 0 -4 -5 -1 -2 0] [ -4 0 -1 -1 -1 -1 2] [ -8 0 -3 -3 0 -3 0] [ 1 0 0 0 0 0 0] [ 12 -1 2 7 0 4 0] [ 4 1 2 1 1 1 0] [ 8 0 4 3 1 3 -1] [-12 0 -4 -5 -1 -2 0] [ -4 0 -1 -1 -1 -1 2] [ -8 0 -3 -3 0 -3 0] |
M = ModularSymbols(389,2,1); M
Modular Symbols space of dimension 33 for Gamma_0(389) of weight 2 with sign 1 over Rational Field Modular Symbols space of dimension 33 for Gamma_0(389) of weight 2 with sign 1 over Rational Field |
time D = M.decomposition(); D
[ Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 33 for Gamma_0(389) of weight 2 with sign 1 over Rational Field, Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 33 for Gamma_0(389) of weight 2 with sign 1 over Rational Field, Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 33 for Gamma_0(389) of weight 2 with sign 1 over Rational Field, Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 33 for Gamma_0(389) of weight 2 with sign 1 over Rational Field, Modular Symbols subspace of dimension 6 of Modular Symbols space of dimension 33 for Gamma_0(389) of weight 2 with sign 1 over Rational Field, Modular Symbols subspace of dimension 20 of Modular Symbols space of dimension 33 for Gamma_0(389) of weight 2 with sign 1 over Rational Field ] CPU time: 0.31 s, Wall time: 0.70 s [ Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 33 for Gamma_0(389) of weight 2 with sign 1 over Rational Field, Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 33 for Gamma_0(389) of weight 2 with sign 1 over Rational Field, Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 33 for Gamma_0(389) of weight 2 with sign 1 over Rational Field, Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 33 for Gamma_0(389) of weight 2 with sign 1 over Rational Field, Modular Symbols subspace of dimension 6 of Modular Symbols space of dimension 33 for Gamma_0(389) of weight 2 with sign 1 over Rational Field, Modular Symbols subspace of dimension 20 of Modular Symbols space of dimension 33 for Gamma_0(389) of weight 2 with sign 1 over Rational Field ] CPU time: 0.31 s, Wall time: 0.70 s |
for A in D:
if A.is_cuspidal():
print A.q_eigenform(4)
q - 2*q^2 - 2*q^3 + O(q^4) q + alpha*q^2 + (alpha - 2)*q^3 + O(q^4) q + alpha*q^2 + -alpha*q^3 + O(q^4) q + alpha*q^2 + (alpha^5 + 3*alpha^4 - 2*alpha^3 - 8*alpha^2 + alpha + 2)*q^3 + O(q^4) q + alpha*q^2 + (-20146763/1097385680*alpha^19 + 20466323/219477136*alpha^18 + 119884773/274346420*alpha^17 - 753611053/274346420*alpha^16 - 381358355/109738568*alpha^15 + 3611475535/109738568*alpha^14 + 6349339639/1097385680*alpha^13 - 56878934241/274346420*alpha^12 + 71555185319/1097385680*alpha^11 + 163330998525/219477136*alpha^10 - 223188336749/548692840*alpha^9 - 169878973265/109738568*alpha^8 + 265944624817/274346420*alpha^7 + 199655892261/109738568*alpha^6 - 1167579836501/1097385680*alpha^5 - 619178000979/548692840*alpha^4 + 261766056911/548692840*alpha^3 + 4410485304/13717321*alpha^2 - 14646077211/274346420*alpha - 1604641167/68586605)*q^3 + O(q^4) q - 2*q^2 - 2*q^3 + O(q^4) q + alpha*q^2 + (alpha - 2)*q^3 + O(q^4) q + alpha*q^2 + -alpha*q^3 + O(q^4) q + alpha*q^2 + (alpha^5 + 3*alpha^4 - 2*alpha^3 - 8*alpha^2 + alpha + 2)*q^3 + O(q^4) q + alpha*q^2 + (-20146763/1097385680*alpha^19 + 20466323/219477136*alpha^18 + 119884773/274346420*alpha^17 - 753611053/274346420*alpha^16 - 381358355/109738568*alpha^15 + 3611475535/109738568*alpha^14 + 6349339639/1097385680*alpha^13 - 56878934241/274346420*alpha^12 + 71555185319/1097385680*alpha^11 + 163330998525/219477136*alpha^10 - 223188336749/548692840*alpha^9 - 169878973265/109738568*alpha^8 + 265944624817/274346420*alpha^7 + 199655892261/109738568*alpha^6 - 1167579836501/1097385680*alpha^5 - 619178000979/548692840*alpha^4 + 261766056911/548692840*alpha^3 + 4410485304/13717321*alpha^2 - 14646077211/274346420*alpha - 1604641167/68586605)*q^3 + O(q^4) |
# what is alpha? f = D[5].q_eigenform(4)
f.parent()
Power Series Ring in q over Univariate Quotient Polynomial Ring in alpha over Rational Field with modulus x^20 - 3*x^19 - 29*x^18 + 91*x^17 + 338*x^16 - 1130*x^15 - 2023*x^14 + 7432*x^13 + 6558*x^12 - 28021*x^11 - 10909*x^10 + 61267*x^9 + 6954*x^8 - 74752*x^7 + 1407*x^6 + 46330*x^5 - 1087*x^4 - 12558*x^3 - 942*x^2 + 960*x + 148 Power Series Ring in q over Univariate Quotient Polynomial Ring in alpha over Rational Field with modulus x^20 - 3*x^19 - 29*x^18 + 91*x^17 + 338*x^16 - 1130*x^15 - 2023*x^14 + 7432*x^13 + 6558*x^12 - 28021*x^11 - 10909*x^10 + 61267*x^9 + 6954*x^8 - 74752*x^7 + 1407*x^6 + 46330*x^5 - 1087*x^4 - 12558*x^3 - 942*x^2 + 960*x + 148 |
M = ModularSymbols(33,2,1); M
Modular Symbols space of dimension 6 for Gamma_0(33) of weight 2 with sign 1 over Rational Field Modular Symbols space of dimension 6 for Gamma_0(33) of weight 2 with sign 1 over Rational Field |
M.atkin_lehner_operator(3).matrix()
[ 0 0 0 0 0 -1] [ 0 0 -1 0 0 0] [ 0 -1 0 0 0 0] [ 0 0 0 -1 0 0] [-1 -1 -1 0 1 -1] [-1 0 0 0 0 0] [ 0 0 0 0 0 -1] [ 0 0 -1 0 0 0] [ 0 -1 0 0 0 0] [ 0 0 0 -1 0 0] [-1 -1 -1 0 1 -1] [-1 0 0 0 0 0] |
S = M.cuspidal_subspace()
S.atkin_lehner_operator(3).matrix()
[ 1 0 0] [ 0 1 0] [ 1 1 -1] [ 1 0 0] [ 0 1 0] [ 1 1 -1] |
S.atkin_lehner_operator(1).matrix()
[1 0 0] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [0 0 1] |
S.atkin_lehner_operator(7).matrix()
Exception (click to the left for traceback): ... ArithmeticError: d (=7) must be a divisor of the level (=33) Traceback (most recent call last): File " |