Modular Forms

Some introductory material on Sage computations for modular forms and modular symbols _Surya Teja Gavva

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Congruence Subgroups

In [8]:
G = SL(2,ZZ)
In [9]:
G
Out[9]:
Special Linear Group of degree 2 over Integer Ring

Generators

In [10]:
G.gens()
Out[10]:
(
[ 0  1]  [1 1]
[-1  0], [0 1]
)
In [11]:
G= Gamma0(10)
In [12]:
G
Out[12]:
Congruence Subgroup Gamma0(10)
In [18]:
G.generators()
Out[18]:
[
[1 1]  [ 3 -1]  [ 19  -7]  [11 -5]  [ 7 -5]
[0 1], [10 -3], [ 30 -11], [20 -9], [10 -7]
]
In [41]:
Gamma(2).generators(algorithm="todd-coxeter")
Out[41]:
[
[1 2]  [-1  0]  [ 1  0]  [-1  0]  [-1  2]  [-1  0]  [1 0]
[0 1], [ 0 -1], [-2  1], [ 0 -1], [-2  3], [ 2 -1], [2 1]
]
In [51]:
[Gamma0(n).index() for n in [1..20]]
Out[51]:
[1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20, 36]
In [52]:
Gamma1(3).image_mod_n()
Out[52]:
Matrix group over Ring of integers modulo 3 with 1 generators (
[1 1]
[0 1]
)
In [42]:
Gamma0(20).is_even()
Out[42]:
True
In [43]:
Gamma(3).is_normal()
Out[43]:
True
In [44]:
Gamma1(3).is_normal()
Out[44]:
False

Coset Representative in SL_2(Z)

In [16]:
list(G.coset_reps())
Out[16]:
[
[1 0]  [ 0 -1]  [1 0]  [ 0 -1]  [ 0 -1]  [ 0 -1]  [ 0 -1]  [ 0 -1]
[0 1], [ 1  0], [1 1], [ 1  2], [ 1  3], [ 1  4], [ 1  5], [ 1  6],

[ 0 -1]  [ 0 -1]  [ 0 -1]  [1 0]  [1 1]  [1 2]  [1 3]  [1 4]  [1 0]
[ 1  7], [ 1  8], [ 1  9], [2 1], [2 3], [2 5], [2 7], [2 9], [5 1],

[-2 -1]
[ 5  2]
]

Dimension of New Cusp Forms

In [17]:
Gamma0(110).dimension_new_cusp_forms()
Out[17]:
5
In [37]:
Gamma1(31).dimension_cusp_forms(2)
26
Out[37]:
26
In [38]:
Gamma1(31).dimension_modular_forms(2)
55
Out[38]:
55
In [39]:
Gamma(13).dimension_modular_forms(1)

---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
<ipython-input-39-ee4b81dc02bb> in <module>
----> 1 Gamma(Integer(13)).dimension_modular_forms(Integer(1))

/var/tmp/sage-9.4-current/local/lib/python3.9/site-packages/sage/modular/arithgroup/arithgroup_generic.py in dimension_modular_forms(self, k)
   1151             NotImplementedError: Computation of dimensions of weight 1 cusp forms spaces not implemented in general
   1152         """
-> 1153         return self.dimension_cusp_forms(k) + self.dimension_eis(k)
   1154 
   1155     def dimension_cusp_forms(self, k=2):

/var/tmp/sage-9.4-current/local/lib/python3.9/site-packages/sage/modular/arithgroup/arithgroup_generic.py in dimension_cusp_forms(self, k)
   1213                         return ZZ(0)
   1214                     else:
-> 1215                         raise NotImplementedError("Computation of dimensions of weight 1 cusp forms spaces not implemented in general")
   1216 
   1217     def dimension_eis(self, k=2):

NotImplementedError: Computation of dimensions of weight 1 cusp forms spaces not implemented in general
In [24]:
G.is_subgroup(Gamma0(10))
Out[24]:
True
In [22]:
G.is_subgroup(Gamma1(20))
Out[22]:
False

Farey Symbol

In [57]:
f=Gamma1(4).farey_symbol()
In [58]:
f.generators()
Out[58]:
[
[1 1]  [-3  1]
[0 1], [-4  1]
]
In [60]:
FareySymbol(Gamma0(11)).fundamental_domain()
Out[60]:

Number of Cusps

In [26]:
[Gamma0(n).ncusps() for n in [1..20]]
Out[26]:
[1, 2, 2, 3, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 4, 6, 2, 8, 2, 6]
In [28]:
[Gamma0(n).ncusps() for n in prime_range(2,100)]
Out[28]:
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
In [36]:
Gamma0(42).cusps()
Out[36]:
[0, 1/21, 1/14, 1/7, 1/6, 1/3, 1/2, Infinity]
In [49]:
G = Gamma1(5).as_permutation_group()
In [50]:
G.cusp_widths()
Out[50]:
[1, 1, 5, 5]
In [33]:
Gamma0(7).are_equivalent(Cusp(1/3), Cusp(1/7))
Out[33]:
False
In [34]:
[Gamma0(100).cusp_width(c) for c in Gamma0(100).cusps()]
Out[34]:
[100, 1, 4, 1, 1, 1, 4, 25, 1, 1, 4, 1, 25, 4, 1, 4, 1, 1]
In [53]:
Gamma1(4).is_regular_cusp(Cusps(1/2))
Out[53]:
False
In [54]:
Gamma0(24).reduce_cusp(Cusps(-1/4))
Out[54]:
1/4
In [56]:
Gamma0(1).reduce_cusp(Cusps(-1/4))
Out[56]:
Infinity

Elliptic Points

In [30]:
Gamma0(7).nu3()
Out[30]:
2
In [31]:
Gamma0(1105).nu2()
Out[31]:
8

Creating Modular Forms

In [1]:
M= ModularForms(Gamma0(4), 4)
In [2]:
M
Out[2]:
Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(4) of weight 4 over Rational Field
In [79]:
M.base_ring()
Out[79]:
Rational Field
In [66]:
M1= ModularForms(Gamma1(4), 10, prec=12)
In [67]:
M1
Out[67]:
Modular Forms space of dimension 6 for Congruence Subgroup Gamma1(4) of weight 10 over Rational Field
In [68]:
M1.basis()
Out[68]:
[
q + 246*q^5 - 3136*q^7 + 15885*q^9 - 45696*q^11 + O(q^12),
q^2 + 16*q^4 - 156*q^6 + 256*q^8 + 870*q^10 + O(q^12),
q^3 - 4*q^5 - 14*q^7 + 72*q^9 + 67*q^11 + O(q^12),
1 - 264*q^4 - 135432*q^8 + O(q^12),
q + 19684*q^3 + 1953126*q^5 + 40353608*q^7 + 387440173*q^9 + 2357947692*q^11 + O(q^12),
q^2 + 512*q^4 + 19684*q^6 + 262144*q^8 + 1953126*q^10 + O(q^12)
]
In [96]:
M1.gens()
Out[96]:
[
q + 246*q^5 - 3136*q^7 + 15885*q^9 - 45696*q^11 + O(q^12),
q^2 + 16*q^4 - 156*q^6 + 256*q^8 + 870*q^10 + O(q^12),
q^3 - 4*q^5 - 14*q^7 + 72*q^9 + 67*q^11 + O(q^12),
1 - 264*q^4 - 135432*q^8 + O(q^12),
q + 19684*q^3 + 1953126*q^5 + 40353608*q^7 + 387440173*q^9 + 2357947692*q^11 + O(q^12),
q^2 + 512*q^4 + 19684*q^6 + 262144*q^8 + 1953126*q^10 + O(q^12)
]
In [75]:
M1.cuspidal_subspace().basis()
Out[75]:
[
q + 246*q^5 - 3136*q^7 + 15885*q^9 - 45696*q^11 + O(q^12),
q^2 + 16*q^4 - 156*q^6 + 256*q^8 + 870*q^10 + O(q^12),
q^3 - 4*q^5 - 14*q^7 + 72*q^9 + 67*q^11 + O(q^12)
]
In [95]:
M1.eisenstein_subspace()
Out[95]:
Eisenstein subspace of dimension 3 of Modular Forms space of dimension 6 for Congruence Subgroup Gamma1(4) of weight 10 over Rational Field
In [115]:
M1.modular_symbols()
Out[115]:
Modular Symbols space of dimension 9 for Gamma_1(4) of weight 10 with sign 0 over Rational Field
In [116]:
M1.new_subspace()
Out[116]:
Modular Forms subspace of dimension 1 of Modular Forms space of dimension 6 for Congruence Subgroup Gamma1(4) of weight 10 over Rational Field
In [86]:
ModularForms(Gamma0(11),2).character()
Out[86]:
Dirichlet character modulo 11 of conductor 1 mapping 2 |--> 1
In [71]:
[ModularForms(Gamma1(7),k).dimension() for k in [2,3,4,5,6]]
Out[71]:
[5, 7, 9, 11, 13]
In [72]:
chi = DirichletGroup(109, CyclotomicField(3)).0
In [73]:
ModularForms(chi, 2, base_ring = CyclotomicField(15))
Out[73]:
Modular Forms space of dimension 10, character [zeta3 + 1] and weight 2 over Cyclotomic Field of order 15 and degree 8
In [ ]:
In [81]:
M2=ModularForms(11,2,base_ring=GF(13))
In [82]:
M2.basis()
Out[82]:
[
q + 11*q^2 + 12*q^3 + 2*q^4 + q^5 + O(q^6),
1 + 12*q^2 + 12*q^3 + 12*q^4 + 12*q^5 + O(q^6)
]
In [83]:
ModularForms(11,2).basis()
Out[83]:
[
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6),
1 + 12/5*q + 36/5*q^2 + 48/5*q^3 + 84/5*q^4 + 72/5*q^5 + O(q^6)
]
In [89]:
M2.cuspidal_submodule()
Out[89]:
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Finite Field of size 13
In [105]:
chi = DirichletGroup(25,QQ).0
In [106]:
n = ModularForms(chi,2)
In [107]:
n.basis()
Out[107]:
[
1 + O(q^6),
q + O(q^6),
q^2 + O(q^6),
q^3 + O(q^6),
q^4 + O(q^6),
q^5 + O(q^6)
]
In [108]:
n.set_precision(20)
In [109]:
n.basis()
Out[109]:
[
1 + 10*q^10 + 20*q^15 + O(q^20),
q + 5*q^6 + q^9 + 12*q^11 - 3*q^14 + 17*q^16 + 8*q^19 + O(q^20),
q^2 + 4*q^7 - q^8 + 8*q^12 + 2*q^13 + 10*q^17 - 5*q^18 + O(q^20),
q^3 + q^7 + 3*q^8 - q^12 + 5*q^13 + 3*q^17 + 6*q^18 + O(q^20),
q^4 - q^6 + 2*q^9 + 3*q^14 - 2*q^16 + 4*q^19 + O(q^20),
q^5 + q^10 + 2*q^15 + O(q^20)
]
In [117]:
e = DirichletGroup(27,CyclotomicField(3)).0**2
In [119]:
M = ModularForms(e,2,prec=10).eisenstein_subspace()
In [120]:
M.eisenstein_series()
Out[120]:
[
-1/3*zeta6 - 1/3 + q + (2*zeta6 - 1)*q^2 + q^3 + (-2*zeta6 - 1)*q^4 + (-5*zeta6 + 1)*q^5 + O(q^6),
-1/3*zeta6 - 1/3 + q^3 + O(q^6),
q + (-2*zeta6 + 1)*q^2 + (-2*zeta6 - 1)*q^4 + (5*zeta6 - 1)*q^5 + O(q^6),
q + (zeta6 + 1)*q^2 + 3*q^3 + (zeta6 + 2)*q^4 + (-zeta6 + 5)*q^5 + O(q^6),
q^3 + O(q^6),
q + (-zeta6 - 1)*q^2 + (zeta6 + 2)*q^4 + (zeta6 - 5)*q^5 + O(q^6)
]
In [121]:
M.eisenstein_subspace().T(2).matrix().fcp()
Out[121]:
(x + 2*zeta3 + 1) * (x + zeta3 + 2) * (x - zeta3 - 2)^2 * (x - 2*zeta3 - 1)^2

Cusp Forms

In [14]:
S= CuspForms(Gamma0(4), 6)
In [15]:
S
Out[15]:
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 4 for Congruence Subgroup Gamma0(4) of weight 6 over Rational Field
In [111]:
ModularForms(Gamma0(27),2).eisenstein_series()
Out[111]:
[
q^3 + O(q^6),
q - 3*q^2 + 7*q^4 - 6*q^5 + O(q^6),
1/12 + q + 3*q^2 + q^3 + 7*q^4 + 6*q^5 + O(q^6),
1/3 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + O(q^6),
13/12 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + O(q^6)
]
In [112]:
ModularForms(Gamma1(5),3).eisenstein_series()
Out[112]:
[
-1/5*zeta4 - 2/5 + q + (4*zeta4 + 1)*q^2 + (-9*zeta4 + 1)*q^3 + (4*zeta4 - 15)*q^4 + q^5 + O(q^6),
q + (zeta4 + 4)*q^2 + (-zeta4 + 9)*q^3 + (4*zeta4 + 15)*q^4 + 25*q^5 + O(q^6),
1/5*zeta4 - 2/5 + q + (-4*zeta4 + 1)*q^2 + (9*zeta4 + 1)*q^3 + (-4*zeta4 - 15)*q^4 + q^5 + O(q^6),
q + (-zeta4 + 4)*q^2 + (zeta4 + 9)*q^3 + (-4*zeta4 + 15)*q^4 + 25*q^5 + O(q^6)
]
In [98]:
M=ModularForms(Gamma0(12), 6, prec=12)
In [99]:
M
Out[99]:
Modular Forms space of dimension 13 for Congruence Subgroup Gamma0(12) of weight 6 over Rational Field
In [100]:
M.cuspidal_subspace().echelon_basis()
Out[100]:
[
q + 39*q^9 - 192*q^11 + O(q^12),
q^2 + 4*q^8 - 30*q^10 + O(q^12),
q^3 - 12*q^9 + O(q^12),
q^4 - 6*q^8 + O(q^12),
q^5 - 15*q^9 + 38*q^11 + O(q^12),
q^6 - 4*q^8 + 4*q^10 + O(q^12),
q^7 - 6*q^9 + 15*q^11 + O(q^12)
]
In [101]:
M.integral_basis()
Out[101]:
[
1 + O(q^12),
q + O(q^12),
q^2 + O(q^12),
q^3 + O(q^12),
q^4 + O(q^12),
q^5 + O(q^12),
q^6 + O(q^12),
q^7 + O(q^12),
q^8 + O(q^12),
q^9 + O(q^12),
q^10 + O(q^12),
q^11 + O(q^12),
O(q^12)
]
In [103]:
S = ModularForms(11,2).cuspidal_submodule()
In [104]:
S.q_expansion_basis(12)
Out[104]:
[
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 - 2*q^10 + q^11 + O(q^12)
]
In [ ]:

Eisenstein Series

In [124]:
E1 = EisensteinForms(25, 4)
In [125]:
E.new_eisenstein_series()
Out[125]:
[q + (-2*zeta6 + 1)*q^2 + (-2*zeta6 - 1)*q^4 + (5*zeta6 - 1)*q^5 + O(q^6),
 q + (-zeta6 - 1)*q^2 + (zeta6 + 2)*q^4 + (zeta6 - 5)*q^5 + O(q^6)]
In [126]:
from sage.modular.modform.eis_series_cython import Ek_ZZ
In [131]:
Ek_ZZ(4,15)
Out[131]:
[1, 1, 9, 28, 73, 126, 252, 344, 585, 757, 1134, 1332, 2044, 2198, 3096]
In [ ]:
In [ ]:

Newforms

In [ ]:
In [ ]:
In [78]:
Newforms(Gamma0(11), 2)
Out[78]:
[q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)]
In [ ]:

Hecke Operators

In [114]:
ModularForms(17,4).basis()
Out[114]:
[
q + 2*q^5 + O(q^6),
q^2 - 3/2*q^5 + O(q^6),
q^3 + O(q^6),
q^4 - 1/2*q^5 + O(q^6),
1 + O(q^6),
q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6)
]
In [113]:
ModularForms(17,4).hecke_polynomial(2)
Out[113]:
x^6 - 16*x^5 + 18*x^4 + 608*x^3 - 1371*x^2 - 4968*x + 7776
In [122]:
E = ModularForms(e,2,prec=10).eisenstein_subspace()
In [132]:
E.eisenstein_subspace().T(2).matrix().fcp()
Out[132]:
(x + 2*zeta3 + 1) * (x + zeta3 + 2) * (x - zeta3 - 2)^2 * (x - 2*zeta3 - 1)^2
In [162]:
hecke_operator_on_basis(ModularForms(1,12).q_expansion_basis(30), 3, 12)
Out[162]:
[   252      0]
[     0 177148]
In [163]:
M = ModularForms(1,12);
hecke_operator_on_qexp(M.basis()[0], 3, 12)
Out[163]:
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + O(q^5)
In [ ]:
In [ ]:

Ring of Modular Forms

In [ ]:
In [ ]:
In [136]:
M=ModularFormsRing(Gamma0(12))
In [137]:
M.generators()
Out[137]:
[(2, 1 + 24*q^6 + O(q^10)),
 (2, q + 6*q^5 + 8*q^7 - 3*q^9 + O(q^10)),
 (2, q^2 + 5*q^6 - 2*q^8 + O(q^10)),
 (2, q^3 + 4*q^9 + O(q^10)),
 (2, q^4 - 2*q^6 + 3*q^8 + O(q^10))]
In [138]:
M2 = ModularFormsRing(1)
In [142]:
E4 = M2.0; E6=M2.1
In [143]:
E4
Out[143]:
q + 11*q^2 + 12*q^3 + 2*q^4 + q^5 + O(q^6)
In [144]:
E6
Out[144]:
1 + 12*q^2 + 12*q^3 + 12*q^4 + 12*q^5 + O(q^6)
In [ ]:

Elliptic Curves

In [145]:
E = EllipticCurve('11a')
In [146]:
E
Out[146]:
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
In [153]:
f2 = E.modular_form()
In [154]:
f2
Out[154]:
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
In [155]:
f2.has_cm()
Out[155]:
False
In [156]:
E = EllipticCurve([-1, 0])
In [157]:
E
Out[157]:
Elliptic Curve defined by y^2 = x^3 - x over Rational Field
In [159]:
f2 = E.modular_form()
In [160]:
f2
Out[160]:
q - 2*q^5 + O(q^6)
In [161]:
f2.has_cm()
Out[161]:
True

Modular Symbols

In [164]:
set_modsym_print_mode ('modular');
In [165]:
M = ModularSymbols(11, 2)
In [166]:
M.basis()
Out[166]:
({Infinity, 0}, {-1/8, 0}, {-1/9, 0})
In [223]:
[ModularSymbols(Gamma1(7),k).dimension() for k in [2,3,4,5]]
Out[223]:
[5, 8, 12, 16]
In [167]:
M = ModularSymbols(23,2,base_ring=QQ)
In [168]:
M.T(2)
Out[168]:
Hecke operator T_2 on Modular Symbols space of dimension 5 for Gamma_0(23) of weight 2 with sign 0 over Rational Field
In [169]:
M.T(2).matrix()
Out[169]:
[ 3  0  0  0 -1]
[ 0  0  1 -1  0]
[ 0  0  1 -1  1]
[ 0 -1  2 -2  1]
[ 0 -1  1  0 -1]
In [170]:
M.T(2).charpoly('x').factor()
Out[170]:
(x - 3) * (x^2 + x - 1)^2
In [171]:
M1 = ModularSymbols(11, 4)
In [172]:
M1.basis()
Out[172]:
(X^2*{0, Infinity},
 36*X^2*{-1/6, 0} + 12*X*Y*{-1/6, 0} + Y^2*{-1/6, 0},
 49*X^2*{-1/7, 0} + 14*X*Y*{-1/7, 0} + Y^2*{-1/7, 0},
 64*X^2*{-1/8, 0} + 16*X*Y*{-1/8, 0} + Y^2*{-1/8, 0},
 81*X^2*{-1/9, 0} + 18*X*Y*{-1/9, 0} + Y^2*{-1/9, 0},
 100*X^2*{-1/10, 0} + 20*X*Y*{-1/10, 0} + Y^2*{-1/10, 0})
In [173]:
M.T(5).matrix()
Out[173]:
[ 6  0  0  0 -2]
[ 0  0  2 -2  0]
[ 0  0  2 -2  2]
[ 0 -2  4 -4  2]
[ 0 -2  2  0 -2]
In [174]:
M1.cuspidal_subspace()
Out[174]:
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 6 for Gamma_0(11) of weight 4 with sign 0 over Rational Field
In [176]:
ModularForms(11, 4).basis()
Out[176]:
[
q + 3*q^3 - 6*q^4 - 7*q^5 + O(q^6),
q^2 - 4*q^3 + 2*q^4 + 8*q^5 + O(q^6),
1 + O(q^6),
q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6)
]
In [ ]:
In [177]:
m = ModularSymbols(Gamma1(3),12); m.dimension()
Out[177]:
8
In [178]:
m.cuspidal_subspace().dimension()
Out[178]:
6
In [179]:
m.cuspidal_subspace().new_subspace().dimension()
Out[179]:
2
In [181]:
CuspForms(Gamma1(3), 12, prec=10).basis()
Out[181]:
[
q - 176*q^4 + 2430*q^5 - 5832*q^6 - 19336*q^7 + 101088*q^8 - 107163*q^9 + O(q^10),
q^2 + 54*q^4 - 100*q^5 - 243*q^6 - 108*q^7 + 692*q^8 + 2916*q^9 + O(q^10),
q^3 - 24*q^6 + 252*q^9 + O(q^10)
]
In [183]:
[m.cuspidal_subspace().T(n).matrix() for n in [2..10]]
Out[183]:
[
[     -24    236/3     50/3   -286/3       14  -623/30]  [       3411       -1364    -14965/8     -7723/8    -6639/40     -521/80]
[       0      462     -243     -243        0 -1701/20]  [          0    -58743/7    -38637/7    -38637/7           0   60507/175]
[       0     -246     -225      447      -81  1671/20]  [     -25920      9528/7     53847/7      6408/7        1242  128259/350]
[       0     -246      447     -225       81    51/20]  [      25920     49848/7    -13752/7     33687/7       -1242 -243441/350]
[       0     1800     2100    -3900      726     -690]  [     207360      -12360      -63900       -7740      -10179       -2790]
[       0     3600    -1800    -1800        0     -654], [          0      -70800      -48600      -48600           0        2673],

[    -1472      4248       900     -5148       756   -5607/5]
[        0     24772    -13122    -13122         0 -45927/10]
[        0    -13284    -12326     24138     -4374  45117/10]
[        0    -13284     24138    -12326      4374   1377/10]
[        0     97200    113400   -210600     39028    -37260]
[        0    194400    -97200    -97200         0    -35492],

[    4830 -23600/3  -5000/3  28600/3    -1400   6230/3]  [       -81864         13620         40845         46335        2907/5       26013/5]
[       0   -43770    24300    24300        0     8505]  [            0      583146/7     1340631/7     1340631/7             0   8658333/700]
[       0    24600    24930   -44700     8100    -8355]  [       622080      189774/7     -950427/7     -914139/7        -10125 -20368287/700]
[       0    24600   -44700    24930    -8100     -255]  [      -622080     -777906/7     -430299/7     -466587/7         10125  11251413/700]
[       0  -180000  -210000   390000   -70170    69000]  [     -4976640       -140760       1023300       1133460         62046        234630]
[       0  -360000   180000   180000        0    67830], [            0        824400       1603800       1603800             0         88938],

[  -16744    -8496    -1800    10296    -1512  11214/5]
[       0   -69232    26244    26244        0  45927/5]
[       0    26568     4964   -48276     8748 -45117/5]
[       0    26568   -48276     4964    -8748  -1377/5]
[       0  -194400  -226800   421200   -97744    74520]
[       0  -388800   194400   194400        0    51296],

[     84480   163312/3    34600/3  -197912/3       9688 -215558/15]  [     682425     -114336   -845595/2  -1042533/2   -10017/10   -248787/4]
[         0     420792    -168156    -168156          0  -294273/5]  [          0     -874719    -2099520    -2099520           0 -4021893/25]
[         0    -170232     -54612     309324     -56052   289083/5]  [   -6531840     -374328     1175229     1534140       76788  8399457/25]
[         0    -170232     309324     -54612      56052     8823/5]  [    6531840     1077192      808380      449469      -76788 -4196043/25]
[         0    1245600    1453200   -2698800     603480    -477480]  [   52254720     2134080    -9979200   -13322880     -555255    -2715120]
[         0    2491200   -1245600   -1245600          0    -351480], [          0    -7344000   -17496000   -17496000           0    -1340631],

[  -115920   -233640    -49500    283140    -41580     61677]
[        0  -1559340    721710    721710         0  505197/2]
[        0    730620    481050  -1327590    240570 -496287/2]
[        0    730620  -1327590    481050   -240570  -15147/2]
[        0  -5346000  -6237000  11583000  -2343420   2049300]
[        0 -10692000   5346000   5346000         0   1755180]
]
In [184]:
ModularSymbols(2, 8)[1].q_eigenform(5, 'a')
Out[184]:
q - 8*q^2 + 12*q^3 + 64*q^4 + O(q^5)
In [185]:
M = ModularSymbols(1, 24).cuspidal_submodule()
In [190]:
M.q_expansion_module(9, QQ)
Out[190]:
Vector space of degree 9 and dimension 2 over Rational Field
Basis matrix:
[          0           1           0      195660    12080128    44656110  -982499328  -147247240 22106234880]
[          0           0           1         -48        1080      -15040      143820     -985824     4857920]
In [186]:
M.q_expansion_basis(8, algorithm='eigen')
Out[186]:
[
q + 195660*q^3 + 12080128*q^4 + 44656110*q^5 - 982499328*q^6 - 147247240*q^7 + O(q^8),
q^2 - 48*q^3 + 1080*q^4 - 15040*q^5 + 143820*q^6 - 985824*q^7 + O(q^8)
]
In [187]:
CuspForms(1, 24).basis()
Out[187]:
[
q + 195660*q^3 + 12080128*q^4 + 44656110*q^5 + O(q^6),
q^2 - 48*q^3 + 1080*q^4 - 15040*q^5 + O(q^6)
]
In [ ]:
In [192]:
M = ModularSymbols(11,4,base_ring=GF(2)); M.basis()
Out[192]:
(X*Y*{Infinity, 0},
 X*Y*{-1/8, 0},
 X^2*{-1/9, 0} + X*Y*{-1/9, 0},
 X^2*{0, Infinity},
 Y^2*{-1/8, 0},
 X^2*{-1/9, 0} + Y^2*{-1/9, 0},
 Y^2*{-1/10, 0})
In [ ]:
In [197]:
M = ModularSymbols(11, 2)
In [198]:
M
Out[198]:
Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
In [199]:
M.basis()
Out[199]:
({Infinity, 0}, {-1/8, 0}, {-1/9, 0})
In [228]:
M.cuspidal_submodule().basis()
Out[228]:
()
In [200]:
M.modular_symbol([2/11, oo])
Out[200]:
-{-1/9, 0}
In [203]:
M.modular_symbol([0, -2/5, 0])
Out[203]:
-{-1/8, 0} + 2*{-1/9, 0}
In [ ]:
In [229]:
M = ModularSymbols(8, 4)
In [230]:
M.basis()
Out[230]:
(X^2*{0, Infinity},
 36*X^2*{-1/6, 0} + 12*X*Y*{-1/6, 0} + Y^2*{-1/6, 0},
 49*X^2*{-1/7, 0} + 14*X*Y*{-1/7, 0} + Y^2*{-1/7, 0},
 X^2*{0, 1/2},
 9*X^2*{1/3, 1/2} - 6*X*Y*{1/3, 1/2} + Y^2*{1/3, 1/2},
 X^2*{0, 1/4})
In [231]:
M.cuspidal_submodule().basis()
Out[231]:
(36*X^2*{-1/6, 0} + 12*X*Y*{-1/6, 0} + Y^2*{-1/6, 0} - 49*X^2*{-1/7, 0} - 14*X*Y*{-1/7, 0} - Y^2*{-1/7, 0},
 X^2*{0, 1/2} - 9*X^2*{1/3, 1/2} + 6*X*Y*{1/3, 1/2} - Y^2*{1/3, 1/2})
In [ ]:
In [232]:
M.modular_symbol([0, -2/5, 0])
Out[232]:
-576*X^2*{-1/6, 0} - 192*X*Y*{-1/6, 0} - 16*Y^2*{-1/6, 0} - 392*X^2*{-1/7, 0} - 112*X*Y*{-1/7, 0} - 8*Y^2*{-1/7, 0} + 88*X^2*{0, 1/2} - 792*X^2*{1/3, 1/2} + 528*X*Y*{1/3, 1/2} - 88*Y^2*{1/3, 1/2}
In [233]:
M.modular_symbol([2, -2/5, 0])
Out[233]:
-72*X^2*{-1/6, 0} - 24*X*Y*{-1/6, 0} - 2*Y^2*{-1/6, 0} - 98*X^2*{-1/7, 0} - 28*X*Y*{-1/7, 0} - 2*Y^2*{-1/7, 0} + 14*X^2*{0, 1/2} - 126*X^2*{1/3, 1/2} + 84*X*Y*{1/3, 1/2} - 14*Y^2*{1/3, 1/2}
In [234]:
set_modsym_print_mode('manin')
In [235]:
M.modular_symbol([2, -2/5, 0])
Out[235]:
-2*[X^2,(1,6)] - 2*[X^2,(1,7)] + 14*[X^2,(2,1)] - 14*[X^2,(2,3)]
In [236]:
M.modular_symbol([0, -2/5, 0])
Out[236]:
-16*[X^2,(1,6)] - 8*[X^2,(1,7)] + 88*[X^2,(2,1)] - 88*[X^2,(2,3)]
In [237]:
set_modsym_print_mode('modular')
In [238]:
M.modular_symbol([2, -2/5, 0])
Out[238]:
-72*X^2*{-1/6, 0} - 24*X*Y*{-1/6, 0} - 2*Y^2*{-1/6, 0} - 98*X^2*{-1/7, 0} - 28*X*Y*{-1/7, 0} - 2*Y^2*{-1/7, 0} + 14*X^2*{0, 1/2} - 126*X^2*{1/3, 1/2} + 84*X*Y*{1/3, 1/2} - 14*Y^2*{1/3, 1/2}
In [ ]:

Manin Symbols

In [239]:
from sage.modular.modsym.manin_symbol import ManinSymbol;
from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma0
In [ ]:
In [224]:
A=ModularSymbols(11,2).manin_symbols()
In [225]:
A
Out[225]:
Manin Symbol List of weight 2 for Gamma0(11)
In [226]:
len(A)
Out[226]:
12
In [194]:
ModularSymbols(11,2).manin_symbols_basis()
Out[194]:
[(1,0), (1,8), (1,9)]
In [ ]:
In [195]:
ModularSymbols(11,6).manin_symbols()
Out[195]:
Manin Symbol List of weight 6 for Gamma0(11)
In [196]:
ModularSymbols(11,6).manin_symbols_basis()
Out[196]:
[[X^4,(0,1)],
 [X^4,(1,2)],
 [X^4,(1,3)],
 [X^4,(1,4)],
 [X^4,(1,5)],
 [X^4,(1,6)],
 [X^4,(1,7)],
 [X^4,(1,8)],
 [X^4,(1,9)],
 [X^4,(1,10)]]
In [ ]: