gp: [N,k,chi] = [729,2,Mod(1,729)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(729, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("729.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [6,3,0,3]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
gp: f = lf[1] \\ Warning: the index may be different
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the q q q -expansion are expressed in terms of a basis 1 , β 1 , … , β 5 1,\beta_1,\ldots,\beta_{5} 1 , β 1 , … , β 5 for the coefficient ring described below.
We also show the integral q q q -expansion of the trace form .
Basis of coefficient ring in terms of a root ν \nu ν of
x 6 − 3 x 5 − 3 x 4 + 10 x 3 + 3 x 2 − 6 x + 1 x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 x 6 − 3 x 5 − 3 x 4 + 1 0 x 3 + 3 x 2 − 6 x + 1
x^6 - 3*x^5 - 3*x^4 + 10*x^3 + 3*x^2 - 6*x + 1
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
ν 2 − ν − 2 \nu^{2} - \nu - 2 ν 2 − ν − 2
v^2 - v - 2
β 3 \beta_{3} β 3 = = =
ν 3 − 2 ν 2 − 2 ν + 2 \nu^{3} - 2\nu^{2} - 2\nu + 2 ν 3 − 2 ν 2 − 2 ν + 2
v^3 - 2*v^2 - 2*v + 2
β 4 \beta_{4} β 4 = = =
ν 4 − 3 ν 3 − ν 2 + 6 ν − 1 \nu^{4} - 3\nu^{3} - \nu^{2} + 6\nu - 1 ν 4 − 3 ν 3 − ν 2 + 6 ν − 1
v^4 - 3*v^3 - v^2 + 6*v - 1
β 5 \beta_{5} β 5 = = =
ν 5 − 3 ν 4 − 3 ν 3 + 9 ν 2 + 4 ν − 3 \nu^{5} - 3\nu^{4} - 3\nu^{3} + 9\nu^{2} + 4\nu - 3 ν 5 − 3 ν 4 − 3 ν 3 + 9 ν 2 + 4 ν − 3
v^5 - 3*v^4 - 3*v^3 + 9*v^2 + 4*v - 3
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 2 + β 1 + 2 \beta_{2} + \beta _1 + 2 β 2 + β 1 + 2
b2 + b1 + 2
ν 3 \nu^{3} ν 3 = = =
β 3 + 2 β 2 + 4 β 1 + 2 \beta_{3} + 2\beta_{2} + 4\beta _1 + 2 β 3 + 2 β 2 + 4 β 1 + 2
b3 + 2*b2 + 4*b1 + 2
ν 4 \nu^{4} ν 4 = = =
β 4 + 3 β 3 + 7 β 2 + 7 β 1 + 9 \beta_{4} + 3\beta_{3} + 7\beta_{2} + 7\beta _1 + 9 β 4 + 3 β 3 + 7 β 2 + 7 β 1 + 9
b4 + 3*b3 + 7*b2 + 7*b1 + 9
ν 5 \nu^{5} ν 5 = = =
β 5 + 3 β 4 + 12 β 3 + 18 β 2 + 20 β 1 + 18 \beta_{5} + 3\beta_{4} + 12\beta_{3} + 18\beta_{2} + 20\beta _1 + 18 β 5 + 3 β 4 + 1 2 β 3 + 1 8 β 2 + 2 0 β 1 + 1 8
b5 + 3*b4 + 12*b3 + 18*b2 + 20*b1 + 18
For each embedding ι m \iota_m ι m of the coefficient field, the values ι m ( a n ) \iota_m(a_n) ι m ( a n ) are shown below.
For more information on an embedded modular form you can click on its label.
gp: mfembed(f)
Refresh table
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
T 2 6 − 3 T 2 5 − 3 T 2 4 + 12 T 2 3 − 9 T 2 + 3 T_{2}^{6} - 3T_{2}^{5} - 3T_{2}^{4} + 12T_{2}^{3} - 9T_{2} + 3 T 2 6 − 3 T 2 5 − 3 T 2 4 + 1 2 T 2 3 − 9 T 2 + 3
T2^6 - 3*T2^5 - 3*T2^4 + 12*T2^3 - 9*T2 + 3
acting on S 2 n e w ( Γ 0 ( 729 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(729)) S 2 n e w ( Γ 0 ( 7 2 9 ) ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 6 − 3 T 5 + ⋯ + 3 T^{6} - 3 T^{5} + \cdots + 3 T 6 − 3 T 5 + ⋯ + 3
T^6 - 3*T^5 - 3*T^4 + 12*T^3 - 9*T + 3
3 3 3
T 6 T^{6} T 6
T^6
5 5 5
T 6 − 6 T 5 + ⋯ + 3 T^{6} - 6 T^{5} + \cdots + 3 T 6 − 6 T 5 + ⋯ + 3
T^6 - 6*T^5 + 6*T^4 + 24*T^3 - 54*T^2 + 27*T + 3
7 7 7
T 6 − 15 T 4 + ⋯ − 17 T^{6} - 15 T^{4} + \cdots - 17 T 6 − 1 5 T 4 + ⋯ − 1 7
T^6 - 15*T^4 + 11*T^3 + 36*T^2 - 15*T - 17
11 11 1 1
T 6 − 12 T 5 + ⋯ + 3 T^{6} - 12 T^{5} + \cdots + 3 T 6 − 1 2 T 5 + ⋯ + 3
T^6 - 12*T^5 + 51*T^4 - 96*T^3 + 81*T^2 - 27*T + 3
13 13 1 3
T 6 − 24 T 4 + ⋯ + 1 T^{6} - 24 T^{4} + \cdots + 1 T 6 − 2 4 T 4 + ⋯ + 1
T^6 - 24*T^4 + 2*T^3 + 90*T^2 + 57*T + 1
17 17 1 7
T 6 − 9 T 5 + ⋯ + 27 T^{6} - 9 T^{5} + \cdots + 27 T 6 − 9 T 5 + ⋯ + 2 7
T^6 - 9*T^5 + 9*T^4 + 54*T^3 - 54*T^2 - 81*T + 27
19 19 1 9
T 6 − 3 T 5 + ⋯ + 19 T^{6} - 3 T^{5} + \cdots + 19 T 6 − 3 T 5 + ⋯ + 1 9
T^6 - 3*T^5 - 30*T^4 + 38*T^3 + 168*T^2 - 120*T + 19
23 23 2 3
T 6 − 15 T 5 + ⋯ + 327 T^{6} - 15 T^{5} + \cdots + 327 T 6 − 1 5 T 5 + ⋯ + 3 2 7
T^6 - 15*T^5 + 60*T^4 + 33*T^3 - 432*T^2 + 45*T + 327
29 29 2 9
T 6 − 12 T 5 + ⋯ − 213 T^{6} - 12 T^{5} + \cdots - 213 T 6 − 1 2 T 5 + ⋯ − 2 1 3
T^6 - 12*T^5 - 3*T^4 + 462*T^3 - 1566*T^2 + 1098*T - 213
31 31 3 1
T 6 − 51 T 4 + ⋯ + 163 T^{6} - 51 T^{4} + \cdots + 163 T 6 − 5 1 T 4 + ⋯ + 1 6 3
T^6 - 51*T^4 + 191*T^3 - 180*T^2 - 105*T + 163
37 37 3 7
T 6 − 3 T 5 + ⋯ + 4933 T^{6} - 3 T^{5} + \cdots + 4933 T 6 − 3 T 5 + ⋯ + 4 9 3 3
T^6 - 3*T^5 - 57*T^4 + 254*T^3 + 492*T^2 - 3873*T + 4933
41 41 4 1
T 6 − 15 T 5 + ⋯ + 3351 T^{6} - 15 T^{5} + \cdots + 3351 T 6 − 1 5 T 5 + ⋯ + 3 3 5 1
T^6 - 15*T^5 + 6*T^4 + 600*T^3 - 1350*T^2 - 2286*T + 3351
43 43 4 3
T 6 − 96 T 4 + ⋯ + 1819 T^{6} - 96 T^{4} + \cdots + 1819 T 6 − 9 6 T 4 + ⋯ + 1 8 1 9
T^6 - 96*T^4 + 173*T^3 + 2358*T^2 - 7602*T + 1819
47 47 4 7
T 6 − 21 T 5 + ⋯ + 6537 T^{6} - 21 T^{5} + \cdots + 6537 T 6 − 2 1 T 5 + ⋯ + 6 5 3 7
T^6 - 21*T^5 + 105*T^4 + 228*T^3 - 2106*T^2 + 279*T + 6537
53 53 5 3
T 6 − 9 T 5 + ⋯ − 12393 T^{6} - 9 T^{5} + \cdots - 12393 T 6 − 9 T 5 + ⋯ − 1 2 3 9 3
T^6 - 9*T^5 - 108*T^4 + 513*T^3 + 4617*T^2 + 2916*T - 12393
59 59 5 9
T 6 − 24 T 5 + ⋯ − 13281 T^{6} - 24 T^{5} + \cdots - 13281 T 6 − 2 4 T 5 + ⋯ − 1 3 2 8 1
T^6 - 24*T^5 + 141*T^4 + 429*T^3 - 5832*T^2 + 15957*T - 13281
61 61 6 1
T 6 + 9 T 5 + ⋯ + 16543 T^{6} + 9 T^{5} + \cdots + 16543 T 6 + 9 T 5 + ⋯ + 1 6 5 4 3
T^6 + 9*T^5 - 159*T^4 - 520*T^3 + 9198*T^2 - 24945*T + 16543
67 67 6 7
T 6 + 9 T 5 + ⋯ − 2879 T^{6} + 9 T^{5} + \cdots - 2879 T 6 + 9 T 5 + ⋯ − 2 8 7 9
T^6 + 9*T^5 - 114*T^4 - 934*T^3 + 1152*T^2 + 3126*T - 2879
71 71 7 1
T 6 − 27 T 5 + ⋯ + 27 T^{6} - 27 T^{5} + \cdots + 27 T 6 − 2 7 T 5 + ⋯ + 2 7
T^6 - 27*T^5 + 225*T^4 - 486*T^3 - 702*T^2 + 243*T + 27
73 73 7 3
T 6 + 6 T 5 + ⋯ − 431 T^{6} + 6 T^{5} + \cdots - 431 T 6 + 6 T 5 + ⋯ − 4 3 1
T^6 + 6*T^5 - 174*T^4 - 250*T^3 + 3984*T^2 + 1437*T - 431
79 79 7 9
T 6 − 177 T 4 + ⋯ + 1873 T^{6} - 177 T^{4} + \cdots + 1873 T 6 − 1 7 7 T 4 + ⋯ + 1 8 7 3
T^6 - 177*T^4 - 70*T^3 + 5085*T^2 - 6279*T + 1873
83 83 8 3
T 6 − 12 T 5 + ⋯ − 83373 T^{6} - 12 T^{5} + \cdots - 83373 T 6 − 1 2 T 5 + ⋯ − 8 3 3 7 3
T^6 - 12*T^5 - 165*T^4 + 1155*T^3 + 10422*T^2 - 3249*T - 83373
89 89 8 9
T 6 − 9 T 5 + ⋯ + 32589 T^{6} - 9 T^{5} + \cdots + 32589 T 6 − 9 T 5 + ⋯ + 3 2 5 8 9
T^6 - 9*T^5 - 180*T^4 + 1026*T^3 + 4968*T^2 - 30456*T + 32589
97 97 9 7
T 6 − 204 T 4 + ⋯ − 8171 T^{6} - 204 T^{4} + \cdots - 8171 T 6 − 2 0 4 T 4 + ⋯ − 8 1 7 1
T^6 - 204*T^4 + 713*T^3 + 5598*T^2 - 21966*T - 8171
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