gp: [N,k,chi] = [720,6,Mod(1,720)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(720, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 6, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("720.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
Newform invariants
sage: traces = [1,0,0,0,25,0,-218,0,0,0,-480]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == 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)
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
p p p
Sign
2 2 2
− 1 -1 − 1
3 3 3
− 1 -1 − 1
5 5 5
− 1 -1 − 1
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S 6 n e w ( Γ 0 ( 720 ) ) S_{6}^{\mathrm{new}}(\Gamma_0(720)) S 6 n e w ( Γ 0 ( 7 2 0 ) ) :
T 7 + 218 T_{7} + 218 T 7 + 2 1 8
T7 + 218
T 11 + 480 T_{11} + 480 T 1 1 + 4 8 0
T11 + 480
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T T T
T
3 3 3
T T T
T
5 5 5
T − 25 T - 25 T − 2 5
T - 25
7 7 7
T + 218 T + 218 T + 2 1 8
T + 218
11 11 1 1
T + 480 T + 480 T + 4 8 0
T + 480
13 13 1 3
T + 622 T + 622 T + 6 2 2
T + 622
17 17 1 7
T + 186 T + 186 T + 1 8 6
T + 186
19 19 1 9
T − 1204 T - 1204 T − 1 2 0 4
T - 1204
23 23 2 3
T + 3186 T + 3186 T + 3 1 8 6
T + 3186
29 29 2 9
T + 5526 T + 5526 T + 5 5 2 6
T + 5526
31 31 3 1
T + 9356 T + 9356 T + 9 3 5 6
T + 9356
37 37 3 7
T − 5618 T - 5618 T − 5 6 1 8
T - 5618
41 41 4 1
T − 14394 T - 14394 T − 1 4 3 9 4
T - 14394
43 43 4 3
T − 370 T - 370 T − 3 7 0
T - 370
47 47 4 7
T − 16146 T - 16146 T − 1 6 1 4 6
T - 16146
53 53 5 3
T − 4374 T - 4374 T − 4 3 7 4
T - 4374
59 59 5 9
T + 11748 T + 11748 T + 1 1 7 4 8
T + 11748
61 61 6 1
T − 13202 T - 13202 T − 1 3 2 0 2
T - 13202
67 67 6 7
T − 11542 T - 11542 T − 1 1 5 4 2
T - 11542
71 71 7 1
T + 29532 T + 29532 T + 2 9 5 3 2
T + 29532
73 73 7 3
T − 33698 T - 33698 T − 3 3 6 9 8
T - 33698
79 79 7 9
T + 31208 T + 31208 T + 3 1 2 0 8
T + 31208
83 83 8 3
T + 38466 T + 38466 T + 3 8 4 6 6
T + 38466
89 89 8 9
T + 119514 T + 119514 T + 1 1 9 5 1 4
T + 119514
97 97 9 7
T − 94658 T - 94658 T − 9 4 6 5 8
T - 94658
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