gp: [N,k,chi] = [338,4,Mod(1,338)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(338, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("338.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [1,2,3,4,-11]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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
13 13 1 3
+ 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 4 n e w ( Γ 0 ( 338 ) ) S_{4}^{\mathrm{new}}(\Gamma_0(338)) S 4 n e w ( Γ 0 ( 3 3 8 ) ) :
T 3 − 3 T_{3} - 3 T 3 − 3
T3 - 3
T 5 + 11 T_{5} + 11 T 5 + 1 1
T5 + 11
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T − 2 T - 2 T − 2
T - 2
3 3 3
T − 3 T - 3 T − 3
T - 3
5 5 5
T + 11 T + 11 T + 1 1
T + 11
7 7 7
T + 19 T + 19 T + 1 9
T + 19
11 11 1 1
T − 38 T - 38 T − 3 8
T - 38
13 13 1 3
T T T
T
17 17 1 7
T + 51 T + 51 T + 5 1
T + 51
19 19 1 9
T + 90 T + 90 T + 9 0
T + 90
23 23 2 3
T + 52 T + 52 T + 5 2
T + 52
29 29 2 9
T + 190 T + 190 T + 1 9 0
T + 190
31 31 3 1
T + 292 T + 292 T + 2 9 2
T + 292
37 37 3 7
T − 441 T - 441 T − 4 4 1
T - 441
41 41 4 1
T + 312 T + 312 T + 3 1 2
T + 312
43 43 4 3
T − 373 T - 373 T − 3 7 3
T - 373
47 47 4 7
T − 41 T - 41 T − 4 1
T - 41
53 53 5 3
T − 468 T - 468 T − 4 6 8
T - 468
59 59 5 9
T + 530 T + 530 T + 5 3 0
T + 530
61 61 6 1
T − 592 T - 592 T − 5 9 2
T - 592
67 67 6 7
T − 206 T - 206 T − 2 0 6
T - 206
71 71 7 1
T − 863 T - 863 T − 8 6 3
T - 863
73 73 7 3
T − 322 T - 322 T − 3 2 2
T - 322
79 79 7 9
T + 460 T + 460 T + 4 6 0
T + 460
83 83 8 3
T + 528 T + 528 T + 5 2 8
T + 528
89 89 8 9
T + 870 T + 870 T + 8 7 0
T + 870
97 97 9 7
T − 346 T - 346 T − 3 4 6
T - 346
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