gp: [N,k,chi] = [63,4,Mod(1,63)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("63.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(63, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
Newform invariants
sage: traces = [1,-4]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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
3 3 3
− 1 -1 − 1
7 7 7
+ 1 +1 + 1
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
T 2 + 4 T_{2} + 4 T 2 + 4
T2 + 4
acting on S 4 n e w ( Γ 0 ( 63 ) ) S_{4}^{\mathrm{new}}(\Gamma_0(63)) S 4 n e w ( Γ 0 ( 6 3 ) ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T + 4 T + 4 T + 4
T + 4
3 3 3
T T T
T
5 5 5
T − 4 T - 4 T − 4
T - 4
7 7 7
T + 7 T + 7 T + 7
T + 7
11 11 1 1
T + 62 T + 62 T + 6 2
T + 62
13 13 1 3
T + 62 T + 62 T + 6 2
T + 62
17 17 1 7
T + 84 T + 84 T + 8 4
T + 84
19 19 1 9
T − 100 T - 100 T − 1 0 0
T - 100
23 23 2 3
T − 42 T - 42 T − 4 2
T - 42
29 29 2 9
T − 10 T - 10 T − 1 0
T - 10
31 31 3 1
T + 48 T + 48 T + 4 8
T + 48
37 37 3 7
T + 246 T + 246 T + 2 4 6
T + 246
41 41 4 1
T − 248 T - 248 T − 2 4 8
T - 248
43 43 4 3
T − 68 T - 68 T − 6 8
T - 68
47 47 4 7
T + 324 T + 324 T + 3 2 4
T + 324
53 53 5 3
T + 258 T + 258 T + 2 5 8
T + 258
59 59 5 9
T + 120 T + 120 T + 1 2 0
T + 120
61 61 6 1
T − 622 T - 622 T − 6 2 2
T - 622
67 67 6 7
T − 904 T - 904 T − 9 0 4
T - 904
71 71 7 1
T − 678 T - 678 T − 6 7 8
T - 678
73 73 7 3
T + 642 T + 642 T + 6 4 2
T + 642
79 79 7 9
T − 740 T - 740 T − 7 4 0
T - 740
83 83 8 3
T + 468 T + 468 T + 4 6 8
T + 468
89 89 8 9
T + 200 T + 200 T + 2 0 0
T + 200
97 97 9 7
T + 1266 T + 1266 T + 1 2 6 6
T + 1266
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