gp: [N,k,chi] = [448,4,Mod(1,448)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(448, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("448.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [1,0,-2,0,16,0,7]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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
7 7 7
− 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 ( 448 ) ) S_{4}^{\mathrm{new}}(\Gamma_0(448)) S 4 n e w ( Γ 0 ( 4 4 8 ) ) :
T 3 + 2 T_{3} + 2 T 3 + 2
T3 + 2
T 5 − 16 T_{5} - 16 T 5 − 1 6
T5 - 16
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T T T
T
3 3 3
T + 2 T + 2 T + 2
T + 2
5 5 5
T − 16 T - 16 T − 1 6
T - 16
7 7 7
T − 7 T - 7 T − 7
T - 7
11 11 1 1
T − 24 T - 24 T − 2 4
T - 24
13 13 1 3
T − 68 T - 68 T − 6 8
T - 68
17 17 1 7
T − 54 T - 54 T − 5 4
T - 54
19 19 1 9
T + 46 T + 46 T + 4 6
T + 46
23 23 2 3
T + 176 T + 176 T + 1 7 6
T + 176
29 29 2 9
T − 174 T - 174 T − 1 7 4
T - 174
31 31 3 1
T − 116 T - 116 T − 1 1 6
T - 116
37 37 3 7
T + 74 T + 74 T + 7 4
T + 74
41 41 4 1
T + 10 T + 10 T + 1 0
T + 10
43 43 4 3
T + 480 T + 480 T + 4 8 0
T + 480
47 47 4 7
T − 572 T - 572 T − 5 7 2
T - 572
53 53 5 3
T − 162 T - 162 T − 1 6 2
T - 162
59 59 5 9
T + 86 T + 86 T + 8 6
T + 86
61 61 6 1
T − 904 T - 904 T − 9 0 4
T - 904
67 67 6 7
T − 660 T - 660 T − 6 6 0
T - 660
71 71 7 1
T + 1024 T + 1024 T + 1 0 2 4
T + 1024
73 73 7 3
T − 770 T - 770 T − 7 7 0
T - 770
79 79 7 9
T − 904 T - 904 T − 9 0 4
T - 904
83 83 8 3
T − 682 T - 682 T − 6 8 2
T - 682
89 89 8 9
T + 102 T + 102 T + 1 0 2
T + 102
97 97 9 7
T + 218 T + 218 T + 2 1 8
T + 218
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