gp: [N,k,chi] = [490,8,Mod(1,490)]
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("490.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(490, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
Newform invariants
sage: traces = [1,8,30]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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
5 5 5
+ 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 3 − 30 T_{3} - 30 T 3 − 3 0
T3 - 30
acting on S 8 n e w ( Γ 0 ( 490 ) ) S_{8}^{\mathrm{new}}(\Gamma_0(490)) S 8 n e w ( Γ 0 ( 4 9 0 ) ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T − 8 T - 8 T − 8
T - 8
3 3 3
T − 30 T - 30 T − 3 0
T - 30
5 5 5
T + 125 T + 125 T + 1 2 5
T + 125
7 7 7
T T T
T
11 11 1 1
T − 3 T - 3 T − 3
T - 3
13 13 1 3
T − 1745 T - 1745 T − 1 7 4 5
T - 1745
17 17 1 7
T + 3786 T + 3786 T + 3 7 8 6
T + 3786
19 19 1 9
T + 1945 T + 1945 T + 1 9 4 5
T + 1945
23 23 2 3
T − 79551 T - 79551 T − 7 9 5 5 1
T - 79551
29 29 2 9
T + 94926 T + 94926 T + 9 4 9 2 6
T + 94926
31 31 3 1
T − 127628 T - 127628 T − 1 2 7 6 2 8
T - 127628
37 37 3 7
T + 128257 T + 128257 T + 1 2 8 2 5 7
T + 128257
41 41 4 1
T + 298077 T + 298077 T + 2 9 8 0 7 7
T + 298077
43 43 4 3
T + 875626 T + 875626 T + 8 7 5 6 2 6
T + 875626
47 47 4 7
T − 611559 T - 611559 T − 6 1 1 5 5 9
T - 611559
53 53 5 3
T + 259137 T + 259137 T + 2 5 9 1 3 7
T + 259137
59 59 5 9
T + 2877336 T + 2877336 T + 2 8 7 7 3 3 6
T + 2877336
61 61 6 1
T + 148564 T + 148564 T + 1 4 8 5 6 4
T + 148564
67 67 6 7
T + 1790884 T + 1790884 T + 1 7 9 0 8 8 4
T + 1790884
71 71 7 1
T + 493236 T + 493236 T + 4 9 3 2 3 6
T + 493236
73 73 7 3
T + 2058052 T + 2058052 T + 2 0 5 8 0 5 2
T + 2058052
79 79 7 9
T + 5867074 T + 5867074 T + 5 8 6 7 0 7 4
T + 5867074
83 83 8 3
T + 921132 T + 921132 T + 9 2 1 1 3 2
T + 921132
89 89 8 9
T + 5123082 T + 5123082 T + 5 1 2 3 0 8 2
T + 5123082
97 97 9 7
T + 5878306 T + 5878306 T + 5 8 7 8 3 0 6
T + 5878306
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