gp: [N,k,chi] = [2368,2,Mod(1,2368)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2368, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2368.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [1,0,3,0,4,0,-1,0,6,0,-3]
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
37 37 3 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 2 n e w ( Γ 0 ( 2368 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(2368)) S 2 n e w ( Γ 0 ( 2 3 6 8 ) ) :
T 3 − 3 T_{3} - 3 T 3 − 3
T3 - 3
T 5 − 4 T_{5} - 4 T 5 − 4
T5 - 4
T 7 + 1 T_{7} + 1 T 7 + 1
T7 + 1
T 11 + 3 T_{11} + 3 T 1 1 + 3
T11 + 3
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T T T
T
3 3 3
T − 3 T - 3 T − 3
T - 3
5 5 5
T − 4 T - 4 T − 4
T - 4
7 7 7
T + 1 T + 1 T + 1
T + 1
11 11 1 1
T + 3 T + 3 T + 3
T + 3
13 13 1 3
T + 2 T + 2 T + 2
T + 2
17 17 1 7
T − 8 T - 8 T − 8
T - 8
19 19 1 9
T − 2 T - 2 T − 2
T - 2
23 23 2 3
T + 6 T + 6 T + 6
T + 6
29 29 2 9
T T T
T
31 31 3 1
T + 8 T + 8 T + 8
T + 8
37 37 3 7
T − 1 T - 1 T − 1
T - 1
41 41 4 1
T + 5 T + 5 T + 5
T + 5
43 43 4 3
T − 2 T - 2 T − 2
T - 2
47 47 4 7
T − 11 T - 11 T − 1 1
T - 11
53 53 5 3
T + 9 T + 9 T + 9
T + 9
59 59 5 9
T − 6 T - 6 T − 6
T - 6
61 61 6 1
T − 6 T - 6 T − 6
T - 6
67 67 6 7
T + 4 T + 4 T + 4
T + 4
71 71 7 1
T − 5 T - 5 T − 5
T - 5
73 73 7 3
T − 11 T - 11 T − 1 1
T - 11
79 79 7 9
T − 8 T - 8 T − 8
T - 8
83 83 8 3
T + 5 T + 5 T + 5
T + 5
89 89 8 9
T + 18 T + 18 T + 1 8
T + 18
97 97 9 7
T + 2 T + 2 T + 2
T + 2
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