gp: [N,k,chi] = [882,4,Mod(1,882)]
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("882.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(882, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
Newform invariants
sage: traces = [2,4,0,8,7,0,0,16,0,14,25,0,49]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == 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)
Coefficients of the q q q β = 1 2 ( 1 + 193 ) \beta = \frac{1}{2}(1 + \sqrt{193}) β = 2 1 ( 1 + 1 9 3 ) q q q trace form .
For each embedding ι m \iota_m ι m ι m ( a n ) \iota_m(a_n) ι m ( a n )
For more information on an embedded modular form you can click on its label.
gp: mfembed(f)
p p p
Sign
2 2 2
− 1 -1 − 1
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 intersection of the kernels of the following linear operators acting on S 4 n e w ( Γ 0 ( 882 ) ) S_{4}^{\mathrm{new}}(\Gamma_0(882)) S 4 n e w ( Γ 0 ( 8 8 2 ) )
T 5 2 − 7 T 5 − 36 T_{5}^{2} - 7T_{5} - 36 T 5 2 − 7 T 5 − 3 6
T 11 2 − 25 T 11 − 2208 T_{11}^{2} - 25T_{11} - 2208 T 1 1 2 − 2 5 T 1 1 − 2 2 0 8
T 13 2 − 49 T 13 − 606 T_{13}^{2} - 49T_{13} - 606 T 1 3 2 − 4 9 T 1 3 − 6 0 6
p p p F p ( T ) F_p(T) F p ( T )
2 2 2
( T − 2 ) 2 (T - 2)^{2} ( T − 2 ) 2
3 3 3
T 2 T^{2} T 2
5 5 5
T 2 − 7 T − 36 T^{2} - 7T - 36 T 2 − 7 T − 3 6
7 7 7
T 2 T^{2} T 2
11 11 1 1
T 2 − 25 T − 2208 T^{2} - 25T - 2208 T 2 − 2 5 T − 2 2 0 8
13 13 1 3
T 2 − 49 T − 606 T^{2} - 49T - 606 T 2 − 4 9 T − 6 0 6
17 17 1 7
T 2 − 98 T − 2424 T^{2} - 98T - 2424 T 2 − 9 8 T − 2 4 2 4
19 19 1 9
T 2 − 119 T + 3106 T^{2} - 119T + 3106 T 2 − 1 1 9 T + 3 1 0 6
23 23 2 3
T 2 + 122 T − 5736 T^{2} + 122T - 5736 T 2 + 1 2 2 T − 5 7 3 6
29 29 2 9
T 2 + 73 T − 1032 T^{2} + 73T - 1032 T 2 + 7 3 T − 1 0 3 2
31 31 3 1
T 2 + 98 T − 35427 T^{2} + 98T - 35427 T 2 + 9 8 T − 3 5 4 2 7
37 37 3 7
T 2 − 289 T − 398 T^{2} - 289T - 398 T 2 − 2 8 9 T − 3 9 8
41 41 4 1
( T − 168 ) 2 (T - 168)^{2} ( T − 1 6 8 ) 2
43 43 4 3
T 2 − 307 T + 2284 T^{2} - 307T + 2284 T 2 − 3 0 7 T + 2 2 8 4
47 47 4 7
T 2 − 672 T + 85104 T^{2} - 672T + 85104 T 2 − 6 7 2 T + 8 5 1 0 4
53 53 5 3
T 2 − 375 T − 156348 T^{2} - 375T - 156348 T 2 − 3 7 5 T − 1 5 6 3 4 8
59 59 5 9
T 2 − 763 T − 33996 T^{2} - 763T - 33996 T 2 − 7 6 3 T − 3 3 9 9 6
61 61 6 1
T 2 − 406 T − 14568 T^{2} - 406T - 14568 T 2 − 4 0 6 T − 1 4 5 6 8
67 67 6 7
T 2 + 1041 T + 211814 T^{2} + 1041 T + 211814 T 2 + 1 0 4 1 T + 2 1 1 8 1 4
71 71 7 1
T 2 + 1652 T + 644448 T^{2} + 1652 T + 644448 T 2 + 1 6 5 2 T + 6 4 4 4 4 8
73 73 7 3
T 2 − 189 T − 445054 T^{2} - 189T - 445054 T 2 − 1 8 9 T − 4 4 5 0 5 4
79 79 7 9
T 2 − 524 T − 394749 T^{2} - 524T - 394749 T 2 − 5 2 4 T − 3 9 4 7 4 9
83 83 8 3
T 2 − 287 T − 361596 T^{2} - 287T - 361596 T 2 − 2 8 7 T − 3 6 1 5 9 6
89 89 8 9
T 2 − 2394 T + 1292112 T^{2} - 2394 T + 1292112 T 2 − 2 3 9 4 T + 1 2 9 2 1 1 2
97 97 9 7
T 2 − 63 T − 1158214 T^{2} - 63T - 1158214 T 2 − 6 3 T − 1 1 5 8 2 1 4
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