gp: [N,k,chi] = [336,4,Mod(1,336)]
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("336.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(336, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 4, names="a")
Newform invariants
sage: traces = [2,0,6,0,-6]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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 = 337 \beta = \sqrt{337} β = 3 3 7 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 ( 336 ) ) S_{4}^{\mathrm{new}}(\Gamma_0(336)) S 4 n e w ( Γ 0 ( 3 3 6 ) )
T 5 2 + 6 T 5 − 328 T_{5}^{2} + 6T_{5} - 328 T 5 2 + 6 T 5 − 3 2 8
T 11 2 + 26 T 11 − 168 T_{11}^{2} + 26T_{11} - 168 T 1 1 2 + 2 6 T 1 1 − 1 6 8
p p p F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 T^{2} T 2
3 3 3
( T − 3 ) 2 (T - 3)^{2} ( T − 3 ) 2
5 5 5
T 2 + 6 T − 328 T^{2} + 6T - 328 T 2 + 6 T − 3 2 8
7 7 7
( T − 7 ) 2 (T - 7)^{2} ( T − 7 ) 2
11 11 1 1
T 2 + 26 T − 168 T^{2} + 26T - 168 T 2 + 2 6 T − 1 6 8
13 13 1 3
T 2 − 96 T + 956 T^{2} - 96T + 956 T 2 − 9 6 T + 9 5 6
17 17 1 7
T 2 − 78 T − 1512 T^{2} - 78T - 1512 T 2 − 7 8 T − 1 5 1 2
19 19 1 9
( T + 20 ) 2 (T + 20)^{2} ( T + 2 0 ) 2
23 23 2 3
T 2 + 22 T − 8304 T^{2} + 22T - 8304 T 2 + 2 2 T − 8 3 0 4
29 29 2 9
( T − 102 ) 2 (T - 102)^{2} ( T − 1 0 2 ) 2
31 31 3 1
T 2 + 96 T − 83968 T^{2} + 96T - 83968 T 2 + 9 6 T − 8 3 9 6 8
37 37 3 7
T 2 − 504 T + 62156 T^{2} - 504T + 62156 T 2 − 5 0 4 T + 6 2 1 5 6
41 41 4 1
T 2 + 102 T − 38176 T^{2} + 102T - 38176 T 2 + 1 0 2 T − 3 8 1 7 6
43 43 4 3
T 2 + 296 T − 64368 T^{2} + 296T - 64368 T 2 + 2 9 6 T − 6 4 3 6 8
47 47 4 7
T 2 − 780 T + 118400 T^{2} - 780T + 118400 T 2 − 7 8 0 T + 1 1 8 4 0 0
53 53 5 3
T 2 − 192 T − 99972 T^{2} - 192T - 99972 T 2 − 1 9 2 T − 9 9 9 7 2
59 59 5 9
T 2 + 212 T − 54816 T^{2} + 212T - 54816 T 2 + 2 1 2 T − 5 4 8 1 6
61 61 6 1
T 2 + 100 T − 83772 T^{2} + 100T - 83772 T 2 + 1 0 0 T − 8 3 7 7 2
67 67 6 7
T 2 + 212 T + 9888 T^{2} + 212T + 9888 T 2 + 2 1 2 T + 9 8 8 8
71 71 7 1
T 2 − 534 T + 30512 T^{2} - 534T + 30512 T 2 − 5 3 4 T + 3 0 5 1 2
73 73 7 3
T 2 − 1128 T + 284396 T^{2} - 1128 T + 284396 T 2 − 1 1 2 8 T + 2 8 4 3 9 6
79 79 7 9
T 2 + 468 T − 1078912 T^{2} + 468 T - 1078912 T 2 + 4 6 8 T − 1 0 7 8 9 1 2
83 83 8 3
T 2 − 824 T − 606704 T^{2} - 824T - 606704 T 2 − 8 2 4 T − 6 0 6 7 0 4
89 89 8 9
T 2 + 2118 T + 875808 T^{2} + 2118 T + 875808 T 2 + 2 1 1 8 T + 8 7 5 8 0 8
97 97 9 7
T 2 − 400 T − 1611300 T^{2} - 400 T - 1611300 T 2 − 4 0 0 T − 1 6 1 1 3 0 0
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