gp: [N,k,chi] = [384,4,Mod(1,384)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(384, 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("384.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [2,0,6,0,8]
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 -expansion are expressed in terms of β = 4 15 \beta = 4\sqrt{15} β = 4 1 5 .
We also show the integral q q q -expansion of the trace form .
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
3 3 3
− 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 ( 384 ) ) S_{4}^{\mathrm{new}}(\Gamma_0(384)) S 4 n e w ( Γ 0 ( 3 8 4 ) ) :
T 5 2 − 8 T 5 − 224 T_{5}^{2} - 8T_{5} - 224 T 5 2 − 8 T 5 − 2 2 4
T5^2 - 8*T5 - 224
T 7 2 − 4 T 7 − 236 T_{7}^{2} - 4T_{7} - 236 T 7 2 − 4 T 7 − 2 3 6
T7^2 - 4*T7 - 236
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 T^{2} T 2
T^2
3 3 3
( T − 3 ) 2 (T - 3)^{2} ( T − 3 ) 2
(T - 3)^2
5 5 5
T 2 − 8 T − 224 T^{2} - 8T - 224 T 2 − 8 T − 2 2 4
T^2 - 8*T - 224
7 7 7
T 2 − 4 T − 236 T^{2} - 4T - 236 T 2 − 4 T − 2 3 6
T^2 - 4*T - 236
11 11 1 1
T 2 − 40 T − 560 T^{2} - 40T - 560 T 2 − 4 0 T − 5 6 0
T^2 - 40*T - 560
13 13 1 3
( T − 2 ) 2 (T - 2)^{2} ( T − 2 ) 2
(T - 2)^2
17 17 1 7
T 2 − 28 T − 8444 T^{2} - 28T - 8444 T 2 − 2 8 T − 8 4 4 4
T^2 - 28*T - 8444
19 19 1 9
T 2 − 112 T + 2176 T^{2} - 112T + 2176 T 2 − 1 1 2 T + 2 1 7 6
T^2 - 112*T + 2176
23 23 2 3
T 2 + 56 T − 7856 T^{2} + 56T - 7856 T 2 + 5 6 T − 7 8 5 6
T^2 + 56*T - 7856
29 29 2 9
T 2 + 144 T − 35376 T^{2} + 144T - 35376 T 2 + 1 4 4 T − 3 5 3 7 6
T^2 + 144*T - 35376
31 31 3 1
T 2 − 276 T + 7284 T^{2} - 276T + 7284 T 2 − 2 7 6 T + 7 2 8 4
T^2 - 276*T + 7284
37 37 3 7
T 2 + 580 T + 83140 T^{2} + 580T + 83140 T 2 + 5 8 0 T + 8 3 1 4 0
T^2 + 580*T + 83140
41 41 4 1
T 2 − 588 T + 85476 T^{2} - 588T + 85476 T 2 − 5 8 8 T + 8 5 4 7 6
T^2 - 588*T + 85476
43 43 4 3
T 2 + 96 T − 113856 T^{2} + 96T - 113856 T 2 + 9 6 T − 1 1 3 8 5 6
T^2 + 96*T - 113856
47 47 4 7
T 2 − 712 T + 125776 T^{2} - 712T + 125776 T 2 − 7 1 2 T + 1 2 5 7 7 6
T^2 - 712*T + 125776
53 53 5 3
T 2 + 1104 T + 304464 T^{2} + 1104 T + 304464 T 2 + 1 1 0 4 T + 3 0 4 4 6 4
T^2 + 1104*T + 304464
59 59 5 9
T 2 − 536 T − 66416 T^{2} - 536T - 66416 T 2 − 5 3 6 T − 6 6 4 1 6
T^2 - 536*T - 66416
61 61 6 1
T 2 + 452 T − 111164 T^{2} + 452T - 111164 T 2 + 4 5 2 T − 1 1 1 1 6 4
T^2 + 452*T - 111164
67 67 6 7
T 2 − 296 T − 730736 T^{2} - 296T - 730736 T 2 − 2 9 6 T − 7 3 0 7 3 6
T^2 - 296*T - 730736
71 71 7 1
T 2 − 216 T + 3024 T^{2} - 216T + 3024 T 2 − 2 1 6 T + 3 0 2 4
T^2 - 216*T + 3024
73 73 7 3
T 2 − 364 T − 615836 T^{2} - 364T - 615836 T 2 − 3 6 4 T − 6 1 5 8 3 6
T^2 - 364*T - 615836
79 79 7 9
T 2 − 1076 T + 235444 T^{2} - 1076 T + 235444 T 2 − 1 0 7 6 T + 2 3 5 4 4 4
T^2 - 1076*T + 235444
83 83 8 3
T 2 − 648 T − 817584 T^{2} - 648T - 817584 T 2 − 6 4 8 T − 8 1 7 5 8 4
T^2 - 648*T - 817584
89 89 8 9
T 2 − 2308 T + 1235716 T^{2} - 2308 T + 1235716 T 2 − 2 3 0 8 T + 1 2 3 5 7 1 6
T^2 - 2308*T + 1235716
97 97 9 7
T 2 + 1900 T + 806500 T^{2} + 1900 T + 806500 T 2 + 1 9 0 0 T + 8 0 6 5 0 0
T^2 + 1900*T + 806500
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