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 = 57 \beta = \sqrt{57} β = 5 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 − 48 T_{5}^{2} - 6T_{5} - 48 T 5 2 − 6 T 5 − 4 8
T 11 2 − 6 T 11 − 1416 T_{11}^{2} - 6T_{11} - 1416 T 1 1 2 − 6 T 1 1 − 1 4 1 6
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 − 48 T^{2} - 6T - 48 T 2 − 6 T − 4 8
7 7 7
( T + 7 ) 2 (T + 7)^{2} ( T + 7 ) 2
11 11 1 1
T 2 − 6 T − 1416 T^{2} - 6T - 1416 T 2 − 6 T − 1 4 1 6
13 13 1 3
T 2 − 16 T − 1988 T^{2} - 16T - 1988 T 2 − 1 6 T − 1 9 8 8
17 17 1 7
T 2 + 6 T − 48 T^{2} + 6T - 48 T 2 + 6 T − 4 8
19 19 1 9
T 2 + 64 T − 7184 T^{2} + 64T - 7184 T 2 + 6 4 T − 7 1 8 4
23 23 2 3
T 2 + 6 T − 16464 T^{2} + 6T - 16464 T 2 + 6 T − 1 6 4 6 4
29 29 2 9
T 2 + 252 T + 7668 T^{2} + 252T + 7668 T 2 + 2 5 2 T + 7 6 6 8
31 31 3 1
T 2 + 40 T − 73472 T^{2} + 40T - 73472 T 2 + 4 0 T − 7 3 4 7 2
37 37 3 7
T 2 + 248 T − 3092 T^{2} + 248T - 3092 T 2 + 2 4 8 T − 3 0 9 2
41 41 4 1
T 2 + 450 T + 37800 T^{2} + 450T + 37800 T 2 + 4 5 0 T + 3 7 8 0 0
43 43 4 3
T 2 + 376 T + 2512 T^{2} + 376T + 2512 T 2 + 3 7 6 T + 2 5 1 2
47 47 4 7
T 2 − 12 T − 65856 T^{2} - 12T - 65856 T 2 − 1 2 T − 6 5 8 5 6
53 53 5 3
T 2 + 1104 T + 304476 T^{2} + 1104 T + 304476 T 2 + 1 1 0 4 T + 3 0 4 4 7 6
59 59 5 9
T 2 + 804 T − 30144 T^{2} + 804T - 30144 T 2 + 8 0 4 T − 3 0 1 4 4
61 61 6 1
T 2 + 428 T − 28076 T^{2} + 428T - 28076 T 2 + 4 2 8 T − 2 8 0 7 6
67 67 6 7
T 2 + 148 T − 160736 T^{2} + 148T - 160736 T 2 + 1 4 8 T − 1 6 0 7 3 6
71 71 7 1
T 2 + 954 T + 214704 T^{2} + 954T + 214704 T 2 + 9 5 4 T + 2 1 4 7 0 4
73 73 7 3
T 2 − 1072 T + 285244 T^{2} - 1072 T + 285244 T 2 − 1 0 7 2 T + 2 8 5 2 4 4
79 79 7 9
T 2 − 572 T − 84416 T^{2} - 572T - 84416 T 2 − 5 7 2 T − 8 4 4 1 6
83 83 8 3
T 2 + 1944 T + 813456 T^{2} + 1944 T + 813456 T 2 + 1 9 4 4 T + 8 1 3 4 5 6
89 89 8 9
T 2 − 366 T − 253848 T^{2} - 366T - 253848 T 2 − 3 6 6 T − 2 5 3 8 4 8
97 97 9 7
T 2 − 808 T − 922292 T^{2} - 808T - 922292 T 2 − 8 0 8 T − 9 2 2 2 9 2
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