gp: [N,k,chi] = [334,2,Mod(1,334)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(334, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("334.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [2,-2,0]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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 = 2 \beta = \sqrt{2} β = 2 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
167 167 1 6 7
− 1 -1 − 1
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
T 3 2 − 8 T_{3}^{2} - 8 T 3 2 − 8 S 2 n e w ( Γ 0 ( 334 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(334)) S 2 n e w ( Γ 0 ( 3 3 4 ) )
p p p F p ( T ) F_p(T) F p ( T )
2 2 2
( T + 1 ) 2 (T + 1)^{2} ( T + 1 ) 2
3 3 3
T 2 − 8 T^{2} - 8 T 2 − 8
5 5 5
T 2 − 2 T − 1 T^{2} - 2T - 1 T 2 − 2 T − 1
7 7 7
( T + 3 ) 2 (T + 3)^{2} ( T + 3 ) 2
11 11 1 1
T 2 − 8 T^{2} - 8 T 2 − 8
13 13 1 3
T 2 − 8 T + 8 T^{2} - 8T + 8 T 2 − 8 T + 8
17 17 1 7
T 2 − 4 T − 4 T^{2} - 4T - 4 T 2 − 4 T − 4
19 19 1 9
T 2 − 4 T − 28 T^{2} - 4T - 28 T 2 − 4 T − 2 8
23 23 2 3
T 2 − 12 T + 28 T^{2} - 12T + 28 T 2 − 1 2 T + 2 8
29 29 2 9
T 2 − 8 T + 8 T^{2} - 8T + 8 T 2 − 8 T + 8
31 31 3 1
T 2 + 10 T + 17 T^{2} + 10T + 17 T 2 + 1 0 T + 1 7
37 37 3 7
T 2 + 6 T − 9 T^{2} + 6T - 9 T 2 + 6 T − 9
41 41 4 1
T 2 − 12 T + 4 T^{2} - 12T + 4 T 2 − 1 2 T + 4
43 43 4 3
( T − 2 ) 2 (T - 2)^{2} ( T − 2 ) 2
47 47 4 7
T 2 + 6 T − 63 T^{2} + 6T - 63 T 2 + 6 T − 6 3
53 53 5 3
T 2 + 22 T + 119 T^{2} + 22T + 119 T 2 + 2 2 T + 1 1 9
59 59 5 9
T 2 − 10 T + 23 T^{2} - 10T + 23 T 2 − 1 0 T + 2 3
61 61 6 1
T 2 − 72 T^{2} - 72 T 2 − 7 2
67 67 6 7
T 2 + 2 T − 1 T^{2} + 2T - 1 T 2 + 2 T − 1
71 71 7 1
T 2 − 12 T − 36 T^{2} - 12T - 36 T 2 − 1 2 T − 3 6
73 73 7 3
T 2 + 16 T + 32 T^{2} + 16T + 32 T 2 + 1 6 T + 3 2
79 79 7 9
T 2 − 72 T^{2} - 72 T 2 − 7 2
83 83 8 3
T 2 − 10 T − 73 T^{2} - 10T - 73 T 2 − 1 0 T − 7 3
89 89 8 9
T 2 − 6 T − 23 T^{2} - 6T - 23 T 2 − 6 T − 2 3
97 97 9 7
T 2 − 2 T − 7 T^{2} - 2T - 7 T 2 − 2 T − 7
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