gp: [N,k,chi] = [5712,2,Mod(1,5712)]
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("5712.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5712, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
N = Newforms(chi, 2, names="a")
Newform invariants
sage: traces = [2,0,2,0,-2,0,2,0,2,0,-2,0,-6,0,-2,0,-2,0,10]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == 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 = 2 \beta = \sqrt{2} β = 2 .
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
7 7 7
− 1 -1 − 1
17 17 1 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 2 n e w ( Γ 0 ( 5712 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(5712)) S 2 n e w ( Γ 0 ( 5 7 1 2 ) ) :
T 5 2 + 2 T 5 − 1 T_{5}^{2} + 2T_{5} - 1 T 5 2 + 2 T 5 − 1
T5^2 + 2*T5 - 1
T 11 + 1 T_{11} + 1 T 1 1 + 1
T11 + 1
T 13 2 + 6 T 13 + 7 T_{13}^{2} + 6T_{13} + 7 T 1 3 2 + 6 T 1 3 + 7
T13^2 + 6*T13 + 7
T 19 2 − 10 T 19 + 23 T_{19}^{2} - 10T_{19} + 23 T 1 9 2 − 1 0 T 1 9 + 2 3
T19^2 - 10*T19 + 23
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 − 1 ) 2 (T - 1)^{2} ( T − 1 ) 2
(T - 1)^2
5 5 5
T 2 + 2 T − 1 T^{2} + 2T - 1 T 2 + 2 T − 1
T^2 + 2*T - 1
7 7 7
( T − 1 ) 2 (T - 1)^{2} ( T − 1 ) 2
(T - 1)^2
11 11 1 1
( T + 1 ) 2 (T + 1)^{2} ( T + 1 ) 2
(T + 1)^2
13 13 1 3
T 2 + 6 T + 7 T^{2} + 6T + 7 T 2 + 6 T + 7
T^2 + 6*T + 7
17 17 1 7
( T + 1 ) 2 (T + 1)^{2} ( T + 1 ) 2
(T + 1)^2
19 19 1 9
T 2 − 10 T + 23 T^{2} - 10T + 23 T 2 − 1 0 T + 2 3
T^2 - 10*T + 23
23 23 2 3
T 2 − 2 T − 7 T^{2} - 2T - 7 T 2 − 2 T − 7
T^2 - 2*T - 7
29 29 2 9
T 2 − 8 T + 8 T^{2} - 8T + 8 T 2 − 8 T + 8
T^2 - 8*T + 8
31 31 3 1
T 2 − 98 T^{2} - 98 T 2 − 9 8
T^2 - 98
37 37 3 7
T 2 + 8 T − 2 T^{2} + 8T - 2 T 2 + 8 T − 2
T^2 + 8*T - 2
41 41 4 1
T 2 − 6 T − 41 T^{2} - 6T - 41 T 2 − 6 T − 4 1
T^2 - 6*T - 41
43 43 4 3
T 2 − 6 T + 1 T^{2} - 6T + 1 T 2 − 6 T + 1
T^2 - 6*T + 1
47 47 4 7
T 2 − 8 T + 14 T^{2} - 8T + 14 T 2 − 8 T + 1 4
T^2 - 8*T + 14
53 53 5 3
T 2 − 4 T − 14 T^{2} - 4T - 14 T 2 − 4 T − 1 4
T^2 - 4*T - 14
59 59 5 9
T 2 − 16 T + 46 T^{2} - 16T + 46 T 2 − 1 6 T + 4 6
T^2 - 16*T + 46
61 61 6 1
T 2 + 4 T − 158 T^{2} + 4T - 158 T 2 + 4 T − 1 5 8
T^2 + 4*T - 158
67 67 6 7
T 2 − 4 T − 28 T^{2} - 4T - 28 T 2 − 4 T − 2 8
T^2 - 4*T - 28
71 71 7 1
( T + 2 ) 2 (T + 2)^{2} ( T + 2 ) 2
(T + 2)^2
73 73 7 3
T 2 + 12 T + 28 T^{2} + 12T + 28 T 2 + 1 2 T + 2 8
T^2 + 12*T + 28
79 79 7 9
T 2 − 4 T − 158 T^{2} - 4T - 158 T 2 − 4 T − 1 5 8
T^2 - 4*T - 158
83 83 8 3
( T − 6 ) 2 (T - 6)^{2} ( T − 6 ) 2
(T - 6)^2
89 89 8 9
T 2 − 16 T + 56 T^{2} - 16T + 56 T 2 − 1 6 T + 5 6
T^2 - 16*T + 56
97 97 9 7
T 2 + 8 T − 16 T^{2} + 8T - 16 T 2 + 8 T − 1 6
T^2 + 8*T - 16
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