gp: [N,k,chi] = [936,2,Mod(181,936)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(936, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("936.181");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [2,2,0,0,4]
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 i = − 1 i = \sqrt{-1} i = − 1 .
We also show the integral q q q -expansion of the trace form .
Character values
We give the values of χ \chi χ on generators for ( Z / 936 Z ) × \left(\mathbb{Z}/936\mathbb{Z}\right)^\times ( Z / 9 3 6 Z ) × .
n n n
145 145 1 4 5
209 209 2 0 9
469 469 4 6 9
703 703 7 0 3
χ ( n ) \chi(n) χ ( n )
− 1 -1 − 1
1 1 1
− 1 -1 − 1
1 1 1
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
This newform subspace can be constructed as the kernel of the linear operator
T 5 − 2 T_{5} - 2 T 5 − 2
T5 - 2
acting on S 2 n e w ( 936 , [ χ ] ) S_{2}^{\mathrm{new}}(936, [\chi]) S 2 n e w ( 9 3 6 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 − 2 T + 2 T^{2} - 2T + 2 T 2 − 2 T + 2
T^2 - 2*T + 2
3 3 3
T 2 T^{2} T 2
T^2
5 5 5
( T − 2 ) 2 (T - 2)^{2} ( T − 2 ) 2
(T - 2)^2
7 7 7
T 2 T^{2} T 2
T^2
11 11 1 1
( T − 4 ) 2 (T - 4)^{2} ( T − 4 ) 2
(T - 4)^2
13 13 1 3
T 2 − 6 T + 13 T^{2} - 6T + 13 T 2 − 6 T + 1 3
T^2 - 6*T + 13
17 17 1 7
( T − 6 ) 2 (T - 6)^{2} ( T − 6 ) 2
(T - 6)^2
19 19 1 9
( T + 6 ) 2 (T + 6)^{2} ( T + 6 ) 2
(T + 6)^2
23 23 2 3
T 2 T^{2} T 2
T^2
29 29 2 9
T 2 + 36 T^{2} + 36 T 2 + 3 6
T^2 + 36
31 31 3 1
T 2 T^{2} T 2
T^2
37 37 3 7
( T + 6 ) 2 (T + 6)^{2} ( T + 6 ) 2
(T + 6)^2
41 41 4 1
T 2 + 4 T^{2} + 4 T 2 + 4
T^2 + 4
43 43 4 3
T 2 + 144 T^{2} + 144 T 2 + 1 4 4
T^2 + 144
47 47 4 7
T 2 + 64 T^{2} + 64 T 2 + 6 4
T^2 + 64
53 53 5 3
T 2 + 36 T^{2} + 36 T 2 + 3 6
T^2 + 36
59 59 5 9
( T + 4 ) 2 (T + 4)^{2} ( T + 4 ) 2
(T + 4)^2
61 61 6 1
T 2 + 64 T^{2} + 64 T 2 + 6 4
T^2 + 64
67 67 6 7
( T + 6 ) 2 (T + 6)^{2} ( T + 6 ) 2
(T + 6)^2
71 71 7 1
T 2 + 16 T^{2} + 16 T 2 + 1 6
T^2 + 16
73 73 7 3
T 2 + 144 T^{2} + 144 T 2 + 1 4 4
T^2 + 144
79 79 7 9
( T − 6 ) 2 (T - 6)^{2} ( T − 6 ) 2
(T - 6)^2
83 83 8 3
( T + 8 ) 2 (T + 8)^{2} ( T + 8 ) 2
(T + 8)^2
89 89 8 9
T 2 + 4 T^{2} + 4 T 2 + 4
T^2 + 4
97 97 9 7
T 2 + 144 T^{2} + 144 T 2 + 1 4 4
T^2 + 144
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