gp: [N,k,chi] = [1134,2,Mod(163,1134)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1134, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1134.163");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [2,1,0,-1,-3,0,5]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 a primitive root of unity ζ 6 \zeta_{6} ζ 6 .
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 / 1134 Z ) × \left(\mathbb{Z}/1134\mathbb{Z}\right)^\times ( Z / 1 1 3 4 Z ) × .
n n n
325 325 3 2 5
407 407 4 0 7
χ ( n ) \chi(n) χ ( n )
− ζ 6 -\zeta_{6} − ζ 6
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 intersection of the kernels of the following linear operators acting on S 2 n e w ( 1134 , [ χ ] ) S_{2}^{\mathrm{new}}(1134, [\chi]) S 2 n e w ( 1 1 3 4 , [ χ ] ) :
T 5 2 + 3 T 5 + 9 T_{5}^{2} + 3T_{5} + 9 T 5 2 + 3 T 5 + 9
T5^2 + 3*T5 + 9
T 11 2 + 3 T 11 + 9 T_{11}^{2} + 3T_{11} + 9 T 1 1 2 + 3 T 1 1 + 9
T11^2 + 3*T11 + 9
T 17 2 − 3 T 17 + 9 T_{17}^{2} - 3T_{17} + 9 T 1 7 2 − 3 T 1 7 + 9
T17^2 - 3*T17 + 9
T 23 2 + 3 T 23 + 9 T_{23}^{2} + 3T_{23} + 9 T 2 3 2 + 3 T 2 3 + 9
T23^2 + 3*T23 + 9
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 − T + 1 T^{2} - T + 1 T 2 − T + 1
T^2 - T + 1
3 3 3
T 2 T^{2} T 2
T^2
5 5 5
T 2 + 3 T + 9 T^{2} + 3T + 9 T 2 + 3 T + 9
T^2 + 3*T + 9
7 7 7
T 2 − 5 T + 7 T^{2} - 5T + 7 T 2 − 5 T + 7
T^2 - 5*T + 7
11 11 1 1
T 2 + 3 T + 9 T^{2} + 3T + 9 T 2 + 3 T + 9
T^2 + 3*T + 9
13 13 1 3
( T − 5 ) 2 (T - 5)^{2} ( T − 5 ) 2
(T - 5)^2
17 17 1 7
T 2 − 3 T + 9 T^{2} - 3T + 9 T 2 − 3 T + 9
T^2 - 3*T + 9
19 19 1 9
T 2 + 5 T + 25 T^{2} + 5T + 25 T 2 + 5 T + 2 5
T^2 + 5*T + 25
23 23 2 3
T 2 + 3 T + 9 T^{2} + 3T + 9 T 2 + 3 T + 9
T^2 + 3*T + 9
29 29 2 9
( T − 3 ) 2 (T - 3)^{2} ( T − 3 ) 2
(T - 3)^2
31 31 3 1
T 2 − 4 T + 16 T^{2} - 4T + 16 T 2 − 4 T + 1 6
T^2 - 4*T + 16
37 37 3 7
T 2 − 7 T + 49 T^{2} - 7T + 49 T 2 − 7 T + 4 9
T^2 - 7*T + 49
41 41 4 1
( T − 9 ) 2 (T - 9)^{2} ( T − 9 ) 2
(T - 9)^2
43 43 4 3
( T − 11 ) 2 (T - 11)^{2} ( T − 1 1 ) 2
(T - 11)^2
47 47 4 7
T 2 T^{2} T 2
T^2
53 53 5 3
T 2 + 3 T + 9 T^{2} + 3T + 9 T 2 + 3 T + 9
T^2 + 3*T + 9
59 59 5 9
T 2 − 12 T + 144 T^{2} - 12T + 144 T 2 − 1 2 T + 1 4 4
T^2 - 12*T + 144
61 61 6 1
T 2 + 2 T + 4 T^{2} + 2T + 4 T 2 + 2 T + 4
T^2 + 2*T + 4
67 67 6 7
T 2 − 4 T + 16 T^{2} - 4T + 16 T 2 − 4 T + 1 6
T^2 - 4*T + 16
71 71 7 1
T 2 T^{2} T 2
T^2
73 73 7 3
T 2 + 11 T + 121 T^{2} + 11T + 121 T 2 + 1 1 T + 1 2 1
T^2 + 11*T + 121
79 79 7 9
T 2 + 8 T + 64 T^{2} + 8T + 64 T 2 + 8 T + 6 4
T^2 + 8*T + 64
83 83 8 3
( T + 3 ) 2 (T + 3)^{2} ( T + 3 ) 2
(T + 3)^2
89 89 8 9
T 2 − 15 T + 225 T^{2} - 15T + 225 T 2 − 1 5 T + 2 2 5
T^2 - 15*T + 225
97 97 9 7
( T + 1 ) 2 (T + 1)^{2} ( T + 1 ) 2
(T + 1)^2
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