gp: [N,k,chi] = [1078,2,Mod(67,1078)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1078, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 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("1078.67");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [2,-1,3,-1,2,-6,0,2,-6,2,1,3,14,0,12,-1,2,-6,0,-4,0,-2,8]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == 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 / 1078 Z ) × \left(\mathbb{Z}/1078\mathbb{Z}\right)^\times ( Z / 1 0 7 8 Z ) × .
n n n
199 199 1 9 9
981 981 9 8 1
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 ( 1078 , [ χ ] ) S_{2}^{\mathrm{new}}(1078, [\chi]) S 2 n e w ( 1 0 7 8 , [ χ ] ) :
T 3 2 − 3 T 3 + 9 T_{3}^{2} - 3T_{3} + 9 T 3 2 − 3 T 3 + 9
T3^2 - 3*T3 + 9
T 5 2 − 2 T 5 + 4 T_{5}^{2} - 2T_{5} + 4 T 5 2 − 2 T 5 + 4
T5^2 - 2*T5 + 4
T 13 − 7 T_{13} - 7 T 1 3 − 7
T13 - 7
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 − 3 T + 9 T^{2} - 3T + 9 T 2 − 3 T + 9
T^2 - 3*T + 9
5 5 5
T 2 − 2 T + 4 T^{2} - 2T + 4 T 2 − 2 T + 4
T^2 - 2*T + 4
7 7 7
T 2 T^{2} T 2
T^2
11 11 1 1
T 2 − T + 1 T^{2} - T + 1 T 2 − T + 1
T^2 - T + 1
13 13 1 3
( T − 7 ) 2 (T - 7)^{2} ( T − 7 ) 2
(T - 7)^2
17 17 1 7
T 2 − 2 T + 4 T^{2} - 2T + 4 T 2 − 2 T + 4
T^2 - 2*T + 4
19 19 1 9
T 2 T^{2} T 2
T^2
23 23 2 3
T 2 − 8 T + 64 T^{2} - 8T + 64 T 2 − 8 T + 6 4
T^2 - 8*T + 64
29 29 2 9
( T + 5 ) 2 (T + 5)^{2} ( T + 5 ) 2
(T + 5)^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 + 4 T + 16 T^{2} + 4T + 16 T 2 + 4 T + 1 6
T^2 + 4*T + 16
41 41 4 1
( T + 4 ) 2 (T + 4)^{2} ( T + 4 ) 2
(T + 4)^2
43 43 4 3
( T + 8 ) 2 (T + 8)^{2} ( T + 8 ) 2
(T + 8)^2
47 47 4 7
T 2 − 2 T + 4 T^{2} - 2T + 4 T 2 − 2 T + 4
T^2 - 2*T + 4
53 53 5 3
T 2 − 6 T + 36 T^{2} - 6T + 36 T 2 − 6 T + 3 6
T^2 - 6*T + 36
59 59 5 9
T 2 − 3 T + 9 T^{2} - 3T + 9 T 2 − 3 T + 9
T^2 - 3*T + 9
61 61 6 1
T 2 − T + 1 T^{2} - T + 1 T 2 − T + 1
T^2 - T + 1
67 67 6 7
T 2 + 9 T + 81 T^{2} + 9T + 81 T 2 + 9 T + 8 1
T^2 + 9*T + 81
71 71 7 1
( T + 2 ) 2 (T + 2)^{2} ( T + 2 ) 2
(T + 2)^2
73 73 7 3
T 2 − 4 T + 16 T^{2} - 4T + 16 T 2 − 4 T + 1 6
T^2 - 4*T + 16
79 79 7 9
T 2 + 9 T + 81 T^{2} + 9T + 81 T 2 + 9 T + 8 1
T^2 + 9*T + 81
83 83 8 3
( T + 6 ) 2 (T + 6)^{2} ( T + 6 ) 2
(T + 6)^2
89 89 8 9
T 2 − 6 T + 36 T^{2} - 6T + 36 T 2 − 6 T + 3 6
T^2 - 6*T + 36
97 97 9 7
( T + 7 ) 2 (T + 7)^{2} ( T + 7 ) 2
(T + 7)^2
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