gp: [N,k,chi] = [1960,2,Mod(361,1960)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1960, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1960.361");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [2,0,1,0,-1,0,0,0,2,0,-2]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == 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 / 1960 Z ) × \left(\mathbb{Z}/1960\mathbb{Z}\right)^\times ( Z / 1 9 6 0 Z ) × .
n n n
981 981 9 8 1
1081 1081 1 0 8 1
1177 1177 1 1 7 7
1471 1471 1 4 7 1
χ ( n ) \chi(n) χ ( n )
1 1 1
− ζ 6 -\zeta_{6} − ζ 6
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 intersection of the kernels of the following linear operators acting on S 2 n e w ( 1960 , [ χ ] ) S_{2}^{\mathrm{new}}(1960, [\chi]) S 2 n e w ( 1 9 6 0 , [ χ ] ) :
T 3 2 − T 3 + 1 T_{3}^{2} - T_{3} + 1 T 3 2 − T 3 + 1
T3^2 - T3 + 1
T 11 2 + 2 T 11 + 4 T_{11}^{2} + 2T_{11} + 4 T 1 1 2 + 2 T 1 1 + 4
T11^2 + 2*T11 + 4
T 13 T_{13} T 1 3
T13
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 2 − T + 1 T^{2} - T + 1 T 2 − T + 1
T^2 - T + 1
5 5 5
T 2 + T + 1 T^{2} + T + 1 T 2 + T + 1
T^2 + T + 1
7 7 7
T 2 T^{2} T 2
T^2
11 11 1 1
T 2 + 2 T + 4 T^{2} + 2T + 4 T 2 + 2 T + 4
T^2 + 2*T + 4
13 13 1 3
T 2 T^{2} T 2
T^2
17 17 1 7
T 2 − 4 T + 16 T^{2} - 4T + 16 T 2 − 4 T + 1 6
T^2 - 4*T + 16
19 19 1 9
T 2 + 2 T + 4 T^{2} + 2T + 4 T 2 + 2 T + 4
T^2 + 2*T + 4
23 23 2 3
T 2 + T + 1 T^{2} + T + 1 T 2 + T + 1
T^2 + T + 1
29 29 2 9
( T − 9 ) 2 (T - 9)^{2} ( T − 9 ) 2
(T - 9)^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 + 1 ) 2 (T + 1)^{2} ( T + 1 ) 2
(T + 1)^2
43 43 4 3
( T − 9 ) 2 (T - 9)^{2} ( T − 9 ) 2
(T - 9)^2
47 47 4 7
T 2 T^{2} T 2
T^2
53 53 5 3
T 2 − 10 T + 100 T^{2} - 10T + 100 T 2 − 1 0 T + 1 0 0
T^2 - 10*T + 100
59 59 5 9
T 2 + 10 T + 100 T^{2} + 10T + 100 T 2 + 1 0 T + 1 0 0
T^2 + 10*T + 100
61 61 6 1
T 2 − 9 T + 81 T^{2} - 9T + 81 T 2 − 9 T + 8 1
T^2 - 9*T + 81
67 67 6 7
T 2 + 5 T + 25 T^{2} + 5T + 25 T 2 + 5 T + 2 5
T^2 + 5*T + 25
71 71 7 1
( T − 14 ) 2 (T - 14)^{2} ( T − 1 4 ) 2
(T - 14)^2
73 73 7 3
T 2 − 12 T + 144 T^{2} - 12T + 144 T 2 − 1 2 T + 1 4 4
T^2 - 12*T + 144
79 79 7 9
T 2 + 14 T + 196 T^{2} + 14T + 196 T 2 + 1 4 T + 1 9 6
T^2 + 14*T + 196
83 83 8 3
( T + 11 ) 2 (T + 11)^{2} ( T + 1 1 ) 2
(T + 11)^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 − 18 ) 2 (T - 18)^{2} ( T − 1 8 ) 2
(T - 18)^2
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