gp: [N,k,chi] = [2016,2,Mod(289,2016)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2016, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 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("2016.289");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [2,0,0,0,3,0,1,0,0,0,-1]
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 / 2016 Z ) × \left(\mathbb{Z}/2016\mathbb{Z}\right)^\times ( Z / 2 0 1 6 Z ) × .
n n n
127 127 1 2 7
577 577 5 7 7
1765 1765 1 7 6 5
1793 1793 1 7 9 3
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 ( 2016 , [ χ ] ) S_{2}^{\mathrm{new}}(2016, [\chi]) S 2 n e w ( 2 0 1 6 , [ 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 + T 11 + 1 T_{11}^{2} + T_{11} + 1 T 1 1 2 + T 1 1 + 1
T11^2 + T11 + 1
T 13 + 4 T_{13} + 4 T 1 3 + 4
T13 + 4
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^{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 − T + 7 T^{2} - T + 7 T 2 − T + 7
T^2 - T + 7
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 + 4 ) 2 (T + 4)^{2} ( T + 4 ) 2
(T + 4)^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 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 − 7 ) 2 (T - 7)^{2} ( T − 7 ) 2
(T - 7)^2
31 31 3 1
T 2 + 11 T + 121 T^{2} + 11T + 121 T 2 + 1 1 T + 1 2 1
T^2 + 11*T + 121
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 + 2 ) 2 (T + 2)^{2} ( T + 2 ) 2
(T + 2)^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 + 11 T + 121 T^{2} + 11T + 121 T 2 + 1 1 T + 1 2 1
T^2 + 11*T + 121
59 59 5 9
T 2 − 7 T + 49 T^{2} - 7T + 49 T 2 − 7 T + 4 9
T^2 - 7*T + 49
61 61 6 1
T 2 + 10 T + 100 T^{2} + 10T + 100 T 2 + 1 0 T + 1 0 0
T^2 + 10*T + 100
67 67 6 7
T 2 + 10 T + 100 T^{2} + 10T + 100 T 2 + 1 0 T + 1 0 0
T^2 + 10*T + 100
71 71 7 1
( T + 6 ) 2 (T + 6)^{2} ( T + 6 ) 2
(T + 6)^2
73 73 7 3
T 2 − 6 T + 36 T^{2} - 6T + 36 T 2 − 6 T + 3 6
T^2 - 6*T + 36
79 79 7 9
T 2 + 11 T + 121 T^{2} + 11T + 121 T 2 + 1 1 T + 1 2 1
T^2 + 11*T + 121
83 83 8 3
( T + 11 ) 2 (T + 11)^{2} ( T + 1 1 ) 2
(T + 11)^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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