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 = [4,0,0,0,2,0,-4,0,0,0,-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 basis 1 , β 1 , β 2 , β 3 1,\beta_1,\beta_2,\beta_3 1 , β 1 , β 2 , β 3 for the coefficient ring described below.
We also show the integral q q q -expansion of the trace form .
Basis of coefficient ring in terms of a root ν \nu ν of
x 4 + 2 x 2 + 4 x^{4} + 2x^{2} + 4 x 4 + 2 x 2 + 4
x^4 + 2*x^2 + 4
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( ν 2 ) / 2 ( \nu^{2} ) / 2 ( ν 2 ) / 2
(v^2) / 2
β 3 \beta_{3} β 3 = = =
( ν 3 ) / 2 ( \nu^{3} ) / 2 ( ν 3 ) / 2
(v^3) / 2
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
2 β 2 2\beta_{2} 2 β 2
2*b2
ν 3 \nu^{3} ν 3 = = =
2 β 3 2\beta_{3} 2 β 3
2*b3
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
− 1 − β 2 -1 - \beta_{2} − 1 − β 2
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 4 − 2 T 5 3 + 11 T 5 2 + 14 T 5 + 49 T_{5}^{4} - 2T_{5}^{3} + 11T_{5}^{2} + 14T_{5} + 49 T 5 4 − 2 T 5 3 + 1 1 T 5 2 + 1 4 T 5 + 4 9
T5^4 - 2*T5^3 + 11*T5^2 + 14*T5 + 49
T 11 4 + 2 T 11 3 + 5 T 11 2 − 2 T 11 + 1 T_{11}^{4} + 2T_{11}^{3} + 5T_{11}^{2} - 2T_{11} + 1 T 1 1 4 + 2 T 1 1 3 + 5 T 1 1 2 − 2 T 1 1 + 1
T11^4 + 2*T11^3 + 5*T11^2 - 2*T11 + 1
T 13 2 − 8 T_{13}^{2} - 8 T 1 3 2 − 8
T13^2 - 8
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 4 T^{4} T 4
T^4
3 3 3
T 4 T^{4} T 4
T^4
5 5 5
T 4 − 2 T 3 + ⋯ + 49 T^{4} - 2 T^{3} + \cdots + 49 T 4 − 2 T 3 + ⋯ + 4 9
T^4 - 2*T^3 + 11*T^2 + 14*T + 49
7 7 7
( T 2 + 2 T + 7 ) 2 (T^{2} + 2 T + 7)^{2} ( T 2 + 2 T + 7 ) 2
(T^2 + 2*T + 7)^2
11 11 1 1
T 4 + 2 T 3 + ⋯ + 1 T^{4} + 2 T^{3} + \cdots + 1 T 4 + 2 T 3 + ⋯ + 1
T^4 + 2*T^3 + 5*T^2 - 2*T + 1
13 13 1 3
( T 2 − 8 ) 2 (T^{2} - 8)^{2} ( T 2 − 8 ) 2
(T^2 - 8)^2
17 17 1 7
T 4 + 6 T 3 + ⋯ + 1 T^{4} + 6 T^{3} + \cdots + 1 T 4 + 6 T 3 + ⋯ + 1
T^4 + 6*T^3 + 35*T^2 + 6*T + 1
19 19 1 9
T 4 + 10 T 3 + ⋯ + 529 T^{4} + 10 T^{3} + \cdots + 529 T 4 + 1 0 T 3 + ⋯ + 5 2 9
T^4 + 10*T^3 + 77*T^2 + 230*T + 529
23 23 2 3
T 4 + 2 T 3 + ⋯ + 289 T^{4} + 2 T^{3} + \cdots + 289 T 4 + 2 T 3 + ⋯ + 2 8 9
T^4 + 2*T^3 + 21*T^2 - 34*T + 289
29 29 2 9
( T 2 − 8 ) 2 (T^{2} - 8)^{2} ( T 2 − 8 ) 2
(T^2 - 8)^2
31 31 3 1
T 4 + 14 T 3 + ⋯ + 2209 T^{4} + 14 T^{3} + \cdots + 2209 T 4 + 1 4 T 3 + ⋯ + 2 2 0 9
T^4 + 14*T^3 + 149*T^2 + 658*T + 2209
37 37 3 7
T 4 + 6 T 3 + ⋯ + 529 T^{4} + 6 T^{3} + \cdots + 529 T 4 + 6 T 3 + ⋯ + 5 2 9
T^4 + 6*T^3 + 59*T^2 - 138*T + 529
41 41 4 1
( T 2 − 8 T + 8 ) 2 (T^{2} - 8 T + 8)^{2} ( T 2 − 8 T + 8 ) 2
(T^2 - 8*T + 8)^2
43 43 4 3
( T 2 − 8 T − 16 ) 2 (T^{2} - 8 T - 16)^{2} ( T 2 − 8 T − 1 6 ) 2
(T^2 - 8*T - 16)^2
47 47 4 7
T 4 − 18 T 3 + ⋯ + 6241 T^{4} - 18 T^{3} + \cdots + 6241 T 4 − 1 8 T 3 + ⋯ + 6 2 4 1
T^4 - 18*T^3 + 245*T^2 - 1422*T + 6241
53 53 5 3
( T 2 + T + 1 ) 2 (T^{2} + T + 1)^{2} ( T 2 + T + 1 ) 2
(T^2 + T + 1)^2
59 59 5 9
T 4 − 2 T 3 + ⋯ + 9409 T^{4} - 2 T^{3} + \cdots + 9409 T 4 − 2 T 3 + ⋯ + 9 4 0 9
T^4 - 2*T^3 + 101*T^2 + 194*T + 9409
61 61 6 1
T 4 + 6 T 3 + ⋯ + 529 T^{4} + 6 T^{3} + \cdots + 529 T 4 + 6 T 3 + ⋯ + 5 2 9
T^4 + 6*T^3 + 59*T^2 - 138*T + 529
67 67 6 7
T 4 − 14 T 3 + ⋯ + 961 T^{4} - 14 T^{3} + \cdots + 961 T 4 − 1 4 T 3 + ⋯ + 9 6 1
T^4 - 14*T^3 + 165*T^2 - 434*T + 961
71 71 7 1
( T 2 + 16 T + 32 ) 2 (T^{2} + 16 T + 32)^{2} ( T 2 + 1 6 T + 3 2 ) 2
(T^2 + 16*T + 32)^2
73 73 7 3
T 4 − 18 T 3 + ⋯ + 2401 T^{4} - 18 T^{3} + \cdots + 2401 T 4 − 1 8 T 3 + ⋯ + 2 4 0 1
T^4 - 18*T^3 + 275*T^2 - 882*T + 2401
79 79 7 9
T 4 − 2 T 3 + ⋯ + 2401 T^{4} - 2 T^{3} + \cdots + 2401 T 4 − 2 T 3 + ⋯ + 2 4 0 1
T^4 - 2*T^3 + 53*T^2 + 98*T + 2401
83 83 8 3
( T 2 + 8 T − 112 ) 2 (T^{2} + 8 T - 112)^{2} ( T 2 + 8 T − 1 1 2 ) 2
(T^2 + 8*T - 112)^2
89 89 8 9
( T 2 + 9 T + 81 ) 2 (T^{2} + 9 T + 81)^{2} ( T 2 + 9 T + 8 1 ) 2
(T^2 + 9*T + 81)^2
97 97 9 7
( T 2 + 8 T + 8 ) 2 (T^{2} + 8 T + 8)^{2} ( T 2 + 8 T + 8 ) 2
(T^2 + 8*T + 8)^2
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