gp: [N,k,chi] = [702,2,Mod(55,702)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(702, 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("702.55");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [4,2,0,-2,2,0,3]
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 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 − x 3 + 4 x 2 + 3 x + 9 x^{4} - x^{3} + 4x^{2} + 3x + 9 x 4 − x 3 + 4 x 2 + 3 x + 9
x^4 - x^3 + 4*x^2 + 3*x + 9
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( − ν 3 + 4 ν 2 − 4 ν − 3 ) / 12 ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 ( − ν 3 + 4 ν 2 − 4 ν − 3 ) / 1 2
(-v^3 + 4*v^2 - 4*v - 3) / 12
β 3 \beta_{3} β 3 = = =
( ν 3 + 7 ) / 4 ( \nu^{3} + 7 ) / 4 ( ν 3 + 7 ) / 4
(v^3 + 7) / 4
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 3 + 3 β 2 + β 1 − 1 \beta_{3} + 3\beta_{2} + \beta _1 - 1 β 3 + 3 β 2 + β 1 − 1
b3 + 3*b2 + b1 - 1
ν 3 \nu^{3} ν 3 = = =
4 β 3 − 7 4\beta_{3} - 7 4 β 3 − 7
4*b3 - 7
Character values
We give the values of χ \chi χ on generators for ( Z / 702 Z ) × \left(\mathbb{Z}/702\mathbb{Z}\right)^\times ( Z / 7 0 2 Z ) × .
n n n
379 379 3 7 9
677 677 6 7 7
χ ( n ) \chi(n) χ ( n )
β 2 \beta_{2} β 2
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 ( 702 , [ χ ] ) S_{2}^{\mathrm{new}}(702, [\chi]) S 2 n e w ( 7 0 2 , [ χ ] ) :
T 5 2 − T 5 − 3 T_{5}^{2} - T_{5} - 3 T 5 2 − T 5 − 3
T5^2 - T5 - 3
T 7 4 − 3 T 7 3 + 10 T 7 2 + 3 T 7 + 1 T_{7}^{4} - 3T_{7}^{3} + 10T_{7}^{2} + 3T_{7} + 1 T 7 4 − 3 T 7 3 + 1 0 T 7 2 + 3 T 7 + 1
T7^4 - 3*T7^3 + 10*T7^2 + 3*T7 + 1
T 11 4 + 5 T 11 3 + 22 T 11 2 + 15 T 11 + 9 T_{11}^{4} + 5T_{11}^{3} + 22T_{11}^{2} + 15T_{11} + 9 T 1 1 4 + 5 T 1 1 3 + 2 2 T 1 1 2 + 1 5 T 1 1 + 9
T11^4 + 5*T11^3 + 22*T11^2 + 15*T11 + 9
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T 2 − T + 1 ) 2 (T^{2} - T + 1)^{2} ( T 2 − T + 1 ) 2
(T^2 - T + 1)^2
3 3 3
T 4 T^{4} T 4
T^4
5 5 5
( T 2 − T − 3 ) 2 (T^{2} - T - 3)^{2} ( T 2 − T − 3 ) 2
(T^2 - T - 3)^2
7 7 7
T 4 − 3 T 3 + ⋯ + 1 T^{4} - 3 T^{3} + \cdots + 1 T 4 − 3 T 3 + ⋯ + 1
T^4 - 3*T^3 + 10*T^2 + 3*T + 1
11 11 1 1
T 4 + 5 T 3 + ⋯ + 9 T^{4} + 5 T^{3} + \cdots + 9 T 4 + 5 T 3 + ⋯ + 9
T^4 + 5*T^3 + 22*T^2 + 15*T + 9
13 13 1 3
T 4 + 13 T 2 + 169 T^{4} + 13T^{2} + 169 T 4 + 1 3 T 2 + 1 6 9
T^4 + 13*T^2 + 169
17 17 1 7
T 4 + 3 T 3 + ⋯ + 729 T^{4} + 3 T^{3} + \cdots + 729 T 4 + 3 T 3 + ⋯ + 7 2 9
T^4 + 3*T^3 + 36*T^2 - 81*T + 729
19 19 1 9
T 4 + 8 T 3 + ⋯ + 9 T^{4} + 8 T^{3} + \cdots + 9 T 4 + 8 T 3 + ⋯ + 9
T^4 + 8*T^3 + 61*T^2 + 24*T + 9
23 23 2 3
T 4 + 4 T 3 + ⋯ + 81 T^{4} + 4 T^{3} + \cdots + 81 T 4 + 4 T 3 + ⋯ + 8 1
T^4 + 4*T^3 + 25*T^2 - 36*T + 81
29 29 2 9
T 4 − 8 T 3 + ⋯ + 9 T^{4} - 8 T^{3} + \cdots + 9 T 4 − 8 T 3 + ⋯ + 9
T^4 - 8*T^3 + 61*T^2 - 24*T + 9
31 31 3 1
( T 2 − 52 ) 2 (T^{2} - 52)^{2} ( T 2 − 5 2 ) 2
(T^2 - 52)^2
37 37 3 7
T 4 + 11 T 3 + ⋯ + 729 T^{4} + 11 T^{3} + \cdots + 729 T 4 + 1 1 T 3 + ⋯ + 7 2 9
T^4 + 11*T^3 + 94*T^2 + 297*T + 729
41 41 4 1
T 4 + 117 T 2 + 13689 T^{4} + 117 T^{2} + 13689 T 4 + 1 1 7 T 2 + 1 3 6 8 9
T^4 + 117*T^2 + 13689
43 43 4 3
T 4 − 6 T 3 + ⋯ + 16 T^{4} - 6 T^{3} + \cdots + 16 T 4 − 6 T 3 + ⋯ + 1 6
T^4 - 6*T^3 + 40*T^2 + 24*T + 16
47 47 4 7
( T 2 − 2 T − 12 ) 2 (T^{2} - 2 T - 12)^{2} ( T 2 − 2 T − 1 2 ) 2
(T^2 - 2*T - 12)^2
53 53 5 3
( T − 3 ) 4 (T - 3)^{4} ( T − 3 ) 4
(T - 3)^4
59 59 5 9
T 4 + 25 T 3 + ⋯ + 23409 T^{4} + 25 T^{3} + \cdots + 23409 T 4 + 2 5 T 3 + ⋯ + 2 3 4 0 9
T^4 + 25*T^3 + 472*T^2 + 3825*T + 23409
61 61 6 1
T 4 + 15 T 3 + ⋯ + 2809 T^{4} + 15 T^{3} + \cdots + 2809 T 4 + 1 5 T 3 + ⋯ + 2 8 0 9
T^4 + 15*T^3 + 172*T^2 + 795*T + 2809
67 67 6 7
T 4 − 9 T 3 + ⋯ + 289 T^{4} - 9 T^{3} + \cdots + 289 T 4 − 9 T 3 + ⋯ + 2 8 9
T^4 - 9*T^3 + 64*T^2 - 153*T + 289
71 71 7 1
T 4 + 17 T 3 + ⋯ + 4761 T^{4} + 17 T^{3} + \cdots + 4761 T 4 + 1 7 T 3 + ⋯ + 4 7 6 1
T^4 + 17*T^3 + 220*T^2 + 1173*T + 4761
73 73 7 3
( T 2 − 13 T + 13 ) 2 (T^{2} - 13 T + 13)^{2} ( T 2 − 1 3 T + 1 3 ) 2
(T^2 - 13*T + 13)^2
79 79 7 9
( T 2 − 7 T − 17 ) 2 (T^{2} - 7 T - 17)^{2} ( T 2 − 7 T − 1 7 ) 2
(T^2 - 7*T - 17)^2
83 83 8 3
T 4 T^{4} T 4
T^4
89 89 8 9
T 4 + 12 T 3 + ⋯ + 6561 T^{4} + 12 T^{3} + \cdots + 6561 T 4 + 1 2 T 3 + ⋯ + 6 5 6 1
T^4 + 12*T^3 + 225*T^2 - 972*T + 6561
97 97 9 7
T 4 − 2 T 3 + ⋯ + 13456 T^{4} - 2 T^{3} + \cdots + 13456 T 4 − 2 T 3 + ⋯ + 1 3 4 5 6
T^4 - 2*T^3 + 120*T^2 + 232*T + 13456
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