gp: [N,k,chi] = [65,2,Mod(16,65)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(65, 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("65.16");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [4,-1]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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 + 2 x 2 + x + 1 x^{4} - x^{3} + 2x^{2} + x + 1 x 4 − x 3 + 2 x 2 + x + 1
x^4 - x^3 + 2*x^2 + x + 1
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( ν 3 + 1 ) / 2 ( \nu^{3} + 1 ) / 2 ( ν 3 + 1 ) / 2
(v^3 + 1) / 2
β 3 \beta_{3} β 3 = = =
( − ν 3 + 2 ν 2 − 2 ν − 1 ) / 2 ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 ( − ν 3 + 2 ν 2 − 2 ν − 1 ) / 2
(-v^3 + 2*v^2 - 2*v - 1) / 2
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 3 + β 2 + β 1 \beta_{3} + \beta_{2} + \beta_1 β 3 + β 2 + β 1
b3 + b2 + b1
ν 3 \nu^{3} ν 3 = = =
2 β 2 − 1 2\beta_{2} - 1 2 β 2 − 1
2*b2 - 1
Character values
We give the values of χ \chi χ on generators for ( Z / 65 Z ) × \left(\mathbb{Z}/65\mathbb{Z}\right)^\times ( Z / 6 5 Z ) × .
n n n
27 27 2 7
41 41 4 1
χ ( n ) \chi(n) χ ( n )
1 1 1
β 3 \beta_{3} β 3
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 kernel of the linear operator
T 2 4 + T 2 3 + 2 T 2 2 − T 2 + 1 T_{2}^{4} + T_{2}^{3} + 2T_{2}^{2} - T_{2} + 1 T 2 4 + T 2 3 + 2 T 2 2 − T 2 + 1
T2^4 + T2^3 + 2*T2^2 - T2 + 1
acting on S 2 n e w ( 65 , [ χ ] ) S_{2}^{\mathrm{new}}(65, [\chi]) S 2 n e w ( 6 5 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 4 + T 3 + 2 T 2 + ⋯ + 1 T^{4} + T^{3} + 2 T^{2} + \cdots + 1 T 4 + T 3 + 2 T 2 + ⋯ + 1
T^4 + T^3 + 2*T^2 - T + 1
3 3 3
T 4 + 5 T 2 + 25 T^{4} + 5T^{2} + 25 T 4 + 5 T 2 + 2 5
T^4 + 5*T^2 + 25
5 5 5
( T − 1 ) 4 (T - 1)^{4} ( T − 1 ) 4
(T - 1)^4
7 7 7
T 4 + 4 T 3 + ⋯ + 1 T^{4} + 4 T^{3} + \cdots + 1 T 4 + 4 T 3 + ⋯ + 1
T^4 + 4*T^3 + 17*T^2 - 4*T + 1
11 11 1 1
T 4 + 4 T 3 + ⋯ + 1 T^{4} + 4 T^{3} + \cdots + 1 T 4 + 4 T 3 + ⋯ + 1
T^4 + 4*T^3 + 17*T^2 - 4*T + 1
13 13 1 3
( T 2 + 2 T + 13 ) 2 (T^{2} + 2 T + 13)^{2} ( T 2 + 2 T + 1 3 ) 2
(T^2 + 2*T + 13)^2
17 17 1 7
T 4 + 2 T 3 + ⋯ + 361 T^{4} + 2 T^{3} + \cdots + 361 T 4 + 2 T 3 + ⋯ + 3 6 1
T^4 + 2*T^3 + 23*T^2 - 38*T + 361
19 19 1 9
T 4 + 4 T 3 + ⋯ + 1 T^{4} + 4 T^{3} + \cdots + 1 T 4 + 4 T 3 + ⋯ + 1
T^4 + 4*T^3 + 17*T^2 - 4*T + 1
23 23 2 3
T 4 + 12 T 3 + ⋯ + 961 T^{4} + 12 T^{3} + \cdots + 961 T 4 + 1 2 T 3 + ⋯ + 9 6 1
T^4 + 12*T^3 + 113*T^2 + 372*T + 961
29 29 2 9
T 4 − 6 T 3 + ⋯ + 121 T^{4} - 6 T^{3} + \cdots + 121 T 4 − 6 T 3 + ⋯ + 1 2 1
T^4 - 6*T^3 + 47*T^2 + 66*T + 121
31 31 3 1
T 4 T^{4} T 4
T^4
37 37 3 7
( T 2 + 3 T + 9 ) 2 (T^{2} + 3 T + 9)^{2} ( T 2 + 3 T + 9 ) 2
(T^2 + 3*T + 9)^2
41 41 4 1
T 4 + 6 T 3 + ⋯ + 5041 T^{4} + 6 T^{3} + \cdots + 5041 T 4 + 6 T 3 + ⋯ + 5 0 4 1
T^4 + 6*T^3 + 107*T^2 - 426*T + 5041
43 43 4 3
T 4 − 8 T 3 + ⋯ + 121 T^{4} - 8 T^{3} + \cdots + 121 T 4 − 8 T 3 + ⋯ + 1 2 1
T^4 - 8*T^3 + 53*T^2 - 88*T + 121
47 47 4 7
( T 2 − 8 T − 64 ) 2 (T^{2} - 8 T - 64)^{2} ( T 2 − 8 T − 6 4 ) 2
(T^2 - 8*T - 64)^2
53 53 5 3
( T − 6 ) 4 (T - 6)^{4} ( T − 6 ) 4
(T - 6)^4
59 59 5 9
T 4 + 12 T 3 + ⋯ + 81 T^{4} + 12 T^{3} + \cdots + 81 T 4 + 1 2 T 3 + ⋯ + 8 1
T^4 + 12*T^3 + 153*T^2 - 108*T + 81
61 61 6 1
T 4 + 2 T 3 + ⋯ + 32041 T^{4} + 2 T^{3} + \cdots + 32041 T 4 + 2 T 3 + ⋯ + 3 2 0 4 1
T^4 + 2*T^3 + 183*T^2 - 358*T + 32041
67 67 6 7
T 4 + 8 T 3 + ⋯ + 841 T^{4} + 8 T^{3} + \cdots + 841 T 4 + 8 T 3 + ⋯ + 8 4 1
T^4 + 8*T^3 + 93*T^2 - 232*T + 841
71 71 7 1
T 4 − 8 T 3 + ⋯ + 121 T^{4} - 8 T^{3} + \cdots + 121 T 4 − 8 T 3 + ⋯ + 1 2 1
T^4 - 8*T^3 + 53*T^2 - 88*T + 121
73 73 7 3
( T + 6 ) 4 (T + 6)^{4} ( T + 6 ) 4
(T + 6)^4
79 79 7 9
T 4 T^{4} T 4
T^4
83 83 8 3
( T 2 − 80 ) 2 (T^{2} - 80)^{2} ( T 2 − 8 0 ) 2
(T^2 - 80)^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 4 + 2 T 3 + ⋯ + 361 T^{4} + 2 T^{3} + \cdots + 361 T 4 + 2 T 3 + ⋯ + 3 6 1
T^4 + 2*T^3 + 23*T^2 - 38*T + 361
show more
show less