gp: [N,k,chi] = [130,3,Mod(77,130)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(130, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1, 2]))
N = Newforms(chi, 3, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("130.77");
S:= CuspForms(chi, 3);
N := Newforms(S);
Newform invariants
sage: traces = [2,2]
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 i = − 1 i = \sqrt{-1} i = − 1 .
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 / 130 Z ) × \left(\mathbb{Z}/130\mathbb{Z}\right)^\times ( Z / 1 3 0 Z ) × .
n n n
27 27 2 7
41 41 4 1
χ ( n ) \chi(n) χ ( n )
i i i
− 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 3 n e w ( 130 , [ χ ] ) S_{3}^{\mathrm{new}}(130, [\chi]) S 3 n e w ( 1 3 0 , [ χ ] ) :
T 3 2 + 4 T 3 + 8 T_{3}^{2} + 4T_{3} + 8 T 3 2 + 4 T 3 + 8
T3^2 + 4*T3 + 8
T 7 2 + 4 T 7 + 8 T_{7}^{2} + 4T_{7} + 8 T 7 2 + 4 T 7 + 8
T7^2 + 4*T7 + 8
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 − 2 T + 2 T^{2} - 2T + 2 T 2 − 2 T + 2
T^2 - 2*T + 2
3 3 3
T 2 + 4 T + 8 T^{2} + 4T + 8 T 2 + 4 T + 8
T^2 + 4*T + 8
5 5 5
T 2 + 25 T^{2} + 25 T 2 + 2 5
T^2 + 25
7 7 7
T 2 + 4 T + 8 T^{2} + 4T + 8 T 2 + 4 T + 8
T^2 + 4*T + 8
11 11 1 1
T 2 + 400 T^{2} + 400 T 2 + 4 0 0
T^2 + 400
13 13 1 3
T 2 + 169 T^{2} + 169 T 2 + 1 6 9
T^2 + 169
17 17 1 7
T 2 − 34 T + 578 T^{2} - 34T + 578 T 2 − 3 4 T + 5 7 8
T^2 - 34*T + 578
19 19 1 9
( T + 8 ) 2 (T + 8)^{2} ( T + 8 ) 2
(T + 8)^2
23 23 2 3
T 2 − 36 T + 648 T^{2} - 36T + 648 T 2 − 3 6 T + 6 4 8
T^2 - 36*T + 648
29 29 2 9
T 2 + 1600 T^{2} + 1600 T 2 + 1 6 0 0
T^2 + 1600
31 31 3 1
T 2 + 1600 T^{2} + 1600 T 2 + 1 6 0 0
T^2 + 1600
37 37 3 7
T 2 − 86 T + 3698 T^{2} - 86T + 3698 T 2 − 8 6 T + 3 6 9 8
T^2 - 86*T + 3698
41 41 4 1
T 2 + 100 T^{2} + 100 T 2 + 1 0 0
T^2 + 100
43 43 4 3
T 2 + 4 T + 8 T^{2} + 4T + 8 T 2 + 4 T + 8
T^2 + 4*T + 8
47 47 4 7
T 2 + 44 T + 968 T^{2} + 44T + 968 T 2 + 4 4 T + 9 6 8
T^2 + 44*T + 968
53 53 5 3
T 2 − 26 T + 338 T^{2} - 26T + 338 T 2 − 2 6 T + 3 3 8
T^2 - 26*T + 338
59 59 5 9
( T − 32 ) 2 (T - 32)^{2} ( T − 3 2 ) 2
(T - 32)^2
61 61 6 1
( T − 112 ) 2 (T - 112)^{2} ( T − 1 1 2 ) 2
(T - 112)^2
67 67 6 7
T 2 − 36 T + 648 T^{2} - 36T + 648 T 2 − 3 6 T + 6 4 8
T^2 - 36*T + 648
71 71 7 1
T 2 + 1600 T^{2} + 1600 T 2 + 1 6 0 0
T^2 + 1600
73 73 7 3
T 2 − 14 T + 98 T^{2} - 14T + 98 T 2 − 1 4 T + 9 8
T^2 - 14*T + 98
79 79 7 9
T 2 + 14400 T^{2} + 14400 T 2 + 1 4 4 0 0
T^2 + 14400
83 83 8 3
T 2 − 44 T + 968 T^{2} - 44T + 968 T 2 − 4 4 T + 9 6 8
T^2 - 44*T + 968
89 89 8 9
( T − 82 ) 2 (T - 82)^{2} ( T − 8 2 ) 2
(T - 82)^2
97 97 9 7
T 2 − 66 T + 2178 T^{2} - 66T + 2178 T 2 − 6 6 T + 2 1 7 8
T^2 - 66*T + 2178
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