gp: [N,k,chi] = [425,2,Mod(49,425)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("425.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(425, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([4, 3]))
N = Newforms(chi, 2, names="a")
Newform invariants
sage: traces = [4,4,4]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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 8 \zeta_{8} ζ 8 .
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 / 425 Z ) × \left(\mathbb{Z}/425\mathbb{Z}\right)^\times ( Z / 4 2 5 Z ) × .
n n n
52 52 5 2
326 326 3 2 6
χ ( n ) \chi(n) χ ( n )
− 1 -1 − 1
ζ 8 \zeta_{8} ζ 8
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 − 4 T 2 3 + 8 T 2 2 − 4 T 2 + 1 T_{2}^{4} - 4T_{2}^{3} + 8T_{2}^{2} - 4T_{2} + 1 T 2 4 − 4 T 2 3 + 8 T 2 2 − 4 T 2 + 1
T2^4 - 4*T2^3 + 8*T2^2 - 4*T2 + 1
acting on S 2 n e w ( 425 , [ χ ] ) S_{2}^{\mathrm{new}}(425, [\chi]) S 2 n e w ( 4 2 5 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 4 − 4 T 3 + ⋯ + 1 T^{4} - 4 T^{3} + \cdots + 1 T 4 − 4 T 3 + ⋯ + 1
T^4 - 4*T^3 + 8*T^2 - 4*T + 1
3 3 3
T 4 − 4 T 3 + ⋯ + 8 T^{4} - 4 T^{3} + \cdots + 8 T 4 − 4 T 3 + ⋯ + 8
T^4 - 4*T^3 + 12*T^2 - 16*T + 8
5 5 5
T 4 T^{4} T 4
T^4
7 7 7
T 4 − 4 T 3 + ⋯ + 8 T^{4} - 4 T^{3} + \cdots + 8 T 4 − 4 T 3 + ⋯ + 8
T^4 - 4*T^3 + 12*T^2 - 16*T + 8
11 11 1 1
T 4 + 4 T 3 + ⋯ + 8 T^{4} + 4 T^{3} + \cdots + 8 T 4 + 4 T 3 + ⋯ + 8
T^4 + 4*T^3 + 12*T^2 + 16*T + 8
13 13 1 3
( T 2 − 2 ) 2 (T^{2} - 2)^{2} ( T 2 − 2 ) 2
(T^2 - 2)^2
17 17 1 7
( T 2 − 6 T + 17 ) 2 (T^{2} - 6 T + 17)^{2} ( T 2 − 6 T + 1 7 ) 2
(T^2 - 6*T + 17)^2
19 19 1 9
T 4 + 8 T 3 + ⋯ + 16 T^{4} + 8 T^{3} + \cdots + 16 T 4 + 8 T 3 + ⋯ + 1 6
T^4 + 8*T^3 + 32*T^2 + 32*T + 16
23 23 2 3
T 4 − 12 T 3 + ⋯ + 392 T^{4} - 12 T^{3} + \cdots + 392 T 4 − 1 2 T 3 + ⋯ + 3 9 2
T^4 - 12*T^3 + 68*T^2 - 224*T + 392
29 29 2 9
T 4 − 4 T 3 + ⋯ + 2 T^{4} - 4 T^{3} + \cdots + 2 T 4 − 4 T 3 + ⋯ + 2
T^4 - 4*T^3 + 22*T^2 - 12*T + 2
31 31 3 1
T 4 + 12 T 3 + ⋯ + 648 T^{4} + 12 T^{3} + \cdots + 648 T 4 + 1 2 T 3 + ⋯ + 6 4 8
T^4 + 12*T^3 + 108*T^2 + 432*T + 648
37 37 3 7
T 4 + 20 T 3 + ⋯ + 1250 T^{4} + 20 T^{3} + \cdots + 1250 T 4 + 2 0 T 3 + ⋯ + 1 2 5 0
T^4 + 20*T^3 + 150*T^2 + 500*T + 1250
41 41 4 1
T 4 + 4 T 3 + ⋯ + 98 T^{4} + 4 T^{3} + \cdots + 98 T 4 + 4 T 3 + ⋯ + 9 8
T^4 + 4*T^3 + 54*T^2 - 140*T + 98
43 43 4 3
T 4 − 8 T 3 + ⋯ + 16 T^{4} - 8 T^{3} + \cdots + 16 T 4 − 8 T 3 + ⋯ + 1 6
T^4 - 8*T^3 + 32*T^2 - 32*T + 16
47 47 4 7
( T 2 − 16 T + 56 ) 2 (T^{2} - 16 T + 56)^{2} ( T 2 − 1 6 T + 5 6 ) 2
(T^2 - 16*T + 56)^2
53 53 5 3
( T 2 + 2 T + 2 ) 2 (T^{2} + 2 T + 2)^{2} ( T 2 + 2 T + 2 ) 2
(T^2 + 2*T + 2)^2
59 59 5 9
T 4 + 1296 T^{4} + 1296 T 4 + 1 2 9 6
T^4 + 1296
61 61 6 1
T 4 + 50 T 2 + ⋯ + 1250 T^{4} + 50 T^{2} + \cdots + 1250 T 4 + 5 0 T 2 + ⋯ + 1 2 5 0
T^4 + 50*T^2 + 500*T + 1250
67 67 6 7
T 4 + 48 T 2 + 64 T^{4} + 48T^{2} + 64 T 4 + 4 8 T 2 + 6 4
T^4 + 48*T^2 + 64
71 71 7 1
T 4 − 20 T 3 + ⋯ + 5000 T^{4} - 20 T^{3} + \cdots + 5000 T 4 − 2 0 T 3 + ⋯ + 5 0 0 0
T^4 - 20*T^3 + 100*T^2 + 5000
73 73 7 3
T 4 + 98 T 2 + ⋯ + 4802 T^{4} + 98 T^{2} + \cdots + 4802 T 4 + 9 8 T 2 + ⋯ + 4 8 0 2
T^4 + 98*T^2 + 1372*T + 4802
79 79 7 9
T 4 − 4 T 3 + ⋯ + 392 T^{4} - 4 T^{3} + \cdots + 392 T 4 − 4 T 3 + ⋯ + 3 9 2
T^4 - 4*T^3 + 12*T^2 - 112*T + 392
83 83 8 3
T 4 − 16 T 3 + ⋯ + 16 T^{4} - 16 T^{3} + \cdots + 16 T 4 − 1 6 T 3 + ⋯ + 1 6
T^4 - 16*T^3 + 128*T^2 + 64*T + 16
89 89 8 9
T 4 + 132 T 2 + 3844 T^{4} + 132T^{2} + 3844 T 4 + 1 3 2 T 2 + 3 8 4 4
T^4 + 132*T^2 + 3844
97 97 9 7
T 4 + 4 T 3 + ⋯ + 4418 T^{4} + 4 T^{3} + \cdots + 4418 T 4 + 4 T 3 + ⋯ + 4 4 1 8
T^4 + 4*T^3 + 54*T^2 + 940*T + 4418
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