gp: [N,k,chi] = [52,2,Mod(7,52)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(52, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 11]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("52.7");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [4]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == 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 12 \zeta_{12} ζ 1 2 .
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 / 52 Z ) × \left(\mathbb{Z}/52\mathbb{Z}\right)^\times ( Z / 5 2 Z ) × .
n n n
27 27 2 7
41 41 4 1
χ ( n ) \chi(n) χ ( n )
− 1 -1 − 1
ζ 12 \zeta_{12} ζ 1 2
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 3 T_{3} T 3
T3
acting on S 2 n e w ( 52 , [ χ ] ) S_{2}^{\mathrm{new}}(52, [\chi]) S 2 n e w ( 5 2 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 4 + 2 T 3 + ⋯ + 4 T^{4} + 2 T^{3} + \cdots + 4 T 4 + 2 T 3 + ⋯ + 4
T^4 + 2*T^3 + 2*T^2 + 4*T + 4
3 3 3
T 4 T^{4} T 4
T^4
5 5 5
T 4 − 6 T 3 + ⋯ + 9 T^{4} - 6 T^{3} + \cdots + 9 T 4 − 6 T 3 + ⋯ + 9
T^4 - 6*T^3 + 18*T^2 - 18*T + 9
7 7 7
T 4 T^{4} T 4
T^4
11 11 1 1
T 4 T^{4} T 4
T^4
13 13 1 3
T 4 − 4 T 3 + ⋯ + 169 T^{4} - 4 T^{3} + \cdots + 169 T 4 − 4 T 3 + ⋯ + 1 6 9
T^4 - 4*T^3 + 3*T^2 - 52*T + 169
17 17 1 7
T 4 + 24 T 3 + ⋯ + 2209 T^{4} + 24 T^{3} + \cdots + 2209 T 4 + 2 4 T 3 + ⋯ + 2 2 0 9
T^4 + 24*T^3 + 239*T^2 + 1128*T + 2209
19 19 1 9
T 4 T^{4} T 4
T^4
23 23 2 3
T 4 T^{4} T 4
T^4
29 29 2 9
T 4 − 4 T 3 + ⋯ + 5041 T^{4} - 4 T^{3} + \cdots + 5041 T 4 − 4 T 3 + ⋯ + 5 0 4 1
T^4 - 4*T^3 + 87*T^2 + 284*T + 5041
31 31 3 1
T 4 T^{4} T 4
T^4
37 37 3 7
T 4 − 26 T 3 + ⋯ + 3721 T^{4} - 26 T^{3} + \cdots + 3721 T 4 − 2 6 T 3 + ⋯ + 3 7 2 1
T^4 - 26*T^3 + 233*T^2 - 976*T + 3721
41 41 4 1
T 4 + 28 T 3 + ⋯ + 14641 T^{4} + 28 T^{3} + \cdots + 14641 T 4 + 2 8 T 3 + ⋯ + 1 4 6 4 1
T^4 + 28*T^3 + 365*T^2 + 3146*T + 14641
43 43 4 3
T 4 T^{4} T 4
T^4
47 47 4 7
T 4 T^{4} T 4
T^4
53 53 5 3
( T 2 − 14 T + 37 ) 2 (T^{2} - 14 T + 37)^{2} ( T 2 − 1 4 T + 3 7 ) 2
(T^2 - 14*T + 37)^2
59 59 5 9
T 4 T^{4} T 4
T^4
61 61 6 1
T 4 − 10 T 3 + ⋯ + 6889 T^{4} - 10 T^{3} + \cdots + 6889 T 4 − 1 0 T 3 + ⋯ + 6 8 8 9
T^4 - 10*T^3 + 183*T^2 + 830*T + 6889
67 67 6 7
T 4 T^{4} T 4
T^4
71 71 7 1
T 4 T^{4} T 4
T^4
73 73 7 3
T 4 − 22 T 3 + ⋯ + 529 T^{4} - 22 T^{3} + \cdots + 529 T 4 − 2 2 T 3 + ⋯ + 5 2 9
T^4 - 22*T^3 + 242*T^2 - 506*T + 529
79 79 7 9
T 4 T^{4} T 4
T^4
83 83 8 3
T 4 T^{4} T 4
T^4
89 89 8 9
T 4 + 6 T 3 + ⋯ + 324 T^{4} + 6 T^{3} + \cdots + 324 T 4 + 6 T 3 + ⋯ + 3 2 4
T^4 + 6*T^3 + 18*T^2 + 108*T + 324
97 97 9 7
T 4 + 10 T 3 + ⋯ + 2500 T^{4} + 10 T^{3} + \cdots + 2500 T 4 + 1 0 T 3 + ⋯ + 2 5 0 0
T^4 + 10*T^3 + 50*T^2 + 500*T + 2500
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