gp: [N,k,chi] = [308,2,Mod(113,308)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(308, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 8]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("308.113");
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 10 \zeta_{10} ζ 1 0 .
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 / 308 Z ) × \left(\mathbb{Z}/308\mathbb{Z}\right)^\times ( Z / 3 0 8 Z ) × .
n n n
45 45 4 5
57 57 5 7
155 155 1 5 5
χ ( n ) \chi(n) χ ( n )
1 1 1
− ζ 10 3 -\zeta_{10}^{3} − ζ 1 0 3
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 kernel of the linear operator
T 3 4 − 4 T 3 3 + 16 T 3 2 − 24 T 3 + 16 T_{3}^{4} - 4T_{3}^{3} + 16T_{3}^{2} - 24T_{3} + 16 T 3 4 − 4 T 3 3 + 1 6 T 3 2 − 2 4 T 3 + 1 6
T3^4 - 4*T3^3 + 16*T3^2 - 24*T3 + 16
acting on S 2 n e w ( 308 , [ χ ] ) S_{2}^{\mathrm{new}}(308, [\chi]) S 2 n e w ( 3 0 8 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 4 T^{4} T 4
T^4
3 3 3
T 4 − 4 T 3 + ⋯ + 16 T^{4} - 4 T^{3} + \cdots + 16 T 4 − 4 T 3 + ⋯ + 1 6
T^4 - 4*T^3 + 16*T^2 - 24*T + 16
5 5 5
T 4 + 2 T 3 + ⋯ + 16 T^{4} + 2 T^{3} + \cdots + 16 T 4 + 2 T 3 + ⋯ + 1 6
T^4 + 2*T^3 + 4*T^2 + 8*T + 16
7 7 7
T 4 − T 3 + T 2 + ⋯ + 1 T^{4} - T^{3} + T^{2} + \cdots + 1 T 4 − T 3 + T 2 + ⋯ + 1
T^4 - T^3 + T^2 - T + 1
11 11 1 1
T 4 − 11 T 3 + ⋯ + 121 T^{4} - 11 T^{3} + \cdots + 121 T 4 − 1 1 T 3 + ⋯ + 1 2 1
T^4 - 11*T^3 + 51*T^2 - 121*T + 121
13 13 1 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 + 16*T^2 + 16*T + 16
17 17 1 7
T 4 − 4 T 3 + ⋯ + 256 T^{4} - 4 T^{3} + \cdots + 256 T 4 − 4 T 3 + ⋯ + 2 5 6
T^4 - 4*T^3 + 16*T^2 - 64*T + 256
19 19 1 9
T 4 − 2 T 3 + ⋯ + 16 T^{4} - 2 T^{3} + \cdots + 16 T 4 − 2 T 3 + ⋯ + 1 6
T^4 - 2*T^3 + 4*T^2 - 8*T + 16
23 23 2 3
( T 2 + T − 1 ) 2 (T^{2} + T - 1)^{2} ( T 2 + T − 1 ) 2
(T^2 + T - 1)^2
29 29 2 9
T 4 + 10 T 3 + ⋯ + 25 T^{4} + 10 T^{3} + \cdots + 25 T 4 + 1 0 T 3 + ⋯ + 2 5
T^4 + 10*T^3 + 40*T^2 + 25*T + 25
31 31 3 1
T 4 − 10 T 3 + ⋯ + 400 T^{4} - 10 T^{3} + \cdots + 400 T 4 − 1 0 T 3 + ⋯ + 4 0 0
T^4 - 10*T^3 + 60*T^2 - 200*T + 400
37 37 3 7
T 4 − T 3 + ⋯ + 961 T^{4} - T^{3} + \cdots + 961 T 4 − T 3 + ⋯ + 9 6 1
T^4 - T^3 + 76*T^2 + 434*T + 961
41 41 4 1
T 4 − 10 T 3 + ⋯ + 400 T^{4} - 10 T^{3} + \cdots + 400 T 4 − 1 0 T 3 + ⋯ + 4 0 0
T^4 - 10*T^3 + 160*T^2 - 400*T + 400
43 43 4 3
( T 2 − 15 T + 45 ) 2 (T^{2} - 15 T + 45)^{2} ( T 2 − 1 5 T + 4 5 ) 2
(T^2 - 15*T + 45)^2
47 47 4 7
T 4 + 10 T 3 + ⋯ + 10000 T^{4} + 10 T^{3} + \cdots + 10000 T 4 + 1 0 T 3 + ⋯ + 1 0 0 0 0
T^4 + 10*T^3 + 100*T^2 + 1000*T + 10000
53 53 5 3
T 4 − 11 T 3 + ⋯ + 5041 T^{4} - 11 T^{3} + \cdots + 5041 T 4 − 1 1 T 3 + ⋯ + 5 0 4 1
T^4 - 11*T^3 + 76*T^2 - 426*T + 5041
59 59 5 9
T 4 + 18 T 3 + ⋯ + 1296 T^{4} + 18 T^{3} + \cdots + 1296 T 4 + 1 8 T 3 + ⋯ + 1 2 9 6
T^4 + 18*T^3 + 144*T^2 + 432*T + 1296
61 61 6 1
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 + 76*T^2 + 56*T + 16
67 67 6 7
( T 2 − 3 T + 1 ) 2 (T^{2} - 3 T + 1)^{2} ( T 2 − 3 T + 1 ) 2
(T^2 - 3*T + 1)^2
71 71 7 1
T 4 − 5 T 3 + ⋯ + 625 T^{4} - 5 T^{3} + \cdots + 625 T 4 − 5 T 3 + ⋯ + 6 2 5
T^4 - 5*T^3 + 150*T^2 + 500*T + 625
73 73 7 3
T 4 + 36 T 3 + ⋯ + 59536 T^{4} + 36 T^{3} + \cdots + 59536 T 4 + 3 6 T 3 + ⋯ + 5 9 5 3 6
T^4 + 36*T^3 + 736*T^2 + 8296*T + 59536
79 79 7 9
T 4 − 13 T 3 + ⋯ + 961 T^{4} - 13 T^{3} + \cdots + 961 T 4 − 1 3 T 3 + ⋯ + 9 6 1
T^4 - 13*T^3 + 94*T^2 - 372*T + 961
83 83 8 3
T 4 + 26 T 3 + ⋯ + 1936 T^{4} + 26 T^{3} + \cdots + 1936 T 4 + 2 6 T 3 + ⋯ + 1 9 3 6
T^4 + 26*T^3 + 276*T^2 + 616*T + 1936
89 89 8 9
( T 2 + 18 T + 76 ) 2 (T^{2} + 18 T + 76)^{2} ( T 2 + 1 8 T + 7 6 ) 2
(T^2 + 18*T + 76)^2
97 97 9 7
T 4 − 32 T 3 + ⋯ + 30976 T^{4} - 32 T^{3} + \cdots + 30976 T 4 − 3 2 T 3 + ⋯ + 3 0 9 7 6
T^4 - 32*T^3 + 544*T^2 - 4928*T + 30976
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