gp: [N,k,chi] = [105,2,Mod(16,105)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(105, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 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("105.16");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [4,2]
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 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 / 105 Z ) × \left(\mathbb{Z}/105\mathbb{Z}\right)^\times ( Z / 1 0 5 Z ) × .
n n n
22 22 2 2
31 31 3 1
71 71 7 1
χ ( n ) \chi(n) χ ( n )
1 1 1
− ζ 12 2 -\zeta_{12}^{2} − ζ 1 2 2
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 2 4 − 2 T 2 3 + 6 T 2 2 + 4 T 2 + 4 T_{2}^{4} - 2T_{2}^{3} + 6T_{2}^{2} + 4T_{2} + 4 T 2 4 − 2 T 2 3 + 6 T 2 2 + 4 T 2 + 4
T2^4 - 2*T2^3 + 6*T2^2 + 4*T2 + 4
acting on S 2 n e w ( 105 , [ χ ] ) S_{2}^{\mathrm{new}}(105, [\chi]) S 2 n e w ( 1 0 5 , [ 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 + 6*T^2 + 4*T + 4
3 3 3
( T 2 − T + 1 ) 2 (T^{2} - T + 1)^{2} ( T 2 − T + 1 ) 2
(T^2 - T + 1)^2
5 5 5
( T 2 + T + 1 ) 2 (T^{2} + T + 1)^{2} ( T 2 + T + 1 ) 2
(T^2 + T + 1)^2
7 7 7
T 4 + 11 T 2 + 49 T^{4} + 11T^{2} + 49 T 4 + 1 1 T 2 + 4 9
T^4 + 11*T^2 + 49
11 11 1 1
T 4 − 2 T 3 + ⋯ + 4 T^{4} - 2 T^{3} + \cdots + 4 T 4 − 2 T 3 + ⋯ + 4
T^4 - 2*T^3 + 6*T^2 + 4*T + 4
13 13 1 3
( T 2 − 8 T + 13 ) 2 (T^{2} - 8 T + 13)^{2} ( T 2 − 8 T + 1 3 ) 2
(T^2 - 8*T + 13)^2
17 17 1 7
T 4 + 10 T 3 + ⋯ + 484 T^{4} + 10 T^{3} + \cdots + 484 T 4 + 1 0 T 3 + ⋯ + 4 8 4
T^4 + 10*T^3 + 78*T^2 + 220*T + 484
19 19 1 9
T 4 + 2 T 3 + ⋯ + 121 T^{4} + 2 T^{3} + \cdots + 121 T 4 + 2 T 3 + ⋯ + 1 2 1
T^4 + 2*T^3 + 15*T^2 - 22*T + 121
23 23 2 3
T 4 − 6 T 3 + ⋯ + 36 T^{4} - 6 T^{3} + \cdots + 36 T 4 − 6 T 3 + ⋯ + 3 6
T^4 - 6*T^3 + 30*T^2 - 36*T + 36
29 29 2 9
( T 2 − 2 T − 26 ) 2 (T^{2} - 2 T - 26)^{2} ( T 2 − 2 T − 2 6 ) 2
(T^2 - 2*T - 26)^2
31 31 3 1
T 4 + 6 T 3 + ⋯ + 9 T^{4} + 6 T^{3} + \cdots + 9 T 4 + 6 T 3 + ⋯ + 9
T^4 + 6*T^3 + 39*T^2 - 18*T + 9
37 37 3 7
T 4 + 4 T 3 + ⋯ + 529 T^{4} + 4 T^{3} + \cdots + 529 T 4 + 4 T 3 + ⋯ + 5 2 9
T^4 + 4*T^3 + 39*T^2 - 92*T + 529
41 41 4 1
( T 2 − 2 T − 2 ) 2 (T^{2} - 2 T - 2)^{2} ( T 2 − 2 T − 2 ) 2
(T^2 - 2*T - 2)^2
43 43 4 3
( T 2 + 4 T − 23 ) 2 (T^{2} + 4 T - 23)^{2} ( T 2 + 4 T − 2 3 ) 2
(T^2 + 4*T - 23)^2
47 47 4 7
( T 2 + 2 T + 4 ) 2 (T^{2} + 2 T + 4)^{2} ( T 2 + 2 T + 4 ) 2
(T^2 + 2*T + 4)^2
53 53 5 3
T 4 + 4 T 3 + ⋯ + 10816 T^{4} + 4 T^{3} + \cdots + 10816 T 4 + 4 T 3 + ⋯ + 1 0 8 1 6
T^4 + 4*T^3 + 120*T^2 - 416*T + 10816
59 59 5 9
T 4 + 10 T 3 + ⋯ + 4 T^{4} + 10 T^{3} + \cdots + 4 T 4 + 1 0 T 3 + ⋯ + 4
T^4 + 10*T^3 + 102*T^2 - 20*T + 4
61 61 6 1
( T 2 + 4 T + 16 ) 2 (T^{2} + 4 T + 16)^{2} ( T 2 + 4 T + 1 6 ) 2
(T^2 + 4*T + 16)^2
67 67 6 7
T 4 − 12 T 3 + ⋯ + 1521 T^{4} - 12 T^{3} + \cdots + 1521 T 4 − 1 2 T 3 + ⋯ + 1 5 2 1
T^4 - 12*T^3 + 183*T^2 + 468*T + 1521
71 71 7 1
( T 2 − 2 T − 26 ) 2 (T^{2} - 2 T - 26)^{2} ( T 2 − 2 T − 2 6 ) 2
(T^2 - 2*T - 26)^2
73 73 7 3
T 4 + 8 T 3 + ⋯ + 3481 T^{4} + 8 T^{3} + \cdots + 3481 T 4 + 8 T 3 + ⋯ + 3 4 8 1
T^4 + 8*T^3 + 123*T^2 - 472*T + 3481
79 79 7 9
T 4 + 6 T 3 + ⋯ + 9801 T^{4} + 6 T^{3} + \cdots + 9801 T 4 + 6 T 3 + ⋯ + 9 8 0 1
T^4 + 6*T^3 + 135*T^2 - 594*T + 9801
83 83 8 3
( T 2 − 6 T − 138 ) 2 (T^{2} - 6 T - 138)^{2} ( T 2 − 6 T − 1 3 8 ) 2
(T^2 - 6*T - 138)^2
89 89 8 9
T 4 + 6 T 3 + ⋯ + 19044 T^{4} + 6 T^{3} + \cdots + 19044 T 4 + 6 T 3 + ⋯ + 1 9 0 4 4
T^4 + 6*T^3 + 174*T^2 - 828*T + 19044
97 97 9 7
( T 2 − 16 T + 16 ) 2 (T^{2} - 16 T + 16)^{2} ( T 2 − 1 6 T + 1 6 ) 2
(T^2 - 16*T + 16)^2
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