gp: [N,k,chi] = [700,3,Mod(101,700)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(700, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 3, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("700.101");
S:= CuspForms(chi, 3);
N := Newforms(S);
Newform invariants
sage: traces = [2,0,-3]
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 6 \zeta_{6} ζ 6 .
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 / 700 Z ) × \left(\mathbb{Z}/700\mathbb{Z}\right)^\times ( Z / 7 0 0 Z ) × .
n n n
101 101 1 0 1
351 351 3 5 1
477 477 4 7 7
χ ( n ) \chi(n) χ ( n )
ζ 6 \zeta_{6} ζ 6
1 1 1
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 2 + 3 T 3 + 3 T_{3}^{2} + 3T_{3} + 3 T 3 2 + 3 T 3 + 3
T3^2 + 3*T3 + 3
acting on S 3 n e w ( 700 , [ χ ] ) S_{3}^{\mathrm{new}}(700, [\chi]) S 3 n e w ( 7 0 0 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 T^{2} T 2
T^2
3 3 3
T 2 + 3 T + 3 T^{2} + 3T + 3 T 2 + 3 T + 3
T^2 + 3*T + 3
5 5 5
T 2 T^{2} T 2
T^2
7 7 7
( T − 7 ) 2 (T - 7)^{2} ( T − 7 ) 2
(T - 7)^2
11 11 1 1
T 2 + 15 T + 225 T^{2} + 15T + 225 T 2 + 1 5 T + 2 2 5
T^2 + 15*T + 225
13 13 1 3
T 2 + 192 T^{2} + 192 T 2 + 1 9 2
T^2 + 192
17 17 1 7
T 2 + 51 T + 867 T^{2} + 51T + 867 T 2 + 5 1 T + 8 6 7
T^2 + 51*T + 867
19 19 1 9
T 2 − 27 T + 243 T^{2} - 27T + 243 T 2 − 2 7 T + 2 4 3
T^2 - 27*T + 243
23 23 2 3
T 2 + 9 T + 81 T^{2} + 9T + 81 T 2 + 9 T + 8 1
T^2 + 9*T + 81
29 29 2 9
( T + 6 ) 2 (T + 6)^{2} ( T + 6 ) 2
(T + 6)^2
31 31 3 1
T 2 + 21 T + 147 T^{2} + 21T + 147 T 2 + 2 1 T + 1 4 7
T^2 + 21*T + 147
37 37 3 7
T 2 − 31 T + 961 T^{2} - 31T + 961 T 2 − 3 1 T + 9 6 1
T^2 - 31*T + 961
41 41 4 1
T 2 + 3072 T^{2} + 3072 T 2 + 3 0 7 2
T^2 + 3072
43 43 4 3
( T + 10 ) 2 (T + 10)^{2} ( T + 1 0 ) 2
(T + 10)^2
47 47 4 7
T 2 + 75 T + 1875 T^{2} + 75T + 1875 T 2 + 7 5 T + 1 8 7 5
T^2 + 75*T + 1875
53 53 5 3
T 2 + 57 T + 3249 T^{2} + 57T + 3249 T 2 + 5 7 T + 3 2 4 9
T^2 + 57*T + 3249
59 59 5 9
T 2 + 141 T + 6627 T^{2} + 141T + 6627 T 2 + 1 4 1 T + 6 6 2 7
T^2 + 141*T + 6627
61 61 6 1
T 2 + 141 T + 6627 T^{2} + 141T + 6627 T 2 + 1 4 1 T + 6 6 2 7
T^2 + 141*T + 6627
67 67 6 7
T 2 + 49 T + 2401 T^{2} + 49T + 2401 T 2 + 4 9 T + 2 4 0 1
T^2 + 49*T + 2401
71 71 7 1
( T + 126 ) 2 (T + 126)^{2} ( T + 1 2 6 ) 2
(T + 126)^2
73 73 7 3
T 2 − 45 T + 675 T^{2} - 45T + 675 T 2 − 4 5 T + 6 7 5
T^2 - 45*T + 675
79 79 7 9
T 2 − 73 T + 5329 T^{2} - 73T + 5329 T 2 − 7 3 T + 5 3 2 9
T^2 - 73*T + 5329
83 83 8 3
T 2 + 192 T^{2} + 192 T 2 + 1 9 2
T^2 + 192
89 89 8 9
T 2 − 99 T + 3267 T^{2} - 99T + 3267 T 2 − 9 9 T + 3 2 6 7
T^2 - 99*T + 3267
97 97 9 7
T 2 + 768 T^{2} + 768 T 2 + 7 6 8
T^2 + 768
show more
show less