gp: [N,k,chi] = [735,2,Mod(374,735)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(735, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 1]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("735.374");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [8,0,0]
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 basis 1 , β 1 , … , β 7 1,\beta_1,\ldots,\beta_{7} 1 , β 1 , … , β 7 for the coefficient ring described below.
We also show the integral q q q -expansion of the trace form .
Basis of coefficient ring in terms of a root ν \nu ν of
x 8 + x 6 − 8 x 4 + 9 x 2 + 81 x^{8} + x^{6} - 8x^{4} + 9x^{2} + 81 x 8 + x 6 − 8 x 4 + 9 x 2 + 8 1
x^8 + x^6 - 8*x^4 + 9*x^2 + 81
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
ν 2 \nu^{2} ν 2
v^2
β 3 \beta_{3} β 3 = = =
( ν 7 − 8 ν 5 + 64 ν 3 + 81 ν ) / 216 ( \nu^{7} - 8\nu^{5} + 64\nu^{3} + 81\nu ) / 216 ( ν 7 − 8 ν 5 + 6 4 ν 3 + 8 1 ν ) / 2 1 6
(v^7 - 8*v^5 + 64*v^3 + 81*v) / 216
β 4 \beta_{4} β 4 = = =
( − ν 6 + 8 ν 4 + 8 ν 2 − 81 ) / 72 ( -\nu^{6} + 8\nu^{4} + 8\nu^{2} - 81 ) / 72 ( − ν 6 + 8 ν 4 + 8 ν 2 − 8 1 ) / 7 2
(-v^6 + 8*v^4 + 8*v^2 - 81) / 72
β 5 \beta_{5} β 5 = = =
( − ν 7 + 8 ν 5 + 8 ν 3 − 81 ν ) / 72 ( -\nu^{7} + 8\nu^{5} + 8\nu^{3} - 81\nu ) / 72 ( − ν 7 + 8 ν 5 + 8 ν 3 − 8 1 ν ) / 7 2
(-v^7 + 8*v^5 + 8*v^3 - 81*v) / 72
β 6 \beta_{6} β 6 = = =
( ν 7 + 17 ν ) / 24 ( \nu^{7} + 17\nu ) / 24 ( ν 7 + 1 7 ν ) / 2 4
(v^7 + 17*v) / 24
β 7 \beta_{7} β 7 = = =
( ν 6 − 8 ν 2 + 9 ) / 8 ( \nu^{6} - 8\nu^{2} + 9 ) / 8 ( ν 6 − 8 ν 2 + 9 ) / 8
(v^6 - 8*v^2 + 9) / 8
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 2 \beta_{2} β 2
b2
ν 3 \nu^{3} ν 3 = = =
β 5 + 3 β 3 \beta_{5} + 3\beta_{3} β 5 + 3 β 3
b5 + 3*b3
ν 4 \nu^{4} ν 4 = = =
β 7 + 9 β 4 + 9 \beta_{7} + 9\beta_{4} + 9 β 7 + 9 β 4 + 9
b7 + 9*b4 + 9
ν 5 \nu^{5} ν 5 = = =
3 β 6 + 8 β 5 − 3 β 3 + 8 β 1 3\beta_{6} + 8\beta_{5} - 3\beta_{3} + 8\beta_1 3 β 6 + 8 β 5 − 3 β 3 + 8 β 1
3*b6 + 8*b5 - 3*b3 + 8*b1
ν 6 \nu^{6} ν 6 = = =
8 β 7 + 8 β 2 − 9 8\beta_{7} + 8\beta_{2} - 9 8 β 7 + 8 β 2 − 9
8*b7 + 8*b2 - 9
ν 7 \nu^{7} ν 7 = = =
24 β 6 − 17 β 1 24\beta_{6} - 17\beta_1 2 4 β 6 − 1 7 β 1
24*b6 - 17*b1
Character values
We give the values of χ \chi χ on generators for ( Z / 735 Z ) × \left(\mathbb{Z}/735\mathbb{Z}\right)^\times ( Z / 7 3 5 Z ) × .
n n n
346 346 3 4 6
442 442 4 4 2
491 491 4 9 1
χ ( n ) \chi(n) χ ( n )
− β 4 -\beta_{4} − β 4
− 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 intersection of the kernels of the following linear operators acting on S 2 n e w ( 735 , [ χ ] ) S_{2}^{\mathrm{new}}(735, [\chi]) S 2 n e w ( 7 3 5 , [ χ ] ) :
T 2 T_{2} T 2
T2
T 13 2 − 7 T_{13}^{2} - 7 T 1 3 2 − 7
T13^2 - 7
T 257 4 − 20 T 257 2 + 400 T_{257}^{4} - 20T_{257}^{2} + 400 T 2 5 7 4 − 2 0 T 2 5 7 2 + 4 0 0
T257^4 - 20*T257^2 + 400
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 8 T^{8} T 8
T^8
3 3 3
T 8 + T 6 + ⋯ + 81 T^{8} + T^{6} + \cdots + 81 T 8 + T 6 + ⋯ + 8 1
T^8 + T^6 - 8*T^4 + 9*T^2 + 81
5 5 5
( T 4 − 5 T 2 + 25 ) 2 (T^{4} - 5 T^{2} + 25)^{2} ( T 4 − 5 T 2 + 2 5 ) 2
(T^4 - 5*T^2 + 25)^2
7 7 7
T 8 T^{8} T 8
T^8
11 11 1 1
( T 4 − 35 T 2 + 1225 ) 2 (T^{4} - 35 T^{2} + 1225)^{2} ( T 4 − 3 5 T 2 + 1 2 2 5 ) 2
(T^4 - 35*T^2 + 1225)^2
13 13 1 3
( T 2 − 7 ) 4 (T^{2} - 7)^{4} ( T 2 − 7 ) 4
(T^2 - 7)^4
17 17 1 7
( T 4 − 5 T 2 + 25 ) 2 (T^{4} - 5 T^{2} + 25)^{2} ( T 4 − 5 T 2 + 2 5 ) 2
(T^4 - 5*T^2 + 25)^2
19 19 1 9
T 8 T^{8} T 8
T^8
23 23 2 3
T 8 T^{8} T 8
T^8
29 29 2 9
( T 2 + 35 ) 4 (T^{2} + 35)^{4} ( T 2 + 3 5 ) 4
(T^2 + 35)^4
31 31 3 1
T 8 T^{8} T 8
T^8
37 37 3 7
T 8 T^{8} T 8
T^8
41 41 4 1
T 8 T^{8} T 8
T^8
43 43 4 3
T 8 T^{8} T 8
T^8
47 47 4 7
( T 4 − 125 T 2 + 15625 ) 2 (T^{4} - 125 T^{2} + 15625)^{2} ( T 4 − 1 2 5 T 2 + 1 5 6 2 5 ) 2
(T^4 - 125*T^2 + 15625)^2
53 53 5 3
T 8 T^{8} T 8
T^8
59 59 5 9
T 8 T^{8} T 8
T^8
61 61 6 1
T 8 T^{8} T 8
T^8
67 67 6 7
T 8 T^{8} T 8
T^8
71 71 7 1
( T 2 + 140 ) 4 (T^{2} + 140)^{4} ( T 2 + 1 4 0 ) 4
(T^2 + 140)^4
73 73 7 3
( T 4 + 112 T 2 + 12544 ) 2 (T^{4} + 112 T^{2} + 12544)^{2} ( T 4 + 1 1 2 T 2 + 1 2 5 4 4 ) 2
(T^4 + 112*T^2 + 12544)^2
79 79 7 9
( T 2 − T + 1 ) 4 (T^{2} - T + 1)^{4} ( T 2 − T + 1 ) 4
(T^2 - T + 1)^4
83 83 8 3
( T 2 + 80 ) 4 (T^{2} + 80)^{4} ( T 2 + 8 0 ) 4
(T^2 + 80)^4
89 89 8 9
T 8 T^{8} T 8
T^8
97 97 9 7
( T 2 − 343 ) 4 (T^{2} - 343)^{4} ( T 2 − 3 4 3 ) 4
(T^2 - 343)^4
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