gp: [N,k,chi] = [975,2,Mod(376,975)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(975, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("975.376");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [6,0,6,-8]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == 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 5 1,\beta_1,\ldots,\beta_{5} 1 , β 1 , … , β 5 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 6 + 10 x 4 + 19 x 2 + 4 x^{6} + 10x^{4} + 19x^{2} + 4 x 6 + 1 0 x 4 + 1 9 x 2 + 4
x^6 + 10*x^4 + 19*x^2 + 4
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
ν 2 + 3 \nu^{2} + 3 ν 2 + 3
v^2 + 3
β 3 \beta_{3} β 3 = = =
( − ν 5 − 8 ν 3 − 5 ν ) / 2 ( -\nu^{5} - 8\nu^{3} - 5\nu ) / 2 ( − ν 5 − 8 ν 3 − 5 ν ) / 2
(-v^5 - 8*v^3 - 5*v) / 2
β 4 \beta_{4} β 4 = = =
( ν 5 + 2 ν 4 + 10 ν 3 + 16 ν 2 + 19 ν + 10 ) / 4 ( \nu^{5} + 2\nu^{4} + 10\nu^{3} + 16\nu^{2} + 19\nu + 10 ) / 4 ( ν 5 + 2 ν 4 + 1 0 ν 3 + 1 6 ν 2 + 1 9 ν + 1 0 ) / 4
(v^5 + 2*v^4 + 10*v^3 + 16*v^2 + 19*v + 10) / 4
β 5 \beta_{5} β 5 = = =
( − ν 5 − 10 ν 3 − 17 ν ) / 2 ( -\nu^{5} - 10\nu^{3} - 17\nu ) / 2 ( − ν 5 − 1 0 ν 3 − 1 7 ν ) / 2
(-v^5 - 10*v^3 - 17*v) / 2
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 2 − 3 \beta_{2} - 3 β 2 − 3
b2 - 3
ν 3 \nu^{3} ν 3 = = =
− β 5 + β 3 − 6 β 1 -\beta_{5} + \beta_{3} - 6\beta_1 − β 5 + β 3 − 6 β 1
-b5 + b3 - 6*b1
ν 4 \nu^{4} ν 4 = = =
β 5 + 2 β 4 − 8 β 2 − β 1 + 19 \beta_{5} + 2\beta_{4} - 8\beta_{2} - \beta _1 + 19 β 5 + 2 β 4 − 8 β 2 − β 1 + 1 9
b5 + 2*b4 - 8*b2 - b1 + 19
ν 5 \nu^{5} ν 5 = = =
8 β 5 − 10 β 3 + 43 β 1 8\beta_{5} - 10\beta_{3} + 43\beta_1 8 β 5 − 1 0 β 3 + 4 3 β 1
8*b5 - 10*b3 + 43*b1
Character values
We give the values of χ \chi χ on generators for ( Z / 975 Z ) × \left(\mathbb{Z}/975\mathbb{Z}\right)^\times ( Z / 9 7 5 Z ) × .
n n n
301 301 3 0 1
326 326 3 2 6
352 352 3 5 2
χ ( n ) \chi(n) χ ( n )
− 1 -1 − 1
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 ( 975 , [ χ ] ) S_{2}^{\mathrm{new}}(975, [\chi]) S 2 n e w ( 9 7 5 , [ χ ] ) :
T 2 6 + 10 T 2 4 + 19 T 2 2 + 4 T_{2}^{6} + 10T_{2}^{4} + 19T_{2}^{2} + 4 T 2 6 + 1 0 T 2 4 + 1 9 T 2 2 + 4
T2^6 + 10*T2^4 + 19*T2^2 + 4
T 17 3 − 35 T 17 + 74 T_{17}^{3} - 35T_{17} + 74 T 1 7 3 − 3 5 T 1 7 + 7 4
T17^3 - 35*T17 + 74
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 6 + 10 T 4 + ⋯ + 4 T^{6} + 10 T^{4} + \cdots + 4 T 6 + 1 0 T 4 + ⋯ + 4
T^6 + 10*T^4 + 19*T^2 + 4
3 3 3
( T − 1 ) 6 (T - 1)^{6} ( T − 1 ) 6
(T - 1)^6
5 5 5
T 6 T^{6} T 6
T^6
7 7 7
T 6 + 17 T 4 + ⋯ + 1 T^{6} + 17 T^{4} + \cdots + 1 T 6 + 1 7 T 4 + ⋯ + 1
T^6 + 17*T^4 + 53*T^2 + 1
11 11 1 1
T 6 + 54 T 4 + ⋯ + 36 T^{6} + 54 T^{4} + \cdots + 36 T 6 + 5 4 T 4 + ⋯ + 3 6
T^6 + 54*T^4 + 667*T^2 + 36
13 13 1 3
T 6 − 3 T 5 + ⋯ + 2197 T^{6} - 3 T^{5} + \cdots + 2197 T 6 − 3 T 5 + ⋯ + 2 1 9 7
T^6 - 3*T^5 + 7*T^4 + 30*T^3 + 91*T^2 - 507*T + 2197
17 17 1 7
( T 3 − 35 T + 74 ) 2 (T^{3} - 35 T + 74)^{2} ( T 3 − 3 5 T + 7 4 ) 2
(T^3 - 35*T + 74)^2
19 19 1 9
T 6 + 71 T 4 + ⋯ + 1444 T^{6} + 71 T^{4} + \cdots + 1444 T 6 + 7 1 T 4 + ⋯ + 1 4 4 4
T^6 + 71*T^4 + 680*T^2 + 1444
23 23 2 3
( T 3 − 2 T 2 − 54 T − 20 ) 2 (T^{3} - 2 T^{2} - 54 T - 20)^{2} ( T 3 − 2 T 2 − 5 4 T − 2 0 ) 2
(T^3 - 2*T^2 - 54*T - 20)^2
29 29 2 9
( T 3 − 8 T 2 + ⋯ + 466 ) 2 (T^{3} - 8 T^{2} + \cdots + 466)^{2} ( T 3 − 8 T 2 + ⋯ + 4 6 6 ) 2
(T^3 - 8*T^2 - 59*T + 466)^2
31 31 3 1
T 6 + 109 T 4 + ⋯ + 29241 T^{6} + 109 T^{4} + \cdots + 29241 T 6 + 1 0 9 T 4 + ⋯ + 2 9 2 4 1
T^6 + 109*T^4 + 3537*T^2 + 29241
37 37 3 7
T 6 + 52 T 4 + ⋯ + 4096 T^{6} + 52 T^{4} + \cdots + 4096 T 6 + 5 2 T 4 + ⋯ + 4 0 9 6
T^6 + 52*T^4 + 844*T^2 + 4096
41 41 4 1
T 6 + 52 T 4 + ⋯ + 4096 T^{6} + 52 T^{4} + \cdots + 4096 T 6 + 5 2 T 4 + ⋯ + 4 0 9 6
T^6 + 52*T^4 + 844*T^2 + 4096
43 43 4 3
( T 3 + 19 T 2 + ⋯ + 178 ) 2 (T^{3} + 19 T^{2} + \cdots + 178)^{2} ( T 3 + 1 9 T 2 + ⋯ + 1 7 8 ) 2
(T^3 + 19*T^2 + 106*T + 178)^2
47 47 4 7
T 6 + 234 T 4 + ⋯ + 206116 T^{6} + 234 T^{4} + \cdots + 206116 T 6 + 2 3 4 T 4 + ⋯ + 2 0 6 1 1 6
T^6 + 234*T^4 + 12867*T^2 + 206116
53 53 5 3
( T 3 + 2 T 2 − 13 T − 24 ) 2 (T^{3} + 2 T^{2} - 13 T - 24)^{2} ( T 3 + 2 T 2 − 1 3 T − 2 4 ) 2
(T^3 + 2*T^2 - 13*T - 24)^2
59 59 5 9
T 6 + 314 T 4 + ⋯ + 93636 T^{6} + 314 T^{4} + \cdots + 93636 T 6 + 3 1 4 T 4 + ⋯ + 9 3 6 3 6
T^6 + 314*T^4 + 23875*T^2 + 93636
61 61 6 1
( T 3 − 7 T 2 + ⋯ + 239 ) 2 (T^{3} - 7 T^{2} + \cdots + 239)^{2} ( T 3 − 7 T 2 + ⋯ + 2 3 9 ) 2
(T^3 - 7*T^2 - 41*T + 239)^2
67 67 6 7
T 6 + 45 T 4 + ⋯ + 729 T^{6} + 45 T^{4} + \cdots + 729 T 6 + 4 5 T 4 + ⋯ + 7 2 9
T^6 + 45*T^4 + 529*T^2 + 729
71 71 7 1
T 6 + 228 T 4 + ⋯ + 64 T^{6} + 228 T^{4} + \cdots + 64 T 6 + 2 2 8 T 4 + ⋯ + 6 4
T^6 + 228*T^4 + 12684*T^2 + 64
73 73 7 3
T 6 + 460 T 4 + ⋯ + 577600 T^{6} + 460 T^{4} + \cdots + 577600 T 6 + 4 6 0 T 4 + ⋯ + 5 7 7 6 0 0
T^6 + 460*T^4 + 53932*T^2 + 577600
79 79 7 9
( T 3 − 12 T 2 + ⋯ + 976 ) 2 (T^{3} - 12 T^{2} + \cdots + 976)^{2} ( T 3 − 1 2 T 2 + ⋯ + 9 7 6 ) 2
(T^3 - 12*T^2 - 98*T + 976)^2
83 83 8 3
T 6 + 362 T 4 + ⋯ + 427716 T^{6} + 362 T^{4} + \cdots + 427716 T 6 + 3 6 2 T 4 + ⋯ + 4 2 7 7 1 6
T^6 + 362*T^4 + 23923*T^2 + 427716
89 89 8 9
T 6 + 360 T 4 + ⋯ + 186624 T^{6} + 360 T^{4} + \cdots + 186624 T 6 + 3 6 0 T 4 + ⋯ + 1 8 6 6 2 4
T^6 + 360*T^4 + 24624*T^2 + 186624
97 97 9 7
T 6 + 187 T 4 + ⋯ + 29584 T^{6} + 187 T^{4} + \cdots + 29584 T 6 + 1 8 7 T 4 + ⋯ + 2 9 5 8 4
T^6 + 187*T^4 + 8872*T^2 + 29584
show more
show less