gp: [N,k,chi] = [1386,2,Mod(307,1386)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1386, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 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("1386.307");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [8,0,0,-8,0,0,0,0,0,0,-4,0,0,-8,0,8,0,0,0,0,0,-4]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(22)] == 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 + 23 x 4 + 1 x^{8} + 23x^{4} + 1 x 8 + 2 3 x 4 + 1
x^8 + 23*x^4 + 1
:
β 1 \beta_{1} β 1 = = =
( ν 6 + 24 ν 2 ) / 5 ( \nu^{6} + 24\nu^{2} ) / 5 ( ν 6 + 2 4 ν 2 ) / 5
(v^6 + 24*v^2) / 5
β 2 \beta_{2} β 2 = = =
( ν 7 + 24 ν 3 + 5 ν ) / 5 ( \nu^{7} + 24\nu^{3} + 5\nu ) / 5 ( ν 7 + 2 4 ν 3 + 5 ν ) / 5
(v^7 + 24*v^3 + 5*v) / 5
β 3 \beta_{3} β 3 = = =
( ν 7 + 24 ν 3 − 5 ν ) / 5 ( \nu^{7} + 24\nu^{3} - 5\nu ) / 5 ( ν 7 + 2 4 ν 3 − 5 ν ) / 5
(v^7 + 24*v^3 - 5*v) / 5
β 4 \beta_{4} β 4 = = =
( − 3 ν 6 − ν 4 − 67 ν 2 − 9 ) / 5 ( -3\nu^{6} - \nu^{4} - 67\nu^{2} - 9 ) / 5 ( − 3 ν 6 − ν 4 − 6 7 ν 2 − 9 ) / 5
(-3*v^6 - v^4 - 67*v^2 - 9) / 5
β 5 \beta_{5} β 5 = = =
( − 3 ν 6 + ν 4 − 67 ν 2 + 9 ) / 5 ( -3\nu^{6} + \nu^{4} - 67\nu^{2} + 9 ) / 5 ( − 3 ν 6 + ν 4 − 6 7 ν 2 + 9 ) / 5
(-3*v^6 + v^4 - 67*v^2 + 9) / 5
β 6 \beta_{6} β 6 = = =
( − 5 ν 7 − ν 5 − 115 ν 3 − 24 ν ) / 5 ( -5\nu^{7} - \nu^{5} - 115\nu^{3} - 24\nu ) / 5 ( − 5 ν 7 − ν 5 − 1 1 5 ν 3 − 2 4 ν ) / 5
(-5*v^7 - v^5 - 115*v^3 - 24*v) / 5
β 7 \beta_{7} β 7 = = =
( − 5 ν 7 + ν 5 − 115 ν 3 + 24 ν ) / 5 ( -5\nu^{7} + \nu^{5} - 115\nu^{3} + 24\nu ) / 5 ( − 5 ν 7 + ν 5 − 1 1 5 ν 3 + 2 4 ν ) / 5
(-5*v^7 + v^5 - 115*v^3 + 24*v) / 5
ν \nu ν = = =
( − β 3 + β 2 ) / 2 ( -\beta_{3} + \beta_{2} ) / 2 ( − β 3 + β 2 ) / 2
(-b3 + b2) / 2
ν 2 \nu^{2} ν 2 = = =
( β 5 + β 4 + 6 β 1 ) / 2 ( \beta_{5} + \beta_{4} + 6\beta_1 ) / 2 ( β 5 + β 4 + 6 β 1 ) / 2
(b5 + b4 + 6*b1) / 2
ν 3 \nu^{3} ν 3 = = =
( β 7 + β 6 + 5 β 3 + 5 β 2 ) / 2 ( \beta_{7} + \beta_{6} + 5\beta_{3} + 5\beta_{2} ) / 2 ( β 7 + β 6 + 5 β 3 + 5 β 2 ) / 2
(b7 + b6 + 5*b3 + 5*b2) / 2
ν 4 \nu^{4} ν 4 = = =
( 5 β 5 − 5 β 4 − 18 ) / 2 ( 5\beta_{5} - 5\beta_{4} - 18 ) / 2 ( 5 β 5 − 5 β 4 − 1 8 ) / 2
(5*b5 - 5*b4 - 18) / 2
ν 5 \nu^{5} ν 5 = = =
( 5 β 7 − 5 β 6 + 24 β 3 − 24 β 2 ) / 2 ( 5\beta_{7} - 5\beta_{6} + 24\beta_{3} - 24\beta_{2} ) / 2 ( 5 β 7 − 5 β 6 + 2 4 β 3 − 2 4 β 2 ) / 2
(5*b7 - 5*b6 + 24*b3 - 24*b2) / 2
ν 6 \nu^{6} ν 6 = = =
− 12 β 5 − 12 β 4 − 67 β 1 -12\beta_{5} - 12\beta_{4} - 67\beta_1 − 1 2 β 5 − 1 2 β 4 − 6 7 β 1
-12*b5 - 12*b4 - 67*b1
ν 7 \nu^{7} ν 7 = = =
( − 24 β 7 − 24 β 6 − 115 β 3 − 115 β 2 ) / 2 ( -24\beta_{7} - 24\beta_{6} - 115\beta_{3} - 115\beta_{2} ) / 2 ( − 2 4 β 7 − 2 4 β 6 − 1 1 5 β 3 − 1 1 5 β 2 ) / 2
(-24*b7 - 24*b6 - 115*b3 - 115*b2) / 2
Character values
We give the values of χ \chi χ on generators for ( Z / 1386 Z ) × \left(\mathbb{Z}/1386\mathbb{Z}\right)^\times ( Z / 1 3 8 6 Z ) × .
n n n
155 155 1 5 5
199 199 1 9 9
1135 1135 1 1 3 5
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 ( 1386 , [ χ ] ) S_{2}^{\mathrm{new}}(1386, [\chi]) S 2 n e w ( 1 3 8 6 , [ χ ] ) :
T 5 4 + 10 T 5 2 + 4 T_{5}^{4} + 10T_{5}^{2} + 4 T 5 4 + 1 0 T 5 2 + 4
T5^4 + 10*T5^2 + 4
T 13 4 − 10 T 13 2 + 4 T_{13}^{4} - 10T_{13}^{2} + 4 T 1 3 4 − 1 0 T 1 3 2 + 4
T13^4 - 10*T13^2 + 4
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T 2 + 1 ) 4 (T^{2} + 1)^{4} ( T 2 + 1 ) 4
(T^2 + 1)^4
3 3 3
T 8 T^{8} T 8
T^8
5 5 5
( T 4 + 10 T 2 + 4 ) 2 (T^{4} + 10 T^{2} + 4)^{2} ( T 4 + 1 0 T 2 + 4 ) 2
(T^4 + 10*T^2 + 4)^2
7 7 7
( T 4 − 10 T 2 + 49 ) 2 (T^{4} - 10 T^{2} + 49)^{2} ( T 4 − 1 0 T 2 + 4 9 ) 2
(T^4 - 10*T^2 + 49)^2
11 11 1 1
( T 4 + 2 T 3 + ⋯ + 121 ) 2 (T^{4} + 2 T^{3} + \cdots + 121)^{2} ( T 4 + 2 T 3 + ⋯ + 1 2 1 ) 2
(T^4 + 2*T^3 + 2*T^2 + 22*T + 121)^2
13 13 1 3
( T 4 − 10 T 2 + 4 ) 2 (T^{4} - 10 T^{2} + 4)^{2} ( T 4 − 1 0 T 2 + 4 ) 2
(T^4 - 10*T^2 + 4)^2
17 17 1 7
( T 2 − 14 ) 4 (T^{2} - 14)^{4} ( T 2 − 1 4 ) 4
(T^2 - 14)^4
19 19 1 9
( T 4 − 34 T 2 + 100 ) 2 (T^{4} - 34 T^{2} + 100)^{2} ( T 4 − 3 4 T 2 + 1 0 0 ) 2
(T^4 - 34*T^2 + 100)^2
23 23 2 3
( T + 4 ) 8 (T + 4)^{8} ( T + 4 ) 8
(T + 4)^8
29 29 2 9
( T 4 + 60 T 2 + 144 ) 2 (T^{4} + 60 T^{2} + 144)^{2} ( T 4 + 6 0 T 2 + 1 4 4 ) 2
(T^4 + 60*T^2 + 144)^2
31 31 3 1
( T 4 + 76 T 2 + 100 ) 2 (T^{4} + 76 T^{2} + 100)^{2} ( T 4 + 7 6 T 2 + 1 0 0 ) 2
(T^4 + 76*T^2 + 100)^2
37 37 3 7
( T 2 + 2 T − 20 ) 4 (T^{2} + 2 T - 20)^{4} ( T 2 + 2 T − 2 0 ) 4
(T^2 + 2*T - 20)^4
41 41 4 1
( T 4 − 124 T 2 + 2500 ) 2 (T^{4} - 124 T^{2} + 2500)^{2} ( T 4 − 1 2 4 T 2 + 2 5 0 0 ) 2
(T^4 - 124*T^2 + 2500)^2
43 43 4 3
( T 4 + 176 T 2 + 6400 ) 2 (T^{4} + 176 T^{2} + 6400)^{2} ( T 4 + 1 7 6 T 2 + 6 4 0 0 ) 2
(T^4 + 176*T^2 + 6400)^2
47 47 4 7
( T 4 + 124 T 2 + 2500 ) 2 (T^{4} + 124 T^{2} + 2500)^{2} ( T 4 + 1 2 4 T 2 + 2 5 0 0 ) 2
(T^4 + 124*T^2 + 2500)^2
53 53 5 3
( T 2 − 14 T + 28 ) 4 (T^{2} - 14 T + 28)^{4} ( T 2 − 1 4 T + 2 8 ) 4
(T^2 - 14*T + 28)^4
59 59 5 9
( T 4 + 10 T 2 + 4 ) 2 (T^{4} + 10 T^{2} + 4)^{2} ( T 4 + 1 0 T 2 + 4 ) 2
(T^4 + 10*T^2 + 4)^2
61 61 6 1
( T 4 − 90 T 2 + 324 ) 2 (T^{4} - 90 T^{2} + 324)^{2} ( T 4 − 9 0 T 2 + 3 2 4 ) 2
(T^4 - 90*T^2 + 324)^2
67 67 6 7
( T 2 + 6 T − 12 ) 4 (T^{2} + 6 T - 12)^{4} ( T 2 + 6 T − 1 2 ) 4
(T^2 + 6*T - 12)^4
71 71 7 1
( T + 2 ) 8 (T + 2)^{8} ( T + 2 ) 8
(T + 2)^8
73 73 7 3
( T 4 − 300 T 2 + 10404 ) 2 (T^{4} - 300 T^{2} + 10404)^{2} ( T 4 − 3 0 0 T 2 + 1 0 4 0 4 ) 2
(T^4 - 300*T^2 + 10404)^2
79 79 7 9
( T 2 + 16 ) 4 (T^{2} + 16)^{4} ( T 2 + 1 6 ) 4
(T^2 + 16)^4
83 83 8 3
( T 4 − 250 T 2 + 13924 ) 2 (T^{4} - 250 T^{2} + 13924)^{2} ( T 4 − 2 5 0 T 2 + 1 3 9 2 4 ) 2
(T^4 - 250*T^2 + 13924)^2
89 89 8 9
( T 2 + 96 ) 4 (T^{2} + 96)^{4} ( T 2 + 9 6 ) 4
(T^2 + 96)^4
97 97 9 7
( T 4 + 328 T 2 + 18496 ) 2 (T^{4} + 328 T^{2} + 18496)^{2} ( T 4 + 3 2 8 T 2 + 1 8 4 9 6 ) 2
(T^4 + 328*T^2 + 18496)^2
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