gp: [N,k,chi] = [1815,2,Mod(364,1815)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1815, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1815.364");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [8,0,0,-4,16,0]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == 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 + 3 x 6 + 5 x 4 + 12 x 2 + 16 x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 x 8 + 3 x 6 + 5 x 4 + 1 2 x 2 + 1 6
x^8 + 3*x^6 + 5*x^4 + 12*x^2 + 16
:
β 1 \beta_{1} β 1 = = =
( ν 7 + 5 ν 5 + 15 ν 3 + 42 ν ) / 20 ( \nu^{7} + 5\nu^{5} + 15\nu^{3} + 42\nu ) / 20 ( ν 7 + 5 ν 5 + 1 5 ν 3 + 4 2 ν ) / 2 0
(v^7 + 5*v^5 + 15*v^3 + 42*v) / 20
β 2 \beta_{2} β 2 = = =
( − ν 6 + 5 ν 4 + 15 ν 2 + 8 ) / 20 ( -\nu^{6} + 5\nu^{4} + 15\nu^{2} + 8 ) / 20 ( − ν 6 + 5 ν 4 + 1 5 ν 2 + 8 ) / 2 0
(-v^6 + 5*v^4 + 15*v^2 + 8) / 20
β 3 \beta_{3} β 3 = = =
( − 3 ν 7 − 5 ν 5 + 5 ν 3 − 16 ν ) / 40 ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40 ( − 3 ν 7 − 5 ν 5 + 5 ν 3 − 1 6 ν ) / 4 0
(-3*v^7 - 5*v^5 + 5*v^3 - 16*v) / 40
β 4 \beta_{4} β 4 = = =
( − ν 6 − 3 ν 4 − ν 2 − 8 ) / 4 ( -\nu^{6} - 3\nu^{4} - \nu^{2} - 8 ) / 4 ( − ν 6 − 3 ν 4 − ν 2 − 8 ) / 4
(-v^6 - 3*v^4 - v^2 - 8) / 4
β 5 \beta_{5} β 5 = = =
( − ν 7 − 3 ν 5 − 5 ν 3 − 4 ν ) / 8 ( -\nu^{7} - 3\nu^{5} - 5\nu^{3} - 4\nu ) / 8 ( − ν 7 − 3 ν 5 − 5 ν 3 − 4 ν ) / 8
(-v^7 - 3*v^5 - 5*v^3 - 4*v) / 8
β 6 \beta_{6} β 6 = = =
( 7 ν 6 + 5 ν 4 + 15 ν 2 + 44 ) / 20 ( 7\nu^{6} + 5\nu^{4} + 15\nu^{2} + 44 ) / 20 ( 7 ν 6 + 5 ν 4 + 1 5 ν 2 + 4 4 ) / 2 0
(7*v^6 + 5*v^4 + 15*v^2 + 44) / 20
β 7 \beta_{7} β 7 = = =
( − ν 7 − ν 5 − 3 ν 3 − 6 ν ) / 4 ( -\nu^{7} - \nu^{5} - 3\nu^{3} - 6\nu ) / 4 ( − ν 7 − ν 5 − 3 ν 3 − 6 ν ) / 4
(-v^7 - v^5 - 3*v^3 - 6*v) / 4
ν \nu ν = = =
( β 5 − β 3 + β 1 ) / 2 ( \beta_{5} - \beta_{3} + \beta_1 ) / 2 ( β 5 − β 3 + β 1 ) / 2
(b5 - b3 + b1) / 2
ν 2 \nu^{2} ν 2 = = =
( β 6 + β 4 + 2 β 2 − 1 ) / 2 ( \beta_{6} + \beta_{4} + 2\beta_{2} - 1 ) / 2 ( β 6 + β 4 + 2 β 2 − 1 ) / 2
(b6 + b4 + 2*b2 - 1) / 2
ν 3 \nu^{3} ν 3 = = =
( − β 7 − β 5 + 5 β 3 ) / 2 ( -\beta_{7} - \beta_{5} + 5\beta_{3} ) / 2 ( − β 7 − β 5 + 5 β 3 ) / 2
(-b7 - b5 + 5*b3) / 2
ν 4 \nu^{4} ν 4 = = =
( − 2 β 6 − 3 β 4 + β 2 − 2 ) / 2 ( -2\beta_{6} - 3\beta_{4} + \beta_{2} - 2 ) / 2 ( − 2 β 6 − 3 β 4 + β 2 − 2 ) / 2
(-2*b6 - 3*b4 + b2 - 2) / 2
ν 5 \nu^{5} ν 5 = = =
( 5 β 7 − 6 β 5 − 6 β 3 + β 1 ) / 2 ( 5\beta_{7} - 6\beta_{5} - 6\beta_{3} + \beta_1 ) / 2 ( 5 β 7 − 6 β 5 − 6 β 3 + β 1 ) / 2
(5*b7 - 6*b5 - 6*b3 + b1) / 2
ν 6 \nu^{6} ν 6 = = =
( 5 β 6 − 5 β 2 − 9 ) / 2 ( 5\beta_{6} - 5\beta_{2} - 9 ) / 2 ( 5 β 6 − 5 β 2 − 9 ) / 2
(5*b6 - 5*b2 - 9) / 2
ν 7 \nu^{7} ν 7 = = =
( − 10 β 7 + 3 β 5 − 3 β 3 − 7 β 1 ) / 2 ( -10\beta_{7} + 3\beta_{5} - 3\beta_{3} - 7\beta_1 ) / 2 ( − 1 0 β 7 + 3 β 5 − 3 β 3 − 7 β 1 ) / 2
(-10*b7 + 3*b5 - 3*b3 - 7*b1) / 2
Character values
We give the values of χ \chi χ on generators for ( Z / 1815 Z ) × \left(\mathbb{Z}/1815\mathbb{Z}\right)^\times ( Z / 1 8 1 5 Z ) × .
n n n
727 727 7 2 7
1211 1211 1 2 1 1
1696 1696 1 6 9 6
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 ( 1815 , [ χ ] ) S_{2}^{\mathrm{new}}(1815, [\chi]) S 2 n e w ( 1 8 1 5 , [ χ ] ) :
T 2 4 + 5 T 2 2 + 1 T_{2}^{4} + 5T_{2}^{2} + 1 T 2 4 + 5 T 2 2 + 1
T2^4 + 5*T2^2 + 1
T 19 2 − 28 T_{19}^{2} - 28 T 1 9 2 − 2 8
T19^2 - 28
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T 4 + 5 T 2 + 1 ) 2 (T^{4} + 5 T^{2} + 1)^{2} ( T 4 + 5 T 2 + 1 ) 2
(T^4 + 5*T^2 + 1)^2
3 3 3
( T 2 + 1 ) 4 (T^{2} + 1)^{4} ( T 2 + 1 ) 4
(T^2 + 1)^4
5 5 5
( T 2 − 4 T + 5 ) 4 (T^{2} - 4 T + 5)^{4} ( T 2 − 4 T + 5 ) 4
(T^2 - 4*T + 5)^4
7 7 7
( T 4 + 20 T 2 + 16 ) 2 (T^{4} + 20 T^{2} + 16)^{2} ( T 4 + 2 0 T 2 + 1 6 ) 2
(T^4 + 20*T^2 + 16)^2
11 11 1 1
T 8 T^{8} T 8
T^8
13 13 1 3
( T 4 + 20 T 2 + 16 ) 2 (T^{4} + 20 T^{2} + 16)^{2} ( T 4 + 2 0 T 2 + 1 6 ) 2
(T^4 + 20*T^2 + 16)^2
17 17 1 7
( T 4 + 62 T 2 + 625 ) 2 (T^{4} + 62 T^{2} + 625)^{2} ( T 4 + 6 2 T 2 + 6 2 5 ) 2
(T^4 + 62*T^2 + 625)^2
19 19 1 9
( T 2 − 28 ) 4 (T^{2} - 28)^{4} ( T 2 − 2 8 ) 4
(T^2 - 28)^4
23 23 2 3
( T 4 + 74 T 2 + 25 ) 2 (T^{4} + 74 T^{2} + 25)^{2} ( T 4 + 7 4 T 2 + 2 5 ) 2
(T^4 + 74*T^2 + 25)^2
29 29 2 9
( T 4 − 20 T 2 + 16 ) 2 (T^{4} - 20 T^{2} + 16)^{2} ( T 4 − 2 0 T 2 + 1 6 ) 2
(T^4 - 20*T^2 + 16)^2
31 31 3 1
( T 2 + 4 T − 17 ) 4 (T^{2} + 4 T - 17)^{4} ( T 2 + 4 T − 1 7 ) 4
(T^2 + 4*T - 17)^4
37 37 3 7
( T 4 + 92 T 2 + 16 ) 2 (T^{4} + 92 T^{2} + 16)^{2} ( T 4 + 9 2 T 2 + 1 6 ) 2
(T^4 + 92*T^2 + 16)^2
41 41 4 1
( T 2 − 48 ) 4 (T^{2} - 48)^{4} ( T 2 − 4 8 ) 4
(T^2 - 48)^4
43 43 4 3
( T 4 + 20 T 2 + 16 ) 2 (T^{4} + 20 T^{2} + 16)^{2} ( T 4 + 2 0 T 2 + 1 6 ) 2
(T^4 + 20*T^2 + 16)^2
47 47 4 7
( T 4 + 50 T 2 + 289 ) 2 (T^{4} + 50 T^{2} + 289)^{2} ( T 4 + 5 0 T 2 + 2 8 9 ) 2
(T^4 + 50*T^2 + 289)^2
53 53 5 3
( T 2 + 25 ) 4 (T^{2} + 25)^{4} ( T 2 + 2 5 ) 4
(T^2 + 25)^4
59 59 5 9
( 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
61 61 6 1
( T 2 − 75 ) 4 (T^{2} - 75)^{4} ( T 2 − 7 5 ) 4
(T^2 - 75)^4
67 67 6 7
( T 4 + 380 T 2 + 35344 ) 2 (T^{4} + 380 T^{2} + 35344)^{2} ( T 4 + 3 8 0 T 2 + 3 5 3 4 4 ) 2
(T^4 + 380*T^2 + 35344)^2
71 71 7 1
( T 2 − 84 ) 4 (T^{2} - 84)^{4} ( T 2 − 8 4 ) 4
(T^2 - 84)^4
73 73 7 3
( T 4 + 272 T 2 + 6400 ) 2 (T^{4} + 272 T^{2} + 6400)^{2} ( T 4 + 2 7 2 T 2 + 6 4 0 0 ) 2
(T^4 + 272*T^2 + 6400)^2
79 79 7 9
( T 2 − 7 ) 4 (T^{2} - 7)^{4} ( T 2 − 7 ) 4
(T^2 - 7)^4
83 83 8 3
( T 4 + 152 T 2 + 400 ) 2 (T^{4} + 152 T^{2} + 400)^{2} ( T 4 + 1 5 2 T 2 + 4 0 0 ) 2
(T^4 + 152*T^2 + 400)^2
89 89 8 9
( T 2 + 6 T − 180 ) 4 (T^{2} + 6 T - 180)^{4} ( T 2 + 6 T − 1 8 0 ) 4
(T^2 + 6*T - 180)^4
97 97 9 7
( T 4 + 44 T 2 + 400 ) 2 (T^{4} + 44 T^{2} + 400)^{2} ( T 4 + 4 4 T 2 + 4 0 0 ) 2
(T^4 + 44*T^2 + 400)^2
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