gp: [N,k,chi] = [1368,2,Mod(379,1368)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1368, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 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("1368.379");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [8,0,0,-16]
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 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 − 12 x 6 + 96 x 4 + 248 x 2 + 900 x^{8} - 12x^{6} + 96x^{4} + 248x^{2} + 900 x 8 − 1 2 x 6 + 9 6 x 4 + 2 4 8 x 2 + 9 0 0
x^8 - 12*x^6 + 96*x^4 + 248*x^2 + 900
:
β 1 \beta_{1} β 1 = = =
( ν 7 − 10 ν 5 + 86 ν 3 + 520 ν ) / 600 ( \nu^{7} - 10\nu^{5} + 86\nu^{3} + 520\nu ) / 600 ( ν 7 − 1 0 ν 5 + 8 6 ν 3 + 5 2 0 ν ) / 6 0 0
(v^7 - 10*v^5 + 86*v^3 + 520*v) / 600
β 2 \beta_{2} β 2 = = =
( ν 6 − 5 ν 4 + 136 ν 2 + 150 ) / 300 ( \nu^{6} - 5\nu^{4} + 136\nu^{2} + 150 ) / 300 ( ν 6 − 5 ν 4 + 1 3 6 ν 2 + 1 5 0 ) / 3 0 0
(v^6 - 5*v^4 + 136*v^2 + 150) / 300
β 3 \beta_{3} β 3 = = =
( − ν 7 + 10 ν 5 − 86 ν 3 + 80 ν ) / 300 ( -\nu^{7} + 10\nu^{5} - 86\nu^{3} + 80\nu ) / 300 ( − ν 7 + 1 0 ν 5 − 8 6 ν 3 + 8 0 ν ) / 3 0 0
(-v^7 + 10*v^5 - 86*v^3 + 80*v) / 300
β 4 \beta_{4} β 4 = = =
( − ν 6 + 5 ν 4 + 14 ν 2 − 600 ) / 150 ( -\nu^{6} + 5\nu^{4} + 14\nu^{2} - 600 ) / 150 ( − ν 6 + 5 ν 4 + 1 4 ν 2 − 6 0 0 ) / 1 5 0
(-v^6 + 5*v^4 + 14*v^2 - 600) / 150
β 5 \beta_{5} β 5 = = =
( − ν 7 + 12 ν 5 − 66 ν 3 − 428 ν ) / 120 ( -\nu^{7} + 12\nu^{5} - 66\nu^{3} - 428\nu ) / 120 ( − ν 7 + 1 2 ν 5 − 6 6 ν 3 − 4 2 8 ν ) / 1 2 0
(-v^7 + 12*v^5 - 66*v^3 - 428*v) / 120
β 6 \beta_{6} β 6 = = =
( 2 ν 7 − 35 ν 5 + 322 ν 3 − 250 ν ) / 300 ( 2\nu^{7} - 35\nu^{5} + 322\nu^{3} - 250\nu ) / 300 ( 2 ν 7 − 3 5 ν 5 + 3 2 2 ν 3 − 2 5 0 ν ) / 3 0 0
(2*v^7 - 35*v^5 + 322*v^3 - 250*v) / 300
β 7 \beta_{7} β 7 = = =
( − 8 ν 6 + 115 ν 4 − 938 ν 2 − 750 ) / 300 ( -8\nu^{6} + 115\nu^{4} - 938\nu^{2} - 750 ) / 300 ( − 8 ν 6 + 1 1 5 ν 4 − 9 3 8 ν 2 − 7 5 0 ) / 3 0 0
(-8*v^6 + 115*v^4 - 938*v^2 - 750) / 300
ν \nu ν = = =
( β 3 + 2 β 1 ) / 2 ( \beta_{3} + 2\beta_1 ) / 2 ( β 3 + 2 β 1 ) / 2
(b3 + 2*b1) / 2
ν 2 \nu^{2} ν 2 = = =
β 4 + 2 β 2 + 3 \beta_{4} + 2\beta_{2} + 3 β 4 + 2 β 2 + 3
b4 + 2*b2 + 3
ν 3 \nu^{3} ν 3 = = =
β 6 + 3 β 5 + β 3 + 13 β 1 \beta_{6} + 3\beta_{5} + \beta_{3} + 13\beta_1 β 6 + 3 β 5 + β 3 + 1 3 β 1
b6 + 3*b5 + b3 + 13*b1
ν 4 \nu^{4} ν 4 = = =
4 β 7 − 2 β 4 + 28 β 2 − 12 4\beta_{7} - 2\beta_{4} + 28\beta_{2} - 12 4 β 7 − 2 β 4 + 2 8 β 2 − 1 2
4*b7 - 2*b4 + 28*b2 - 12
ν 5 \nu^{5} ν 5 = = =
− 10 β 6 + 30 β 5 − 33 β 3 + 124 β 1 -10\beta_{6} + 30\beta_{5} - 33\beta_{3} + 124\beta_1 − 1 0 β 6 + 3 0 β 5 − 3 3 β 3 + 1 2 4 β 1
-10*b6 + 30*b5 - 33*b3 + 124*b1
ν 6 \nu^{6} ν 6 = = =
20 β 7 − 146 β 4 + 168 β 2 − 618 20\beta_{7} - 146\beta_{4} + 168\beta_{2} - 618 2 0 β 7 − 1 4 6 β 4 + 1 6 8 β 2 − 6 1 8
20*b7 - 146*b4 + 168*b2 - 618
ν 7 \nu^{7} ν 7 = = =
− 186 β 6 + 42 β 5 − 676 β 3 + 202 β 1 -186\beta_{6} + 42\beta_{5} - 676\beta_{3} + 202\beta_1 − 1 8 6 β 6 + 4 2 β 5 − 6 7 6 β 3 + 2 0 2 β 1
-186*b6 + 42*b5 - 676*b3 + 202*b1
Character values
We give the values of χ \chi χ on generators for ( Z / 1368 Z ) × \left(\mathbb{Z}/1368\mathbb{Z}\right)^\times ( Z / 1 3 6 8 Z ) × .
n n n
343 343 3 4 3
685 685 6 8 5
1009 1009 1 0 0 9
1217 1217 1 2 1 7
χ ( n ) \chi(n) χ ( n )
− 1 -1 − 1
− 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 ( 1368 , [ χ ] ) S_{2}^{\mathrm{new}}(1368, [\chi]) S 2 n e w ( 1 3 6 8 , [ χ ] ) :
T 5 4 + 20 T 5 2 + 24 T_{5}^{4} + 20T_{5}^{2} + 24 T 5 4 + 2 0 T 5 2 + 2 4
T5^4 + 20*T5^2 + 24
T 7 T_{7} T 7
T7
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T 2 + 2 ) 4 (T^{2} + 2)^{4} ( T 2 + 2 ) 4
(T^2 + 2)^4
3 3 3
T 8 T^{8} T 8
T^8
5 5 5
( T 4 + 20 T 2 + 24 ) 2 (T^{4} + 20 T^{2} + 24)^{2} ( T 4 + 2 0 T 2 + 2 4 ) 2
(T^4 + 20*T^2 + 24)^2
7 7 7
T 8 T^{8} T 8
T^8
11 11 1 1
T 8 T^{8} T 8
T^8
13 13 1 3
( T 4 − 52 T 2 + 600 ) 2 (T^{4} - 52 T^{2} + 600)^{2} ( T 4 − 5 2 T 2 + 6 0 0 ) 2
(T^4 - 52*T^2 + 600)^2
17 17 1 7
T 8 T^{8} T 8
T^8
19 19 1 9
( T 2 − 19 ) 4 (T^{2} - 19)^{4} ( T 2 − 1 9 ) 4
(T^2 - 19)^4
23 23 2 3
( T 4 + 92 T 2 + 216 ) 2 (T^{4} + 92 T^{2} + 216)^{2} ( T 4 + 9 2 T 2 + 2 1 6 ) 2
(T^4 + 92*T^2 + 216)^2
29 29 2 9
T 8 T^{8} T 8
T^8
31 31 3 1
( T 4 − 124 T 2 + 1944 ) 2 (T^{4} - 124 T^{2} + 1944)^{2} ( T 4 − 1 2 4 T 2 + 1 9 4 4 ) 2
(T^4 - 124*T^2 + 1944)^2
37 37 3 7
( T 4 − 148 T 2 + 5400 ) 2 (T^{4} - 148 T^{2} + 5400)^{2} ( T 4 − 1 4 8 T 2 + 5 4 0 0 ) 2
(T^4 - 148*T^2 + 5400)^2
41 41 4 1
( T 2 + 152 ) 4 (T^{2} + 152)^{4} ( T 2 + 1 5 2 ) 4
(T^2 + 152)^4
43 43 4 3
( T 2 − 76 ) 4 (T^{2} - 76)^{4} ( T 2 − 7 6 ) 4
(T^2 - 76)^4
47 47 4 7
( T 4 + 188 T 2 + 6936 ) 2 (T^{4} + 188 T^{2} + 6936)^{2} ( T 4 + 1 8 8 T 2 + 6 9 3 6 ) 2
(T^4 + 188*T^2 + 6936)^2
53 53 5 3
T 8 T^{8} T 8
T^8
59 59 5 9
( T 2 + 8 ) 4 (T^{2} + 8)^{4} ( T 2 + 8 ) 4
(T^2 + 8)^4
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 8 T^{8} T 8
T^8
73 73 7 3
( T 2 − 76 ) 4 (T^{2} - 76)^{4} ( T 2 − 7 6 ) 4
(T^2 - 76)^4
79 79 7 9
( T 4 − 316 T 2 + 23064 ) 2 (T^{4} - 316 T^{2} + 23064)^{2} ( T 4 − 3 1 6 T 2 + 2 3 0 6 4 ) 2
(T^4 - 316*T^2 + 23064)^2
83 83 8 3
T 8 T^{8} T 8
T^8
89 89 8 9
( T 2 + 128 ) 4 (T^{2} + 128)^{4} ( T 2 + 1 2 8 ) 4
(T^2 + 128)^4
97 97 9 7
T 8 T^{8} T 8
T^8
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