gp: [N,k,chi] = [208,2,Mod(77,208)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(208, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 2]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("208.77");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [8]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == 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 + 7 x 4 + 81 x^{8} + 7x^{4} + 81 x 8 + 7 x 4 + 8 1
x^8 + 7*x^4 + 81
:
β 1 \beta_{1} β 1 = = =
( ν 5 + ν ) / 15 ( \nu^{5} + \nu ) / 15 ( ν 5 + ν ) / 1 5
(v^5 + v) / 15
β 2 \beta_{2} β 2 = = =
( ν 6 + 16 ν 2 ) / 45 ( \nu^{6} + 16\nu^{2} ) / 45 ( ν 6 + 1 6 ν 2 ) / 4 5
(v^6 + 16*v^2) / 45
β 3 \beta_{3} β 3 = = =
( − ν 6 + 3 ν 4 − ν 2 + 3 ) / 15 ( -\nu^{6} + 3\nu^{4} - \nu^{2} + 3 ) / 15 ( − ν 6 + 3 ν 4 − ν 2 + 3 ) / 1 5
(-v^6 + 3*v^4 - v^2 + 3) / 15
β 4 \beta_{4} β 4 = = =
( − 2 ν 7 + 13 ν 3 ) / 135 ( -2\nu^{7} + 13\nu^{3} ) / 135 ( − 2 ν 7 + 1 3 ν 3 ) / 1 3 5
(-2*v^7 + 13*v^3) / 135
β 5 \beta_{5} β 5 = = =
( ν 7 + 16 ν 3 + 45 ν ) / 45 ( \nu^{7} + 16\nu^{3} + 45\nu ) / 45 ( ν 7 + 1 6 ν 3 + 4 5 ν ) / 4 5
(v^7 + 16*v^3 + 45*v) / 45
β 6 \beta_{6} β 6 = = =
( − ν 6 − 3 ν 4 − ν 2 − 3 ) / 15 ( -\nu^{6} - 3\nu^{4} - \nu^{2} - 3 ) / 15 ( − ν 6 − 3 ν 4 − ν 2 − 3 ) / 1 5
(-v^6 - 3*v^4 - v^2 - 3) / 15
β 7 \beta_{7} β 7 = = =
( − ν 7 − 16 ν 3 + 45 ν ) / 45 ( -\nu^{7} - 16\nu^{3} + 45\nu ) / 45 ( − ν 7 − 1 6 ν 3 + 4 5 ν ) / 4 5
(-v^7 - 16*v^3 + 45*v) / 45
ν \nu ν = = =
( β 7 + β 5 ) / 2 ( \beta_{7} + \beta_{5} ) / 2 ( β 7 + β 5 ) / 2
(b7 + b5) / 2
ν 2 \nu^{2} ν 2 = = =
( β 6 + β 3 + 6 β 2 ) / 2 ( \beta_{6} + \beta_{3} + 6\beta_{2} ) / 2 ( β 6 + β 3 + 6 β 2 ) / 2
(b6 + b3 + 6*b2) / 2
ν 3 \nu^{3} ν 3 = = =
− β 7 + β 5 + 3 β 4 -\beta_{7} + \beta_{5} + 3\beta_{4} − β 7 + β 5 + 3 β 4
-b7 + b5 + 3*b4
ν 4 \nu^{4} ν 4 = = =
( − 5 β 6 + 5 β 3 − 2 ) / 2 ( -5\beta_{6} + 5\beta_{3} - 2 ) / 2 ( − 5 β 6 + 5 β 3 − 2 ) / 2
(-5*b6 + 5*b3 - 2) / 2
ν 5 \nu^{5} ν 5 = = =
( − β 7 − β 5 + 30 β 1 ) / 2 ( -\beta_{7} - \beta_{5} + 30\beta_1 ) / 2 ( − β 7 − β 5 + 3 0 β 1 ) / 2
(-b7 - b5 + 30*b1) / 2
ν 6 \nu^{6} ν 6 = = =
− 8 β 6 − 8 β 3 − 3 β 2 -8\beta_{6} - 8\beta_{3} - 3\beta_{2} − 8 β 6 − 8 β 3 − 3 β 2
-8*b6 - 8*b3 - 3*b2
ν 7 \nu^{7} ν 7 = = =
( − 13 β 7 + 13 β 5 − 96 β 4 ) / 2 ( -13\beta_{7} + 13\beta_{5} - 96\beta_{4} ) / 2 ( − 1 3 β 7 + 1 3 β 5 − 9 6 β 4 ) / 2
(-13*b7 + 13*b5 - 96*b4) / 2
Character values
We give the values of χ \chi χ on generators for ( Z / 208 Z ) × \left(\mathbb{Z}/208\mathbb{Z}\right)^\times ( Z / 2 0 8 Z ) × .
n n n
53 53 5 3
79 79 7 9
145 145 1 4 5
χ ( n ) \chi(n) χ ( n )
− β 2 -\beta_{2} − β 2
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 kernel of the linear operator
T 3 4 − 2 T 3 3 + 2 T 3 2 + 10 T 3 + 25 T_{3}^{4} - 2T_{3}^{3} + 2T_{3}^{2} + 10T_{3} + 25 T 3 4 − 2 T 3 3 + 2 T 3 2 + 1 0 T 3 + 2 5
T3^4 - 2*T3^3 + 2*T3^2 + 10*T3 + 25
acting on S 2 n e w ( 208 , [ χ ] ) S_{2}^{\mathrm{new}}(208, [\chi]) S 2 n e w ( 2 0 8 , [ χ ] ) .
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 4 − 2 T 3 + 2 T 2 + ⋯ + 25 ) 2 (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 25)^{2} ( T 4 − 2 T 3 + 2 T 2 + ⋯ + 2 5 ) 2
(T^4 - 2*T^3 + 2*T^2 + 10*T + 25)^2
5 5 5
( T 4 + 1 ) 2 (T^{4} + 1)^{2} ( T 4 + 1 ) 2
(T^4 + 1)^2
7 7 7
( T 4 − 20 T 2 + 1 ) 2 (T^{4} - 20 T^{2} + 1)^{2} ( T 4 − 2 0 T 2 + 1 ) 2
(T^4 - 20*T^2 + 1)^2
11 11 1 1
( T 4 + 16 ) 2 (T^{4} + 16)^{2} ( T 4 + 1 6 ) 2
(T^4 + 16)^2
13 13 1 3
( T 4 + 4 T 3 + ⋯ + 169 ) 2 (T^{4} + 4 T^{3} + \cdots + 169)^{2} ( T 4 + 4 T 3 + ⋯ + 1 6 9 ) 2
(T^4 + 4*T^3 + 8*T^2 + 52*T + 169)^2
17 17 1 7
( T − 3 ) 8 (T - 3)^{8} ( T − 3 ) 8
(T - 3)^8
19 19 1 9
T 8 + 1592 T 4 + 16 T^{8} + 1592T^{4} + 16 T 8 + 1 5 9 2 T 4 + 1 6
T^8 + 1592*T^4 + 16
23 23 2 3
( T 4 + 40 T 2 + 4 ) 2 (T^{4} + 40 T^{2} + 4)^{2} ( T 4 + 4 0 T 2 + 4 ) 2
(T^4 + 40*T^2 + 4)^2
29 29 2 9
( T 4 + 12 T 3 + ⋯ + 16 ) 2 (T^{4} + 12 T^{3} + \cdots + 16)^{2} ( T 4 + 1 2 T 3 + ⋯ + 1 6 ) 2
(T^4 + 12*T^3 + 72*T^2 - 48*T + 16)^2
31 31 3 1
( T 2 + 18 ) 4 (T^{2} + 18)^{4} ( T 2 + 1 8 ) 4
(T^2 + 18)^4
37 37 3 7
T 8 + 7586 T 4 + 390625 T^{8} + 7586 T^{4} + 390625 T 8 + 7 5 8 6 T 4 + 3 9 0 6 2 5
T^8 + 7586*T^4 + 390625
41 41 4 1
T 8 T^{8} T 8
T^8
43 43 4 3
( T 4 + 2 T 3 + ⋯ + 2401 ) 2 (T^{4} + 2 T^{3} + \cdots + 2401)^{2} ( T 4 + 2 T 3 + ⋯ + 2 4 0 1 ) 2
(T^4 + 2*T^3 + 2*T^2 - 98*T + 2401)^2
47 47 4 7
( T 4 + 124 T 2 + 1369 ) 2 (T^{4} + 124 T^{2} + 1369)^{2} ( T 4 + 1 2 4 T 2 + 1 3 6 9 ) 2
(T^4 + 124*T^2 + 1369)^2
53 53 5 3
( T 4 + 12 T 3 + ⋯ + 4900 ) 2 (T^{4} + 12 T^{3} + \cdots + 4900)^{2} ( T 4 + 1 2 T 3 + ⋯ + 4 9 0 0 ) 2
(T^4 + 12*T^3 + 72*T^2 - 840*T + 4900)^2
59 59 5 9
T 8 + 20792 T 4 + 92236816 T^{8} + 20792 T^{4} + 92236816 T 8 + 2 0 7 9 2 T 4 + 9 2 2 3 6 8 1 6
T^8 + 20792*T^4 + 92236816
61 61 6 1
( T 2 − 8 T + 32 ) 4 (T^{2} - 8 T + 32)^{4} ( T 2 − 8 T + 3 2 ) 4
(T^2 - 8*T + 32)^4
67 67 6 7
( T 4 + 1936 ) 2 (T^{4} + 1936)^{2} ( T 4 + 1 9 3 6 ) 2
(T^4 + 1936)^2
71 71 7 1
( T 4 − 180 T 2 + 81 ) 2 (T^{4} - 180 T^{2} + 81)^{2} ( T 4 − 1 8 0 T 2 + 8 1 ) 2
(T^4 - 180*T^2 + 81)^2
73 73 7 3
( T 4 − 188 T 2 + 2500 ) 2 (T^{4} - 188 T^{2} + 2500)^{2} ( T 4 − 1 8 8 T 2 + 2 5 0 0 ) 2
(T^4 - 188*T^2 + 2500)^2
79 79 7 9
( T 2 + 2 T − 98 ) 4 (T^{2} + 2 T - 98)^{4} ( T 2 + 2 T − 9 8 ) 4
(T^2 + 2*T - 98)^4
83 83 8 3
T 8 + 82616 T 4 + 6250000 T^{8} + 82616 T^{4} + 6250000 T 8 + 8 2 6 1 6 T 4 + 6 2 5 0 0 0 0
T^8 + 82616*T^4 + 6250000
89 89 8 9
( T 2 − 198 ) 4 (T^{2} - 198)^{4} ( T 2 − 1 9 8 ) 4
(T^2 - 198)^4
97 97 9 7
( T 2 + 72 ) 4 (T^{2} + 72)^{4} ( T 2 + 7 2 ) 4
(T^2 + 72)^4
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