gp: [N,k,chi] = [612,2,Mod(217,612)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(612, base_ring=CyclotomicField(4))
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("612.217");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [4,0,0,0,0]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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 2 , β 3 1,\beta_1,\beta_2,\beta_3 1 , β 1 , β 2 , β 3 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 4 + 289 x^{4} + 289 x 4 + 2 8 9
x^4 + 289
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( ν 2 ) / 17 ( \nu^{2} ) / 17 ( ν 2 ) / 1 7
(v^2) / 17
β 3 \beta_{3} β 3 = = =
( ν 3 ) / 17 ( \nu^{3} ) / 17 ( ν 3 ) / 1 7
(v^3) / 17
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
17 β 2 17\beta_{2} 1 7 β 2
17*b2
ν 3 \nu^{3} ν 3 = = =
17 β 3 17\beta_{3} 1 7 β 3
17*b3
Character values
We give the values of χ \chi χ on generators for ( Z / 612 Z ) × \left(\mathbb{Z}/612\mathbb{Z}\right)^\times ( Z / 6 1 2 Z ) × .
n n n
37 37 3 7
137 137 1 3 7
307 307 3 0 7
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 intersection of the kernels of the following linear operators acting on S 2 n e w ( 612 , [ χ ] ) S_{2}^{\mathrm{new}}(612, [\chi]) S 2 n e w ( 6 1 2 , [ χ ] ) :
T 5 4 + 289 T_{5}^{4} + 289 T 5 4 + 2 8 9
T5^4 + 289
T 7 2 − 6 T 7 + 18 T_{7}^{2} - 6T_{7} + 18 T 7 2 − 6 T 7 + 1 8
T7^2 - 6*T7 + 18
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 4 T^{4} T 4
T^4
3 3 3
T 4 T^{4} T 4
T^4
5 5 5
T 4 + 289 T^{4} + 289 T 4 + 2 8 9
T^4 + 289
7 7 7
( T 2 − 6 T + 18 ) 2 (T^{2} - 6 T + 18)^{2} ( T 2 − 6 T + 1 8 ) 2
(T^2 - 6*T + 18)^2
11 11 1 1
T 4 + 289 T^{4} + 289 T 4 + 2 8 9
T^4 + 289
13 13 1 3
( T + 5 ) 4 (T + 5)^{4} ( T + 5 ) 4
(T + 5)^4
17 17 1 7
T 4 + 289 T^{4} + 289 T 4 + 2 8 9
T^4 + 289
19 19 1 9
( T 2 + 1 ) 2 (T^{2} + 1)^{2} ( T 2 + 1 ) 2
(T^2 + 1)^2
23 23 2 3
T 4 + 289 T^{4} + 289 T 4 + 2 8 9
T^4 + 289
29 29 2 9
T 4 T^{4} T 4
T^4
31 31 3 1
( T 2 + 8 T + 32 ) 2 (T^{2} + 8 T + 32)^{2} ( T 2 + 8 T + 3 2 ) 2
(T^2 + 8*T + 32)^2
37 37 3 7
( T 2 + 6 T + 18 ) 2 (T^{2} + 6 T + 18)^{2} ( T 2 + 6 T + 1 8 ) 2
(T^2 + 6*T + 18)^2
41 41 4 1
T 4 + 289 T^{4} + 289 T 4 + 2 8 9
T^4 + 289
43 43 4 3
( T 2 + 25 ) 2 (T^{2} + 25)^{2} ( T 2 + 2 5 ) 2
(T^2 + 25)^2
47 47 4 7
( T 2 − 136 ) 2 (T^{2} - 136)^{2} ( T 2 − 1 3 6 ) 2
(T^2 - 136)^2
53 53 5 3
( T 2 + 136 ) 2 (T^{2} + 136)^{2} ( T 2 + 1 3 6 ) 2
(T^2 + 136)^2
59 59 5 9
( T 2 + 34 ) 2 (T^{2} + 34)^{2} ( T 2 + 3 4 ) 2
(T^2 + 34)^2
61 61 6 1
( T 2 − 6 T + 18 ) 2 (T^{2} - 6 T + 18)^{2} ( T 2 − 6 T + 1 8 ) 2
(T^2 - 6*T + 18)^2
67 67 6 7
( T + 6 ) 4 (T + 6)^{4} ( T + 6 ) 4
(T + 6)^4
71 71 7 1
T 4 + 4624 T^{4} + 4624 T 4 + 4 6 2 4
T^4 + 4624
73 73 7 3
( T 2 + 8 T + 32 ) 2 (T^{2} + 8 T + 32)^{2} ( T 2 + 8 T + 3 2 ) 2
(T^2 + 8*T + 32)^2
79 79 7 9
( T 2 − 10 T + 50 ) 2 (T^{2} - 10 T + 50)^{2} ( T 2 − 1 0 T + 5 0 ) 2
(T^2 - 10*T + 50)^2
83 83 8 3
( T 2 + 136 ) 2 (T^{2} + 136)^{2} ( T 2 + 1 3 6 ) 2
(T^2 + 136)^2
89 89 8 9
( T 2 − 306 ) 2 (T^{2} - 306)^{2} ( T 2 − 3 0 6 ) 2
(T^2 - 306)^2
97 97 9 7
( T 2 − 12 T + 72 ) 2 (T^{2} - 12 T + 72)^{2} ( T 2 − 1 2 T + 7 2 ) 2
(T^2 - 12*T + 72)^2
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