gp: [N,k,chi] = [300,3,Mod(107,300)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(300, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 2, 1]))
N = Newforms(chi, 3, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("300.107");
S:= CuspForms(chi, 3);
N := Newforms(S);
Newform invariants
sage: traces = [8,0,0,0,0,12]
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 + 17 x 4 + 256 x^{8} + 17x^{4} + 256 x 8 + 1 7 x 4 + 2 5 6
x^8 + 17*x^4 + 256
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( ν 4 + 5 ) / 7 ( \nu^{4} + 5 ) / 7 ( ν 4 + 5 ) / 7
(v^4 + 5) / 7
β 3 \beta_{3} β 3 = = =
( ν 5 + 5 ν ) / 28 ( \nu^{5} + 5\nu ) / 28 ( ν 5 + 5 ν ) / 2 8
(v^5 + 5*v) / 28
β 4 \beta_{4} β 4 = = =
( ν 6 + 33 ν 2 ) / 112 ( \nu^{6} + 33\nu^{2} ) / 112 ( ν 6 + 3 3 ν 2 ) / 1 1 2
(v^6 + 33*v^2) / 112
β 5 \beta_{5} β 5 = = =
( − 3 ν 7 + 13 ν 3 ) / 448 ( -3\nu^{7} + 13\nu^{3} ) / 448 ( − 3 ν 7 + 1 3 ν 3 ) / 4 4 8
(-3*v^7 + 13*v^3) / 448
β 6 \beta_{6} β 6 = = =
( − ν 6 − 5 ν 2 ) / 28 ( -\nu^{6} - 5\nu^{2} ) / 28 ( − ν 6 − 5 ν 2 ) / 2 8
(-v^6 - 5*v^2) / 28
β 7 \beta_{7} β 7 = = =
( ν 7 + 33 ν 3 ) / 112 ( \nu^{7} + 33\nu^{3} ) / 112 ( ν 7 + 3 3 ν 3 ) / 1 1 2
(v^7 + 33*v^3) / 112
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 6 + 4 β 4 \beta_{6} + 4\beta_{4} β 6 + 4 β 4
b6 + 4*b4
ν 3 \nu^{3} ν 3 = = =
3 β 7 + 4 β 5 3\beta_{7} + 4\beta_{5} 3 β 7 + 4 β 5
3*b7 + 4*b5
ν 4 \nu^{4} ν 4 = = =
7 β 2 − 5 7\beta_{2} - 5 7 β 2 − 5
7*b2 - 5
ν 5 \nu^{5} ν 5 = = =
28 β 3 − 5 β 1 28\beta_{3} - 5\beta_1 2 8 β 3 − 5 β 1
28*b3 - 5*b1
ν 6 \nu^{6} ν 6 = = =
− 33 β 6 − 20 β 4 -33\beta_{6} - 20\beta_{4} − 3 3 β 6 − 2 0 β 4
-33*b6 - 20*b4
ν 7 \nu^{7} ν 7 = = =
13 β 7 − 132 β 5 13\beta_{7} - 132\beta_{5} 1 3 β 7 − 1 3 2 β 5
13*b7 - 132*b5
Character values
We give the values of χ \chi χ on generators for ( Z / 300 Z ) × \left(\mathbb{Z}/300\mathbb{Z}\right)^\times ( Z / 3 0 0 Z ) × .
n n n
101 101 1 0 1
151 151 1 5 1
277 277 2 7 7
χ ( n ) \chi(n) χ ( n )
− 1 -1 − 1
− 1 -1 − 1
− β 4 -\beta_{4} − β 4
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 3 n e w ( 300 , [ χ ] ) S_{3}^{\mathrm{new}}(300, [\chi]) S 3 n e w ( 3 0 0 , [ χ ] ) :
T 7 T_{7} T 7
T7
T 17 4 + 921600 T_{17}^{4} + 921600 T 1 7 4 + 9 2 1 6 0 0
T17^4 + 921600
T 19 2 − 960 T_{19}^{2} - 960 T 1 9 2 − 9 6 0
T19^2 - 960
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 8 + 17 T 4 + 256 T^{8} + 17T^{4} + 256 T 8 + 1 7 T 4 + 2 5 6
T^8 + 17*T^4 + 256
3 3 3
( T 4 + 81 ) 2 (T^{4} + 81)^{2} ( T 4 + 8 1 ) 2
(T^4 + 81)^2
5 5 5
T 8 T^{8} T 8
T^8
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 8 T^{8} T 8
T^8
17 17 1 7
( T 4 + 921600 ) 2 (T^{4} + 921600)^{2} ( T 4 + 9 2 1 6 0 0 ) 2
(T^4 + 921600)^2
19 19 1 9
( T 2 − 960 ) 4 (T^{2} - 960)^{4} ( T 2 − 9 6 0 ) 4
(T^2 - 960)^4
23 23 2 3
( T 4 + 1336336 ) 2 (T^{4} + 1336336)^{2} ( T 4 + 1 3 3 6 3 3 6 ) 2
(T^4 + 1336336)^2
29 29 2 9
T 8 T^{8} T 8
T^8
31 31 3 1
( T 2 + 3840 ) 4 (T^{2} + 3840)^{4} ( T 2 + 3 8 4 0 ) 4
(T^2 + 3840)^4
37 37 3 7
T 8 T^{8} T 8
T^8
41 41 4 1
T 8 T^{8} T 8
T^8
43 43 4 3
T 8 T^{8} T 8
T^8
47 47 4 7
( T 4 + 38416 ) 2 (T^{4} + 38416)^{2} ( T 4 + 3 8 4 1 6 ) 2
(T^4 + 38416)^2
53 53 5 3
( T 4 + 14745600 ) 2 (T^{4} + 14745600)^{2} ( T 4 + 1 4 7 4 5 6 0 0 ) 2
(T^4 + 14745600)^2
59 59 5 9
T 8 T^{8} T 8
T^8
61 61 6 1
( T − 118 ) 8 (T - 118)^{8} ( T − 1 1 8 ) 8
(T - 118)^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 8 T^{8} T 8
T^8
79 79 7 9
( T 2 − 15360 ) 4 (T^{2} - 15360)^{4} ( T 2 − 1 5 3 6 0 ) 4
(T^2 - 15360)^4
83 83 8 3
( T 4 + 562448656 ) 2 (T^{4} + 562448656)^{2} ( T 4 + 5 6 2 4 4 8 6 5 6 ) 2
(T^4 + 562448656)^2
89 89 8 9
T 8 T^{8} T 8
T^8
97 97 9 7
T 8 T^{8} T 8
T^8
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