gp: [N,k,chi] = [576,8,Mod(1,576)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(576, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("576.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
Newform invariants
sage: traces = [2,0,0,0,-280,0,0,0,0,0,0]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == 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 β = 48 435 \beta = 48\sqrt{435} β = 4 8 4 3 5 .
We also show the integral q q q -expansion of the trace form .
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
p p p
Sign
2 2 2
− 1 -1 − 1
3 3 3
+ 1 +1 + 1
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S 8 n e w ( Γ 0 ( 576 ) ) S_{8}^{\mathrm{new}}(\Gamma_0(576)) S 8 n e w ( Γ 0 ( 5 7 6 ) ) :
T 5 + 140 T_{5} + 140 T 5 + 1 4 0
T5 + 140
T 7 2 − 1002240 T_{7}^{2} - 1002240 T 7 2 − 1 0 0 2 2 4 0
T7^2 - 1002240
T 11 2 − 16035840 T_{11}^{2} - 16035840 T 1 1 2 − 1 6 0 3 5 8 4 0
T11^2 - 16035840
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 T^{2} T 2
T^2
3 3 3
T 2 T^{2} T 2
T^2
5 5 5
( T + 140 ) 2 (T + 140)^{2} ( T + 1 4 0 ) 2
(T + 140)^2
7 7 7
T 2 − 1002240 T^{2} - 1002240 T 2 − 1 0 0 2 2 4 0
T^2 - 1002240
11 11 1 1
T 2 − 16035840 T^{2} - 16035840 T 2 − 1 6 0 3 5 8 4 0
T^2 - 16035840
13 13 1 3
( T − 2238 ) 2 (T - 2238)^{2} ( T − 2 2 3 8 ) 2
(T - 2238)^2
17 17 1 7
( T − 1144 ) 2 (T - 1144)^{2} ( T − 1 1 4 4 ) 2
(T - 1144)^2
19 19 1 9
T 2 − 196439040 T^{2} - 196439040 T 2 − 1 9 6 4 3 9 0 4 0
T^2 - 196439040
23 23 2 3
T 2 − 7761346560 T^{2} - 7761346560 T 2 − 7 7 6 1 3 4 6 5 6 0
T^2 - 7761346560
29 29 2 9
( T − 41324 ) 2 (T - 41324)^{2} ( T − 4 1 3 2 4 ) 2
(T - 41324)^2
31 31 3 1
T 2 − 33564015360 T^{2} - 33564015360 T 2 − 3 3 5 6 4 0 1 5 3 6 0
T^2 - 33564015360
37 37 3 7
( T − 137594 ) 2 (T - 137594)^{2} ( T − 1 3 7 5 9 4 ) 2
(T - 137594)^2
41 41 4 1
( T − 123848 ) 2 (T - 123848)^{2} ( T − 1 2 3 8 4 8 ) 2
(T - 123848)^2
43 43 4 3
T 2 − 407955778560 T^{2} - 407955778560 T 2 − 4 0 7 9 5 5 7 7 8 5 6 0
T^2 - 407955778560
47 47 4 7
T 2 − 603524874240 T^{2} - 603524874240 T 2 − 6 0 3 5 2 4 8 7 4 2 4 0
T^2 - 603524874240
53 53 5 3
( T − 981068 ) 2 (T - 981068)^{2} ( T − 9 8 1 0 6 8 ) 2
(T - 981068)^2
59 59 5 9
T 2 − 323346677760 T^{2} - 323346677760 T 2 − 3 2 3 3 4 6 6 7 7 7 6 0
T^2 - 323346677760
61 61 6 1
( T + 1004750 ) 2 (T + 1004750)^{2} ( T + 1 0 0 4 7 5 0 ) 2
(T + 1004750)^2
67 67 6 7
T 2 − 1177688125440 T^{2} - 1177688125440 T 2 − 1 1 7 7 6 8 8 1 2 5 4 4 0
T^2 - 1177688125440
71 71 7 1
T 2 − 5469119447040 T^{2} - 5469119447040 T 2 − 5 4 6 9 1 1 9 4 4 7 0 4 0
T^2 - 5469119447040
73 73 7 3
( T − 603798 ) 2 (T - 603798)^{2} ( T − 6 0 3 7 9 8 ) 2
(T - 603798)^2
79 79 7 9
T 2 − 3367415151360 T^{2} - 3367415151360 T 2 − 3 3 6 7 4 1 5 1 5 1 3 6 0
T^2 - 3367415151360
83 83 8 3
T 2 − 5228341309440 T^{2} - 5228341309440 T 2 − 5 2 2 8 3 4 1 3 0 9 4 4 0
T^2 - 5228341309440
89 89 8 9
( T + 6185104 ) 2 (T + 6185104)^{2} ( T + 6 1 8 5 1 0 4 ) 2
(T + 6185104)^2
97 97 9 7
( T + 6619986 ) 2 (T + 6619986)^{2} ( T + 6 6 1 9 9 8 6 ) 2
(T + 6619986)^2
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