gp: [N,k,chi] = [1008,4,Mod(1,1008)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1008, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 4, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1008.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [2,0,0,0,4,0,-14,0,0,0,-36,0,-12]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == 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 = 2 30 \beta = 2\sqrt{30} β = 2 3 0 .
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
7 7 7
+ 1 +1 + 1
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S 4 n e w ( Γ 0 ( 1008 ) ) S_{4}^{\mathrm{new}}(\Gamma_0(1008)) S 4 n e w ( Γ 0 ( 1 0 0 8 ) ) :
T 5 2 − 4 T 5 − 116 T_{5}^{2} - 4T_{5} - 116 T 5 2 − 4 T 5 − 1 1 6
T5^2 - 4*T5 - 116
T 11 2 + 36 T 11 − 756 T_{11}^{2} + 36T_{11} - 756 T 1 1 2 + 3 6 T 1 1 − 7 5 6
T11^2 + 36*T11 - 756
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 2 − 4 T − 116 T^{2} - 4T - 116 T 2 − 4 T − 1 1 6
T^2 - 4*T - 116
7 7 7
( T + 7 ) 2 (T + 7)^{2} ( T + 7 ) 2
(T + 7)^2
11 11 1 1
T 2 + 36 T − 756 T^{2} + 36T - 756 T 2 + 3 6 T − 7 5 6
T^2 + 36*T - 756
13 13 1 3
T 2 + 12 T − 1884 T^{2} + 12T - 1884 T 2 + 1 2 T − 1 8 8 4
T^2 + 12*T - 1884
17 17 1 7
T 2 − 44 T − 596 T^{2} - 44T - 596 T 2 − 4 4 T − 5 9 6
T^2 - 44*T - 596
19 19 1 9
T 2 + 8 T − 1904 T^{2} + 8T - 1904 T 2 + 8 T − 1 9 0 4
T^2 + 8*T - 1904
23 23 2 3
T 2 + 76 T − 1556 T^{2} + 76T - 1556 T 2 + 7 6 T − 1 5 5 6
T^2 + 76*T - 1556
29 29 2 9
T 2 − 160 T + 4480 T^{2} - 160T + 4480 T 2 − 1 6 0 T + 4 4 8 0
T^2 - 160*T + 4480
31 31 3 1
T 2 − 88 T + 16 T^{2} - 88T + 16 T 2 − 8 8 T + 1 6
T^2 - 88*T + 16
37 37 3 7
T 2 − 180 T − 85980 T^{2} - 180T - 85980 T 2 − 1 8 0 T − 8 5 9 8 0
T^2 - 180*T - 85980
41 41 4 1
T 2 − 356 T + 21964 T^{2} - 356T + 21964 T 2 − 3 5 6 T + 2 1 9 6 4
T^2 - 356*T + 21964
43 43 4 3
T 2 − 56 T − 122096 T^{2} - 56T - 122096 T 2 − 5 6 T − 1 2 2 0 9 6
T^2 - 56*T - 122096
47 47 4 7
T 2 + 616 T + 36784 T^{2} + 616T + 36784 T 2 + 6 1 6 T + 3 6 7 8 4
T^2 + 616*T + 36784
53 53 5 3
T 2 − 808 T + 124336 T^{2} - 808T + 124336 T 2 − 8 0 8 T + 1 2 4 3 3 6
T^2 - 808*T + 124336
59 59 5 9
T 2 + 504 T + 5424 T^{2} + 504T + 5424 T 2 + 5 0 4 T + 5 4 2 4
T^2 + 504*T + 5424
61 61 6 1
T 2 − 572 T + 79876 T^{2} - 572T + 79876 T 2 − 5 7 2 T + 7 9 8 7 6
T^2 - 572*T + 79876
67 67 6 7
T 2 − 504 T − 128496 T^{2} - 504T - 128496 T 2 − 5 0 4 T − 1 2 8 4 9 6
T^2 - 504*T - 128496
71 71 7 1
T 2 + 1188 T + 352716 T^{2} + 1188 T + 352716 T 2 + 1 1 8 8 T + 3 5 2 7 1 6
T^2 + 1188*T + 352716
73 73 7 3
T 2 − 572 T − 540284 T^{2} - 572T - 540284 T 2 − 5 7 2 T − 5 4 0 2 8 4
T^2 - 572*T - 540284
79 79 7 9
T 2 − 784 T − 468416 T^{2} - 784T - 468416 T 2 − 7 8 4 T − 4 6 8 4 1 6
T^2 - 784*T - 468416
83 83 8 3
T 2 + 1344 T + 420864 T^{2} + 1344 T + 420864 T 2 + 1 3 4 4 T + 4 2 0 8 6 4
T^2 + 1344*T + 420864
89 89 8 9
T 2 − 308 T − 422804 T^{2} - 308T - 422804 T 2 − 3 0 8 T − 4 2 2 8 0 4
T^2 - 308*T - 422804
97 97 9 7
T 2 − 1052 T − 652604 T^{2} - 1052 T - 652604 T 2 − 1 0 5 2 T − 6 5 2 6 0 4
T^2 - 1052*T - 652604
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