gp: [N,k,chi] = [90,18,Mod(1,90)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(90, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 18, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("90.1");
S:= CuspForms(chi, 18);
N := Newforms(S);
Newform invariants
sage: traces = [2,512,0,131072,781250,0,1059712]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 β = 60 88769641 \beta = 60\sqrt{88769641} β = 6 0 8 8 7 6 9 6 4 1 q q q trace form .
For each embedding ι m \iota_m ι m ι m ( a n ) \iota_m(a_n) ι m ( a n )
For more information on an embedded modular form you can click on its label.
gp: mfembed(f)
p p p
Sign
2 2 2
− 1 -1 − 1
3 3 3
− 1 -1 − 1
5 5 5
− 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 18 n e w ( Γ 0 ( 90 ) ) S_{18}^{\mathrm{new}}(\Gamma_0(90)) S 1 8 n e w ( Γ 0 ( 9 0 ) )
T 7 2 − 1059712 T 7 − 590605490971664 T_{7}^{2} - 1059712T_{7} - 590605490971664 T 7 2 − 1 0 5 9 7 1 2 T 7 − 5 9 0 6 0 5 4 9 0 9 7 1 6 6 4
T 11 2 − 1570139040 T 11 + 366603704354764800 T_{11}^{2} - 1570139040T_{11} + 366603704354764800 T 1 1 2 − 1 5 7 0 1 3 9 0 4 0 T 1 1 + 3 6 6 6 0 3 7 0 4 3 5 4 7 6 4 8 0 0
p p p F p ( T ) F_p(T) F p ( T )
2 2 2
( T − 256 ) 2 (T - 256)^{2} ( T − 2 5 6 ) 2
3 3 3
T 2 T^{2} T 2
5 5 5
( T − 390625 ) 2 (T - 390625)^{2} ( T − 3 9 0 6 2 5 ) 2
7 7 7
T 2 + ⋯ − 590605490971664 T^{2} + \cdots - 590605490971664 T 2 + ⋯ − 5 9 0 6 0 5 4 9 0 9 7 1 6 6 4
11 11 1 1
T 2 + ⋯ + 36 ⋯ 00 T^{2} + \cdots + 36\!\cdots\!00 T 2 + ⋯ + 3 6 ⋯ 0 0
13 13 1 3
T 2 + ⋯ − 13 ⋯ 96 T^{2} + \cdots - 13\!\cdots\!96 T 2 + ⋯ − 1 3 ⋯ 9 6
17 17 1 7
T 2 + ⋯ − 53 ⋯ 04 T^{2} + \cdots - 53\!\cdots\!04 T 2 + ⋯ − 5 3 ⋯ 0 4
19 19 1 9
T 2 + ⋯ + 74 ⋯ 96 T^{2} + \cdots + 74\!\cdots\!96 T 2 + ⋯ + 7 4 ⋯ 9 6
23 23 2 3
T 2 + ⋯ − 12 ⋯ 00 T^{2} + \cdots - 12\!\cdots\!00 T 2 + ⋯ − 1 2 ⋯ 0 0
29 29 2 9
T 2 + ⋯ − 55 ⋯ 24 T^{2} + \cdots - 55\!\cdots\!24 T 2 + ⋯ − 5 5 ⋯ 2 4
31 31 3 1
T 2 + ⋯ + 25 ⋯ 24 T^{2} + \cdots + 25\!\cdots\!24 T 2 + ⋯ + 2 5 ⋯ 2 4
37 37 3 7
T 2 + ⋯ + 12 ⋯ 04 T^{2} + \cdots + 12\!\cdots\!04 T 2 + ⋯ + 1 2 ⋯ 0 4
41 41 4 1
T 2 + ⋯ − 68 ⋯ 44 T^{2} + \cdots - 68\!\cdots\!44 T 2 + ⋯ − 6 8 ⋯ 4 4
43 43 4 3
T 2 + ⋯ − 37 ⋯ 04 T^{2} + \cdots - 37\!\cdots\!04 T 2 + ⋯ − 3 7 ⋯ 0 4
47 47 4 7
T 2 + ⋯ − 73 ⋯ 00 T^{2} + \cdots - 73\!\cdots\!00 T 2 + ⋯ − 7 3 ⋯ 0 0
53 53 5 3
T 2 + ⋯ + 12 ⋯ 36 T^{2} + \cdots + 12\!\cdots\!36 T 2 + ⋯ + 1 2 ⋯ 3 6
59 59 5 9
T 2 + ⋯ + 42 ⋯ 00 T^{2} + \cdots + 42\!\cdots\!00 T 2 + ⋯ + 4 2 ⋯ 0 0
61 61 6 1
T 2 + ⋯ + 62 ⋯ 00 T^{2} + \cdots + 62\!\cdots\!00 T 2 + ⋯ + 6 2 ⋯ 0 0
67 67 6 7
T 2 + ⋯ + 85 ⋯ 56 T^{2} + \cdots + 85\!\cdots\!56 T 2 + ⋯ + 8 5 ⋯ 5 6
71 71 7 1
T 2 + ⋯ − 83 ⋯ 00 T^{2} + \cdots - 83\!\cdots\!00 T 2 + ⋯ − 8 3 ⋯ 0 0
73 73 7 3
T 2 + ⋯ − 16 ⋯ 56 T^{2} + \cdots - 16\!\cdots\!56 T 2 + ⋯ − 1 6 ⋯ 5 6
79 79 7 9
T 2 + ⋯ + 89 ⋯ 04 T^{2} + \cdots + 89\!\cdots\!04 T 2 + ⋯ + 8 9 ⋯ 0 4
83 83 8 3
T 2 + ⋯ − 29 ⋯ 36 T^{2} + \cdots - 29\!\cdots\!36 T 2 + ⋯ − 2 9 ⋯ 3 6
89 89 8 9
T 2 + ⋯ + 96 ⋯ 24 T^{2} + \cdots + 96\!\cdots\!24 T 2 + ⋯ + 9 6 ⋯ 2 4
97 97 9 7
T 2 + ⋯ − 27 ⋯ 16 T^{2} + \cdots - 27\!\cdots\!16 T 2 + ⋯ − 2 7 ⋯ 1 6
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