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gp:[N,k,chi] = [450,8,Mod(1,450)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(450, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("450.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
Newform invariants
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sage:traces = [2,16,0,128,0,0,-408]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
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gp:f = lf[1] \\ Warning: the index may be different
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sage:f.q_expansion() # note that sage often uses an isomorphic number field
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gp:mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 10\sqrt{31}\).
We also show the integral \(q\)-expansion of the trace form.
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
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gp:mfembed(f)
| \( p \) |
Sign
|
| \(2\) |
\( -1 \) |
| \(3\) |
\( -1 \) |
| \(5\) |
\( -1 \) |
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 \(S_{8}^{\mathrm{new}}(\Gamma_0(450))\):
\( T_{7}^{2} + 408T_{7} - 110284 \)
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\( T_{11}^{2} + 8904T_{11} + 19026704 \)
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\( T_{17}^{2} - 43648T_{17} + 388792576 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T - 8)^{2} \)
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| $3$ |
\( T^{2} \)
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|
| $5$ |
\( T^{2} \)
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|
| $7$ |
\( T^{2} + 408T - 110284 \)
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|
| $11$ |
\( T^{2} + 8904 T + 19026704 \)
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|
| $13$ |
\( T^{2} - 3504 T - 15790896 \)
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|
| $17$ |
\( T^{2} - 43648 T + 388792576 \)
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|
| $19$ |
\( T^{2} + 31800 T - 954255600 \)
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|
| $23$ |
\( T^{2} - 71656 T - 950837516 \)
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|
| $29$ |
\( T^{2} + \cdots - 19030326300 \)
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|
| $31$ |
\( T^{2} + \cdots + 4879504384 \)
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| $37$ |
\( T^{2} + \cdots - 51511044784 \)
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| $41$ |
\( T^{2} + \cdots - 183681487516 \)
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| $43$ |
\( T^{2} + \cdots + 17685717524 \)
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| $47$ |
\( T^{2} + \cdots - 599256039404 \)
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| $53$ |
\( T^{2} + \cdots + 504522481104 \)
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| $59$ |
\( T^{2} + \cdots - 159475001200 \)
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| $61$ |
\( T^{2} + \cdots + 1716951910564 \)
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| $67$ |
\( T^{2} + \cdots - 2422233726604 \)
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| $71$ |
\( T^{2} + \cdots - 4934358737856 \)
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| $73$ |
\( T^{2} + \cdots + 296239095104 \)
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| $79$ |
\( T^{2} + \cdots - 3963839328000 \)
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| $83$ |
\( T^{2} + \cdots - 36935334540236 \)
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| $89$ |
\( T^{2} + \cdots + 66278240560100 \)
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| $97$ |
\( T^{2} + \cdots - 1192948931584 \)
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