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gp:[N,k,chi] = [240,8,Mod(1,240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("240.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(240, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 8, names="a")
Newform invariants
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sage:traces = [2,0,54,0,250,0,704]
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 \(\beta = 48\sqrt{106}\).
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 kernel of the linear operator
\( T_{7}^{2} - 704T_{7} - 120320 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(240))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} \)
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|
| $3$ |
\( (T - 27)^{2} \)
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|
| $5$ |
\( (T - 125)^{2} \)
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|
| $7$ |
\( T^{2} - 704T - 120320 \)
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|
| $11$ |
\( T^{2} - 2072 T - 46794608 \)
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|
| $13$ |
\( T^{2} - 4844 T - 35407772 \)
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|
| $17$ |
\( T^{2} - 15620 T - 46706684 \)
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|
| $19$ |
\( T^{2} - 9064 T - 948786032 \)
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|
| $23$ |
\( T^{2} + \cdots - 1657218560 \)
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|
| $29$ |
\( T^{2} + \cdots - 11018435996 \)
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|
| $31$ |
\( T^{2} + \cdots - 35635873280 \)
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|
| $37$ |
\( T^{2} + \cdots + 49103499780 \)
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|
| $41$ |
\( T^{2} + \cdots - 36337609692 \)
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|
| $43$ |
\( T^{2} + \cdots + 369109388944 \)
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|
| $47$ |
\( T^{2} + \cdots + 769971772480 \)
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|
| $53$ |
\( T^{2} + \cdots - 438173549820 \)
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|
| $59$ |
\( T^{2} + \cdots - 545520451952 \)
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|
| $61$ |
\( T^{2} + \cdots - 4595089220252 \)
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|
| $67$ |
\( T^{2} + \cdots + 47470004752 \)
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|
| $71$ |
\( T^{2} + \cdots - 5585408644032 \)
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|
| $73$ |
\( T^{2} + \cdots - 3209657409756 \)
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|
| $79$ |
\( T^{2} + \cdots + 12943601293824 \)
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|
| $83$ |
\( T^{2} + \cdots + 12823450770448 \)
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|
| $89$ |
\( T^{2} + \cdots - 24972500131740 \)
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|
| $97$ |
\( T^{2} + \cdots - 9015739434236 \)
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