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gp:[N,k,chi] = [45,8,Mod(1,45)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45, 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("45.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
Newform invariants
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sage:traces = [2,-20]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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 \(\beta = 2\sqrt{19}\).
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
|
| \(3\) |
\( -1 \) |
| \(5\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 20T_{2} + 24 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(45))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 20T + 24 \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( (T - 125)^{2} \)
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|
| $7$ |
\( T^{2} + 100T - 235836 \)
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|
| $11$ |
\( T^{2} + 4544 T - 6998016 \)
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|
| $13$ |
\( T^{2} - 3540 T - 24961564 \)
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|
| $17$ |
\( T^{2} - 27340 T + 80327844 \)
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|
| $19$ |
\( T^{2} - 38760 T + 367802000 \)
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|
| $23$ |
\( T^{2} + \cdots + 3840033636 \)
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|
| $29$ |
\( T^{2} + \cdots - 27652933500 \)
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|
| $31$ |
\( T^{2} + \cdots + 22939401744 \)
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|
| $37$ |
\( T^{2} + \cdots - 45775154396 \)
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|
| $41$ |
\( T^{2} + \cdots - 227722158876 \)
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|
| $43$ |
\( T^{2} + \cdots - 96985991164 \)
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|
| $47$ |
\( T^{2} + \cdots - 154530884316 \)
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|
| $53$ |
\( T^{2} + \cdots + 1213130224836 \)
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|
| $59$ |
\( T^{2} + \cdots - 3614968086000 \)
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|
| $61$ |
\( T^{2} + \cdots - 672038095516 \)
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|
| $67$ |
\( T^{2} + \cdots + 4620664454244 \)
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|
| $71$ |
\( T^{2} + \cdots - 275746164336 \)
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|
| $73$ |
\( T^{2} + \cdots + 1330152816836 \)
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|
| $79$ |
\( T^{2} + \cdots - 12272229720000 \)
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|
| $83$ |
\( T^{2} + \cdots + 5699002341636 \)
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|
| $89$ |
\( T^{2} + \cdots + 1403196358500 \)
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|
| $97$ |
\( T^{2} + \cdots - 18666217374716 \)
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