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gp:[N,k,chi] = [45,6,Mod(1,45)]
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("45.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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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, 6, names="a")
Newform invariants
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sage:traces = [2,5]
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 \(\beta = \frac{1}{2}(1 + \sqrt{145})\).
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
\( T_{2}^{2} - 5T_{2} - 30 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(45))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} - 5T - 30 \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( (T - 25)^{2} \)
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|
| $7$ |
\( T^{2} - 80T - 12900 \)
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|
| $11$ |
\( T^{2} - 800T + 145500 \)
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|
| $13$ |
\( T^{2} + 120T - 228400 \)
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|
| $17$ |
\( T^{2} - 1940 T + 903780 \)
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|
| $19$ |
\( T^{2} + 1512 T - 3140464 \)
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|
| $23$ |
\( T^{2} + 1320 T - 8772480 \)
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|
| $29$ |
\( T^{2} - 1300 T - 58969500 \)
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|
| $31$ |
\( T^{2} + 5824 T + 1461744 \)
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|
| $37$ |
\( T^{2} + 12560 T - 35729600 \)
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|
| $41$ |
\( T^{2} - 400 T - 67008000 \)
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|
| $43$ |
\( T^{2} - 25680 T + 105473600 \)
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|
| $47$ |
\( T^{2} - 18920 T + 88264320 \)
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|
| $53$ |
\( T^{2} - 49460 T + 598172580 \)
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|
| $59$ |
\( T^{2} - 63200 T + 963745500 \)
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|
| $61$ |
\( T^{2} + 49116 T - 469672636 \)
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|
| $67$ |
\( T^{2} + \cdots - 2380880400 \)
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|
| $71$ |
\( T^{2} - 65200 T + 983358000 \)
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|
| $73$ |
\( T^{2} + \cdots + 1607828900 \)
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|
| $79$ |
\( T^{2} + 46288 T + 275282736 \)
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|
| $83$ |
\( T^{2} + \cdots - 3349469520 \)
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|
| $89$ |
\( T^{2} + 87000 T - 665550000 \)
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|
| $97$ |
\( T^{2} - 10180 T - 274763900 \)
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