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gp:[N,k,chi] = [44,6,Mod(1,44)]
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("44.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44, 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]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == 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 \(\beta = 4\sqrt{31}\).
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
|
| \(2\) |
\( -1 \) |
| \(11\) |
\( +1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 6T_{3} - 487 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(44))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} \)
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|
| $3$ |
\( T^{2} + 6T - 487 \)
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|
| $5$ |
\( T^{2} + 22T - 1863 \)
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|
| $7$ |
\( T^{2} - 268T + 13492 \)
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|
| $11$ |
\( (T + 121)^{2} \)
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|
| $13$ |
\( T^{2} - 1232 T + 367056 \)
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|
| $17$ |
\( T^{2} + 124 T - 1091820 \)
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|
| $19$ |
\( T^{2} - 1944 T + 468128 \)
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|
| $23$ |
\( T^{2} - 3346 T + 1072353 \)
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|
| $29$ |
\( T^{2} + 6576 T + 7793280 \)
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|
| $31$ |
\( T^{2} - 2498 T - 66709935 \)
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|
| $37$ |
\( T^{2} + 14674 T + 49448913 \)
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|
| $41$ |
\( T^{2} + 6504 T - 102278880 \)
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|
| $43$ |
\( T^{2} - 11628 T - 171906460 \)
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|
| $47$ |
\( T^{2} - 36816 T + 203702400 \)
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|
| $53$ |
\( T^{2} + 3292 T - 653250684 \)
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|
| $59$ |
\( T^{2} - 12126 T - 132983631 \)
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|
| $61$ |
\( T^{2} + \cdots + 1180267472 \)
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|
| $67$ |
\( T^{2} + \cdots - 1274510455 \)
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|
| $71$ |
\( T^{2} + 46122 T + 83226825 \)
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|
| $73$ |
\( T^{2} - 8240 T - 299672496 \)
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|
| $79$ |
\( T^{2} + 14780 T - 305436284 \)
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|
| $83$ |
\( T^{2} + \cdots + 1070918340 \)
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|
| $89$ |
\( T^{2} + 33698 T + 74328801 \)
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|
| $97$ |
\( T^{2} + \cdots - 1061806063 \)
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