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gp:[N,k,chi] = [33,6,Mod(1,33)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("33.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\).
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
|
| \(3\) |
\( +1 \) |
| \(11\) |
\( +1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 5T_{2} - 38 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(33))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 5T - 38 \)
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|
| $3$ |
\( (T + 9)^{2} \)
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|
| $5$ |
\( T^{2} - 58T - 3584 \)
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|
| $7$ |
\( T^{2} + 286T + 16024 \)
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|
| $11$ |
\( (T + 121)^{2} \)
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|
| $13$ |
\( T^{2} + 166T - 57008 \)
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|
| $17$ |
\( T^{2} + 800T + 700 \)
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|
| $19$ |
\( T^{2} + 1476 T + 538272 \)
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|
| $23$ |
\( T^{2} + 3370 T - 3088328 \)
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|
| $29$ |
\( T^{2} - 6600 T + 3378828 \)
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|
| $31$ |
\( T^{2} + 7528 T + 8931328 \)
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|
| $37$ |
\( T^{2} + 29916 T + 222923316 \)
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|
| $41$ |
\( T^{2} + 5780 T - 14080172 \)
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|
| $43$ |
\( T^{2} + 16656 T - 29676624 \)
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|
| $47$ |
\( T^{2} - 7850 T - 266939288 \)
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|
| $53$ |
\( T^{2} - 14178 T - 165085872 \)
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|
| $59$ |
\( T^{2} - 17300 T - 21579488 \)
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|
| $61$ |
\( T^{2} + 2946 T - 238059096 \)
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|
| $67$ |
\( T^{2} - 31336 T - 207633776 \)
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|
| $71$ |
\( T^{2} + 33810 T - 138914952 \)
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|
| $73$ |
\( T^{2} + \cdots - 1593591164 \)
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|
| $79$ |
\( T^{2} + \cdots - 3977569112 \)
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|
| $83$ |
\( T^{2} + 58296 T + 552313872 \)
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|
| $89$ |
\( T^{2} + \cdots - 7896843132 \)
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|
| $97$ |
\( T^{2} + \cdots - 20063108900 \)
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