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gp:[N,k,chi] = [338,6,Mod(1,338)]
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("338.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(338, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
Newform invariants
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sage:traces = [2,-8,9]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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 \(\beta = \frac{1}{2}(1 + \sqrt{849})\).
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
|
| \(2\) |
\( +1 \) |
| \(13\) |
\( +1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(338))\):
\( T_{3}^{2} - 9T_{3} - 192 \)
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|
\( T_{5}^{2} + 73T_{5} - 578 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T + 4)^{2} \)
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|
| $3$ |
\( T^{2} - 9T - 192 \)
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|
| $5$ |
\( T^{2} + 73T - 578 \)
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|
| $7$ |
\( T^{2} + 155T - 11186 \)
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|
| $11$ |
\( T^{2} - 220T - 110156 \)
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|
| $13$ |
\( T^{2} \)
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|
| $17$ |
\( T^{2} + 189 T - 3523122 \)
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|
| $19$ |
\( T^{2} - 2496 T + 793404 \)
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|
| $23$ |
\( T^{2} + 3044 T - 159200 \)
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| $29$ |
\( T^{2} - 1900 T - 13890476 \)
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|
| $31$ |
\( T^{2} + 2798 T - 28369928 \)
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| $37$ |
\( T^{2} + 17805 T + 72604926 \)
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| $41$ |
\( T^{2} + 11634 T - 11466000 \)
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| $43$ |
\( T^{2} + 4069 T - 6040532 \)
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| $47$ |
\( T^{2} - 25489 T + 127607974 \)
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| $53$ |
\( T^{2} + 4614 T - 129839400 \)
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|
| $59$ |
\( T^{2} - 23420 T + 92989684 \)
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|
| $61$ |
\( T^{2} + \cdots + 2309711776 \)
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|
| $67$ |
\( T^{2} + \cdots + 1295720044 \)
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| $71$ |
\( T^{2} + \cdots - 1862988962 \)
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| $73$ |
\( T^{2} + \cdots - 1462893860 \)
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| $79$ |
\( T^{2} - 52024 T + 439936528 \)
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| $83$ |
\( T^{2} - 37758 T + 83472480 \)
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| $89$ |
\( T^{2} + \cdots + 1072134396 \)
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| $97$ |
\( T^{2} + \cdots - 4229773940 \)
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