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gp:[N,k,chi] = [98,16,Mod(67,98)]
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("98.67");
S:= CuspForms(chi, 16);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(98, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 16, names="a")
Newform invariants
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sage:traces = [2,128,1350]
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 a primitive root of unity \(\zeta_{6}\).
We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).
| \(n\) |
\(3\) |
| \(\chi(n)\) |
\(-\zeta_{6}\) |
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)
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 1350T_{3} + 1822500 \)
acting on \(S_{16}^{\mathrm{new}}(98, [\chi])\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} - 128T + 16384 \)
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|
| $3$ |
\( T^{2} - 1350 T + 1822500 \)
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|
| $5$ |
\( T^{2} + \cdots + 6570723600 \)
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|
| $7$ |
\( T^{2} \)
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|
| $11$ |
\( T^{2} + \cdots + 49\!\cdots\!56 \)
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|
| $13$ |
\( (T + 151469552)^{2} \)
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|
| $17$ |
\( T^{2} + \cdots + 62\!\cdots\!16 \)
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|
| $19$ |
\( T^{2} + \cdots + 41\!\cdots\!00 \)
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|
| $23$ |
\( T^{2} + \cdots + 44\!\cdots\!00 \)
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|
| $29$ |
\( (T - 7794825354)^{2} \)
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|
| $31$ |
\( T^{2} + \cdots + 90\!\cdots\!44 \)
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| $37$ |
\( T^{2} + \cdots + 75\!\cdots\!00 \)
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| $41$ |
\( (T + 1007666657262)^{2} \)
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| $43$ |
\( (T - 155007585272)^{2} \)
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| $47$ |
\( T^{2} + \cdots + 65\!\cdots\!16 \)
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| $53$ |
\( T^{2} + \cdots + 16\!\cdots\!76 \)
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| $59$ |
\( T^{2} + \cdots + 15\!\cdots\!04 \)
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| $61$ |
\( T^{2} + \cdots + 15\!\cdots\!36 \)
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| $67$ |
\( T^{2} + \cdots + 23\!\cdots\!76 \)
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| $71$ |
\( (T - 37693101366144)^{2} \)
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| $73$ |
\( T^{2} + \cdots + 19\!\cdots\!36 \)
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| $79$ |
\( T^{2} + \cdots + 61\!\cdots\!84 \)
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| $83$ |
\( (T + 2788789610034)^{2} \)
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| $89$ |
\( T^{2} + \cdots + 34\!\cdots\!00 \)
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| $97$ |
\( (T + 278027158065374)^{2} \)
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