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gp:[N,k,chi] = [98,6,Mod(1,98)]
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("98.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(98, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
Newform invariants
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sage:traces = [2,8,14,32,70,56,0,128,244]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == 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 \(\beta = 2\sqrt{79}\).
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
|
| \(2\) |
\( -1 \) |
| \(7\) |
\( +1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 14T_{3} - 267 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(98))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( (T - 4)^{2} \)
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|
| $3$ |
\( T^{2} - 14T - 267 \)
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|
| $5$ |
\( T^{2} - 70T - 3831 \)
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|
| $7$ |
\( T^{2} \)
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|
| $11$ |
\( T^{2} - 62T - 14523 \)
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|
| $13$ |
\( T^{2} - 1820 T + 766164 \)
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|
| $17$ |
\( T^{2} - 1694 T + 564465 \)
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|
| $19$ |
\( T^{2} - 826T - 310067 \)
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|
| $23$ |
\( T^{2} + 2734 T - 2606187 \)
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|
| $29$ |
\( T^{2} + 2852 T - 15866028 \)
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|
| $31$ |
\( T^{2} + 2674 T + 831669 \)
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|
| $37$ |
\( T^{2} + 9146 T + 15895513 \)
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|
| $41$ |
\( T^{2} - 6132 T + 8842932 \)
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|
| $43$ |
\( T^{2} + 16040 T - 78380144 \)
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|
| $47$ |
\( T^{2} + 25326 T + 157959765 \)
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|
| $53$ |
\( T^{2} - 14958 T - 292454559 \)
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|
| $59$ |
\( T^{2} + 1106 T - 29476875 \)
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|
| $61$ |
\( T^{2} + 28042 T + 195919785 \)
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|
| $67$ |
\( T^{2} + \cdots + 2564337365 \)
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|
| $71$ |
\( T^{2} + 11056 T - 177793920 \)
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|
| $73$ |
\( T^{2} - 35070 T - 477396991 \)
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|
| $79$ |
\( T^{2} - 101762 T - 788013915 \)
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|
| $83$ |
\( T^{2} + \cdots - 5608638000 \)
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|
| $89$ |
\( T^{2} + \cdots - 1917778095 \)
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|
| $97$ |
\( T^{2} + 8316 T - 349929580 \)
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