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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,-8,14,32,42,-56,0,-128,652]
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 \(\beta = 2\sqrt{130}\).
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
\( T_{3}^{2} - 14T_{3} - 471 \)
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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|
| $3$ |
\( T^{2} - 14T - 471 \)
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|
| $5$ |
\( (T - 21)^{2} \)
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|
| $7$ |
\( T^{2} \)
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|
| $11$ |
\( T^{2} + 294T - 207711 \)
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|
| $13$ |
\( T^{2} - 140T - 13820 \)
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|
| $17$ |
\( T^{2} + 1302 T + 255321 \)
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|
| $19$ |
\( T^{2} - 1442 T - 832679 \)
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|
| $23$ |
\( T^{2} - 2646 T - 3982671 \)
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|
| $29$ |
\( T^{2} - 1668 T - 221724 \)
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|
| $31$ |
\( T^{2} - 14798 T + 51820201 \)
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|
| $37$ |
\( T^{2} - 5182 T - 67586399 \)
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|
| $41$ |
\( T^{2} - 5124 T - 207761436 \)
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|
| $43$ |
\( T^{2} + 4520 T - 53598320 \)
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|
| $47$ |
\( T^{2} - 14994 T - 31633911 \)
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|
| $53$ |
\( T^{2} - 24006 T + 99125289 \)
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|
| $59$ |
\( T^{2} + 38850 T - 104901255 \)
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|
| $61$ |
\( T^{2} - 23618 T + 45084961 \)
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|
| $67$ |
\( T^{2} + \cdots - 1560411719 \)
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|
| $71$ |
\( T^{2} + \cdots + 1817230464 \)
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|
| $73$ |
\( T^{2} - 47138 T + 285929761 \)
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|
| $79$ |
\( T^{2} + 40970 T - 130962095 \)
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|
| $83$ |
\( T^{2} + \cdots + 1030366224 \)
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|
| $89$ |
\( T^{2} + \cdots + 2383102881 \)
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|
| $97$ |
\( T^{2} + \cdots - 8307517724 \)
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