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gp:[N,k,chi] = [99,8,Mod(1,99)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(99, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("99.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
Newform invariants
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sage:traces = [2,-1]
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 \(\beta = \frac{1}{2}(1 + 3\sqrt{97})\).
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
\( T_{2}^{2} + T_{2} - 218 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(99))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + T - 218 \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( T^{2} - 194T - 33368 \)
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|
| $7$ |
\( T^{2} + 418T - 341312 \)
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|
| $11$ |
\( (T + 1331)^{2} \)
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|
| $13$ |
\( T^{2} + 13246 T + 43548976 \)
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|
| $17$ |
\( T^{2} - 10256 T + 19225084 \)
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|
| $19$ |
\( T^{2} + \cdots - 1177576704 \)
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|
| $23$ |
\( T^{2} - 13666 T - 870505736 \)
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|
| $29$ |
\( T^{2} + \cdots - 4487554116 \)
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|
| $31$ |
\( T^{2} + 48040 T + 569571328 \)
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|
| $37$ |
\( T^{2} + \cdots - 18695152476 \)
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|
| $41$ |
\( T^{2} + \cdots - 49849727804 \)
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|
| $43$ |
\( T^{2} + \cdots + 714621373632 \)
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|
| $47$ |
\( T^{2} + \cdots - 47516184584 \)
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|
| $53$ |
\( T^{2} + \cdots + 1006243830216 \)
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|
| $59$ |
\( T^{2} + \cdots - 1058263475744 \)
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|
| $61$ |
\( T^{2} + \cdots - 6224547468744 \)
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|
| $67$ |
\( T^{2} + \cdots + 2655573007408 \)
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|
| $71$ |
\( T^{2} + \cdots + 7547380719912 \)
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|
| $73$ |
\( T^{2} + \cdots + 4341768952708 \)
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|
| $79$ |
\( T^{2} + \cdots + 177407563168 \)
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|
| $83$ |
\( T^{2} + \cdots + 5354532740304 \)
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|
| $89$ |
\( T^{2} + \cdots + 10431675116388 \)
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|
| $97$ |
\( T^{2} + \cdots - 26135282253956 \)
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