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gp:[N,k,chi] = [49,16,Mod(18,49)]
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("49.18");
S:= CuspForms(chi, 16);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, 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,-216,-3348]
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 \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).
| \(n\) |
\(3\) |
| \(\chi(n)\) |
\(-\zeta_{6}\) |
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 intersection of the kernels of the following linear operators acting on \(S_{16}^{\mathrm{new}}(49, [\chi])\):
\( T_{2}^{2} + 216T_{2} + 46656 \)
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\( T_{3}^{2} + 3348T_{3} + 11209104 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 216T + 46656 \)
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|
| $3$ |
\( T^{2} + 3348 T + 11209104 \)
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| $5$ |
\( T^{2} + \cdots + 2715452100 \)
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|
| $7$ |
\( T^{2} \)
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|
| $11$ |
\( T^{2} + \cdots + 423818475269904 \)
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| $13$ |
\( (T - 190073338)^{2} \)
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| $17$ |
\( T^{2} + \cdots + 27\!\cdots\!96 \)
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| $19$ |
\( T^{2} + \cdots + 24\!\cdots\!00 \)
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| $23$ |
\( T^{2} + \cdots + 89\!\cdots\!84 \)
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| $29$ |
\( (T + 36902568330)^{2} \)
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| $31$ |
\( T^{2} + \cdots + 51\!\cdots\!04 \)
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| $37$ |
\( T^{2} + \cdots + 10\!\cdots\!16 \)
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| $41$ |
\( (T + 1641974018202)^{2} \)
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| $43$ |
\( (T + 492403109308)^{2} \)
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| $47$ |
\( T^{2} + \cdots + 11\!\cdots\!76 \)
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| $53$ |
\( T^{2} + \cdots + 46\!\cdots\!04 \)
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| $59$ |
\( T^{2} + \cdots + 97\!\cdots\!00 \)
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| $61$ |
\( T^{2} + \cdots + 24\!\cdots\!04 \)
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| $67$ |
\( T^{2} + \cdots + 83\!\cdots\!96 \)
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| $71$ |
\( (T - 125050114914552)^{2} \)
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| $73$ |
\( T^{2} + \cdots + 67\!\cdots\!84 \)
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| $79$ |
\( T^{2} + \cdots + 64\!\cdots\!00 \)
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| $83$ |
\( (T - 281736730890468)^{2} \)
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| $89$ |
\( T^{2} + \cdots + 51\!\cdots\!00 \)
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| $97$ |
\( (T + 612786136081826)^{2} \)
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