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gp:[N,k,chi] = [783,2,Mod(82,783)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(783, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("783.82");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
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sage:traces = [6,-2,0,-2,-8,0,-8,-7,0,-9]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == 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 \(\zeta_{14}\).
We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/783\mathbb{Z}\right)^\times\).
| \(n\) |
\(379\) |
\(407\) |
| \(\chi(n)\) |
\(-\zeta_{14}\) |
\(1\) |
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
\( T_{2}^{6} + 2T_{2}^{5} + 4T_{2}^{4} + T_{2}^{3} + 2T_{2}^{2} - 3T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(783, [\chi])\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{6} + 2 T^{5} + \cdots + 1 \)
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|
| $3$ |
\( T^{6} \)
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|
| $5$ |
\( T^{6} + 8 T^{5} + \cdots + 169 \)
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| $7$ |
\( T^{6} + 8 T^{5} + \cdots + 169 \)
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|
| $11$ |
\( T^{6} + 5 T^{5} + \cdots + 169 \)
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| $13$ |
\( T^{6} + 5 T^{5} + \cdots + 1 \)
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|
| $17$ |
\( (T^{3} - 4 T^{2} - 4 T + 8)^{2} \)
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|
| $19$ |
\( T^{6} - 4 T^{5} + \cdots + 169 \)
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| $23$ |
\( T^{6} + T^{5} + T^{4} + \cdots + 1 \)
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| $29$ |
\( T^{6} - 27 T^{5} + \cdots + 24389 \)
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| $31$ |
\( T^{6} - 17 T^{5} + \cdots + 28561 \)
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| $37$ |
\( T^{6} + 13 T^{5} + \cdots + 1849 \)
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| $41$ |
\( (T^{3} + 23 T^{2} + \cdots + 433)^{2} \)
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| $43$ |
\( T^{6} + 11 T^{5} + \cdots + 78961 \)
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| $47$ |
\( T^{6} - 20 T^{5} + \cdots + 4096 \)
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| $53$ |
\( T^{6} - 6 T^{5} + \cdots + 53824 \)
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| $59$ |
\( (T^{3} + 19 T^{2} + \cdots + 197)^{2} \)
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| $61$ |
\( T^{6} + 3 T^{5} + \cdots + 1 \)
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| $67$ |
\( T^{6} + 2 T^{5} + \cdots + 841 \)
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| $71$ |
\( T^{6} + 63 T^{4} + \cdots + 35721 \)
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| $73$ |
\( T^{6} + 25 T^{5} + \cdots + 175561 \)
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| $79$ |
\( T^{6} + 30 T^{5} + \cdots + 1225449 \)
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| $83$ |
\( T^{6} + T^{5} + 15 T^{4} + \cdots + 1 \)
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| $89$ |
\( T^{6} + 29 T^{5} + \cdots + 2247001 \)
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| $97$ |
\( T^{6} - 19 T^{5} + \cdots + 7912969 \)
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