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gp:[N,k,chi] = [882,2,Mod(127,882)]
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("882.127");
S:= CuspForms(chi, 2);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(882, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 6]))
N = Newforms(chi, 2, names="a")
Newform invariants
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sage:traces = [6,1,0,-1,-4]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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 \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).
| \(n\) |
\(199\) |
\(785\) |
| \(\chi(n)\) |
\(-\zeta_{14}\) |
\(1\) |
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
\( T_{5}^{6} + 4T_{5}^{5} + 16T_{5}^{4} + 64T_{5}^{3} + 144T_{5}^{2} + 128T_{5} + 64 \)
acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{6} - T^{5} + T^{4} + \cdots + 1 \)
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|
| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} + 4 T^{5} + \cdots + 64 \)
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| $7$ |
\( T^{6} + 7 T^{5} + \cdots + 343 \)
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| $11$ |
\( T^{6} + 10 T^{5} + \cdots + 64 \)
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| $13$ |
\( T^{6} - 12 T^{5} + \cdots + 1681 \)
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| $17$ |
\( T^{6} - 6 T^{5} + \cdots + 64 \)
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| $19$ |
\( (T^{3} - 13 T^{2} + \cdots - 71)^{2} \)
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| $23$ |
\( T^{6} - 22 T^{5} + \cdots + 10816 \)
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| $29$ |
\( T^{6} - 2 T^{5} + \cdots + 10816 \)
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| $31$ |
\( (T^{3} + 7 T^{2} + \cdots - 301)^{2} \)
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| $37$ |
\( T^{6} + 25 T^{5} + \cdots + 19321 \)
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| $41$ |
\( T^{6} - 280 T^{3} + \cdots + 3136 \)
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| $43$ |
\( T^{6} - 18 T^{5} + \cdots + 9409 \)
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| $47$ |
\( T^{6} + 30 T^{5} + \cdots + 322624 \)
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| $53$ |
\( T^{6} - 32 T^{5} + \cdots + 118336 \)
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| $59$ |
\( T^{6} - 32 T^{5} + \cdots + 692224 \)
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| $61$ |
\( T^{6} + 17 T^{5} + \cdots + 121801 \)
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| $67$ |
\( (T^{3} - 13 T^{2} + \cdots + 13)^{2} \)
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| $71$ |
\( T^{6} + 28 T^{5} + \cdots + 3136 \)
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| $73$ |
\( T^{6} + 2 T^{5} + \cdots + 5041 \)
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| $79$ |
\( (T^{3} + 13 T^{2} + \cdots - 167)^{2} \)
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| $83$ |
\( T^{6} - 16 T^{5} + \cdots + 4096 \)
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| $89$ |
\( T^{6} - 22 T^{5} + \cdots + 3655744 \)
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| $97$ |
\( (T^{3} + 11 T^{2} + \cdots + 13)^{2} \)
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