![Copy content]()
gp:[N,k,chi] = [197,2,Mod(6,197)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
![Copy content]()
magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("197.6");
S:= CuspForms(chi, 2);
N := Newforms(S);
![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(197, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([13]))
N = Newforms(chi, 2, names="a")
Newform invariants
![Copy content]()
sage:traces = [6]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
![Copy content]()
gp:f = lf[1] \\ Warning: the index may be different
![Copy content]()
sage:f.q_expansion() # note that sage often uses an isomorphic number field
![Copy content]()
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 \(\left(\mathbb{Z}/197\mathbb{Z}\right)^\times\).
| \(n\) |
\(2\) |
| \(\chi(n)\) |
\(-\zeta_{14}^{2}\) |
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.
![Copy content]()
gp:mfembed(f)
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 7T_{2}^{3} - 7T_{2} + 7 \)
acting on \(S_{2}^{\mathrm{new}}(197, [\chi])\).
![Copy content]()
![Toggle raw display]()
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{6} + 7 T^{3} + \cdots + 7 \)
![Copy content]()
![Toggle raw display]()
|
| $3$ |
\( T^{6} + 7 T^{5} + \cdots + 7 \)
![Copy content]()
![Toggle raw display]()
|
| $5$ |
\( T^{6} - 7 T^{5} + \cdots + 1183 \)
![Copy content]()
![Toggle raw display]()
|
| $7$ |
\( T^{6} + T^{5} + 15 T^{4} + \cdots + 1 \)
![Copy content]()
![Toggle raw display]()
|
| $11$ |
\( T^{6} + 7 T^{5} + \cdots + 1183 \)
![Copy content]()
![Toggle raw display]()
|
| $13$ |
\( T^{6} + 7 T^{5} + \cdots + 7 \)
![Copy content]()
![Toggle raw display]()
|
| $17$ |
\( T^{6} + 7 T^{5} + \cdots + 1183 \)
![Copy content]()
![Toggle raw display]()
|
| $19$ |
\( (T^{3} + 4 T^{2} - 32 T - 64)^{2} \)
![Copy content]()
![Toggle raw display]()
|
| $23$ |
\( T^{6} - 25 T^{5} + \cdots + 15625 \)
![Copy content]()
![Toggle raw display]()
|
| $29$ |
\( T^{6} - 13 T^{5} + \cdots + 841 \)
![Copy content]()
![Toggle raw display]()
|
| $31$ |
\( T^{6} + 7 T^{5} + \cdots + 12943 \)
![Copy content]()
![Toggle raw display]()
|
| $37$ |
\( T^{6} - 13 T^{5} + \cdots + 1 \)
![Copy content]()
![Toggle raw display]()
|
| $41$ |
\( T^{6} + 19 T^{5} + \cdots + 6889 \)
![Copy content]()
![Toggle raw display]()
|
| $43$ |
\( T^{6} + 29 T^{5} + \cdots + 142129 \)
![Copy content]()
![Toggle raw display]()
|
| $47$ |
\( T^{6} + 29 T^{5} + \cdots + 12769 \)
![Copy content]()
![Toggle raw display]()
|
| $53$ |
\( T^{6} - 19 T^{5} + \cdots + 12769 \)
![Copy content]()
![Toggle raw display]()
|
| $59$ |
\( T^{6} + 11 T^{5} + \cdots + 841 \)
![Copy content]()
![Toggle raw display]()
|
| $61$ |
\( T^{6} - 21 T^{5} + \cdots + 8281 \)
![Copy content]()
![Toggle raw display]()
|
| $67$ |
\( T^{6} + 35 T^{5} + \cdots + 112903 \)
![Copy content]()
![Toggle raw display]()
|
| $71$ |
\( T^{6} + 7 T^{5} + \cdots + 7 \)
![Copy content]()
![Toggle raw display]()
|
| $73$ |
\( T^{6} + 35 T^{5} + \cdots + 109375 \)
![Copy content]()
![Toggle raw display]()
|
| $79$ |
\( T^{6} - 35 T^{5} + \cdots + 1183 \)
![Copy content]()
![Toggle raw display]()
|
| $83$ |
\( (T^{3} - 6 T^{2} + \cdots + 216)^{2} \)
![Copy content]()
![Toggle raw display]()
|
| $89$ |
\( T^{6} + 21 T^{5} + \cdots + 48223 \)
![Copy content]()
![Toggle raw display]()
|
| $97$ |
\( T^{6} + 37 T^{5} + \cdots + 253009 \)
![Copy content]()
![Toggle raw display]()
|
|
show more
|
|
show less
|
