![Copy content]()
gp:[N,k,chi] = [847,6,Mod(1,847)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
![Copy content]()
magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("847.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
Newform invariants
![Copy content]()
sage:traces = [2,-9]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\).
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.
![Copy content]()
gp:mfembed(f)
| \( p \) |
Sign
|
| \(7\) |
\( +1 \) |
| \(11\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 9T_{2} + 6 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(847))\).
![Copy content]()
![Toggle raw display]()
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 9T + 6 \)
![Copy content]()
![Toggle raw display]()
|
| $3$ |
\( T^{2} + 6T - 504 \)
![Copy content]()
![Toggle raw display]()
|
| $5$ |
\( T^{2} + 18T - 1344 \)
![Copy content]()
![Toggle raw display]()
|
| $7$ |
\( (T + 49)^{2} \)
![Copy content]()
![Toggle raw display]()
|
| $11$ |
\( T^{2} \)
![Copy content]()
![Toggle raw display]()
|
| $13$ |
\( T^{2} - 350T - 195608 \)
![Copy content]()
![Toggle raw display]()
|
| $17$ |
\( T^{2} + 1800 T + 727692 \)
![Copy content]()
![Toggle raw display]()
|
| $19$ |
\( T^{2} - 3266 T + 2662072 \)
![Copy content]()
![Toggle raw display]()
|
| $23$ |
\( T^{2} - 2088 T - 3507456 \)
![Copy content]()
![Toggle raw display]()
|
| $29$ |
\( T^{2} + 6696 T + 10304172 \)
![Copy content]()
![Toggle raw display]()
|
| $31$ |
\( T^{2} + 20T - 4155200 \)
![Copy content]()
![Toggle raw display]()
|
| $37$ |
\( T^{2} - 6232 T + 5554156 \)
![Copy content]()
![Toggle raw display]()
|
| $41$ |
\( T^{2} - 6048 T - 7848036 \)
![Copy content]()
![Toggle raw display]()
|
| $43$ |
\( T^{2} - 3020 T - 324400352 \)
![Copy content]()
![Toggle raw display]()
|
| $47$ |
\( T^{2} - 11700 T - 165954432 \)
![Copy content]()
![Toggle raw display]()
|
| $53$ |
\( T^{2} - 9468 T + 21794244 \)
![Copy content]()
![Toggle raw display]()
|
| $59$ |
\( T^{2} + 43938 T + 422751336 \)
![Copy content]()
![Toggle raw display]()
|
| $61$ |
\( T^{2} - 64754 T + 719128816 \)
![Copy content]()
![Toggle raw display]()
|
| $67$ |
\( T^{2} - 24784 T + 99708976 \)
![Copy content]()
![Toggle raw display]()
|
| $71$ |
\( T^{2} + \cdots + 2121099264 \)
![Copy content]()
![Toggle raw display]()
|
| $73$ |
\( T^{2} + 17452 T - 317520812 \)
![Copy content]()
![Toggle raw display]()
|
| $79$ |
\( T^{2} + \cdots - 2508546944 \)
![Copy content]()
![Toggle raw display]()
|
| $83$ |
\( T^{2} + 117558 T - 79919784 \)
![Copy content]()
![Toggle raw display]()
|
| $89$ |
\( T^{2} + \cdots - 5252421468 \)
![Copy content]()
![Toggle raw display]()
|
| $97$ |
\( T^{2} + \cdots - 1000631156 \)
![Copy content]()
![Toggle raw display]()
|
|
show more
|
|
show less
|
