![Copy content]()
gp:[N,k,chi] = [448,6,Mod(1,448)]
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("448.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(448, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
Newform invariants
![Copy content]()
sage:traces = [2,0,-14,0,-42]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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 \(\beta = \sqrt{193}\).
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
|
| \(2\) |
\( -1 \) |
| \(7\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(448))\):
\( T_{3}^{2} + 14T_{3} - 144 \)
![Copy content]()
![Toggle raw display]()
|
\( T_{5}^{2} + 42T_{5} - 4384 \)
![Copy content]()
![Toggle raw display]()
|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} \)
![Copy content]()
![Toggle raw display]()
|
| $3$ |
\( T^{2} + 14T - 144 \)
![Copy content]()
![Toggle raw display]()
|
| $5$ |
\( T^{2} + 42T - 4384 \)
![Copy content]()
![Toggle raw display]()
|
| $7$ |
\( (T - 49)^{2} \)
![Copy content]()
![Toggle raw display]()
|
| $11$ |
\( T^{2} + 716T + 90336 \)
![Copy content]()
![Toggle raw display]()
|
| $13$ |
\( T^{2} - 714T + 71672 \)
![Copy content]()
![Toggle raw display]()
|
| $17$ |
\( T^{2} + 1344 T - 111204 \)
![Copy content]()
![Toggle raw display]()
|
| $19$ |
\( T^{2} + 1946 T - 722528 \)
![Copy content]()
![Toggle raw display]()
|
| $23$ |
\( T^{2} - 1792 T - 1618176 \)
![Copy content]()
![Toggle raw display]()
|
| $29$ |
\( T^{2} - 1200 T - 57176388 \)
![Copy content]()
![Toggle raw display]()
|
| $31$ |
\( T^{2} - 6804 T - 26817184 \)
![Copy content]()
![Toggle raw display]()
|
| $37$ |
\( T^{2} + 14640 T + 49005212 \)
![Copy content]()
![Toggle raw display]()
|
| $41$ |
\( T^{2} - 7896 T - 22118548 \)
![Copy content]()
![Toggle raw display]()
|
| $43$ |
\( T^{2} - 524 T - 51717888 \)
![Copy content]()
![Toggle raw display]()
|
| $47$ |
\( T^{2} - 18396 T - 22804384 \)
![Copy content]()
![Toggle raw display]()
|
| $53$ |
\( T^{2} + 45132 T + 313124004 \)
![Copy content]()
![Toggle raw display]()
|
| $59$ |
\( T^{2} - 22582 T + 101774256 \)
![Copy content]()
![Toggle raw display]()
|
| $61$ |
\( T^{2} - 52822 T + 456736944 \)
![Copy content]()
![Toggle raw display]()
|
| $67$ |
\( T^{2} - 9848 T + 14561808 \)
![Copy content]()
![Toggle raw display]()
|
| $71$ |
\( T^{2} + \cdots - 3181158400 \)
![Copy content]()
![Toggle raw display]()
|
| $73$ |
\( T^{2} + \cdots + 3444396788 \)
![Copy content]()
![Toggle raw display]()
|
| $79$ |
\( T^{2} + 31704 T - 112014208 \)
![Copy content]()
![Toggle raw display]()
|
| $83$ |
\( T^{2} + \cdots - 3038476256 \)
![Copy content]()
![Toggle raw display]()
|
| $89$ |
\( T^{2} + \cdots + 10645600644 \)
![Copy content]()
![Toggle raw display]()
|
| $97$ |
\( T^{2} + \cdots - 14736460932 \)
![Copy content]()
![Toggle raw display]()
|
|
show more
|
|
show less
|
