Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [400,3,Mod(17,400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(400, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([0, 0, 13]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("400.17");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 400.bg (of order \(20\), degree \(8\), not minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(10.8992105744\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{20})\) |
Twist minimal: | no (minimal twist has level 200) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | 0 | −0.811497 | + | 5.12359i | 0 | 1.63338 | − | 4.72568i | 0 | 4.59566 | + | 4.59566i | 0 | −17.0331 | − | 5.53440i | 0 | ||||||||||
17.2 | 0 | −0.583470 | + | 3.68389i | 0 | 3.77915 | + | 3.27383i | 0 | −8.52562 | − | 8.52562i | 0 | −4.67107 | − | 1.51772i | 0 | ||||||||||
17.3 | 0 | −0.526150 | + | 3.32198i | 0 | −1.34260 | + | 4.81637i | 0 | 6.50894 | + | 6.50894i | 0 | −2.19922 | − | 0.714570i | 0 | ||||||||||
17.4 | 0 | −0.0258720 | + | 0.163349i | 0 | −1.84061 | − | 4.64889i | 0 | −3.03617 | − | 3.03617i | 0 | 8.53350 | + | 2.77270i | 0 | ||||||||||
17.5 | 0 | −0.00512856 | + | 0.0323805i | 0 | −4.99090 | + | 0.301564i | 0 | 0.994182 | + | 0.994182i | 0 | 8.55849 | + | 2.78082i | 0 | ||||||||||
17.6 | 0 | 0.298963 | − | 1.88758i | 0 | 4.96497 | + | 0.590862i | 0 | 0.603173 | + | 0.603173i | 0 | 5.08593 | + | 1.65252i | 0 | ||||||||||
17.7 | 0 | 0.754209 | − | 4.76189i | 0 | −3.14938 | + | 3.88348i | 0 | −8.90117 | − | 8.90117i | 0 | −13.5472 | − | 4.40177i | 0 | ||||||||||
17.8 | 0 | 0.898945 | − | 5.67572i | 0 | −0.177358 | − | 4.99685i | 0 | 8.20346 | + | 8.20346i | 0 | −22.8462 | − | 7.42316i | 0 | ||||||||||
33.1 | 0 | −5.60705 | − | 0.888069i | 0 | −0.904406 | − | 4.91752i | 0 | 3.60781 | − | 3.60781i | 0 | 22.0908 | + | 7.17774i | 0 | ||||||||||
33.2 | 0 | −3.88880 | − | 0.615926i | 0 | 2.05707 | + | 4.55724i | 0 | −5.11305 | + | 5.11305i | 0 | 6.18393 | + | 2.00928i | 0 | ||||||||||
33.3 | 0 | −1.33419 | − | 0.211315i | 0 | −4.90670 | + | 0.961386i | 0 | 5.78356 | − | 5.78356i | 0 | −6.82409 | − | 2.21728i | 0 | ||||||||||
33.4 | 0 | −0.535050 | − | 0.0847436i | 0 | 2.05203 | − | 4.55951i | 0 | −4.47022 | + | 4.47022i | 0 | −8.28041 | − | 2.69047i | 0 | ||||||||||
33.5 | 0 | 0.0475930 | + | 0.00753798i | 0 | 4.31624 | + | 2.52389i | 0 | 8.24576 | − | 8.24576i | 0 | −8.55730 | − | 2.78044i | 0 | ||||||||||
33.6 | 0 | 2.61980 | + | 0.414935i | 0 | −2.28576 | + | 4.44694i | 0 | −4.29378 | + | 4.29378i | 0 | −1.86835 | − | 0.607064i | 0 | ||||||||||
33.7 | 0 | 3.58254 | + | 0.567419i | 0 | −3.75338 | − | 3.30335i | 0 | −1.35870 | + | 1.35870i | 0 | 3.95314 | + | 1.28445i | 0 | ||||||||||
33.8 | 0 | 5.11517 | + | 0.810163i | 0 | 4.93022 | − | 0.832424i | 0 | 0.392224 | − | 0.392224i | 0 | 16.9490 | + | 5.50708i | 0 | ||||||||||
97.1 | 0 | −5.60705 | + | 0.888069i | 0 | −0.904406 | + | 4.91752i | 0 | 3.60781 | + | 3.60781i | 0 | 22.0908 | − | 7.17774i | 0 | ||||||||||
97.2 | 0 | −3.88880 | + | 0.615926i | 0 | 2.05707 | − | 4.55724i | 0 | −5.11305 | − | 5.11305i | 0 | 6.18393 | − | 2.00928i | 0 | ||||||||||
97.3 | 0 | −1.33419 | + | 0.211315i | 0 | −4.90670 | − | 0.961386i | 0 | 5.78356 | + | 5.78356i | 0 | −6.82409 | + | 2.21728i | 0 | ||||||||||
97.4 | 0 | −0.535050 | + | 0.0847436i | 0 | 2.05203 | + | 4.55951i | 0 | −4.47022 | − | 4.47022i | 0 | −8.28041 | + | 2.69047i | 0 | ||||||||||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.f | odd | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 400.3.bg.f | 64 | |
4.b | odd | 2 | 1 | 200.3.u.b | ✓ | 64 | |
25.f | odd | 20 | 1 | inner | 400.3.bg.f | 64 | |
100.l | even | 20 | 1 | 200.3.u.b | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
200.3.u.b | ✓ | 64 | 4.b | odd | 2 | 1 | |
200.3.u.b | ✓ | 64 | 100.l | even | 20 | 1 | |
400.3.bg.f | 64 | 1.a | even | 1 | 1 | trivial | |
400.3.bg.f | 64 | 25.f | odd | 20 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} + 20 T_{3}^{62} + 30 T_{3}^{61} - 1175 T_{3}^{60} + 852 T_{3}^{59} - 31530 T_{3}^{58} + \cdots + 49\!\cdots\!00 \) acting on \(S_{3}^{\mathrm{new}}(400, [\chi])\).