Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [770,2,Mod(57,770)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(770, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([5, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("770.57");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 770.bh (of order \(20\), degree \(8\), 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: | \(6.14848095564\) |
Analytic rank: | \(0\) |
Dimension: | \(144\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
57.1 | −0.453990 | − | 0.891007i | −0.533991 | + | 3.37148i | −0.587785 | + | 0.809017i | −2.17647 | − | 0.512797i | 3.24644 | − | 1.05483i | 0.987688 | − | 0.156434i | 0.987688 | + | 0.156434i | −8.22859 | − | 2.67363i | 0.531193 | + | 2.17206i |
57.2 | −0.453990 | − | 0.891007i | −0.243764 | + | 1.53906i | −0.587785 | + | 0.809017i | −1.06097 | − | 1.96833i | 1.48198 | − | 0.481525i | 0.987688 | − | 0.156434i | 0.987688 | + | 0.156434i | 0.543876 | + | 0.176716i | −1.27213 | + | 1.83894i |
57.3 | −0.453990 | − | 0.891007i | −0.228528 | + | 1.44287i | −0.587785 | + | 0.809017i | 1.22219 | + | 1.87250i | 1.38936 | − | 0.451429i | 0.987688 | − | 0.156434i | 0.987688 | + | 0.156434i | 0.823523 | + | 0.267579i | 1.11355 | − | 1.93908i |
57.4 | −0.453990 | − | 0.891007i | 0.0651291 | − | 0.411209i | −0.587785 | + | 0.809017i | 2.19488 | − | 0.427198i | −0.395958 | + | 0.128655i | 0.987688 | − | 0.156434i | 0.987688 | + | 0.156434i | 2.68832 | + | 0.873488i | −1.37709 | − | 1.76171i |
57.5 | −0.453990 | − | 0.891007i | 0.0894891 | − | 0.565012i | −0.587785 | + | 0.809017i | −1.43548 | + | 1.71446i | −0.544056 | + | 0.176775i | 0.987688 | − | 0.156434i | 0.987688 | + | 0.156434i | 2.54194 | + | 0.825926i | 2.17929 | + | 0.500676i |
57.6 | −0.453990 | − | 0.891007i | 0.0900897 | − | 0.568804i | −0.587785 | + | 0.809017i | −1.98573 | − | 1.02805i | −0.547708 | + | 0.177961i | 0.987688 | − | 0.156434i | 0.987688 | + | 0.156434i | 2.53775 | + | 0.824564i | −0.0144993 | + | 2.23602i |
57.7 | −0.453990 | − | 0.891007i | 0.284566 | − | 1.79668i | −0.587785 | + | 0.809017i | 2.23599 | − | 0.0183689i | −1.73004 | + | 0.562125i | 0.987688 | − | 0.156434i | 0.987688 | + | 0.156434i | −0.293913 | − | 0.0954981i | −1.03149 | − | 1.98394i |
57.8 | −0.453990 | − | 0.891007i | 0.374414 | − | 2.36396i | −0.587785 | + | 0.809017i | 0.303006 | + | 2.21544i | −2.27628 | + | 0.739608i | 0.987688 | − | 0.156434i | 0.987688 | + | 0.156434i | −2.59493 | − | 0.843144i | 1.83641 | − | 1.27577i |
57.9 | −0.453990 | − | 0.891007i | 0.507726 | − | 3.20565i | −0.587785 | + | 0.809017i | 1.06586 | − | 1.96569i | −3.08676 | + | 1.00295i | 0.987688 | − | 0.156434i | 0.987688 | + | 0.156434i | −7.16527 | − | 2.32814i | −2.23533 | − | 0.0572840i |
57.10 | 0.453990 | + | 0.891007i | −0.444700 | + | 2.80772i | −0.587785 | + | 0.809017i | 0.570480 | − | 2.16207i | −2.70359 | + | 0.878449i | −0.987688 | + | 0.156434i | −0.987688 | − | 0.156434i | −4.83238 | − | 1.57014i | 2.18541 | − | 0.473259i |
57.11 | 0.453990 | + | 0.891007i | −0.364482 | + | 2.30125i | −0.587785 | + | 0.809017i | −0.503171 | + | 2.17872i | −2.21590 | + | 0.719988i | −0.987688 | + | 0.156434i | −0.987688 | − | 0.156434i | −2.30972 | − | 0.750473i | −2.16969 | + | 0.540789i |
57.12 | 0.453990 | + | 0.891007i | −0.226529 | + | 1.43025i | −0.587785 | + | 0.809017i | 1.67803 | + | 1.47791i | −1.37720 | + | 0.447481i | −0.987688 | + | 0.156434i | −0.987688 | − | 0.156434i | 0.858868 | + | 0.279063i | −0.555020 | + | 2.16609i |
57.13 | 0.453990 | + | 0.891007i | −0.198154 | + | 1.25110i | −0.587785 | + | 0.809017i | 2.00045 | − | 0.999108i | −1.20470 | + | 0.391429i | −0.987688 | + | 0.156434i | −0.987688 | − | 0.156434i | 1.32719 | + | 0.431231i | 1.79840 | + | 1.32882i |
57.14 | 0.453990 | + | 0.891007i | −0.0604460 | + | 0.381641i | −0.587785 | + | 0.809017i | −0.949664 | − | 2.02439i | −0.367486 | + | 0.119404i | −0.987688 | + | 0.156434i | −0.987688 | − | 0.156434i | 2.71117 | + | 0.880914i | 1.37260 | − | 1.76521i |
57.15 | 0.453990 | + | 0.891007i | 0.189968 | − | 1.19941i | −0.587785 | + | 0.809017i | −2.22313 | + | 0.240204i | 1.15492 | − | 0.375257i | −0.987688 | + | 0.156434i | −0.987688 | − | 0.156434i | 1.45068 | + | 0.471354i | −1.22330 | − | 1.87177i |
57.16 | 0.453990 | + | 0.891007i | 0.362144 | − | 2.28648i | −0.587785 | + | 0.809017i | −1.26466 | − | 1.84408i | 2.20168 | − | 0.715370i | −0.987688 | + | 0.156434i | −0.987688 | − | 0.156434i | −2.24370 | − | 0.729021i | 1.06895 | − | 1.96401i |
57.17 | 0.453990 | + | 0.891007i | 0.379834 | − | 2.39817i | −0.587785 | + | 0.809017i | −0.949803 | + | 2.02432i | 2.30923 | − | 0.750314i | −0.987688 | + | 0.156434i | −0.987688 | − | 0.156434i | −2.75380 | − | 0.894764i | −2.23488 | + | 0.0727419i |
57.18 | 0.453990 | + | 0.891007i | 0.399698 | − | 2.52360i | −0.587785 | + | 0.809017i | 2.00474 | + | 0.990458i | 2.43000 | − | 0.789555i | −0.987688 | + | 0.156434i | −0.987688 | − | 0.156434i | −3.35561 | − | 1.09030i | 0.0276292 | + | 2.23590i |
127.1 | −0.891007 | − | 0.453990i | −3.37148 | + | 0.533991i | 0.587785 | + | 0.809017i | 1.45939 | + | 1.69416i | 3.24644 | + | 1.05483i | −0.156434 | + | 0.987688i | −0.156434 | − | 0.987688i | 8.22859 | − | 2.67363i | −0.531193 | − | 2.17206i |
127.2 | −0.891007 | − | 0.453990i | −1.53906 | + | 0.243764i | 0.587785 | + | 0.809017i | −0.298611 | + | 2.21604i | 1.48198 | + | 0.481525i | −0.156434 | + | 0.987688i | −0.156434 | − | 0.987688i | −0.543876 | + | 0.176716i | 1.27213 | − | 1.83894i |
See next 80 embeddings (of 144 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
11.d | odd | 10 | 1 | inner |
55.l | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 770.2.bh.b | ✓ | 144 |
5.c | odd | 4 | 1 | inner | 770.2.bh.b | ✓ | 144 |
11.d | odd | 10 | 1 | inner | 770.2.bh.b | ✓ | 144 |
55.l | even | 20 | 1 | inner | 770.2.bh.b | ✓ | 144 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
770.2.bh.b | ✓ | 144 | 1.a | even | 1 | 1 | trivial |
770.2.bh.b | ✓ | 144 | 5.c | odd | 4 | 1 | inner |
770.2.bh.b | ✓ | 144 | 11.d | odd | 10 | 1 | inner |
770.2.bh.b | ✓ | 144 | 55.l | even | 20 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{144} - 4 T_{3}^{143} + 8 T_{3}^{142} + 24 T_{3}^{141} - 385 T_{3}^{140} + 964 T_{3}^{139} + \cdots + 95\!\cdots\!25 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\).