Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [760,2,Mod(217,760)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(760, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("760.217");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 760 = 2^{3} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 760.bv (of order \(12\), degree \(4\), 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: | \(6.06863055362\) |
Analytic rank: | \(0\) |
Dimension: | \(120\) |
Relative dimension: | \(30\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
217.1 | 0 | −3.33013 | + | 0.892307i | 0 | −2.14938 | + | 0.616566i | 0 | −1.42706 | − | 1.42706i | 0 | 7.69551 | − | 4.44300i | 0 | ||||||||||
217.2 | 0 | −3.02016 | + | 0.809249i | 0 | 1.75403 | − | 1.38686i | 0 | −0.311202 | − | 0.311202i | 0 | 5.86840 | − | 3.38812i | 0 | ||||||||||
217.3 | 0 | −2.83879 | + | 0.760651i | 0 | 1.80776 | + | 1.31605i | 0 | 1.74814 | + | 1.74814i | 0 | 4.88206 | − | 2.81866i | 0 | ||||||||||
217.4 | 0 | −2.32691 | + | 0.623494i | 0 | −2.16867 | + | 0.544849i | 0 | 1.17522 | + | 1.17522i | 0 | 2.42770 | − | 1.40163i | 0 | ||||||||||
217.5 | 0 | −2.17220 | + | 0.582039i | 0 | −0.102898 | + | 2.23370i | 0 | 2.97765 | + | 2.97765i | 0 | 1.78161 | − | 1.02861i | 0 | ||||||||||
217.6 | 0 | −2.05819 | + | 0.551489i | 0 | 0.917131 | + | 2.03933i | 0 | −1.67591 | − | 1.67591i | 0 | 1.33391 | − | 0.770133i | 0 | ||||||||||
217.7 | 0 | −1.84434 | + | 0.494188i | 0 | −0.484665 | − | 2.18291i | 0 | −2.21971 | − | 2.21971i | 0 | 0.559277 | − | 0.322899i | 0 | ||||||||||
217.8 | 0 | −1.62580 | + | 0.435631i | 0 | −2.08443 | − | 0.809424i | 0 | −3.36316 | − | 3.36316i | 0 | −0.144631 | + | 0.0835026i | 0 | ||||||||||
217.9 | 0 | −1.53637 | + | 0.411668i | 0 | 1.82525 | − | 1.29170i | 0 | 1.29575 | + | 1.29575i | 0 | −0.407123 | + | 0.235053i | 0 | ||||||||||
217.10 | 0 | −1.48571 | + | 0.398094i | 0 | −1.68119 | − | 1.47432i | 0 | 1.98038 | + | 1.98038i | 0 | −0.549231 | + | 0.317098i | 0 | ||||||||||
217.11 | 0 | −1.28291 | + | 0.343755i | 0 | 0.643100 | + | 2.14159i | 0 | −2.68803 | − | 2.68803i | 0 | −1.07038 | + | 0.617986i | 0 | ||||||||||
217.12 | 0 | −0.683312 | + | 0.183093i | 0 | 1.03740 | − | 1.98086i | 0 | 3.66185 | + | 3.66185i | 0 | −2.16468 | + | 1.24978i | 0 | ||||||||||
217.13 | 0 | −0.653014 | + | 0.174975i | 0 | 2.17367 | − | 0.524542i | 0 | −1.74329 | − | 1.74329i | 0 | −2.20227 | + | 1.27148i | 0 | ||||||||||
217.14 | 0 | −0.389775 | + | 0.104440i | 0 | 2.23580 | + | 0.0346815i | 0 | −1.53896 | − | 1.53896i | 0 | −2.45706 | + | 1.41858i | 0 | ||||||||||
217.15 | 0 | −0.0239568 | + | 0.00641922i | 0 | −0.526583 | + | 2.17318i | 0 | 2.97754 | + | 2.97754i | 0 | −2.59754 | + | 1.49969i | 0 | ||||||||||
217.16 | 0 | 0.238684 | − | 0.0639551i | 0 | 0.0369332 | − | 2.23576i | 0 | −0.846919 | − | 0.846919i | 0 | −2.54520 | + | 1.46947i | 0 | ||||||||||
217.17 | 0 | 0.304604 | − | 0.0816185i | 0 | −1.67426 | + | 1.48218i | 0 | 0.297573 | + | 0.297573i | 0 | −2.51195 | + | 1.45028i | 0 | ||||||||||
217.18 | 0 | 0.701895 | − | 0.188072i | 0 | −2.06254 | − | 0.863675i | 0 | 1.65863 | + | 1.65863i | 0 | −2.14079 | + | 1.23599i | 0 | ||||||||||
217.19 | 0 | 0.711431 | − | 0.190627i | 0 | −2.23605 | + | 0.00824860i | 0 | 0.639892 | + | 0.639892i | 0 | −2.12828 | + | 1.22876i | 0 | ||||||||||
217.20 | 0 | 0.882247 | − | 0.236397i | 0 | 0.464974 | + | 2.18719i | 0 | −1.04303 | − | 1.04303i | 0 | −1.87560 | + | 1.08288i | 0 | ||||||||||
See next 80 embeddings (of 120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
19.d | odd | 6 | 1 | inner |
95.l | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 760.2.bv.a | ✓ | 120 |
5.c | odd | 4 | 1 | inner | 760.2.bv.a | ✓ | 120 |
19.d | odd | 6 | 1 | inner | 760.2.bv.a | ✓ | 120 |
95.l | even | 12 | 1 | inner | 760.2.bv.a | ✓ | 120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
760.2.bv.a | ✓ | 120 | 1.a | even | 1 | 1 | trivial |
760.2.bv.a | ✓ | 120 | 5.c | odd | 4 | 1 | inner |
760.2.bv.a | ✓ | 120 | 19.d | odd | 6 | 1 | inner |
760.2.bv.a | ✓ | 120 | 95.l | even | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(760, [\chi])\).