Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [819,2,Mod(131,819)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(819, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5, 5, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("819.131");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 819.be (of order \(6\), degree \(2\), 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: | \(6.53974792554\) |
Analytic rank: | \(0\) |
Dimension: | \(192\) |
Relative dimension: | \(96\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
131.1 | − | 2.75013i | −0.525766 | + | 1.65032i | −5.56324 | −0.680031 | − | 1.17785i | 4.53861 | + | 1.44593i | 2.64066 | − | 0.164126i | 9.79938i | −2.44714 | − | 1.73537i | −3.23924 | + | 1.87018i | |||||
131.2 | − | 2.74898i | 1.63715 | − | 0.565445i | −5.55690 | −1.29542 | − | 2.24373i | −1.55440 | − | 4.50050i | −1.51202 | − | 2.17113i | 9.77785i | 2.36054 | − | 1.85144i | −6.16798 | + | 3.56109i | |||||
131.3 | − | 2.73882i | −1.09155 | − | 1.34481i | −5.50114 | 1.60660 | + | 2.78271i | −3.68320 | + | 2.98955i | 0.236348 | − | 2.63517i | 9.58898i | −0.617044 | + | 2.93586i | 7.62134 | − | 4.40018i | |||||
131.4 | − | 2.71616i | −1.40234 | − | 1.01658i | −5.37750 | −0.333015 | − | 0.576798i | −2.76118 | + | 3.80899i | −0.171395 | + | 2.64019i | 9.17383i | 0.933138 | + | 2.85118i | −1.56667 | + | 0.904520i | |||||
131.5 | − | 2.68351i | 0.690276 | + | 1.58856i | −5.20121 | 1.56627 | + | 2.71286i | 4.26291 | − | 1.85236i | −2.21901 | − | 1.44084i | 8.59048i | −2.04704 | + | 2.19309i | 7.27998 | − | 4.20310i | |||||
131.6 | − | 2.54597i | −1.52580 | + | 0.819716i | −4.48196 | 0.609884 | + | 1.05635i | 2.08697 | + | 3.88464i | −1.82539 | + | 1.91519i | 6.31900i | 1.65613 | − | 2.50145i | 2.68943 | − | 1.55275i | |||||
131.7 | − | 2.45977i | 1.40603 | − | 1.01147i | −4.05047 | 0.451648 | + | 0.782277i | −2.48798 | − | 3.45852i | 2.51399 | − | 0.824541i | 5.04369i | 0.953860 | − | 2.84432i | 1.92422 | − | 1.11095i | |||||
131.8 | − | 2.45370i | −1.69378 | + | 0.362108i | −4.02064 | −1.04334 | − | 1.80712i | 0.888504 | + | 4.15602i | −0.243984 | − | 2.63448i | 4.95804i | 2.73776 | − | 1.22666i | −4.43414 | + | 2.56005i | |||||
131.9 | − | 2.42664i | 1.65199 | + | 0.520509i | −3.88858 | 0.293231 | + | 0.507891i | 1.26309 | − | 4.00878i | −0.999638 | + | 2.44964i | 4.58290i | 2.45814 | + | 1.71975i | 1.23247 | − | 0.711565i | |||||
131.10 | − | 2.40217i | 1.19422 | + | 1.25453i | −3.77044 | −1.77637 | − | 3.07677i | 3.01360 | − | 2.86871i | 1.13710 | + | 2.38893i | 4.25291i | −0.147698 | + | 2.99636i | −7.39093 | + | 4.26715i | |||||
131.11 | − | 2.36062i | 0.342821 | − | 1.69778i | −3.57253 | 0.0272649 | + | 0.0472242i | −4.00783 | − | 0.809270i | −2.55198 | − | 0.698138i | 3.71214i | −2.76495 | − | 1.16407i | 0.111478 | − | 0.0643620i | |||||
131.12 | − | 2.27732i | 1.58919 | + | 0.688832i | −3.18620 | 0.872025 | + | 1.51039i | 1.56869 | − | 3.61909i | 2.08014 | − | 1.63493i | 2.70135i | 2.05102 | + | 2.18936i | 3.43965 | − | 1.98588i | |||||
131.13 | − | 2.20642i | −0.106553 | − | 1.72877i | −2.86827 | 0.566894 | + | 0.981889i | −3.81438 | + | 0.235101i | 2.63987 | − | 0.176277i | 1.91576i | −2.97729 | + | 0.368412i | 2.16646 | − | 1.25080i | |||||
131.14 | − | 2.18169i | −0.805241 | + | 1.53349i | −2.75975 | −2.02931 | − | 3.51487i | 3.34559 | + | 1.75678i | −2.62703 | + | 0.314218i | 1.65754i | −1.70317 | − | 2.46965i | −7.66833 | + | 4.42731i | |||||
131.15 | − | 2.14803i | −0.0525246 | + | 1.73125i | −2.61402 | 1.40260 | + | 2.42937i | 3.71878 | + | 0.112824i | 1.44779 | + | 2.21447i | 1.31894i | −2.99448 | − | 0.181867i | 5.21836 | − | 3.01282i | |||||
131.16 | − | 2.06287i | −1.17177 | − | 1.27552i | −2.25542 | −1.56599 | − | 2.71237i | −2.63123 | + | 2.41720i | −2.52851 | + | 0.778876i | 0.526900i | −0.253916 | + | 2.98924i | −5.59525 | + | 3.23042i | |||||
131.17 | − | 2.03908i | −1.26173 | + | 1.18661i | −2.15784 | 1.34835 | + | 2.33541i | 2.41959 | + | 2.57276i | −1.96236 | − | 1.77459i | 0.321848i | 0.183910 | − | 2.99436i | 4.76208 | − | 2.74939i | |||||
131.18 | − | 1.94462i | 1.40048 | − | 1.01915i | −1.78156 | −1.83006 | − | 3.16976i | −1.98186 | − | 2.72340i | 2.51888 | + | 0.809472i | − | 0.424777i | 0.922674 | − | 2.85459i | −6.16400 | + | 3.55879i | ||||
131.19 | − | 1.93848i | 0.0119177 | + | 1.73201i | −1.75770 | −0.441415 | − | 0.764554i | 3.35746 | − | 0.0231021i | −0.111737 | − | 2.64339i | − | 0.469696i | −2.99972 | + | 0.0412830i | −1.48207 | + | 0.855674i | ||||
131.20 | − | 1.82356i | −1.62712 | − | 0.593694i | −1.32536 | 0.468964 | + | 0.812270i | −1.08263 | + | 2.96715i | 0.0300109 | − | 2.64558i | − | 1.23025i | 2.29505 | + | 1.93203i | 1.48122 | − | 0.855183i | ||||
See next 80 embeddings (of 192 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.i | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 819.2.be.a | ✓ | 192 |
7.d | odd | 6 | 1 | 819.2.du.a | yes | 192 | |
9.d | odd | 6 | 1 | 819.2.du.a | yes | 192 | |
63.i | even | 6 | 1 | inner | 819.2.be.a | ✓ | 192 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
819.2.be.a | ✓ | 192 | 1.a | even | 1 | 1 | trivial |
819.2.be.a | ✓ | 192 | 63.i | even | 6 | 1 | inner |
819.2.du.a | yes | 192 | 7.d | odd | 6 | 1 | |
819.2.du.a | yes | 192 | 9.d | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(819, [\chi])\).