Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [671,2,Mod(102,671)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(671, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([8, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("671.102");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 671 = 11 \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 671.bd (of order \(10\), 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: | \(5.35796197563\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(60\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
102.1 | −1.62186 | + | 2.23229i | 0.327661 | − | 0.238060i | −1.73468 | − | 5.33881i | −2.62339 | 1.11754i | 2.05763 | − | 2.83209i | 9.48276 | + | 3.08114i | −0.876362 | + | 2.69716i | 4.25477 | − | 5.85618i | ||||
102.2 | −1.53609 | + | 2.11425i | 2.00887 | − | 1.45953i | −1.49243 | − | 4.59324i | −0.0405784 | 6.48921i | −1.49840 | + | 2.06237i | 7.03287 | + | 2.28512i | 0.978276 | − | 3.01082i | 0.0623321 | − | 0.0857928i | ||||
102.3 | −1.52483 | + | 2.09875i | −1.26756 | + | 0.920933i | −1.46160 | − | 4.49834i | 2.66995 | − | 4.06454i | 0.972481 | − | 1.33851i | 6.73513 | + | 2.18838i | −0.168472 | + | 0.518503i | −4.07122 | + | 5.60356i | |||
102.4 | −1.49902 | + | 2.06323i | 2.00706 | − | 1.45822i | −1.39181 | − | 4.28355i | 3.60749 | 6.32693i | 1.54400 | − | 2.12513i | 6.07335 | + | 1.97335i | 0.974855 | − | 3.00029i | −5.40771 | + | 7.44307i | ||||
102.5 | −1.45402 | + | 2.00129i | 0.314204 | − | 0.228283i | −1.27295 | − | 3.91773i | 2.04936 | 0.960741i | −0.786859 | + | 1.08302i | 4.98608 | + | 1.62008i | −0.880440 | + | 2.70971i | −2.97982 | + | 4.10137i | ||||
102.6 | −1.45197 | + | 1.99847i | −2.09315 | + | 1.52076i | −1.26762 | − | 3.90133i | −1.20752 | − | 6.39120i | 1.89653 | − | 2.61035i | 4.93856 | + | 1.60464i | 1.14150 | − | 3.51319i | 1.75329 | − | 2.41320i | |||
102.7 | −1.42256 | + | 1.95798i | −2.47668 | + | 1.79941i | −1.19200 | − | 3.66859i | −1.88066 | − | 7.40907i | −2.72530 | + | 3.75106i | 4.27522 | + | 1.38910i | 1.96900 | − | 6.05996i | 2.67535 | − | 3.68230i | |||
102.8 | −1.40937 | + | 1.93983i | 0.438999 | − | 0.318952i | −1.15858 | − | 3.56575i | −4.27798 | 1.30110i | −1.80714 | + | 2.48731i | 3.98901 | + | 1.29611i | −0.836061 | + | 2.57313i | 6.02924 | − | 8.29854i | ||||
102.9 | −1.25583 | + | 1.72851i | 1.68142 | − | 1.22162i | −0.792581 | − | 2.43931i | −1.35471 | 4.44049i | −0.563963 | + | 0.776229i | 1.14775 | + | 0.372928i | 0.407758 | − | 1.25495i | 1.70129 | − | 2.34162i | ||||
102.10 | −1.22626 | + | 1.68780i | −1.09421 | + | 0.794989i | −0.726927 | − | 2.23725i | 0.967708 | − | 2.82167i | −0.537161 | + | 0.739339i | 0.699180 | + | 0.227177i | −0.361767 | + | 1.11340i | −1.18666 | + | 1.63330i | |||
102.11 | −1.18673 | + | 1.63340i | 0.570475 | − | 0.414474i | −0.641616 | − | 1.97469i | −0.927575 | 1.42368i | 1.04506 | − | 1.43840i | 0.146532 | + | 0.0476111i | −0.773398 | + | 2.38028i | 1.10078 | − | 1.51510i | ||||
102.12 | −1.18452 | + | 1.63036i | −2.40841 | + | 1.74981i | −0.636935 | − | 1.96029i | 4.34190 | − | 5.99927i | −0.657519 | + | 0.904997i | 0.117230 | + | 0.0380904i | 1.81155 | − | 5.57538i | −5.14309 | + | 7.07886i | |||
102.13 | −1.10903 | + | 1.52644i | 2.26374 | − | 1.64471i | −0.482058 | − | 1.48362i | −0.790541 | 5.27950i | 2.48480 | − | 3.42003i | −0.789602 | − | 0.256557i | 1.49243 | − | 4.59322i | 0.876731 | − | 1.20672i | ||||
102.14 | −1.01923 | + | 1.40285i | 0.332528 | − | 0.241596i | −0.311121 | − | 0.957531i | 2.08625 | 0.712728i | −2.32616 | + | 3.20168i | −1.63792 | − | 0.532192i | −0.874845 | + | 2.69249i | −2.12637 | + | 2.92669i | ||||
102.15 | −0.937766 | + | 1.29072i | −1.39323 | + | 1.01224i | −0.168530 | − | 0.518682i | −1.42360 | − | 2.74753i | 0.165676 | − | 0.228034i | −2.20716 | − | 0.717149i | −0.0105864 | + | 0.0325816i | 1.33501 | − | 1.83748i | |||
102.16 | −0.921245 | + | 1.26799i | −0.946289 | + | 0.687519i | −0.141060 | − | 0.434137i | −3.39867 | − | 1.83325i | 1.02358 | − | 1.40883i | −2.30078 | − | 0.747569i | −0.504270 | + | 1.55198i | 3.13101 | − | 4.30946i | |||
102.17 | −0.821787 | + | 1.13109i | 2.50821 | − | 1.82232i | 0.0139969 | + | 0.0430781i | 1.93065 | 4.33457i | −2.03656 | + | 2.80309i | −2.71959 | − | 0.883647i | 2.04321 | − | 6.28834i | −1.58658 | + | 2.18374i | ||||
102.18 | −0.811356 | + | 1.11674i | −0.683410 | + | 0.496527i | 0.0292341 | + | 0.0899733i | 2.32240 | − | 1.16605i | 2.50690 | − | 3.45045i | −2.74980 | − | 0.893464i | −0.706540 | + | 2.17451i | −1.88429 | + | 2.59350i | |||
102.19 | −0.680409 | + | 0.936503i | −1.18938 | + | 0.864135i | 0.203953 | + | 0.627703i | −1.32564 | − | 1.70182i | −2.96018 | + | 4.07434i | −2.92847 | − | 0.951517i | −0.259156 | + | 0.797600i | 0.901977 | − | 1.24146i | |||
102.20 | −0.646017 | + | 0.889167i | 0.881531 | − | 0.640469i | 0.244755 | + | 0.753279i | 3.06178 | 1.19758i | 0.560983 | − | 0.772127i | −2.91846 | − | 0.948266i | −0.560156 | + | 1.72398i | −1.97796 | + | 2.72243i | ||||
See next 80 embeddings (of 240 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
671.bd | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 671.2.bd.a | ✓ | 240 |
11.c | even | 5 | 1 | 671.2.bf.a | yes | 240 | |
61.g | even | 10 | 1 | 671.2.bf.a | yes | 240 | |
671.bd | even | 10 | 1 | inner | 671.2.bd.a | ✓ | 240 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
671.2.bd.a | ✓ | 240 | 1.a | even | 1 | 1 | trivial |
671.2.bd.a | ✓ | 240 | 671.bd | even | 10 | 1 | inner |
671.2.bf.a | yes | 240 | 11.c | even | 5 | 1 | |
671.2.bf.a | yes | 240 | 61.g | even | 10 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).