Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [241,4,Mod(3,241)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(241, base_ring=CyclotomicField(120))
chi = DirichletCharacter(H, H._module([91]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("241.3");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 241 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 241.s (of order \(120\), degree \(32\), 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: | \(14.2194603114\) |
Analytic rank: | \(0\) |
Dimension: | \(1888\) |
Relative dimension: | \(59\) over \(\Q(\zeta_{120})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{120}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −5.45583 | − | 1.46189i | −5.86831 | − | 0.307545i | 20.7008 | + | 11.9516i | 3.34193 | − | 6.55890i | 31.5670 | + | 10.2567i | 14.7378 | − | 13.9857i | −63.5167 | − | 63.5167i | 7.49043 | + | 0.787276i | −27.8214 | + | 30.8988i |
3.2 | −5.40671 | − | 1.44872i | 8.67744 | + | 0.454765i | 20.2055 | + | 11.6657i | 4.12759 | − | 8.10085i | −46.2576 | − | 15.0300i | −26.2321 | + | 24.8933i | −60.6814 | − | 60.6814i | 48.2391 | + | 5.07013i | −34.0526 | + | 37.8193i |
3.3 | −5.05874 | − | 1.35549i | 4.70412 | + | 0.246532i | 16.8253 | + | 9.71410i | −0.400189 | + | 0.785414i | −23.4627 | − | 7.62351i | 20.8543 | − | 19.7900i | −42.3216 | − | 42.3216i | −4.78414 | − | 0.502833i | 3.08907 | − | 3.43076i |
3.4 | −4.89020 | − | 1.31033i | −3.05889 | − | 0.160310i | 15.2689 | + | 8.81552i | −3.30014 | + | 6.47689i | 14.7485 | + | 4.79209i | −10.5852 | + | 10.0450i | −34.4779 | − | 34.4779i | −17.5210 | − | 1.84153i | 24.6252 | − | 27.3490i |
3.5 | −4.59862 | − | 1.23220i | 5.43488 | + | 0.284830i | 12.7008 | + | 7.33280i | −6.98441 | + | 13.7077i | −24.6420 | − | 8.00666i | 4.61565 | − | 4.38008i | −22.4392 | − | 22.4392i | 2.60470 | + | 0.273765i | 49.0092 | − | 54.4302i |
3.6 | −4.47910 | − | 1.20017i | −6.60498 | − | 0.346153i | 11.6937 | + | 6.75136i | −9.65223 | + | 18.9436i | 29.1689 | + | 9.47755i | 11.4575 | − | 10.8727i | −18.0430 | − | 18.0430i | 16.6539 | + | 1.75039i | 65.9688 | − | 73.2657i |
3.7 | −4.45304 | − | 1.19319i | −9.12845 | − | 0.478402i | 11.4776 | + | 6.62662i | 0.431636 | − | 0.847134i | 40.0785 | + | 13.0223i | −19.8889 | + | 18.8739i | −17.1248 | − | 17.1248i | 56.2477 | + | 5.91187i | −2.93288 | + | 3.25730i |
3.8 | −4.42372 | − | 1.18533i | −1.23361 | − | 0.0646508i | 11.2361 | + | 6.48714i | 5.75748 | − | 11.2997i | 5.38051 | + | 1.74823i | −9.77533 | + | 9.27645i | −16.1086 | − | 16.1086i | −25.3345 | − | 2.66276i | −38.8633 | + | 43.1621i |
3.9 | −4.32461 | − | 1.15878i | 0.352706 | + | 0.0184845i | 10.4313 | + | 6.02252i | 8.71129 | − | 17.0969i | −1.50390 | − | 0.488645i | 1.36770 | − | 1.29790i | −12.8059 | − | 12.8059i | −26.7280 | − | 2.80923i | −57.4844 | + | 63.8429i |
3.10 | −4.30950 | − | 1.15473i | 2.77153 | + | 0.145250i | 10.3102 | + | 5.95259i | −0.410992 | + | 0.806617i | −11.7762 | − | 3.82631i | −7.29875 | + | 6.92626i | −12.3200 | − | 12.3200i | −19.1918 | − | 2.01714i | 2.70259 | − | 3.00153i |
3.11 | −3.73936 | − | 1.00196i | −7.99140 | − | 0.418811i | 6.05069 | + | 3.49336i | 6.11073 | − | 11.9930i | 29.4631 | + | 9.57313i | 13.1400 | − | 12.4694i | 2.77373 | + | 2.77373i | 36.8349 | + | 3.87151i | −34.8667 | + | 38.7234i |
3.12 | −3.69998 | − | 0.991406i | 8.86257 | + | 0.464468i | 5.77875 | + | 3.33636i | 5.01594 | − | 9.84434i | −32.3308 | − | 10.5049i | 12.7714 | − | 12.1196i | 3.59502 | + | 3.59502i | 51.4774 | + | 5.41049i | −28.3186 | + | 31.4510i |
3.13 | −3.56129 | − | 0.954244i | 9.05328 | + | 0.474462i | 4.84399 | + | 2.79668i | −6.57630 | + | 12.9067i | −31.7886 | − | 10.3287i | −10.9085 | + | 10.3518i | 6.27423 | + | 6.27423i | 54.8846 | + | 5.76861i | 35.7362 | − | 39.6891i |
3.14 | −3.19486 | − | 0.856060i | −3.62401 | − | 0.189926i | 2.54607 | + | 1.46998i | −2.92117 | + | 5.73312i | 11.4156 | + | 3.70916i | 13.4020 | − | 12.7180i | 11.8344 | + | 11.8344i | −13.7547 | − | 1.44568i | 14.2406 | − | 15.8158i |
3.15 | −2.94380 | − | 0.788790i | −0.721247 | − | 0.0377990i | 1.11559 | + | 0.644088i | 0.0967234 | − | 0.189830i | 2.09340 | + | 0.680186i | 22.4174 | − | 21.2733i | 14.4641 | + | 14.4641i | −26.3333 | − | 2.76774i | −0.434471 | + | 0.482529i |
3.16 | −2.80709 | − | 0.752159i | 0.707572 | + | 0.0370823i | 0.385833 | + | 0.222761i | −6.46030 | + | 12.6791i | −1.95833 | − | 0.636300i | −21.5036 | + | 20.4062i | 15.5240 | + | 15.5240i | −26.3528 | − | 2.76979i | 27.6713 | − | 30.7321i |
3.17 | −2.73130 | − | 0.731849i | 5.39299 | + | 0.282635i | −0.00381879 | − | 0.00220478i | 5.62183 | − | 11.0335i | −14.5230 | − | 4.71881i | −9.28388 | + | 8.81007i | 16.0044 | + | 16.0044i | 2.15235 | + | 0.226221i | −23.4297 | + | 26.0214i |
3.18 | −2.68145 | − | 0.718492i | −6.29664 | − | 0.329993i | −0.254269 | − | 0.146802i | −3.49067 | + | 6.85084i | 16.6470 | + | 5.40894i | −10.0835 | + | 9.56891i | 16.2800 | + | 16.2800i | 12.6867 | + | 1.33343i | 14.2823 | − | 15.8621i |
3.19 | −2.14885 | − | 0.575782i | 3.61279 | + | 0.189338i | −2.64218 | − | 1.52546i | −0.925061 | + | 1.81553i | −7.65432 | − | 2.48704i | 3.61723 | − | 3.43262i | 17.3838 | + | 17.3838i | −13.8357 | − | 1.45419i | 3.03317 | − | 3.36867i |
3.20 | −1.98415 | − | 0.531653i | 5.99839 | + | 0.314362i | −3.27399 | − | 1.89024i | 4.80556 | − | 9.43143i | −11.7346 | − | 3.81280i | −19.1648 | + | 18.1867i | 17.1112 | + | 17.1112i | 9.02972 | + | 0.949062i | −14.5492 | + | 16.1585i |
See next 80 embeddings (of 1888 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
241.s | even | 120 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 241.4.s.a | ✓ | 1888 |
241.s | even | 120 | 1 | inner | 241.4.s.a | ✓ | 1888 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
241.4.s.a | ✓ | 1888 | 1.a | even | 1 | 1 | trivial |
241.4.s.a | ✓ | 1888 | 241.s | even | 120 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(241, [\chi])\).