Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [71,5,Mod(14,71)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(71, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("71.14");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 71 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 71.e (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: | \(7.33926737895\) |
Analytic rank: | \(0\) |
Dimension: | \(92\) |
Relative dimension: | \(23\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
14.1 | −2.33798 | + | 7.19556i | −5.29880 | + | 16.3080i | −33.3656 | − | 24.2415i | 22.9444 | − | 16.6701i | −104.957 | − | 76.2556i | 28.6513 | + | 9.30937i | 154.505 | − | 112.254i | −172.344 | − | 125.215i | 66.3071 | + | 204.072i |
14.2 | −2.21238 | + | 6.80899i | −0.750926 | + | 2.31111i | −28.5235 | − | 20.7235i | −38.4378 | + | 27.9267i | −14.0750 | − | 10.2261i | 18.6593 | + | 6.06278i | 111.538 | − | 81.0369i | 60.7530 | + | 44.1397i | −105.114 | − | 323.507i |
14.3 | −2.21097 | + | 6.80468i | 1.63361 | − | 5.02773i | −28.4709 | − | 20.6853i | 16.3761 | − | 11.8979i | 30.6002 | + | 22.2324i | −21.7377 | − | 7.06300i | 111.091 | − | 80.7124i | 42.9210 | + | 31.1839i | 44.7543 | + | 137.740i |
14.4 | −1.78313 | + | 5.48790i | 4.10332 | − | 12.6287i | −13.9932 | − | 10.1667i | −10.6624 | + | 7.74667i | 61.9883 | + | 45.0372i | −11.6810 | − | 3.79537i | 6.05266 | − | 4.39751i | −77.1168 | − | 56.0286i | −23.5006 | − | 72.3274i |
14.5 | −1.48247 | + | 4.56257i | −2.26569 | + | 6.97309i | −5.67503 | − | 4.12315i | 11.4019 | − | 8.28397i | −28.4564 | − | 20.6748i | −88.7423 | − | 28.8341i | −34.8732 | + | 25.3369i | 22.0398 | + | 16.0128i | 20.8932 | + | 64.3027i |
14.6 | −1.42593 | + | 4.38858i | −0.426217 | + | 1.31176i | −4.28203 | − | 3.11108i | 15.9930 | − | 11.6196i | −5.14900 | − | 3.74097i | 71.2142 | + | 23.1389i | −39.9712 | + | 29.0408i | 63.9913 | + | 46.4924i | 28.1885 | + | 86.7552i |
14.7 | −1.38037 | + | 4.24834i | −3.77624 | + | 11.6221i | −3.19872 | − | 2.32401i | −17.7367 | + | 12.8865i | −44.1620 | − | 32.0856i | −1.89994 | − | 0.617330i | −43.5331 | + | 31.6286i | −55.2825 | − | 40.1651i | −30.2630 | − | 93.1399i |
14.8 | −0.903467 | + | 2.78059i | 3.64933 | − | 11.2315i | 6.02886 | + | 4.38023i | −21.7356 | + | 15.7918i | 27.9331 | + | 20.2946i | 36.5121 | + | 11.8635i | −55.4714 | + | 40.3023i | −47.2983 | − | 34.3642i | −24.2731 | − | 74.7051i |
14.9 | −0.750757 | + | 2.31059i | 4.38800 | − | 13.5049i | 8.16908 | + | 5.93518i | 34.8891 | − | 25.3484i | 27.9099 | + | 20.2777i | −32.0797 | − | 10.4233i | −51.2949 | + | 37.2679i | −97.5967 | − | 70.9081i | 32.3766 | + | 99.6450i |
14.10 | −0.395056 | + | 1.21586i | 0.615850 | − | 1.89539i | 11.6220 | + | 8.44390i | −20.6238 | + | 14.9841i | 2.06123 | + | 1.49757i | −51.4846 | − | 16.7284i | −31.4063 | + | 22.8180i | 62.3171 | + | 45.2761i | −10.0710 | − | 30.9952i |
14.11 | −0.0189892 | + | 0.0584427i | −3.28390 | + | 10.1068i | 12.9412 | + | 9.40234i | 33.4497 | − | 24.3027i | −0.528310 | − | 0.383839i | −1.40375 | − | 0.456106i | −1.59067 | + | 1.15569i | −25.8329 | − | 18.7687i | 0.785130 | + | 2.41638i |
14.12 | 0.00803474 | − | 0.0247284i | −4.10998 | + | 12.6492i | 12.9437 | + | 9.40417i | −17.3417 | + | 12.5995i | 0.279772 | + | 0.203266i | 44.2278 | + | 14.3705i | 0.673113 | − | 0.489045i | −77.5801 | − | 56.3652i | 0.172229 | + | 0.530066i |
14.13 | 0.0929586 | − | 0.286097i | 1.01964 | − | 3.13814i | 12.8711 | + | 9.35137i | 5.77327 | − | 4.19453i | −0.803027 | − | 0.583433i | −0.0874858 | − | 0.0284258i | 7.76577 | − | 5.64216i | 56.7221 | + | 41.2110i | −0.663367 | − | 2.04163i |
14.14 | 0.681743 | − | 2.09819i | 3.85088 | − | 11.8518i | 9.00665 | + | 6.54371i | 7.43329 | − | 5.40060i | −22.2420 | − | 16.1597i | 84.8852 | + | 27.5809i | 48.4274 | − | 35.1845i | −60.1053 | − | 43.6691i | −6.26389 | − | 19.2783i |
14.15 | 1.13409 | − | 3.49038i | 4.67733 | − | 14.3953i | 2.04766 | + | 1.48771i | −34.6068 | + | 25.1433i | −44.9407 | − | 32.6513i | −52.0590 | − | 16.9150i | 55.0205 | − | 39.9747i | −119.818 | − | 87.0528i | 48.5124 | + | 149.306i |
14.16 | 1.22495 | − | 3.77000i | −1.45133 | + | 4.46673i | 0.231852 | + | 0.168450i | 1.12547 | − | 0.817703i | 15.0618 | + | 10.9430i | 1.80874 | + | 0.587695i | 52.2304 | − | 37.9476i | 47.6851 | + | 34.6452i | −1.70410 | − | 5.24468i |
14.17 | 1.24405 | − | 3.82878i | 3.15116 | − | 9.69828i | −0.167663 | − | 0.121814i | 18.0016 | − | 13.0789i | −33.2124 | − | 24.1302i | −71.2215 | − | 23.1413i | 51.4364 | − | 37.3707i | −18.5964 | − | 13.5111i | −27.6815 | − | 85.1950i |
14.18 | 1.28524 | − | 3.95557i | −4.83631 | + | 14.8846i | −1.05042 | − | 0.763175i | −4.79770 | + | 3.48573i | 52.6613 | + | 38.2607i | −85.4311 | − | 27.7582i | 49.4681 | − | 35.9407i | −132.632 | − | 96.3625i | 7.62186 | + | 23.4577i |
14.19 | 1.30403 | − | 4.01340i | −0.422249 | + | 1.29955i | −1.46264 | − | 1.06267i | −37.2304 | + | 27.0495i | 4.66499 | + | 3.38931i | 56.2511 | + | 18.2771i | 48.4518 | − | 35.2023i | 64.0198 | + | 46.5131i | 60.0108 | + | 184.694i |
14.20 | 1.78194 | − | 5.48424i | −0.705335 | + | 2.17080i | −13.9573 | − | 10.1406i | 36.0884 | − | 26.2198i | 10.6483 | + | 7.73646i | 16.7832 | + | 5.45319i | −5.84178 | + | 4.24430i | 61.3155 | + | 44.5483i | −79.4882 | − | 244.639i |
See all 92 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
71.e | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 71.5.e.a | ✓ | 92 |
71.e | odd | 10 | 1 | inner | 71.5.e.a | ✓ | 92 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
71.5.e.a | ✓ | 92 | 1.a | even | 1 | 1 | trivial |
71.5.e.a | ✓ | 92 | 71.e | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(71, [\chi])\).