# Properties

 Label 80.6.j.a Level 80 Weight 6 Character orbit 80.j Analytic conductor 12.831 Analytic rank 0 Dimension 116 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 80.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.8307055850$$ Analytic rank: $$0$$ Dimension: $$116$$ Relative dimension: $$58$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$116q - 2q^{2} - 20q^{4} - 2q^{5} - 4q^{6} - 4q^{7} - 248q^{8} - 8748q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$116q - 2q^{2} - 20q^{4} - 2q^{5} - 4q^{6} - 4q^{7} - 248q^{8} - 8748q^{9} - 66q^{10} - 4q^{11} - 308q^{12} - 4q^{13} + 972q^{15} - 1224q^{16} - 4q^{17} + 4214q^{18} - 2360q^{19} - 836q^{20} - 4q^{21} - 2440q^{22} - 4q^{23} + 972q^{24} - 884q^{26} - 12416q^{28} + 15336q^{30} - 17612q^{32} - 4q^{33} - 12520q^{34} - 8640q^{35} + 2380q^{36} - 4q^{37} + 15108q^{38} + 11864q^{40} - 41092q^{42} - 1316q^{43} + 8200q^{44} - 5766q^{45} - 35924q^{46} + 65256q^{47} + 5180q^{48} - 44378q^{50} + 10436q^{51} + 63080q^{52} + 33820q^{54} - 4q^{55} - 64684q^{56} + 972q^{57} - 66940q^{58} - 14480q^{59} + 136260q^{60} + 48076q^{61} + 109524q^{62} + 972q^{63} + 71920q^{64} - 4q^{65} + 72436q^{66} - 89260q^{67} + 36360q^{68} - 21348q^{69} + 59552q^{70} - 143848q^{71} + 179728q^{72} - 10072q^{73} - 82508q^{74} - 32272q^{75} - 128004q^{76} + 111388q^{78} + 313732q^{80} + 551116q^{81} - 282876q^{82} - 80928q^{84} - 6252q^{85} - 85324q^{86} - 282188q^{87} + 80224q^{88} + 115550q^{90} - 164724q^{91} + 474536q^{92} + 968q^{93} - 106060q^{94} + 204760q^{95} - 62264q^{96} - 4q^{97} + 50214q^{98} - 168788q^{99} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1 −5.64987 0.281110i 12.1374i 31.8420 + 3.17647i −16.5299 53.4019i −3.41195 + 68.5748i 122.686 122.686i −179.010 26.8977i 95.6829 78.3800 + 306.360i
43.2 −5.55053 + 1.09161i 28.2900i 29.6168 12.1180i −13.1634 + 54.3298i 30.8817 + 157.024i 49.8932 49.8932i −151.161 + 99.5915i −557.323 13.7568 315.928i
43.3 −5.53664 + 1.15999i 18.7987i 29.3088 12.8449i 22.2139 51.2985i −21.8063 104.082i −39.1700 + 39.1700i −147.373 + 105.116i −110.391 −63.4848 + 309.790i
43.4 −5.50602 + 1.29758i 15.6304i 28.6326 14.2890i −43.9107 + 34.5956i −20.2816 86.0613i 159.957 159.957i −139.111 + 115.828i −1.30901 196.883 247.462i
43.5 −5.49306 1.35140i 25.4114i 28.3474 + 14.8466i −55.1294 + 9.26025i 34.3409 139.586i −100.866 + 100.866i −135.651 119.862i −402.739 315.343 + 23.6347i
43.6 −5.48456 + 1.38551i 7.86510i 28.1607 15.1978i 34.2061 + 44.2148i −10.8972 43.1366i −94.0660 + 94.0660i −133.392 + 122.370i 181.140 −248.865 195.105i
43.7 −5.42078 1.61714i 1.90847i 26.7697 + 17.5323i 50.7820 + 23.3706i 3.08626 10.3454i 36.1463 36.1463i −116.761 138.329i 239.358 −237.485 208.808i
43.8 −5.39225 1.70986i 25.2540i 26.1528 + 18.4400i 44.1288 34.3168i −43.1808 + 136.176i −154.305 + 154.305i −109.492 144.151i −394.763 −296.631 + 109.591i
43.9 −5.15225 + 2.33545i 11.2637i 21.0913 24.0657i −52.1852 20.0427i 26.3059 + 58.0336i −164.495 + 164.495i −52.4636 + 173.250i 116.128 315.680 18.6111i
43.10 −5.11599 2.41384i 8.69655i 20.3467 + 24.6984i −39.9809 + 39.0708i −20.9921 + 44.4915i −29.8131 + 29.8131i −44.4755 175.471i 167.370 298.853 103.378i
43.11 −4.74956 3.07273i 27.2231i 13.1167 + 29.1882i 52.9572 17.9036i 83.6491 129.298i 106.164 106.164i 27.3891 178.935i −498.095 −306.536 77.6888i
43.12 −4.58344 + 3.31543i 14.5939i 10.0158 30.3922i 52.9491 17.9275i 48.3851 + 66.8903i 47.3215 47.3215i 54.8563 + 172.507i 30.0178 −183.251 + 257.719i
43.13 −4.35770 3.60700i 8.79961i 5.97913 + 31.4364i −29.5991 47.4225i 31.7402 38.3461i −6.66435 + 6.66435i 87.3359 158.557i 165.567 −42.0688 + 313.417i
43.14 −3.90055 + 4.09704i 12.0492i −1.57141 31.9614i −42.8214 35.9351i −49.3660 46.9985i 63.2042 63.2042i 137.076 + 118.229i 97.8169 314.254 35.2743i
43.15 −3.73115 + 4.25189i 30.7007i −4.15710 31.7288i 29.9068 + 47.2290i −130.536 114.549i −7.67296 + 7.67296i 150.418 + 100.709i −699.530 −312.399 49.0579i
43.16 −3.72428 + 4.25790i 2.96317i −4.25947 31.7152i −13.9016 + 54.1456i 12.6169 + 11.0357i 6.13739 6.13739i 150.904 + 99.9801i 234.220 −178.773 260.845i
43.17 −3.41694 4.50828i 19.9039i −8.64910 + 30.8090i 37.7217 + 41.2562i −89.7323 + 68.0104i 100.956 100.956i 168.449 66.2797i −153.166 57.1017 311.030i
43.18 −3.17219 4.68372i 25.5536i −11.8744 + 29.7153i −47.2997 29.7950i −119.686 + 81.0609i 22.5590 22.5590i 176.846 38.6460i −409.988 10.4922 + 316.054i
43.19 −2.87799 4.87003i 18.3965i −15.4344 + 28.0318i 18.3385 + 52.8081i 89.5915 52.9449i −170.706 + 170.706i 180.935 5.50899i −95.4313 204.399 241.290i
43.20 −2.81360 4.90751i 1.97498i −16.1674 + 27.6155i 42.5691 36.2336i −9.69225 + 5.55680i −85.3503 + 85.3503i 181.012 + 1.64266i 239.099 −297.589 106.962i
See next 80 embeddings (of 116 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 67.58 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.j even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.6.j.a 116
5.c odd 4 1 80.6.s.a yes 116
16.f odd 4 1 80.6.s.a yes 116
80.j even 4 1 inner 80.6.j.a 116

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.6.j.a 116 1.a even 1 1 trivial
80.6.j.a 116 80.j even 4 1 inner
80.6.s.a yes 116 5.c odd 4 1
80.6.s.a yes 116 16.f odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(80, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database