# Properties

 Label 32.6.g.a Level 32 Weight 6 Character orbit 32.g Analytic conductor 5.132 Analytic rank 0 Dimension 76 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$32 = 2^{5}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 32.g (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.13228223402$$ Analytic rank: $$0$$ Dimension: $$76$$ Relative dimension: $$19$$ over $$\Q(\zeta_{8})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$76q - 4q^{2} - 4q^{3} - 4q^{4} - 4q^{5} - 4q^{6} - 4q^{7} - 4q^{8} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$76q - 4q^{2} - 4q^{3} - 4q^{4} - 4q^{5} - 4q^{6} - 4q^{7} - 4q^{8} - 4q^{9} - 204q^{10} - 4q^{11} - 1588q^{12} - 4q^{13} + 2476q^{14} + 4176q^{16} - 1624q^{18} - 4q^{19} - 7604q^{20} - 4q^{21} - 12384q^{22} + 1668q^{23} + 21456q^{24} - 4q^{25} + 12976q^{26} - 7468q^{27} - 2184q^{28} - 4q^{29} - 32316q^{30} + 23056q^{31} - 18584q^{32} - 8q^{33} - 6256q^{34} - 4780q^{35} + 65584q^{36} - 4q^{37} + 404q^{38} - 44908q^{39} - 28952q^{40} - 4q^{41} - 53624q^{42} + 32068q^{43} + 3716q^{44} + 968q^{45} + 63324q^{46} + 112368q^{48} - 4524q^{50} - 19912q^{51} + 18468q^{52} - 49460q^{53} + 1312q^{54} + 110044q^{55} - 80984q^{56} - 4q^{57} - 84576q^{58} - 28964q^{59} - 46088q^{60} + 96156q^{61} + 63480q^{62} - 158768q^{63} - 49192q^{64} - 8q^{65} - 145012q^{66} - 61164q^{67} - 151216q^{68} - 44644q^{69} - 56296q^{70} + 143836q^{71} + 27092q^{72} - 4q^{73} + 213100q^{74} + 205744q^{75} + 255996q^{76} - 14900q^{77} + 83508q^{78} - 97096q^{80} + 435196q^{82} - 329244q^{83} + 597472q^{84} + 12496q^{85} + 269888q^{86} - 282188q^{87} - 199840q^{88} - 4q^{89} - 706840q^{90} + 200108q^{91} - 650328q^{92} - 976q^{93} - 261592q^{94} + 577592q^{95} - 501376q^{96} - 8q^{97} - 395624q^{98} + 338544q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −5.53109 + 1.18618i 7.44181 + 17.9661i 29.1860 13.1217i 29.7710 + 12.3316i −62.4723 90.5449i 32.2015 + 32.2015i −145.866 + 107.197i −95.5738 + 95.5738i −179.294 32.8933i
5.2 −5.42606 + 1.59933i −9.61436 23.2111i 26.8843 17.3561i −17.5778 7.28095i 89.2903 + 110.568i −10.1181 10.1181i −118.118 + 137.172i −274.493 + 274.493i 107.023 + 11.3943i
5.3 −5.36185 1.80293i 0.399918 + 0.965487i 25.4989 + 19.3341i −41.3553 17.1299i −0.403590 5.89782i 125.552 + 125.552i −101.863 149.639i 171.055 171.055i 190.857 + 166.409i
5.4 −4.96462 2.71156i −3.34607 8.07812i 17.2949 + 26.9237i 90.7112 + 37.5738i −5.29232 + 49.1779i −136.759 136.759i −12.8576 180.562i 117.767 117.767i −348.463 432.508i
5.5 −3.77244 + 4.21529i 1.59693 + 3.85534i −3.53735 31.8039i −32.3386 13.3951i −22.2757 7.81250i −71.7275 71.7275i 147.407 + 105.067i 159.514 159.514i 178.459 85.7843i
5.6 −3.71766 4.26369i 10.1367 + 24.4721i −4.35808 + 31.7018i −58.4422 24.2075i 66.6567 134.198i −163.765 163.765i 151.369 99.2751i −324.305 + 324.305i 114.054 + 339.175i
5.7 −2.59560 5.02622i −6.55463 15.8243i −18.5257 + 26.0921i −38.0662 15.7675i −62.5230 + 74.0185i 29.7916 + 29.7916i 179.230 + 25.3893i −35.6173 + 35.6173i 19.5537 + 232.255i
5.8 −1.94841 + 5.31072i −5.15883 12.4545i −24.4074 20.6949i 50.0856 + 20.7461i 76.1938 3.13064i 106.306 + 106.306i 157.460 89.2990i 43.3257 43.3257i −207.764 + 225.568i
5.9 −0.578533 5.62719i 4.90532 + 11.8425i −31.3306 + 6.51103i 56.4155 + 23.3681i 63.8021 34.4545i 77.5935 + 77.5935i 54.7646 + 172.536i 55.6443 55.6443i 98.8584 330.980i
5.10 0.0731871 + 5.65638i 9.04699 + 21.8414i −31.9893 + 0.827948i 83.7809 + 34.7032i −122.881 + 52.7718i −34.8236 34.8236i −7.02439 180.883i −223.371 + 223.371i −190.163 + 476.437i
5.11 1.01888 + 5.56434i 4.68811 + 11.3181i −29.9238 + 11.3388i −98.5667 40.8277i −58.2012 + 37.6180i 104.908 + 104.908i −93.5814 154.953i 65.7060 65.7060i 126.752 590.057i
5.12 2.01099 + 5.28734i −7.16078 17.2876i −23.9119 + 21.2655i −8.84628 3.66425i 77.0054 72.6267i −158.442 158.442i −160.525 83.6652i −75.7590 + 75.7590i 1.58435 54.1420i
5.13 2.44286 5.10220i 0.355593 + 0.858478i −20.0649 24.9279i −47.1761 19.5410i 5.24879 + 0.282835i −64.1149 64.1149i −176.203 + 41.4795i 171.216 171.216i −214.946 + 192.966i
5.14 2.73793 4.95012i −11.3810 27.4761i −17.0075 27.1062i 62.9000 + 26.0541i −167.170 18.8905i 32.1730 + 32.1730i −180.744 + 9.97404i −453.581 + 453.581i 301.187 240.029i
5.15 4.48012 + 3.45378i −4.24511 10.2486i 8.14287 + 30.9466i 36.3338 + 15.0500i 16.3778 60.5765i 153.750 + 153.750i −70.4017 + 166.768i 84.8142 84.8142i 110.801 + 192.914i
5.16 4.85839 2.89759i 10.7913 + 26.0525i 15.2079 28.1553i −26.5895 11.0137i 127.918 + 95.3043i 131.646 + 131.646i −7.69630 180.856i −390.452 + 390.452i −161.096 + 23.5365i
5.17 4.94686 + 2.74382i 6.19399 + 14.9536i 16.9429 + 27.1466i −2.09949 0.869637i −10.3893 + 90.9686i −63.4847 63.4847i 9.32842 + 180.779i −13.4180 + 13.4180i −7.99975 10.0626i
5.18 5.38021 1.74739i −0.512391 1.23702i 25.8932 18.8027i 50.7130 + 21.0060i −4.91834 5.76009i −40.9895 40.9895i 106.455 146.408i 170.559 170.559i 309.552 + 24.4011i
5.19 5.65395 0.181169i −7.87640 19.0153i 31.9344 2.04864i −91.3601 37.8426i −47.9778 106.085i 18.5982 + 18.5982i 180.184 17.3684i −127.718 + 127.718i −523.402 197.409i
13.1 −5.53109 1.18618i 7.44181 17.9661i 29.1860 + 13.1217i 29.7710 12.3316i −62.4723 + 90.5449i 32.2015 32.2015i −145.866 107.197i −95.5738 95.5738i −179.294 + 32.8933i
See all 76 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.19 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.6.g.a 76
4.b odd 2 1 128.6.g.a 76
32.g even 8 1 inner 32.6.g.a 76
32.h odd 8 1 128.6.g.a 76

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.6.g.a 76 1.a even 1 1 trivial
32.6.g.a 76 32.g even 8 1 inner
128.6.g.a 76 4.b odd 2 1
128.6.g.a 76 32.h odd 8 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(32, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database