# Properties

 Label 192.3.q.a Level $192$ Weight $3$ Character orbit 192.q Analytic conductor $5.232$ Analytic rank $0$ Dimension $496$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 192.q (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.23162107572$$ Analytic rank: $$0$$ Dimension: $$496$$ Relative dimension: $$62$$ over $$\Q(\zeta_{16})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$496q - 8q^{3} - 16q^{4} - 8q^{6} - 16q^{7} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$496q - 8q^{3} - 16q^{4} - 8q^{6} - 16q^{7} - 8q^{9} - 16q^{10} - 8q^{12} - 16q^{13} - 8q^{15} - 16q^{16} - 8q^{18} - 16q^{19} - 8q^{21} - 16q^{22} + 272q^{24} - 16q^{25} - 8q^{27} - 16q^{28} + 72q^{30} - 16q^{34} - 408q^{36} - 16q^{37} - 8q^{39} - 16q^{40} - 448q^{42} - 16q^{43} - 8q^{45} - 16q^{46} - 8q^{48} - 16q^{49} - 8q^{51} - 544q^{52} - 8q^{54} + 496q^{55} - 8q^{57} - 736q^{58} - 8q^{60} - 16q^{61} - 16q^{63} + 80q^{64} - 40q^{66} - 528q^{67} - 8q^{69} + 656q^{70} - 8q^{72} - 16q^{73} - 8q^{75} + 1440q^{76} - 416q^{78} - 528q^{79} - 8q^{81} - 1056q^{82} - 1240q^{84} - 16q^{85} - 8q^{87} - 576q^{88} - 728q^{90} - 16q^{91} + 64q^{93} - 112q^{94} + 128q^{96} - 8q^{99} + O(q^{100})$$

## 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 −1.99952 0.0439169i −0.469761 + 2.96299i 3.99614 + 0.175625i −1.47453 + 7.41296i 1.06942 5.90393i −4.15461 + 10.0301i −7.98265 0.526664i −8.55865 2.78380i 3.27390 14.7576i
5.2 −1.99227 0.175619i −1.83061 2.37674i 3.93832 + 0.699764i 0.541397 2.72179i 3.22967 + 5.05660i −4.07794 + 9.84503i −7.72331 2.08577i −2.29776 + 8.70174i −1.55661 + 5.32746i
5.3 −1.98950 + 0.204660i −0.243775 + 2.99008i 3.91623 0.814343i 1.66868 8.38903i −0.126959 5.99866i 0.401965 0.970430i −7.62468 + 2.42163i −8.88115 1.45781i −1.60294 + 17.0315i
5.4 −1.94960 0.446149i −2.81493 + 1.03739i 3.60190 + 1.73963i −0.199626 + 1.00359i 5.95082 0.766618i 2.15070 5.19226i −6.24615 4.99856i 6.84765 5.84035i 0.836939 1.86753i
5.5 −1.94227 0.477050i 2.38867 1.81500i 3.54485 + 1.85312i 0.875646 4.40217i −5.50530 + 2.38571i 0.768038 1.85421i −6.00102 5.29034i 2.41153 8.67090i −3.80080 + 8.13248i
5.6 −1.93172 + 0.518145i 2.81729 + 1.03097i 3.46305 2.00182i −0.547841 + 2.75418i −5.97639 0.531779i 4.75360 11.4762i −5.65240 + 5.66130i 6.87420 + 5.80908i −0.368792 5.60416i
5.7 −1.86334 + 0.726623i 1.85384 2.35866i 2.94404 2.70788i −1.54249 + 7.75462i −1.74046 + 5.74202i −1.28266 + 3.09661i −3.51812 + 7.18490i −2.12657 8.74515i −2.76051 15.5703i
5.8 −1.83712 + 0.790557i 2.82801 + 1.00117i 2.75004 2.90470i 0.733309 3.68659i −5.98689 + 0.396429i −3.52641 + 8.51350i −2.75583 + 7.51035i 6.99531 + 5.66265i 1.56728 + 7.35245i
5.9 −1.82442 0.819447i −0.525462 2.95362i 2.65701 + 2.99003i −1.44434 + 7.26117i −1.46167 + 5.81924i 3.15020 7.60525i −2.39734 7.63235i −8.44778 + 3.10403i 8.58522 12.0639i
5.10 −1.79658 + 0.878814i −2.85707 0.914956i 2.45537 3.15771i −0.757135 + 3.80637i 5.93702 0.867047i 0.246285 0.594584i −1.63622 + 7.83089i 7.32571 + 5.22819i −1.98484 7.50382i
5.11 −1.74186 + 0.982818i −0.766801 2.90035i 2.06814 3.42386i 1.45744 7.32705i 4.18617 + 4.29837i 4.63081 11.1798i −0.237375 + 7.99648i −7.82403 + 4.44798i 4.66250 + 14.1951i
5.12 −1.73173 1.00056i 2.76570 + 1.16228i 1.99775 + 3.46540i −0.965897 + 4.85589i −3.62650 4.78001i −1.32825 + 3.20667i 0.00779347 8.00000i 6.29820 + 6.42905i 6.53129 7.44264i
5.13 −1.55855 1.25336i 1.01740 + 2.82222i 0.858181 + 3.90686i 0.512850 2.57827i 1.95158 5.67374i 1.92886 4.65669i 3.55917 7.16466i −6.92980 + 5.74263i −4.03080 + 3.37559i
5.14 −1.53172 + 1.28601i −1.90665 + 2.31618i 0.692348 3.93963i 0.262424 1.31929i −0.0581871 5.99972i 0.595494 1.43765i 4.00592 + 6.92478i −1.72941 8.83228i 1.29467 + 2.35827i
5.15 −1.40959 1.41882i −2.45976 + 1.71744i −0.0261177 + 3.99991i 1.04139 5.23542i 5.90399 + 1.06907i −4.45319 + 10.7510i 5.71199 5.60118i 3.10080 8.44897i −8.89607 + 5.90224i
5.16 −1.37275 1.45450i −2.57644 1.53687i −0.231128 + 3.99332i 1.34532 6.76339i 1.30142 + 5.85716i 3.66985 8.85981i 6.12555 5.14564i 4.27605 + 7.91930i −11.6841 + 7.32766i
5.17 −1.32849 + 1.49503i 1.59078 2.54350i −0.470240 3.97226i 0.758965 3.81558i 1.68928 + 5.75729i −2.88939 + 6.97561i 6.56337 + 4.57408i −3.93881 8.09233i 4.69613 + 6.20363i
5.18 −1.26686 1.54760i 0.233068 2.99093i −0.790124 + 3.92119i 0.0472744 0.237664i −4.92403 + 3.42840i −2.66133 + 6.42502i 7.06940 3.74481i −8.89136 1.39418i −0.427699 + 0.227926i
5.19 −1.15682 + 1.63149i 1.21225 + 2.74417i −1.32354 3.77468i −0.581021 + 2.92099i −5.87944 1.19674i −0.312379 + 0.754149i 7.68947 + 2.20728i −6.06092 + 6.65321i −4.09344 4.32699i
5.20 −0.963600 1.75256i −1.28898 + 2.70897i −2.14295 + 3.37754i −1.30107 + 6.54093i 5.98970 0.351339i 2.26828 5.47612i 7.98429 + 0.501060i −5.67704 6.98364i 12.7171 4.02263i
See next 80 embeddings (of 496 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 173.62 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
64.i even 16 1 inner
192.q odd 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.3.q.a 496
3.b odd 2 1 inner 192.3.q.a 496
64.i even 16 1 inner 192.3.q.a 496
192.q odd 16 1 inner 192.3.q.a 496

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.3.q.a 496 1.a even 1 1 trivial
192.3.q.a 496 3.b odd 2 1 inner
192.3.q.a 496 64.i even 16 1 inner
192.3.q.a 496 192.q odd 16 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(192, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database