# Properties

 Label 189.4.g.a Level $189$ Weight $4$ Character orbit 189.g Analytic conductor $11.151$ Analytic rank $0$ Dimension $44$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 189.g (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1513609911$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$22$$ over $$\Q(\zeta_{3})$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$44q - q^{2} - 79q^{4} - 38q^{5} - 7q^{7} + 24q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q - q^{2} - 79q^{4} - 38q^{5} - 7q^{7} + 24q^{8} - 18q^{10} + 10q^{11} - 14q^{13} + 79q^{14} - 247q^{16} + 162q^{17} + 58q^{19} + 362q^{20} - 18q^{22} - 186q^{23} + 698q^{25} + 266q^{26} - 172q^{28} - 248q^{29} + 61q^{31} + 163q^{32} + 6q^{34} - 289q^{35} - 86q^{37} - 1522q^{38} + 36q^{40} + 692q^{41} - 86q^{43} + 443q^{44} - 270q^{46} + 1005q^{47} - 277q^{49} - 239q^{50} + 670q^{52} - 258q^{53} - 870q^{55} - 714q^{56} - 474q^{58} + 1665q^{59} + 439q^{61} - 1812q^{62} + 872q^{64} + 613q^{65} + 295q^{67} - 2748q^{68} - 1044q^{70} - 636q^{71} - 338q^{73} + 2238q^{74} + 1006q^{76} + 2909q^{77} + 133q^{79} + 4817q^{80} + 6q^{82} + 1356q^{83} + 483q^{85} - 6686q^{86} - 738q^{88} + 2200q^{89} + 1552q^{91} + 396q^{92} - 1191q^{94} - 3083q^{95} - 266q^{97} - 3601q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
100.1 −2.54715 4.41178i 0 −8.97590 + 15.5467i −6.84046 0 16.1584 9.05014i 50.6973 0 17.4236 + 30.1786i
100.2 −2.53999 4.39939i 0 −8.90312 + 15.4206i −18.4700 0 −9.68740 + 15.7846i 49.8155 0 46.9136 + 81.2567i
100.3 −2.43564 4.21866i 0 −7.86471 + 13.6221i 6.21080 0 −4.64741 17.9277i 37.6522 0 −15.1273 26.2012i
100.4 −2.15287 3.72888i 0 −5.26968 + 9.12735i 15.9966 0 −16.8821 + 7.61550i 10.9338 0 −34.4385 59.6493i
100.5 −1.68671 2.92146i 0 −1.68996 + 2.92710i 9.74532 0 0.158327 + 18.5196i −15.5854 0 −16.4375 28.4706i
100.6 −1.46729 2.54141i 0 −0.305860 + 0.529765i 3.69711 0 12.3500 + 13.8013i −21.6814 0 −5.42471 9.39588i
100.7 −1.32738 2.29909i 0 0.476130 0.824682i −7.35561 0 16.6607 8.08840i −23.7661 0 9.76368 + 16.9112i
100.8 −1.18941 2.06012i 0 1.17060 2.02754i −18.4675 0 −16.1997 8.97605i −24.5999 0 21.9654 + 38.0453i
100.9 −0.904546 1.56672i 0 2.36359 4.09386i 2.09779 0 −11.3205 14.6576i −23.0246 0 −1.89755 3.28665i
100.10 −0.295387 0.511626i 0 3.82549 6.62595i −10.9845 0 1.43976 + 18.4642i −9.24621 0 3.24467 + 5.61993i
100.11 −0.219258 0.379765i 0 3.90385 6.76167i 16.0932 0 −1.97176 18.4150i −6.93192 0 −3.52855 6.11163i
100.12 0.267129 + 0.462682i 0 3.85728 6.68101i 1.39324 0 16.9882 7.37568i 8.39564 0 0.372176 + 0.644627i
100.13 0.491847 + 0.851904i 0 3.51617 6.09019i 18.7142 0 10.0203 + 15.5754i 14.7872 0 9.20454 + 15.9427i
100.14 0.667800 + 1.15666i 0 3.10809 5.38337i −9.00469 0 −14.2234 + 11.8615i 18.9871 0 −6.01333 10.4154i
100.15 0.724044 + 1.25408i 0 2.95152 5.11218i −4.43275 0 −18.5126 0.531848i 20.1328 0 −3.20951 5.55903i
100.16 1.33146 + 2.30616i 0 0.454424 0.787086i −19.2381 0 18.0786 4.02053i 23.7236 0 −25.6147 44.3660i
100.17 1.67658 + 2.90391i 0 −1.62181 + 2.80905i 8.70652 0 −14.4030 + 11.6428i 15.9489 0 14.5971 + 25.2830i
100.18 1.83489 + 3.17813i 0 −2.73366 + 4.73484i −14.7742 0 0.242329 18.5187i 9.29438 0 −27.1090 46.9542i
100.19 1.93332 + 3.34860i 0 −3.47543 + 6.01962i 13.3562 0 1.72288 18.4399i 4.05663 0 25.8218 + 44.7246i
100.20 2.10738 + 3.65009i 0 −4.88211 + 8.45607i 7.82402 0 16.1743 + 9.02169i −7.43579 0 16.4882 + 28.5584i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 172.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.4.g.a 44
3.b odd 2 1 63.4.g.a 44
7.c even 3 1 189.4.h.a 44
9.c even 3 1 189.4.h.a 44
9.d odd 6 1 63.4.h.a yes 44
21.h odd 6 1 63.4.h.a yes 44
63.g even 3 1 inner 189.4.g.a 44
63.n odd 6 1 63.4.g.a 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.g.a 44 3.b odd 2 1
63.4.g.a 44 63.n odd 6 1
63.4.h.a yes 44 9.d odd 6 1
63.4.h.a yes 44 21.h odd 6 1
189.4.g.a 44 1.a even 1 1 trivial
189.4.g.a 44 63.g even 3 1 inner
189.4.h.a 44 7.c even 3 1
189.4.h.a 44 9.c even 3 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(189, [\chi])$$.