# Properties

 Label 189.4.h.a Level $189$ Weight $4$ Character orbit 189.h 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.h (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.

 $$\operatorname{Tr}(f)(q) =$$ $$44q + 2q^{2} + 158q^{4} + 19q^{5} - 7q^{7} + 24q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q + 2q^{2} + 158q^{4} + 19q^{5} - 7q^{7} + 24q^{8} - 18q^{10} - 5q^{11} - 14q^{13} + 52q^{14} + 494q^{16} + 162q^{17} + 58q^{19} + 362q^{20} - 18q^{22} + 93q^{23} - 349q^{25} + 266q^{26} - 172q^{28} - 248q^{29} - 122q^{31} - 326q^{32} + 6q^{34} - 289q^{35} - 86q^{37} + 761q^{38} - 18q^{40} + 692q^{41} - 86q^{43} + 443q^{44} - 270q^{46} - 2010q^{47} + 317q^{49} - 239q^{50} - 335q^{52} - 258q^{53} - 870q^{55} + 1752q^{56} + 237q^{58} - 3330q^{59} - 878q^{61} - 1812q^{62} + 872q^{64} - 1226q^{65} - 590q^{67} + 1374q^{68} + 1251q^{70} - 636q^{71} - 338q^{73} - 1119q^{74} + 1006q^{76} - 2269q^{77} - 266q^{79} + 4817q^{80} + 6q^{82} + 1356q^{83} + 483q^{85} + 3343q^{86} + 369q^{88} + 2200q^{89} + 1552q^{91} + 396q^{92} + 2382q^{94} + 6166q^{95} - 266q^{97} - 3601q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 −5.43050 0 21.4904 6.55850 + 11.3597i 0 −18.0987 + 3.92907i −73.2595 0 −35.6160 61.6887i
37.2 −5.03185 0 17.3195 0.0751526 + 0.130168i 0 12.4545 13.7072i −46.8942 0 −0.378156 0.654986i
37.3 −4.21476 0 9.76423 −3.91201 6.77580i 0 −15.9002 + 9.49654i −7.43579 0 16.4882 + 28.5584i
37.4 −3.86663 0 6.95086 −6.67810 11.5668i 0 15.1080 + 10.7120i 4.05663 0 25.8218 + 44.7246i
37.5 −3.66978 0 5.46732 7.38708 + 12.7948i 0 15.9165 + 9.46920i 9.29438 0 −27.1090 46.9542i
37.6 −3.35315 0 3.24362 −4.35326 7.54007i 0 −2.88142 18.2947i 15.9489 0 14.5971 + 25.2830i
37.7 −2.66292 0 −0.908849 9.61903 + 16.6607i 0 −5.55741 + 17.6668i 23.7236 0 −25.6147 44.3660i
37.8 −1.44809 0 −5.90304 2.21638 + 3.83887i 0 9.71690 15.7665i 20.1328 0 −3.20951 5.55903i
37.9 −1.33560 0 −6.21617 4.50235 + 7.79829i 0 −3.16069 18.2486i 18.9871 0 −6.01333 10.4154i
37.10 −0.983694 0 −7.03235 −9.35711 16.2070i 0 −18.4989 + 0.890133i 14.7872 0 9.20454 + 15.9427i
37.11 −0.534259 0 −7.71457 −0.696621 1.20658i 0 −2.10659 + 18.4001i 8.39564 0 0.372176 + 0.644627i
37.12 0.438515 0 −7.80770 −8.04659 13.9371i 0 16.9337 + 7.49990i −6.93192 0 −3.52855 6.11163i
37.13 0.590775 0 −7.65099 5.49223 + 9.51282i 0 −16.7104 7.98524i −9.24621 0 3.24467 + 5.61993i
37.14 1.80909 0 −4.72719 −1.04890 1.81674i 0 18.3541 2.47507i −23.0246 0 −1.89755 3.28665i
37.15 2.37882 0 −2.34120 9.23374 + 15.9933i 0 15.8733 9.54133i −24.5999 0 21.9654 + 38.0453i
37.16 2.65476 0 −0.952261 3.67781 + 6.37015i 0 −1.32557 + 18.4728i −23.7661 0 9.76368 + 16.9112i
37.17 2.93457 0 0.611720 −1.84855 3.20179i 0 −18.1273 + 3.79476i −21.6814 0 −5.42471 9.39588i
37.18 3.37342 0 3.37993 −4.87266 8.43970i 0 −16.1176 9.12268i −15.5854 0 −16.4375 28.4706i
37.19 4.30573 0 10.5394 −7.99829 13.8535i 0 1.84582 18.4280i 10.9338 0 −34.4385 59.6493i
37.20 4.87128 0 15.7294 −3.10540 5.37871i 0 17.8495 + 4.93906i 37.6522 0 −15.1273 26.2012i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 46.22 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.h 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.h.a 44
3.b odd 2 1 63.4.h.a yes 44
7.c even 3 1 189.4.g.a 44
9.c even 3 1 189.4.g.a 44
9.d odd 6 1 63.4.g.a 44
21.h odd 6 1 63.4.g.a 44
63.h even 3 1 inner 189.4.h.a 44
63.j odd 6 1 63.4.h.a yes 44

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

## Hecke kernels

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