# Properties

 Label 189.4.s.a Level $189$ Weight $4$ Character orbit 189.s 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.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1513609911$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$22$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$44q + 3q^{2} + 81q^{4} + 6q^{5} + 5q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q + 3q^{2} + 81q^{4} + 6q^{5} + 5q^{7} - 6q^{10} + 36q^{13} - 129q^{14} - 263q^{16} - 72q^{17} - 6q^{19} + 24q^{20} + 14q^{22} + 698q^{25} - 96q^{26} - 156q^{28} + 132q^{29} + 177q^{31} + 501q^{32} - 24q^{34} + 765q^{35} + 82q^{37} + 1746q^{38} + 618q^{41} + 82q^{43} + 603q^{44} + 266q^{46} + 201q^{47} + 515q^{49} + 1845q^{50} + 564q^{53} - 3600q^{56} - 538q^{58} - 747q^{59} - 1209q^{61} - 2904q^{62} - 1144q^{64} + 831q^{65} + 295q^{67} - 7008q^{68} - 390q^{70} - 6q^{73} + 144q^{76} + 1203q^{77} - 551q^{79} - 4239q^{80} + 18q^{82} + 1830q^{83} - 237q^{85} + 1246q^{88} + 4266q^{89} - 1140q^{91} - 7896q^{92} - 3q^{94} + 1053q^{95} + 792q^{97} + 5667q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 −4.66014 2.69053i 0 10.4779 + 18.1483i 11.1598 0 18.4983 + 0.902321i 69.7163i 0 −52.0063 30.0259i
17.2 −4.39687 2.53853i 0 8.88829 + 15.3950i −8.42670 0 −13.4514 12.7303i 49.6363i 0 37.0511 + 21.3915i
17.3 −3.68213 2.12588i 0 5.03872 + 8.72732i 2.97507 0 −4.05014 18.0720i 8.83278i 0 −10.9546 6.32464i
17.4 −3.55689 2.05357i 0 4.43430 + 7.68044i 7.61183 0 4.76690 + 17.8963i 3.56749i 0 −27.0744 15.6314i
17.5 −3.22215 1.86031i 0 2.92152 + 5.06021i −13.6667 0 −10.0995 + 15.5242i 8.02526i 0 44.0363 + 25.4243i
17.6 −2.52419 1.45734i 0 0.247680 + 0.428994i −17.2113 0 17.7231 + 5.37495i 21.8736i 0 43.4446 + 25.0828i
17.7 −2.09278 1.20827i 0 −1.08019 1.87094i 10.7193 0 −17.6173 5.71220i 24.5529i 0 −22.4331 12.9518i
17.8 −1.57448 0.909026i 0 −2.34735 4.06572i 10.3247 0 15.1453 10.6593i 23.0796i 0 −16.2560 9.38543i
17.9 −0.958607 0.553452i 0 −3.38738 5.86712i −12.4738 0 −18.2811 + 2.96677i 16.3542i 0 11.9574 + 6.90363i
17.10 −0.725355 0.418784i 0 −3.64924 6.32067i 21.9169 0 2.19637 + 18.3896i 12.8135i 0 −15.8975 9.17842i
17.11 −0.647627 0.373907i 0 −3.72039 6.44390i −8.70220 0 18.2993 2.85258i 11.5468i 0 5.63577 + 3.25382i
17.12 0.223110 + 0.128812i 0 −3.96681 6.87072i −6.38772 0 −1.54394 18.4558i 4.10490i 0 −1.42516 0.822818i
17.13 0.998155 + 0.576285i 0 −3.33579 5.77776i −0.274718 0 −9.15344 + 16.1001i 16.9100i 0 −0.274212 0.158316i
17.14 1.54833 + 0.893930i 0 −2.40178 4.16000i 16.9434 0 −9.83512 15.6930i 22.8910i 0 26.2340 + 15.1462i
17.15 1.59189 + 0.919076i 0 −2.31060 4.00207i −0.414554 0 12.7471 + 13.4355i 23.1997i 0 −0.659923 0.381007i
17.16 2.65116 + 1.53065i 0 0.685763 + 1.18778i −5.50223 0 −18.4916 1.02989i 20.2917i 0 −14.5873 8.42197i
17.17 3.00186 + 1.73312i 0 2.00744 + 3.47699i −18.0540 0 12.2090 13.9263i 13.8134i 0 −54.1956 31.2898i
17.18 3.16085 + 1.82492i 0 2.66064 + 4.60836i 12.2314 0 4.78838 17.8905i 9.77690i 0 38.6615 + 22.3212i
17.19 3.22249 + 1.86051i 0 2.92296 + 5.06272i −2.66012 0 9.53109 + 15.8795i 8.01534i 0 −8.57221 4.94917i
17.20 4.26829 + 2.46430i 0 8.14552 + 14.1085i 9.24869 0 17.8986 4.75801i 40.8632i 0 39.4761 + 22.7915i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 89.22 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.s even 6 1 inner

## Twists

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

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

## Hecke kernels

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