# Properties

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

 $$\operatorname{Tr}(f)(q) =$$ $$44q - 162q^{4} + 3q^{5} + 5q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q - 162q^{4} + 3q^{5} + 5q^{7} - 6q^{10} - 9q^{11} - 36q^{13} - 54q^{14} + 526q^{16} + 72q^{17} - 6q^{19} - 24q^{20} + 14q^{22} + 285q^{23} - 349q^{25} + 96q^{26} - 156q^{28} + 132q^{29} + 24q^{34} - 765q^{35} + 82q^{37} + 873q^{38} + 420q^{40} - 618q^{41} + 82q^{43} - 603q^{44} + 266q^{46} + 402q^{47} - 79q^{49} + 1845q^{50} + 189q^{52} - 564q^{53} - 66q^{56} + 269q^{58} - 1494q^{59} + 2904q^{62} - 1144q^{64} - 590q^{67} - 3504q^{68} - 105q^{70} - 6q^{73} - 1515q^{74} - 144q^{76} + 4443q^{77} + 1102q^{79} + 4239q^{80} + 18q^{82} - 1830q^{83} - 237q^{85} - 1209q^{86} - 623q^{88} - 4266q^{89} - 1140q^{91} - 7896q^{92} - 792q^{97} - 5667q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1 5.38106i 0 −20.9559 5.57991 9.66469i 0 −8.46770 16.4711i 69.7163i 0 −52.0063 30.0259i
143.2 5.07706i 0 −17.7766 −4.21335 + 7.29774i 0 −4.29909 + 18.0144i 49.6363i 0 37.0511 + 21.3915i
143.3 4.25176i 0 −10.0774 1.48754 2.57649i 0 −13.6257 + 12.5435i 8.83278i 0 −10.9546 6.32464i
143.4 4.10714i 0 −8.86861 3.80591 6.59204i 0 13.1152 13.0764i 3.56749i 0 −27.0744 15.6314i
143.5 3.72062i 0 −5.84303 −6.83336 + 11.8357i 0 18.4941 + 0.984293i 8.02526i 0 44.0363 + 25.4243i
143.6 2.91468i 0 −0.495360 −8.60567 + 14.9055i 0 −4.20673 18.0362i 21.8736i 0 43.4446 + 25.0828i
143.7 2.41653i 0 2.16037 5.35965 9.28319i 0 3.86176 + 18.1132i 24.5529i 0 −22.4331 12.9518i
143.8 1.81805i 0 4.69469 5.16236 8.94146i 0 −16.8039 7.78656i 23.0796i 0 −16.2560 9.38543i
143.9 1.10690i 0 6.77476 −6.23688 + 10.8026i 0 11.7098 + 14.3485i 16.3542i 0 11.9574 + 6.90363i
143.10 0.837567i 0 7.29848 10.9584 18.9806i 0 14.8276 11.0969i 12.8135i 0 −15.8975 9.17842i
143.11 0.747815i 0 7.44077 −4.35110 + 7.53632i 0 −11.6200 14.4213i 11.5468i 0 5.63577 + 3.25382i
143.12 0.257625i 0 7.93363 −3.19386 + 5.53193i 0 −15.2112 + 10.5650i 4.10490i 0 −1.42516 0.822818i
143.13 1.15257i 0 6.67158 −0.137359 + 0.237913i 0 18.5199 0.122959i 16.9100i 0 −0.274212 0.158316i
143.14 1.78786i 0 4.80356 8.47168 14.6734i 0 −8.67298 + 16.3640i 22.8910i 0 26.2340 + 15.1462i
143.15 1.83815i 0 4.62119 −0.207277 + 0.359014i 0 5.26194 17.7570i 23.1997i 0 −0.659923 0.381007i
143.16 3.06129i 0 −1.37153 −2.75111 + 4.76507i 0 8.35389 + 16.5291i 20.2917i 0 −14.5873 8.42197i
143.17 3.46625i 0 −4.01488 −9.02701 + 15.6352i 0 −18.1650 3.61017i 13.8134i 0 −54.1956 31.2898i
143.18 3.64983i 0 −5.32127 6.11568 10.5927i 0 −17.8879 + 4.79841i 9.77690i 0 38.6615 + 22.3212i
143.19 3.72101i 0 −5.84592 −1.33006 + 2.30373i 0 8.98650 16.1939i 8.01534i 0 −8.57221 4.94917i
143.20 4.92859i 0 −16.2910 4.62434 8.00960i 0 −13.0699 13.1217i 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 152.22 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.i 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.i.a 44
3.b odd 2 1 63.4.i.a 44
7.d odd 6 1 189.4.s.a 44
9.c even 3 1 63.4.s.a yes 44
9.d odd 6 1 189.4.s.a 44
21.g even 6 1 63.4.s.a yes 44
63.i even 6 1 inner 189.4.i.a 44
63.t odd 6 1 63.4.i.a 44

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

## Hecke kernels

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