# Properties

 Label 189.2.w.a Level 189 Weight 2 Character orbit 189.w Analytic conductor 1.509 Analytic rank 0 Dimension 132 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 189.w (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.50917259820$$ Analytic rank: $$0$$ Dimension: $$132$$ Relative dimension: $$22$$ over $$\Q(\zeta_{9})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$132q - 3q^{2} - 3q^{3} - 3q^{4} - 3q^{5} - 18q^{6} - 6q^{7} - 6q^{8} + 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$132q - 3q^{2} - 3q^{3} - 3q^{4} - 3q^{5} - 18q^{6} - 6q^{7} - 6q^{8} + 3q^{9} - 6q^{10} + 3q^{11} - 3q^{12} - 12q^{13} + 15q^{14} - 9q^{16} - 54q^{17} - 3q^{18} - 6q^{19} - 18q^{20} - 21q^{21} - 12q^{22} + 45q^{24} - 3q^{25} + 30q^{26} - 12q^{27} - 12q^{28} - 30q^{29} - 57q^{30} - 3q^{31} + 51q^{32} + 15q^{33} - 18q^{34} - 12q^{35} - 60q^{36} + 3q^{37} - 57q^{38} - 66q^{39} - 66q^{40} + 33q^{42} - 12q^{43} + 3q^{44} + 33q^{45} + 3q^{46} - 21q^{47} + 90q^{48} + 12q^{49} - 39q^{50} - 48q^{51} + 9q^{52} + 9q^{53} - 63q^{54} - 24q^{55} + 57q^{56} - 18q^{57} - 3q^{58} - 18q^{59} + 81q^{60} + 33q^{61} + 75q^{62} + 63q^{63} - 30q^{64} + 81q^{65} + 69q^{66} - 3q^{67} + 6q^{68} - 6q^{69} - 42q^{70} - 18q^{71} - 105q^{72} + 21q^{73} - 93q^{74} + 18q^{75} - 24q^{76} + 87q^{77} - 30q^{78} + 15q^{79} + 102q^{80} + 39q^{81} - 6q^{82} - 42q^{83} - 36q^{84} - 63q^{85} + 159q^{86} + 30q^{87} + 57q^{88} - 150q^{89} - 39q^{90} + 6q^{91} - 66q^{92} - 27q^{93} + 33q^{94} - 147q^{95} + 81q^{96} - 12q^{97} + 99q^{98} + 24q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1 −2.58171 + 0.939667i −1.71864 + 0.215106i 4.25019 3.56633i 0.335317 + 0.122045i 4.23491 2.17029i −0.921726 + 2.48000i −4.87420 + 8.44237i 2.90746 0.739379i −0.980375
25.2 −2.28724 + 0.832486i 1.36942 1.06052i 3.00633 2.52261i 4.14290 + 1.50789i −2.24932 + 3.56567i −1.33708 2.28303i −2.34212 + 4.05668i 0.750612 2.90458i −10.7311
25.3 −2.11851 + 0.771075i 1.70797 + 0.287798i 2.36145 1.98149i −1.51793 0.552482i −3.84028 + 0.707271i 2.02397 + 1.70398i −1.22040 + 2.11380i 2.83434 + 0.983104i 3.64176
25.4 −2.06527 + 0.751697i −0.131292 + 1.72707i 2.16820 1.81934i −1.77012 0.644271i −1.02708 3.66555i −0.610341 2.57439i −0.912517 + 1.58053i −2.96552 0.453502i 4.14007
25.5 −1.47150 + 0.535583i 0.0586596 + 1.73106i 0.346382 0.290649i 3.26769 + 1.18934i −1.01344 2.51584i −0.172863 + 2.64010i 1.21191 2.09908i −2.99312 + 0.203086i −5.44540
25.6 −1.36957 + 0.498483i 0.717145 1.57661i 0.0951481 0.0798388i −1.88092 0.684598i −0.196268 + 2.51676i −1.86025 + 1.88135i 1.36695 2.36763i −1.97140 2.26132i 2.91731
25.7 −1.30907 + 0.476464i −1.39125 1.03171i −0.0454330 + 0.0381228i 1.40139 + 0.510065i 2.31282 + 0.687701i −2.64009 0.173054i 1.43440 2.48445i 0.871157 + 2.87073i −2.07755
25.8 −0.928908 + 0.338095i −1.69366 + 0.362654i −0.783527 + 0.657458i −3.08847 1.12411i 1.45064 0.909490i 2.54716 + 0.715520i 1.49406 2.58780i 2.73696 1.22843i 3.24896
25.9 −0.606828 + 0.220867i 1.44289 + 0.958162i −1.21263 + 1.01752i 1.46866 + 0.534550i −1.08721 0.262753i 1.88474 1.85681i 1.15690 2.00380i 1.16385 + 2.76504i −1.00929
25.10 −0.332947 + 0.121183i −1.35881 + 1.07408i −1.43592 + 1.20488i 0.672463 + 0.244756i 0.322250 0.522275i −2.22176 1.43659i 0.686389 1.18886i 0.692708 2.91893i −0.253554
25.11 −0.328890 + 0.119706i 0.434559 1.67665i −1.43825 + 1.20683i 2.13988 + 0.778854i 0.0577835 + 0.603453i 2.62809 + 0.305198i 0.678558 1.17530i −2.62232 1.45721i −0.797020
25.12 0.0303128 0.0110329i 0.919170 + 1.46804i −1.53129 + 1.28491i −3.70820 1.34968i 0.0440593 + 0.0343591i −2.47092 + 0.945798i −0.0644996 + 0.111717i −1.31025 + 2.69875i −0.127297
25.13 0.456477 0.166144i 1.72709 0.131048i −1.35132 + 1.13389i 1.44421 + 0.525651i 0.766603 0.346766i −1.09247 + 2.40967i −0.914231 + 1.58349i 2.96565 0.452662i 0.746585
25.14 0.562011 0.204555i −0.692899 1.58742i −1.25807 + 1.05565i −2.61250 0.950872i −0.714132 0.750410i −0.229775 2.63575i −1.08919 + 1.88654i −2.03978 + 2.19984i −1.66276
25.15 1.00414 0.365478i −0.803391 + 1.53446i −0.657360 + 0.551590i 0.354797 + 0.129136i −0.245908 + 1.83444i 1.55771 + 2.13859i −1.52708 + 2.64497i −1.70913 2.46554i 0.403463
25.16 1.17974 0.429388i −1.72810 0.116945i −0.324689 + 0.272446i 3.35074 + 1.21957i −2.08891 + 0.604061i 1.13056 2.39204i −1.52151 + 2.63533i 2.97265 + 0.404186i 4.47665
25.17 1.45469 0.529465i 1.56527 0.741574i 0.303710 0.254843i 0.0676746 + 0.0246315i 1.88435 1.90752i −1.36843 2.26438i −1.24118 + 2.14978i 1.90014 2.32153i 0.111487
25.18 1.68235 0.612324i 1.10447 + 1.33422i 0.923259 0.774706i −1.38133 0.502765i 2.67508 + 1.56832i 2.29659 1.31364i −0.711445 + 1.23226i −0.560272 + 2.94722i −2.63174
25.19 1.90772 0.694353i 0.482660 1.66344i 1.62518 1.36369i −2.38896 0.869509i −0.234236 3.50852i 1.31108 + 2.29806i 0.123353 0.213653i −2.53408 1.60575i −5.16121
25.20 2.03199 0.739583i −0.528499 1.64945i 2.04991 1.72008i 3.15692 + 1.14902i −2.29381 2.96080i −2.22776 + 1.42726i 0.730849 1.26587i −2.44138 + 1.74347i 7.26462
See next 80 embeddings (of 132 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 184.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.w even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.w.a yes 132
3.b odd 2 1 567.2.w.a 132
7.c even 3 1 189.2.u.a 132
21.h odd 6 1 567.2.u.a 132
27.e even 9 1 189.2.u.a 132
27.f odd 18 1 567.2.u.a 132
189.w even 9 1 inner 189.2.w.a yes 132
189.bf odd 18 1 567.2.w.a 132

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.u.a 132 7.c even 3 1
189.2.u.a 132 27.e even 9 1
189.2.w.a yes 132 1.a even 1 1 trivial
189.2.w.a yes 132 189.w even 9 1 inner
567.2.u.a 132 21.h odd 6 1
567.2.u.a 132 27.f odd 18 1
567.2.w.a 132 3.b odd 2 1
567.2.w.a 132 189.bf odd 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(189, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database