# Properties

 Label 189.2.v.a Level 189 Weight 2 Character orbit 189.v Analytic conductor 1.509 Analytic rank 0 Dimension 54 CM No

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$1.5091725982$$ Analytic rank: $$0$$ Dimension: $$54$$ Relative dimension: $$9$$ over $$\Q(\zeta_{9})$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$54q$$ $$\mathstrut -\mathstrut 3q^{3}$$ $$\mathstrut -\mathstrut 3q^{5}$$ $$\mathstrut -\mathstrut 27q^{8}$$ $$\mathstrut -\mathstrut 9q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$54q$$ $$\mathstrut -\mathstrut 3q^{3}$$ $$\mathstrut -\mathstrut 3q^{5}$$ $$\mathstrut -\mathstrut 27q^{8}$$ $$\mathstrut -\mathstrut 9q^{9}$$ $$\mathstrut -\mathstrut 6q^{11}$$ $$\mathstrut +\mathstrut 24q^{12}$$ $$\mathstrut -\mathstrut 9q^{13}$$ $$\mathstrut -\mathstrut 9q^{15}$$ $$\mathstrut -\mathstrut 30q^{17}$$ $$\mathstrut -\mathstrut 27q^{18}$$ $$\mathstrut -\mathstrut 12q^{20}$$ $$\mathstrut -\mathstrut 3q^{21}$$ $$\mathstrut -\mathstrut 9q^{22}$$ $$\mathstrut -\mathstrut 12q^{23}$$ $$\mathstrut +\mathstrut 36q^{24}$$ $$\mathstrut +\mathstrut 27q^{25}$$ $$\mathstrut +\mathstrut 18q^{26}$$ $$\mathstrut +\mathstrut 63q^{27}$$ $$\mathstrut +\mathstrut 54q^{28}$$ $$\mathstrut +\mathstrut 6q^{29}$$ $$\mathstrut -\mathstrut 72q^{30}$$ $$\mathstrut -\mathstrut 9q^{31}$$ $$\mathstrut -\mathstrut 9q^{32}$$ $$\mathstrut -\mathstrut 36q^{33}$$ $$\mathstrut -\mathstrut 9q^{34}$$ $$\mathstrut -\mathstrut 12q^{35}$$ $$\mathstrut +\mathstrut 54q^{38}$$ $$\mathstrut +\mathstrut 12q^{39}$$ $$\mathstrut -\mathstrut 45q^{40}$$ $$\mathstrut -\mathstrut 15q^{41}$$ $$\mathstrut -\mathstrut 18q^{42}$$ $$\mathstrut -\mathstrut 9q^{43}$$ $$\mathstrut -\mathstrut 42q^{44}$$ $$\mathstrut -\mathstrut 9q^{45}$$ $$\mathstrut -\mathstrut 45q^{47}$$ $$\mathstrut -\mathstrut 93q^{48}$$ $$\mathstrut +\mathstrut 18q^{50}$$ $$\mathstrut +\mathstrut 72q^{51}$$ $$\mathstrut -\mathstrut 63q^{52}$$ $$\mathstrut +\mathstrut 132q^{53}$$ $$\mathstrut +\mathstrut 54q^{54}$$ $$\mathstrut -\mathstrut 9q^{56}$$ $$\mathstrut +\mathstrut 3q^{57}$$ $$\mathstrut -\mathstrut 27q^{58}$$ $$\mathstrut -\mathstrut 9q^{60}$$ $$\mathstrut -\mathstrut 36q^{62}$$ $$\mathstrut -\mathstrut 9q^{63}$$ $$\mathstrut -\mathstrut 27q^{64}$$ $$\mathstrut +\mathstrut 66q^{65}$$ $$\mathstrut +\mathstrut 153q^{66}$$ $$\mathstrut +\mathstrut 45q^{67}$$ $$\mathstrut +\mathstrut 87q^{68}$$ $$\mathstrut -\mathstrut 72q^{71}$$ $$\mathstrut -\mathstrut 45q^{72}$$ $$\mathstrut -\mathstrut 72q^{74}$$ $$\mathstrut -\mathstrut 39q^{75}$$ $$\mathstrut +\mathstrut 54q^{76}$$ $$\mathstrut +\mathstrut 3q^{77}$$ $$\mathstrut -\mathstrut 54q^{78}$$ $$\mathstrut -\mathstrut 36q^{79}$$ $$\mathstrut +\mathstrut 42q^{80}$$ $$\mathstrut +\mathstrut 27q^{81}$$ $$\mathstrut +\mathstrut 24q^{83}$$ $$\mathstrut -\mathstrut 12q^{84}$$ $$\mathstrut +\mathstrut 18q^{85}$$ $$\mathstrut -\mathstrut 90q^{86}$$ $$\mathstrut -\mathstrut 99q^{87}$$ $$\mathstrut +\mathstrut 54q^{88}$$ $$\mathstrut -\mathstrut 42q^{89}$$ $$\mathstrut -\mathstrut 9q^{90}$$ $$\mathstrut +\mathstrut 87q^{92}$$ $$\mathstrut +\mathstrut 93q^{93}$$ $$\mathstrut -\mathstrut 90q^{94}$$ $$\mathstrut +\mathstrut 12q^{95}$$ $$\mathstrut +\mathstrut 108q^{96}$$ $$\mathstrut -\mathstrut 18q^{97}$$ $$\mathstrut -\mathstrut 9q^{98}$$ $$\mathstrut -\mathstrut 117q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1 −1.88664 + 1.58308i 0.279041 1.70943i 0.705975 4.00378i −1.28676 + 0.468343i 2.17970 + 3.66681i 0.173648 + 0.984808i 2.54355 + 4.40555i −2.84427 0.954001i 1.68623 2.92064i
22.2 −1.17963 + 0.989828i 0.853595 + 1.50711i 0.0644735 0.365648i −1.99174 + 0.724932i −2.49871 0.932919i 0.173648 + 0.984808i −1.25403 2.17204i −1.54275 + 2.57292i 1.63195 2.82663i
22.3 −1.09798 + 0.921318i −1.39977 1.02012i 0.00944557 0.0535685i 1.35272 0.492351i 2.47678 0.169559i 0.173648 + 0.984808i −1.39433 2.41506i 0.918713 + 2.85587i −1.03166 + 1.78688i
22.4 0.0808704 0.0678583i 1.71585 0.236354i −0.345361 + 1.95864i 0.0449304 0.0163533i 0.122723 0.135549i 0.173648 + 0.984808i 0.210549 + 0.364682i 2.88827 0.811094i 0.00252383 0.00437140i
22.5 0.731503 0.613803i −1.57520 0.720230i −0.188955 + 1.07162i −3.78665 + 1.37823i −1.59435 + 0.440016i 0.173648 + 0.984808i 1.47445 + 2.55382i 1.96254 + 2.26902i −1.92398 + 3.33244i
22.6 0.786541 0.659987i −0.244375 1.71472i −0.164231 + 0.931402i 3.56827 1.29874i −1.32391 1.18742i 0.173648 + 0.984808i 1.51229 + 2.61937i −2.88056 + 0.838073i 1.94944 3.37653i
22.7 0.969537 0.813538i −0.430185 + 1.67778i −0.0691389 + 0.392106i −0.855069 + 0.311220i 0.947857 + 1.97664i 0.173648 + 0.984808i 1.51760 + 2.62856i −2.62988 1.44351i −0.575832 + 0.997370i
22.8 1.89179 1.58740i −1.45321 + 0.942429i 0.711733 4.03644i 2.16637 0.788496i −1.25316 + 4.08971i 0.173648 + 0.984808i −2.59144 4.48850i 1.22366 2.73910i 2.84667 4.93057i
22.9 2.00215 1.68000i 1.58061 + 0.708280i 0.838893 4.75760i −3.29720 + 1.20008i 4.35453 1.23735i 0.173648 + 0.984808i −3.69957 6.40784i 1.99668 + 2.23903i −4.58534 + 7.94204i
43.1 −1.88664 1.58308i 0.279041 + 1.70943i 0.705975 + 4.00378i −1.28676 0.468343i 2.17970 3.66681i 0.173648 0.984808i 2.54355 4.40555i −2.84427 + 0.954001i 1.68623 + 2.92064i
43.2 −1.17963 0.989828i 0.853595 1.50711i 0.0644735 + 0.365648i −1.99174 0.724932i −2.49871 + 0.932919i 0.173648 0.984808i −1.25403 + 2.17204i −1.54275 2.57292i 1.63195 + 2.82663i
43.3 −1.09798 0.921318i −1.39977 + 1.02012i 0.00944557 + 0.0535685i 1.35272 + 0.492351i 2.47678 + 0.169559i 0.173648 0.984808i −1.39433 + 2.41506i 0.918713 2.85587i −1.03166 1.78688i
43.4 0.0808704 + 0.0678583i 1.71585 + 0.236354i −0.345361 1.95864i 0.0449304 + 0.0163533i 0.122723 + 0.135549i 0.173648 0.984808i 0.210549 0.364682i 2.88827 + 0.811094i 0.00252383 + 0.00437140i
43.5 0.731503 + 0.613803i −1.57520 + 0.720230i −0.188955 1.07162i −3.78665 1.37823i −1.59435 0.440016i 0.173648 0.984808i 1.47445 2.55382i 1.96254 2.26902i −1.92398 3.33244i
43.6 0.786541 + 0.659987i −0.244375 + 1.71472i −0.164231 0.931402i 3.56827 + 1.29874i −1.32391 + 1.18742i 0.173648 0.984808i 1.51229 2.61937i −2.88056 0.838073i 1.94944 + 3.37653i
43.7 0.969537 + 0.813538i −0.430185 1.67778i −0.0691389 0.392106i −0.855069 0.311220i 0.947857 1.97664i 0.173648 0.984808i 1.51760 2.62856i −2.62988 + 1.44351i −0.575832 0.997370i
43.8 1.89179 + 1.58740i −1.45321 0.942429i 0.711733 + 4.03644i 2.16637 + 0.788496i −1.25316 4.08971i 0.173648 0.984808i −2.59144 + 4.48850i 1.22366 + 2.73910i 2.84667 + 4.93057i
43.9 2.00215 + 1.68000i 1.58061 0.708280i 0.838893 + 4.75760i −3.29720 1.20008i 4.35453 + 1.23735i 0.173648 0.984808i −3.69957 + 6.40784i 1.99668 2.23903i −4.58534 7.94204i
85.1 −2.52562 0.919249i 1.69786 + 0.342450i 4.00163 + 3.35776i 0.203200 1.15240i −3.97334 2.42565i 0.766044 0.642788i −4.33224 7.50367i 2.76546 + 1.16287i −1.57255 + 2.72374i
85.2 −2.17921 0.793168i −1.72903 0.102202i 2.58775 + 2.17138i 0.443161 2.51329i 3.68686 + 1.59413i 0.766044 0.642788i −1.59792 2.76768i 2.97911 + 0.353421i −2.95920 + 5.12549i
See all 54 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 169.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not have CM; other inner twists have not been computed.

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{54} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.