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

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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:

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 \(S_{2}^{\mathrm{new}}(189, [\chi])\).