# Properties

 Label 189.2.bd.a Level 189 Weight 2 Character orbit 189.bd 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.bd (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.50917259820$$ Analytic rank: $$0$$ Dimension: $$132$$ Relative dimension: $$22$$ over $$\Q(\zeta_{18})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$132q - 3q^{2} - 9q^{3} - 3q^{4} - 9q^{5} + 18q^{6} - 6q^{7} - 18q^{8} - 15q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$132q - 3q^{2} - 9q^{3} - 3q^{4} - 9q^{5} + 18q^{6} - 6q^{7} - 18q^{8} - 15q^{9} - 9q^{10} + 9q^{11} - 9q^{12} - 42q^{14} - 24q^{15} - 15q^{16} - 9q^{17} - 3q^{18} - 9q^{19} - 18q^{20} + 15q^{21} - 12q^{22} + 30q^{23} - 36q^{24} - 3q^{25} - 12q^{28} + 6q^{29} - 3q^{30} - 9q^{31} - 51q^{32} - 9q^{33} + 18q^{34} - 9q^{35} - 6q^{37} - 9q^{38} - 9q^{39} - 9q^{40} + 27q^{42} - 12q^{43} - 63q^{45} - 6q^{46} + 45q^{47} + 30q^{49} - 9q^{50} + 33q^{51} - 9q^{52} + 45q^{53} + 117q^{54} - 51q^{56} - 3q^{58} - 9q^{59} - 15q^{60} - 63q^{61} + 99q^{62} - 33q^{63} + 18q^{64} - 102q^{65} + 63q^{66} - 3q^{67} + 144q^{68} - 108q^{69} - 15q^{70} + 18q^{71} + 15q^{72} - 33q^{74} - 9q^{75} - 36q^{76} - 57q^{77} + 66q^{78} - 21q^{79} - 72q^{80} + 57q^{81} - 18q^{82} + 90q^{83} + 51q^{84} + 9q^{85} - 33q^{86} - 9q^{87} + 45q^{88} - 9q^{89} - 81q^{90} - 21q^{91} + 150q^{92} - 87q^{93} - 9q^{94} + 27q^{95} - 9q^{96} - 180q^{98} + 96q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1 −2.68189 0.472890i 0.605371 + 1.62281i 5.08954 + 1.85244i 2.55610 + 0.930344i −0.856127 4.63849i 2.58147 0.579686i −8.05677 4.65158i −2.26705 + 1.96481i −6.41523 3.70384i
47.2 −2.47573 0.436538i −1.50729 + 0.853281i 4.05929 + 1.47746i −2.58032 0.939158i 4.10412 1.45451i −2.62404 + 0.338290i −5.05050 2.91591i 1.54382 2.57228i 5.97819 + 3.45151i
47.3 −2.26715 0.399760i 0.415691 1.68143i 3.10078 + 1.12859i −1.73217 0.630457i −1.61460 + 3.64587i 0.678971 2.55715i −2.59137 1.49613i −2.65440 1.39791i 3.67505 + 2.12179i
47.4 −2.06025 0.363278i −1.46652 0.921586i 2.23328 + 0.812849i 0.338534 + 0.123216i 2.68661 + 2.43145i 1.84499 + 1.89632i −0.682329 0.393943i 1.30136 + 2.70305i −0.652704 0.376839i
47.5 −1.63190 0.287749i 1.35485 + 1.07906i 0.700923 + 0.255115i −3.60525 1.31220i −1.90049 2.15078i 2.03118 + 1.69538i 1.79971 + 1.03907i 0.671243 + 2.92394i 5.50583 + 3.17879i
47.6 −1.57155 0.277106i 1.72554 + 0.149982i 0.513586 + 0.186930i 1.78318 + 0.649025i −2.67021 0.713863i −1.28547 2.31248i 2.00866 + 1.15970i 2.95501 + 0.517603i −2.62250 1.51410i
47.7 −1.38665 0.244504i −0.458387 + 1.67029i −0.0163608 0.00595485i 1.98299 + 0.721749i 1.04402 2.20404i −2.18605 + 1.49037i 2.46004 + 1.42030i −2.57976 1.53128i −2.57325 1.48566i
47.8 −0.910598 0.160563i −1.05237 1.37569i −1.07598 0.391623i −0.473927 0.172495i 0.737399 + 1.42167i −2.46079 + 0.971863i 2.51844 + 1.45402i −0.785044 + 2.89546i 0.403861 + 0.233169i
47.9 −0.877614 0.154747i −0.824649 + 1.52314i −1.13313 0.412424i −1.30288 0.474210i 0.959425 1.20912i 1.81190 1.92796i 2.47415 + 1.42845i −1.63991 2.51211i 1.07004 + 0.617790i
47.10 −0.313923 0.0553531i 1.24200 1.20725i −1.78390 0.649287i −2.68563 0.977491i −0.456717 + 0.310234i −2.48320 + 0.913082i 1.07619 + 0.621337i 0.0851152 2.99879i 0.788975 + 0.455515i
47.11 −0.0147002 0.00259205i −1.71867 + 0.214901i −1.87918 0.683964i 1.54651 + 0.562885i 0.0258218 + 0.00129578i 2.21011 + 1.45445i 0.0517058 + 0.0298524i 2.90764 0.738686i −0.0212751 0.0122832i
47.12 0.0159182 + 0.00280680i −0.402650 1.68460i −1.87914 0.683951i 3.75147 + 1.36542i −0.00168110 0.0279459i 0.157938 2.64103i −0.0559891 0.0323253i −2.67575 + 1.35661i 0.0558840 + 0.0322646i
47.13 0.245063 + 0.0432112i 1.36915 + 1.06086i −1.82120 0.662861i 1.99870 + 0.727467i 0.289688 + 0.319140i 0.302346 + 2.62842i −0.848674 0.489982i 0.749157 + 2.90496i 0.458373 + 0.264642i
47.14 0.882178 + 0.155552i 1.65967 0.495491i −1.12534 0.409592i 0.123522 + 0.0449584i 1.54119 0.178947i 1.69913 2.02805i −2.48059 1.43217i 2.50898 1.64470i 0.101975 + 0.0588754i
47.15 0.892534 + 0.157378i −1.71459 + 0.245341i −1.10754 0.403110i −1.14226 0.415747i −1.56894 0.0508626i −1.98483 1.74942i −2.49483 1.44039i 2.87962 0.841317i −0.954073 0.550834i
47.16 1.10401 + 0.194666i −0.867806 1.49897i −0.698446 0.254214i −3.85196 1.40200i −0.666265 1.82381i 2.61259 + 0.417595i −2.66330 1.53766i −1.49383 + 2.60163i −3.97967 2.29767i
47.17 1.65422 + 0.291684i −0.549790 + 1.64248i 0.771989 + 0.280981i 3.52188 + 1.28186i −1.38856 + 2.55666i −2.17331 1.50888i −1.71431 0.989760i −2.39546 1.80604i 5.45209 + 3.14776i
47.18 1.66985 + 0.294440i 0.685602 1.59058i 0.822331 + 0.299304i 1.39543 + 0.507895i 1.61319 2.45417i −0.356869 + 2.62157i −1.65184 0.953692i −2.05990 2.18101i 2.18062 + 1.25898i
47.19 1.87320 + 0.330296i 0.583821 + 1.63069i 1.52040 + 0.553380i −0.848304 0.308757i 0.555004 + 3.24744i 2.48993 0.894564i −0.629295 0.363324i −2.31831 + 1.90406i −1.48706 0.858555i
47.20 2.22112 + 0.391643i 1.67689 + 0.433617i 2.90060 + 1.05573i −3.20807 1.16764i 3.55476 + 1.61986i −2.62457 + 0.334082i 2.12267 + 1.22552i 2.62395 + 1.45426i −6.66821 3.84989i
See next 80 embeddings (of 132 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 185.22 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.bd even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.bd.a yes 132
3.b odd 2 1 567.2.bd.a 132
7.d odd 6 1 189.2.ba.a 132
21.g even 6 1 567.2.ba.a 132
27.e even 9 1 567.2.ba.a 132
27.f odd 18 1 189.2.ba.a 132
189.z odd 18 1 567.2.bd.a 132
189.bd even 18 1 inner 189.2.bd.a yes 132

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.ba.a 132 7.d odd 6 1
189.2.ba.a 132 27.f odd 18 1
189.2.bd.a yes 132 1.a even 1 1 trivial
189.2.bd.a yes 132 189.bd even 18 1 inner
567.2.ba.a 132 21.g even 6 1
567.2.ba.a 132 27.e even 9 1
567.2.bd.a 132 3.b odd 2 1
567.2.bd.a 132 189.z 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