# Properties

 Label 189.2.v.b Level 189 Weight 2 Character orbit 189.v Analytic conductor 1.509 Analytic rank 0 Dimension 54 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.50917259820$$ Analytic rank: $$0$$ Dimension: $$54$$ Relative dimension: $$9$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$54q + 3q^{3} - 3q^{5} + 9q^{8} + 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$54q + 3q^{3} - 3q^{5} + 9q^{8} + 3q^{9} - 6q^{11} - 60q^{12} + 9q^{13} - 9q^{15} + 30q^{17} - 3q^{18} + 18q^{20} + 3q^{21} - 9q^{22} + 36q^{24} - 45q^{25} - 54q^{26} - 57q^{27} - 54q^{28} + 30q^{29} + 24q^{30} - 9q^{31} + 51q^{32} - 12q^{33} - 9q^{34} - 12q^{35} + 48q^{36} - 78q^{38} - 36q^{39} + 45q^{40} - 51q^{41} - 12q^{42} - 9q^{43} + 30q^{44} + 51q^{45} - 9q^{47} + 15q^{48} + 126q^{50} - 12q^{51} + 9q^{52} - 60q^{53} - 90q^{54} + 9q^{56} + 39q^{57} - 27q^{58} + 42q^{59} + 135q^{60} + 36q^{62} + 9q^{63} - 27q^{64} - 18q^{65} - 147q^{66} - 27q^{67} - 81q^{68} + 48q^{69} + 75q^{72} + 84q^{74} + 15q^{75} + 54q^{76} - 3q^{77} - 66q^{78} + 72q^{79} - 222q^{80} - 69q^{81} - 54q^{83} - 12q^{84} + 18q^{85} + 66q^{86} + 3q^{87} + 54q^{88} + 90q^{89} + 15q^{90} - 129q^{92} + 21q^{93} + 36q^{94} - 48q^{95} + 36q^{96} + 3q^{98} + 51q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1 −2.12375 + 1.78204i 1.53024 + 0.811393i 0.987362 5.59961i 0.371253 0.135125i −4.69579 + 1.00376i −0.173648 0.984808i 5.10945 + 8.84983i 1.68328 + 2.48326i −0.547652 + 0.948561i
22.2 −1.67142 + 1.40248i −1.72870 0.107685i 0.479372 2.71866i −2.82618 + 1.02865i 3.04040 2.24449i −0.173648 0.984808i 0.829763 + 1.43719i 2.97681 + 0.372309i 3.28107 5.68297i
22.3 −1.18344 + 0.993024i 0.993131 1.41905i 0.0671374 0.380755i 2.98981 1.08820i 0.233835 + 2.66556i −0.173648 0.984808i −1.24623 2.15853i −1.02738 2.81860i −2.45765 + 4.25678i
22.4 −0.461316 + 0.387090i 1.33551 + 1.10291i −0.284323 + 1.61247i 2.68051 0.975627i −1.04302 + 0.00817329i −0.173648 0.984808i −1.09521 1.89697i 0.567181 + 2.94590i −0.858907 + 1.48767i
22.5 −0.246848 + 0.207130i −0.915107 + 1.47057i −0.329265 + 1.86736i −1.61217 + 0.586782i −0.0787071 0.552555i −0.173648 0.984808i −0.627745 1.08729i −1.32516 2.69146i 0.276422 0.478776i
22.6 0.554554 0.465326i −1.73195 0.0184677i −0.256295 + 1.45352i 2.40653 0.875906i −0.969054 + 0.795681i −0.173648 0.984808i 1.25815 + 2.17918i 2.99932 + 0.0639705i 0.926970 1.60556i
22.7 1.19955 1.00654i 1.37519 1.05302i 0.0784988 0.445189i −0.920293 + 0.334959i 0.589706 2.64734i −0.173648 0.984808i 1.21197 + 2.09919i 0.782305 2.89620i −0.766788 + 1.32812i
22.8 1.35330 1.13555i 0.722934 + 1.57397i 0.194644 1.10388i 0.893231 0.325110i 2.76567 + 1.30912i −0.173648 0.984808i 0.776506 + 1.34495i −1.95473 + 2.27575i 0.839631 1.45428i
22.9 1.81332 1.52156i −1.25490 1.19383i 0.625702 3.54853i −0.550279 + 0.200285i −4.09202 0.255396i −0.173648 0.984808i −1.89757 3.28669i 0.149542 + 2.99627i −0.693087 + 1.20046i
43.1 −2.12375 1.78204i 1.53024 0.811393i 0.987362 + 5.59961i 0.371253 + 0.135125i −4.69579 1.00376i −0.173648 + 0.984808i 5.10945 8.84983i 1.68328 2.48326i −0.547652 0.948561i
43.2 −1.67142 1.40248i −1.72870 + 0.107685i 0.479372 + 2.71866i −2.82618 1.02865i 3.04040 + 2.24449i −0.173648 + 0.984808i 0.829763 1.43719i 2.97681 0.372309i 3.28107 + 5.68297i
43.3 −1.18344 0.993024i 0.993131 + 1.41905i 0.0671374 + 0.380755i 2.98981 + 1.08820i 0.233835 2.66556i −0.173648 + 0.984808i −1.24623 + 2.15853i −1.02738 + 2.81860i −2.45765 4.25678i
43.4 −0.461316 0.387090i 1.33551 1.10291i −0.284323 1.61247i 2.68051 + 0.975627i −1.04302 0.00817329i −0.173648 + 0.984808i −1.09521 + 1.89697i 0.567181 2.94590i −0.858907 1.48767i
43.5 −0.246848 0.207130i −0.915107 1.47057i −0.329265 1.86736i −1.61217 0.586782i −0.0787071 + 0.552555i −0.173648 + 0.984808i −0.627745 + 1.08729i −1.32516 + 2.69146i 0.276422 + 0.478776i
43.6 0.554554 + 0.465326i −1.73195 + 0.0184677i −0.256295 1.45352i 2.40653 + 0.875906i −0.969054 0.795681i −0.173648 + 0.984808i 1.25815 2.17918i 2.99932 0.0639705i 0.926970 + 1.60556i
43.7 1.19955 + 1.00654i 1.37519 + 1.05302i 0.0784988 + 0.445189i −0.920293 0.334959i 0.589706 + 2.64734i −0.173648 + 0.984808i 1.21197 2.09919i 0.782305 + 2.89620i −0.766788 1.32812i
43.8 1.35330 + 1.13555i 0.722934 1.57397i 0.194644 + 1.10388i 0.893231 + 0.325110i 2.76567 1.30912i −0.173648 + 0.984808i 0.776506 1.34495i −1.95473 2.27575i 0.839631 + 1.45428i
43.9 1.81332 + 1.52156i −1.25490 + 1.19383i 0.625702 + 3.54853i −0.550279 0.200285i −4.09202 + 0.255396i −0.173648 + 0.984808i −1.89757 + 3.28669i 0.149542 2.99627i −0.693087 1.20046i
85.1 −2.47328 0.900200i −0.969270 + 1.43545i 3.77466 + 3.16732i −0.697772 + 3.95726i 3.68947 2.67773i −0.766044 + 0.642788i −3.85257 6.67284i −1.12103 2.78268i 5.28812 9.15929i
85.2 −1.78358 0.649172i 0.189618 + 1.72164i 1.22766 + 1.03013i 0.611519 3.46809i 0.779441 3.19379i −0.766044 + 0.642788i 0.377143 + 0.653231i −2.92809 + 0.652907i −3.34208 + 5.78866i
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

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e 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.v.b 54
3.b odd 2 1 567.2.v.a 54
27.e even 9 1 inner 189.2.v.b 54
27.e even 9 1 5103.2.a.g 27
27.f odd 18 1 567.2.v.a 54
27.f odd 18 1 5103.2.a.h 27

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.v.b 54 1.a even 1 1 trivial
189.2.v.b 54 27.e even 9 1 inner
567.2.v.a 54 3.b odd 2 1
567.2.v.a 54 27.f odd 18 1
5103.2.a.g 27 27.e even 9 1
5103.2.a.h 27 27.f odd 18 1

## Hecke kernels

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

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database