# 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

# 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.

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

## 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 −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

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