# Properties

 Label 189.2.be.a Level 189 Weight 2 Character orbit 189.be Analytic conductor 1.509 Analytic rank 0 Dimension 132 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 189.be (of order $$18$$ and degree $$6$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.5091725982$$ Analytic rank: $$0$$ Dimension: $$132$$ Relative dimension: $$22$$ over $$\Q(\zeta_{18})$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$132q$$ $$\mathstrut -\mathstrut 12q^{2}$$ $$\mathstrut -\mathstrut 12q^{4}$$ $$\mathstrut -\mathstrut 6q^{7}$$ $$\mathstrut -\mathstrut 18q^{8}$$ $$\mathstrut -\mathstrut 6q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$132q$$ $$\mathstrut -\mathstrut 12q^{2}$$ $$\mathstrut -\mathstrut 12q^{4}$$ $$\mathstrut -\mathstrut 6q^{7}$$ $$\mathstrut -\mathstrut 18q^{8}$$ $$\mathstrut -\mathstrut 6q^{9}$$ $$\mathstrut -\mathstrut 18q^{11}$$ $$\mathstrut +\mathstrut 3q^{14}$$ $$\mathstrut -\mathstrut 24q^{15}$$ $$\mathstrut -\mathstrut 24q^{16}$$ $$\mathstrut -\mathstrut 12q^{18}$$ $$\mathstrut -\mathstrut 12q^{21}$$ $$\mathstrut -\mathstrut 12q^{22}$$ $$\mathstrut +\mathstrut 12q^{23}$$ $$\mathstrut -\mathstrut 12q^{25}$$ $$\mathstrut -\mathstrut 12q^{28}$$ $$\mathstrut -\mathstrut 48q^{29}$$ $$\mathstrut +\mathstrut 42q^{30}$$ $$\mathstrut -\mathstrut 6q^{32}$$ $$\mathstrut -\mathstrut 36q^{35}$$ $$\mathstrut -\mathstrut 36q^{36}$$ $$\mathstrut -\mathstrut 6q^{37}$$ $$\mathstrut -\mathstrut 18q^{39}$$ $$\mathstrut -\mathstrut 12q^{43}$$ $$\mathstrut -\mathstrut 18q^{44}$$ $$\mathstrut -\mathstrut 6q^{46}$$ $$\mathstrut -\mathstrut 24q^{49}$$ $$\mathstrut +\mathstrut 18q^{50}$$ $$\mathstrut +\mathstrut 24q^{51}$$ $$\mathstrut +\mathstrut 57q^{56}$$ $$\mathstrut -\mathstrut 12q^{58}$$ $$\mathstrut -\mathstrut 6q^{60}$$ $$\mathstrut +\mathstrut 21q^{63}$$ $$\mathstrut +\mathstrut 18q^{64}$$ $$\mathstrut +\mathstrut 78q^{65}$$ $$\mathstrut -\mathstrut 12q^{67}$$ $$\mathstrut -\mathstrut 69q^{70}$$ $$\mathstrut +\mathstrut 18q^{71}$$ $$\mathstrut +\mathstrut 114q^{72}$$ $$\mathstrut -\mathstrut 6q^{74}$$ $$\mathstrut -\mathstrut 57q^{77}$$ $$\mathstrut +\mathstrut 12q^{78}$$ $$\mathstrut +\mathstrut 24q^{79}$$ $$\mathstrut -\mathstrut 42q^{81}$$ $$\mathstrut -\mathstrut 48q^{84}$$ $$\mathstrut +\mathstrut 54q^{85}$$ $$\mathstrut -\mathstrut 42q^{86}$$ $$\mathstrut -\mathstrut 72q^{88}$$ $$\mathstrut +\mathstrut 6q^{91}$$ $$\mathstrut -\mathstrut 120q^{92}$$ $$\mathstrut -\mathstrut 60q^{93}$$ $$\mathstrut +\mathstrut 126q^{95}$$ $$\mathstrut +\mathstrut 126q^{98}$$ $$\mathstrut -\mathstrut 192q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
20.1 −0.878867 2.41467i −1.25404 1.19473i −3.52612 + 2.95877i −0.588152 3.33557i −1.78274 + 4.07810i −0.566293 2.58444i 5.79269 + 3.34441i 0.145238 + 2.99648i −7.53740 + 4.35172i
20.2 −0.878867 2.41467i 1.25404 + 1.19473i −3.52612 + 2.95877i 0.588152 + 3.33557i 1.78274 4.07810i −2.64351 0.108907i 5.79269 + 3.34441i 0.145238 + 2.99648i 7.53740 4.35172i
20.3 −0.752422 2.06726i −1.63905 0.559928i −2.17534 + 1.82533i 0.562637 + 3.19087i 0.0757386 + 3.80964i 1.24057 + 2.33688i 1.59981 + 0.923650i 2.37296 + 1.83550i 6.17302 3.56400i
20.4 −0.752422 2.06726i 1.63905 + 0.559928i −2.17534 + 1.82533i −0.562637 3.19087i −0.0757386 3.80964i 2.51680 + 0.815925i 1.59981 + 0.923650i 2.37296 + 1.83550i −6.17302 + 3.56400i
20.5 −0.681310 1.87188i −0.917364 + 1.46916i −1.50768 + 1.26509i −0.291735 1.65451i 3.37511 + 0.716242i −2.47296 + 0.940471i −0.0549773 0.0317412i −1.31689 2.69552i −2.89829 + 1.67333i
20.6 −0.681310 1.87188i 0.917364 1.46916i −1.50768 + 1.26509i 0.291735 + 1.65451i −3.37511 0.716242i 0.496759 2.59870i −0.0549773 0.0317412i −1.31689 2.69552i 2.89829 1.67333i
20.7 −0.422526 1.16088i −1.51095 + 0.846773i 0.362974 0.304571i 0.253991 + 1.44045i 1.62142 + 1.39625i 1.07486 2.41758i −2.64668 1.52806i 1.56595 2.55887i 1.56488 0.903482i
20.8 −0.422526 1.16088i 1.51095 0.846773i 0.362974 0.304571i −0.253991 1.44045i −1.62142 1.39625i −2.19420 + 1.47834i −2.64668 1.52806i 1.56595 2.55887i −1.56488 + 0.903482i
20.9 −0.234777 0.645045i −0.440723 1.67504i 1.17113 0.982692i −0.231854 1.31491i −0.977005 + 0.677547i 2.46223 + 0.968204i −2.09779 1.21116i −2.61153 + 1.47646i −0.793741 + 0.458267i
20.10 −0.234777 0.645045i 0.440723 + 1.67504i 1.17113 0.982692i 0.231854 + 1.31491i 0.977005 0.677547i 1.38106 + 2.25670i −2.09779 1.21116i −2.61153 + 1.47646i 0.793741 0.458267i
20.11 −0.0448460 0.123213i −1.62941 0.587383i 1.51892 1.27452i −0.231324 1.31190i 0.000699206 0.227107i −2.62614 0.321557i −0.452264 0.261115i 2.30996 + 1.91418i −0.151270 + 0.0873359i
20.12 −0.0448460 0.123213i 1.62941 + 0.587383i 1.51892 1.27452i 0.231324 + 1.31190i −0.000699206 0.227107i −0.772696 2.53040i −0.452264 0.261115i 2.30996 + 1.91418i 0.151270 0.0873359i
20.13 0.157445 + 0.432576i −0.888001 + 1.48710i 1.36976 1.14936i −0.725040 4.11191i −0.783093 0.149993i 2.17886 1.50085i 1.51017 + 0.871900i −1.42291 2.64109i 1.66456 0.961033i
20.14 0.157445 + 0.432576i 0.888001 1.48710i 1.36976 1.14936i 0.725040 + 4.11191i 0.783093 + 0.149993i −1.09969 + 2.40638i 1.51017 + 0.871900i −1.42291 2.64109i −1.66456 + 0.961033i
20.15 0.471430 + 1.29524i −1.37746 1.05005i 0.0766803 0.0643424i 0.459149 + 2.60396i 0.710690 2.27917i 1.90985 1.83097i 2.50689 + 1.44736i 0.794802 + 2.89280i −3.15631 + 1.82230i
20.16 0.471430 + 1.29524i 1.37746 + 1.05005i 0.0766803 0.0643424i −0.459149 2.60396i −0.710690 + 2.27917i −1.47152 + 2.19878i 2.50689 + 1.44736i 0.794802 + 2.89280i 3.15631 1.82230i
20.17 0.492726 + 1.35375i −0.333137 + 1.69971i −0.0577819 + 0.0484848i 0.440273 + 2.49691i −2.46514 + 0.386507i −1.60923 2.10009i 2.40115 + 1.38630i −2.77804 1.13247i −3.16327 + 1.82631i
20.18 0.492726 + 1.35375i 0.333137 1.69971i −0.0577819 + 0.0484848i −0.440273 2.49691i 2.46514 0.386507i −2.34762 1.22011i 2.40115 + 1.38630i −2.77804 1.13247i 3.16327 1.82631i
20.19 0.735125 + 2.01974i −1.28358 + 1.16294i −2.00685 + 1.68395i 0.0677816 + 0.384409i −3.29242 1.73760i 0.618456 + 2.57245i −1.15362 0.666045i 0.295160 2.98544i −0.726578 + 0.419490i
20.20 0.735125 + 2.01974i 1.28358 1.16294i −2.00685 + 1.68395i −0.0677816 0.384409i 3.29242 + 1.73760i 2.64076 + 0.162359i −1.15362 0.666045i 0.295160 2.98544i 0.726578 0.419490i
See next 80 embeddings (of 132 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 167.22 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

There are no other newforms in $$S_{2}^{\mathrm{new}}(189, [\chi])$$.