# Properties

 Label 189.2.o.a Level 189 Weight 2 Character orbit 189.o Analytic conductor 1.509 Analytic rank 0 Dimension 12 CM No Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$1.5091725982$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{2}$$ $$+ ( -\beta_{3} - \beta_{5} - \beta_{8} ) q^{4}$$ $$+ \beta_{10} q^{5}$$ $$+ ( \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} ) q^{7}$$ $$+ ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{8} ) q^{8}$$ $$+O(q^{10})$$ $$q$$ $$+ ( \beta_{1} - \beta_{3} - \beta_{5} ) q^{2}$$ $$+ ( -\beta_{3} - \beta_{5} - \beta_{8} ) q^{4}$$ $$+ \beta_{10} q^{5}$$ $$+ ( \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} ) q^{7}$$ $$+ ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{8} ) q^{8}$$ $$+ ( \beta_{2} + \beta_{6} - \beta_{9} - \beta_{11} ) q^{10}$$ $$+ ( \beta_{3} - \beta_{4} + \beta_{5} + \beta_{8} ) q^{11}$$ $$+ ( -\beta_{2} - 2 \beta_{7} - \beta_{10} ) q^{13}$$ $$+ ( 1 - \beta_{1} - \beta_{2} + \beta_{5} + 2 \beta_{9} - \beta_{10} ) q^{14}$$ $$+ ( -\beta_{3} + \beta_{4} + \beta_{8} ) q^{16}$$ $$+ ( \beta_{6} + \beta_{11} ) q^{17}$$ $$+ ( \beta_{2} - \beta_{6} + \beta_{7} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{19}$$ $$+ ( 2 \beta_{2} - \beta_{9} ) q^{20}$$ $$+ ( -1 + 2 \beta_{3} + 3 \beta_{5} + 2 \beta_{8} ) q^{22}$$ $$+ ( -1 - \beta_{1} - \beta_{5} ) q^{23}$$ $$+ ( \beta_{1} + \beta_{3} ) q^{25}$$ $$+ ( -2 \beta_{2} - \beta_{6} - \beta_{7} - 2 \beta_{9} - \beta_{11} ) q^{26}$$ $$+ ( -\beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{28}$$ $$+ ( -4 + \beta_{1} - \beta_{3} + \beta_{5} ) q^{29}$$ $$+ ( 2 \beta_{6} + 2 \beta_{7} + \beta_{10} ) q^{31}$$ $$+ ( 1 + \beta_{1} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} + 2 \beta_{8} ) q^{32}$$ $$+ ( \beta_{7} - \beta_{10} ) q^{34}$$ $$+ ( 1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{11} ) q^{35}$$ $$+ ( -1 + \beta_{1} - \beta_{4} ) q^{37}$$ $$+ ( 4 \beta_{2} + 2 \beta_{6} - 2 \beta_{9} - \beta_{11} ) q^{38}$$ $$+ ( \beta_{2} - 2 \beta_{6} + 2 \beta_{7} + \beta_{10} ) q^{40}$$ $$+ ( -\beta_{2} + 2 \beta_{9} - 2 \beta_{10} ) q^{41}$$ $$+ ( -2 \beta_{1} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{8} ) q^{43}$$ $$+ ( -2 + 3 \beta_{3} + 7 \beta_{5} ) q^{44}$$ $$+ ( -2 - 3 \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{46}$$ $$+ ( -2 \beta_{2} - 2 \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{47}$$ $$+ ( -1 + 2 \beta_{1} - 4 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} ) q^{49}$$ $$+ ( 2 + \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{8} ) q^{50}$$ $$+ ( -\beta_{7} - 5 \beta_{9} + \beta_{10} - \beta_{11} ) q^{52}$$ $$+ ( -1 - 2 \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 4 \beta_{8} ) q^{53}$$ $$+ ( -2 \beta_{2} + \beta_{6} + 2 \beta_{9} - \beta_{11} ) q^{55}$$ $$+ ( -6 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{56}$$ $$+ ( 2 - 2 \beta_{1} + 3 \beta_{3} + \beta_{5} - \beta_{8} ) q^{58}$$ $$+ ( 2 \beta_{2} - \beta_{6} - 4 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{59}$$ $$+ ( \beta_{9} + \beta_{11} ) q^{61}$$ $$+ ( \beta_{2} + \beta_{6} + 2 \beta_{7} + \beta_{9} + \beta_{11} ) q^{62}$$ $$+ ( -2 + 7 \beta_{1} - 4 \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{64}$$ $$+ ( 10 - 2 \beta_{1} + \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{8} ) q^{65}$$ $$+ ( 1 + 2 \beta_{1} - \beta_{3} - 2 \beta_{5} + 3 \beta_{8} ) q^{67}$$ $$+ ( -\beta_{2} + \beta_{6} + 2 \beta_{9} - 2 \beta_{11} ) q^{68}$$ $$+ ( 2 \beta_{1} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{70}$$ $$+ ( 2 + 2 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} - 7 \beta_{5} + 4 \beta_{8} ) q^{71}$$ $$+ ( 3 \beta_{2} + 3 \beta_{6} - \beta_{7} - 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{73}$$ $$+ ( 2 - \beta_{1} + \beta_{3} ) q^{74}$$ $$+ ( 4 \beta_{2} + 2 \beta_{6} + 2 \beta_{7} + \beta_{10} ) q^{76}$$ $$+ ( 1 + \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{77}$$ $$+ ( -\beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{8} ) q^{79}$$ $$+ ( \beta_{6} - \beta_{7} + \beta_{11} ) q^{80}$$ $$+ ( -5 \beta_{2} - 2 \beta_{6} - \beta_{7} + 5 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{82}$$ $$+ ( -4 \beta_{2} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{83}$$ $$+ ( 3 - 6 \beta_{1} + 6 \beta_{3} + 3 \beta_{5} - 6 \beta_{8} ) q^{85}$$ $$+ ( -5 - 3 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - 2 \beta_{8} ) q^{86}$$ $$+ ( \beta_{1} + 2 \beta_{3} - \beta_{4} + 6 \beta_{5} - \beta_{8} ) q^{88}$$ $$+ ( 3 \beta_{2} + 3 \beta_{9} ) q^{89}$$ $$+ ( -5 \beta_{1} + 4 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{10} + 2 \beta_{11} ) q^{91}$$ $$+ ( -2 - 2 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} + 2 \beta_{8} ) q^{92}$$ $$+ ( -2 \beta_{2} + \beta_{6} - 4 \beta_{7} - 2 \beta_{10} ) q^{94}$$ $$+ ( -6 + 3 \beta_{1} + 3 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} - 3 \beta_{8} ) q^{95}$$ $$+ ( -\beta_{7} + \beta_{10} - \beta_{11} ) q^{97}$$ $$+ ( 4 - 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - 9 \beta_{5} - 2 \beta_{7} - 4 \beta_{8} ) q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q$$ $$\mathstrut +\mathstrut 6q^{2}$$ $$\mathstrut +\mathstrut 2q^{4}$$ $$\mathstrut -\mathstrut 2q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$12q$$ $$\mathstrut +\mathstrut 6q^{2}$$ $$\mathstrut +\mathstrut 2q^{4}$$ $$\mathstrut -\mathstrut 2q^{7}$$ $$\mathstrut +\mathstrut 12q^{14}$$ $$\mathstrut +\mathstrut 2q^{16}$$ $$\mathstrut -\mathstrut 10q^{22}$$ $$\mathstrut -\mathstrut 24q^{23}$$ $$\mathstrut -\mathstrut 8q^{28}$$ $$\mathstrut -\mathstrut 30q^{29}$$ $$\mathstrut +\mathstrut 12q^{32}$$ $$\mathstrut -\mathstrut 4q^{37}$$ $$\mathstrut -\mathstrut 10q^{43}$$ $$\mathstrut -\mathstrut 40q^{46}$$ $$\mathstrut +\mathstrut 6q^{49}$$ $$\mathstrut +\mathstrut 36q^{50}$$ $$\mathstrut -\mathstrut 42q^{56}$$ $$\mathstrut +\mathstrut 2q^{58}$$ $$\mathstrut +\mathstrut 16q^{64}$$ $$\mathstrut +\mathstrut 78q^{65}$$ $$\mathstrut +\mathstrut 12q^{67}$$ $$\mathstrut +\mathstrut 18q^{70}$$ $$\mathstrut +\mathstrut 12q^{74}$$ $$\mathstrut +\mathstrut 24q^{77}$$ $$\mathstrut -\mathstrut 6q^{79}$$ $$\mathstrut -\mathstrut 6q^{85}$$ $$\mathstrut -\mathstrut 96q^{86}$$ $$\mathstrut +\mathstrut 34q^{88}$$ $$\mathstrut -\mathstrut 24q^{91}$$ $$\mathstrut -\mathstrut 30q^{92}$$ $$\mathstrut -\mathstrut 72q^{95}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12}\mathstrut -\mathstrut$$ $$7$$ $$x^{10}\mathstrut +\mathstrut$$ $$37$$ $$x^{8}\mathstrut -\mathstrut$$ $$78$$ $$x^{6}\mathstrut +\mathstrut$$ $$123$$ $$x^{4}\mathstrut -\mathstrut$$ $$36$$ $$x^{2}\mathstrut +\mathstrut$$ $$9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-28 \nu^{10} + 148 \nu^{8} + 446 \nu^{6} - 3807 \nu^{4} + 17052 \nu^{2} + 4446$$$$)/12897$$ $$\beta_{2}$$ $$=$$ $$($$$$-49 \nu^{11} + 259 \nu^{9} - 1369 \nu^{7} + 861 \nu^{5} - 252 \nu^{3} - 15864 \nu$$$$)/4299$$ $$\beta_{3}$$ $$=$$ $$($$$$-164 \nu^{10} + 1481 \nu^{8} - 7214 \nu^{6} + 17007 \nu^{4} - 16197 \nu^{2} - 3438$$$$)/12897$$ $$\beta_{4}$$ $$=$$ $$($$$$-175 \nu^{10} + 925 \nu^{8} - 3661 \nu^{6} - 1224 \nu^{4} + 16296 \nu^{2} - 30249$$$$)/12897$$ $$\beta_{5}$$ $$=$$ $$($$$$148 \nu^{10} - 987 \nu^{8} + 5217 \nu^{6} - 10175 \nu^{4} + 17343 \nu^{2} - 777$$$$)/4299$$ $$\beta_{6}$$ $$=$$ $$($$$$70 \nu^{11} - 370 \nu^{9} + 1751 \nu^{7} - 1230 \nu^{5} + 360 \nu^{3} + 8947 \nu$$$$)/1433$$ $$\beta_{7}$$ $$=$$ $$($$$$-730 \nu^{11} + 5701 \nu^{9} - 30748 \nu^{7} + 76698 \nu^{5} - 122898 \nu^{3} + 66168 \nu$$$$)/12897$$ $$\beta_{8}$$ $$=$$ $$($$$$-877 \nu^{10} + 6478 \nu^{8} - 34855 \nu^{6} + 79281 \nu^{4} - 123654 \nu^{2} + 31473$$$$)/12897$$ $$\beta_{9}$$ $$=$$ $$($$$$543 \nu^{11} - 3689 \nu^{9} + 19499 \nu^{7} - 39839 \nu^{5} + 64821 \nu^{3} - 18972 \nu$$$$)/4299$$ $$\beta_{10}$$ $$=$$ $$($$$$2237 \nu^{11} - 15509 \nu^{9} + 81362 \nu^{7} - 167049 \nu^{5} + 249792 \nu^{3} - 42912 \nu$$$$)/12897$$ $$\beta_{11}$$ $$=$$ $$($$$$-890 \nu^{11} + 6342 \nu^{9} - 33522 \nu^{7} + 71935 \nu^{5} - 111438 \nu^{3} + 32616 \nu$$$$)/4299$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-$$$$\beta_{10}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$4$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$8$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{2}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$4$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$12$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$19$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$16$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$16$$ $$\beta_{7}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$-$$$$7$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$16$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$30$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$51$$ $$\nu^{7}$$ $$=$$ $$($$$$67$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$134$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$88$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$23$$ $$\beta_{2}$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$-$$$$67$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$118$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$67$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$104$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$37$$ $$\beta_{1}$$ $$\nu^{9}$$ $$=$$ $$($$$$400$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$578$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$134$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$289$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$400$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$134$$ $$\beta_{2}$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$-$$$$289$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$333$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$645$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$467$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$978$$ $$\nu^{11}$$ $$=$$ $$($$$$1801$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$1267$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$668$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$1267$$ $$\beta_{7}$$$$)/3$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$1 - \beta_{5}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
62.1
 −1.82904 + 1.05600i 1.82904 − 1.05600i 0.474636 − 0.274031i −0.474636 + 0.274031i −1.29589 + 0.748185i 1.29589 − 0.748185i −1.82904 − 1.05600i 1.82904 + 1.05600i 0.474636 + 0.274031i −0.474636 − 0.274031i −1.29589 − 0.748185i 1.29589 + 0.748185i
−1.02704 0.592963i 0 −0.296790 0.514055i −1.41899 2.45776i 0 −2.07253 + 1.64457i 3.07579i 0 3.36562i
62.2 −1.02704 0.592963i 0 −0.296790 0.514055i 1.41899 + 2.45776i 0 −0.387972 + 2.61715i 3.07579i 0 3.36562i
62.3 0.555632 + 0.320794i 0 −0.794182 1.37556i −1.10552 1.91482i 0 −0.906161 2.48573i 2.30225i 0 1.41858i
62.4 0.555632 + 0.320794i 0 −0.794182 1.37556i 1.10552 + 1.91482i 0 2.60579 0.458109i 2.30225i 0 1.41858i
62.5 1.97141 + 1.13819i 0 1.59097 + 2.75564i −0.717144 1.24213i 0 2.16235 + 1.52455i 2.69056i 0 3.26499i
62.6 1.97141 + 1.13819i 0 1.59097 + 2.75564i 0.717144 + 1.24213i 0 −2.40147 1.11037i 2.69056i 0 3.26499i
125.1 −1.02704 + 0.592963i 0 −0.296790 + 0.514055i −1.41899 + 2.45776i 0 −2.07253 1.64457i 3.07579i 0 3.36562i
125.2 −1.02704 + 0.592963i 0 −0.296790 + 0.514055i 1.41899 2.45776i 0 −0.387972 2.61715i 3.07579i 0 3.36562i
125.3 0.555632 0.320794i 0 −0.794182 + 1.37556i −1.10552 + 1.91482i 0 −0.906161 + 2.48573i 2.30225i 0 1.41858i
125.4 0.555632 0.320794i 0 −0.794182 + 1.37556i 1.10552 1.91482i 0 2.60579 + 0.458109i 2.30225i 0 1.41858i
125.5 1.97141 1.13819i 0 1.59097 2.75564i −0.717144 + 1.24213i 0 2.16235 1.52455i 2.69056i 0 3.26499i
125.6 1.97141 1.13819i 0 1.59097 2.75564i 0.717144 1.24213i 0 −2.40147 + 1.11037i 2.69056i 0 3.26499i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 125.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.b Odd 1 yes
9.d Odd 1 yes
63.o Even 1 yes

## Hecke kernels

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