# Properties

 Label 189.2.f.b Level 189 Weight 2 Character orbit 189.f Analytic conductor 1.509 Analytic rank 0 Dimension 6 CM No Inner twists 2

# Learn more about

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$1.5091725982$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{18})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{2}$$ $$+ ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{4}$$ $$+ ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{5}$$ $$-\beta_{1} q^{7}$$ $$+ ( -2 - \beta_{3} + 2 \beta_{4} ) q^{8}$$ $$+O(q^{10})$$ $$q$$ $$+ ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{2}$$ $$+ ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{4}$$ $$+ ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{5}$$ $$-\beta_{1} q^{7}$$ $$+ ( -2 - \beta_{3} + 2 \beta_{4} ) q^{8}$$ $$+ ( -\beta_{3} + \beta_{4} ) q^{10}$$ $$+ ( 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{11}$$ $$+ ( 1 - \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} ) q^{13}$$ $$+ ( 1 - \beta_{1} - \beta_{5} ) q^{14}$$ $$+ ( -\beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{16}$$ $$+ ( -2 + \beta_{3} ) q^{17}$$ $$+ ( -1 - \beta_{3} - 2 \beta_{4} ) q^{19}$$ $$+ ( -2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{20}$$ $$+ ( -3 + 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{5} ) q^{22}$$ $$+ ( 4 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{23}$$ $$+ ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{25}$$ $$+ ( 1 - \beta_{3} - 7 \beta_{4} ) q^{26}$$ $$+ ( 1 - \beta_{3} - 2 \beta_{4} ) q^{28}$$ $$+ ( 3 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} + 5 \beta_{4} - 5 \beta_{5} ) q^{29}$$ $$+ ( 1 - \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{5} ) q^{31}$$ $$-3 \beta_{5} q^{32}$$ $$+ ( -3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{34}$$ $$+ ( -1 - \beta_{4} ) q^{35}$$ $$+ ( -1 + 6 \beta_{3} + 3 \beta_{4} ) q^{37}$$ $$+ ( 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{38}$$ $$+ ( 3 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{40}$$ $$+ ( -\beta_{2} + \beta_{3} + \beta_{5} ) q^{41}$$ $$+ ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{43}$$ $$+ ( -5 + 3 \beta_{3} + 7 \beta_{4} ) q^{44}$$ $$+ ( -4 \beta_{3} - 5 \beta_{4} ) q^{46}$$ $$+ ( \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{47}$$ $$+ ( -1 + \beta_{1} ) q^{49}$$ $$+ ( -2 + 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 5 \beta_{5} ) q^{50}$$ $$+ ( 7 \beta_{1} + 5 \beta_{2} - 10 \beta_{3} - 10 \beta_{4} + 10 \beta_{5} ) q^{52}$$ $$+ ( -2 + \beta_{3} + 3 \beta_{4} ) q^{53}$$ $$+ ( \beta_{3} + 2 \beta_{4} ) q^{55}$$ $$+ ( 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{56}$$ $$+ ( 3 - 3 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} - 6 \beta_{5} ) q^{58}$$ $$+ ( -1 + \beta_{1} - 5 \beta_{5} ) q^{59}$$ $$+ ( -2 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{61}$$ $$+ ( 10 - \beta_{3} - \beta_{4} ) q^{62}$$ $$+ ( 4 + 3 \beta_{4} ) q^{64}$$ $$+ ( 5 \beta_{1} - 5 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{65}$$ $$+ ( 4 - 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{67}$$ $$+ ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} ) q^{68}$$ $$+ ( 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{70}$$ $$+ ( -3 + 3 \beta_{3} - 3 \beta_{4} ) q^{71}$$ $$+ ( -7 + 5 \beta_{3} + 4 \beta_{4} ) q^{73}$$ $$+ ( -10 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{74}$$ $$+ ( -5 + 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} ) q^{76}$$ $$+ ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{77}$$ $$+ ( 7 \beta_{1} - 3 \beta_{2} ) q^{79}$$ $$+ ( 5 - 3 \beta_{3} - \beta_{4} ) q^{80}$$ $$-3 q^{82}$$ $$+ ( -6 \beta_{1} - \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{83}$$ $$+ ( -3 + 3 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} ) q^{85}$$ $$+ ( 2 - 2 \beta_{1} + \beta_{5} ) q^{86}$$ $$+ ( -9 \beta_{1} - 7 \beta_{2} + 8 \beta_{3} + 8 \beta_{4} - 8 \beta_{5} ) q^{88}$$ $$+ ( -4 - 4 \beta_{3} + 3 \beta_{4} ) q^{89}$$ $$+ ( -1 - 2 \beta_{3} + 2 \beta_{4} ) q^{91}$$ $$+ ( \beta_{1} + 2 \beta_{2} - 8 \beta_{3} - 8 \beta_{4} + 8 \beta_{5} ) q^{92}$$ $$+ ( 6 - 6 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{94}$$ $$+ ( -4 + 4 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{95}$$ $$+ ( \beta_{1} - 8 \beta_{2} + 7 \beta_{3} + 7 \beta_{4} - 7 \beta_{5} ) q^{97}$$ $$+ ( -1 + \beta_{3} + \beta_{4} ) q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q$$ $$\mathstrut +\mathstrut 3q^{2}$$ $$\mathstrut -\mathstrut 3q^{4}$$ $$\mathstrut +\mathstrut 3q^{5}$$ $$\mathstrut -\mathstrut 3q^{7}$$ $$\mathstrut -\mathstrut 12q^{8}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$6q$$ $$\mathstrut +\mathstrut 3q^{2}$$ $$\mathstrut -\mathstrut 3q^{4}$$ $$\mathstrut +\mathstrut 3q^{5}$$ $$\mathstrut -\mathstrut 3q^{7}$$ $$\mathstrut -\mathstrut 12q^{8}$$ $$\mathstrut +\mathstrut 6q^{11}$$ $$\mathstrut +\mathstrut 3q^{13}$$ $$\mathstrut +\mathstrut 3q^{14}$$ $$\mathstrut -\mathstrut 3q^{16}$$ $$\mathstrut -\mathstrut 12q^{17}$$ $$\mathstrut -\mathstrut 6q^{19}$$ $$\mathstrut -\mathstrut 6q^{20}$$ $$\mathstrut -\mathstrut 9q^{22}$$ $$\mathstrut +\mathstrut 12q^{23}$$ $$\mathstrut +\mathstrut 6q^{25}$$ $$\mathstrut +\mathstrut 6q^{26}$$ $$\mathstrut +\mathstrut 6q^{28}$$ $$\mathstrut +\mathstrut 9q^{29}$$ $$\mathstrut +\mathstrut 3q^{31}$$ $$\mathstrut -\mathstrut 9q^{34}$$ $$\mathstrut -\mathstrut 6q^{35}$$ $$\mathstrut -\mathstrut 6q^{37}$$ $$\mathstrut +\mathstrut 6q^{38}$$ $$\mathstrut +\mathstrut 9q^{40}$$ $$\mathstrut +\mathstrut 3q^{43}$$ $$\mathstrut -\mathstrut 30q^{44}$$ $$\mathstrut +\mathstrut 3q^{47}$$ $$\mathstrut -\mathstrut 3q^{49}$$ $$\mathstrut -\mathstrut 6q^{50}$$ $$\mathstrut +\mathstrut 21q^{52}$$ $$\mathstrut -\mathstrut 12q^{53}$$ $$\mathstrut +\mathstrut 6q^{56}$$ $$\mathstrut +\mathstrut 9q^{58}$$ $$\mathstrut -\mathstrut 3q^{59}$$ $$\mathstrut -\mathstrut 6q^{61}$$ $$\mathstrut +\mathstrut 60q^{62}$$ $$\mathstrut +\mathstrut 24q^{64}$$ $$\mathstrut +\mathstrut 15q^{65}$$ $$\mathstrut +\mathstrut 12q^{67}$$ $$\mathstrut +\mathstrut 6q^{68}$$ $$\mathstrut -\mathstrut 18q^{71}$$ $$\mathstrut -\mathstrut 42q^{73}$$ $$\mathstrut -\mathstrut 30q^{74}$$ $$\mathstrut -\mathstrut 15q^{76}$$ $$\mathstrut +\mathstrut 6q^{77}$$ $$\mathstrut +\mathstrut 21q^{79}$$ $$\mathstrut +\mathstrut 30q^{80}$$ $$\mathstrut -\mathstrut 18q^{82}$$ $$\mathstrut -\mathstrut 18q^{83}$$ $$\mathstrut -\mathstrut 9q^{85}$$ $$\mathstrut +\mathstrut 6q^{86}$$ $$\mathstrut -\mathstrut 27q^{88}$$ $$\mathstrut -\mathstrut 24q^{89}$$ $$\mathstrut -\mathstrut 6q^{91}$$ $$\mathstrut +\mathstrut 3q^{92}$$ $$\mathstrut +\mathstrut 18q^{94}$$ $$\mathstrut -\mathstrut 12q^{95}$$ $$\mathstrut +\mathstrut 3q^{97}$$ $$\mathstrut -\mathstrut 6q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\zeta_{18}^{3}$$ $$\beta_{2}$$ $$=$$ $$\zeta_{18}^{5} + \zeta_{18}$$ $$\beta_{3}$$ $$=$$ $$-\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18}$$ $$\beta_{4}$$ $$=$$ $$-\zeta_{18}^{5} + \zeta_{18}^{4}$$ $$\beta_{5}$$ $$=$$ $$-\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}$$
 $$1$$ $$=$$ $$\beta_0$$ $$\zeta_{18}$$ $$=$$ $$($$$$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{2}$$$$)/3$$ $$\zeta_{18}^{2}$$ $$=$$ $$($$$$-$$$$2$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$\beta_{2}$$$$)/3$$ $$\zeta_{18}^{3}$$ $$=$$ $$\beta_{1}$$ $$\zeta_{18}^{4}$$ $$=$$ $$($$$$-$$$$\beta_{5}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{2}$$$$)/3$$ $$\zeta_{18}^{5}$$ $$=$$ $$($$$$-$$$$\beta_{5}\mathstrut -\mathstrut$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{2}$$$$)/3$$

## Character Values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 0.939693 + 0.342020i −0.173648 − 0.984808i −0.766044 + 0.642788i 0.939693 − 0.342020i −0.173648 + 0.984808i −0.766044 − 0.642788i
−0.439693 0.761570i 0 0.613341 1.06234i 0.673648 1.16679i 0 −0.500000 0.866025i −2.83750 0 −1.18479
64.2 0.673648 + 1.16679i 0 0.0923963 0.160035i 1.26604 2.19285i 0 −0.500000 0.866025i 2.94356 0 3.41147
64.3 1.26604 + 2.19285i 0 −2.20574 + 3.82045i −0.439693 + 0.761570i 0 −0.500000 0.866025i −6.10607 0 −2.22668
127.1 −0.439693 + 0.761570i 0 0.613341 + 1.06234i 0.673648 + 1.16679i 0 −0.500000 + 0.866025i −2.83750 0 −1.18479
127.2 0.673648 1.16679i 0 0.0923963 + 0.160035i 1.26604 + 2.19285i 0 −0.500000 + 0.866025i 2.94356 0 3.41147
127.3 1.26604 2.19285i 0 −2.20574 3.82045i −0.439693 0.761570i 0 −0.500000 + 0.866025i −6.10607 0 −2.22668
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 127.3 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
9.c Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{6}$$ $$\mathstrut -\mathstrut 3 T_{2}^{5}$$ $$\mathstrut +\mathstrut 9 T_{2}^{4}$$ $$\mathstrut -\mathstrut 6 T_{2}^{3}$$ $$\mathstrut +\mathstrut 9 T_{2}^{2}$$ $$\mathstrut +\mathstrut 9$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.