# Properties

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

# Related objects

## 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: 6.0.309123.1 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( \beta_{1} - \beta_{5} ) q^{2}$$ $$+ ( -1 + \beta_{2} + \beta_{4} + \beta_{5} ) q^{4}$$ $$+ ( -2 + \beta_{2} + 2 \beta_{4} ) q^{5}$$ $$+ \beta_{4} q^{7}$$ $$+ ( 2 - \beta_{1} + \beta_{3} ) q^{8}$$ $$+O(q^{10})$$ $$q$$ $$+ ( \beta_{1} - \beta_{5} ) q^{2}$$ $$+ ( -1 + \beta_{2} + \beta_{4} + \beta_{5} ) q^{4}$$ $$+ ( -2 + \beta_{2} + 2 \beta_{4} ) q^{5}$$ $$+ \beta_{4} q^{7}$$ $$+ ( 2 - \beta_{1} + \beta_{3} ) q^{8}$$ $$+ ( -1 - 3 \beta_{1} ) q^{10}$$ $$+ ( -2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{11}$$ $$+ ( -1 + \beta_{4} ) q^{13}$$ $$-\beta_{5} q^{14}$$ $$+ ( 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{16}$$ $$+ ( 4 + \beta_{1} - \beta_{3} ) q^{17}$$ $$+ ( -1 + \beta_{1} - \beta_{3} ) q^{19}$$ $$+ ( 2 \beta_{1} - \beta_{2} - \beta_{3} - 5 \beta_{4} - 2 \beta_{5} ) q^{20}$$ $$+ ( 5 - 2 \beta_{2} - 5 \beta_{4} - 2 \beta_{5} ) q^{22}$$ $$+ ( 1 - 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{23}$$ $$+ ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{25}$$ $$-\beta_{1} q^{26}$$ $$+ ( -1 + \beta_{1} - \beta_{3} ) q^{28}$$ $$+ ( -\beta_{1} + \beta_{5} ) q^{29}$$ $$+ ( 2 - \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{31}$$ $$+ ( -3 + 3 \beta_{4} - \beta_{5} ) q^{32}$$ $$+ ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{34}$$ $$+ ( -2 - \beta_{3} ) q^{35}$$ $$+ 3 \beta_{3} q^{37}$$ $$+ ( -3 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{38}$$ $$+ ( -7 + 2 \beta_{2} + 7 \beta_{4} + 2 \beta_{5} ) q^{40}$$ $$+ ( -7 - \beta_{2} + 7 \beta_{4} ) q^{41}$$ $$+ ( -4 \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{5} ) q^{43}$$ $$+ ( -6 + 5 \beta_{1} ) q^{44}$$ $$+ ( 5 + 2 \beta_{1} + \beta_{3} ) q^{46}$$ $$+ ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{47}$$ $$+ ( -1 + \beta_{4} ) q^{49}$$ $$+ ( 5 - \beta_{2} - 5 \beta_{4} + 4 \beta_{5} ) q^{50}$$ $$+ ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{52}$$ $$+ ( 5 - \beta_{1} - 2 \beta_{3} ) q^{53}$$ $$+ ( 5 \beta_{1} + \beta_{3} ) q^{55}$$ $$+ ( -\beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{56}$$ $$+ ( 3 - \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{58}$$ $$+ ( -4 + 2 \beta_{2} + 4 \beta_{4} - \beta_{5} ) q^{59}$$ $$+ ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{61}$$ $$+ ( 7 + \beta_{1} + 2 \beta_{3} ) q^{62}$$ $$+ ( -3 + 2 \beta_{1} + \beta_{3} ) q^{64}$$ $$+ ( -\beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{65}$$ $$+ ( -2 - \beta_{2} + 2 \beta_{4} - 7 \beta_{5} ) q^{67}$$ $$+ ( 1 + 4 \beta_{2} - \beta_{4} + \beta_{5} ) q^{68}$$ $$+ ( -3 \beta_{1} - \beta_{4} + 3 \beta_{5} ) q^{70}$$ $$+ ( -2 + \beta_{1} + 2 \beta_{3} ) q^{71}$$ $$+ ( -1 - \beta_{1} - 5 \beta_{3} ) q^{73}$$ $$+ ( 3 \beta_{1} + 3 \beta_{4} - 3 \beta_{5} ) q^{74}$$ $$+ ( 6 - \beta_{2} - 6 \beta_{4} - 4 \beta_{5} ) q^{76}$$ $$+ ( 1 + \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{77}$$ $$+ ( 5 \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} - 5 \beta_{5} ) q^{79}$$ $$+ ( -6 - 7 \beta_{1} ) q^{80}$$ $$+ ( 1 - 6 \beta_{1} ) q^{82}$$ $$+ ( -\beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{83}$$ $$+ ( -5 + 4 \beta_{2} + 5 \beta_{4} - 2 \beta_{5} ) q^{85}$$ $$+ ( 11 - 4 \beta_{2} - 11 \beta_{4} - 5 \beta_{5} ) q^{86}$$ $$+ ( -7 \beta_{1} + \beta_{2} + \beta_{3} + 5 \beta_{4} + 7 \beta_{5} ) q^{88}$$ $$+ ( -1 - 6 \beta_{1} + \beta_{3} ) q^{89}$$ $$- q^{91}$$ $$+ ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{92}$$ $$+ ( -6 + 3 \beta_{2} + 6 \beta_{4} + 9 \beta_{5} ) q^{94}$$ $$+ ( 5 - \beta_{2} - 5 \beta_{4} - 2 \beta_{5} ) q^{95}$$ $$+ ( -5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} ) q^{97}$$ $$-\beta_{1} q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q$$ $$\mathstrut -\mathstrut q^{2}$$ $$\mathstrut -\mathstrut 3q^{4}$$ $$\mathstrut -\mathstrut 5q^{5}$$ $$\mathstrut +\mathstrut 3q^{7}$$ $$\mathstrut +\mathstrut 12q^{8}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$6q$$ $$\mathstrut -\mathstrut q^{2}$$ $$\mathstrut -\mathstrut 3q^{4}$$ $$\mathstrut -\mathstrut 5q^{5}$$ $$\mathstrut +\mathstrut 3q^{7}$$ $$\mathstrut +\mathstrut 12q^{8}$$ $$\mathstrut -\mathstrut 2q^{11}$$ $$\mathstrut -\mathstrut 3q^{13}$$ $$\mathstrut +\mathstrut q^{14}$$ $$\mathstrut -\mathstrut 3q^{16}$$ $$\mathstrut +\mathstrut 24q^{17}$$ $$\mathstrut -\mathstrut 6q^{19}$$ $$\mathstrut -\mathstrut 16q^{20}$$ $$\mathstrut +\mathstrut 15q^{22}$$ $$\mathstrut -\mathstrut 6q^{25}$$ $$\mathstrut +\mathstrut 2q^{26}$$ $$\mathstrut -\mathstrut 6q^{28}$$ $$\mathstrut +\mathstrut q^{29}$$ $$\mathstrut +\mathstrut 3q^{31}$$ $$\mathstrut -\mathstrut 8q^{32}$$ $$\mathstrut +\mathstrut 3q^{34}$$ $$\mathstrut -\mathstrut 10q^{35}$$ $$\mathstrut -\mathstrut 6q^{37}$$ $$\mathstrut +\mathstrut 8q^{38}$$ $$\mathstrut -\mathstrut 21q^{40}$$ $$\mathstrut -\mathstrut 22q^{41}$$ $$\mathstrut +\mathstrut 3q^{43}$$ $$\mathstrut -\mathstrut 46q^{44}$$ $$\mathstrut +\mathstrut 24q^{46}$$ $$\mathstrut -\mathstrut 9q^{47}$$ $$\mathstrut -\mathstrut 3q^{49}$$ $$\mathstrut +\mathstrut 10q^{50}$$ $$\mathstrut -\mathstrut 3q^{52}$$ $$\mathstrut +\mathstrut 36q^{53}$$ $$\mathstrut -\mathstrut 12q^{55}$$ $$\mathstrut +\mathstrut 6q^{56}$$ $$\mathstrut +\mathstrut 9q^{58}$$ $$\mathstrut -\mathstrut 9q^{59}$$ $$\mathstrut +\mathstrut 6q^{61}$$ $$\mathstrut +\mathstrut 36q^{62}$$ $$\mathstrut -\mathstrut 24q^{64}$$ $$\mathstrut -\mathstrut 5q^{65}$$ $$\mathstrut +\mathstrut 6q^{68}$$ $$\mathstrut -\mathstrut 18q^{71}$$ $$\mathstrut +\mathstrut 6q^{73}$$ $$\mathstrut +\mathstrut 6q^{74}$$ $$\mathstrut +\mathstrut 21q^{76}$$ $$\mathstrut +\mathstrut 2q^{77}$$ $$\mathstrut -\mathstrut 15q^{79}$$ $$\mathstrut -\mathstrut 22q^{80}$$ $$\mathstrut +\mathstrut 18q^{82}$$ $$\mathstrut -\mathstrut 12q^{83}$$ $$\mathstrut -\mathstrut 9q^{85}$$ $$\mathstrut +\mathstrut 34q^{86}$$ $$\mathstrut +\mathstrut 21q^{88}$$ $$\mathstrut +\mathstrut 4q^{89}$$ $$\mathstrut -\mathstrut 6q^{91}$$ $$\mathstrut +\mathstrut 15q^{92}$$ $$\mathstrut -\mathstrut 24q^{94}$$ $$\mathstrut +\mathstrut 16q^{95}$$ $$\mathstrut -\mathstrut 3q^{97}$$ $$\mathstrut +\mathstrut 2q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6}\mathstrut -\mathstrut$$ $$3$$ $$x^{5}\mathstrut +\mathstrut$$ $$10$$ $$x^{4}\mathstrut -\mathstrut$$ $$15$$ $$x^{3}\mathstrut +\mathstrut$$ $$19$$ $$x^{2}\mathstrut -\mathstrut$$ $$12$$ $$x\mathstrut +\mathstrut$$ $$3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 8 \nu^{3} + 5 \nu^{2} - 18 \nu + 6$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} + 6 \nu^{2} - 5 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 9$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 30 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-$$$$2$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-$$$$2$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$4$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$7$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$10$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$16$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$11$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$8$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$10$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$17$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$5$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$14$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$16$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$23$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$47$$$$)/3$$

## Character Values

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

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$-1 + \beta_{4}$$ $$1$$

## 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
 0.5 − 2.05195i 0.5 + 1.41036i 0.5 − 0.224437i 0.5 + 2.05195i 0.5 − 1.41036i 0.5 + 0.224437i
−1.23025 2.13086i 0 −2.02704 + 3.51094i −1.29679 + 2.24611i 0 0.500000 + 0.866025i 5.05408 0 6.38151
64.2 −0.119562 0.207087i 0 0.971410 1.68253i 0.590972 1.02359i 0 0.500000 + 0.866025i −0.942820 0 −0.282630
64.3 0.849814 + 1.47192i 0 −0.444368 + 0.769668i −1.79418 + 3.10761i 0 0.500000 + 0.866025i 1.88874 0 −6.09888
127.1 −1.23025 + 2.13086i 0 −2.02704 3.51094i −1.29679 2.24611i 0 0.500000 0.866025i 5.05408 0 6.38151
127.2 −0.119562 + 0.207087i 0 0.971410 + 1.68253i 0.590972 + 1.02359i 0 0.500000 0.866025i −0.942820 0 −0.282630
127.3 0.849814 1.47192i 0 −0.444368 0.769668i −1.79418 3.10761i 0 0.500000 0.866025i 1.88874 0 −6.09888
 $$n$$: 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 $$T_{2}^{6}$$ $$\mathstrut +\mathstrut T_{2}^{5}$$ $$\mathstrut +\mathstrut 5 T_{2}^{4}$$ $$\mathstrut -\mathstrut 2 T_{2}^{3}$$ $$\mathstrut +\mathstrut 17 T_{2}^{2}$$ $$\mathstrut +\mathstrut 4 T_{2}$$ $$\mathstrut +\mathstrut 1$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.