Properties

 Label 189.2.e.f Level 189 Weight 2 Character orbit 189.e 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.e (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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 1 - \beta_{4} + \beta_{5} ) q^{2}$$ $$+ ( -\beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{4}$$ $$+ \beta_{2} q^{5}$$ $$+ ( 1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{7}$$ $$+ ( -4 - \beta_{1} - 2 \beta_{3} ) q^{8}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 1 - \beta_{4} + \beta_{5} ) q^{2}$$ $$+ ( -\beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{4}$$ $$+ \beta_{2} q^{5}$$ $$+ ( 1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{7}$$ $$+ ( -4 - \beta_{1} - 2 \beta_{3} ) q^{8}$$ $$+ ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{10}$$ $$+ ( -\beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{11}$$ $$+ ( 1 + 2 \beta_{1} - \beta_{3} ) q^{13}$$ $$+ ( -2 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{14}$$ $$+ ( -5 + \beta_{2} + 5 \beta_{4} - 4 \beta_{5} ) q^{16}$$ $$+ ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{17}$$ $$+ ( -1 - \beta_{2} + \beta_{4} + \beta_{5} ) q^{19}$$ $$+ ( 5 + 2 \beta_{1} ) q^{20}$$ $$+ ( -1 + 2 \beta_{1} - \beta_{3} ) q^{22}$$ $$+ ( 2 - \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{23}$$ $$+ ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{25}$$ $$+ ( 6 - \beta_{2} - 6 \beta_{4} ) q^{26}$$ $$+ ( -6 - 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{28}$$ $$+ ( -4 - \beta_{1} + 2 \beta_{3} ) q^{29}$$ $$+ ( 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{31}$$ $$+ ( 4 \beta_{1} + \beta_{2} + \beta_{3} + 10 \beta_{4} - 4 \beta_{5} ) q^{32}$$ $$+ ( 6 + 3 \beta_{1} + 3 \beta_{3} ) q^{34}$$ $$+ ( -3 + \beta_{1} - \beta_{2} + \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{35}$$ $$+ ( 3 - 2 \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{37}$$ $$+ ( -2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{38}$$ $$+ ( 9 - 9 \beta_{4} + 3 \beta_{5} ) q^{40}$$ $$+ ( -2 - 5 \beta_{1} + \beta_{3} ) q^{41}$$ $$+ ( -4 - 3 \beta_{1} ) q^{43}$$ $$-\beta_{2} q^{44}$$ $$+ ( -3 \beta_{1} + 3 \beta_{5} ) q^{46}$$ $$+ ( 2 - \beta_{2} - 2 \beta_{4} - 4 \beta_{5} ) q^{47}$$ $$+ ( 1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{5} ) q^{49}$$ $$+ ( 3 \beta_{1} + \beta_{3} ) q^{50}$$ $$+ ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} ) q^{52}$$ $$+ ( -2 \beta_{1} - \beta_{2} - \beta_{3} + 7 \beta_{4} + 2 \beta_{5} ) q^{53}$$ $$+ ( 1 + \beta_{1} - 2 \beta_{3} ) q^{55}$$ $$+ ( -13 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 9 \beta_{4} - 6 \beta_{5} ) q^{56}$$ $$+ ( -5 - \beta_{2} + 5 \beta_{4} - 2 \beta_{5} ) q^{58}$$ $$+ ( \beta_{1} - \beta_{2} - \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{59}$$ $$+ ( -1 + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{61}$$ $$+ ( 8 + 2 \beta_{1} + \beta_{3} ) q^{62}$$ $$+ ( 13 + 3 \beta_{1} + 3 \beta_{3} ) q^{64}$$ $$+ ( 2 + 3 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{65}$$ $$+ ( \beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{67}$$ $$+ ( 16 - 2 \beta_{2} - 16 \beta_{4} + 7 \beta_{5} ) q^{68}$$ $$+ ( 5 + 3 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{70}$$ $$+ ( 3 + 3 \beta_{1} + 3 \beta_{3} ) q^{71}$$ $$+ ( 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 5 \beta_{5} ) q^{73}$$ $$+ ( -5 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 5 \beta_{5} ) q^{74}$$ $$+ ( -7 - 3 \beta_{1} ) q^{76}$$ $$+ ( -1 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{77}$$ $$+ ( -2 + 3 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} ) q^{79}$$ $$+ ( -5 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 8 \beta_{4} + 5 \beta_{5} ) q^{80}$$ $$+ ( -16 + 4 \beta_{2} + 16 \beta_{4} - \beta_{5} ) q^{82}$$ $$+ ( 6 \beta_{1} - 3 \beta_{3} ) q^{83}$$ $$+ ( -9 - 3 \beta_{1} + 3 \beta_{3} ) q^{85}$$ $$+ ( -13 + 3 \beta_{2} + 13 \beta_{4} - 4 \beta_{5} ) q^{86}$$ $$+ ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{88}$$ $$+ ( -3 + 4 \beta_{2} + 3 \beta_{4} ) q^{89}$$ $$+ ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} - 6 \beta_{4} ) q^{91}$$ $$+ ( -5 - 2 \beta_{1} - \beta_{3} ) q^{92}$$ $$+ ( -3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 9 \beta_{4} + 3 \beta_{5} ) q^{94}$$ $$+ ( 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{95}$$ $$+ ( -6 - 2 \beta_{1} - 2 \beta_{3} ) q^{97}$$ $$+ ( -10 - \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q$$ $$\mathstrut +\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut q^{5}$$ $$\mathstrut +\mathstrut 2q^{7}$$ $$\mathstrut -\mathstrut 18q^{8}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$6q$$ $$\mathstrut +\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut q^{5}$$ $$\mathstrut +\mathstrut 2q^{7}$$ $$\mathstrut -\mathstrut 18q^{8}$$ $$\mathstrut +\mathstrut q^{10}$$ $$\mathstrut +\mathstrut 7q^{11}$$ $$\mathstrut +\mathstrut 4q^{13}$$ $$\mathstrut -\mathstrut 17q^{14}$$ $$\mathstrut -\mathstrut 10q^{16}$$ $$\mathstrut -\mathstrut 5q^{19}$$ $$\mathstrut +\mathstrut 26q^{20}$$ $$\mathstrut -\mathstrut 8q^{22}$$ $$\mathstrut +\mathstrut 6q^{23}$$ $$\mathstrut +\mathstrut 2q^{25}$$ $$\mathstrut +\mathstrut 17q^{26}$$ $$\mathstrut -\mathstrut 30q^{28}$$ $$\mathstrut -\mathstrut 26q^{29}$$ $$\mathstrut +\mathstrut 8q^{31}$$ $$\mathstrut +\mathstrut 25q^{32}$$ $$\mathstrut +\mathstrut 24q^{34}$$ $$\mathstrut -\mathstrut 10q^{35}$$ $$\mathstrut +\mathstrut 8q^{37}$$ $$\mathstrut -\mathstrut 7q^{38}$$ $$\mathstrut +\mathstrut 24q^{40}$$ $$\mathstrut -\mathstrut 4q^{41}$$ $$\mathstrut -\mathstrut 18q^{43}$$ $$\mathstrut -\mathstrut q^{44}$$ $$\mathstrut +\mathstrut 3q^{46}$$ $$\mathstrut +\mathstrut 9q^{47}$$ $$\mathstrut +\mathstrut 12q^{49}$$ $$\mathstrut -\mathstrut 8q^{50}$$ $$\mathstrut -\mathstrut 9q^{52}$$ $$\mathstrut +\mathstrut 24q^{53}$$ $$\mathstrut +\mathstrut 8q^{55}$$ $$\mathstrut -\mathstrut 48q^{56}$$ $$\mathstrut -\mathstrut 14q^{58}$$ $$\mathstrut -\mathstrut 15q^{59}$$ $$\mathstrut +\mathstrut q^{61}$$ $$\mathstrut +\mathstrut 42q^{62}$$ $$\mathstrut +\mathstrut 66q^{64}$$ $$\mathstrut +\mathstrut 10q^{65}$$ $$\mathstrut -\mathstrut 14q^{67}$$ $$\mathstrut +\mathstrut 39q^{68}$$ $$\mathstrut +\mathstrut 26q^{70}$$ $$\mathstrut +\mathstrut 6q^{71}$$ $$\mathstrut -\mathstrut 7q^{73}$$ $$\mathstrut -\mathstrut 36q^{76}$$ $$\mathstrut -\mathstrut q^{77}$$ $$\mathstrut -\mathstrut 6q^{79}$$ $$\mathstrut -\mathstrut 16q^{80}$$ $$\mathstrut -\mathstrut 43q^{82}$$ $$\mathstrut -\mathstrut 6q^{83}$$ $$\mathstrut -\mathstrut 54q^{85}$$ $$\mathstrut -\mathstrut 32q^{86}$$ $$\mathstrut -\mathstrut 9q^{88}$$ $$\mathstrut -\mathstrut 5q^{89}$$ $$\mathstrut -\mathstrut 33q^{91}$$ $$\mathstrut -\mathstrut 24q^{92}$$ $$\mathstrut +\mathstrut 27q^{94}$$ $$\mathstrut +\mathstrut 16q^{95}$$ $$\mathstrut -\mathstrut 28q^{97}$$ $$\mathstrut -\mathstrut 49q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

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 $$\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$$.

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.5 + 2.05195i 0.5 − 1.41036i 0.5 + 0.224437i 0.5 − 2.05195i 0.5 + 1.41036i 0.5 − 0.224437i
−0.730252 1.26483i 0 −0.0665372 + 0.115246i −0.296790 0.514055i 0 2.32383 1.26483i −2.72665 0 −0.433463 + 0.750780i
109.2 0.380438 + 0.658939i 0 0.710533 1.23068i 1.59097 + 2.75564i 0 −2.56238 + 0.658939i 2.60301 0 −1.21053 + 2.09671i
109.3 1.34981 + 2.33795i 0 −2.64400 + 4.57954i −0.794182 1.37556i 0 1.23855 + 2.33795i −8.87636 0 2.14400 3.71351i
163.1 −0.730252 + 1.26483i 0 −0.0665372 0.115246i −0.296790 + 0.514055i 0 2.32383 + 1.26483i −2.72665 0 −0.433463 0.750780i
163.2 0.380438 0.658939i 0 0.710533 + 1.23068i 1.59097 2.75564i 0 −2.56238 0.658939i 2.60301 0 −1.21053 2.09671i
163.3 1.34981 2.33795i 0 −2.64400 4.57954i −0.794182 + 1.37556i 0 1.23855 2.33795i −8.87636 0 2.14400 + 3.71351i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 163.3 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
7.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{6}$$ $$\mathstrut -\mathstrut 2 T_{2}^{5}$$ $$\mathstrut +\mathstrut 7 T_{2}^{4}$$ $$\mathstrut +\mathstrut 15 T_{2}^{2}$$ $$\mathstrut -\mathstrut 9 T_{2}$$ $$\mathstrut +\mathstrut 9$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.