# Properties

 Label 189.2.s.b Level 189 Weight 2 Character orbit 189.s Analytic conductor 1.509 Analytic rank 0 Dimension 10 CM No Inner twists 2

# Learn more about

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$1.5091725982$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{6})$$ Coefficient field: 10.0.288778218147.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -\beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} ) q^{2}$$ $$+ ( \beta_{2} + \beta_{4} - \beta_{6} + \beta_{8} - \beta_{9} ) q^{4}$$ $$+ ( \beta_{2} + \beta_{8} + \beta_{9} ) q^{5}$$ $$+ ( 1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{7}$$ $$+ ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{8}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -\beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} ) q^{2}$$ $$+ ( \beta_{2} + \beta_{4} - \beta_{6} + \beta_{8} - \beta_{9} ) q^{4}$$ $$+ ( \beta_{2} + \beta_{8} + \beta_{9} ) q^{5}$$ $$+ ( 1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{7}$$ $$+ ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{8}$$ $$+ ( -1 - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{10}$$ $$+ ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{11}$$ $$+ ( -\beta_{1} + \beta_{3} + \beta_{7} ) q^{13}$$ $$+ ( 1 + \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{14}$$ $$+ ( -1 - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{16}$$ $$+ ( 3 + \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{6} - \beta_{7} + \beta_{9} ) q^{17}$$ $$+ ( 2 \beta_{4} + \beta_{5} + 3 \beta_{7} + \beta_{8} - \beta_{9} ) q^{19}$$ $$+ ( -\beta_{2} + 4 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{20}$$ $$+ ( 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{22}$$ $$+ ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - \beta_{8} + \beta_{9} ) q^{23}$$ $$+ ( -2 - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{7} - 2 \beta_{9} ) q^{25}$$ $$+ ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{26}$$ $$+ ( 2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - 4 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{28}$$ $$+ ( 2 - \beta_{1} + 2 \beta_{2} + 3 \beta_{4} + 4 \beta_{5} + \beta_{6} + 4 \beta_{7} - \beta_{8} - \beta_{9} ) q^{29}$$ $$+ ( -2 \beta_{1} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{31}$$ $$+ ( -6 - \beta_{1} - \beta_{2} - 3 \beta_{6} ) q^{32}$$ $$+ ( \beta_{1} + \beta_{2} - 3 \beta_{4} - 3 \beta_{7} - 3 \beta_{8} ) q^{34}$$ $$+ ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - 3 \beta_{6} - \beta_{7} - \beta_{8} ) q^{35}$$ $$+ ( 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} ) q^{37}$$ $$+ ( -3 - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{7} - 2 \beta_{8} ) q^{38}$$ $$+ ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{40}$$ $$+ ( 5 \beta_{1} - 4 \beta_{2} - 4 \beta_{5} + 4 \beta_{8} - 4 \beta_{9} ) q^{41}$$ $$+ ( 6 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 3 \beta_{8} - 3 \beta_{9} ) q^{43}$$ $$+ ( -2 - \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{44}$$ $$+ ( -3 - \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{46}$$ $$+ ( -3 - 2 \beta_{1} - \beta_{2} - \beta_{4} - 3 \beta_{6} - \beta_{9} ) q^{47}$$ $$+ ( -2 - 2 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} ) q^{49}$$ $$+ ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} ) q^{50}$$ $$+ ( 1 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{5} + 2 \beta_{6} ) q^{52}$$ $$+ ( 1 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{53}$$ $$+ ( -1 + 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{6} + \beta_{8} - \beta_{9} ) q^{55}$$ $$+ ( 6 + 3 \beta_{1} + 3 \beta_{2} + \beta_{5} - 4 \beta_{7} - \beta_{8} ) q^{56}$$ $$+ ( -3 \beta_{3} - 3 \beta_{4} - 6 \beta_{7} - 7 \beta_{8} - \beta_{9} ) q^{58}$$ $$+ ( -2 \beta_{1} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} - 2 \beta_{8} + 3 \beta_{9} ) q^{59}$$ $$+ ( 1 + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - \beta_{6} - 4 \beta_{7} - 3 \beta_{8} ) q^{61}$$ $$+ ( -3 + \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + \beta_{5} + 6 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} ) q^{62}$$ $$+ ( -1 - \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + \beta_{5} + 6 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} ) q^{64}$$ $$+ ( -2 \beta_{1} + 4 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{8} + 2 \beta_{9} ) q^{65}$$ $$+ ( -\beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{67}$$ $$+ ( 6 + 3 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{8} - 3 \beta_{9} ) q^{68}$$ $$+ ( -1 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{70}$$ $$+ ( 2 \beta_{1} - \beta_{2} - 5 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{8} - 2 \beta_{9} ) q^{71}$$ $$+ ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{7} - 3 \beta_{8} ) q^{73}$$ $$+ ( 2 - 4 \beta_{3} - 4 \beta_{5} + 4 \beta_{6} ) q^{74}$$ $$+ ( -3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{5} + 3 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{76}$$ $$+ ( -3 + \beta_{2} - \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{77}$$ $$+ ( 4 - 3 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - \beta_{4} + 3 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} ) q^{79}$$ $$+ ( 6 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 6 \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{80}$$ $$+ ( 2 - \beta_{1} - 3 \beta_{2} - \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{82}$$ $$+ ( 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} - 3 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{83}$$ $$+ ( 3 - 2 \beta_{2} - \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{85}$$ $$+ ( 5 - 6 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 10 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{86}$$ $$+ ( 3 + 3 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} + \beta_{8} + \beta_{9} ) q^{88}$$ $$+ ( -3 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + 5 \beta_{4} + \beta_{5} + 6 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} ) q^{89}$$ $$+ ( -4 - 3 \beta_{1} - 2 \beta_{3} - 6 \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{91}$$ $$+ ( -6 + 2 \beta_{1} - 4 \beta_{2} - 6 \beta_{5} - 3 \beta_{6} + 6 \beta_{8} ) q^{92}$$ $$+ ( -2 + 2 \beta_{1} - 2 \beta_{2} + 6 \beta_{4} - 2 \beta_{5} - \beta_{6} + 8 \beta_{7} + 8 \beta_{8} - 2 \beta_{9} ) q^{94}$$ $$+ ( \beta_{1} + 4 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{95}$$ $$+ ( 3 \beta_{1} - 2 \beta_{2} - \beta_{4} - 6 \beta_{5} - 2 \beta_{7} + 5 \beta_{8} + \beta_{9} ) q^{97}$$ $$+ ( -9 - 2 \beta_{1} - \beta_{2} + 5 \beta_{3} + \beta_{4} + 3 \beta_{5} - 6 \beta_{6} + 5 \beta_{7} + 4 \beta_{8} ) q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q$$ $$\mathstrut +\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut 3q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$10q$$ $$\mathstrut +\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut 3q^{7}$$ $$\mathstrut -\mathstrut 15q^{10}$$ $$\mathstrut +\mathstrut 6q^{13}$$ $$\mathstrut +\mathstrut 6q^{14}$$ $$\mathstrut -\mathstrut 6q^{16}$$ $$\mathstrut +\mathstrut 12q^{17}$$ $$\mathstrut +\mathstrut 3q^{19}$$ $$\mathstrut +\mathstrut 3q^{20}$$ $$\mathstrut +\mathstrut 5q^{22}$$ $$\mathstrut -\mathstrut 14q^{25}$$ $$\mathstrut -\mathstrut 3q^{26}$$ $$\mathstrut +\mathstrut 2q^{28}$$ $$\mathstrut +\mathstrut 15q^{29}$$ $$\mathstrut -\mathstrut 9q^{31}$$ $$\mathstrut -\mathstrut 48q^{32}$$ $$\mathstrut +\mathstrut 3q^{34}$$ $$\mathstrut +\mathstrut 15q^{35}$$ $$\mathstrut +\mathstrut 6q^{37}$$ $$\mathstrut -\mathstrut 36q^{38}$$ $$\mathstrut +\mathstrut 9q^{41}$$ $$\mathstrut +\mathstrut 3q^{43}$$ $$\mathstrut -\mathstrut 24q^{44}$$ $$\mathstrut -\mathstrut 13q^{46}$$ $$\mathstrut -\mathstrut 15q^{47}$$ $$\mathstrut -\mathstrut 23q^{49}$$ $$\mathstrut -\mathstrut 3q^{50}$$ $$\mathstrut +\mathstrut 9q^{53}$$ $$\mathstrut +\mathstrut 51q^{56}$$ $$\mathstrut -\mathstrut 16q^{58}$$ $$\mathstrut +\mathstrut 18q^{59}$$ $$\mathstrut +\mathstrut 12q^{61}$$ $$\mathstrut -\mathstrut 12q^{62}$$ $$\mathstrut +\mathstrut 6q^{64}$$ $$\mathstrut +\mathstrut 3q^{65}$$ $$\mathstrut -\mathstrut 10q^{67}$$ $$\mathstrut +\mathstrut 54q^{68}$$ $$\mathstrut +\mathstrut 9q^{70}$$ $$\mathstrut +\mathstrut 3q^{73}$$ $$\mathstrut +\mathstrut 9q^{76}$$ $$\mathstrut -\mathstrut 45q^{77}$$ $$\mathstrut +\mathstrut 20q^{79}$$ $$\mathstrut +\mathstrut 30q^{80}$$ $$\mathstrut +\mathstrut 9q^{82}$$ $$\mathstrut +\mathstrut 15q^{83}$$ $$\mathstrut +\mathstrut 18q^{85}$$ $$\mathstrut +\mathstrut 16q^{88}$$ $$\mathstrut -\mathstrut 24q^{89}$$ $$\mathstrut -\mathstrut 24q^{91}$$ $$\mathstrut -\mathstrut 39q^{92}$$ $$\mathstrut -\mathstrut 3q^{94}$$ $$\mathstrut +\mathstrut 6q^{97}$$ $$\mathstrut -\mathstrut 45q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10}\mathstrut -\mathstrut$$ $$x^{9}\mathstrut +\mathstrut$$ $$7$$ $$x^{8}\mathstrut -\mathstrut$$ $$4$$ $$x^{7}\mathstrut +\mathstrut$$ $$34$$ $$x^{6}\mathstrut -\mathstrut$$ $$19$$ $$x^{5}\mathstrut +\mathstrut$$ $$64$$ $$x^{4}\mathstrut -\mathstrut$$ $$x^{3}\mathstrut +\mathstrut$$ $$64$$ $$x^{2}\mathstrut -\mathstrut$$ $$21$$ $$x\mathstrut +\mathstrut$$ $$9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-339 \nu^{9} + 1348 \nu^{8} - 4381 \nu^{7} + 7882 \nu^{6} - 19883 \nu^{5} + 36059 \nu^{4} - 75410 \nu^{3} + 44484 \nu^{2} - 15165 \nu + 29709$$$$)/72795$$ $$\beta_{3}$$ $$=$$ $$($$$$658 \nu^{9} + 2394 \nu^{8} + 4352 \nu^{7} + 10326 \nu^{6} + 25351 \nu^{5} + 51907 \nu^{4} + 47450 \nu^{3} + 30472 \nu^{2} + 130790 \nu + 98232$$$$)/72795$$ $$\beta_{4}$$ $$=$$ $$($$$$-4192 \nu^{9} - 796 \nu^{8} - 21678 \nu^{7} - 20279 \nu^{6} - 85319 \nu^{5} - 118353 \nu^{4} - 2560 \nu^{3} - 414508 \nu^{2} + 81750 \nu - 398583$$$$)/218385$$ $$\beta_{5}$$ $$=$$ $$($$$$8236 \nu^{9} - 9272 \nu^{8} + 54399 \nu^{7} - 28438 \nu^{6} + 233822 \nu^{5} - 150966 \nu^{4} + 361225 \nu^{3} + 82264 \nu^{2} + 31515 \nu - 336546$$$$)/218385$$ $$\beta_{6}$$ $$=$$ $$($$$$3301 \nu^{9} - 2962 \nu^{8} + 21759 \nu^{7} - 8823 \nu^{6} + 104352 \nu^{5} - 42836 \nu^{4} + 175205 \nu^{3} + 72109 \nu^{2} + 166780 \nu - 54156$$$$)/72795$$ $$\beta_{7}$$ $$=$$ $$($$$$-840 \nu^{9} + 248 \nu^{8} - 5659 \nu^{7} - 998 \nu^{6} - 27923 \nu^{5} - 3072 \nu^{4} - 51488 \nu^{3} - 30640 \nu^{2} - 51320 \nu + 11514$$$$)/14559$$ $$\beta_{8}$$ $$=$$ $$($$$$3085 \nu^{9} - 1373 \nu^{8} + 17808 \nu^{7} + 1181 \nu^{6} + 84554 \nu^{5} + 5736 \nu^{4} + 111910 \nu^{3} + 124546 \nu^{2} + 106440 \nu + 17856$$$$)/43677$$ $$\beta_{9}$$ $$=$$ $$($$$$-18476 \nu^{9} + 18997 \nu^{8} - 128469 \nu^{7} + 65033 \nu^{6} - 601717 \nu^{5} + 295851 \nu^{4} - 1019855 \nu^{3} - 222374 \nu^{2} - 668685 \nu + 178101$$$$)/218385$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{2}$$ $$\nu^{4}$$ $$=$$ $$-$$$$5$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$12$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$12$$ $$\nu^{5}$$ $$=$$ $$-$$$$5$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$11$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$11$$ $$\beta_{1}$$ $$\nu^{6}$$ $$=$$ $$6$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$8$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$14$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$16$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$9$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$22$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$51$$ $$\nu^{7}$$ $$=$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$31$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$8$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$31$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$30$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$8$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$43$$ $$\beta_{1}$$ $$\nu^{8}$$ $$=$$ $$75$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$66$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$38$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$222$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$47$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$37$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$76$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$8$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$112$$ $$\beta_{1}$$ $$\nu^{9}$$ $$=$$ $$95$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$189$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$94$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$37$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$84$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$47$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$194$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$3$$

## Character Values

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

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$1 + \beta_{6}$$ $$-\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.827154 − 1.43267i −1.04536 + 1.81062i −0.539982 + 0.935277i 0.187540 − 0.324828i 1.07065 − 1.85442i 0.827154 + 1.43267i −1.04536 − 1.81062i −0.539982 − 0.935277i 0.187540 + 0.324828i 1.07065 + 1.85442i
−1.81474 1.04774i 0 1.19552 + 2.07070i 2.08983 0 −0.879217 + 2.49539i 0.819421i 0 −3.79250 2.18960i
17.2 −1.30778 0.755047i 0 0.140193 + 0.242822i 0.775876 0 2.05881 1.66171i 2.59678i 0 −1.01468 0.585823i
17.3 0.254498 + 0.146935i 0 −0.956820 1.65726i −3.06027 0 −1.22581 2.34465i 1.15010i 0 −0.778834 0.449660i
17.4 0.621951 + 0.359083i 0 −0.742118 1.28539i 1.44755 0 2.19442 + 1.47801i 2.50226i 0 0.900304 + 0.519791i
17.5 2.24607 + 1.29677i 0 2.36322 + 4.09323i −1.25299 0 −0.648211 2.56512i 7.07116i 0 −2.81429 1.62483i
89.1 −1.81474 + 1.04774i 0 1.19552 2.07070i 2.08983 0 −0.879217 2.49539i 0.819421i 0 −3.79250 + 2.18960i
89.2 −1.30778 + 0.755047i 0 0.140193 0.242822i 0.775876 0 2.05881 + 1.66171i 2.59678i 0 −1.01468 + 0.585823i
89.3 0.254498 0.146935i 0 −0.956820 + 1.65726i −3.06027 0 −1.22581 + 2.34465i 1.15010i 0 −0.778834 + 0.449660i
89.4 0.621951 0.359083i 0 −0.742118 + 1.28539i 1.44755 0 2.19442 1.47801i 2.50226i 0 0.900304 0.519791i
89.5 2.24607 1.29677i 0 2.36322 4.09323i −1.25299 0 −0.648211 + 2.56512i 7.07116i 0 −2.81429 + 1.62483i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 89.5 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
63.s Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{10} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.