# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$1.5091725982$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: 10.0.991381711347.1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q$$ $$-\beta_{1} q^{2}$$ $$+ ( -\beta_{3} + \beta_{6} + \beta_{7} ) q^{4}$$ $$+ ( 1 - \beta_{2} + \beta_{9} ) q^{5}$$ $$+ ( 1 - \beta_{1} - \beta_{4} + \beta_{6} - \beta_{8} ) q^{7}$$ $$+ ( 1 - \beta_{4} - \beta_{8} ) q^{8}$$ $$+O(q^{10})$$ $$q$$ $$-\beta_{1} q^{2}$$ $$+ ( -\beta_{3} + \beta_{6} + \beta_{7} ) q^{4}$$ $$+ ( 1 - \beta_{2} + \beta_{9} ) q^{5}$$ $$+ ( 1 - \beta_{1} - \beta_{4} + \beta_{6} - \beta_{8} ) q^{7}$$ $$+ ( 1 - \beta_{4} - \beta_{8} ) q^{8}$$ $$+ ( -2 + \beta_{2} + \beta_{4} - 2 \beta_{6} ) q^{10}$$ $$+ ( 1 - \beta_{3} - \beta_{4} - \beta_{8} ) q^{11}$$ $$+ ( -1 - \beta_{1} - \beta_{2} - \beta_{6} ) q^{13}$$ $$+ ( -2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{14}$$ $$+ ( \beta_{2} + \beta_{4} + \beta_{7} ) q^{16}$$ $$+ ( -3 + \beta_{1} - 3 \beta_{6} - \beta_{7} ) q^{17}$$ $$+ ( \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{19}$$ $$+ ( -\beta_{3} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{20}$$ $$+ ( -1 + \beta_{1} + \beta_{2} - \beta_{6} - \beta_{7} ) q^{22}$$ $$+ ( \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{8} - \beta_{9} ) q^{23}$$ $$+ ( \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{9} ) q^{25}$$ $$+ ( -\beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{26}$$ $$+ ( 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{9} ) q^{28}$$ $$+ ( 2 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{29}$$ $$+ ( \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{31}$$ $$+ ( -\beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{32}$$ $$+ ( \beta_{3} + 4 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{34}$$ $$+ ( -1 + \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{35}$$ $$+ ( -2 \beta_{5} - 2 \beta_{8} ) q^{37}$$ $$+ ( 5 - 2 \beta_{1} + 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{8} ) q^{38}$$ $$+ ( 1 + \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} ) q^{40}$$ $$+ ( -\beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{7} ) q^{41}$$ $$+ ( \beta_{3} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{43}$$ $$+ ( -\beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{44}$$ $$+ ( 4 - 5 \beta_{1} - 3 \beta_{2} - \beta_{4} + 4 \beta_{6} + 2 \beta_{7} ) q^{46}$$ $$+ ( -4 - \beta_{1} - \beta_{2} - \beta_{4} - 4 \beta_{6} + 2 \beta_{7} ) q^{47}$$ $$+ ( 1 + 3 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{5} - \beta_{7} + \beta_{9} ) q^{49}$$ $$+ ( -6 + 3 \beta_{1} + 2 \beta_{2} + \beta_{4} - 6 \beta_{6} - \beta_{7} ) q^{50}$$ $$+ ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{8} ) q^{52}$$ $$+ ( 5 - \beta_{1} - 2 \beta_{2} + 5 \beta_{6} ) q^{53}$$ $$+ ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} ) q^{55}$$ $$+ ( 4 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{56}$$ $$+ ( 3 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{8} + \beta_{9} ) q^{58}$$ $$+ ( -\beta_{3} + \beta_{5} + 6 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{59}$$ $$+ ( -2 - \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{61}$$ $$+ ( 3 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{8} + 2 \beta_{9} ) q^{62}$$ $$+ ( -6 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{9} ) q^{64}$$ $$+ ( 1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{65}$$ $$+ ( -2 \beta_{3} + 4 \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{67}$$ $$+ ( 7 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{5} + \beta_{9} ) q^{68}$$ $$+ ( -5 - 2 \beta_{1} - \beta_{3} + 2 \beta_{4} - 3 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{70}$$ $$+ ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{8} - 2 \beta_{9} ) q^{71}$$ $$+ ( 4 - 3 \beta_{4} + 4 \beta_{6} - \beta_{7} ) q^{73}$$ $$+ ( -10 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{8} - 2 \beta_{9} ) q^{74}$$ $$+ ( 3 - 5 \beta_{1} + \beta_{2} + 3 \beta_{6} + \beta_{7} ) q^{76}$$ $$+ ( 3 + \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{77}$$ $$+ ( -3 + 4 \beta_{1} + \beta_{2} - 3 \beta_{6} - 2 \beta_{7} ) q^{79}$$ $$+ ( -3 - 2 \beta_{1} + \beta_{2} - \beta_{4} - 3 \beta_{6} ) q^{80}$$ $$+ ( 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 3 \beta_{9} ) q^{82}$$ $$+ ( -2 \beta_{3} + \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} ) q^{83}$$ $$+ ( -2 + \beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{85}$$ $$+ ( 1 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{8} + 4 \beta_{9} ) q^{86}$$ $$+ ( 4 + \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{8} - \beta_{9} ) q^{88}$$ $$+ ( -2 \beta_{5} + 7 \beta_{6} + 2 \beta_{8} + \beta_{9} ) q^{89}$$ $$+ ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{91}$$ $$+ ( -2 \beta_{5} + 5 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{92}$$ $$+ ( 4 \beta_{5} - 3 \beta_{6} - 4 \beta_{8} + 2 \beta_{9} ) q^{94}$$ $$+ ( -2 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{95}$$ $$+ ( -4 \beta_{3} + \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{97}$$ $$+ ( -10 + \beta_{1} + \beta_{2} - \beta_{5} - 5 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} ) q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut 8q^{5}$$ $$\mathstrut -\mathstrut q^{7}$$ $$\mathstrut +\mathstrut 6q^{8}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$10q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut 8q^{5}$$ $$\mathstrut -\mathstrut q^{7}$$ $$\mathstrut +\mathstrut 6q^{8}$$ $$\mathstrut -\mathstrut 7q^{10}$$ $$\mathstrut +\mathstrut 8q^{11}$$ $$\mathstrut -\mathstrut 8q^{13}$$ $$\mathstrut -\mathstrut 16q^{14}$$ $$\mathstrut +\mathstrut 2q^{16}$$ $$\mathstrut -\mathstrut 12q^{17}$$ $$\mathstrut +\mathstrut q^{19}$$ $$\mathstrut -\mathstrut 5q^{20}$$ $$\mathstrut -\mathstrut q^{22}$$ $$\mathstrut +\mathstrut 6q^{23}$$ $$\mathstrut +\mathstrut 2q^{25}$$ $$\mathstrut -\mathstrut 11q^{26}$$ $$\mathstrut -\mathstrut 2q^{28}$$ $$\mathstrut -\mathstrut 7q^{29}$$ $$\mathstrut -\mathstrut 3q^{31}$$ $$\mathstrut +\mathstrut 2q^{32}$$ $$\mathstrut +\mathstrut 3q^{34}$$ $$\mathstrut -\mathstrut 5q^{35}$$ $$\mathstrut +\mathstrut 40q^{38}$$ $$\mathstrut +\mathstrut 6q^{40}$$ $$\mathstrut -\mathstrut 5q^{41}$$ $$\mathstrut -\mathstrut 7q^{43}$$ $$\mathstrut +\mathstrut 10q^{44}$$ $$\mathstrut +\mathstrut 3q^{46}$$ $$\mathstrut -\mathstrut 27q^{47}$$ $$\mathstrut +\mathstrut 25q^{49}$$ $$\mathstrut -\mathstrut 19q^{50}$$ $$\mathstrut +\mathstrut 20q^{52}$$ $$\mathstrut +\mathstrut 21q^{53}$$ $$\mathstrut +\mathstrut 4q^{55}$$ $$\mathstrut +\mathstrut 45q^{56}$$ $$\mathstrut +\mathstrut 20q^{58}$$ $$\mathstrut -\mathstrut 30q^{59}$$ $$\mathstrut -\mathstrut 14q^{61}$$ $$\mathstrut +\mathstrut 12q^{62}$$ $$\mathstrut -\mathstrut 50q^{64}$$ $$\mathstrut +\mathstrut 11q^{65}$$ $$\mathstrut -\mathstrut 2q^{67}$$ $$\mathstrut +\mathstrut 54q^{68}$$ $$\mathstrut -\mathstrut 29q^{70}$$ $$\mathstrut +\mathstrut 6q^{71}$$ $$\mathstrut +\mathstrut 15q^{73}$$ $$\mathstrut -\mathstrut 72q^{74}$$ $$\mathstrut +\mathstrut 5q^{76}$$ $$\mathstrut +\mathstrut 31q^{77}$$ $$\mathstrut -\mathstrut 4q^{79}$$ $$\mathstrut -\mathstrut 20q^{80}$$ $$\mathstrut -\mathstrut 5q^{82}$$ $$\mathstrut -\mathstrut 9q^{83}$$ $$\mathstrut -\mathstrut 6q^{85}$$ $$\mathstrut -\mathstrut 16q^{86}$$ $$\mathstrut +\mathstrut 36q^{88}$$ $$\mathstrut -\mathstrut 28q^{89}$$ $$\mathstrut -\mathstrut 4q^{91}$$ $$\mathstrut -\mathstrut 27q^{92}$$ $$\mathstrut -\mathstrut 3q^{94}$$ $$\mathstrut +\mathstrut 14q^{95}$$ $$\mathstrut -\mathstrut 12q^{97}$$ $$\mathstrut -\mathstrut 59q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10}\mathstrut -\mathstrut$$ $$2$$ $$x^{9}\mathstrut +\mathstrut$$ $$9$$ $$x^{8}\mathstrut -\mathstrut$$ $$8$$ $$x^{7}\mathstrut +\mathstrut$$ $$40$$ $$x^{6}\mathstrut -\mathstrut$$ $$36$$ $$x^{5}\mathstrut +\mathstrut$$ $$90$$ $$x^{4}\mathstrut -\mathstrut$$ $$3$$ $$x^{3}\mathstrut +\mathstrut$$ $$36$$ $$x^{2}\mathstrut -\mathstrut$$ $$9$$ $$x\mathstrut +\mathstrut$$ $$9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{9} - 9 \nu^{8} - 3 \nu^{7} - 61 \nu^{6} - 72 \nu^{5} - 282 \nu^{4} - 204 \nu^{3} - 387 \nu^{2} - 873 \nu - 117$$$$)/189$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{9} - 12 \nu^{8} + 48 \nu^{7} - 23 \nu^{6} + 204 \nu^{5} - 240 \nu^{4} + 303 \nu^{3} - 108 \nu^{2} + 36 \nu - 1557$$$$)/567$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{9} - \nu^{8} + 12 \nu^{7} + 8 \nu^{6} + 68 \nu^{5} + 30 \nu^{4} + 123 \nu^{3} + 204 \nu^{2} + 270 \nu + 63$$$$)/63$$ $$\beta_{5}$$ $$=$$ $$($$$$16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} - 351 \nu^{2} + 684 \nu - 180$$$$)/567$$ $$\beta_{6}$$ $$=$$ $$($$$$20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + 1620 \nu^{2} + 369 \nu - 63$$$$)/567$$ $$\beta_{7}$$ $$=$$ $$($$$$-53 \nu^{9} + 60 \nu^{8} - 375 \nu^{7} - 11 \nu^{6} - 1668 \nu^{5} - 69 \nu^{4} - 2757 \nu^{3} - 4401 \nu^{2} - 1071 \nu - 1368$$$$)/567$$ $$\beta_{8}$$ $$=$$ $$($$$$-82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} - 432 \nu^{2} - 2898 \nu + 720$$$$)/567$$ $$\beta_{9}$$ $$=$$ $$($$$$-91 \nu^{9} + 174 \nu^{8} - 813 \nu^{7} + 704 \nu^{6} - 3633 \nu^{5} + 3174 \nu^{4} - 8070 \nu^{3} + 648 \nu^{2} - 3222 \nu + 801$$$$)/567$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{3}$$ $$\nu^{3}$$ $$=$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1$$ $$\nu^{4}$$ $$=$$ $$-$$$$5$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$14$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$14$$ $$\nu^{5}$$ $$=$$ $$2$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$9$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$17$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{3}$$ $$\nu^{6}$$ $$=$$ $$9$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$10$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$10$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$24$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$9$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$70$$ $$\nu^{7}$$ $$=$$ $$11$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$65$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$43$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$19$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$75$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$65$$ $$\nu^{8}$$ $$=$$ $$-$$$$62$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$73$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$118$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$360$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$14$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$118$$ $$\beta_{3}$$ $$\nu^{9}$$ $$=$$ $$-$$$$135$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$253$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$343$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$253$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$87$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$135$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$343$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$430$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
100.1
 1.19343 + 2.06709i 0.920620 + 1.59456i 0.247934 + 0.429435i −0.335166 − 0.580525i −1.02682 − 1.77851i 1.19343 − 2.06709i 0.920620 − 1.59456i 0.247934 − 0.429435i −0.335166 + 0.580525i −1.02682 + 1.77851i
−1.19343 2.06709i 0 −1.84857 + 3.20182i 2.92087 0 2.35742 1.20106i 4.05086 0 −3.48586 6.03769i
100.2 −0.920620 1.59456i 0 −0.695084 + 1.20392i −1.33475 0 −2.54347 0.728536i −1.12285 0 1.22880 + 2.12835i
100.3 −0.247934 0.429435i 0 0.877057 1.51911i 3.69258 0 −2.60948 + 0.436591i −1.86155 0 −0.915516 1.58572i
100.4 0.335166 + 0.580525i 0 0.775327 1.34291i −1.42494 0 2.21529 + 1.44655i 2.38012 0 −0.477591 0.827212i
100.5 1.02682 + 1.77851i 0 −1.10873 + 1.92038i 0.146246 0 0.0802402 + 2.64453i −0.446582 0 0.150168 + 0.260099i
172.1 −1.19343 + 2.06709i 0 −1.84857 3.20182i 2.92087 0 2.35742 + 1.20106i 4.05086 0 −3.48586 + 6.03769i
172.2 −0.920620 + 1.59456i 0 −0.695084 1.20392i −1.33475 0 −2.54347 + 0.728536i −1.12285 0 1.22880 2.12835i
172.3 −0.247934 + 0.429435i 0 0.877057 + 1.51911i 3.69258 0 −2.60948 0.436591i −1.86155 0 −0.915516 + 1.58572i
172.4 0.335166 0.580525i 0 0.775327 + 1.34291i −1.42494 0 2.21529 1.44655i 2.38012 0 −0.477591 + 0.827212i
172.5 1.02682 1.77851i 0 −1.10873 1.92038i 0.146246 0 0.0802402 2.64453i −0.446582 0 0.150168 0.260099i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 172.5 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.g Even 1 no

## Hecke kernels

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