# Properties

 Label 162.2.e.b Level $162$ Weight $2$ Character orbit 162.e Analytic conductor $1.294$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 162.e (of order $$9$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.29357651274$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: no (minimal twist has level 54) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{4} + \beta_{8} ) q^{2} -\beta_{6} q^{4} + ( 1 - \beta_{2} + \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{7} + ( 1 - \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( \beta_{4} + \beta_{8} ) q^{2} -\beta_{6} q^{4} + ( 1 - \beta_{2} + \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{7} + ( 1 - \beta_{2} ) q^{8} + ( -1 + \beta_{1} - \beta_{3} ) q^{10} + ( 1 + \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{11} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{9} - \beta_{10} ) q^{13} + ( -\beta_{2} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{14} + \beta_{8} q^{16} + ( 1 + \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{17} + ( -2 + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{8} ) q^{19} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{9} + \beta_{10} ) q^{20} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{8} - \beta_{9} ) q^{22} + ( -3 - \beta_{1} + \beta_{2} - \beta_{4} - 3 \beta_{8} - \beta_{10} ) q^{23} + ( 1 + \beta_{1} - 4 \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{10} + \beta_{11} ) q^{25} + ( -1 - \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{26} + ( \beta_{1} + \beta_{2} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{28} + ( -3 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{29} + ( -2 - \beta_{1} + 3 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{31} + \beta_{5} q^{32} + ( -2 - \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{34} + ( -1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{35} + ( -\beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} + 5 \beta_{5} + 3 \beta_{6} - \beta_{7} + 3 \beta_{8} - \beta_{10} + \beta_{11} ) q^{37} + ( -1 + 2 \beta_{2} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{38} + ( -\beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{11} ) q^{40} + ( 1 + \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{8} + \beta_{9} ) q^{41} + ( -1 + \beta_{1} + 3 \beta_{2} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{11} ) q^{43} + ( -\beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{44} + ( 1 - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{9} - \beta_{11} ) q^{46} + ( -1 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{47} + ( -3 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + 4 \beta_{8} + \beta_{9} + \beta_{10} ) q^{49} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{50} + ( 2 - \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{11} ) q^{52} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{53} + ( 3 - 2 \beta_{1} - \beta_{2} + 5 \beta_{4} - 3 \beta_{5} - 5 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{55} + ( 1 + \beta_{4} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{56} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{58} + ( -2 + \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{59} + ( -4 + \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{61} + ( 3 - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{62} -\beta_{2} q^{64} + ( 4 - 6 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{65} + ( 2 + 4 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{67} + ( 1 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{68} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{5} + 5 \beta_{6} + \beta_{11} ) q^{70} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} + 3 \beta_{6} + 3 \beta_{8} + \beta_{9} + \beta_{11} ) q^{71} + ( -3 + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{9} + \beta_{11} ) q^{73} + ( -1 - 2 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{11} ) q^{74} + ( \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{8} ) q^{76} + ( 4 + 2 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + 3 \beta_{10} - 2 \beta_{11} ) q^{77} + ( 3 - \beta_{2} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} - 5 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{79} + ( -1 + \beta_{1} ) q^{80} + ( 1 - \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{82} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 5 \beta_{4} - 3 \beta_{5} + \beta_{6} + 3 \beta_{7} - 5 \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{83} + ( 3 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{85} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{86} + ( 2 - \beta_{1} - \beta_{2} + \beta_{6} - \beta_{11} ) q^{88} + ( -2 - 2 \beta_{1} - \beta_{2} + 4 \beta_{3} + 6 \beta_{4} + 7 \beta_{5} - \beta_{6} - \beta_{7} + 7 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{89} + ( 3 - \beta_{2} - 3 \beta_{3} - 7 \beta_{4} - \beta_{5} - 5 \beta_{6} + 3 \beta_{7} - 5 \beta_{8} + 3 \beta_{10} - 3 \beta_{11} ) q^{91} + ( -1 + 3 \beta_{2} - \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{92} + ( 4 - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{94} + ( -2 \beta_{1} + 4 \beta_{2} + 3 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} + 4 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{95} + ( 4 - \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} + 4 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + \beta_{11} ) q^{97} + ( 2 \beta_{2} + \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 3q^{5} - 3q^{7} + 6q^{8} + O(q^{10})$$ $$12q + 3q^{5} - 3q^{7} + 6q^{8} - 3q^{10} + 12q^{11} + 12q^{13} + 3q^{14} + 6q^{17} - 9q^{19} - 6q^{20} - 12q^{22} - 30q^{23} - 9q^{25} - 18q^{26} + 12q^{28} - 15q^{29} - 15q^{34} - 3q^{35} - 15q^{37} - 3q^{38} - 3q^{40} + 12q^{41} + 9q^{43} + 3q^{44} + 3q^{46} + 9q^{47} - 39q^{49} + 27q^{50} + 12q^{52} + 12q^{53} + 18q^{55} + 3q^{56} - 3q^{58} - 12q^{59} - 36q^{61} + 12q^{62} - 6q^{64} + 15q^{65} + 36q^{67} - 3q^{68} + 39q^{70} - 12q^{71} - 21q^{73} - 33q^{74} + 3q^{76} - 3q^{77} + 39q^{79} - 6q^{80} + 6q^{82} - 18q^{83} + 45q^{85} - 9q^{86} + 6q^{88} - 12q^{89} - 6q^{91} + 6q^{92} + 36q^{94} + 15q^{95} + 39q^{97} + 12q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} - 3 \nu^{5} + 14 \nu^{4} - 23 \nu^{3} + 41 \nu^{2} - 30 \nu + 11$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$6 \nu^{11} - 33 \nu^{10} + 127 \nu^{9} - 324 \nu^{8} + 438 \nu^{7} - 252 \nu^{6} - 1278 \nu^{5} + 3234 \nu^{4} - 5701 \nu^{3} + 5358 \nu^{2} - 3477 \nu + 1060$$$$)/218$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{11} - 11 \nu^{10} + 115 \nu^{9} - 435 \nu^{8} + 1781 \nu^{7} - 4226 \nu^{6} + 9493 \nu^{5} - 13637 \nu^{4} + 16775 \nu^{3} - 12275 \nu^{2} + 5163 \nu - 882$$$$)/218$$ $$\beta_{4}$$ $$=$$ $$($$$$-27 \nu^{11} + 94 \nu^{10} - 408 \nu^{9} + 586 \nu^{8} - 445 \nu^{7} - 2572 \nu^{6} + 9021 \nu^{5} - 18150 \nu^{4} + 24401 \nu^{3} - 20623 \nu^{2} + 10469 \nu - 2263$$$$)/218$$ $$\beta_{5}$$ $$=$$ $$($$$$-26 \nu^{11} + 34 \nu^{10} - 187 \nu^{9} - 449 \nu^{8} + 1590 \nu^{7} - 6865 \nu^{6} + 12623 \nu^{5} - 20118 \nu^{4} + 19981 \nu^{3} - 12972 \nu^{2} + 4712 \nu - 524$$$$)/218$$ $$\beta_{6}$$ $$=$$ $$($$$$-27 \nu^{11} + 203 \nu^{10} - 953 \nu^{9} + 3311 \nu^{8} - 8075 \nu^{7} + 16285 \nu^{6} - 23134 \nu^{5} + 26758 \nu^{4} - 19635 \nu^{3} + 9570 \nu^{2} - 1957 \nu - 83$$$$)/218$$ $$\beta_{7}$$ $$=$$ $$($$$$-39 \nu^{11} - 58 \nu^{10} + 210 \nu^{9} - 3126 \nu^{8} + 8816 \nu^{7} - 26048 \nu^{6} + 44604 \nu^{5} - 65711 \nu^{4} + 63925 \nu^{3} - 42784 \nu^{2} + 16878 \nu - 3184$$$$)/218$$ $$\beta_{8}$$ $$=$$ $$($$$$-26 \nu^{11} + 252 \nu^{10} - 1277 \nu^{9} + 4892 \nu^{8} - 13234 \nu^{7} + 28887 \nu^{6} - 47327 \nu^{5} + 60760 \nu^{4} - 56973 \nu^{3} + 36296 \nu^{2} - 13927 \nu + 2201$$$$)/218$$ $$\beta_{9}$$ $$=$$ $$($$$$-19 \nu^{11} + 268 \nu^{10} - 1365 \nu^{9} + 5713 \nu^{8} - 15666 \nu^{7} + 35787 \nu^{6} - 59173 \nu^{5} + 78267 \nu^{4} - 73416 \nu^{3} + 47888 \nu^{2} - 18038 \nu + 3256$$$$)/218$$ $$\beta_{10}$$ $$=$$ $$($$$$106 \nu^{11} - 583 \nu^{10} + 3043 \nu^{9} - 9321 \nu^{8} + 24960 \nu^{7} - 47943 \nu^{6} + 77593 \nu^{5} - 93068 \nu^{4} + 88252 \nu^{3} - 58487 \nu^{2} + 25228 \nu - 5544$$$$)/218$$ $$\beta_{11}$$ $$=$$ $$($$$$-106 \nu^{11} + 583 \nu^{10} - 3043 \nu^{9} + 9321 \nu^{8} - 24960 \nu^{7} + 47943 \nu^{6} - 77593 \nu^{5} + 92850 \nu^{4} - 87816 \nu^{3} + 56961 \nu^{2} - 23920 \nu + 4454$$$$)/218$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{7} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 9$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{11} - 2 \beta_{10} + 2 \beta_{9} - \beta_{7} - 6 \beta_{6} + 4 \beta_{5} - 3 \beta_{4} + 7 \beta_{3} - 8 \beta_{2} - 5 \beta_{1} - 4$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{11} - 3 \beta_{9} + 8 \beta_{8} + 5 \beta_{7} - 6 \beta_{6} - 5 \beta_{5} - 14 \beta_{4} + 16 \beta_{3} - 24 \beta_{2} - 4 \beta_{1} + 40$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$10 \beta_{10} - 13 \beta_{9} + 21 \beta_{8} + 12 \beta_{7} + 28 \beta_{6} - 16 \beta_{5} + 3 \beta_{4} - 23 \beta_{3} + 29 \beta_{2} + 24 \beta_{1} + 49$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-10 \beta_{11} + 25 \beta_{10} + 8 \beta_{9} + 3 \beta_{8} - 16 \beta_{7} + 60 \beta_{6} + 43 \beta_{5} + 84 \beta_{4} - 110 \beta_{3} + 187 \beta_{2} + 49 \beta_{1} - 169$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$23 \beta_{11} - 9 \beta_{10} + 87 \beta_{9} - 142 \beta_{8} - 88 \beta_{7} - 105 \beta_{6} + 121 \beta_{5} + 70 \beta_{4} + 25 \beta_{3} + 12 \beta_{2} - 79 \beta_{1} - 395$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$108 \beta_{11} - 188 \beta_{10} + 32 \beta_{9} - 297 \beta_{8} + 6 \beta_{7} - 431 \beta_{6} - 265 \beta_{5} - 429 \beta_{4} + 652 \beta_{3} - 1153 \beta_{2} - 363 \beta_{1} + 607$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$-133 \beta_{11} - 263 \beta_{10} - 511 \beta_{9} + 483 \beta_{8} + 533 \beta_{7} + 261 \beta_{6} - 1040 \beta_{5} - 891 \beta_{4} + 478 \beta_{3} - 1265 \beta_{2} + 49 \beta_{1} + 2735$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$-976 \beta_{11} + 849 \beta_{10} - 717 \beta_{9} + 2720 \beta_{8} + 476 \beta_{7} + 2778 \beta_{6} + 1021 \beta_{5} + 1633 \beta_{4} - 3353 \beta_{3} + 5760 \beta_{2} + 2135 \beta_{1} - 1121$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$45 \beta_{11} + 2914 \beta_{10} + 2441 \beta_{9} + 558 \beta_{8} - 2784 \beta_{7} + 742 \beta_{6} + 7907 \beta_{5} + 7221 \beta_{4} - 6041 \beta_{3} + 13586 \beta_{2} + 2025 \beta_{1} - 16958$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$$.

 $$n$$ $$83$$ $$\chi(n)$$ $$-\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.5 − 0.677980i 0.5 + 1.96356i 0.5 + 0.168222i 0.5 + 1.80139i 0.5 − 1.74095i 0.5 + 2.42499i 0.5 + 1.74095i 0.5 − 2.42499i 0.5 − 0.168222i 0.5 − 1.80139i 0.5 + 0.677980i 0.5 − 1.96356i
0.939693 0.342020i 0 0.766044 0.642788i −0.617090 3.49969i 0 −0.244752 0.205371i 0.500000 0.866025i 0 −1.77684 3.07758i
19.2 0.939693 0.342020i 0 0.766044 0.642788i 0.177398 + 1.00607i 0 2.04289 + 1.71418i 0.500000 0.866025i 0 0.510796 + 0.884725i
37.1 −0.766044 + 0.642788i 0 0.173648 0.984808i −0.696050 + 0.253341i 0 0.717657 + 4.07003i 0.500000 + 0.866025i 0 0.370360 0.641483i
37.2 −0.766044 + 0.642788i 0 0.173648 0.984808i 1.96209 0.714144i 0 −0.696712 3.95125i 0.500000 + 0.866025i 0 −1.04401 + 1.80828i
73.1 −0.173648 + 0.984808i 0 −0.939693 0.342020i −2.42692 + 2.03643i 0 −3.46344 + 1.26059i 0.500000 0.866025i 0 −1.58406 2.74367i
73.2 −0.173648 + 0.984808i 0 −0.939693 0.342020i 3.10057 2.60168i 0 0.144365 0.0525446i 0.500000 0.866025i 0 2.02375 + 3.50524i
91.1 −0.173648 0.984808i 0 −0.939693 + 0.342020i −2.42692 2.03643i 0 −3.46344 1.26059i 0.500000 + 0.866025i 0 −1.58406 + 2.74367i
91.2 −0.173648 0.984808i 0 −0.939693 + 0.342020i 3.10057 + 2.60168i 0 0.144365 + 0.0525446i 0.500000 + 0.866025i 0 2.02375 3.50524i
127.1 −0.766044 0.642788i 0 0.173648 + 0.984808i −0.696050 0.253341i 0 0.717657 4.07003i 0.500000 0.866025i 0 0.370360 + 0.641483i
127.2 −0.766044 0.642788i 0 0.173648 + 0.984808i 1.96209 + 0.714144i 0 −0.696712 + 3.95125i 0.500000 0.866025i 0 −1.04401 1.80828i
145.1 0.939693 + 0.342020i 0 0.766044 + 0.642788i −0.617090 + 3.49969i 0 −0.244752 + 0.205371i 0.500000 + 0.866025i 0 −1.77684 + 3.07758i
145.2 0.939693 + 0.342020i 0 0.766044 + 0.642788i 0.177398 1.00607i 0 2.04289 1.71418i 0.500000 + 0.866025i 0 0.510796 0.884725i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 145.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.2.e.b 12
3.b odd 2 1 54.2.e.b 12
9.c even 3 1 486.2.e.e 12
9.c even 3 1 486.2.e.g 12
9.d odd 6 1 486.2.e.f 12
9.d odd 6 1 486.2.e.h 12
12.b even 2 1 432.2.u.b 12
27.e even 9 1 inner 162.2.e.b 12
27.e even 9 1 486.2.e.e 12
27.e even 9 1 486.2.e.g 12
27.e even 9 1 1458.2.a.f 6
27.e even 9 2 1458.2.c.g 12
27.f odd 18 1 54.2.e.b 12
27.f odd 18 1 486.2.e.f 12
27.f odd 18 1 486.2.e.h 12
27.f odd 18 1 1458.2.a.g 6
27.f odd 18 2 1458.2.c.f 12
108.l even 18 1 432.2.u.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.e.b 12 3.b odd 2 1
54.2.e.b 12 27.f odd 18 1
162.2.e.b 12 1.a even 1 1 trivial
162.2.e.b 12 27.e even 9 1 inner
432.2.u.b 12 12.b even 2 1
432.2.u.b 12 108.l even 18 1
486.2.e.e 12 9.c even 3 1
486.2.e.e 12 27.e even 9 1
486.2.e.f 12 9.d odd 6 1
486.2.e.f 12 27.f odd 18 1
486.2.e.g 12 9.c even 3 1
486.2.e.g 12 27.e even 9 1
486.2.e.h 12 9.d odd 6 1
486.2.e.h 12 27.f odd 18 1
1458.2.a.f 6 27.e even 9 1
1458.2.a.g 6 27.f odd 18 1
1458.2.c.f 12 27.f odd 18 2
1458.2.c.g 12 27.e even 9 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{12} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(162, [\chi])$$.