# Properties

 Label 81.2.e.a Level $81$ Weight $2$ Character orbit 81.e Analytic conductor $0.647$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 81.e (of order $$9$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.646788256372$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{9})$$ Coefficient field: 12.0.1952986685049.1 Defining polynomial: $$x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 27) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{3} - \beta_{8} ) q^{2} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{4} + ( 1 + \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{5} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{7} + ( \beta_{2} - \beta_{4} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} ) q^{8} +O(q^{10})$$ $$q + ( 1 + \beta_{3} - \beta_{8} ) q^{2} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{4} + ( 1 + \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{5} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{7} + ( \beta_{2} - \beta_{4} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} ) q^{8} + ( -2 - \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{10} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{11} + ( \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{13} + ( -3 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{8} + \beta_{11} ) q^{14} + ( -2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{16} + ( -1 + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{17} + ( 1 + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{19} + ( -2 - \beta_{2} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{10} + 3 \beta_{11} ) q^{20} + ( -\beta_{1} + \beta_{3} - \beta_{5} - 2 \beta_{7} - \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{22} + ( -1 - \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{8} + 3 \beta_{10} + 2 \beta_{11} ) q^{23} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{11} ) q^{25} + ( 3 - \beta_{4} - \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} ) q^{26} + ( -1 + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} ) q^{28} + ( 1 - 2 \beta_{3} + 3 \beta_{7} - \beta_{8} + 3 \beta_{10} - 3 \beta_{11} ) q^{29} + ( -4 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{31} + ( 1 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{32} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} ) q^{34} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - 4 \beta_{11} ) q^{35} + ( 2 + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{37} + ( 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{7} - \beta_{8} + \beta_{9} ) q^{38} + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} + \beta_{8} + 3 \beta_{9} - \beta_{11} ) q^{40} + ( 3 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - \beta_{5} + 5 \beta_{7} - 3 \beta_{9} ) q^{41} + ( 4 + 4 \beta_{1} + 3 \beta_{2} - \beta_{4} - 4 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{43} + ( -2 - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} - 4 \beta_{8} + \beta_{9} + 3 \beta_{10} ) q^{44} + ( 1 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} - 4 \beta_{11} ) q^{46} + ( 3 - \beta_{1} + 2 \beta_{2} - 3 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} + 6 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{47} + ( 1 - \beta_{4} + 2 \beta_{5} + \beta_{6} + 4 \beta_{7} - \beta_{8} + \beta_{9} + 4 \beta_{10} - \beta_{11} ) q^{49} + ( 5 - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{8} - 3 \beta_{9} - \beta_{10} - \beta_{11} ) q^{50} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{5} + 2 \beta_{6} - 5 \beta_{7} - 2 \beta_{8} - \beta_{10} + 5 \beta_{11} ) q^{52} + ( 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{7} - 3 \beta_{8} ) q^{53} + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{55} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{56} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{6} + 2 \beta_{8} - 5 \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{58} + ( -2 - 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} + 2 \beta_{8} - 5 \beta_{9} - \beta_{10} + 5 \beta_{11} ) q^{59} + ( 1 - \beta_{2} + 5 \beta_{3} - 5 \beta_{4} + \beta_{5} + \beta_{6} + 4 \beta_{7} - 5 \beta_{9} - 2 \beta_{10} + 6 \beta_{11} ) q^{61} + ( -6 - 3 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} ) q^{62} + ( 3 - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 3 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} ) q^{64} + ( -4 - 2 \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} + 4 \beta_{10} + 4 \beta_{11} ) q^{65} + ( -4 - 3 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + \beta_{8} - 3 \beta_{9} - \beta_{11} ) q^{67} + ( -1 - 2 \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{68} + ( -1 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{70} + ( -4 - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} - 3 \beta_{10} + 2 \beta_{11} ) q^{71} + ( -6 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{8} + 3 \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{73} + ( 1 + 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{6} + 3 \beta_{7} - \beta_{9} + 3 \beta_{10} ) q^{74} + ( -2 + 4 \beta_{1} - 4 \beta_{3} - 2 \beta_{5} + 2 \beta_{8} + 3 \beta_{9} - 3 \beta_{11} ) q^{76} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{77} + ( -6 - 5 \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{8} - \beta_{10} ) q^{79} + ( -4 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - 3 \beta_{7} + 3 \beta_{8} - \beta_{9} - \beta_{10} ) q^{80} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} - 4 \beta_{5} - 3 \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} ) q^{82} + ( -5 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - \beta_{8} - 3 \beta_{10} + 3 \beta_{11} ) q^{83} + ( -3 - \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{6} + 3 \beta_{9} + 3 \beta_{11} ) q^{85} + ( -2 + 4 \beta_{1} - 4 \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} ) q^{86} + ( -4 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{9} + 5 \beta_{10} - 3 \beta_{11} ) q^{88} + ( 1 + \beta_{1} + 4 \beta_{2} - 4 \beta_{4} - \beta_{6} - \beta_{7} + 5 \beta_{8} - 6 \beta_{9} + 5 \beta_{10} + 2 \beta_{11} ) q^{89} + ( 1 + \beta_{2} + \beta_{3} + \beta_{6} + 4 \beta_{7} + 3 \beta_{8} - 4 \beta_{9} + \beta_{10} ) q^{91} + ( 6 - 3 \beta_{2} + \beta_{4} - 3 \beta_{6} - \beta_{8} + \beta_{9} - 6 \beta_{10} - 6 \beta_{11} ) q^{92} + ( -4 - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} - 5 \beta_{8} + 5 \beta_{11} ) q^{94} + ( -4 - 5 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + 4 \beta_{9} ) q^{95} + ( 1 - 2 \beta_{1} - 3 \beta_{2} + 5 \beta_{4} + 2 \beta_{5} - \beta_{6} - 5 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 8 \beta_{10} - 8 \beta_{11} ) q^{97} + ( 8 - \beta_{2} + 3 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} + \beta_{10} - 5 \beta_{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 6 q^{2} - 6 q^{4} + 3 q^{5} - 6 q^{7} - 6 q^{8} + O(q^{10})$$ $$12 q + 6 q^{2} - 6 q^{4} + 3 q^{5} - 6 q^{7} - 6 q^{8} - 3 q^{10} - 3 q^{11} - 6 q^{13} - 15 q^{14} - 9 q^{17} - 3 q^{19} + 3 q^{20} + 3 q^{22} + 12 q^{23} + 3 q^{25} + 30 q^{26} - 12 q^{28} + 6 q^{29} + 3 q^{31} + 9 q^{34} - 12 q^{35} - 3 q^{37} - 42 q^{38} + 21 q^{40} - 15 q^{41} + 3 q^{43} - 3 q^{44} - 3 q^{46} + 15 q^{47} + 12 q^{49} + 33 q^{50} + 9 q^{52} + 18 q^{53} - 12 q^{55} + 33 q^{56} + 21 q^{58} + 12 q^{59} + 12 q^{61} + 12 q^{62} + 12 q^{64} - 3 q^{65} - 15 q^{67} - 9 q^{68} - 15 q^{70} - 27 q^{71} + 6 q^{73} - 33 q^{74} - 48 q^{76} - 15 q^{77} - 42 q^{79} - 42 q^{80} - 12 q^{82} - 39 q^{83} - 27 q^{85} - 51 q^{86} - 30 q^{88} - 9 q^{89} + 6 q^{91} + 39 q^{92} - 15 q^{94} + 33 q^{95} + 3 q^{97} + 45 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{10} - 5 \nu^{9} + 22 \nu^{8} - 58 \nu^{7} + 127 \nu^{6} - 199 \nu^{5} + 249 \nu^{4} - 224 \nu^{3} + 145 \nu^{2} - 58 \nu + 9$$ $$\beta_{2}$$ $$=$$ $$3 \nu^{11} - 16 \nu^{10} + 71 \nu^{9} - 197 \nu^{8} + 445 \nu^{7} - 747 \nu^{6} + 1006 \nu^{5} - 1030 \nu^{4} + 803 \nu^{3} - 445 \nu^{2} + 155 \nu - 25$$ $$\beta_{3}$$ $$=$$ $$-6 \nu^{11} + 32 \nu^{10} - 140 \nu^{9} + 384 \nu^{8} - 849 \nu^{7} + 1390 \nu^{6} - 1805 \nu^{5} + 1762 \nu^{4} - 1285 \nu^{3} + 649 \nu^{2} - 195 \nu + 25$$ $$\beta_{4}$$ $$=$$ $$-9 \nu^{11} + 49 \nu^{10} - 216 \nu^{9} + 601 \nu^{8} - 1344 \nu^{7} + 2232 \nu^{6} - 2942 \nu^{5} + 2918 \nu^{4} - 2170 \nu^{3} + 1118 \nu^{2} - 348 \nu + 49$$ $$\beta_{5}$$ $$=$$ $$9 \nu^{11} - 50 \nu^{10} + 221 \nu^{9} - 623 \nu^{8} + 1402 \nu^{7} - 2360 \nu^{6} + 3144 \nu^{5} - 3178 \nu^{4} + 2411 \nu^{3} - 1286 \nu^{2} + 421 \nu - 62$$ $$\beta_{6}$$ $$=$$ $$11 \nu^{11} - 60 \nu^{10} + 265 \nu^{9} - 739 \nu^{8} + 1657 \nu^{7} - 2761 \nu^{6} + 3653 \nu^{5} - 3643 \nu^{4} + 2724 \nu^{3} - 1417 \nu^{2} + 442 \nu - 61$$ $$\beta_{7}$$ $$=$$ $$-16 \nu^{11} + 87 \nu^{10} - 383 \nu^{9} + 1064 \nu^{8} - 2375 \nu^{7} + 3936 \nu^{6} - 5176 \nu^{5} + 5122 \nu^{4} - 3802 \nu^{3} + 1958 \nu^{2} - 610 \nu + 85$$ $$\beta_{8}$$ $$=$$ $$-16 \nu^{11} + 89 \nu^{10} - 393 \nu^{9} + 1108 \nu^{8} - 2491 \nu^{7} + 4191 \nu^{6} - 5577 \nu^{5} + 5631 \nu^{4} - 4267 \nu^{3} + 2272 \nu^{2} - 742 \nu + 110$$ $$\beta_{9}$$ $$=$$ $$36 \nu^{11} - 198 \nu^{10} + 873 \nu^{9} - 2443 \nu^{8} + 5472 \nu^{7} - 9134 \nu^{6} + 12076 \nu^{5} - 12058 \nu^{4} + 9024 \nu^{3} - 4708 \nu^{2} + 1486 \nu - 209$$ $$\beta_{10}$$ $$=$$ $$-36 \nu^{11} + 198 \nu^{10} - 873 \nu^{9} + 2444 \nu^{8} - 5476 \nu^{7} + 9150 \nu^{6} - 12110 \nu^{5} + 12120 \nu^{4} - 9096 \nu^{3} + 4772 \nu^{2} - 1519 \nu + 217$$ $$\beta_{11}$$ $$=$$ $$-42 \nu^{11} + 231 \nu^{10} - 1019 \nu^{9} + 2853 \nu^{8} - 6396 \nu^{7} + 10689 \nu^{6} - 14157 \nu^{5} + 14172 \nu^{4} - 10648 \nu^{3} + 5589 \nu^{2} - 1785 \nu + 257$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} + 3$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} - \beta_{10} + \beta_{9} + 4 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_{1} - 6$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{11} + 5 \beta_{10} - 5 \beta_{9} + \beta_{8} - 8 \beta_{7} + 7 \beta_{6} + 4 \beta_{5} + 10 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} - 8 \beta_{1} - 18$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-11 \beta_{11} + 17 \beta_{10} - 5 \beta_{9} - 20 \beta_{8} + 4 \beta_{7} + 16 \beta_{6} - 14 \beta_{5} + 4 \beta_{4} - 14 \beta_{3} - 8 \beta_{2} + \beta_{1} + 6$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$19 \beta_{11} - \beta_{10} + 31 \beta_{9} - 32 \beta_{8} + 43 \beta_{7} - 20 \beta_{6} - 41 \beta_{5} - 44 \beta_{4} + 10 \beta_{3} + 25 \beta_{2} + 40 \beta_{1} + 87$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$85 \beta_{11} - 97 \beta_{10} + 55 \beta_{9} + 64 \beta_{8} + 10 \beta_{7} - 101 \beta_{6} + 19 \beta_{5} - 62 \beta_{4} + 91 \beta_{3} + 70 \beta_{2} + 31 \beta_{1} + 60$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-20 \beta_{11} - 118 \beta_{10} - 134 \beta_{9} + 244 \beta_{8} - 218 \beta_{7} + \beta_{6} + 232 \beta_{5} + 157 \beta_{4} + 52 \beta_{3} - 74 \beta_{2} - 179 \beta_{1} - 357$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$-503 \beta_{11} + 386 \beta_{10} - 440 \beta_{9} - 47 \beta_{8} - 233 \beta_{7} + 514 \beta_{6} + 163 \beta_{5} + 466 \beta_{4} - 431 \beta_{3} - 461 \beta_{2} - 329 \beta_{1} - 639$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$-425 \beta_{11} + 1076 \beta_{10} + 319 \beta_{9} - 1313 \beta_{8} + 955 \beta_{7} + 502 \beta_{6} - 1013 \beta_{5} - 332 \beta_{4} - 743 \beta_{3} - 59 \beta_{2} + 631 \beta_{1} + 1164$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$2299 \beta_{11} - 862 \beta_{10} + 2725 \beta_{9} - 1193 \beta_{8} + 2104 \beta_{7} - 2135 \beta_{6} - 1907 \beta_{5} - 2705 \beta_{4} + 1495 \beta_{3} + 2425 \beta_{2} + 2344 \beta_{1} + 4356$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$4708 \beta_{11} - 6628 \beta_{10} + 985 \beta_{9} + 5506 \beta_{8} - 3107 \beta_{7} - 4679 \beta_{6} + 3238 \beta_{5} - 992 \beta_{4} + 5476 \beta_{3} + 2770 \beta_{2} - 1043 \beta_{1} - 1698$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 0.5 + 1.00210i 0.5 − 1.68614i 0.5 + 0.0126039i 0.5 + 1.27297i 0.5 + 2.22827i 0.5 − 0.258654i 0.5 − 2.22827i 0.5 + 0.258654i 0.5 − 0.0126039i 0.5 − 1.27297i 0.5 − 1.00210i 0.5 + 1.68614i
−0.183082 1.03831i 0 0.834822 0.303850i 1.33735 + 1.12217i 0 −2.31094 0.841112i −1.52266 2.63732i 0 0.920313 1.59403i
10.2 0.417037 + 2.36514i 0 −3.54056 + 1.28866i −0.0713060 0.0598329i 0 0.544891 + 0.198324i −2.12277 3.67675i 0 0.111776 0.193601i
19.1 −0.753189 + 0.274138i 0 −1.03995 + 0.872619i 0.477505 + 2.70806i 0 1.82076 + 1.52780i 1.34559 2.33062i 0 −1.10204 1.90878i
19.2 1.57954 0.574906i 0 0.632343 0.530599i 0.196143 + 1.11238i 0 −2.99441 2.51261i −0.987144 + 1.70978i 0 0.949332 + 1.64429i
37.1 0.318266 0.267057i 0 −0.317323 + 1.79963i 2.08159 0.757639i 0 −0.229151 1.29958i 0.795075 + 1.37711i 0 0.460168 0.797034i
37.2 1.62143 1.36054i 0 0.430663 2.44241i −2.52129 + 0.917674i 0 0.168844 + 0.957561i −0.508086 0.880031i 0 −2.83955 + 4.91825i
46.1 0.318266 + 0.267057i 0 −0.317323 1.79963i 2.08159 + 0.757639i 0 −0.229151 + 1.29958i 0.795075 1.37711i 0 0.460168 + 0.797034i
46.2 1.62143 + 1.36054i 0 0.430663 + 2.44241i −2.52129 0.917674i 0 0.168844 0.957561i −0.508086 + 0.880031i 0 −2.83955 4.91825i
64.1 −0.753189 0.274138i 0 −1.03995 0.872619i 0.477505 2.70806i 0 1.82076 1.52780i 1.34559 + 2.33062i 0 −1.10204 + 1.90878i
64.2 1.57954 + 0.574906i 0 0.632343 + 0.530599i 0.196143 1.11238i 0 −2.99441 + 2.51261i −0.987144 1.70978i 0 0.949332 1.64429i
73.1 −0.183082 + 1.03831i 0 0.834822 + 0.303850i 1.33735 1.12217i 0 −2.31094 + 0.841112i −1.52266 + 2.63732i 0 0.920313 + 1.59403i
73.2 0.417037 2.36514i 0 −3.54056 1.28866i −0.0713060 + 0.0598329i 0 0.544891 0.198324i −2.12277 + 3.67675i 0 0.111776 + 0.193601i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 73.2 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 81.2.e.a 12
3.b odd 2 1 27.2.e.a 12
9.c even 3 1 243.2.e.a 12
9.c even 3 1 243.2.e.b 12
9.d odd 6 1 243.2.e.c 12
9.d odd 6 1 243.2.e.d 12
12.b even 2 1 432.2.u.c 12
15.d odd 2 1 675.2.l.c 12
15.e even 4 2 675.2.u.b 24
27.e even 9 1 inner 81.2.e.a 12
27.e even 9 1 243.2.e.a 12
27.e even 9 1 243.2.e.b 12
27.e even 9 1 729.2.a.d 6
27.e even 9 2 729.2.c.b 12
27.f odd 18 1 27.2.e.a 12
27.f odd 18 1 243.2.e.c 12
27.f odd 18 1 243.2.e.d 12
27.f odd 18 1 729.2.a.a 6
27.f odd 18 2 729.2.c.e 12
108.l even 18 1 432.2.u.c 12
135.n odd 18 1 675.2.l.c 12
135.q even 36 2 675.2.u.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.e.a 12 3.b odd 2 1
27.2.e.a 12 27.f odd 18 1
81.2.e.a 12 1.a even 1 1 trivial
81.2.e.a 12 27.e even 9 1 inner
243.2.e.a 12 9.c even 3 1
243.2.e.a 12 27.e even 9 1
243.2.e.b 12 9.c even 3 1
243.2.e.b 12 27.e even 9 1
243.2.e.c 12 9.d odd 6 1
243.2.e.c 12 27.f odd 18 1
243.2.e.d 12 9.d odd 6 1
243.2.e.d 12 27.f odd 18 1
432.2.u.c 12 12.b even 2 1
432.2.u.c 12 108.l even 18 1
675.2.l.c 12 15.d odd 2 1
675.2.l.c 12 135.n odd 18 1
675.2.u.b 24 15.e even 4 2
675.2.u.b 24 135.q even 36 2
729.2.a.a 6 27.f odd 18 1
729.2.a.d 6 27.e even 9 1
729.2.c.b 12 27.e even 9 2
729.2.c.e 12 27.f odd 18 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(81, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 - 27 T + 27 T^{2} + 45 T^{3} - 18 T^{4} + 9 T^{5} + 6 T^{6} - 54 T^{7} + 72 T^{8} - 48 T^{9} + 21 T^{10} - 6 T^{11} + T^{12}$$
$3$ $$T^{12}$$
$5$ $$9 + 135 T + 837 T^{2} - 1287 T^{3} + 1521 T^{4} - 1008 T^{5} + 303 T^{6} + 63 T^{7} - 63 T^{8} + 12 T^{9} + 3 T^{10} - 3 T^{11} + T^{12}$$
$7$ $$289 - 816 T + 888 T^{2} - 758 T^{3} + 639 T^{4} + 225 T^{5} + 273 T^{6} + 225 T^{7} + 18 T^{8} - 11 T^{9} + 12 T^{10} + 6 T^{11} + T^{12}$$
$11$ $$9 + 27 T + 108 T^{2} + 531 T^{3} + 495 T^{4} - 1341 T^{5} + 1491 T^{6} - 657 T^{7} + 261 T^{8} + 6 T^{9} - 15 T^{10} + 3 T^{11} + T^{12}$$
$13$ $$1 + 84 T + 2994 T^{2} - 2675 T^{3} - 3546 T^{4} + 873 T^{5} + 3729 T^{6} + 2223 T^{7} + 747 T^{8} + 214 T^{9} + 48 T^{10} + 6 T^{11} + T^{12}$$
$17$ $$729 + 2187 T + 8019 T^{2} - 1458 T^{3} + 7047 T^{4} + 1701 T^{5} + 2727 T^{6} + 567 T^{7} + 621 T^{8} + 189 T^{9} + 72 T^{10} + 9 T^{11} + T^{12}$$
$19$ $$361 + 2280 T + 11208 T^{2} + 21604 T^{3} + 33354 T^{4} + 873 T^{5} + 6162 T^{6} + 252 T^{7} + 846 T^{8} - 14 T^{9} + 39 T^{10} + 3 T^{11} + T^{12}$$
$23$ $$106929 - 264870 T + 322812 T^{2} - 288216 T^{3} + 170838 T^{4} - 32220 T^{5} + 22713 T^{6} + 774 T^{7} - 1440 T^{8} + 192 T^{9} + 48 T^{10} - 12 T^{11} + T^{12}$$
$29$ $$45369 + 203202 T + 341037 T^{2} + 619236 T^{3} + 1278324 T^{4} - 188802 T^{5} - 4044 T^{6} + 10206 T^{7} + 423 T^{8} + 96 T^{9} + 21 T^{10} - 6 T^{11} + T^{12}$$
$31$ $$26569 - 77262 T + 73617 T^{2} - 6158 T^{3} - 29034 T^{4} + 14481 T^{5} + 3810 T^{6} - 4365 T^{7} + 1881 T^{8} - 434 T^{9} + 84 T^{10} - 3 T^{11} + T^{12}$$
$37$ $$24334489 + 19105509 T + 12573093 T^{2} + 4411480 T^{3} + 1506987 T^{4} + 331353 T^{5} + 90807 T^{6} + 15399 T^{7} + 3519 T^{8} + 337 T^{9} + 66 T^{10} + 3 T^{11} + T^{12}$$
$41$ $$11229201 - 15350931 T + 16475103 T^{2} - 10695744 T^{3} + 2682513 T^{4} + 178299 T^{5} - 57801 T^{6} - 8208 T^{7} + 4797 T^{8} + 705 T^{9} + 93 T^{10} + 15 T^{11} + T^{12}$$
$43$ $$3308761 - 4709391 T + 49059204 T^{2} - 4380266 T^{3} - 386325 T^{4} + 386325 T^{5} + 10965 T^{6} - 963 T^{7} + 2799 T^{8} + 16 T^{9} - 60 T^{10} - 3 T^{11} + T^{12}$$
$47$ $$42732369 + 17296902 T + 4549149 T^{2} - 1182393 T^{3} + 1192761 T^{4} - 676449 T^{5} + 29409 T^{6} + 30357 T^{7} - 927 T^{8} + 114 T^{9} + 111 T^{10} - 15 T^{11} + T^{12}$$
$53$ $$( -12393 + 2916 T + 4617 T^{2} + 513 T^{3} - 108 T^{4} - 9 T^{5} + T^{6} )^{2}$$
$59$ $$176384961 + 170448354 T + 29313684 T^{2} - 10115676 T^{3} + 2890305 T^{4} + 2031741 T^{5} + 256911 T^{6} + 36189 T^{7} + 5796 T^{8} - 933 T^{9} + 192 T^{10} - 12 T^{11} + T^{12}$$
$61$ $$273670849 - 12754653 T + 226908411 T^{2} + 81354967 T^{3} - 9276372 T^{4} - 1547091 T^{5} + 1238790 T^{6} - 134964 T^{7} + 19512 T^{8} + 34 T^{9} - 51 T^{10} - 12 T^{11} + T^{12}$$
$67$ $$8288641 - 13275069 T + 5116533 T^{2} + 7604980 T^{3} - 2980269 T^{4} - 1780641 T^{5} + 660099 T^{6} + 86274 T^{7} + 23913 T^{8} + 2365 T^{9} + 255 T^{10} + 15 T^{11} + T^{12}$$
$71$ $$729 - 6561 T + 78003 T^{2} + 144342 T^{3} + 604827 T^{4} - 231093 T^{5} + 400761 T^{6} + 147015 T^{7} + 38205 T^{8} + 5103 T^{9} + 504 T^{10} + 27 T^{11} + T^{12}$$
$73$ $$185761 + 619347 T + 3782073 T^{2} - 5509508 T^{3} + 16156512 T^{4} + 498510 T^{5} + 746232 T^{6} + 2871 T^{7} + 27792 T^{8} + 544 T^{9} + 210 T^{10} - 6 T^{11} + T^{12}$$
$79$ $$3508129 - 1848651 T - 6043107 T^{2} + 15614746 T^{3} + 26778888 T^{4} + 7412832 T^{5} + 2309097 T^{6} + 487872 T^{7} + 78291 T^{8} + 9520 T^{9} + 813 T^{10} + 42 T^{11} + T^{12}$$
$83$ $$6951057129 + 10578532986 T + 7328037465 T^{2} + 3047714604 T^{3} + 853146414 T^{4} + 170563383 T^{5} + 25416024 T^{6} + 2898477 T^{7} + 256059 T^{8} + 17196 T^{9} + 912 T^{10} + 39 T^{11} + T^{12}$$
$89$ $$1062042921 + 992530584 T + 765665784 T^{2} + 218178036 T^{3} + 61794900 T^{4} + 6160293 T^{5} + 1737990 T^{6} + 125712 T^{7} + 36666 T^{8} + 432 T^{9} + 261 T^{10} + 9 T^{11} + T^{12}$$
$97$ $$66765241 + 21154719 T + 532756806 T^{2} - 381413396 T^{3} + 112050423 T^{4} - 14303619 T^{5} + 35103 T^{6} + 185823 T^{7} - 11349 T^{8} - 1010 T^{9} + 102 T^{10} - 3 T^{11} + T^{12}$$