# Properties

 Label 1089.2.e.i Level 1089 Weight 2 Character orbit 1089.e Analytic conductor 8.696 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1089.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.69570878012$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.508277025.1 Defining polynomial: $$x^{8} - 3 x^{7} + 5 x^{6} - 15 x^{5} + 21 x^{4} + 3 x^{3} - 22 x^{2} + 3 x + 19$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 99) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} + \beta_{7} ) q^{2} + ( 1 + \beta_{7} ) q^{3} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{4} + ( -1 - \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{5} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{6} + ( -\beta_{1} + \beta_{4} + \beta_{6} ) q^{7} + ( -3 + 3 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{8} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} + \beta_{7} ) q^{2} + ( 1 + \beta_{7} ) q^{3} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{4} + ( -1 - \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{5} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{6} + ( -\beta_{1} + \beta_{4} + \beta_{6} ) q^{7} + ( -3 + 3 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{8} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{9} + ( 2 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{6} + 3 \beta_{7} ) q^{10} + ( -3 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{12} + ( 4 - 3 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{13} + ( -1 + \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{7} ) q^{14} + ( 1 - 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{15} + ( -\beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} - 6 \beta_{6} ) q^{16} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{17} + ( -2 - \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{18} + ( -3 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{19} + ( -3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 6 \beta_{6} ) q^{20} + ( 1 + \beta_{2} + \beta_{3} + \beta_{6} ) q^{21} + ( -2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{23} + ( -4 + \beta_{1} + 7 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} - 8 \beta_{6} - 5 \beta_{7} ) q^{24} + ( \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{25} + ( 4 - \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{26} + ( 1 - \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{27} + ( -2 + 4 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{6} - 4 \beta_{7} ) q^{28} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{29} + ( 8 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} ) q^{30} + ( 2 - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{7} ) q^{31} + ( -4 - 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} - 7 \beta_{4} - \beta_{5} + 10 \beta_{6} + 3 \beta_{7} ) q^{32} + ( -3 \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{34} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{35} + ( -4 + 6 \beta_{1} + 8 \beta_{2} - \beta_{3} + 6 \beta_{4} - 7 \beta_{6} - 6 \beta_{7} ) q^{36} + ( 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{37} + ( 4 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{38} + ( 5 + 2 \beta_{1} - 4 \beta_{2} + \beta_{3} - 6 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{39} + ( 5 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{40} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + \beta_{6} + 3 \beta_{7} ) q^{41} + ( -4 + 5 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{42} + ( -\beta_{2} - 5 \beta_{6} - \beta_{7} ) q^{43} + ( 8 - 3 \beta_{1} + 3 \beta_{5} - \beta_{7} ) q^{45} + ( -5 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{46} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{7} ) q^{47} + ( -1 - 4 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} - 7 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{48} + ( 3 + \beta_{1} + \beta_{2} + \beta_{4} - 4 \beta_{6} ) q^{49} + ( 2 - 3 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} - 7 \beta_{6} - 6 \beta_{7} ) q^{50} + ( 7 - \beta_{1} + 2 \beta_{3} - \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{51} + ( -3 \beta_{1} + 7 \beta_{2} + 3 \beta_{4} - 2 \beta_{6} + 7 \beta_{7} ) q^{52} + ( -3 + 3 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{53} + ( -4 + 5 \beta_{1} + 6 \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} + 3 \beta_{6} - 4 \beta_{7} ) q^{54} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 7 \beta_{6} - \beta_{7} ) q^{56} + ( -6 + 4 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{7} ) q^{57} + ( 3 - 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} ) q^{58} + ( 1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{59} + ( 4 + 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{60} + ( \beta_{1} - \beta_{4} + 3 \beta_{6} ) q^{61} + ( 2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{62} + ( -1 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{63} + ( -2 + 6 \beta_{1} - \beta_{2} - 2 \beta_{3} + 7 \beta_{4} - 2 \beta_{5} + \beta_{6} - 6 \beta_{7} ) q^{64} + ( -2 \beta_{1} + \beta_{2} - 4 \beta_{3} + 6 \beta_{4} + 8 \beta_{5} + 3 \beta_{6} - 7 \beta_{7} ) q^{65} + ( -2 - 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{67} + ( -1 + 2 \beta_{1} - \beta_{2} + 8 \beta_{3} - 5 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{68} + ( -2 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{69} + ( \beta_{1} + 3 \beta_{2} - \beta_{4} + 8 \beta_{6} + 3 \beta_{7} ) q^{70} + ( -\beta_{1} - 3 \beta_{2} + 2 \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{71} + ( -6 - 3 \beta_{1} + 5 \beta_{2} - \beta_{3} + 2 \beta_{4} + 9 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} ) q^{72} + ( -4 - 3 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{73} + ( 3 \beta_{1} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 12 \beta_{6} - 2 \beta_{7} ) q^{74} + ( -1 + 3 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 7 \beta_{6} - 2 \beta_{7} ) q^{75} + ( 12 - 5 \beta_{1} - 2 \beta_{2} - 10 \beta_{3} + 3 \beta_{4} + 5 \beta_{5} - 10 \beta_{6} + 2 \beta_{7} ) q^{76} + ( 3 - 7 \beta_{1} + 5 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} + 2 \beta_{5} + 4 \beta_{7} ) q^{78} + ( -\beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} ) q^{79} + ( -7 + 2 \beta_{3} + 2 \beta_{5} ) q^{80} + ( -4 + 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 6 \beta_{6} + 2 \beta_{7} ) q^{81} + ( -2 - 5 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 6 \beta_{6} + 5 \beta_{7} ) q^{82} + ( \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} + 7 \beta_{6} + 2 \beta_{7} ) q^{83} + ( -6 + 6 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{84} + ( 1 - 7 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - 7 \beta_{7} ) q^{85} + ( 9 - 6 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - 5 \beta_{6} - \beta_{7} ) q^{86} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{5} - 5 \beta_{6} - 2 \beta_{7} ) q^{87} + ( -6 + 6 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} - 4 \beta_{6} - 6 \beta_{7} ) q^{89} + ( 8 + 4 \beta_{1} + \beta_{2} + 2 \beta_{3} - 8 \beta_{4} - 5 \beta_{5} + 3 \beta_{6} + 7 \beta_{7} ) q^{90} + ( 7 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{6} + 3 \beta_{7} ) q^{91} + ( -6 \beta_{1} + 7 \beta_{2} - 4 \beta_{3} + 10 \beta_{4} + 8 \beta_{5} - 17 \beta_{6} - \beta_{7} ) q^{92} + ( 2 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 4 \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{93} + ( -4 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} + 5 \beta_{6} + 3 \beta_{7} ) q^{94} + ( -9 + 5 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} + 7 \beta_{6} + \beta_{7} ) q^{95} + ( \beta_{1} - 8 \beta_{2} - \beta_{3} + 2 \beta_{4} - 5 \beta_{5} + 10 \beta_{6} - 7 \beta_{7} ) q^{96} + ( 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 9 \beta_{6} ) q^{97} + ( -4 \beta_{1} + \beta_{3} - 4 \beta_{4} + \beta_{5} + 4 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + q^{2} + 5q^{3} - 11q^{4} - 4q^{5} - 17q^{6} + q^{7} - 5q^{9} + O(q^{10})$$ $$8q + q^{2} + 5q^{3} - 11q^{4} - 4q^{5} - 17q^{6} + q^{7} - 5q^{9} - 2q^{10} - 2q^{12} + 7q^{13} - q^{14} - q^{15} - 17q^{16} + 10q^{17} + 2q^{18} - 18q^{19} + 10q^{20} + 13q^{21} - 14q^{23} - 18q^{24} - 14q^{25} + 44q^{26} + 5q^{27} + 2q^{28} - 6q^{29} + 37q^{30} + 2q^{31} - 34q^{32} - 16q^{34} + 16q^{35} + 11q^{36} + 6q^{37} - 3q^{38} + 22q^{39} + 12q^{40} - 2q^{41} - q^{42} - 21q^{43} + 49q^{45} - 4q^{46} + 7q^{47} - 59q^{48} + 15q^{49} + 23q^{50} + 31q^{51} - 10q^{52} - 12q^{53} + 37q^{54} - 18q^{56} - 33q^{57} + 21q^{58} - 2q^{59} + 73q^{60} + 15q^{61} + 40q^{62} + 5q^{63} + 32q^{64} + 19q^{65} - 14q^{67} - 7q^{68} - 2q^{69} + 38q^{70} - 6q^{71} - 75q^{72} - 44q^{73} - 36q^{74} + 10q^{75} + 42q^{76} + 29q^{78} + 11q^{79} - 68q^{80} + 7q^{81} - 34q^{82} + 18q^{83} - 34q^{84} + 13q^{85} + 24q^{86} + 9q^{87} - 12q^{89} + 80q^{90} + 38q^{91} - 67q^{92} + 20q^{93} - 19q^{94} - 30q^{95} + 50q^{96} - 26q^{97} - 30q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 5 x^{6} - 15 x^{5} + 21 x^{4} + 3 x^{3} - 22 x^{2} + 3 x + 19$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{7} + 79 \nu^{6} - 177 \nu^{5} + 459 \nu^{4} - 1008 \nu^{3} + 1011 \nu^{2} - 752 \nu - 478$$$$)/933$$ $$\beta_{3}$$ $$=$$ $$($$$$35 \nu^{7} + 164 \nu^{6} - 395 \nu^{5} + 260 \nu^{4} - 2687 \nu^{3} + 2894 \nu^{2} + 1604 \nu - 2193$$$$)/1866$$ $$\beta_{4}$$ $$=$$ $$($$$$217 \nu^{7} + 146 \nu^{6} + 39 \nu^{5} - 876 \nu^{4} - 4095 \nu^{3} + 6498 \nu^{2} + 1610 \nu - 2525$$$$)/5598$$ $$\beta_{5}$$ $$=$$ $$($$$$241 \nu^{7} - 328 \nu^{6} + 1101 \nu^{5} - 3630 \nu^{4} + 1953 \nu^{3} - 5166 \nu^{2} + 6122 \nu + 343$$$$)/5598$$ $$\beta_{6}$$ $$=$$ $$($$$$-145 \nu^{7} + 298 \nu^{6} - 585 \nu^{5} + 1944 \nu^{4} - 2019 \nu^{3} - 438 \nu^{2} + 730 \nu + 1799$$$$)/1866$$ $$\beta_{7}$$ $$=$$ $$($$$$701 \nu^{7} - 1016 \nu^{6} + 1863 \nu^{5} - 6966 \nu^{4} + 2181 \nu^{3} + 10122 \nu^{2} - 6296 \nu - 6265$$$$)/5598$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} - \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{7} - 5 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{2} + \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$-\beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{3} + 6 \beta_{2} + 7 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$5 \beta_{7} + 12 \beta_{6} - \beta_{5} + 12 \beta_{4} - 8 \beta_{3} - 8 \beta_{2} - \beta_{1} - 14$$ $$\nu^{6}$$ $$=$$ $$-29 \beta_{7} - 35 \beta_{6} - 7 \beta_{5} + 32 \beta_{4} - \beta_{3} + 2 \beta_{2} - 18 \beta_{1} + 16$$ $$\nu^{7}$$ $$=$$ $$-38 \beta_{7} - 77 \beta_{6} + 20 \beta_{5} - 13 \beta_{4} - 10 \beta_{3} + 92 \beta_{2} + 52 \beta_{1} + 60$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$$.

 $$n$$ $$244$$ $$848$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
364.1
 1.86526 − 0.199842i 0.947217 − 0.807294i −0.577806 + 2.22188i −0.734668 − 0.348716i 1.86526 + 0.199842i 0.947217 + 0.807294i −0.577806 − 2.22188i −0.734668 + 0.348716i
−1.36526 2.36469i 0.240440 1.71528i −2.72785 + 4.72478i 0.468293 0.811107i −4.38438 + 1.77323i −0.259560 0.449571i 9.43585 −2.88438 0.824844i −2.55736
364.2 −0.447217 0.774602i 1.22553 1.22396i 0.599994 1.03922i −1.87447 + 3.24667i −1.49616 0.401921i 0.725528 + 1.25665i −2.86218 0.00384004 3.00000i 3.35317
364.3 1.07781 + 1.86682i −0.635299 + 1.61133i −1.32333 + 2.29208i −1.81197 + 3.13842i −3.69279 + 0.550720i −1.13530 1.96640i −1.39396 −2.19279 2.04736i −7.81179
364.4 1.23467 + 2.13851i 1.66933 + 0.461883i −2.04881 + 3.54864i 1.21814 2.10988i 1.07333 + 4.14015i 1.16933 + 2.02534i −5.17972 2.57333 + 1.54207i 6.01598
727.1 −1.36526 + 2.36469i 0.240440 + 1.71528i −2.72785 4.72478i 0.468293 + 0.811107i −4.38438 1.77323i −0.259560 + 0.449571i 9.43585 −2.88438 + 0.824844i −2.55736
727.2 −0.447217 + 0.774602i 1.22553 + 1.22396i 0.599994 + 1.03922i −1.87447 3.24667i −1.49616 + 0.401921i 0.725528 1.25665i −2.86218 0.00384004 + 3.00000i 3.35317
727.3 1.07781 1.86682i −0.635299 1.61133i −1.32333 2.29208i −1.81197 3.13842i −3.69279 0.550720i −1.13530 + 1.96640i −1.39396 −2.19279 + 2.04736i −7.81179
727.4 1.23467 2.13851i 1.66933 0.461883i −2.04881 3.54864i 1.21814 + 2.10988i 1.07333 4.14015i 1.16933 2.02534i −5.17972 2.57333 1.54207i 6.01598
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 727.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.2.e.i 8
9.c even 3 1 inner 1089.2.e.i 8
9.c even 3 1 9801.2.a.bi 4
9.d odd 6 1 9801.2.a.bl 4
11.b odd 2 1 99.2.e.e 8
33.d even 2 1 297.2.e.e 8
99.g even 6 1 297.2.e.e 8
99.g even 6 1 891.2.a.p 4
99.h odd 6 1 99.2.e.e 8
99.h odd 6 1 891.2.a.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.e.e 8 11.b odd 2 1
99.2.e.e 8 99.h odd 6 1
297.2.e.e 8 33.d even 2 1
297.2.e.e 8 99.g even 6 1
891.2.a.p 4 99.g even 6 1
891.2.a.q 4 99.h odd 6 1
1089.2.e.i 8 1.a even 1 1 trivial
1089.2.e.i 8 9.c even 3 1 inner
9801.2.a.bi 4 9.c even 3 1
9801.2.a.bl 4 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1089, [\chi])$$:

 $$T_{2}^{8} - \cdots$$ $$T_{5}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 2 T^{2} - 3 T^{3} + 2 T^{4} + 4 T^{5} + T^{6} + 6 T^{7} - 11 T^{8} + 12 T^{9} + 4 T^{10} + 32 T^{11} + 32 T^{12} - 96 T^{13} + 128 T^{14} - 128 T^{15} + 256 T^{16}$$
$3$ $$1 - 5 T + 15 T^{2} - 35 T^{3} + 67 T^{4} - 105 T^{5} + 135 T^{6} - 135 T^{7} + 81 T^{8}$$
$5$ $$1 + 4 T + 5 T^{2} - 18 T^{3} - 94 T^{4} - 232 T^{5} - 110 T^{6} + 1011 T^{7} + 3826 T^{8} + 5055 T^{9} - 2750 T^{10} - 29000 T^{11} - 58750 T^{12} - 56250 T^{13} + 78125 T^{14} + 312500 T^{15} + 390625 T^{16}$$
$7$ $$1 - T - 21 T^{2} + 10 T^{3} + 254 T^{4} - 36 T^{5} - 2408 T^{6} + 83 T^{7} + 18573 T^{8} + 581 T^{9} - 117992 T^{10} - 12348 T^{11} + 609854 T^{12} + 168070 T^{13} - 2470629 T^{14} - 823543 T^{15} + 5764801 T^{16}$$
$11$ 1
$13$ $$1 - 7 T + 12 T^{2} - 23 T^{3} + 77 T^{4} + 624 T^{5} - 1550 T^{6} - 622 T^{7} - 8472 T^{8} - 8086 T^{9} - 261950 T^{10} + 1370928 T^{11} + 2199197 T^{12} - 8539739 T^{13} + 57921708 T^{14} - 439239619 T^{15} + 815730721 T^{16}$$
$17$ $$( 1 - 5 T + 44 T^{2} - 86 T^{3} + 682 T^{4} - 1462 T^{5} + 12716 T^{6} - 24565 T^{7} + 83521 T^{8} )^{2}$$
$19$ $$( 1 + 9 T + 76 T^{2} + 432 T^{3} + 2112 T^{4} + 8208 T^{5} + 27436 T^{6} + 61731 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$1 + 14 T + 53 T^{2} + 96 T^{3} + 1775 T^{4} + 8902 T^{5} - 9839 T^{6} + 41958 T^{7} + 1213519 T^{8} + 965034 T^{9} - 5204831 T^{10} + 108310634 T^{11} + 496717775 T^{12} + 617888928 T^{13} + 7845902117 T^{14} + 47667556258 T^{15} + 78310985281 T^{16}$$
$29$ $$1 + 6 T - 71 T^{2} - 192 T^{3} + 4573 T^{4} + 4224 T^{5} - 186713 T^{6} - 77688 T^{7} + 5509213 T^{8} - 2252952 T^{9} - 157025633 T^{10} + 103019136 T^{11} + 3234396013 T^{12} - 3938140608 T^{13} - 42232455791 T^{14} + 103499257854 T^{15} + 500246412961 T^{16}$$
$31$ $$1 - 2 T - 99 T^{2} + 20 T^{3} + 5978 T^{4} + 2478 T^{5} - 249986 T^{6} - 33695 T^{7} + 8127744 T^{8} - 1044545 T^{9} - 240236546 T^{10} + 73822098 T^{11} + 5520808538 T^{12} + 572583020 T^{13} - 87862864419 T^{14} - 55025228222 T^{15} + 852891037441 T^{16}$$
$37$ $$( 1 - 3 T + 67 T^{2} - 189 T^{3} + 2277 T^{4} - 6993 T^{5} + 91723 T^{6} - 151959 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$1 + 2 T - 97 T^{2} - 612 T^{3} + 4493 T^{4} + 38218 T^{5} - 22457 T^{6} - 901512 T^{7} - 2262647 T^{8} - 36961992 T^{9} - 37750217 T^{10} + 2634022778 T^{11} + 12696144173 T^{12} - 70903995012 T^{13} - 460760111377 T^{14} + 389508547762 T^{15} + 7984925229121 T^{16}$$
$43$ $$1 + 21 T + 113 T^{2} + 516 T^{3} + 15670 T^{4} + 131670 T^{5} + 283958 T^{6} + 4210359 T^{7} + 56707993 T^{8} + 181045437 T^{9} + 525038342 T^{10} + 10468686690 T^{11} + 53572611670 T^{12} + 75856356588 T^{13} + 714314024537 T^{14} + 5708190833247 T^{15} + 11688200277601 T^{16}$$
$47$ $$1 - 7 T - 124 T^{2} + 759 T^{3} + 11189 T^{4} - 50870 T^{5} - 638711 T^{6} + 948885 T^{7} + 33885262 T^{8} + 44597595 T^{9} - 1410912599 T^{10} - 5281476010 T^{11} + 54598750709 T^{12} + 174072860313 T^{13} - 1336622700796 T^{14} - 3546361843241 T^{15} + 23811286661761 T^{16}$$
$53$ $$( 1 + 6 T + 167 T^{2} + 789 T^{3} + 11961 T^{4} + 41817 T^{5} + 469103 T^{6} + 893262 T^{7} + 7890481 T^{8} )^{2}$$
$59$ $$1 + 2 T - 211 T^{2} - 180 T^{3} + 27194 T^{4} + 9886 T^{5} - 2406650 T^{6} - 211857 T^{7} + 162461212 T^{8} - 12499563 T^{9} - 8377548650 T^{10} + 2030376794 T^{11} + 329519515034 T^{12} - 128686373820 T^{13} - 8900092598251 T^{14} + 4977302969638 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 - 15 T - 97 T^{2} + 990 T^{3} + 28072 T^{4} - 155670 T^{5} - 2089990 T^{6} - 70125 T^{7} + 207502789 T^{8} - 4277625 T^{9} - 7776852790 T^{10} - 35334132270 T^{11} + 388680448552 T^{12} + 836150337990 T^{13} - 4997476313017 T^{14} - 47141142540315 T^{15} + 191707312997281 T^{16}$$
$67$ $$1 + 14 T - 51 T^{2} - 500 T^{3} + 11354 T^{4} + 6834 T^{5} - 1143698 T^{6} - 2330227 T^{7} + 43235628 T^{8} - 156125209 T^{9} - 5134060322 T^{10} + 2055414342 T^{11} + 228795827834 T^{12} - 675062553500 T^{13} - 4613377490619 T^{14} + 84849962474522 T^{15} + 406067677556641 T^{16}$$
$71$ $$( 1 + 3 T + 197 T^{2} + 909 T^{3} + 17697 T^{4} + 64539 T^{5} + 993077 T^{6} + 1073733 T^{7} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 22 T + 421 T^{2} + 4807 T^{3} + 49780 T^{4} + 350911 T^{5} + 2243509 T^{6} + 8558374 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$1 - 11 T - 171 T^{2} + 1952 T^{3} + 23168 T^{4} - 216654 T^{5} - 1770938 T^{6} + 6655723 T^{7} + 156847173 T^{8} + 525802117 T^{9} - 11052424058 T^{10} - 106818871506 T^{11} + 902395476608 T^{12} + 6006414090848 T^{13} - 41567954894091 T^{14} - 211242998847749 T^{15} + 1517108809906561 T^{16}$$
$83$ $$1 - 18 T - 5 T^{2} + 1620 T^{3} - 2021 T^{4} - 27828 T^{5} - 824807 T^{6} - 4969494 T^{7} + 193494259 T^{8} - 412468002 T^{9} - 5682095423 T^{10} - 15911688636 T^{11} - 95913266741 T^{12} + 6381245841660 T^{13} - 1634701866845 T^{14} - 488448917813286 T^{15} + 2252292232139041 T^{16}$$
$89$ $$( 1 + 6 T + 224 T^{2} + 1554 T^{3} + 26094 T^{4} + 138306 T^{5} + 1774304 T^{6} + 4229814 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$1 + 26 T + 132 T^{2} - 776 T^{3} + 16487 T^{4} + 336432 T^{5} + 568456 T^{6} + 7259774 T^{7} + 271106712 T^{8} + 704198078 T^{9} + 5348602504 T^{10} + 307052402736 T^{11} + 1459582255847 T^{12} - 6663776039432 T^{13} + 109952304650628 T^{14} + 2100755396430938 T^{15} + 7837433594376961 T^{16}$$