# Properties

 Label 1089.2.e.k Level $1089$ Weight $2$ Character orbit 1089.e Analytic conductor $8.696$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# 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: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 3 x^{14} + 5 x^{12} + 15 x^{10} + 45 x^{8} + 60 x^{6} + 80 x^{4} + 192 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} - \beta_{6} ) q^{2} -\beta_{13} q^{3} + ( 2 \beta_{5} - \beta_{14} ) q^{4} + ( \beta_{4} + \beta_{8} ) q^{5} + ( -\beta_{1} + \beta_{2} + 2 \beta_{10} + \beta_{12} ) q^{6} + ( \beta_{2} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{12} + \beta_{15} ) q^{7} + ( -\beta_{1} + \beta_{2} - \beta_{7} + 2 \beta_{10} - 2 \beta_{11} - \beta_{15} ) q^{8} + ( -1 + \beta_{8} + \beta_{14} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} - \beta_{6} ) q^{2} -\beta_{13} q^{3} + ( 2 \beta_{5} - \beta_{14} ) q^{4} + ( \beta_{4} + \beta_{8} ) q^{5} + ( -\beta_{1} + \beta_{2} + 2 \beta_{10} + \beta_{12} ) q^{6} + ( \beta_{2} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{12} + \beta_{15} ) q^{7} + ( -\beta_{1} + \beta_{2} - \beta_{7} + 2 \beta_{10} - 2 \beta_{11} - \beta_{15} ) q^{8} + ( -1 + \beta_{8} + \beta_{14} ) q^{9} + ( 2 \beta_{7} + \beta_{10} - \beta_{15} ) q^{10} + ( -1 + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} + 2 \beta_{13} + \beta_{14} ) q^{12} + ( -\beta_{2} - \beta_{6} + \beta_{10} - \beta_{11} ) q^{13} + ( 4 \beta_{3} + 3 \beta_{4} + 3 \beta_{8} + 4 \beta_{13} ) q^{14} + ( -4 - \beta_{3} - \beta_{5} + \beta_{8} + \beta_{9} + \beta_{14} ) q^{15} + ( -2 + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{9} - \beta_{13} ) q^{16} + ( -\beta_{1} + \beta_{2} - 2 \beta_{7} - \beta_{10} + 2 \beta_{11} + 2 \beta_{15} ) q^{17} + ( \beta_{1} - \beta_{2} + \beta_{6} + 2 \beta_{7} - 2 \beta_{10} + 2 \beta_{11} - \beta_{15} ) q^{18} + ( 2 \beta_{7} - 2 \beta_{11} ) q^{19} + ( -1 + \beta_{4} - \beta_{5} + \beta_{8} + 2 \beta_{9} + \beta_{13} ) q^{20} + ( -\beta_{1} - 2 \beta_{2} + \beta_{7} - \beta_{10} + 3 \beta_{11} - \beta_{12} ) q^{21} + ( -\beta_{3} - \beta_{4} - 6 \beta_{5} - \beta_{8} - \beta_{13} ) q^{23} + ( 3 \beta_{1} - 2 \beta_{2} - \beta_{6} - 3 \beta_{10} + \beta_{11} + \beta_{12} + 3 \beta_{15} ) q^{24} + ( -2 - \beta_{3} - 2 \beta_{5} + \beta_{8} + 2 \beta_{9} + \beta_{13} ) q^{25} + ( -4 + \beta_{3} - \beta_{4} + 2 \beta_{8} - \beta_{9} + 2 \beta_{13} - \beta_{14} ) q^{26} + ( -2 - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} + \beta_{13} - \beta_{14} ) q^{27} + ( 6 \beta_{1} - 6 \beta_{2} - 7 \beta_{10} + 2 \beta_{11} + \beta_{15} ) q^{28} + ( 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{6} + 3 \beta_{7} + \beta_{10} - 3 \beta_{12} + \beta_{15} ) q^{29} + ( 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{6} + 2 \beta_{7} - \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{15} ) q^{30} + ( \beta_{3} - 2 \beta_{5} + \beta_{13} ) q^{31} + ( 3 \beta_{1} + 3 \beta_{12} + 3 \beta_{15} ) q^{32} + ( 6 + 4 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} + \beta_{8} - \beta_{13} ) q^{34} + ( -\beta_{7} + \beta_{10} + 3 \beta_{11} - \beta_{15} ) q^{35} + ( 5 - 5 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} - \beta_{13} + \beta_{14} ) q^{36} + ( -1 - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{8} + \beta_{9} - \beta_{13} + \beta_{14} ) q^{37} + ( -2 \beta_{3} - 2 \beta_{4} ) q^{38} + ( -\beta_{1} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{10} - \beta_{11} - \beta_{12} ) q^{39} + ( -2 \beta_{1} - 2 \beta_{2} - 3 \beta_{6} + 2 \beta_{10} - 3 \beta_{11} + \beta_{12} - 2 \beta_{15} ) q^{40} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{6} - \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{15} ) q^{41} + ( 4 - 3 \beta_{3} + 9 \beta_{5} - \beta_{8} + 3 \beta_{9} - \beta_{14} ) q^{42} + ( -\beta_{1} + \beta_{2} + 2 \beta_{6} - \beta_{7} + 2 \beta_{10} + \beta_{12} + 2 \beta_{15} ) q^{43} + ( -3 - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{8} - \beta_{9} + 2 \beta_{13} ) q^{45} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{7} - 4 \beta_{10} + 6 \beta_{11} + 6 \beta_{15} ) q^{46} + ( 2 + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} - \beta_{9} + \beta_{13} ) q^{47} + ( 4 - 2 \beta_{3} - 2 \beta_{4} + 7 \beta_{5} + 5 \beta_{8} + 2 \beta_{9} + \beta_{13} + 2 \beta_{14} ) q^{48} + ( -\beta_{3} - 3 \beta_{4} + 4 \beta_{5} - 3 \beta_{8} - \beta_{13} - 2 \beta_{14} ) q^{49} + ( -\beta_{1} + \beta_{2} - 2 \beta_{6} - \beta_{10} - 2 \beta_{11} - 4 \beta_{12} - \beta_{15} ) q^{50} + ( -3 \beta_{1} - 2 \beta_{2} + \beta_{6} - 2 \beta_{7} + 2 \beta_{11} + 3 \beta_{12} + \beta_{15} ) q^{51} + ( 2 \beta_{1} - \beta_{2} + 4 \beta_{6} - 3 \beta_{10} - 3 \beta_{15} ) q^{52} + ( -7 - \beta_{3} + \beta_{4} + \beta_{8} - \beta_{9} + 3 \beta_{13} - \beta_{14} ) q^{53} + ( \beta_{1} + 4 \beta_{2} + 4 \beta_{6} + \beta_{10} - \beta_{11} - 2 \beta_{12} - 3 \beta_{15} ) q^{54} + ( 5 - 6 \beta_{3} - 9 \beta_{4} + 5 \beta_{5} + 5 \beta_{8} + 2 \beta_{9} - 3 \beta_{13} ) q^{56} + ( -4 \beta_{6} - 2 \beta_{7} - 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{15} ) q^{57} + ( \beta_{3} - \beta_{4} - 7 \beta_{5} - \beta_{8} + \beta_{13} + \beta_{14} ) q^{58} + ( -2 \beta_{4} - 7 \beta_{5} - 2 \beta_{8} + \beta_{14} ) q^{59} + ( 1 - \beta_{3} - \beta_{4} - \beta_{5} - 4 \beta_{8} + \beta_{9} - 2 \beta_{13} + 2 \beta_{14} ) q^{60} + ( -\beta_{1} - 3 \beta_{2} - 4 \beta_{7} - 2 \beta_{10} + 4 \beta_{12} - 2 \beta_{15} ) q^{61} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{7} - 5 \beta_{10} + 2 \beta_{11} + 3 \beta_{15} ) q^{62} + ( -2 \beta_{1} + 5 \beta_{2} + 4 \beta_{6} + 2 \beta_{7} + 4 \beta_{10} - \beta_{11} - \beta_{15} ) q^{63} + ( -1 + 2 \beta_{3} - 2 \beta_{4} + 6 \beta_{8} + \beta_{9} + 3 \beta_{13} + \beta_{14} ) q^{64} + ( \beta_{1} + 2 \beta_{2} + 3 \beta_{7} + \beta_{10} - 3 \beta_{12} + \beta_{15} ) q^{65} + ( \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{8} + \beta_{13} + \beta_{14} ) q^{67} + ( \beta_{1} - 3 \beta_{2} - 2 \beta_{6} + 3 \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{15} ) q^{68} + ( \beta_{3} - 6 \beta_{4} - 2 \beta_{5} - \beta_{9} - 6 \beta_{13} ) q^{69} + ( -1 + \beta_{3} + 4 \beta_{4} - \beta_{5} - \beta_{8} + 2 \beta_{9} + 3 \beta_{13} ) q^{70} + ( 4 + \beta_{3} - \beta_{4} + \beta_{8} - 2 \beta_{9} + 2 \beta_{13} - 2 \beta_{14} ) q^{71} + ( -8 \beta_{1} - 2 \beta_{2} - 5 \beta_{6} + 7 \beta_{10} - 4 \beta_{11} - 2 \beta_{12} - 6 \beta_{15} ) q^{72} + ( \beta_{1} - \beta_{2} - 2 \beta_{7} - \beta_{10} + 3 \beta_{11} ) q^{73} + ( 2 \beta_{1} + 7 \beta_{2} - \beta_{6} + 2 \beta_{7} + 5 \beta_{10} - 2 \beta_{12} + 5 \beta_{15} ) q^{74} + ( -1 - 2 \beta_{4} - 3 \beta_{5} - 5 \beta_{8} - 2 \beta_{13} + \beta_{14} ) q^{75} + ( -4 \beta_{1} - 4 \beta_{6} - 4 \beta_{11} + 2 \beta_{12} - 4 \beta_{15} ) q^{76} + ( -3 + 3 \beta_{3} + \beta_{4} + 3 \beta_{5} + 5 \beta_{13} - 3 \beta_{14} ) q^{78} + ( -\beta_{2} + 2 \beta_{6} - \beta_{7} - \beta_{10} + \beta_{12} - \beta_{15} ) q^{79} + ( -10 - 2 \beta_{3} + 2 \beta_{4} + \beta_{8} + 2 \beta_{9} + \beta_{13} + 2 \beta_{14} ) q^{80} + ( -4 + 3 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} + \beta_{8} + 4 \beta_{13} - 2 \beta_{14} ) q^{81} + ( 6 - \beta_{3} + \beta_{4} + 5 \beta_{8} + 3 \beta_{9} + 3 \beta_{13} + 3 \beta_{14} ) q^{82} + ( -2 \beta_{1} - 3 \beta_{6} - 5 \beta_{7} + 2 \beta_{10} + 5 \beta_{12} + 2 \beta_{15} ) q^{83} + ( -8 \beta_{1} + \beta_{2} - 4 \beta_{6} + 13 \beta_{10} - 11 \beta_{11} - 6 \beta_{12} - 7 \beta_{15} ) q^{84} + ( -5 \beta_{1} - 6 \beta_{2} + 6 \beta_{10} + 5 \beta_{12} - 5 \beta_{15} ) q^{85} + ( 3 \beta_{3} + 5 \beta_{4} + 5 \beta_{8} + 3 \beta_{13} + 3 \beta_{14} ) q^{86} + ( \beta_{1} + \beta_{2} - 3 \beta_{6} + \beta_{7} - 5 \beta_{10} - 3 \beta_{11} + \beta_{15} ) q^{87} + ( -2 - 3 \beta_{3} + 3 \beta_{4} - 4 \beta_{8} - 2 \beta_{9} + \beta_{13} - 2 \beta_{14} ) q^{89} + ( 5 \beta_{1} + 4 \beta_{2} + 5 \beta_{6} - 2 \beta_{7} - 7 \beta_{10} + 4 \beta_{11} - 3 \beta_{12} + 4 \beta_{15} ) q^{90} + ( -6 + 2 \beta_{3} - 2 \beta_{4} + \beta_{8} - 3 \beta_{9} + 2 \beta_{13} - 3 \beta_{14} ) q^{91} + ( 11 + 3 \beta_{3} + 4 \beta_{4} + 11 \beta_{5} + 4 \beta_{8} + 5 \beta_{9} + \beta_{13} ) q^{92} + ( 4 - 2 \beta_{4} + 3 \beta_{5} - \beta_{8} - 2 \beta_{13} - \beta_{14} ) q^{93} + ( 5 \beta_{1} + \beta_{12} + 5 \beta_{15} ) q^{94} + ( 2 \beta_{2} - 6 \beta_{6} - 2 \beta_{10} - 6 \beta_{11} + 4 \beta_{12} ) q^{95} + ( -6 \beta_{1} - 6 \beta_{2} - 6 \beta_{6} + 3 \beta_{7} + 3 \beta_{10} - 3 \beta_{11} - 3 \beta_{12} - 9 \beta_{15} ) q^{96} + ( -\beta_{3} - 4 \beta_{4} + 5 \beta_{8} + 2 \beta_{9} - 3 \beta_{13} ) q^{97} + ( -4 \beta_{1} + 4 \beta_{2} - 7 \beta_{7} + 8 \beta_{10} - 8 \beta_{11} - 4 \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 2q^{3} - 16q^{4} + 4q^{5} - 14q^{9} + O(q^{10})$$ $$16q - 2q^{3} - 16q^{4} + 4q^{5} - 14q^{9} - 6q^{12} + 4q^{14} - 52q^{15} - 24q^{16} - 6q^{20} + 46q^{23} - 12q^{25} - 60q^{26} - 32q^{27} + 14q^{31} + 38q^{34} + 54q^{36} - 12q^{37} + 4q^{38} - 4q^{42} - 28q^{45} + 16q^{47} + 20q^{48} - 42q^{49} - 96q^{53} + 46q^{56} + 50q^{58} + 48q^{59} + 12q^{60} - 12q^{64} - 22q^{67} - 10q^{69} - 4q^{70} + 68q^{71} - 10q^{75} - 72q^{78} - 148q^{80} - 14q^{81} + 112q^{82} + 14q^{86} - 16q^{89} - 96q^{91} + 84q^{92} + 30q^{93} - 4q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 3 x^{14} + 5 x^{12} + 15 x^{10} + 45 x^{8} + 60 x^{6} + 80 x^{4} + 192 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{15} + 25 \nu^{13} + 255 \nu^{11} + 525 \nu^{9} + 615 \nu^{7} + 1920 \nu^{5} + 1840 \nu^{3} + 4928 \nu$$$$)/5760$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{15} + 5 \nu^{13} + 15 \nu^{11} - 75 \nu^{9} + 15 \nu^{7} - 30 \nu^{5} + 140 \nu^{3} + 352 \nu$$$$)/1440$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{14} + 5 \nu^{12} + 15 \nu^{10} - 75 \nu^{8} + 15 \nu^{6} - 30 \nu^{4} - 580 \nu^{2} - 1088$$$$)/720$$ $$\beta_{4}$$ $$=$$ $$($$$$-7 \nu^{14} - 65 \nu^{12} - 135 \nu^{10} - 165 \nu^{8} - 495 \nu^{6} - 1920 \nu^{4} + 160 \nu^{2} - 64$$$$)/2880$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{14} + 25 \nu^{12} + 15 \nu^{10} + 45 \nu^{8} + 135 \nu^{6} + 180 \nu^{4} - 64$$$$)/960$$ $$\beta_{6}$$ $$=$$ $$($$$$2 \nu^{15} - 5 \nu^{13} - 15 \nu^{11} + 15 \nu^{9} + 45 \nu^{7} - 15 \nu^{5} + 100 \nu^{3} + 224 \nu$$$$)/480$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{13} - 7 \nu^{11} - 29 \nu^{9} - 55 \nu^{7} - 26 \nu^{5} - 192 \nu^{3} - 256 \nu$$$$)/192$$ $$\beta_{8}$$ $$=$$ $$($$$$17 \nu^{14} + 25 \nu^{12} + 15 \nu^{10} + 405 \nu^{8} + 735 \nu^{6} + 450 \nu^{4} + 1600 \nu^{2} + 5024$$$$)/1440$$ $$\beta_{9}$$ $$=$$ $$($$$$-37 \nu^{14} - 95 \nu^{12} - 345 \nu^{10} - 1035 \nu^{8} - 2145 \nu^{6} - 2700 \nu^{4} - 8480 \nu^{2} - 10624$$$$)/2880$$ $$\beta_{10}$$ $$=$$ $$($$$$\nu^{15} - \nu^{13} + 3 \nu^{11} + 9 \nu^{9} + 3 \nu^{7} - 18 \nu^{5} + 2 \nu^{3} - 8 \nu$$$$)/144$$ $$\beta_{11}$$ $$=$$ $$($$$$-\nu^{15} - 3 \nu^{13} - 5 \nu^{11} - 15 \nu^{9} - 45 \nu^{7} - 60 \nu^{5} - 80 \nu^{3} - 64 \nu$$$$)/128$$ $$\beta_{12}$$ $$=$$ $$($$$$17 \nu^{15} + 15 \nu^{13} + 25 \nu^{11} + 155 \nu^{9} + 145 \nu^{7} + 440 \nu^{5} + 1040 \nu^{3} + 1664 \nu$$$$)/1920$$ $$\beta_{13}$$ $$=$$ $$($$$$14 \nu^{14} + 25 \nu^{12} + 75 \nu^{10} + 165 \nu^{8} + 255 \nu^{6} + 435 \nu^{4} + 700 \nu^{2} + 608$$$$)/720$$ $$\beta_{14}$$ $$=$$ $$($$$$31 \nu^{14} + 45 \nu^{12} + 75 \nu^{10} + 225 \nu^{8} + 675 \nu^{6} + 660 \nu^{4} + 1520 \nu^{2} + 2112$$$$)/960$$ $$\beta_{15}$$ $$=$$ $$($$$$5 \nu^{15} + 10 \nu^{13} + 6 \nu^{11} + 54 \nu^{9} + 114 \nu^{7} + 63 \nu^{5} + 64 \nu^{3} + 368 \nu$$$$)/288$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} + 2 \beta_{11} - \beta_{10} + \beta_{6} + 2 \beta_{2} + 2 \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{14} - \beta_{13} - \beta_{9} - 2 \beta_{8} + \beta_{5} + \beta_{4} - 2 \beta_{3} - 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{15} + 3 \beta_{12} - \beta_{11} - \beta_{10} - 3 \beta_{7} - 2 \beta_{6} - \beta_{2} - 4 \beta_{1}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{14} - 2 \beta_{13} + \beta_{9} - \beta_{8} - 4 \beta_{5} - 7 \beta_{4} - 4 \beta_{3} - 2$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-2 \beta_{15} + 3 \beta_{12} - 4 \beta_{11} - \beta_{10} + 6 \beta_{7} + 4 \beta_{6} - \beta_{2} + 5 \beta_{1}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{13} - \beta_{9} + 2 \beta_{8} - \beta_{5} + 4 \beta_{3} - 3$$ $$\nu^{7}$$ $$=$$ $$($$$$\beta_{15} - 15 \beta_{12} - 7 \beta_{11} + 2 \beta_{10} + 10 \beta_{6} + 8 \beta_{2} + 2 \beta_{1}$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$-11 \beta_{14} + 14 \beta_{13} + 5 \beta_{9} + 13 \beta_{8} + \beta_{5} + \beta_{4} - 5 \beta_{3} - 22$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$4 \beta_{15} + 3 \beta_{12} + 8 \beta_{11} - \beta_{10} - 9 \beta_{7} - 11 \beta_{6} - 40 \beta_{2} - \beta_{1}$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$-22 \beta_{14} + 31 \beta_{13} - 14 \beta_{9} - 4 \beta_{8} - 13 \beta_{5} + 17 \beta_{4} + 41 \beta_{3} + 46$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$-5 \beta_{15} - 18 \beta_{12} - 4 \beta_{11} + 41 \beta_{10} - 23 \beta_{6} + 44 \beta_{2} + 26 \beta_{1}$$$$)/3$$ $$\nu^{12}$$ $$=$$ $$\beta_{14} - \beta_{13} - \beta_{9} - 14 \beta_{8} + 61 \beta_{5} + 13 \beta_{4} - 14 \beta_{3} + 27$$ $$\nu^{13}$$ $$=$$ $$($$$$46 \beta_{15} + 45 \beta_{12} + 29 \beta_{11} - 91 \beta_{10} - 45 \beta_{7} - 134 \beta_{6} - 43 \beta_{2} - 88 \beta_{1}$$$$)/3$$ $$\nu^{14}$$ $$=$$ $$($$$$130 \beta_{14} + 50 \beta_{13} + 95 \beta_{9} - 35 \beta_{8} - 140 \beta_{5} - 5 \beta_{4} - 80 \beta_{3} + 14$$$$)/3$$ $$\nu^{15}$$ $$=$$ $$($$$$-2 \beta_{15} + 165 \beta_{12} - 64 \beta_{11} + 197 \beta_{10} + 150 \beta_{7} + 88 \beta_{6} + 161 \beta_{2} - 49 \beta_{1}$$$$)/3$$

## Character values

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

 $$n$$ $$244$$ $$848$$ $$\chi(n)$$ $$1$$ $$\beta_{5}$$

## 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
 −0.263711 + 1.38941i −1.27069 + 0.620769i 0.485097 + 1.32841i 1.15347 + 0.818235i −1.15347 − 0.818235i −0.485097 − 1.32841i 1.27069 − 0.620769i 0.263711 − 1.38941i −0.263711 − 1.38941i −1.27069 − 0.620769i 0.485097 − 1.32841i 1.15347 − 0.818235i −1.15347 + 0.818235i −0.485097 + 1.32841i 1.27069 + 0.620769i 0.263711 + 1.38941i
−1.33512 2.31249i −1.60832 + 0.642882i −2.56508 + 4.44285i 0.247410 0.428526i 3.63396 + 2.86091i 2.31660 + 4.01248i 8.35829 2.17341 2.06792i −1.32129
364.2 −1.17295 2.03160i 0.949292 1.44874i −1.75160 + 3.03386i 0.779999 1.35100i −4.05673 0.229291i −1.95114 3.37948i 3.52635 −1.19769 2.75055i −3.65958
364.3 −0.907891 1.57251i 0.691682 + 1.58795i −0.648531 + 1.12329i −1.72104 + 2.98093i 1.86910 2.52936i −0.575887 0.997466i −1.27638 −2.04315 + 2.19671i 6.25008
364.4 −0.131877 0.228418i −0.532651 1.64811i 0.965217 1.67180i 1.69363 2.93346i −0.306215 + 0.339016i 1.65673 + 2.86954i −1.03667 −2.43257 + 1.75574i −0.893408
364.5 0.131877 + 0.228418i −0.532651 1.64811i 0.965217 1.67180i 1.69363 2.93346i 0.306215 0.339016i −1.65673 2.86954i 1.03667 −2.43257 + 1.75574i 0.893408
364.6 0.907891 + 1.57251i 0.691682 + 1.58795i −0.648531 + 1.12329i −1.72104 + 2.98093i −1.86910 + 2.52936i 0.575887 + 0.997466i 1.27638 −2.04315 + 2.19671i −6.25008
364.7 1.17295 + 2.03160i 0.949292 1.44874i −1.75160 + 3.03386i 0.779999 1.35100i 4.05673 + 0.229291i 1.95114 + 3.37948i −3.52635 −1.19769 2.75055i 3.65958
364.8 1.33512 + 2.31249i −1.60832 + 0.642882i −2.56508 + 4.44285i 0.247410 0.428526i −3.63396 2.86091i −2.31660 4.01248i −8.35829 2.17341 2.06792i 1.32129
727.1 −1.33512 + 2.31249i −1.60832 0.642882i −2.56508 4.44285i 0.247410 + 0.428526i 3.63396 2.86091i 2.31660 4.01248i 8.35829 2.17341 + 2.06792i −1.32129
727.2 −1.17295 + 2.03160i 0.949292 + 1.44874i −1.75160 3.03386i 0.779999 + 1.35100i −4.05673 + 0.229291i −1.95114 + 3.37948i 3.52635 −1.19769 + 2.75055i −3.65958
727.3 −0.907891 + 1.57251i 0.691682 1.58795i −0.648531 1.12329i −1.72104 2.98093i 1.86910 + 2.52936i −0.575887 + 0.997466i −1.27638 −2.04315 2.19671i 6.25008
727.4 −0.131877 + 0.228418i −0.532651 + 1.64811i 0.965217 + 1.67180i 1.69363 + 2.93346i −0.306215 0.339016i 1.65673 2.86954i −1.03667 −2.43257 1.75574i −0.893408
727.5 0.131877 0.228418i −0.532651 + 1.64811i 0.965217 + 1.67180i 1.69363 + 2.93346i 0.306215 + 0.339016i −1.65673 + 2.86954i 1.03667 −2.43257 1.75574i 0.893408
727.6 0.907891 1.57251i 0.691682 1.58795i −0.648531 1.12329i −1.72104 2.98093i −1.86910 2.52936i 0.575887 0.997466i 1.27638 −2.04315 2.19671i −6.25008
727.7 1.17295 2.03160i 0.949292 + 1.44874i −1.75160 3.03386i 0.779999 + 1.35100i 4.05673 0.229291i 1.95114 3.37948i −3.52635 −1.19769 + 2.75055i 3.65958
727.8 1.33512 2.31249i −1.60832 0.642882i −2.56508 4.44285i 0.247410 + 0.428526i −3.63396 + 2.86091i −2.31660 + 4.01248i −8.35829 2.17341 + 2.06792i 1.32129
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 727.8 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
11.b odd 2 1 inner
99.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.2.e.k 16
9.c even 3 1 inner 1089.2.e.k 16
9.c even 3 1 9801.2.a.bz 8
9.d odd 6 1 9801.2.a.ca 8
11.b odd 2 1 inner 1089.2.e.k 16
99.g even 6 1 9801.2.a.ca 8
99.h odd 6 1 inner 1089.2.e.k 16
99.h odd 6 1 9801.2.a.bz 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.2.e.k 16 1.a even 1 1 trivial
1089.2.e.k 16 9.c even 3 1 inner
1089.2.e.k 16 11.b odd 2 1 inner
1089.2.e.k 16 99.h odd 6 1 inner
9801.2.a.bz 8 9.c even 3 1
9801.2.a.bz 8 99.h odd 6 1
9801.2.a.ca 8 9.d odd 6 1
9801.2.a.ca 8 99.g even 6 1

## Hecke kernels

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

 $$T_{2}^{16} + \cdots$$ $$T_{5}^{8} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$81 + 1215 T^{2} + 17487 T^{4} + 10782 T^{6} + 4555 T^{8} + 1042 T^{10} + 174 T^{12} + 16 T^{14} + T^{16}$$
$3$ $$( 81 + 27 T + 36 T^{2} + 27 T^{3} + 15 T^{4} + 9 T^{5} + 4 T^{6} + T^{7} + T^{8} )^{2}$$
$5$ $$( 81 - 216 T + 477 T^{2} - 300 T^{3} + 178 T^{4} - 26 T^{5} + 15 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$7$ $$22667121 + 21695877 T^{2} + 16990776 T^{4} + 3147123 T^{6} + 400795 T^{8} + 29743 T^{10} + 1608 T^{12} + 49 T^{14} + T^{16}$$
$11$ $$T^{16}$$
$13$ $$6561 + 19683 T^{2} + 43011 T^{4} + 42282 T^{6} + 30375 T^{8} + 6642 T^{10} + 1098 T^{12} + 36 T^{14} + T^{16}$$
$17$ $$( 12321 - 8514 T^{2} + 1603 T^{4} - 82 T^{6} + T^{8} )^{2}$$
$19$ $$( 2304 - 2304 T^{2} + 736 T^{4} - 76 T^{6} + T^{8} )^{2}$$
$23$ $$( 641601 - 528660 T + 283410 T^{2} - 88554 T^{3} + 20119 T^{4} - 3050 T^{5} + 339 T^{6} - 23 T^{7} + T^{8} )^{2}$$
$29$ $$39923636481 + 21302436726 T^{2} + 9840204045 T^{4} + 748087758 T^{6} + 40456588 T^{8} + 1054846 T^{10} + 19917 T^{12} + 166 T^{14} + T^{16}$$
$31$ $$( 81 + 270 T + 864 T^{2} + 246 T^{3} + 217 T^{4} - 88 T^{5} + 45 T^{6} - 7 T^{7} + T^{8} )^{2}$$
$37$ $$( -365 - 369 T - 71 T^{2} + 3 T^{3} + T^{4} )^{4}$$
$41$ $$2313441 + 3791853 T^{2} + 4552596 T^{4} + 2466279 T^{6} + 981223 T^{8} + 87919 T^{10} + 6132 T^{12} + 85 T^{14} + T^{16}$$
$43$ $$9116621361 + 13942613025 T^{2} + 20537205552 T^{4} + 1172242791 T^{6} + 44760883 T^{8} + 1000531 T^{10} + 16416 T^{12} + 157 T^{14} + T^{16}$$
$47$ $$( 363609 - 240597 T + 136287 T^{2} - 24810 T^{3} + 5239 T^{4} - 494 T^{5} + 102 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$53$ $$( -1053 - 81 T + 144 T^{2} + 24 T^{3} + T^{4} )^{4}$$
$59$ $$( 5774409 - 583929 T + 390663 T^{2} - 81810 T^{3} + 27279 T^{4} - 3798 T^{5} + 438 T^{6} - 24 T^{7} + T^{8} )^{2}$$
$61$ $$4193325113121 + 1177446192912 T^{2} + 281350765926 T^{4} + 12624978546 T^{6} + 407116963 T^{8} + 5947126 T^{10} + 62967 T^{12} + 295 T^{14} + T^{16}$$
$67$ $$( 114921 + 41697 T + 21570 T^{2} + 5121 T^{3} + 2053 T^{4} + 455 T^{5} + 102 T^{6} + 11 T^{7} + T^{8} )^{2}$$
$71$ $$( -2367 + 672 T + 22 T^{2} - 17 T^{3} + T^{4} )^{4}$$
$73$ $$( 9 - 348 T^{2} + 442 T^{4} - 127 T^{6} + T^{8} )^{2}$$
$79$ $$4100625 + 11263050 T^{2} + 27673569 T^{4} + 8498682 T^{6} + 1959228 T^{8} + 172530 T^{10} + 11385 T^{12} + 114 T^{14} + T^{16}$$
$83$ $$622503747500625 + 46128903621300 T^{2} + 2414115011754 T^{4} + 57293180442 T^{6} + 960634255 T^{8} + 10106674 T^{10} + 77403 T^{12} + 343 T^{14} + T^{16}$$
$89$ $$( 5193 - 840 T - 215 T^{2} + 4 T^{3} + T^{4} )^{4}$$
$97$ $$( 58079641 - 9953026 T + 3969073 T^{2} + 357398 T^{3} + 83200 T^{4} + 2018 T^{5} + 301 T^{6} + 2 T^{7} + T^{8} )^{2}$$