# Properties

 Label 90.2.l.b Level $90$ Weight $2$ Character orbit 90.l Analytic conductor $0.719$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 90.l (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.718653618192$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: 16.0.9349208943630483456.9 Defining polynomial: $$x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + 9297 x^{8} - 11276 x^{7} + 11224 x^{6} - 9024 x^{5} + 5736 x^{4} - 2780 x^{3} + 972 x^{2} - 220 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{11} q^{2} + ( -\beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{3} -\beta_{9} q^{4} + ( 1 + \beta_{2} - \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{5} + ( -\beta_{2} - \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{12} ) q^{6} + ( -\beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{7} -\beta_{15} q^{8} + ( -\beta_{5} + 2 \beta_{6} + \beta_{9} + 2 \beta_{11} + \beta_{13} + 2 \beta_{15} ) q^{9} +O(q^{10})$$ $$q -\beta_{11} q^{2} + ( -\beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{3} -\beta_{9} q^{4} + ( 1 + \beta_{2} - \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{5} + ( -\beta_{2} - \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{12} ) q^{6} + ( -\beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{7} -\beta_{15} q^{8} + ( -\beta_{5} + 2 \beta_{6} + \beta_{9} + 2 \beta_{11} + \beta_{13} + 2 \beta_{15} ) q^{9} + ( -1 - \beta_{1} - \beta_{3} + \beta_{5} + \beta_{10} + 2 \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{10} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{11} + ( -1 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{12} + ( \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - 2 \beta_{15} ) q^{13} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{14} + ( -1 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - 2 \beta_{9} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{15} ) q^{15} -\beta_{2} q^{16} + ( -1 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{17} + ( 1 + 2 \beta_{2} - \beta_{6} + \beta_{9} + 2 \beta_{13} + \beta_{15} ) q^{18} + ( -1 - \beta_{2} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + 4 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{19} + ( -2 - \beta_{1} + 2 \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{13} - \beta_{15} ) q^{20} + ( 3 + 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{21} + ( 2 + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} ) q^{22} + ( -2 - \beta_{1} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - 3 \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{23} + ( \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{24} + ( -3 - \beta_{1} + 4 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} + 5 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} + 4 \beta_{11} + \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{25} + ( -1 - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{14} ) q^{26} + ( -1 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - \beta_{8} + 3 \beta_{9} + 3 \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} + 3 \beta_{14} - \beta_{15} ) q^{27} + ( \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{8} - 2 \beta_{9} - 2 \beta_{11} + 3 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{28} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} + 6 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} ) q^{29} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{11} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{30} + ( -4 - 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} - 5 \beta_{11} - 4 \beta_{12} + 3 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{31} -\beta_{5} q^{32} + ( -2 - \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{11} - \beta_{12} - 5 \beta_{13} + \beta_{14} - 4 \beta_{15} ) q^{33} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{12} - 4 \beta_{13} - 3 \beta_{15} ) q^{34} + ( 4 + 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} - 2 \beta_{9} - 4 \beta_{10} - \beta_{11} + 2 \beta_{12} - 3 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{35} + ( 2 + \beta_{2} + 2 \beta_{5} - \beta_{11} - \beta_{13} + \beta_{15} ) q^{36} + ( 4 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - 4 \beta_{9} - 4 \beta_{10} - 4 \beta_{11} - 2 \beta_{13} - 2 \beta_{14} ) q^{37} + ( 2 + 2 \beta_{2} + \beta_{3} + 2 \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{11} + 2 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{38} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} - 4 \beta_{11} - 2 \beta_{13} - 2 \beta_{14} - 3 \beta_{15} ) q^{39} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} + 2 \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{40} + ( -1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{11} - 2 \beta_{12} + \beta_{13} + 4 \beta_{14} ) q^{41} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} - 3 \beta_{8} - 2 \beta_{9} - \beta_{10} - 4 \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{42} + ( 4 + 4 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 7 \beta_{6} + 3 \beta_{7} - 2 \beta_{9} - 4 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} - 4 \beta_{13} - 3 \beta_{14} + 4 \beta_{15} ) q^{43} + ( 1 - \beta_{2} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{14} ) q^{44} + ( 1 - \beta_{1} + 5 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - 8 \beta_{11} - 3 \beta_{12} + 6 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{45} + ( -3 + \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{46} + ( 1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - \beta_{7} - 5 \beta_{9} - \beta_{10} - \beta_{11} - 4 \beta_{12} + 5 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{47} + ( -\beta_{3} - \beta_{4} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{48} + ( -2 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + \beta_{9} - 4 \beta_{11} + \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{49} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{50} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} + 5 \beta_{15} ) q^{51} + ( \beta_{3} - 2 \beta_{5} + \beta_{7} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{52} + ( -3 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{53} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + \beta_{7} + 4 \beta_{8} + \beta_{9} + \beta_{10} + 5 \beta_{11} - 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{54} + ( -1 - 4 \beta_{1} + \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - \beta_{7} + 2 \beta_{8} + 4 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} + \beta_{12} - 6 \beta_{13} + 3 \beta_{14} - 5 \beta_{15} ) q^{55} + ( 3 + 2 \beta_{2} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{56} + ( 2 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + 4 \beta_{12} - 4 \beta_{13} - 3 \beta_{15} ) q^{57} + ( 2 - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{13} ) q^{58} + ( 2 + \beta_{2} - \beta_{4} - \beta_{6} - \beta_{8} - 2 \beta_{10} - \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{59} + ( 2 + \beta_{2} + \beta_{3} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{12} - \beta_{13} ) q^{60} + ( 3 + 2 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} - \beta_{8} - 4 \beta_{9} - 3 \beta_{10} - 2 \beta_{11} - \beta_{12} - 6 \beta_{14} - 2 \beta_{15} ) q^{61} + ( 1 + 3 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} - 3 \beta_{13} - 4 \beta_{14} - \beta_{15} ) q^{62} + ( -7 - 3 \beta_{1} + \beta_{2} - \beta_{3} - 4 \beta_{4} + \beta_{5} + 4 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + \beta_{10} + 4 \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{63} + ( -\beta_{6} - \beta_{9} ) q^{64} + ( 1 - \beta_{1} + \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - \beta_{10} + 5 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + \beta_{15} ) q^{65} + ( -6 - 5 \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{7} + \beta_{8} - \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{66} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 9 \beta_{11} - 2 \beta_{13} - 2 \beta_{14} ) q^{67} + ( -4 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} + 4 \beta_{11} - \beta_{12} + \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{68} + ( 1 + 4 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 10 \beta_{6} + 3 \beta_{7} - \beta_{8} - 4 \beta_{9} - \beta_{10} - 8 \beta_{11} - 3 \beta_{12} + 6 \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{69} + ( -3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 5 \beta_{5} - \beta_{7} - 3 \beta_{8} - 3 \beta_{9} - 4 \beta_{11} - 3 \beta_{12} + 3 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{70} + ( 2 - 2 \beta_{1} + 10 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 5 \beta_{6} - \beta_{7} - \beta_{8} + 3 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 4 \beta_{14} - 2 \beta_{15} ) q^{71} + ( -1 + \beta_{2} + \beta_{5} + 2 \beta_{6} + \beta_{9} - 2 \beta_{11} ) q^{72} + ( 3 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} + 2 \beta_{9} - 3 \beta_{10} - 4 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} - 5 \beta_{14} ) q^{73} + ( -2 - 4 \beta_{2} + 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 2 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + 6 \beta_{13} + 4 \beta_{14} + 2 \beta_{15} ) q^{74} + ( -3 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 5 \beta_{5} + 7 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 4 \beta_{11} - 2 \beta_{12} + 3 \beta_{13} - 3 \beta_{14} ) q^{75} + ( 3 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{11} + \beta_{12} + 2 \beta_{15} ) q^{76} + ( -8 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + 9 \beta_{6} - \beta_{7} + 5 \beta_{8} + 6 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} - 2 \beta_{12} + 4 \beta_{13} + 2 \beta_{14} + 4 \beta_{15} ) q^{77} + ( -1 - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{78} + ( -5 - 4 \beta_{5} + 2 \beta_{6} + 5 \beta_{8} + 5 \beta_{10} + 7 \beta_{11} - 5 \beta_{12} + 3 \beta_{13} + 7 \beta_{15} ) q^{79} + ( 1 + \beta_{4} - 2 \beta_{6} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{80} + ( -3 \beta_{2} + 6 \beta_{11} - 6 \beta_{13} - 6 \beta_{15} ) q^{81} + ( -5 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{8} + \beta_{9} - 3 \beta_{11} - 4 \beta_{12} + 6 \beta_{13} + 2 \beta_{14} + 4 \beta_{15} ) q^{82} + ( 6 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 4 \beta_{11} + 4 \beta_{13} + 2 \beta_{15} ) q^{83} + ( -3 - \beta_{1} - \beta_{3} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} + 3 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + \beta_{14} ) q^{84} + ( 1 + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - 6 \beta_{8} + \beta_{9} + 4 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + 8 \beta_{13} + 5 \beta_{14} + 7 \beta_{15} ) q^{85} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} - 4 \beta_{8} - 4 \beta_{9} - 3 \beta_{11} - \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{86} + ( -6 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 8 \beta_{9} + 4 \beta_{10} + 8 \beta_{11} + 2 \beta_{12} + 2 \beta_{14} ) q^{87} + ( -2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{7} - 2 \beta_{9} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{88} + ( \beta_{5} - 2 \beta_{6} + 2 \beta_{9} - \beta_{11} - \beta_{15} ) q^{89} + ( 2 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{8} - 3 \beta_{9} + 2 \beta_{10} + 2 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{90} + ( 1 + \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} - 4 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{91} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} + \beta_{9} - \beta_{10} + 4 \beta_{11} + 2 \beta_{12} - 4 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{92} + ( 7 + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} + 3 \beta_{7} + 6 \beta_{9} + 2 \beta_{11} - \beta_{12} - 3 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{93} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - 7 \beta_{11} - 2 \beta_{12} - \beta_{13} - 3 \beta_{14} - 4 \beta_{15} ) q^{94} + ( 7 + 3 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} - 5 \beta_{11} + \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{95} + ( -1 - \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{96} + ( 1 - \beta_{1} + 6 \beta_{2} + 3 \beta_{3} + \beta_{4} + 7 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} - 5 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{97} + ( -3 - 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} - \beta_{9} + 2 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{98} + ( 4 + 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 8 \beta_{6} - \beta_{7} - 7 \beta_{8} - 2 \beta_{9} - 4 \beta_{10} - 10 \beta_{11} + 4 \beta_{13} - 2 \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 12q^{5} + 8q^{7} + O(q^{10})$$ $$16q - 12q^{5} + 8q^{7} - 8q^{10} - 24q^{15} + 8q^{16} - 12q^{20} + 24q^{21} + 8q^{22} - 24q^{23} - 16q^{25} - 16q^{28} - 12q^{30} - 8q^{31} + 24q^{36} + 24q^{38} - 4q^{40} + 24q^{41} + 24q^{42} + 36q^{45} - 32q^{46} + 48q^{47} + 24q^{50} - 48q^{51} + 24q^{55} + 24q^{56} + 24q^{57} + 16q^{58} + 12q^{60} - 24q^{61} - 48q^{63} - 48q^{66} - 16q^{67} - 24q^{68} + 16q^{70} - 24q^{72} + 16q^{73} + 16q^{76} - 72q^{77} + 24q^{81} - 16q^{82} + 48q^{83} - 4q^{85} - 48q^{86} - 48q^{87} + 8q^{88} + 12q^{90} - 24q^{92} + 72q^{93} + 84q^{95} - 48q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + 9297 x^{8} - 11276 x^{7} + 11224 x^{6} - 9024 x^{5} + 5736 x^{4} - 2780 x^{3} + 972 x^{2} - 220 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-4341 \nu^{15} + 25062 \nu^{14} - 134821 \nu^{13} + 417198 \nu^{12} - 1075015 \nu^{11} + 1783997 \nu^{10} - 1878159 \nu^{9} - 718793 \nu^{8} + 7025637 \nu^{7} - 16554394 \nu^{6} + 24694320 \nu^{5} - 26670871 \nu^{4} + 20953396 \nu^{3} - 11605609 \nu^{2} + 4090718 \nu - 740270$$$$)/17095$$ $$\beta_{2}$$ $$=$$ $$($$$$3456 \nu^{15} - 25920 \nu^{14} + 151876 \nu^{13} - 594074 \nu^{12} + 1879372 \nu^{11} - 4666596 \nu^{10} + 9554736 \nu^{9} - 15945783 \nu^{8} + 21928484 \nu^{7} - 24527176 \nu^{6} + 22138664 \nu^{5} - 15739188 \nu^{4} + 8598668 \nu^{3} - 3418428 \nu^{2} + 929924 \nu - 142555$$$$)/17095$$ $$\beta_{3}$$ $$=$$ $$($$$$-5033 \nu^{15} + 21310 \nu^{14} - 104426 \nu^{13} + 187861 \nu^{12} - 172588 \nu^{11} - 1167140 \nu^{10} + 5261069 \nu^{9} - 15006590 \nu^{8} + 29556405 \nu^{7} - 45536907 \nu^{6} + 53902394 \nu^{5} - 49692993 \nu^{4} + 34301505 \nu^{3} - 17033716 \nu^{2} + 5420814 \nu - 887140$$$$)/17095$$ $$\beta_{4}$$ $$=$$ $$($$$$-2793 \nu^{15} + 28180 \nu^{14} - 172311 \nu^{13} + 766484 \nu^{12} - 2581266 \nu^{11} + 7044831 \nu^{10} - 15479146 \nu^{9} + 28225918 \nu^{8} - 42069895 \nu^{7} + 51829741 \nu^{6} - 51541475 \nu^{5} + 41221233 \nu^{4} - 25401631 \nu^{3} + 11716753 \nu^{2} - 3615388 \nu + 618310$$$$)/17095$$ $$\beta_{5}$$ $$=$$ $$($$$$15977 \nu^{15} - 141262 \nu^{14} + 856169 \nu^{13} - 3642449 \nu^{12} + 12106306 \nu^{11} - 32236863 \nu^{10} + 70027507 \nu^{9} - 125576015 \nu^{8} + 185376534 \nu^{7} - 225133368 \nu^{6} + 221440773 \nu^{5} - 173497381 \nu^{4} + 104335813 \nu^{3} - 45658062 \nu^{2} + 12999216 \nu - 1883725$$$$)/17095$$ $$\beta_{6}$$ $$=$$ $$($$$$26630 \nu^{15} - 182630 \nu^{14} + 1056740 \nu^{13} - 3923413 \nu^{12} + 12097498 \nu^{11} - 28666706 \nu^{10} + 56590650 \nu^{9} - 89528958 \nu^{8} + 116711850 \nu^{7} - 121073088 \nu^{6} + 99711814 \nu^{5} - 61611091 \nu^{4} + 27082796 \nu^{3} - 6949514 \nu^{2} + 510382 \nu + 218885$$$$)/17095$$ $$\beta_{7}$$ $$=$$ $$($$$$26630 \nu^{15} - 210508 \nu^{14} + 1251886 \nu^{13} - 5050631 \nu^{12} + 16323908 \nu^{11} - 41702827 \nu^{10} + 87680143 \nu^{9} - 150958920 \nu^{8} + 214412142 \nu^{7} - 248665697 \nu^{6} + 232995743 \nu^{5} - 172095024 \nu^{4} + 96732560 \nu^{3} - 38723333 \nu^{2} + 9836888 \nu - 1160550$$$$)/17095$$ $$\beta_{8}$$ $$=$$ $$($$$$-26630 \nu^{15} + 212086 \nu^{14} - 1262932 \nu^{13} + 5122693 \nu^{12} - 16612682 \nu^{11} + 42720111 \nu^{10} - 90382731 \nu^{9} + 157187286 \nu^{8} - 225835284 \nu^{7} + 266547330 \nu^{6} - 255307874 \nu^{5} + 195077805 \nu^{4} - 114781987 \nu^{3} + 49547624 \nu^{2} - 14057775 \nu + 2084995$$$$)/17095$$ $$\beta_{9}$$ $$=$$ $$($$$$26630 \nu^{15} - 216820 \nu^{14} + 1296070 \nu^{13} - 5311527 \nu^{12} + 17314892 \nu^{11} - 44845414 \nu^{10} + 95362110 \nu^{9} - 166733397 \nu^{8} + 240513840 \nu^{7} - 284726942 \nu^{6} + 273082466 \nu^{5} - 208344314 \nu^{4} + 122001074 \nu^{3} - 52073476 \nu^{2} + 14507768 \nu - 2071845$$$$)/17095$$ $$\beta_{10}$$ $$=$$ $$($$$$36479 \nu^{15} - 258470 \nu^{14} + 1499462 \nu^{13} - 5670854 \nu^{12} + 17608509 \nu^{11} - 42377599 \nu^{10} + 84494792 \nu^{9} - 135892302 \nu^{8} + 179666959 \nu^{7} - 190504608 \nu^{6} + 160694617 \nu^{5} - 103510253 \nu^{4} + 48537906 \nu^{3} - 14757489 \nu^{2} + 2298926 \nu + 18365$$$$)/17095$$ $$\beta_{11}$$ $$=$$ $$($$$$-38207 \nu^{15} + 272482 \nu^{14} - 1582764 \nu^{13} + 6010234 \nu^{12} - 18706521 \nu^{11} + 45195343 \nu^{10} - 90418577 \nu^{9} + 146094250 \nu^{8} - 194109639 \nu^{7} + 207156088 \nu^{6} - 176142373 \nu^{5} + 114735201 \nu^{4} - 54727158 \nu^{3} + 17186797 \nu^{2} - 2916691 \nu + 68035$$$$)/17095$$ $$\beta_{12}$$ $$=$$ $$($$$$-35414 \nu^{15} + 286908 \nu^{14} - 1708695 \nu^{13} + 6974027 \nu^{12} - 22629771 \nu^{11} + 58331291 \nu^{10} - 123326697 \nu^{9} + 214327527 \nu^{8} - 306917288 \nu^{7} + 360531571 \nu^{6} - 342616589 \nu^{5} + 258909506 \nu^{4} - 149985982 \nu^{3} + 63408155 \nu^{2} - 17509319 \nu + 2511045$$$$)/17095$$ $$\beta_{13}$$ $$=$$ $$($$$$-38207 \nu^{15} + 300623 \nu^{14} - 1779751 \nu^{13} + 7154021 \nu^{12} - 23008412 \nu^{11} + 58552587 \nu^{10} - 122465653 \nu^{9} + 210008510 \nu^{8} - 296756698 \nu^{7} + 343052922 \nu^{6} - 320295829 \nu^{5} + 236754314 \nu^{4} - 133562723 \nu^{3} + 54462313 \nu^{2} - 14369552 \nu + 1923500$$$$)/17095$$ $$\beta_{14}$$ $$=$$ $$($$$$39935 \nu^{15} - 314635 \nu^{14} + 1863053 \nu^{13} - 7493401 \nu^{12} + 24106424 \nu^{11} - 61370331 \nu^{10} + 128389438 \nu^{9} - 220210458 \nu^{8} + 311199378 \nu^{7} - 359704402 \nu^{6} + 335743585 \nu^{5} - 247979262 \nu^{4} + 139751975 \nu^{3} - 56891621 \nu^{2} + 15004412 \nu - 2009900$$$$)/17095$$ $$\beta_{15}$$ $$=$$ $$($$$$54184 \nu^{15} - 399016 \nu^{14} + 2335837 \nu^{13} - 9056462 \nu^{12} + 28575749 \nu^{11} - 70523459 \nu^{10} + 143951776 \nu^{9} - 239041170 \nu^{8} + 327539991 \nu^{7} - 364427829 \nu^{6} + 326718848 \nu^{5} - 229677473 \nu^{4} + 122374471 \nu^{3} - 46393946 \nu^{2} + 11232929 \nu - 1312670$$$$)/17095$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{14} + \beta_{13} + \beta_{11} + \beta_{10}$$ $$\nu^{2}$$ $$=$$ $$\beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} - 3 \beta_{14} - 5 \beta_{13} + 3 \beta_{12} - 3 \beta_{10} + 3 \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - 3 \beta_{3} + 2 \beta_{2} - \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$-8 \beta_{15} - 6 \beta_{14} - 8 \beta_{13} - 2 \beta_{12} - 6 \beta_{11} - 8 \beta_{10} + 5 \beta_{9} + 6 \beta_{8} - 6 \beta_{7} + 13 \beta_{6} + 8 \beta_{5} - 8 \beta_{4} - 6 \beta_{3} + 3 \beta_{2} - 2 \beta_{1} + 5$$ $$\nu^{5}$$ $$=$$ $$-17 \beta_{15} + 13 \beta_{14} + 27 \beta_{13} - 20 \beta_{12} - 21 \beta_{11} + 8 \beta_{10} - 21 \beta_{9} - 11 \beta_{7} + 15 \beta_{6} + 15 \beta_{5} - 10 \beta_{4} + 20 \beta_{3} - 14 \beta_{2} + 8 \beta_{1} - 4$$ $$\nu^{6}$$ $$=$$ $$43 \beta_{15} + 40 \beta_{14} + 64 \beta_{13} - 2 \beta_{12} + 11 \beta_{11} + 59 \beta_{10} - 63 \beta_{9} - 44 \beta_{8} + 34 \beta_{7} - 81 \beta_{6} - 49 \beta_{5} + 43 \beta_{4} + 75 \beta_{3} - 35 \beta_{2} + 29 \beta_{1} - 32$$ $$\nu^{7}$$ $$=$$ $$170 \beta_{15} - 68 \beta_{14} - 154 \beta_{13} + 133 \beta_{12} + 208 \beta_{11} + 16 \beta_{10} + 104 \beta_{9} - 21 \beta_{8} + 103 \beta_{7} - 167 \beta_{6} - 155 \beta_{5} + 119 \beta_{4} - 77 \beta_{3} + 93 \beta_{2} - 39 \beta_{1} + 9$$ $$\nu^{8}$$ $$=$$ $$-160 \beta_{15} - 316 \beta_{14} - 556 \beta_{13} + 108 \beta_{12} + 196 \beta_{11} - 388 \beta_{10} + 590 \beta_{9} + 324 \beta_{8} - 172 \beta_{7} + 472 \beta_{6} + 248 \beta_{5} - 172 \beta_{4} - 672 \beta_{3} + 385 \beta_{2} - 296 \beta_{1} + 231$$ $$\nu^{9}$$ $$=$$ $$-1442 \beta_{15} + 303 \beta_{14} + 759 \beta_{13} - 888 \beta_{12} - 1474 \beta_{11} - 546 \beta_{10} - 254 \beta_{9} + 396 \beta_{8} - 906 \beta_{7} + 1654 \beta_{6} + 1414 \beta_{5} - 1044 \beta_{4} - 96 \beta_{3} - 402 \beta_{2} + 6 \beta_{1} + 116$$ $$\nu^{10}$$ $$=$$ $$-163 \beta_{15} + 2609 \beta_{14} + 4787 \beta_{13} - 1507 \beta_{12} - 3305 \beta_{11} + 2222 \beta_{10} - 4829 \beta_{9} - 2201 \beta_{8} + 557 \beta_{7} - 2176 \beta_{6} - 679 \beta_{5} + 158 \beta_{4} + 5115 \beta_{3} - 3570 \beta_{2} + 2463 \beta_{1} - 1611$$ $$\nu^{11}$$ $$=$$ $$10909 \beta_{15} - 253 \beta_{14} - 1879 \beta_{13} + 5489 \beta_{12} + 8068 \beta_{11} + 6380 \beta_{10} - 2613 \beta_{9} - 4950 \beta_{8} + 7438 \beta_{7} - 14737 \beta_{6} - 11667 \beta_{5} + 7920 \beta_{4} + 5896 \beta_{3} - 376 \beta_{2} + 2589 \beta_{1} - 2314$$ $$\nu^{12}$$ $$=$$ $$11866 \beta_{15} - 20444 \beta_{14} - 38548 \beta_{13} + 16364 \beta_{12} + 34826 \beta_{11} - 9970 \beta_{10} + 35121 \beta_{9} + 12788 \beta_{8} + 2360 \beta_{7} + 3142 \beta_{6} - 5568 \beta_{5} + 6926 \beta_{4} - 33726 \beta_{3} + 28215 \beta_{2} - 17114 \beta_{1} + 10052$$ $$\nu^{13}$$ $$=$$ $$-72687 \beta_{15} - 16328 \beta_{14} - 20114 \beta_{13} - 27664 \beta_{12} - 26597 \beta_{11} - 58357 \beta_{10} + 54341 \beta_{9} + 50414 \beta_{8} - 55565 \beta_{7} + 117537 \beta_{6} + 86075 \beta_{5} - 52962 \beta_{4} - 79378 \beta_{3} + 31749 \beta_{2} - 38629 \beta_{1} + 27130$$ $$\nu^{14}$$ $$=$$ $$-163323 \beta_{15} + 144445 \beta_{14} + 280994 \beta_{13} - 152817 \beta_{12} - 300247 \beta_{11} + 16587 \beta_{10} - 220979 \beta_{9} - 52573 \beta_{8} - 70669 \beta_{7} + 88965 \beta_{6} + 126532 \beta_{5} - 106225 \beta_{4} + 182546 \beta_{3} - 190606 \beta_{2} + 95560 \beta_{1} - 51022$$ $$\nu^{15}$$ $$=$$ $$402584 \beta_{15} + 264336 \beta_{14} + 422112 \beta_{13} + 70486 \beta_{12} - 100959 \beta_{11} + 463038 \beta_{10} - 640520 \beta_{9} - 444317 \beta_{8} + 365424 \beta_{7} - 828756 \beta_{6} - 550507 \beta_{5} + 299218 \beta_{4} + 798216 \beta_{3} - 443118 \beta_{2} + 403389 \beta_{1} - 258101$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/90\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$1 + \beta_{2}$$ $$-\beta_{6} - \beta_{9}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 0.5 − 0.410882i 0.5 − 2.00333i 0.5 − 0.589118i 0.5 + 1.00333i 0.5 + 0.410882i 0.5 + 2.00333i 0.5 + 0.589118i 0.5 − 1.00333i 0.5 − 1.74530i 0.5 + 1.33108i 0.5 − 0.331082i 0.5 + 2.74530i 0.5 + 1.74530i 0.5 − 1.33108i 0.5 + 0.331082i 0.5 − 2.74530i
−0.258819 + 0.965926i −0.0795432 1.73022i −0.866025 0.500000i 0.661570 2.13596i 1.69185 + 0.370982i 3.75574 + 1.00635i 0.707107 0.707107i −2.98735 + 0.275255i 1.89195 + 1.19185i
23.2 −0.258819 + 0.965926i 0.0795432 + 1.73022i −0.866025 0.500000i −2.00265 + 0.994679i −1.69185 0.370982i 2.32238 + 0.622279i 0.707107 0.707107i −2.98735 + 0.275255i −0.442462 2.19185i
23.3 0.258819 0.965926i −1.45865 0.933998i −0.866025 0.500000i 1.36868 1.76825i −1.27970 + 1.16721i −2.56188 0.686453i −0.707107 + 0.707107i 1.25529 + 2.72474i −1.35376 1.77970i
23.4 0.258819 0.965926i 1.45865 + 0.933998i −0.866025 0.500000i −1.29554 1.82252i 1.27970 1.16721i 1.94786 + 0.521929i −0.707107 + 0.707107i 1.25529 + 2.72474i −2.09573 + 0.779698i
47.1 −0.258819 0.965926i −0.0795432 + 1.73022i −0.866025 + 0.500000i 0.661570 + 2.13596i 1.69185 0.370982i 3.75574 1.00635i 0.707107 + 0.707107i −2.98735 0.275255i 1.89195 1.19185i
47.2 −0.258819 0.965926i 0.0795432 1.73022i −0.866025 + 0.500000i −2.00265 0.994679i −1.69185 + 0.370982i 2.32238 0.622279i 0.707107 + 0.707107i −2.98735 0.275255i −0.442462 + 2.19185i
47.3 0.258819 + 0.965926i −1.45865 + 0.933998i −0.866025 + 0.500000i 1.36868 + 1.76825i −1.27970 1.16721i −2.56188 + 0.686453i −0.707107 0.707107i 1.25529 2.72474i −1.35376 + 1.77970i
47.4 0.258819 + 0.965926i 1.45865 0.933998i −0.866025 + 0.500000i −1.29554 + 1.82252i 1.27970 + 1.16721i 1.94786 0.521929i −0.707107 0.707107i 1.25529 2.72474i −2.09573 0.779698i
77.1 −0.965926 0.258819i −1.73022 + 0.0795432i 0.866025 + 0.500000i −1.51901 + 1.64092i 1.69185 + 0.370982i −1.00635 + 3.75574i −0.707107 0.707107i 2.98735 0.275255i 1.89195 1.19185i
77.2 −0.965926 0.258819i 1.73022 0.0795432i 0.866025 + 0.500000i −0.139908 2.23169i −1.69185 0.370982i −0.622279 + 2.32238i −0.707107 0.707107i 2.98735 0.275255i −0.442462 + 2.19185i
77.3 0.965926 + 0.258819i −0.933998 + 1.45865i 0.866025 + 0.500000i −0.847015 + 2.06944i −1.27970 + 1.16721i 0.686453 2.56188i 0.707107 + 0.707107i −1.25529 2.72474i −1.35376 + 1.77970i
77.4 0.965926 + 0.258819i 0.933998 1.45865i 0.866025 + 0.500000i −2.22612 0.210717i 1.27970 1.16721i −0.521929 + 1.94786i 0.707107 + 0.707107i −1.25529 2.72474i −2.09573 0.779698i
83.1 −0.965926 + 0.258819i −1.73022 0.0795432i 0.866025 0.500000i −1.51901 1.64092i 1.69185 0.370982i −1.00635 3.75574i −0.707107 + 0.707107i 2.98735 + 0.275255i 1.89195 + 1.19185i
83.2 −0.965926 + 0.258819i 1.73022 + 0.0795432i 0.866025 0.500000i −0.139908 + 2.23169i −1.69185 + 0.370982i −0.622279 2.32238i −0.707107 + 0.707107i 2.98735 + 0.275255i −0.442462 2.19185i
83.3 0.965926 0.258819i −0.933998 1.45865i 0.866025 0.500000i −0.847015 2.06944i −1.27970 1.16721i 0.686453 + 2.56188i 0.707107 0.707107i −1.25529 + 2.72474i −1.35376 1.77970i
83.4 0.965926 0.258819i 0.933998 + 1.45865i 0.866025 0.500000i −2.22612 + 0.210717i 1.27970 + 1.16721i −0.521929 1.94786i 0.707107 0.707107i −1.25529 + 2.72474i −2.09573 + 0.779698i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 83.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.d odd 6 1 inner
45.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.2.l.b 16
3.b odd 2 1 270.2.m.b 16
4.b odd 2 1 720.2.cu.b 16
5.b even 2 1 450.2.p.h 16
5.c odd 4 1 inner 90.2.l.b 16
5.c odd 4 1 450.2.p.h 16
9.c even 3 1 270.2.m.b 16
9.c even 3 1 810.2.f.c 16
9.d odd 6 1 inner 90.2.l.b 16
9.d odd 6 1 810.2.f.c 16
15.d odd 2 1 1350.2.q.h 16
15.e even 4 1 270.2.m.b 16
15.e even 4 1 1350.2.q.h 16
20.e even 4 1 720.2.cu.b 16
36.h even 6 1 720.2.cu.b 16
45.h odd 6 1 450.2.p.h 16
45.j even 6 1 1350.2.q.h 16
45.k odd 12 1 270.2.m.b 16
45.k odd 12 1 810.2.f.c 16
45.k odd 12 1 1350.2.q.h 16
45.l even 12 1 inner 90.2.l.b 16
45.l even 12 1 450.2.p.h 16
45.l even 12 1 810.2.f.c 16
180.v odd 12 1 720.2.cu.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.l.b 16 1.a even 1 1 trivial
90.2.l.b 16 5.c odd 4 1 inner
90.2.l.b 16 9.d odd 6 1 inner
90.2.l.b 16 45.l even 12 1 inner
270.2.m.b 16 3.b odd 2 1
270.2.m.b 16 9.c even 3 1
270.2.m.b 16 15.e even 4 1
270.2.m.b 16 45.k odd 12 1
450.2.p.h 16 5.b even 2 1
450.2.p.h 16 5.c odd 4 1
450.2.p.h 16 45.h odd 6 1
450.2.p.h 16 45.l even 12 1
720.2.cu.b 16 4.b odd 2 1
720.2.cu.b 16 20.e even 4 1
720.2.cu.b 16 36.h even 6 1
720.2.cu.b 16 180.v odd 12 1
810.2.f.c 16 9.c even 3 1
810.2.f.c 16 9.d odd 6 1
810.2.f.c 16 45.k odd 12 1
810.2.f.c 16 45.l even 12 1
1350.2.q.h 16 15.d odd 2 1
1350.2.q.h 16 15.e even 4 1
1350.2.q.h 16 45.j even 6 1
1350.2.q.h 16 45.k odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{16} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(90, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{4} + T^{8} )^{2}$$
$3$ $$6561 - 486 T^{4} - 45 T^{8} - 6 T^{12} + T^{16}$$
$5$ $$390625 + 937500 T + 1250000 T^{2} + 1200000 T^{3} + 925000 T^{4} + 607500 T^{5} + 351200 T^{6} + 182100 T^{7} + 85471 T^{8} + 36420 T^{9} + 14048 T^{10} + 4860 T^{11} + 1480 T^{12} + 384 T^{13} + 80 T^{14} + 12 T^{15} + T^{16}$$
$7$ $$6250000 - 7000000 T + 3920000 T^{2} - 2390400 T^{3} + 1088624 T^{4} - 207200 T^{5} + 6400 T^{6} + 26016 T^{7} - 20276 T^{8} + 4656 T^{9} - 128 T^{10} - 592 T^{11} + 476 T^{12} - 144 T^{13} + 32 T^{14} - 8 T^{15} + T^{16}$$
$11$ $$( 1849 - 1032 T - 754 T^{2} + 528 T^{3} + 441 T^{4} - 22 T^{6} + T^{8} )^{2}$$
$13$ $$1296 - 5184 T + 10368 T^{2} - 20736 T^{3} + 35856 T^{4} - 38880 T^{5} + 34560 T^{6} - 31104 T^{7} + 17388 T^{8} - 3888 T^{9} + 1152 T^{10} - 288 T^{11} - 156 T^{12} + 48 T^{13} + T^{16}$$
$17$ $$390625 + 467222500 T^{4} + 4269606 T^{8} + 4132 T^{12} + T^{16}$$
$19$ $$( 10201 + 5308 T^{2} + 858 T^{4} + 52 T^{6} + T^{8} )^{2}$$
$23$ $$82538991616 + 103481720832 T + 64869138432 T^{2} + 27109490688 T^{3} + 7528268800 T^{4} + 1234913280 T^{5} + 83607552 T^{6} - 20752896 T^{7} - 9066816 T^{8} - 1621248 T^{9} - 110592 T^{10} + 27840 T^{11} + 11920 T^{12} + 2304 T^{13} + 288 T^{14} + 24 T^{15} + T^{16}$$
$29$ $$40960000 + 37683200 T^{2} + 25452544 T^{4} + 7454720 T^{6} + 1596160 T^{8} + 103424 T^{10} + 4960 T^{12} + 80 T^{14} + T^{16}$$
$31$ $$( 448900 - 85760 T + 64624 T^{2} + 3856 T^{3} + 5026 T^{4} - 32 T^{5} + 88 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$37$ $$( 82944 - 165888 T + 165888 T^{2} - 55296 T^{3} + 9792 T^{4} - 576 T^{5} + T^{8} )^{2}$$
$41$ $$( 555025 + 384420 T + 45542 T^{2} - 29928 T^{3} + 555 T^{4} + 696 T^{5} - 10 T^{6} - 12 T^{7} + T^{8} )^{2}$$
$43$ $$35806100625 + 3951018000 T + 217987200 T^{2} + 1658957760 T^{3} - 1276568046 T^{4} - 315984240 T^{5} + 11335680 T^{6} - 31123008 T^{7} + 51876315 T^{8} - 326160 T^{9} + 4608 T^{10} - 4320 T^{11} - 7230 T^{12} + 96 T^{13} + T^{16}$$
$47$ $$2981133747216 - 1796847738048 T + 541515756672 T^{2} - 108797686272 T^{3} - 460693008 T^{4} + 6702297696 T^{5} - 1970749440 T^{6} + 435253824 T^{7} - 5582196 T^{8} - 27126576 T^{9} + 8709120 T^{10} - 1625184 T^{11} + 207612 T^{12} - 18432 T^{13} + 1152 T^{14} - 48 T^{15} + T^{16}$$
$53$ $$3906250000 + 418925536 T^{4} + 7672152 T^{8} + 10552 T^{12} + T^{16}$$
$59$ $$625 + 37700 T^{2} + 2262214 T^{4} + 712592 T^{6} + 158299 T^{8} + 17840 T^{10} + 1462 T^{12} + 44 T^{14} + T^{16}$$
$61$ $$( 45887076 + 13331232 T + 3141432 T^{2} + 375120 T^{3} + 42054 T^{4} + 2640 T^{5} + 252 T^{6} + 12 T^{7} + T^{8} )^{2}$$
$67$ $$3418801 - 85231504 T + 1062420608 T^{2} - 26745489504 T^{3} + 333377741378 T^{4} + 40651903312 T^{5} + 2456667136 T^{6} + 274915632 T^{7} - 57912821 T^{8} - 8720688 T^{9} - 518144 T^{10} - 64528 T^{11} + 11858 T^{12} + 2016 T^{13} + 128 T^{14} + 16 T^{15} + T^{16}$$
$71$ $$( 14032516 + 1023488 T^{2} + 25980 T^{4} + 272 T^{6} + T^{8} )^{2}$$
$73$ $$( 966289 - 1140280 T + 672800 T^{2} - 154024 T^{3} + 17842 T^{4} - 152 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$79$ $$1730768006250000 - 138985632000000 T^{2} + 8558458650000 T^{4} - 173042524800 T^{6} + 2428425036 T^{8} - 20342592 T^{10} + 124068 T^{12} - 432 T^{14} + T^{16}$$
$83$ $$21743271936 - 1472200704 T^{4} + 679477248 T^{5} - 176947200 T^{7} + 127844352 T^{8} - 46006272 T^{9} + 10616832 T^{10} - 1751040 T^{11} + 211200 T^{12} - 18432 T^{13} + 1152 T^{14} - 48 T^{15} + T^{16}$$
$89$ $$( 100 - 28 T^{2} + T^{4} )^{4}$$
$97$ $$19559470366881 - 10714460229504 T + 2934631047168 T^{2} - 5985304288416 T^{3} + 1103454012942 T^{4} + 983023623072 T^{5} + 211720241664 T^{6} + 37134895824 T^{7} + 6126178419 T^{8} + 738652608 T^{9} + 70972416 T^{10} + 6446736 T^{11} + 473262 T^{12} + 24768 T^{13} + 1152 T^{14} + 48 T^{15} + T^{16}$$