# Properties

 Label 930.2.bg.b Level $930$ Weight $2$ Character orbit 930.bg Analytic conductor $7.426$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.bg (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{15})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 22 x^{14} + 195 x^{12} + 892 x^{10} + 2229 x^{8} + 2923 x^{6} + 1685 x^{4} + 213 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 22 x^{14} + 195 x^{12} + 892 x^{10} + 2229 x^{8} + 2923 x^{6} + 1685 x^{4} + 213 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$228 \nu^{15} - 67 \nu^{14} + 3838 \nu^{13} - 1173 \nu^{12} + 22914 \nu^{11} - 7411 \nu^{10} + 54093 \nu^{9} - 20706 \nu^{8} + 20197 \nu^{7} - 28270 \nu^{6} - 90972 \nu^{5} - 30719 \nu^{4} - 88065 \nu^{3} - 26763 \nu^{2} - 8759 \nu + 293$$$$)/10298$$ $$\beta_{2}$$ $$=$$ $$($$$$-216 \nu^{15} + 27 \nu^{14} - 7701 \nu^{13} + 319 \nu^{12} - 98130 \nu^{11} + 681 \nu^{10} - 600021 \nu^{9} - 2876 \nu^{8} - 1897977 \nu^{7} - 2825 \nu^{6} - 3004300 \nu^{5} + 43427 \nu^{4} - 1984571 \nu^{3} + 78798 \nu^{2} - 256198 \nu + 2418$$$$)/10298$$ $$\beta_{3}$$ $$=$$ $$($$$$-228 \nu^{15} - 67 \nu^{14} - 3838 \nu^{13} - 1173 \nu^{12} - 22914 \nu^{11} - 7411 \nu^{10} - 54093 \nu^{9} - 20706 \nu^{8} - 20197 \nu^{7} - 28270 \nu^{6} + 90972 \nu^{5} - 30719 \nu^{4} + 88065 \nu^{3} - 26763 \nu^{2} + 8759 \nu + 293$$$$)/10298$$ $$\beta_{4}$$ $$=$$ $$($$$$293 \nu^{15} - 228 \nu^{14} + 6513 \nu^{13} - 3838 \nu^{12} + 58308 \nu^{11} - 22914 \nu^{10} + 268767 \nu^{9} - 54093 \nu^{8} + 673803 \nu^{7} - 20197 \nu^{6} + 884709 \nu^{5} + 90972 \nu^{4} + 524424 \nu^{3} + 88065 \nu^{2} + 89172 \nu + 8759$$$$)/10298$$ $$\beta_{5}$$ $$=$$ $$($$$$-293 \nu^{15} - 228 \nu^{14} - 6513 \nu^{13} - 3838 \nu^{12} - 58308 \nu^{11} - 22914 \nu^{10} - 268767 \nu^{9} - 54093 \nu^{8} - 673803 \nu^{7} - 20197 \nu^{6} - 884709 \nu^{5} + 90972 \nu^{4} - 524424 \nu^{3} + 88065 \nu^{2} - 89172 \nu + 8759$$$$)/10298$$ $$\beta_{6}$$ $$=$$ $$($$$$-46 \nu^{15} - 955 \nu^{13} - 7875 \nu^{11} - 33000 \nu^{9} - 74580 \nu^{7} - 88420 \nu^{5} - 47950 \nu^{3} - 6950 \nu + 271$$$$)/542$$ $$\beta_{7}$$ $$=$$ $$($$$$-72 \nu^{15} + 1003 \nu^{14} - 2567 \nu^{13} + 19097 \nu^{12} - 32710 \nu^{11} + 139148 \nu^{10} - 200007 \nu^{9} + 483962 \nu^{8} - 632659 \nu^{7} + 807383 \nu^{6} - 999717 \nu^{5} + 540715 \nu^{4} - 647793 \nu^{3} + 55202 \nu^{2} - 63087 \nu - 9151$$$$)/10298$$ $$\beta_{8}$$ $$=$$ $$($$$$57 \nu^{14} + 1095 \nu^{12} + 8032 \nu^{10} + 27954 \nu^{8} + 46038 \nu^{6} + 29560 \nu^{4} + 2848 \nu^{2} - 271 \nu + 46$$$$)/542$$ $$\beta_{9}$$ $$=$$ $$($$$$57 \nu^{14} + 1095 \nu^{12} + 8032 \nu^{10} + 27954 \nu^{8} + 46038 \nu^{6} + 29560 \nu^{4} + 2848 \nu^{2} + 271 \nu + 46$$$$)/542$$ $$\beta_{10}$$ $$=$$ $$($$$$-365 \nu^{15} - 775 \nu^{14} - 9080 \nu^{13} - 15259 \nu^{12} - 91018 \nu^{11} - 116234 \nu^{10} - 468774 \nu^{9} - 429869 \nu^{8} - 1306462 \nu^{7} - 787186 \nu^{6} - 1884426 \nu^{5} - 631687 \nu^{4} - 1172217 \nu^{3} - 143267 \nu^{2} - 152259 \nu + 392$$$$)/10298$$ $$\beta_{11}$$ $$=$$ $$($$$$-345 \nu^{15} + 1010 \nu^{14} - 8653 \nu^{13} + 18989 \nu^{12} - 87653 \nu^{11} + 136464 \nu^{10} - 456983 \nu^{9} + 469295 \nu^{8} - 1288340 \nu^{7} + 786436 \nu^{6} - 1865035 \nu^{5} + 569900 \nu^{4} - 1128723 \nu^{3} + 126549 \nu^{2} - 114997 \nu + 7495$$$$)/10298$$ $$\beta_{12}$$ $$=$$ $$($$$$-345 \nu^{15} - 1010 \nu^{14} - 8653 \nu^{13} - 18989 \nu^{12} - 87653 \nu^{11} - 136464 \nu^{10} - 456983 \nu^{9} - 469295 \nu^{8} - 1288340 \nu^{7} - 786436 \nu^{6} - 1865035 \nu^{5} - 569900 \nu^{4} - 1128723 \nu^{3} - 126549 \nu^{2} - 114997 \nu - 7495$$$$)/10298$$ $$\beta_{13}$$ $$=$$ $$($$$$-117 \nu^{15} - 1544 \nu^{14} - 4815 \nu^{13} - 29875 \nu^{12} - 64739 \nu^{11} - 223735 \nu^{10} - 402890 \nu^{9} - 818613 \nu^{8} - 1268143 \nu^{7} - 1514356 \nu^{6} - 1956007 \nu^{5} - 1302353 \nu^{4} - 1216788 \nu^{3} - 396032 \nu^{2} - 118607 \nu - 26140$$$$)/10298$$ $$\beta_{14}$$ $$=$$ $$($$$$775 \nu^{15} + 1050 \nu^{14} + 15259 \nu^{13} + 19843 \nu^{12} + 116234 \nu^{11} + 143194 \nu^{10} + 429869 \nu^{9} + 492877 \nu^{8} + 787186 \nu^{7} + 817531 \nu^{6} + 631687 \nu^{5} + 557192 \nu^{4} + 143267 \nu^{3} + 74514 \nu^{2} - 392 \nu - 365$$$$)/10298$$ $$\beta_{15}$$ $$=$$ $$($$$$2294 \nu^{15} + 214 \nu^{14} + 48462 \nu^{13} + 4054 \nu^{12} + 409136 \nu^{11} + 28282 \nu^{10} + 1767952 \nu^{9} + 83427 \nu^{8} + 4145402 \nu^{7} + 62091 \nu^{6} + 5090596 \nu^{5} - 144193 \nu^{4} + 2778811 \nu^{3} - 199865 \nu^{2} + 352473 \nu - 11157$$$$)/10298$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{9} - \beta_{8}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{13} + \beta_{12} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{5} + \beta_{1} - 2$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{15} + \beta_{14} + \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - 3 \beta_{9} + 4 \beta_{8} + 2 \beta_{7} - 3 \beta_{6} + 2 \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{14} + 7 \beta_{13} - 8 \beta_{12} + 2 \beta_{11} - 4 \beta_{10} - 6 \beta_{9} + \beta_{8} + 4 \beta_{7} + \beta_{6} + 7 \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} - 5 \beta_{1} + 6$$ $$\nu^{5}$$ $$=$$ $$16 \beta_{15} - 9 \beta_{14} - 8 \beta_{13} + 15 \beta_{12} + 16 \beta_{11} - 13 \beta_{10} + 11 \beta_{9} - 19 \beta_{8} - 13 \beta_{7} + 23 \beta_{6} - 13 \beta_{4} - 8 \beta_{3} + 7 \beta_{2} - 9 \beta_{1} - 16$$ $$\nu^{6}$$ $$=$$ $$-11 \beta_{14} - 43 \beta_{13} + 50 \beta_{12} - 18 \beta_{11} + 17 \beta_{10} + 35 \beta_{9} - 8 \beta_{8} - 17 \beta_{7} - 11 \beta_{6} - 43 \beta_{5} - 26 \beta_{4} - 7 \beta_{3} + 11 \beta_{2} + 25 \beta_{1} - 17$$ $$\nu^{7}$$ $$=$$ $$-106 \beta_{15} + 64 \beta_{14} + 53 \beta_{13} - 92 \beta_{12} - 103 \beta_{11} + 70 \beta_{10} - 45 \beta_{9} + 98 \beta_{8} + 70 \beta_{7} - 146 \beta_{6} - 5 \beta_{5} + 75 \beta_{4} + 52 \beta_{3} - 42 \beta_{2} + 65 \beta_{1} + 105$$ $$\nu^{8}$$ $$=$$ $$87 \beta_{14} + 256 \beta_{13} - 297 \beta_{12} + 128 \beta_{11} - 76 \beta_{10} - 204 \beta_{9} + 52 \beta_{8} + 76 \beta_{7} + 87 \beta_{6} + 257 \beta_{5} + 181 \beta_{4} + 37 \beta_{3} - 87 \beta_{2} - 132 \beta_{1} + 34$$ $$\nu^{9}$$ $$=$$ $$664 \beta_{15} - 423 \beta_{14} - 332 \beta_{13} + 538 \beta_{12} + 629 \beta_{11} - 360 \beta_{10} + 201 \beta_{9} - 533 \beta_{8} - 360 \beta_{7} + 875 \beta_{6} + 69 \beta_{5} - 429 \beta_{4} - 320 \beta_{3} + 241 \beta_{2} - 435 \beta_{1} - 649$$ $$\nu^{10}$$ $$=$$ $$-613 \beta_{14} - 1518 \beta_{13} + 1747 \beta_{12} - 842 \beta_{11} + 351 \beta_{10} + 1194 \beta_{9} - 324 \beta_{8} - 351 \beta_{7} - 613 \beta_{6} - 1530 \beta_{5} - 1179 \beta_{4} - 172 \beta_{3} + 613 \beta_{2} + 733 \beta_{1} + 61$$ $$\nu^{11}$$ $$=$$ $$-4072 \beta_{15} + 2707 \beta_{14} + 2036 \beta_{13} - 3113 \beta_{12} - 3784 \beta_{11} + 1830 \beta_{10} - 961 \beta_{9} + 2997 \beta_{8} + 1830 \beta_{7} - 5135 \beta_{6} - 650 \beta_{5} + 2480 \beta_{4} + 1929 \beta_{3} - 1365 \beta_{2} + 2814 \beta_{1} + 3921$$ $$\nu^{12}$$ $$=$$ $$4097 \beta_{14} + 9031 \beta_{13} - 10289 \beta_{12} + 5355 \beta_{11} - 1655 \beta_{10} - 7025 \beta_{9} + 2006 \beta_{8} + 1655 \beta_{7} + 4097 \beta_{6} + 9138 \beta_{5} + 7483 \beta_{4} + 715 \beta_{3} - 4097 \beta_{2} - 4219 \beta_{1} - 1481$$ $$\nu^{13}$$ $$=$$ $$24760 \beta_{15} - 17035 \beta_{14} - 12380 \beta_{13} + 18029 \beta_{12} + 22684 \beta_{11} - 9292 \beta_{10} + 4829 \beta_{9} - 17209 \beta_{8} - 9292 \beta_{7} + 29933 \beta_{6} + 5216 \beta_{5} - 14508 \beta_{4} - 11525 \beta_{3} + 7725 \beta_{2} - 17890 \beta_{1} - 23484$$ $$\nu^{14}$$ $$=$$ $$-26627 \beta_{14} - 53983 \beta_{13} + 60853 \beta_{12} - 33497 \beta_{11} + 7899 \beta_{10} + 41548 \beta_{9} - 12435 \beta_{8} - 7899 \beta_{7} - 26627 \beta_{6} - 54838 \beta_{5} - 46939 \beta_{4} - 2509 \beta_{3} + 26627 \beta_{2} + 24847 \beta_{1} + 13899$$ $$\nu^{15}$$ $$=$$ $$-150090 \beta_{15} + 106185 \beta_{14} + 75045 \beta_{13} - 104910 \beta_{12} - 136050 \beta_{11} + 47295 \beta_{10} - 25140 \beta_{9} + 100185 \beta_{8} + 47295 \beta_{7} - 174447 \beta_{6} - 38405 \beta_{5} + 85700 \beta_{4} + 68625 \beta_{3} - 43905 \beta_{2} + 112605 \beta_{1} + 140316$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$$.

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{10} + \beta_{12}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 0.0698697i − 2.47474i 2.25624i − 1.75349i 1.86029i − 1.18748i − 1.86029i 1.18748i − 2.25624i 1.75349i − 0.0698697i 2.47474i − 0.404926i 1.63422i 0.404926i − 1.63422i
−0.809017 + 0.587785i 0.104528 0.994522i 0.309017 0.951057i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.272042 0.0578243i 0.309017 + 0.951057i −0.978148 0.207912i −0.104528 0.994522i
121.2 −0.809017 + 0.587785i 0.104528 0.994522i 0.309017 0.951057i −0.500000 + 0.866025i 0.500000 + 0.866025i 3.97142 0.844151i 0.309017 + 0.951057i −0.978148 0.207912i −0.104528 0.994522i
361.1 0.309017 + 0.951057i −0.669131 0.743145i −0.809017 + 0.587785i −0.500000 0.866025i 0.500000 0.866025i 0.0818083 + 0.778354i −0.809017 0.587785i −0.104528 + 0.994522i 0.669131 0.743145i
361.2 0.309017 + 0.951057i −0.669131 0.743145i −0.809017 + 0.587785i −0.500000 0.866025i 0.500000 0.866025i 0.422760 + 4.02229i −0.809017 0.587785i −0.104528 + 0.994522i 0.669131 0.743145i
391.1 −0.809017 0.587785i −0.913545 0.406737i 0.309017 + 0.951057i −0.500000 + 0.866025i 0.500000 + 0.866025i −0.773194 + 0.858718i 0.309017 0.951057i 0.669131 + 0.743145i 0.913545 0.406737i
391.2 −0.809017 0.587785i −0.913545 0.406737i 0.309017 + 0.951057i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.0748188 0.0830946i 0.309017 0.951057i 0.669131 + 0.743145i 0.913545 0.406737i
421.1 −0.809017 + 0.587785i −0.913545 + 0.406737i 0.309017 0.951057i −0.500000 0.866025i 0.500000 0.866025i −0.773194 0.858718i 0.309017 + 0.951057i 0.669131 0.743145i 0.913545 + 0.406737i
421.2 −0.809017 + 0.587785i −0.913545 + 0.406737i 0.309017 0.951057i −0.500000 0.866025i 0.500000 0.866025i 0.0748188 + 0.0830946i 0.309017 + 0.951057i 0.669131 0.743145i 0.913545 + 0.406737i
541.1 0.309017 0.951057i −0.669131 + 0.743145i −0.809017 0.587785i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.0818083 0.778354i −0.809017 + 0.587785i −0.104528 0.994522i 0.669131 + 0.743145i
541.2 0.309017 0.951057i −0.669131 + 0.743145i −0.809017 0.587785i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.422760 4.02229i −0.809017 + 0.587785i −0.104528 0.994522i 0.669131 + 0.743145i
661.1 −0.809017 0.587785i 0.104528 + 0.994522i 0.309017 + 0.951057i −0.500000 0.866025i 0.500000 0.866025i 0.272042 + 0.0578243i 0.309017 0.951057i −0.978148 + 0.207912i −0.104528 + 0.994522i
661.2 −0.809017 0.587785i 0.104528 + 0.994522i 0.309017 + 0.951057i −0.500000 0.866025i 0.500000 0.866025i 3.97142 + 0.844151i 0.309017 0.951057i −0.978148 + 0.207912i −0.104528 + 0.994522i
691.1 0.309017 0.951057i 0.978148 + 0.207912i −0.809017 0.587785i −0.500000 0.866025i 0.500000 0.866025i −3.12748 + 1.39244i −0.809017 + 0.587785i 0.913545 + 0.406737i −0.978148 + 0.207912i
691.2 0.309017 0.951057i 0.978148 + 0.207912i −0.809017 0.587785i −0.500000 0.866025i 0.500000 0.866025i 0.577823 0.257263i −0.809017 + 0.587785i 0.913545 + 0.406737i −0.978148 + 0.207912i
751.1 0.309017 + 0.951057i 0.978148 0.207912i −0.809017 + 0.587785i −0.500000 + 0.866025i 0.500000 + 0.866025i −3.12748 1.39244i −0.809017 0.587785i 0.913545 0.406737i −0.978148 0.207912i
751.2 0.309017 + 0.951057i 0.978148 0.207912i −0.809017 + 0.587785i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.577823 + 0.257263i −0.809017 0.587785i 0.913545 0.406737i −0.978148 0.207912i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 751.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.bg.b 16
31.g even 15 1 inner 930.2.bg.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bg.b 16 1.a even 1 1 trivial
930.2.bg.b 16 31.g even 15 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{16} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$
$3$ $$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$
$5$ $$( 1 + T + T^{2} )^{8}$$
$7$ $$1 - 21 T + 217 T^{2} - 1110 T^{3} + 2850 T^{4} - 3834 T^{5} + 3656 T^{6} - 2955 T^{7} + 1409 T^{8} - 1305 T^{9} + 3074 T^{10} + 78 T^{11} - 165 T^{12} - 2 T^{14} - 3 T^{15} + T^{16}$$
$11$ $$3481 - 8496 T - 5483 T^{2} + 15865 T^{3} + 41655 T^{4} + 14176 T^{5} - 3929 T^{6} + 11725 T^{7} + 14894 T^{8} + 2875 T^{9} + 479 T^{10} - 427 T^{11} - 370 T^{12} - 5 T^{13} + 18 T^{14} + 2 T^{15} + T^{16}$$
$13$ $$23338561 + 29053634 T + 37678339 T^{2} + 28994464 T^{3} + 12989311 T^{4} + 4177897 T^{5} + 457846 T^{6} - 519468 T^{7} - 300028 T^{8} + 10143 T^{9} + 46356 T^{10} - 3232 T^{11} - 2259 T^{12} + 266 T^{13} + 54 T^{14} - 14 T^{15} + T^{16}$$
$17$ $$923521 - 834148 T + 16676233 T^{2} - 31217930 T^{3} + 13631365 T^{4} - 3514112 T^{5} + 2406604 T^{6} + 4108500 T^{7} + 1209719 T^{8} - 50370 T^{9} - 69054 T^{10} + 1466 T^{11} + 3030 T^{12} - 10 T^{13} - 48 T^{14} + 4 T^{15} + T^{16}$$
$19$ $$961 - 18538 T + 136166 T^{2} - 448772 T^{3} + 1154201 T^{4} - 1957979 T^{5} + 1854639 T^{6} - 909616 T^{7} + 193722 T^{8} + 11074 T^{9} - 5271 T^{10} - 2389 T^{11} + 116 T^{12} + 53 T^{13} + 26 T^{14} + 7 T^{15} + T^{16}$$
$23$ $$70576801 + 127266749 T + 850606846 T^{2} - 144641684 T^{3} + 716740718 T^{4} + 192993809 T^{5} + 28380614 T^{6} - 3028712 T^{7} + 2590363 T^{8} - 152486 T^{9} + 70436 T^{10} - 2137 T^{11} + 2258 T^{12} - 62 T^{13} + 34 T^{14} - 7 T^{15} + T^{16}$$
$29$ $$746983561 - 1955068423 T + 3154593481 T^{2} - 3193532952 T^{3} + 2083684071 T^{4} - 836783524 T^{5} + 212426924 T^{6} - 38623531 T^{7} + 8774452 T^{8} - 1371871 T^{9} + 237274 T^{10} - 37149 T^{11} + 8606 T^{12} - 1252 T^{13} + 196 T^{14} - 13 T^{15} + T^{16}$$
$31$ $$852891037441 + 577764896331 T + 217438401845 T^{2} + 65274464280 T^{3} + 16480232245 T^{4} + 3596012028 T^{5} + 735028538 T^{6} + 141565065 T^{7} + 25788685 T^{8} + 4566615 T^{9} + 764858 T^{10} + 120708 T^{11} + 17845 T^{12} + 2280 T^{13} + 245 T^{14} + 21 T^{15} + T^{16}$$
$37$ $$4871285995801 - 1242470932357 T + 2321513523494 T^{2} + 1286949838629 T^{3} + 712520344962 T^{4} + 174396112648 T^{5} + 39042918043 T^{6} + 5330307194 T^{7} + 811782091 T^{8} + 85373406 T^{9} + 10878063 T^{10} + 867872 T^{11} + 85332 T^{12} + 4881 T^{13} + 424 T^{14} + 17 T^{15} + T^{16}$$
$41$ $$255648121 - 662024545 T + 849357688 T^{2} - 317558060 T^{3} - 31652725 T^{4} + 3340355 T^{5} + 6732158 T^{6} + 3949750 T^{7} + 270844 T^{8} - 465005 T^{9} - 125398 T^{10} - 310 T^{11} + 6395 T^{12} + 1555 T^{13} + 217 T^{14} + 20 T^{15} + T^{16}$$
$43$ $$7775235904921 - 4048123072237 T - 1573810903271 T^{2} + 916506268069 T^{3} + 291233475692 T^{4} + 1361654813 T^{5} - 1824190957 T^{6} + 24220534 T^{7} - 31132154 T^{8} - 2007224 T^{9} + 648608 T^{10} - 108838 T^{11} + 7907 T^{12} - 464 T^{13} + 109 T^{14} + 2 T^{15} + T^{16}$$
$47$ $$12655528766521 - 1547150461283 T + 3333503620666 T^{2} - 675917652852 T^{3} + 380664140046 T^{4} - 64562821319 T^{5} + 9812319059 T^{6} - 1461855566 T^{7} + 217117027 T^{8} - 18074996 T^{9} + 2056669 T^{10} - 275859 T^{11} + 29816 T^{12} - 2327 T^{13} + 271 T^{14} - 23 T^{15} + T^{16}$$
$53$ $$491865502597681 - 135615127872358 T - 4885200913847 T^{2} + 3412767021975 T^{3} + 7974437835 T^{4} + 3543426323 T^{5} + 8595387029 T^{6} - 2066413690 T^{7} - 185724896 T^{8} + 9466565 T^{9} + 2166676 T^{10} + 211746 T^{11} + 14150 T^{12} + 1175 T^{13} + 112 T^{14} + 14 T^{15} + T^{16}$$
$59$ $$19002982096081 + 2909252821616 T + 1438386394966 T^{2} + 674327165287 T^{3} + 205809941587 T^{4} + 67319256191 T^{5} + 18413124062 T^{6} + 3727777447 T^{7} + 606412631 T^{8} + 76600287 T^{9} + 5401967 T^{10} - 167814 T^{11} - 73398 T^{12} - 5228 T^{13} + 66 T^{14} + 26 T^{15} + T^{16}$$
$61$ $$( 14762331 + 9235863 T - 1997757 T^{2} - 349029 T^{3} + 43605 T^{4} + 3726 T^{5} - 357 T^{6} - 12 T^{7} + T^{8} )^{2}$$
$67$ $$68475625 + 354376875 T + 2289726250 T^{2} - 2353621875 T^{3} + 2920819625 T^{4} - 1325027625 T^{5} + 1078152625 T^{6} - 569472000 T^{7} + 264763450 T^{8} - 79014900 T^{9} + 17919275 T^{10} - 2671875 T^{11} + 296210 T^{12} - 22695 T^{13} + 1280 T^{14} - 45 T^{15} + T^{16}$$
$71$ $$11515221481 - 10823293049 T + 3350335759 T^{2} - 2155369379 T^{3} - 757891129 T^{4} + 807238123 T^{5} + 459882451 T^{6} + 80232443 T^{7} + 3385882 T^{8} - 2269968 T^{9} - 423149 T^{10} + 28662 T^{11} + 29581 T^{12} + 6169 T^{13} + 664 T^{14} + 39 T^{15} + T^{16}$$
$73$ $$77490413641 + 28908549979 T - 38642746529 T^{2} - 28162165872 T^{3} + 102785487 T^{4} + 5708109529 T^{5} + 2955982907 T^{6} + 761675998 T^{7} + 139452571 T^{8} + 14697598 T^{9} + 689317 T^{10} - 81381 T^{11} - 8413 T^{12} + 208 T^{13} + 121 T^{14} + 19 T^{15} + T^{16}$$
$79$ $$2559280050625 + 4318912567500 T + 3812837701250 T^{2} + 881063266875 T^{3} + 8809395125 T^{4} - 13019232750 T^{5} - 512283000 T^{6} - 486074875 T^{7} - 26311375 T^{8} + 6375725 T^{9} + 277500 T^{10} + 56950 T^{11} + 16055 T^{12} + 695 T^{13} + 90 T^{14} + 20 T^{15} + T^{16}$$
$83$ $$9271117612201 - 8141435263287 T + 735497138933 T^{2} - 581488033245 T^{3} + 423933043970 T^{4} + 189999041952 T^{5} + 73721786724 T^{6} + 17996897510 T^{7} + 3199946249 T^{8} + 375332690 T^{9} + 37960086 T^{10} + 5438854 T^{11} + 732080 T^{12} + 62690 T^{13} + 3147 T^{14} + 86 T^{15} + T^{16}$$
$89$ $$233548926361 + 274507907187 T + 112609489300 T^{2} - 13428460570 T^{3} + 6710272680 T^{4} + 11547774941 T^{5} + 7016917342 T^{6} + 2240150900 T^{7} + 516344870 T^{8} + 76252070 T^{9} + 10768007 T^{10} + 1107499 T^{11} + 106475 T^{12} + 7145 T^{13} + 420 T^{14} + 13 T^{15} + T^{16}$$
$97$ $$105910629281521 - 139550640395386 T + 146800312671051 T^{2} - 83549257842134 T^{3} + 30230370952578 T^{4} - 7727591342466 T^{5} + 1496132669159 T^{6} - 228844549152 T^{7} + 28480595978 T^{8} - 2944150286 T^{9} + 256825811 T^{10} - 19013812 T^{11} + 1186483 T^{12} - 60707 T^{13} + 2419 T^{14} - 67 T^{15} + T^{16}$$