# Properties

 Label 930.2.bg.f 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: 16.0.1669788916259765625.1 Defining polynomial: $$x^{16} + 16 x^{14} + 90 x^{12} + 239 x^{10} + 329 x^{8} + 239 x^{6} + 90 x^{4} + 16 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 16 x^{14} + 90 x^{12} + 239 x^{10} + 329 x^{8} + 239 x^{6} + 90 x^{4} + 16 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{14} + 29 \nu^{12} + 135 \nu^{10} + 253 \nu^{8} + 166 \nu^{6} - 16 \nu^{4} - 33 \nu^{2} + \nu - 4$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{15} + 2 \nu^{14} + 15 \nu^{13} + 30 \nu^{12} + 75 \nu^{11} + 149 \nu^{10} + 164 \nu^{9} + 314 \nu^{8} + 165 \nu^{7} + 269 \nu^{6} + 74 \nu^{5} + 45 \nu^{4} + 16 \nu^{3} - 29 \nu^{2} - 6$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$5 \nu^{15} + 2 \nu^{14} + 77 \nu^{13} + 33 \nu^{12} + 404 \nu^{11} + 194 \nu^{10} + 956 \nu^{9} + 539 \nu^{8} + 1092 \nu^{7} + 760 \nu^{6} + 597 \nu^{5} + 528 \nu^{4} + 166 \nu^{3} + 156 \nu^{2} + 21 \nu + 13$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{15} + 6 \nu^{14} + 46 \nu^{13} + 92 \nu^{12} + 239 \nu^{11} + 479 \nu^{10} + 553 \nu^{9} + 1119 \nu^{8} + 598 \nu^{7} + 1244 \nu^{6} + 284 \nu^{5} + 623 \nu^{4} + 60 \nu^{3} + 126 \nu^{2} + 6 \nu + 8$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$8 \nu^{15} - \nu^{14} + 125 \nu^{13} - 15 \nu^{12} + 673 \nu^{11} - 75 \nu^{10} + 1658 \nu^{9} - 164 \nu^{8} + 2004 \nu^{7} - 165 \nu^{6} + 1151 \nu^{5} - 74 \nu^{4} + 282 \nu^{3} - 17 \nu^{2} + 21 \nu - 2$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$($$$$2 \nu^{15} - 8 \nu^{14} + 31 \nu^{13} - 125 \nu^{12} + 165 \nu^{11} - 673 \nu^{10} + 403 \nu^{9} - 1658 \nu^{8} + 494 \nu^{7} - 2004 \nu^{6} + 313 \nu^{5} - 1151 \nu^{4} + 106 \nu^{3} - 282 \nu^{2} + 15 \nu - 21$$$$)/2$$ $$\beta_{8}$$ $$=$$ $$($$$$8 \nu^{15} + \nu^{14} + 125 \nu^{13} + 15 \nu^{12} + 673 \nu^{11} + 75 \nu^{10} + 1658 \nu^{9} + 164 \nu^{8} + 2004 \nu^{7} + 165 \nu^{6} + 1151 \nu^{5} + 74 \nu^{4} + 282 \nu^{3} + 17 \nu^{2} + 21 \nu + 2$$$$)/2$$ $$\beta_{9}$$ $$=$$ $$($$$$8 \nu^{15} + 3 \nu^{14} + 122 \nu^{13} + 46 \nu^{12} + 628 \nu^{11} + 239 \nu^{10} + 1433 \nu^{9} + 553 \nu^{8} + 1513 \nu^{7} + 598 \nu^{6} + 668 \nu^{5} + 284 \nu^{4} + 97 \nu^{3} + 60 \nu^{2} + 2 \nu + 6$$$$)/2$$ $$\beta_{10}$$ $$=$$ $$($$$$12 \nu^{14} + 185 \nu^{12} + 973 \nu^{10} + 2313 \nu^{8} + 2652 \nu^{6} + 1410 \nu^{4} + 317 \nu^{2} + \nu + 22$$$$)/2$$ $$\beta_{11}$$ $$=$$ $$($$$$-10 \nu^{15} + 11 \nu^{14} - 153 \nu^{13} + 171 \nu^{12} - 793 \nu^{11} + 912 \nu^{10} - 1836 \nu^{9} + 2211 \nu^{8} - 2007 \nu^{7} + 2602 \nu^{6} - 981 \nu^{5} + 1435 \nu^{4} - 203 \nu^{3} + 342 \nu^{2} - 17 \nu + 27$$$$)/2$$ $$\beta_{12}$$ $$=$$ $$($$$$-13 \nu^{15} - 6 \nu^{14} - 201 \nu^{13} - 93 \nu^{12} - 1061 \nu^{11} - 494 \nu^{10} - 2525 \nu^{9} - 1194 \nu^{8} - 2871 \nu^{7} - 1408 \nu^{6} - 1480 \nu^{5} - 787 \nu^{4} - 313 \nu^{3} - 192 \nu^{2} - 20 \nu - 15$$$$)/2$$ $$\beta_{13}$$ $$=$$ $$($$$$10 \nu^{15} + 11 \nu^{14} + 153 \nu^{13} + 171 \nu^{12} + 793 \nu^{11} + 912 \nu^{10} + 1836 \nu^{9} + 2211 \nu^{8} + 2007 \nu^{7} + 2602 \nu^{6} + 981 \nu^{5} + 1435 \nu^{4} + 203 \nu^{3} + 342 \nu^{2} + 17 \nu + 27$$$$)/2$$ $$\beta_{14}$$ $$=$$ $$($$$$22 \nu^{15} + 340 \nu^{13} + 1795 \nu^{11} + 4285 \nu^{9} + 4925 \nu^{7} + 2606 \nu^{5} + 570 \nu^{3} + 35 \nu - 1$$$$)/2$$ $$\beta_{15}$$ $$=$$ $$($$$$14 \nu^{15} + 11 \nu^{14} + 219 \nu^{13} + 171 \nu^{12} + 1182 \nu^{11} + 912 \nu^{10} + 2927 \nu^{9} + 2211 \nu^{8} + 3576 \nu^{7} + 2602 \nu^{6} + 2103 \nu^{5} + 1435 \nu^{4} + 547 \nu^{3} + 342 \nu^{2} + 48 \nu + 26$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{13} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} - \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - \beta_{11} - \beta_{10} - 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{5} - 2 \beta_{4} + \beta_{2} - 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{15} + \beta_{14} - 4 \beta_{13} + 2 \beta_{11} + 5 \beta_{10} + 5 \beta_{9} - 8 \beta_{8} + 2 \beta_{7} + 8 \beta_{6} - 5 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} + 7 \beta_{2} - 6 \beta_{1} + 3$$ $$\nu^{5}$$ $$=$$ $$-7 \beta_{15} + 5 \beta_{14} - 20 \beta_{13} + 20 \beta_{12} + 5 \beta_{11} + 10 \beta_{10} + 22 \beta_{9} + 21 \beta_{8} - 3 \beta_{7} + \beta_{6} + 19 \beta_{5} + 19 \beta_{4} - 10 \beta_{2} + 6 \beta_{1} - 1$$ $$\nu^{6}$$ $$=$$ $$10 \beta_{15} - 10 \beta_{14} + 19 \beta_{13} - 19 \beta_{11} - 29 \beta_{10} - 28 \beta_{9} + 62 \beta_{8} - 18 \beta_{7} - 62 \beta_{6} + 30 \beta_{5} - 30 \beta_{4} + 20 \beta_{3} - 53 \beta_{2} + 41 \beta_{1} - 16$$ $$\nu^{7}$$ $$=$$ $$53 \beta_{15} - 29 \beta_{14} + 163 \beta_{13} - 166 \beta_{12} - 30 \beta_{11} - 83 \beta_{10} - 186 \beta_{9} - 179 \beta_{8} + 32 \beta_{7} - 13 \beta_{6} - 154 \beta_{5} - 154 \beta_{4} + 83 \beta_{2} - 24 \beta_{1} + 12$$ $$\nu^{8}$$ $$=$$ $$-83 \beta_{15} + 83 \beta_{14} - 112 \beta_{13} + 154 \beta_{11} + 194 \beta_{10} + 183 \beta_{9} - 477 \beta_{8} + 141 \beta_{7} + 477 \beta_{6} - 208 \beta_{5} + 208 \beta_{4} - 166 \beta_{3} + 410 \beta_{2} - 302 \beta_{1} + 107$$ $$\nu^{9}$$ $$=$$ $$-410 \beta_{15} + 194 \beta_{14} - 1269 \beta_{13} + 1308 \beta_{12} + 208 \beta_{11} + 654 \beta_{10} + 1471 \beta_{9} + 1429 \beta_{8} - 269 \beta_{7} + 121 \beta_{6} + 1202 \beta_{5} + 1202 \beta_{4} - 654 \beta_{2} + 124 \beta_{1} - 108$$ $$\nu^{10}$$ $$=$$ $$654 \beta_{15} - 654 \beta_{14} + 759 \beta_{13} - 1202 \beta_{11} - 1399 \beta_{10} - 1307 \beta_{9} + 3659 \beta_{8} - 1081 \beta_{7} - 3659 \beta_{6} + 1534 \beta_{5} - 1534 \beta_{4} + 1308 \beta_{3} - 3165 \beta_{2} + 2282 \beta_{1} - 778$$ $$\nu^{11}$$ $$=$$ $$3165 \beta_{15} - 1399 \beta_{14} + 9762 \beta_{13} - 10122 \beta_{12} - 1534 \beta_{11} - 5061 \beta_{10} - 11393 \beta_{9} - 11126 \beta_{8} + 2125 \beta_{7} - 1004 \beta_{6} - 9268 \beta_{5} - 9268 \beta_{4} + 5061 \beta_{2} - 775 \beta_{1} + 883$$ $$\nu^{12}$$ $$=$$ $$-5061 \beta_{15} + 5061 \beta_{14} - 5512 \beta_{13} + 9268 \beta_{11} + 10438 \beta_{10} + 9719 \beta_{9} - 28031 \beta_{8} + 8264 \beta_{7} + 28031 \beta_{6} - 11577 \beta_{5} + 11577 \beta_{4} - 10122 \beta_{3} + 24334 \beta_{2} - 17386 \beta_{1} + 5836$$ $$\nu^{13}$$ $$=$$ $$-24334 \beta_{15} + 10438 \beta_{14} - 74836 \beta_{13} + 77802 \beta_{12} + 11577 \beta_{11} + 38901 \beta_{10} + 87593 \beta_{9} + 85754 \beta_{8} - 16454 \beta_{7} + 7952 \beta_{6} + 71139 \beta_{5} + 71139 \beta_{4} - 38901 \beta_{2} + 5404 \beta_{1} - 6948$$ $$\nu^{14}$$ $$=$$ $$38901 \beta_{15} - 38901 \beta_{14} + 41267 \beta_{13} - 71139 \beta_{11} - 79029 \beta_{10} - 73505 \beta_{9} + 214614 \beta_{8} - 63187 \beta_{7} - 214614 \beta_{6} + 88120 \beta_{5} - 88120 \beta_{4} + 77802 \beta_{3} - 186631 \beta_{2} + 132830 \beta_{1} - 44305$$ $$\nu^{15}$$ $$=$$ $$186631 \beta_{15} - 79029 \beta_{14} + 573060 \beta_{13} - 596450 \beta_{12} - 88120 \beta_{11} - 298225 \beta_{10} - 671571 \beta_{9} - 658203 \beta_{8} + 126494 \beta_{7} - 61753 \beta_{6} - 545077 \beta_{5} - 545077 \beta_{4} + 298225 \beta_{2} - 39723 \beta_{1} + 53801$$

## Character values

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

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{7}$$

## 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
 0.559146i − 1.78844i − 0.361487i 2.76635i 0.779734i − 1.28249i − 0.779734i 1.28249i 0.361487i − 2.76635i − 0.559146i 1.78844i − 1.39148i 0.718661i 1.39148i − 0.718661i
0.809017 0.587785i −0.104528 + 0.994522i 0.309017 0.951057i −0.500000 + 0.866025i 0.500000 + 0.866025i −2.95386 + 0.627863i −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 1.41394 0.300541i −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 −0.0662721 0.630537i 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 0.318079 + 3.02632i 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 −1.88789 + 2.09671i −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 0.736831 0.818334i −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 −1.88789 2.09671i −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 0.736831 + 0.818334i −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 −0.0662721 + 0.630537i 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 0.318079 3.02632i 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 −2.95386 0.627863i −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 1.41394 + 0.300541i −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 −3.16349 + 1.40848i 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 −0.897334 + 0.399519i 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 −3.16349 1.40848i 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 −0.897334 0.399519i 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.f 16
31.g even 15 1 inner 930.2.bg.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bg.f 16 1.a even 1 1 trivial
930.2.bg.f 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

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$
$3$ $$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$
$5$ $$( 1 + T + T^{2} )^{8}$$
$7$ $$7921 + 9523 T + 16431 T^{2} + 12332 T^{3} - 7674 T^{4} - 3426 T^{5} + 15284 T^{6} + 1581 T^{7} - 8203 T^{8} - 4039 T^{9} + 374 T^{10} + 1174 T^{11} + 691 T^{12} + 272 T^{13} + 76 T^{14} + 13 T^{15} + T^{16}$$
$11$ $$29474041 + 60196752 T + 46304921 T^{2} + 16737003 T^{3} + 1980791 T^{4} - 1451424 T^{5} - 980571 T^{6} - 345481 T^{7} - 868 T^{8} + 37529 T^{9} + 11859 T^{10} - 89 T^{11} - 514 T^{12} - 67 T^{13} + 6 T^{14} + 2 T^{15} + T^{16}$$
$13$ $$1 + 14 T + 129 T^{2} + 1004 T^{3} + 5261 T^{4} + 15757 T^{5} + 18816 T^{6} - 16078 T^{7} - 1888 T^{8} + 4553 T^{9} + 126 T^{10} - 432 T^{11} + 131 T^{12} + 66 T^{13} + 14 T^{14} + 6 T^{15} + T^{16}$$
$17$ $$208253761 + 351019644 T + 202850811 T^{2} + 44102008 T^{3} + 19910517 T^{4} - 6341846 T^{5} - 939218 T^{6} + 1051668 T^{7} - 103979 T^{8} - 70992 T^{9} + 12802 T^{10} - 1376 T^{11} - 528 T^{12} + 298 T^{13} + 6 T^{14} - 6 T^{15} + T^{16}$$
$19$ $$5517801 - 4946994 T + 4675806 T^{2} + 11822922 T^{3} + 11403747 T^{4} + 7362171 T^{5} + 2891457 T^{6} + 496692 T^{7} - 108864 T^{8} - 83268 T^{9} - 18333 T^{10} - 459 T^{11} + 1332 T^{12} + 297 T^{13} - 24 T^{14} - 9 T^{15} + T^{16}$$
$23$ $$8288641 + 91169293 T + 404989176 T^{2} + 385017982 T^{3} + 185075126 T^{4} + 48440009 T^{5} + 19077744 T^{6} + 5043286 T^{7} + 1029647 T^{8} + 72976 T^{9} + 103194 T^{10} + 3609 T^{11} + 5186 T^{12} + 132 T^{13} + 116 T^{14} + 3 T^{15} + T^{16}$$
$29$ $$407821454881 + 292382660387 T + 309077380469 T^{2} + 93004110952 T^{3} + 39298278863 T^{4} + 1132545292 T^{5} + 185099586 T^{6} + 62960411 T^{7} + 44619348 T^{8} + 2065537 T^{9} + 860544 T^{10} + 3191 T^{11} + 9188 T^{12} + 134 T^{13} + 26 T^{14} + T^{15} + T^{16}$$
$31$ $$852891037441 + 522739668109 T + 126913026383 T^{2} + 8703261904 T^{3} - 2459336423 T^{4} - 538621280 T^{5} + 3509572 T^{6} + 17031927 T^{7} + 3990975 T^{8} + 549417 T^{9} + 3652 T^{10} - 18080 T^{11} - 2663 T^{12} + 304 T^{13} + 143 T^{14} + 19 T^{15} + T^{16}$$
$37$ $$41383358041 + 36684148141 T + 28217855752 T^{2} + 11535313945 T^{3} + 4750591100 T^{4} + 1256822224 T^{5} + 422552371 T^{6} + 86502290 T^{7} + 23958529 T^{8} + 2972640 T^{9} + 729079 T^{10} + 59172 T^{11} + 16420 T^{12} + 535 T^{13} + 148 T^{14} + 3 T^{15} + T^{16}$$
$41$ $$1 - 885 T + 438528 T^{2} + 5071560 T^{3} + 34844585 T^{4} - 156725775 T^{5} + 212615938 T^{6} - 107321460 T^{7} + 32682564 T^{8} - 7097175 T^{9} + 954622 T^{10} - 40650 T^{11} - 5955 T^{12} + 945 T^{13} - 83 T^{14} + T^{16}$$
$43$ $$679697041 + 315276603 T + 8606483671 T^{2} - 2283448603 T^{3} + 1876002846 T^{4} + 159474359 T^{5} - 79600751 T^{6} + 95074406 T^{7} + 16287092 T^{8} - 4451734 T^{9} - 804856 T^{10} + 45694 T^{11} + 20201 T^{12} + 2 T^{13} - 189 T^{14} - 2 T^{15} + T^{16}$$
$47$ $$1907161 - 4616683 T + 7619204 T^{2} - 10211828 T^{3} + 12726758 T^{4} - 11104493 T^{5} + 7895061 T^{6} - 4380334 T^{7} + 2316333 T^{8} - 661058 T^{9} + 261549 T^{10} + 44111 T^{11} + 2678 T^{12} - 901 T^{13} + 101 T^{14} + T^{15} + T^{16}$$
$53$ $$12952801 + 9731696 T + 148829749 T^{2} + 6954541 T^{3} + 12687941 T^{4} - 10016797 T^{5} - 13340629 T^{6} + 1538878 T^{7} + 1941742 T^{8} - 113963 T^{9} - 82474 T^{10} + 26462 T^{11} + 6146 T^{12} - 341 T^{13} + 4 T^{14} + 14 T^{15} + T^{16}$$
$59$ $$253796761 - 673785714 T + 7765786324 T^{2} - 11356856239 T^{3} + 8613268081 T^{4} - 2646458457 T^{5} + 3657526 T^{6} + 268467243 T^{7} - 24486473 T^{8} - 11243183 T^{9} + 1006051 T^{10} + 188432 T^{11} - 14224 T^{12} + 14 T^{13} + 94 T^{14} - 16 T^{15} + T^{16}$$
$61$ $$( 15601 - 41521 T + 3275 T^{2} + 19301 T^{3} - 471 T^{4} - 1374 T^{5} - 75 T^{6} + 14 T^{7} + T^{8} )^{2}$$
$67$ $$6132304369801 - 8839033069969 T + 8097819141612 T^{2} - 4576274685105 T^{3} + 1911634977995 T^{4} - 566107760181 T^{5} + 131283399071 T^{6} - 23353288780 T^{7} + 3516526224 T^{8} - 443314820 T^{9} + 51442049 T^{10} - 5078253 T^{11} + 437750 T^{12} - 28575 T^{13} + 1458 T^{14} - 47 T^{15} + T^{16}$$
$71$ $$9771124801 - 23919777567 T + 29850611769 T^{2} - 26214768011 T^{3} + 14679740907 T^{4} - 3329717837 T^{5} - 654062537 T^{6} + 357988149 T^{7} + 15915916 T^{8} - 22078764 T^{9} + 4261573 T^{10} - 262778 T^{11} + 15177 T^{12} - 2279 T^{13} + 144 T^{14} - 3 T^{15} + T^{16}$$
$73$ $$96448721530561 + 51240608038233 T + 29348568164739 T^{2} + 8432005772564 T^{3} + 1365666830787 T^{4} - 13050513907 T^{5} - 32164725977 T^{6} - 4291657176 T^{7} + 9218341 T^{8} + 50604396 T^{9} + 5261683 T^{10} + 119717 T^{11} - 41853 T^{12} - 4354 T^{13} + 69 T^{14} + 27 T^{15} + T^{16}$$
$79$ $$37181713577761 + 38496574824834 T + 17103269089594 T^{2} + 4306927550629 T^{3} + 719730248361 T^{4} + 93232290172 T^{5} + 9380622826 T^{6} + 485232067 T^{7} - 9105313 T^{8} - 1706147 T^{9} - 200554 T^{10} - 26602 T^{11} + 341 T^{12} - 59 T^{13} - 36 T^{14} - 4 T^{15} + T^{16}$$
$83$ $$199218502921 - 451975815231 T + 731041082851 T^{2} - 830451980157 T^{3} + 574006078842 T^{4} - 240713227446 T^{5} + 64241746252 T^{6} - 11505198222 T^{7} + 1641245181 T^{8} - 266916282 T^{9} + 48692452 T^{10} - 6730926 T^{11} + 634262 T^{12} - 42372 T^{13} + 2061 T^{14} - 66 T^{15} + T^{16}$$
$89$ $$18312984183975625 + 3876834656031875 T + 1243111016831250 T^{2} + 295904924442500 T^{3} + 54627417674750 T^{4} + 7111877602375 T^{5} + 839826116250 T^{6} + 81186351500 T^{7} + 7114005950 T^{8} + 522527150 T^{9} + 39222375 T^{10} + 3153675 T^{11} + 288365 T^{12} + 21375 T^{13} + 1220 T^{14} + 45 T^{15} + T^{16}$$
$97$ $$5042562121 + 34436784450 T + 518345038969 T^{2} + 357096561580 T^{3} + 131145165082 T^{4} + 27948146340 T^{5} + 4080436447 T^{6} + 290318250 T^{7} + 18607480 T^{8} + 2368250 T^{9} + 1164463 T^{10} + 108850 T^{11} + 16037 T^{12} - 935 T^{13} + 91 T^{14} - 5 T^{15} + T^{16}$$