# Properties

 Label 930.2.bg.c 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: $$x^{16} + 40 x^{14} + 654 x^{12} + 5650 x^{10} + 27831 x^{8} + 78445 x^{6} + 119819 x^{4} + 87030 x^{2} + 22801$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 40 x^{14} + 654 x^{12} + 5650 x^{10} + 27831 x^{8} + 78445 x^{6} + 119819 x^{4} + 87030 x^{2} + 22801$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-995 \nu^{14} - 34078 \nu^{12} - 465200 \nu^{10} - 3223672 \nu^{8} - 11868388 \nu^{6} - 21934333 \nu^{4} - 17488620 \nu^{2} - 399691 \nu - 5305687$$$$)/799382$$ $$\beta_{2}$$ $$=$$ $$($$$$-995 \nu^{14} - 34078 \nu^{12} - 465200 \nu^{10} - 3223672 \nu^{8} - 11868388 \nu^{6} - 21934333 \nu^{4} - 17488620 \nu^{2} + 399691 \nu - 5305687$$$$)/799382$$ $$\beta_{3}$$ $$=$$ $$($$$$-1718215 \nu^{14} - 60936406 \nu^{12} - 848372834 \nu^{10} - 5894779186 \nu^{8} - 21531538578 \nu^{6} - 39892235189 \nu^{4} - 33191196434 \nu^{2} - 57955195 \nu - 9519995679$$$$)/ 115910390$$ $$\beta_{4}$$ $$=$$ $$($$$$-35137 \nu^{15} - 1255235 \nu^{13} - 17833820 \nu^{11} - 128278850 \nu^{9} - 491123375 \nu^{7} - 964195377 \nu^{5} - 897995920 \nu^{3} - 417191490 \nu - 60353341$$$$)/ 120706682$$ $$\beta_{5}$$ $$=$$ $$($$$$4706363 \nu^{15} + 122061303 \nu^{14} + 212701722 \nu^{13} + 4317955532 \nu^{12} + 3886690373 \nu^{11} + 60033458523 \nu^{10} + 36673149074 \nu^{9} + 417331391399 \nu^{8} + 189862300797 \nu^{7} + 1528473900512 \nu^{6} + 530052448287 \nu^{5} + 2839828170627 \nu^{4} + 716941913112 \nu^{3} + 2341805119727 \nu^{2} + 333142485860 \nu + 629548325710$$$$)/ 17502468890$$ $$\beta_{6}$$ $$=$$ $$($$$$9650638 \nu^{15} - 28483734 \nu^{14} + 352789665 \nu^{13} - 965113480 \nu^{12} + 5071614425 \nu^{11} - 12631799300 \nu^{10} + 36341854502 \nu^{9} - 80082068726 \nu^{8} + 135681134706 \nu^{7} - 251203570253 \nu^{6} + 250592861739 \nu^{5} - 345988251222 \nu^{4} + 197852507369 \nu^{3} - 129164149267 \nu^{2} + 58124761677 \nu + 19056870289$$$$)/ 17502468890$$ $$\beta_{7}$$ $$=$$ $$($$$$9650638 \nu^{15} + 28483734 \nu^{14} + 352789665 \nu^{13} + 965113480 \nu^{12} + 5071614425 \nu^{11} + 12631799300 \nu^{10} + 36341854502 \nu^{9} + 80082068726 \nu^{8} + 135681134706 \nu^{7} + 251203570253 \nu^{6} + 250592861739 \nu^{5} + 345988251222 \nu^{4} + 197852507369 \nu^{3} + 129164149267 \nu^{2} + 58124761677 \nu - 19056870289$$$$)/ 17502468890$$ $$\beta_{8}$$ $$=$$ $$($$$$-14497489 \nu^{15} - 23520213 \nu^{14} - 527379797 \nu^{13} - 810462753 \nu^{12} - 7648870428 \nu^{11} - 10871633372 \nu^{10} - 56809024793 \nu^{9} - 71548827545 \nu^{8} - 231449324654 \nu^{7} - 239145005005 \nu^{6} - 511836087780 \nu^{5} - 379074973941 \nu^{4} - 555179622305 \nu^{3} - 237999896656 \nu^{2} - 208311686164 \nu - 37865539764$$$$)/ 17502468890$$ $$\beta_{9}$$ $$=$$ $$($$$$-14478243 \nu^{15} + 56597065 \nu^{14} - 512817013 \nu^{13} + 1965702296 \nu^{12} - 7073628552 \nu^{11} + 26544967084 \nu^{10} - 47944489040 \nu^{9} + 176377443691 \nu^{8} - 165209593315 \nu^{7} + 602976117018 \nu^{6} - 265463223951 \nu^{5} + 1010440320854 \nu^{4} - 144925963951 \nu^{3} + 726054749449 \nu^{2} + 1032393521 \nu + 182944699149$$$$)/ 17502468890$$ $$\beta_{10}$$ $$=$$ $$($$$$14478243 \nu^{15} + 56597065 \nu^{14} + 512817013 \nu^{13} + 1965702296 \nu^{12} + 7073628552 \nu^{11} + 26544967084 \nu^{10} + 47944489040 \nu^{9} + 176377443691 \nu^{8} + 165209593315 \nu^{7} + 602976117018 \nu^{6} + 265463223951 \nu^{5} + 1010440320854 \nu^{4} + 144925963951 \nu^{3} + 726054749449 \nu^{2} - 1032393521 \nu + 182944699149$$$$)/ 17502468890$$ $$\beta_{11}$$ $$=$$ $$($$$$14361573 \nu^{15} - 122061303 \nu^{14} + 532756418 \nu^{13} - 4317955532 \nu^{12} + 7890718627 \nu^{11} - 60033458523 \nu^{10} + 59878418150 \nu^{9} - 417331391399 \nu^{8} + 248919218015 \nu^{7} - 1528473900512 \nu^{6} + 559793172711 \nu^{5} - 2839828170627 \nu^{4} + 611088826276 \nu^{3} - 2341805119727 \nu^{2} + 197325706574 \nu - 629548325710$$$$)/ 17502468890$$ $$\beta_{12}$$ $$=$$ $$($$$$17698863 \nu^{15} - 93061753 \nu^{14} + 583469063 \nu^{13} - 3295930907 \nu^{12} + 7234550077 \nu^{11} - 45803303838 \nu^{10} + 41420862070 \nu^{9} - 316849927239 \nu^{8} + 105575851065 \nu^{7} - 1142199468692 \nu^{6} + 78462317246 \nu^{5} - 2037008132382 \nu^{4} - 51333701789 \nu^{3} - 1535151469322 \nu^{2} - 16918969901 \nu - 380612791010$$$$)/ 17502468890$$ $$\beta_{13}$$ $$=$$ $$($$$$-155763 \nu^{15} + 347813 \nu^{14} - 5367303 \nu^{13} + 12135678 \nu^{12} - 71997572 \nu^{11} + 166237007 \nu^{10} - 473833295 \nu^{9} + 1139273455 \nu^{8} - 1583741755 \nu^{7} + 4141850575 \nu^{6} - 2510430291 \nu^{5} + 7827119286 \nu^{4} - 1576158256 \nu^{3} + 6976190606 \nu^{2} - 250765164 \nu + 2189120839$$$$)/ 115910390$$ $$\beta_{14}$$ $$=$$ $$($$$$24148127 \nu^{15} + 4963521 \nu^{14} + 880169462 \nu^{13} + 154650727 \nu^{12} + 12720484853 \nu^{11} + 1760165928 \nu^{10} + 93150879295 \nu^{9} + 8533241181 \nu^{8} + 367130459360 \nu^{7} + 12058565248 \nu^{6} + 762428949519 \nu^{5} - 33086722719 \nu^{4} + 753032129674 \nu^{3} - 108835747389 \nu^{2} + 266436447841 \nu - 56922410053$$$$)/ 17502468890$$ $$\beta_{15}$$ $$=$$ $$($$$$51769460 \nu^{15} + 38199376 \nu^{14} + 1805674948 \nu^{13} + 1394553554 \nu^{12} + 24542952957 \nu^{11} + 19944416126 \nu^{10} + 164774809153 \nu^{9} + 141439012653 \nu^{8} + 573447658409 \nu^{7} + 518510001419 \nu^{6} + 995569958642 \nu^{5} + 925390564434 \nu^{4} + 778981292867 \nu^{3} + 664183281629 \nu^{2} + 242101854347 \nu + 128116849205$$$$)/ 17502468890$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{2} - \beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{14} + \beta_{11} - \beta_{10} + \beta_{9} + 2 \beta_{8} - \beta_{5} - \beta_{3} - \beta_{1} - 6$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{14} - 2 \beta_{11} - 3 \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 7 \beta_{2} + 7 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{15} - 27 \beta_{14} - \beta_{12} - 11 \beta_{11} + 15 \beta_{10} - 6 \beta_{9} - 26 \beta_{8} - 2 \beta_{7} + 7 \beta_{6} + 11 \beta_{5} + \beta_{4} + 13 \beta_{3} + 8 \beta_{1} + 53$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{15} - 32 \beta_{14} + 4 \beta_{13} + 29 \beta_{11} + 40 \beta_{10} - 36 \beta_{9} + 32 \beta_{8} - 24 \beta_{7} - 52 \beta_{6} + 29 \beta_{5} - 40 \beta_{4} + 2 \beta_{3} + 61 \beta_{2} - 63 \beta_{1} - 20$$ $$\nu^{6}$$ $$=$$ $$-18 \beta_{15} + 292 \beta_{14} + 18 \beta_{12} + 113 \beta_{11} - 175 \beta_{10} + 33 \beta_{9} + 274 \beta_{8} + 37 \beta_{7} - 103 \beta_{6} - 113 \beta_{5} - 18 \beta_{4} - 145 \beta_{3} - 63 \beta_{1} - 505$$ $$\nu^{7}$$ $$=$$ $$-67 \beta_{15} + 382 \beta_{14} - 74 \beta_{13} + 7 \beta_{12} - 329 \beta_{11} - 457 \beta_{10} + 390 \beta_{9} - 375 \beta_{8} + 269 \beta_{7} + 577 \beta_{6} - 329 \beta_{5} + 527 \beta_{4} - 37 \beta_{3} - 575 \beta_{2} + 605 \beta_{1} + 260$$ $$\nu^{8}$$ $$=$$ $$225 \beta_{15} - 2979 \beta_{14} - 225 \beta_{12} - 1152 \beta_{11} + 1893 \beta_{10} - 186 \beta_{9} - 2754 \beta_{8} - 500 \beta_{7} + 1175 \beta_{6} + 1152 \beta_{5} + 225 \beta_{4} + 1534 \beta_{3} - 5 \beta_{2} + 540 \beta_{1} + 4929$$ $$\nu^{9}$$ $$=$$ $$849 \beta_{15} - 4149 \beta_{14} + 990 \beta_{13} - 141 \beta_{12} + 3474 \beta_{11} + 4922 \beta_{10} - 4073 \beta_{9} + 4008 \beta_{8} - 2924 \beta_{7} - 6083 \beta_{6} + 3474 \beta_{5} - 6001 \beta_{4} + 495 \beta_{3} + 5605 \beta_{2} - 5959 \beta_{1} - 2930$$ $$\nu^{10}$$ $$=$$ $$-2493 \beta_{15} + 29863 \beta_{14} + 2493 \beta_{12} + 11688 \beta_{11} - 19823 \beta_{10} + 1060 \beta_{9} + 27370 \beta_{8} + 5933 \beta_{7} - 12420 \beta_{6} - 11688 \beta_{5} - 2493 \beta_{4} - 15837 \beta_{3} + 83 \beta_{2} - 4963 \beta_{1} - 48652$$ $$\nu^{11}$$ $$=$$ $$-9753 \beta_{15} + 43382 \beta_{14} - 11700 \beta_{13} + 1947 \beta_{12} - 35713 \beta_{11} - 51435 \beta_{10} + 41682 \beta_{9} - 41435 \beta_{8} + 30888 \beta_{7} + 62570 \beta_{6} - 35713 \beta_{5} + 63951 \beta_{4} - 5850 \beta_{3} - 55479 \beta_{2} + 59382 \beta_{1} + 31002$$ $$\nu^{12}$$ $$=$$ $$26397 \beta_{15} - 297924 \beta_{14} - 26397 \beta_{12} - 118049 \beta_{11} + 204028 \beta_{10} - 5673 \beta_{9} - 271527 \beta_{8} - 65860 \beta_{7} + 127686 \beta_{6} + 118049 \beta_{5} + 26397 \beta_{4} + 161431 \beta_{3} - 702 \beta_{2} + 47568 \beta_{1} + 483077$$ $$\nu^{13}$$ $$=$$ $$107053 \beta_{15} - 446010 \beta_{14} + 130316 \beta_{13} - 23263 \beta_{12} + 363082 \beta_{11} + 528646 \beta_{10} - 421593 \beta_{9} + 422747 \beta_{8} - 318987 \beta_{7} - 634681 \beta_{6} + 363082 \beta_{5} - 658757 \beta_{4} + 65158 \beta_{3} + 553206 \beta_{2} - 595101 \beta_{1} - 317747$$ $$\nu^{14}$$ $$=$$ $$-274817 \beta_{15} + 2970237 \beta_{14} + 274817 \beta_{12} + 1187887 \beta_{11} - 2078569 \beta_{10} + 22388 \beta_{9} + 2695420 \beta_{8} + 704442 \beta_{7} - 1298905 \beta_{6} - 1187887 \beta_{5} - 274817 \beta_{4} - 1633897 \beta_{3} + 1035 \beta_{2} - 466025 \beta_{1} - 4812286$$ $$\nu^{15}$$ $$=$$ $$-1147044 \beta_{15} + 4549582 \beta_{14} - 1406814 \beta_{13} + 259770 \beta_{12} - 3673644 \beta_{11} - 5382115 \beta_{10} + 4235071 \beta_{9} - 4289812 \beta_{8} + 3240814 \beta_{7} + 6383582 \beta_{6} - 3673644 \beta_{5} + 6660634 \beta_{4} - 703407 \beta_{3} - 5537046 \beta_{2} + 5980683 \beta_{1} + 3200432$$

## Character values

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

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{6}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 − 3.17690i 0.772036i − 1.63922i 2.14198i − 2.47367i 3.14649i 2.47367i − 3.14649i 1.63922i − 2.14198i 3.17690i − 0.772036i 2.23655i − 1.00725i − 2.23655i 1.00725i
−0.809017 + 0.587785i 0.104528 0.994522i 0.309017 0.951057i 0.500000 0.866025i 0.500000 + 0.866025i −1.89605 + 0.403018i 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 3.84496 0.817271i 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 −0.426549 4.05835i −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 −0.105030 0.999294i −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 1.01621 1.12862i 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 2.57997 2.86534i 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 1.01621 + 1.12862i 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 2.57997 + 2.86534i 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 −0.426549 + 4.05835i −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 −0.105030 + 0.999294i −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 −1.89605 0.403018i 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 3.84496 + 0.817271i 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 −2.70387 + 1.20384i −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 3.19037 1.42044i −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 −2.70387 1.20384i −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 3.19037 + 1.42044i −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.c 16
31.g even 15 1 inner 930.2.bg.c 16

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

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$
$3$ $$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$
$5$ $$( 1 - T + T^{2} )^{8}$$
$7$ $$3575881 - 1299117 T + 2547497 T^{2} - 776964 T^{3} - 488632 T^{4} + 1080114 T^{5} - 169206 T^{6} - 206897 T^{7} + 78833 T^{8} + 1771 T^{9} - 5634 T^{10} + 1172 T^{11} - 7 T^{12} - 98 T^{13} + 48 T^{14} - 11 T^{15} + T^{16}$$
$11$ $$24025 - 165850 T + 379775 T^{2} - 222875 T^{3} + 303605 T^{4} + 39130 T^{5} - 65295 T^{6} + 210185 T^{7} + 156186 T^{8} - 101003 T^{9} + 29385 T^{10} - 4243 T^{11} - 226 T^{12} + 233 T^{13} - 40 T^{14} - 2 T^{15} + T^{16}$$
$13$ $$3575881 - 7900598 T + 6329497 T^{2} - 2717106 T^{3} + 566163 T^{4} + 532201 T^{5} - 189586 T^{6} - 130898 T^{7} + 124438 T^{8} - 47621 T^{9} + 12946 T^{10} - 3792 T^{11} + 1223 T^{12} - 362 T^{13} + 88 T^{14} - 14 T^{15} + T^{16}$$
$17$ $$( 961 + 2573 T + 1725 T^{2} - 92 T^{3} + 149 T^{4} + 232 T^{5} + 60 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$19$ $$32560398025 - 918465050 T + 3018589050 T^{2} + 403326050 T^{3} + 372031605 T^{4} + 121277915 T^{5} + 32074755 T^{6} - 5871120 T^{7} - 2589694 T^{8} - 172146 T^{9} + 45895 T^{10} - 4559 T^{11} + 4114 T^{12} + 241 T^{13} - 110 T^{14} - T^{15} + T^{16}$$
$23$ $$21335161 + 36014343 T + 105982030 T^{2} + 70575450 T^{3} + 91687590 T^{4} - 2474451 T^{5} + 9619472 T^{6} - 2443470 T^{7} + 2925575 T^{8} + 1177440 T^{9} + 191822 T^{10} + 3621 T^{11} + 2070 T^{12} + 300 T^{13} + 10 T^{14} - 3 T^{15} + T^{16}$$
$29$ $$31488857401 - 46923545381 T + 38614189673 T^{2} - 21212340252 T^{3} + 8638798231 T^{4} - 2665918236 T^{5} + 648700130 T^{6} - 124987301 T^{7} + 20061994 T^{8} - 2753137 T^{9} + 382190 T^{10} - 49953 T^{11} + 7366 T^{12} - 894 T^{13} + 122 T^{14} - 13 T^{15} + T^{16}$$
$31$ $$852891037441 - 247613526999 T + 134013055831 T^{2} - 33152556858 T^{3} + 10747013877 T^{4} - 2272576644 T^{5} + 554777612 T^{6} - 102261963 T^{7} + 20771621 T^{8} - 3298773 T^{9} + 577292 T^{10} - 76284 T^{11} + 11637 T^{12} - 1158 T^{13} + 151 T^{14} - 9 T^{15} + T^{16}$$
$37$ $$880480078921 - 103073724133 T + 475903282872 T^{2} + 328294227499 T^{3} + 232563248498 T^{4} + 73605146496 T^{5} + 19730458969 T^{6} + 3190126472 T^{7} + 499405383 T^{8} + 51336044 T^{9} + 6983341 T^{10} + 556548 T^{11} + 62768 T^{12} + 3013 T^{13} + 318 T^{14} + 11 T^{15} + T^{16}$$
$41$ $$923521 - 4632981 T + 56706268 T^{2} - 334727238 T^{3} + 761829453 T^{4} - 403499973 T^{5} + 112780286 T^{6} - 4454574 T^{7} - 7177132 T^{8} + 2945583 T^{9} - 466666 T^{10} - 24216 T^{11} + 29073 T^{12} - 6489 T^{13} + 727 T^{14} - 42 T^{15} + T^{16}$$
$43$ $$30684879241 - 16949721131 T + 40284405483 T^{2} - 13870862323 T^{3} + 6307142288 T^{4} - 1776264173 T^{5} + 140382761 T^{6} + 82815086 T^{7} - 13178932 T^{8} - 3013642 T^{9} + 441904 T^{10} + 108524 T^{11} - 5697 T^{12} - 2194 T^{13} - 23 T^{14} + 18 T^{15} + T^{16}$$
$47$ $$15205780492681 + 3889012349339 T + 2588783212548 T^{2} + 613248735318 T^{3} + 203913404126 T^{4} + 39552934359 T^{5} + 7617960395 T^{6} + 1119401474 T^{7} + 158624319 T^{8} + 21835078 T^{9} + 3131045 T^{10} + 400227 T^{11} + 54506 T^{12} + 6921 T^{13} + 657 T^{14} + 37 T^{15} + T^{16}$$
$53$ $$281966310025 - 94662261350 T + 10958394175 T^{2} - 14386240225 T^{3} + 3720758555 T^{4} - 1097075915 T^{5} + 529820885 T^{6} - 28471960 T^{7} + 15335116 T^{8} - 2299669 T^{9} + 281900 T^{10} - 100016 T^{11} + 4994 T^{12} + 1529 T^{13} - 80 T^{14} - 14 T^{15} + T^{16}$$
$59$ $$748624321 + 450854558 T + 286981316 T^{2} + 302404507 T^{3} + 71487341 T^{4} - 57472361 T^{5} + 70509894 T^{6} - 9936439 T^{7} + 4168377 T^{8} - 2001899 T^{9} + 15159 T^{10} - 80836 T^{11} + 35396 T^{12} + 1322 T^{13} - 304 T^{14} - 2 T^{15} + T^{16}$$
$61$ $$( 84361 - 70251 T - 71367 T^{2} + 11125 T^{3} + 6931 T^{4} - 720 T^{5} - 143 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$67$ $$781641651025 - 1486503203325 T + 1897833692950 T^{2} - 1276714320575 T^{3} + 615220584805 T^{4} - 202209035775 T^{5} + 51505744135 T^{6} - 9991554840 T^{7} + 1625801326 T^{8} - 219940096 T^{9} + 27061255 T^{10} - 2859089 T^{11} + 267764 T^{12} - 19439 T^{13} + 1120 T^{14} - 41 T^{15} + T^{16}$$
$71$ $$729751605025 + 179423448925 T + 1688811282475 T^{2} - 111171376075 T^{3} - 73472140145 T^{4} + 89423311505 T^{5} + 17494621135 T^{6} - 1668044535 T^{7} - 279493084 T^{8} - 10606986 T^{9} + 1376085 T^{10} + 269476 T^{11} + 18999 T^{12} - 1679 T^{13} - 150 T^{14} - T^{15} + T^{16}$$
$73$ $$1253230801 - 10132368017 T + 2085358192211 T^{2} + 952800044802 T^{3} - 838862002739 T^{4} - 465896979531 T^{5} + 130953915359 T^{6} + 42233804296 T^{7} + 4954631467 T^{8} + 398315036 T^{9} + 37414829 T^{10} + 3963999 T^{11} + 331621 T^{12} + 22302 T^{13} + 1301 T^{14} + 53 T^{15} + T^{16}$$
$79$ $$13653688801 - 9626020620 T - 4768019484 T^{2} + 2610424765 T^{3} + 588073605 T^{4} - 291554840 T^{5} + 985260874 T^{6} - 658689735 T^{7} + 197752369 T^{8} - 34871205 T^{9} + 4147294 T^{10} - 405050 T^{11} + 41235 T^{12} - 3755 T^{13} + 396 T^{14} - 30 T^{15} + T^{16}$$
$83$ $$4877765281 - 20704015245 T + 39047924131 T^{2} - 41296742415 T^{3} + 25878278670 T^{4} - 9156127050 T^{5} + 1781242754 T^{6} - 348507510 T^{7} + 84424139 T^{8} + 16870020 T^{9} + 3365744 T^{10} + 573990 T^{11} + 23280 T^{12} - 2880 T^{13} - 299 T^{14} + T^{16}$$
$89$ $$59150634993025 + 28081524476525 T + 10407983968100 T^{2} + 1670712205300 T^{3} + 292415418590 T^{4} + 144452569615 T^{5} + 80580495060 T^{6} + 22062134390 T^{7} + 4349711826 T^{8} + 615893806 T^{9} + 68538237 T^{10} + 6104363 T^{11} + 454205 T^{12} + 27707 T^{13} + 1382 T^{14} + 49 T^{15} + T^{16}$$
$97$ $$1536605680801 + 857267001232 T - 50308871733 T^{2} - 224802313684 T^{3} + 119095923056 T^{4} - 7239996038 T^{5} + 2202599135 T^{6} - 237372502 T^{7} + 131070194 T^{8} - 23060746 T^{9} + 5053255 T^{10} - 585136 T^{11} + 86571 T^{12} - 6487 T^{13} + 613 T^{14} - 21 T^{15} + T^{16}$$