# Properties

 Label 847.2.f.w Level 847 Weight 2 Character orbit 847.f Analytic conductor 6.763 Analytic rank 0 Dimension 16 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 847.f (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.76332905120$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$5$$ Twist minimal: no (minimal twist has level 77) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 86 x^{12} - 145 x^{11} + 245 x^{10} - 245 x^{9} + 640 x^{8} - 1175 x^{7} + 2135 x^{6} - 2300 x^{5} + 1850 x^{4} - 925 x^{3} + 700 x^{2} - 200 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-33722118880975 \nu^{15} - 215936836813390 \nu^{14} + 154302737507927 \nu^{13} - 3234633146529285 \nu^{12} + 4494612950061958 \nu^{11} - 20835947263746904 \nu^{10} + 25956864481307657 \nu^{9} - 65098048812157320 \nu^{8} + 30869277708222543 \nu^{7} - 160665000409949210 \nu^{6} + 100524107363786610 \nu^{5} - 560336279122558890 \nu^{4} + 546963632352958070 \nu^{3} - 434474896057227745 \nu^{2} + 120133073008741205 \nu + 88528161377575350$$$$)/ 454580475630153760$$ $$\beta_{2}$$ $$=$$ $$($$$$-686277383720070 \nu^{15} - 1360333162376645 \nu^{14} + 444742620384542 \nu^{13} - 24553779304461089 \nu^{12} + 45808084826141647 \nu^{11} - 179218800902324756 \nu^{10} + 292393929070812298 \nu^{9} - 592228758909440833 \nu^{8} + 274072637813433960 \nu^{7} - 1226946693965489055 \nu^{6} + 2313198962618101340 \nu^{5} - 5164313069616781590 \nu^{4} + 5505326380247646890 \nu^{3} - 4265354995237286060 \nu^{2} + 1164454349991851775 \nu - 1569079812936964435$$$$)/ 454580475630153760$$ $$\beta_{3}$$ $$=$$ $$($$$$-874936745197272 \nu^{15} - 514706296938285 \nu^{14} - 5527222230184928 \nu^{13} - 8597109141630243 \nu^{12} - 9687442744709873 \nu^{11} - 69609203132183412 \nu^{10} + 41469078152400808 \nu^{9} - 248365491806816213 \nu^{8} - 292096370375405502 \nu^{7} - 583922916320059255 \nu^{6} + 395306407846606840 \nu^{5} - 2070346335621533450 \nu^{4} + 1021707328353270230 \nu^{3} - 1173049633586345730 \nu^{2} + 368306587037923705 \nu - 928639412535219835$$$$)/ 454580475630153760$$ $$\beta_{4}$$ $$=$$ $$($$$$-1565624412275039 \nu^{15} + 4832412997019746 \nu^{14} - 21672367495495401 \nu^{13} + 49577176484430299 \nu^{12} - 129795369286809034 \nu^{11} + 214474805906670840 \nu^{10} - 347621794857624183 \nu^{9} + 312652728515985000 \nu^{8} - 887685879699924177 \nu^{7} + 1770758763681867190 \nu^{6} - 3159142418183242750 \nu^{5} + 2957115228664641670 \nu^{4} - 1979053855141622330 \nu^{3} + 366777001484809055 \nu^{2} - 739274040704445675 \nu + 95214593589332550$$$$)/ 454580475630153760$$ $$\beta_{5}$$ $$=$$ $$($$$$1676337845170790 \nu^{15} - 5194753922189191 \nu^{14} + 23996414540642882 \nu^{13} - 55793399026072099 \nu^{12} + 149577475159066613 \nu^{11} - 254981761232364844 \nu^{10} + 433123005758956374 \nu^{9} - 436889234085257675 \nu^{8} + 1101433166283370416 \nu^{7} - 2030901870569891045 \nu^{6} + 3769863137668677300 \nu^{5} - 4085868252644216690 \nu^{4} + 3585858111334770990 \nu^{3} - 1476295694818421740 \nu^{2} + 1479871527393288525 \nu - 189874590198959025$$$$)/ 454580475630153760$$ $$\beta_{6}$$ $$=$$ $$($$$$-1676337845170790 \nu^{15} + 5194753922189191 \nu^{14} - 23996414540642882 \nu^{13} + 55793399026072099 \nu^{12} - 149577475159066613 \nu^{11} + 254981761232364844 \nu^{10} - 433123005758956374 \nu^{9} + 436889234085257675 \nu^{8} - 1101433166283370416 \nu^{7} + 2030901870569891045 \nu^{6} - 3769863137668677300 \nu^{5} + 4085868252644216690 \nu^{4} - 3585858111334770990 \nu^{3} + 1476295694818421740 \nu^{2} - 1025291051763134765 \nu + 189874590198959025$$$$)/ 454580475630153760$$ $$\beta_{7}$$ $$=$$ $$($$$$-3541126455103014 \nu^{15} + 10589657246428067 \nu^{14} - 49791707208255586 \nu^{13} + 113470349300804375 \nu^{12} - 307771508285388489 \nu^{11} + 517957948939998988 \nu^{10} - 888411928763985334 \nu^{9} + 893532845981546087 \nu^{8} - 2331418980078086280 \nu^{7} + 4191692862454263993 \nu^{6} - 7720969982054884100 \nu^{5} + 8245114954100718810 \nu^{4} - 7111420221063134790 \nu^{3} + 3822505603323246020 \nu^{2} - 2913263414629337545 \nu + 828358364029344005$$$$)/ 454580475630153760$$ $$\beta_{8}$$ $$=$$ $$($$$$-3808583743573302 \nu^{15} + 9860126818444867 \nu^{14} - 48487759413006482 \nu^{13} + 100202312298850263 \nu^{12} - 277961025462873673 \nu^{11} + 422449273531319756 \nu^{10} - 718628211268788150 \nu^{9} + 585481222317834807 \nu^{8} - 2124840867370928280 \nu^{7} + 3587400018998705673 \nu^{6} - 6360567528847132580 \nu^{5} + 5600600192035351850 \nu^{4} - 4088764696945967030 \nu^{3} + 1543886107663682020 \nu^{2} - 2299231619016502345 \nu + 22442708010214725$$$$)/ 454580475630153760$$ $$\beta_{9}$$ $$=$$ $$($$$$4147870816217860 \nu^{15} - 9098179878471885 \nu^{14} + 48642002600512068 \nu^{13} - 88958591156351927 \nu^{12} + 262233539631762351 \nu^{11} - 349871456206210068 \nu^{10} + 623455379026927044 \nu^{9} - 381247122554708893 \nu^{8} + 2150910817199786310 \nu^{7} - 3112048315103081495 \nu^{6} + 5535211606668989440 \nu^{5} - 3712550967753264370 \nu^{4} + 2767110524781279550 \nu^{3} - 1335895153189756830 \nu^{2} + 2975718288695695725 \nu - 29045982804284875$$$$)/ 454580475630153760$$ $$\beta_{10}$$ $$=$$ $$($$$$7327526319488073 \nu^{15} - 21838143406687493 \nu^{14} + 102438964384476967 \nu^{13} - 232311097915978782 \nu^{12} + 627185674080861763 \nu^{11} - 1047203823403574964 \nu^{10} + 1775572940212630785 \nu^{9} - 1735699601854553351 \nu^{8} + 4630478387715251365 \nu^{7} - 8426965416523262079 \nu^{6} + 15544790585628961210 \nu^{5} - 16343113922700919960 \nu^{4} + 12987506946195918920 \nu^{3} - 5718121221686276935 \nu^{2} + 4454224626365266160 \nu - 845040850217512955$$$$)/ 454580475630153760$$ $$\beta_{11}$$ $$=$$ $$($$$$-5481281688546819 \nu^{15} + 16493950936678582 \nu^{14} - 76461534295292341 \nu^{13} + 174584775915853027 \nu^{12} - 466393479528723526 \nu^{11} + 782325573976058616 \nu^{10} - 1307780222870847083 \nu^{9} + 1276995934654789444 \nu^{8} - 3395227103875767205 \nu^{7} + 6340299488823316290 \nu^{6} - 11480502622702818310 \nu^{5} + 12158198218393584390 \nu^{4} - 9161804501671007210 \nu^{3} + 3888216659798809835 \nu^{2} - 2965788298433688075 \nu + 864066394282347650$$$$)/ 227290237815076880$$ $$\beta_{12}$$ $$=$$ $$($$$$-11668286039317543 \nu^{15} + 32209452556198329 \nu^{14} - 155422622027191961 \nu^{13} + 335017774059801892 \nu^{12} - 918061503999498391 \nu^{11} + 1456783386223745044 \nu^{10} - 2469198487594261567 \nu^{9} + 2179375478325730279 \nu^{8} - 6793193349504836855 \nu^{7} + 11856118418058363915 \nu^{6} - 21771993760224740470 \nu^{5} + 21003963276182506660 \nu^{4} - 15299497235328621900 \nu^{3} + 5270338292583504165 \nu^{2} - 5226852994856502950 \nu + 59418569180970775$$$$)/ 454580475630153760$$ $$\beta_{13}$$ $$=$$ $$($$$$-16296167766296901 \nu^{15} + 46657645630438849 \nu^{14} - 220693813391828059 \nu^{13} + 490255203026234886 \nu^{12} - 1324081568271118439 \nu^{11} + 2171662124721910708 \nu^{10} - 3646287371621077981 \nu^{9} + 3462410165728884427 \nu^{8} - 9848220683260194529 \nu^{7} + 17797250626139187779 \nu^{6} - 31784433905471092610 \nu^{5} + 32902654497697219800 \nu^{4} - 24876887445936951080 \nu^{3} + 11522918100224582795 \nu^{2} - 8819030598088243120 \nu + 1667167131638041055$$$$)/ 454580475630153760$$ $$\beta_{14}$$ $$=$$ $$($$$$16576781827677814 \nu^{15} - 45453688208119129 \nu^{14} + 219284175557671186 \nu^{13} - 470575312000685073 \nu^{12} + 1289205279920317419 \nu^{11} - 2038073676823909444 \nu^{10} + 3445236405584454006 \nu^{9} - 3032211154151816061 \nu^{8} + 9585093068819660548 \nu^{7} - 16816185325336642299 \nu^{6} + 30431971572347844100 \nu^{5} - 29120858031490839590 \nu^{4} + 20842243045981534410 \nu^{3} - 8417551544734639360 \nu^{2} + 7973236770827366535 \nu - 1025213227739747135$$$$)/ 454580475630153760$$ $$\beta_{15}$$ $$=$$ $$($$$$-31742657223153564 \nu^{15} + 90930749048576029 \nu^{14} - 431347940936808556 \nu^{13} + 955931937354509231 \nu^{12} - 2590293343054165503 \nu^{11} + 4236554416589668948 \nu^{10} - 7143632052250437628 \nu^{9} + 6742540241983255677 \nu^{8} - 19234203982822940782 \nu^{7} + 34610408690995495111 \nu^{6} - 62595008084751532400 \nu^{5} + 64110835668533354210 \nu^{4} - 48322167102593217550 \nu^{3} + 21980961042064417270 \nu^{2} - 18197077383887239845 \nu + 3492178872565443195$$$$)/ 454580475630153760$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{6} + \beta_{5}$$ $$\nu^{2}$$ $$=$$ $$\beta_{11} + 2 \beta_{10} - 2 \beta_{8} + 3 \beta_{7} + \beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + \beta_{8} - 5 \beta_{6} - \beta_{5} - \beta_{2} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{15} - \beta_{14} - \beta_{11} - 8 \beta_{10} - 6 \beta_{9} - 15 \beta_{7} + \beta_{5} + \beta_{4} - 6 \beta_{2} - \beta_{1} + 1$$ $$\nu^{5}$$ $$=$$ $$-7 \beta_{14} - 7 \beta_{13} - 7 \beta_{12} + 7 \beta_{11} - 8 \beta_{8} + 8 \beta_{7} + 10 \beta_{6} - 10 \beta_{4} - \beta_{3} + 8 \beta_{2} - 18 \beta_{1} - 3$$ $$\nu^{6}$$ $$=$$ $$7 \beta_{15} + \beta_{14} - 7 \beta_{13} + 11 \beta_{12} + 54 \beta_{10} + 35 \beta_{9} + 40 \beta_{8} + 47 \beta_{7} - 10 \beta_{6} - 10 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} + 2 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$42 \beta_{15} + 54 \beta_{14} + 13 \beta_{12} - 45 \beta_{11} + 12 \beta_{10} - 13 \beta_{9} + 30 \beta_{8} - 86 \beta_{7} + 76 \beta_{5} + 165 \beta_{4} + 42 \beta_{3} - 58 \beta_{2} + 28$$ $$\nu^{8}$$ $$=$$ $$-9 \beta_{15} + 38 \beta_{13} - 210 \beta_{12} + 89 \beta_{11} - 234 \beta_{10} - 89 \beta_{9} - 308 \beta_{8} + 80 \beta_{6} + 28 \beta_{5} + 210 \beta_{2} - 28 \beta_{1} - 98$$ $$\nu^{9}$$ $$=$$ $$-346 \beta_{15} - 346 \beta_{14} + 248 \beta_{13} + 117 \beta_{11} + 38 \beta_{10} + 290 \beta_{9} + 614 \beta_{7} - 524 \beta_{6} - 1000 \beta_{5} - 1000 \beta_{4} - 248 \beta_{3} + 290 \beta_{2} + 476 \beta_{1} - 94$$ $$\nu^{10}$$ $$=$$ $$-56 \beta_{14} - 56 \beta_{13} + 1290 \beta_{12} - 1290 \beta_{11} + 1992 \beta_{8} - 1992 \beta_{7} - 272 \beta_{6} + 272 \beta_{4} - 131 \beta_{3} - 1931 \beta_{2} + 328 \beta_{1} + 1406$$ $$\nu^{11}$$ $$=$$ $$2174 \beta_{15} + 1477 \beta_{14} - 2174 \beta_{13} - 913 \beta_{12} - 447 \beta_{10} - 1890 \beta_{9} - 1625 \beta_{8} - 2621 \beta_{7} + 6160 \beta_{6} + 6160 \beta_{5} + 3461 \beta_{4} + 697 \beta_{3} - 3461 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$284 \beta_{15} + 848 \beta_{14} - 4374 \beta_{12} + 8050 \beta_{11} + 8579 \beta_{10} + 4374 \beta_{9} - 8295 \beta_{8} + 21093 \beta_{7} - 2273 \beta_{5} - 4375 \beta_{4} + 284 \beta_{3} + 12424 \beta_{2} - 8669$$ $$\nu^{13}$$ $$=$$ $$-8898 \beta_{15} + 13556 \beta_{13} + 12425 \beta_{12} - 6647 \beta_{11} + 2237 \beta_{10} + 6647 \beta_{9} + 17819 \beta_{8} - 38325 \beta_{6} - 22394 \beta_{5} - 12425 \beta_{2} + 22394 \beta_{1} + 5394$$ $$\nu^{14}$$ $$=$$ $$-3382 \beta_{15} - 3382 \beta_{14} + 1131 \beta_{13} - 29041 \beta_{11} - 85364 \beta_{10} - 50750 \beta_{9} - 133987 \beta_{7} + 17557 \beta_{6} + 31350 \beta_{5} + 31350 \beta_{4} - 1131 \beta_{3} - 50750 \beta_{2} - 13793 \beta_{1} + 31232$$ $$\nu^{15}$$ $$=$$ $$-54132 \beta_{14} - 54132 \beta_{13} - 82100 \beta_{12} + 82100 \beta_{11} - 120596 \beta_{8} + 120596 \beta_{7} + 143446 \beta_{6} - 143446 \beta_{4} - 30172 \beta_{3} + 128698 \beta_{2} - 96554 \beta_{1} - 67213$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/847\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$365$$ $$\chi(n)$$ $$1$$ $$\beta_{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
148.1
 −1.38112 + 1.00344i 0.183009 − 0.132964i 0.901622 − 0.655067i 1.60551 − 1.16647i −0.788594 − 2.42704i −0.206962 − 0.636964i 0.435488 + 1.34029i 0.751051 + 2.31150i −1.38112 − 1.00344i 0.183009 + 0.132964i 0.901622 + 0.655067i 1.60551 + 1.16647i −0.788594 + 2.42704i −0.206962 + 0.636964i 0.435488 − 1.34029i 0.751051 − 2.31150i
−0.527541 1.62360i −1.85391 1.34694i −0.739758 + 0.537466i 1.25658 3.86735i −1.20889 + 3.72058i −0.809017 + 0.587785i −1.49936 1.08935i 0.695665 + 2.14104i −6.94194
148.2 0.0699031 + 0.215140i 0.177280 + 0.128801i 1.57664 1.14549i −0.771159 + 2.37338i −0.0153178 + 0.0471435i −0.809017 + 0.587785i 0.722670 + 0.525051i −0.912213 2.80750i −0.564516
148.3 0.344389 + 1.05992i 2.31283 + 1.68037i 0.613206 0.445520i 1.06799 3.28693i −0.984546 + 3.03012i −0.809017 + 0.587785i 2.48664 + 1.80665i 1.59850 + 4.91966i 3.85168
148.4 0.613249 + 1.88739i −2.25424 1.63780i −1.56812 + 1.13930i −0.00832008 + 0.0256066i 1.70875 5.25900i −0.809017 + 0.587785i 0.0990607 + 0.0719718i 1.47215 + 4.53082i −0.0534317
323.1 −2.06456 1.49999i −0.126882 + 0.390502i 1.39441 + 4.29156i −2.77410 + 2.01550i 0.847707 0.615895i 0.309017 + 0.951057i 1.98127 6.09772i 2.29066 + 1.66426i 8.75055
323.2 −0.541834 0.393666i 0.970243 2.98610i −0.479422 1.47551i −1.73435 + 1.26008i −1.70124 + 1.23602i 0.309017 + 0.951057i −0.735015 + 2.26214i −5.54838 4.03113i 1.43578
323.3 1.14012 + 0.828347i −0.668522 + 2.05750i −0.00431527 0.0132810i 1.48162 1.07646i −2.46652 + 1.79203i 0.309017 + 0.951057i 0.877057 2.69930i −1.35932 0.987607i 2.58091
323.4 1.96628 + 1.42858i 0.443194 1.36401i 1.20736 + 3.71587i −1.01825 + 0.739805i 2.82005 2.04888i 0.309017 + 0.951057i −1.43233 + 4.40826i 0.762946 + 0.554312i −3.05904
372.1 −0.527541 + 1.62360i −1.85391 + 1.34694i −0.739758 0.537466i 1.25658 + 3.86735i −1.20889 3.72058i −0.809017 0.587785i −1.49936 + 1.08935i 0.695665 2.14104i −6.94194
372.2 0.0699031 0.215140i 0.177280 0.128801i 1.57664 + 1.14549i −0.771159 2.37338i −0.0153178 0.0471435i −0.809017 0.587785i 0.722670 0.525051i −0.912213 + 2.80750i −0.564516
372.3 0.344389 1.05992i 2.31283 1.68037i 0.613206 + 0.445520i 1.06799 + 3.28693i −0.984546 3.03012i −0.809017 0.587785i 2.48664 1.80665i 1.59850 4.91966i 3.85168
372.4 0.613249 1.88739i −2.25424 + 1.63780i −1.56812 1.13930i −0.00832008 0.0256066i 1.70875 + 5.25900i −0.809017 0.587785i 0.0990607 0.0719718i 1.47215 4.53082i −0.0534317
729.1 −2.06456 + 1.49999i −0.126882 0.390502i 1.39441 4.29156i −2.77410 2.01550i 0.847707 + 0.615895i 0.309017 0.951057i 1.98127 + 6.09772i 2.29066 1.66426i 8.75055
729.2 −0.541834 + 0.393666i 0.970243 + 2.98610i −0.479422 + 1.47551i −1.73435 1.26008i −1.70124 1.23602i 0.309017 0.951057i −0.735015 2.26214i −5.54838 + 4.03113i 1.43578
729.3 1.14012 0.828347i −0.668522 2.05750i −0.00431527 + 0.0132810i 1.48162 + 1.07646i −2.46652 1.79203i 0.309017 0.951057i 0.877057 + 2.69930i −1.35932 + 0.987607i 2.58091
729.4 1.96628 1.42858i 0.443194 + 1.36401i 1.20736 3.71587i −1.01825 0.739805i 2.82005 + 2.04888i 0.309017 0.951057i −1.43233 4.40826i 0.762946 0.554312i −3.05904
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 729.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.f.w 16
11.b odd 2 1 847.2.f.v 16
11.c even 5 2 77.2.f.b 16
11.c even 5 1 847.2.a.p 8
11.c even 5 1 inner 847.2.f.w 16
11.d odd 10 1 847.2.a.o 8
11.d odd 10 1 847.2.f.v 16
11.d odd 10 2 847.2.f.x 16
33.f even 10 1 7623.2.a.cw 8
33.h odd 10 2 693.2.m.i 16
33.h odd 10 1 7623.2.a.ct 8
77.j odd 10 2 539.2.f.e 16
77.j odd 10 1 5929.2.a.bt 8
77.l even 10 1 5929.2.a.bs 8
77.m even 15 4 539.2.q.g 32
77.p odd 30 4 539.2.q.f 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.b 16 11.c even 5 2
539.2.f.e 16 77.j odd 10 2
539.2.q.f 32 77.p odd 30 4
539.2.q.g 32 77.m even 15 4
693.2.m.i 16 33.h odd 10 2
847.2.a.o 8 11.d odd 10 1
847.2.a.p 8 11.c even 5 1
847.2.f.v 16 11.b odd 2 1
847.2.f.v 16 11.d odd 10 1
847.2.f.w 16 1.a even 1 1 trivial
847.2.f.w 16 11.c even 5 1 inner
847.2.f.x 16 11.d odd 10 2
5929.2.a.bs 8 77.l even 10 1
5929.2.a.bt 8 77.j odd 10 1
7623.2.a.ct 8 33.h odd 10 1
7623.2.a.cw 8 33.f even 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(847, [\chi])$$:

 $$T_{2}^{16} - \cdots$$ $$T_{3}^{16} + \cdots$$ $$T_{13}^{16} - \cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T - 4 T^{2} + 7 T^{3} + 13 T^{4} - 18 T^{5} - 25 T^{6} + 39 T^{7} + 52 T^{8} - 109 T^{9} - 115 T^{10} + 198 T^{11} + 303 T^{12} - 287 T^{13} - 418 T^{14} + 148 T^{15} + 625 T^{16} + 296 T^{17} - 1672 T^{18} - 2296 T^{19} + 4848 T^{20} + 6336 T^{21} - 7360 T^{22} - 13952 T^{23} + 13312 T^{24} + 19968 T^{25} - 25600 T^{26} - 36864 T^{27} + 53248 T^{28} + 57344 T^{29} - 65536 T^{30} - 65536 T^{31} + 65536 T^{32}$$
$3$ $$1 + 2 T - 3 T^{2} - 4 T^{3} + 2 T^{4} - 6 T^{5} - 8 T^{6} - 86 T^{7} - 114 T^{8} + 228 T^{9} + 367 T^{10} + 18 T^{11} - 333 T^{12} + 1016 T^{13} + 2720 T^{14} - 2368 T^{15} - 10592 T^{16} - 7104 T^{17} + 24480 T^{18} + 27432 T^{19} - 26973 T^{20} + 4374 T^{21} + 267543 T^{22} + 498636 T^{23} - 747954 T^{24} - 1692738 T^{25} - 472392 T^{26} - 1062882 T^{27} + 1062882 T^{28} - 6377292 T^{29} - 14348907 T^{30} + 28697814 T^{31} + 43046721 T^{32}$$
$5$ $$1 + 5 T + 14 T^{2} + 44 T^{3} + 109 T^{4} + 292 T^{5} + 994 T^{6} + 2416 T^{7} + 5986 T^{8} + 16176 T^{9} + 35286 T^{10} + 91101 T^{11} + 234456 T^{12} + 492396 T^{13} + 1194866 T^{14} + 2681212 T^{15} + 5341741 T^{16} + 13406060 T^{17} + 29871650 T^{18} + 61549500 T^{19} + 146535000 T^{20} + 284690625 T^{21} + 551343750 T^{22} + 1263750000 T^{23} + 2338281250 T^{24} + 4718750000 T^{25} + 9707031250 T^{26} + 14257812500 T^{27} + 26611328125 T^{28} + 53710937500 T^{29} + 85449218750 T^{30} + 152587890625 T^{31} + 152587890625 T^{32}$$
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$
$11$ 
$13$ $$1 - 13 T + 72 T^{2} - 244 T^{3} + 737 T^{4} - 2006 T^{5} + 3492 T^{6} - 7936 T^{7} + 40586 T^{8} - 125222 T^{9} + 55652 T^{10} + 2894553 T^{11} - 22953788 T^{12} + 102379926 T^{13} - 373476350 T^{14} + 1282762402 T^{15} - 4399561547 T^{16} + 16675911226 T^{17} - 63117503150 T^{18} + 224928697422 T^{19} - 655583139068 T^{20} + 1074727267029 T^{21} + 268621574468 T^{22} - 7857494795774 T^{23} + 33107247042506 T^{24} - 84157307024128 T^{25} + 481401853536708 T^{26} - 3595073750438222 T^{27} + 17170688735268497 T^{28} - 73901526008509732 T^{29} + 283491099770348808 T^{30} - 665416609183179841 T^{31} + 665416609183179841 T^{32}$$
$17$ $$1 - 10 T - 37 T^{2} + 633 T^{3} - 168 T^{4} - 13271 T^{5} + 17626 T^{6} - 16005 T^{7} + 351109 T^{8} + 5653873 T^{9} - 25003323 T^{10} - 95686688 T^{11} + 315645412 T^{12} + 305068606 T^{13} + 5822669428 T^{14} + 3664060760 T^{15} - 218663265475 T^{16} + 62289032920 T^{17} + 1682751464692 T^{18} + 1498802061278 T^{19} + 26363020455652 T^{20} - 135861413763616 T^{21} - 603519434141787 T^{22} + 2320002744130529 T^{23} + 2449251219352069 T^{24} - 1897998963334485 T^{25} + 35533908489314074 T^{26} - 454822335898597543 T^{27} - 97880535854599848 T^{28} + 6269597894829458121 T^{29} - 6229979582697834373 T^{30} - 28624230515098157930 T^{31} + 48661191875666868481 T^{32}$$
$19$ $$1 + 6 T + 3 T^{2} + 85 T^{3} + 1184 T^{4} - 2399 T^{5} - 34320 T^{6} + 47453 T^{7} + 19149 T^{8} - 5712061 T^{9} - 3409115 T^{10} + 100225110 T^{11} - 227341598 T^{12} - 1375692546 T^{13} + 11005678798 T^{14} + 19570163952 T^{15} - 191338098209 T^{16} + 371833115088 T^{17} + 3973050046078 T^{18} - 9435875173014 T^{19} - 29627384392958 T^{20} + 248167294645890 T^{21} - 160384818605315 T^{22} - 5105849899344079 T^{23} + 325218248672109 T^{24} + 15312499322706887 T^{25} - 210418193967730320 T^{26} - 279460131096827381 T^{27} + 2620564864174334624 T^{28} + 3574503594291850015 T^{29} + 2397020057348652363 T^{30} + 91086762179248789794 T^{31} +$$$$28\!\cdots\!81$$$$T^{32}$$
$23$ $$( 1 - 16 T + 224 T^{2} - 2234 T^{3} + 19567 T^{4} - 141712 T^{5} + 926670 T^{6} - 5230592 T^{7} + 26816927 T^{8} - 120303616 T^{9} + 490208430 T^{10} - 1724209904 T^{11} + 5475648847 T^{12} - 14378790262 T^{13} + 33160039136 T^{14} - 54477207152 T^{15} + 78310985281 T^{16} )^{2}$$
$29$ $$1 + 12 T - 70 T^{2} - 1390 T^{3} - 774 T^{4} + 64772 T^{5} + 304755 T^{6} - 772470 T^{7} - 15507999 T^{8} - 61115248 T^{9} + 297254044 T^{10} + 3599964238 T^{11} + 3985868241 T^{12} - 93662943228 T^{13} - 421310359901 T^{14} + 1028919750178 T^{15} + 15133672321246 T^{16} + 29838672755162 T^{17} - 354322012676741 T^{18} - 2284345522387692 T^{19} + 2819128875362721 T^{20} + 73839402880289462 T^{21} + 176813637632760124 T^{22} - 1054230468593859632 T^{23} - 7757820871952775039 T^{24} - 11206335051979526430 T^{25} +$$$$12\!\cdots\!55$$$$T^{26} +$$$$79\!\cdots\!88$$$$T^{27} -$$$$27\!\cdots\!34$$$$T^{28} -$$$$14\!\cdots\!10$$$$T^{29} -$$$$20\!\cdots\!70$$$$T^{30} +$$$$10\!\cdots\!88$$$$T^{31} +$$$$25\!\cdots\!21$$$$T^{32}$$
$31$ $$1 + 2 T - 75 T^{2} - 352 T^{3} + 2587 T^{4} + 15784 T^{5} - 75864 T^{6} - 435376 T^{7} + 3782312 T^{8} + 14561418 T^{9} - 150890523 T^{10} - 457844234 T^{11} + 4912277406 T^{12} + 5133296146 T^{13} - 182512228746 T^{14} + 17734057088 T^{15} + 6474843737997 T^{16} + 549755769728 T^{17} - 175394251824906 T^{18} + 152926025485486 T^{19} + 4536591342266526 T^{20} - 13107691709665334 T^{21} - 133915894590515163 T^{22} + 400622674342969398 T^{23} + 3225900005605543592 T^{24} - 11511176937824297296 T^{25} - 62180280363511487064 T^{26} +$$$$40\!\cdots\!04$$$$T^{27} +$$$$20\!\cdots\!07$$$$T^{28} -$$$$85\!\cdots\!32$$$$T^{29} -$$$$56\!\cdots\!75$$$$T^{30} +$$$$46\!\cdots\!02$$$$T^{31} +$$$$72\!\cdots\!81$$$$T^{32}$$
$37$ $$1 + 11 T - 74 T^{2} - 1323 T^{3} + 1923 T^{4} + 97285 T^{5} + 90914 T^{6} - 5713113 T^{7} - 17349512 T^{8} + 265295915 T^{9} + 1411267662 T^{10} - 9905347845 T^{11} - 84217332664 T^{12} + 276882360296 T^{13} + 4072917985018 T^{14} - 3748231783796 T^{15} - 163433617792703 T^{16} - 138684576000452 T^{17} + 5575824721489642 T^{18} + 14024922196073288 T^{19} - 157836840402894904 T^{20} - 686876015033722665 T^{21} + 3620926710561085758 T^{22} + 25185039206666811695 T^{23} - 60939804435555836552 T^{24} -$$$$74\!\cdots\!01$$$$T^{25} +$$$$43\!\cdots\!86$$$$T^{26} +$$$$17\!\cdots\!05$$$$T^{27} +$$$$12\!\cdots\!63$$$$T^{28} -$$$$32\!\cdots\!31$$$$T^{29} -$$$$66\!\cdots\!86$$$$T^{30} +$$$$36\!\cdots\!23$$$$T^{31} +$$$$12\!\cdots\!41$$$$T^{32}$$
$41$ $$1 - 20 T + 7 T^{2} + 2422 T^{3} - 13592 T^{4} - 89190 T^{5} + 972960 T^{6} - 955362 T^{7} - 17703016 T^{8} + 127968546 T^{9} - 596716619 T^{10} - 1461391100 T^{11} + 26017347843 T^{12} - 43282378760 T^{13} - 65439843920 T^{14} + 655546700040 T^{15} - 12524609697280 T^{16} + 26877414701640 T^{17} - 110004377629520 T^{18} - 2983064826517960 T^{19} + 73518806858183523 T^{20} - 169311221021211100 T^{21} - 2834466142587081179 T^{22} + 24922421255837347026 T^{23} -$$$$14\!\cdots\!36$$$$T^{24} -$$$$31\!\cdots\!82$$$$T^{25} +$$$$13\!\cdots\!60$$$$T^{26} -$$$$49\!\cdots\!90$$$$T^{27} -$$$$30\!\cdots\!52$$$$T^{28} +$$$$22\!\cdots\!62$$$$T^{29} +$$$$26\!\cdots\!27$$$$T^{30} -$$$$31\!\cdots\!20$$$$T^{31} +$$$$63\!\cdots\!41$$$$T^{32}$$
$43$ $$( 1 + 4 T + 239 T^{2} + 936 T^{3} + 27462 T^{4} + 98508 T^{5} + 2007760 T^{6} + 6271668 T^{7} + 102278657 T^{8} + 269681724 T^{9} + 3712348240 T^{10} + 7832075556 T^{11} + 93887113062 T^{12} + 137599902648 T^{13} + 1510805768711 T^{14} + 1087274444428 T^{15} + 11688200277601 T^{16} )^{2}$$
$47$ $$1 - 7 T - 89 T^{2} + 202 T^{3} + 4968 T^{4} + 21827 T^{5} + 11495 T^{6} - 2393311 T^{7} - 19649733 T^{8} + 35705166 T^{9} + 1391558095 T^{10} + 7499381673 T^{11} - 30821621912 T^{12} - 590826226892 T^{13} - 1975676885408 T^{14} + 14803300228298 T^{15} + 164951174929905 T^{16} + 695755110730006 T^{17} - 4364270239866272 T^{18} - 61341351354608116 T^{19} - 150399682833170072 T^{20} + 1719945742289856711 T^{21} + 14999904348818038255 T^{22} + 18089062615569411858 T^{23} -$$$$46\!\cdots\!13$$$$T^{24} -$$$$26\!\cdots\!37$$$$T^{25} +$$$$60\!\cdots\!55$$$$T^{26} +$$$$53\!\cdots\!81$$$$T^{27} +$$$$57\!\cdots\!88$$$$T^{28} +$$$$11\!\cdots\!54$$$$T^{29} -$$$$22\!\cdots\!41$$$$T^{30} -$$$$84\!\cdots\!01$$$$T^{31} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 + 41 T + 698 T^{2} + 5865 T^{3} + 18415 T^{4} - 17787 T^{5} + 1184038 T^{6} + 25764329 T^{7} + 182959840 T^{8} + 399954485 T^{9} - 220252678 T^{10} + 21803507967 T^{11} + 208503614596 T^{12} - 375622782850 T^{13} - 10836301461310 T^{14} - 2007212797106 T^{15} + 469856653928949 T^{16} - 106382278246618 T^{17} - 30439170804819790 T^{18} - 55921593042359450 T^{19} + 1645193809401060676 T^{20} + 9118128763388992731 T^{21} - 4881759894821353462 T^{22} +$$$$46\!\cdots\!45$$$$T^{23} +$$$$11\!\cdots\!40$$$$T^{24} +$$$$85\!\cdots\!57$$$$T^{25} +$$$$20\!\cdots\!62$$$$T^{26} -$$$$16\!\cdots\!39$$$$T^{27} +$$$$90\!\cdots\!15$$$$T^{28} +$$$$15\!\cdots\!45$$$$T^{29} +$$$$96\!\cdots\!62$$$$T^{30} +$$$$29\!\cdots\!37$$$$T^{31} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$1 + 18 T + 25 T^{2} - 860 T^{3} + 7311 T^{4} + 177528 T^{5} + 280250 T^{6} - 9281880 T^{7} - 6546264 T^{8} + 769859398 T^{9} + 2676500529 T^{10} - 31642972128 T^{11} - 147191476614 T^{12} + 1958975281938 T^{13} + 18358900572134 T^{14} + 6395455699032 T^{15} - 550800860184419 T^{16} + 377331886242888 T^{17} + 63907332891598454 T^{18} + 402332384429144502 T^{19} - 1783572258254895654 T^{20} - 22622329666886938272 T^{21} +$$$$11\!\cdots\!89$$$$T^{22} +$$$$19\!\cdots\!62$$$$T^{23} -$$$$96\!\cdots\!44$$$$T^{24} -$$$$80\!\cdots\!20$$$$T^{25} +$$$$14\!\cdots\!50$$$$T^{26} +$$$$53\!\cdots\!52$$$$T^{27} +$$$$13\!\cdots\!91$$$$T^{28} -$$$$90\!\cdots\!40$$$$T^{29} +$$$$15\!\cdots\!25$$$$T^{30} +$$$$65\!\cdots\!82$$$$T^{31} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$1 + 12 T - 195 T^{2} - 3592 T^{3} + 9457 T^{4} + 507244 T^{5} + 1595716 T^{6} - 40490936 T^{7} - 318233418 T^{8} + 1680265148 T^{9} + 27894939097 T^{10} + 2641977826 T^{11} - 1448376512614 T^{12} - 4191432114084 T^{13} + 46058803271574 T^{14} + 147290895640268 T^{15} - 1443941235745343 T^{16} + 8984744634056348 T^{17} + 171384806973526854 T^{18} - 951375452686900404 T^{19} - 20053990901787938374 T^{20} + 2231404699163621626 T^{21} +$$$$14\!\cdots\!17$$$$T^{22} +$$$$52\!\cdots\!08$$$$T^{23} -$$$$61\!\cdots\!58$$$$T^{24} -$$$$47\!\cdots\!76$$$$T^{25} +$$$$11\!\cdots\!16$$$$T^{26} +$$$$22\!\cdots\!84$$$$T^{27} +$$$$25\!\cdots\!97$$$$T^{28} -$$$$58\!\cdots\!52$$$$T^{29} -$$$$19\!\cdots\!95$$$$T^{30} +$$$$72\!\cdots\!12$$$$T^{31} +$$$$36\!\cdots\!61$$$$T^{32}$$
$67$ $$( 1 + 19 T + 520 T^{2} + 7751 T^{3} + 119005 T^{4} + 1415171 T^{5} + 15702408 T^{6} + 150444400 T^{7} + 1308159665 T^{8} + 10079774800 T^{9} + 70488109512 T^{10} + 425631075473 T^{11} + 2398084154605 T^{12} + 10464819704357 T^{13} + 47038358727880 T^{14} + 115153520501137 T^{15} + 406067677556641 T^{16} )^{2}$$
$71$ $$1 - T - 355 T^{2} - 347 T^{3} + 62162 T^{4} + 107731 T^{5} - 7363645 T^{6} - 8830711 T^{7} + 730398853 T^{8} + 111771627 T^{9} - 68716020299 T^{10} + 17483084955 T^{11} + 5903862179484 T^{12} - 219949697282 T^{13} - 448386307599448 T^{14} - 29021605851186 T^{15} + 32028099319201737 T^{16} - 2060534015434206 T^{17} - 2260315376608817368 T^{18} - 78722416103897902 T^{19} +$$$$15\!\cdots\!04$$$$T^{20} + 31543495021837514205 T^{21} -$$$$88\!\cdots\!79$$$$T^{22} +$$$$10\!\cdots\!57$$$$T^{23} +$$$$47\!\cdots\!33$$$$T^{24} -$$$$40\!\cdots\!41$$$$T^{25} -$$$$23\!\cdots\!45$$$$T^{26} +$$$$24\!\cdots\!01$$$$T^{27} +$$$$10\!\cdots\!42$$$$T^{28} -$$$$40\!\cdots\!17$$$$T^{29} -$$$$29\!\cdots\!55$$$$T^{30} -$$$$58\!\cdots\!51$$$$T^{31} +$$$$41\!\cdots\!21$$$$T^{32}$$
$73$ $$1 - 60 T + 1487 T^{2} - 18394 T^{3} + 89657 T^{4} + 411322 T^{5} - 5441626 T^{6} - 37899490 T^{7} + 846703154 T^{8} - 1296063306 T^{9} - 89886244257 T^{10} + 1038415881434 T^{11} - 2204058435198 T^{12} - 53506579646092 T^{13} + 424723915258082 T^{14} + 2140532344138680 T^{15} - 48779036759889075 T^{16} + 156258861122123640 T^{17} + 2263353744410318978 T^{18} - 20814969094183771564 T^{19} - 62591382620835686718 T^{20} +$$$$21\!\cdots\!62$$$$T^{21} -$$$$13\!\cdots\!73$$$$T^{22} -$$$$14\!\cdots\!82$$$$T^{23} +$$$$68\!\cdots\!74$$$$T^{24} -$$$$22\!\cdots\!70$$$$T^{25} -$$$$23\!\cdots\!74$$$$T^{26} +$$$$12\!\cdots\!94$$$$T^{27} +$$$$20\!\cdots\!97$$$$T^{28} -$$$$30\!\cdots\!02$$$$T^{29} +$$$$18\!\cdots\!83$$$$T^{30} -$$$$53\!\cdots\!20$$$$T^{31} +$$$$65\!\cdots\!61$$$$T^{32}$$
$79$ $$1 + 15 T - 206 T^{2} - 3010 T^{3} + 29105 T^{4} + 124250 T^{5} - 5253080 T^{6} + 18691300 T^{7} + 581808430 T^{8} - 4760179440 T^{9} - 24079531838 T^{10} + 673703703905 T^{11} - 1814209779892 T^{12} - 52895538994030 T^{13} + 475566405446690 T^{14} + 1654552484719050 T^{15} - 51014747327008815 T^{16} + 130709646292804950 T^{17} + 2968009936392792290 T^{18} - 26079563650077557170 T^{19} - 70663617877785571252 T^{20} +$$$$20\!\cdots\!95$$$$T^{21} -$$$$58\!\cdots\!98$$$$T^{22} -$$$$91\!\cdots\!60$$$$T^{23} +$$$$88\!\cdots\!30$$$$T^{24} +$$$$22\!\cdots\!00$$$$T^{25} -$$$$49\!\cdots\!80$$$$T^{26} +$$$$92\!\cdots\!50$$$$T^{27} +$$$$17\!\cdots\!05$$$$T^{28} -$$$$14\!\cdots\!90$$$$T^{29} -$$$$75\!\cdots\!86$$$$T^{30} +$$$$43\!\cdots\!85$$$$T^{31} +$$$$23\!\cdots\!21$$$$T^{32}$$
$83$ $$1 - 20 T + 57 T^{2} - 124 T^{3} + 26767 T^{4} - 258208 T^{5} + 403724 T^{6} - 19732220 T^{7} + 380894284 T^{8} - 2094980066 T^{9} + 16014449293 T^{10} - 312895475826 T^{11} + 2857383169742 T^{12} - 20223947588302 T^{13} + 234825461586902 T^{14} - 2222853955193260 T^{15} + 17391213981381345 T^{16} - 184496878281040580 T^{17} + 1617712604872167878 T^{18} - 11563790319672435674 T^{19} +$$$$13\!\cdots\!82$$$$T^{20} -$$$$12\!\cdots\!18$$$$T^{21} +$$$$52\!\cdots\!17$$$$T^{22} -$$$$56\!\cdots\!82$$$$T^{23} +$$$$85\!\cdots\!44$$$$T^{24} -$$$$36\!\cdots\!60$$$$T^{25} +$$$$62\!\cdots\!76$$$$T^{26} -$$$$33\!\cdots\!36$$$$T^{27} +$$$$28\!\cdots\!87$$$$T^{28} -$$$$11\!\cdots\!12$$$$T^{29} +$$$$41\!\cdots\!53$$$$T^{30} -$$$$12\!\cdots\!40$$$$T^{31} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$( 1 - 37 T + 1032 T^{2} - 20901 T^{3} + 357823 T^{4} - 5152639 T^{5} + 65606528 T^{6} - 731972999 T^{7} + 7339853952 T^{8} - 65145596911 T^{9} + 519669308288 T^{10} - 3632450763191 T^{11} + 22450616901343 T^{12} - 116712426543549 T^{13} + 512884692271752 T^{14} - 1636559391134573 T^{15} + 3936588805702081 T^{16} )^{2}$$
$97$ $$1 + 35 T + 248 T^{2} - 5577 T^{3} - 95113 T^{4} + 5579 T^{5} + 8611936 T^{6} + 29293955 T^{7} - 337756866 T^{8} + 1241258413 T^{9} + 46602892672 T^{10} - 183890451593 T^{11} - 7991146758268 T^{12} - 52842171229544 T^{13} + 128069756523188 T^{14} + 4904533562796750 T^{15} + 55268578137339225 T^{16} + 475739755591284750 T^{17} + 1205008339126675892 T^{18} - 48227622942581611112 T^{19} -$$$$70\!\cdots\!08$$$$T^{20} -$$$$15\!\cdots\!01$$$$T^{21} +$$$$38\!\cdots\!88$$$$T^{22} +$$$$10\!\cdots\!69$$$$T^{23} -$$$$26\!\cdots\!26$$$$T^{24} +$$$$22\!\cdots\!35$$$$T^{25} +$$$$63\!\cdots\!64$$$$T^{26} +$$$$39\!\cdots\!87$$$$T^{27} -$$$$65\!\cdots\!33$$$$T^{28} -$$$$37\!\cdots\!29$$$$T^{29} +$$$$16\!\cdots\!12$$$$T^{30} +$$$$22\!\cdots\!55$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$