# Properties

 Label 5239.2.a.k Level 5239 Weight 2 Character orbit 5239.a Self dual yes Analytic conductor 41.834 Analytic rank 1 Dimension 16 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$5239 = 13^{2} \cdot 31$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 5239.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$41.8336256189$$ Analytic rank: $$1$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 403) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} -\beta_{6} q^{3} + ( 1 + \beta_{2} ) q^{4} + \beta_{9} q^{5} + ( -1 - \beta_{1} + \beta_{2} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{14} ) q^{6} -\beta_{4} q^{7} + ( -1 - \beta_{1} - \beta_{8} + \beta_{13} - \beta_{15} ) q^{8} + ( 1 - \beta_{1} - \beta_{5} - \beta_{9} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} -\beta_{6} q^{3} + ( 1 + \beta_{2} ) q^{4} + \beta_{9} q^{5} + ( -1 - \beta_{1} + \beta_{2} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{14} ) q^{6} -\beta_{4} q^{7} + ( -1 - \beta_{1} - \beta_{8} + \beta_{13} - \beta_{15} ) q^{8} + ( 1 - \beta_{1} - \beta_{5} - \beta_{9} ) q^{9} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{13} + \beta_{15} ) q^{10} + ( -1 - \beta_{3} ) q^{11} + ( \beta_{4} - \beta_{6} - \beta_{9} + \beta_{12} ) q^{12} + ( 2 \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} ) q^{14} + ( \beta_{1} - \beta_{2} + \beta_{6} - \beta_{12} ) q^{15} + ( 2 \beta_{2} + \beta_{10} + \beta_{12} + \beta_{14} + \beta_{15} ) q^{16} + ( 1 - \beta_{5} - \beta_{13} + \beta_{15} ) q^{17} + ( -\beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{13} - \beta_{15} ) q^{18} + ( -2 - \beta_{7} + \beta_{11} + \beta_{12} ) q^{19} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{20} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} - \beta_{14} ) q^{21} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{22} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{12} - \beta_{14} ) q^{23} + ( -3 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - 3 \beta_{8} + 2 \beta_{10} - \beta_{11} + \beta_{14} ) q^{24} + ( -1 + \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{15} ) q^{25} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} ) q^{27} + ( -\beta_{1} - 3 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{28} + ( -1 + \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{9} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{29} + ( -1 + 4 \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{13} - 2 \beta_{14} ) q^{30} + q^{31} + ( -3 - 2 \beta_{2} - \beta_{6} - \beta_{9} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{32} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{6} + \beta_{8} - \beta_{10} - \beta_{13} + \beta_{15} ) q^{33} + ( -2 \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} - \beta_{13} ) q^{34} + ( -2 \beta_{1} + \beta_{4} - \beta_{9} + \beta_{12} - \beta_{14} ) q^{35} + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{11} + \beta_{12} - \beta_{15} ) q^{36} + ( -2 + 2 \beta_{1} + \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{37} + ( -1 + 2 \beta_{1} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{38} + ( 1 + \beta_{1} - 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} - 2 \beta_{13} ) q^{40} + ( -4 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} - \beta_{14} ) q^{41} + ( 1 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{42} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{11} - \beta_{13} ) q^{43} + ( 4 \beta_{1} - 3 \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{44} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{12} + \beta_{15} ) q^{45} + ( -1 - 2 \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{46} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{47} + ( -1 + 2 \beta_{4} + 3 \beta_{5} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{48} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{10} - \beta_{11} + \beta_{14} + \beta_{15} ) q^{49} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} ) q^{50} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{51} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{53} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{9} + 2 \beta_{12} + \beta_{13} ) q^{54} + ( \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{55} + ( 6 \beta_{1} - 4 \beta_{2} - \beta_{3} + 5 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} + 3 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} + 2 \beta_{13} - 3 \beta_{14} + \beta_{15} ) q^{56} + ( \beta_{1} - \beta_{3} + 2 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} ) q^{57} + ( -3 + 2 \beta_{1} - 3 \beta_{2} + \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{15} ) q^{58} + ( -1 - \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{10} + \beta_{11} - \beta_{13} + 2 \beta_{14} ) q^{59} + ( -2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} + 2 \beta_{9} + \beta_{10} - 2 \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{60} + ( 1 + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{61} -\beta_{1} q^{62} + ( -3 + 3 \beta_{1} + 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{63} + ( 2 + 3 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{14} + \beta_{15} ) q^{64} + ( 1 + 4 \beta_{1} - 4 \beta_{2} - \beta_{4} - 2 \beta_{5} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{12} - 2 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{66} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{67} + ( 2 + 3 \beta_{1} + \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{15} ) q^{68} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} + \beta_{14} ) q^{69} + ( 5 - 4 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + 3 \beta_{10} - 2 \beta_{11} + 2 \beta_{14} ) q^{70} + ( -3 - \beta_{1} + \beta_{6} - \beta_{11} - \beta_{14} - \beta_{15} ) q^{71} + ( 3 - 3 \beta_{1} + 4 \beta_{2} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{72} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{7} - \beta_{9} + 2 \beta_{12} + \beta_{14} + \beta_{15} ) q^{73} + ( -5 \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{7} - 3 \beta_{8} - \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + \beta_{13} + 2 \beta_{14} ) q^{74} + ( -2 - 3 \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{75} + ( -2 + 3 \beta_{1} - 4 \beta_{2} + \beta_{4} - \beta_{6} + 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{76} + ( -1 - 4 \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} - 3 \beta_{8} + 2 \beta_{10} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{77} + ( 2 + 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{79} + ( -4 + 3 \beta_{1} - 4 \beta_{2} + 3 \beta_{4} + 4 \beta_{5} - \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{80} + ( -2 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + 2 \beta_{13} - \beta_{15} ) q^{81} + ( -1 + \beta_{2} + \beta_{3} - 3 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{14} - \beta_{15} ) q^{82} + ( 3 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{13} + 2 \beta_{15} ) q^{83} + ( -7 + 6 \beta_{1} - 7 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} - \beta_{12} - 3 \beta_{14} ) q^{84} + ( 3 + \beta_{1} - \beta_{2} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{15} ) q^{85} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{86} + ( -2 + 3 \beta_{1} - 3 \beta_{2} + \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{8} - 3 \beta_{10} + 3 \beta_{11} - 2 \beta_{13} - 3 \beta_{14} ) q^{87} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} + 2 \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{88} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{89} + ( -2 + 7 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{5} + 3 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{90} + ( -1 + 3 \beta_{2} - \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + 4 \beta_{9} + 2 \beta_{10} + \beta_{12} + 3 \beta_{13} + \beta_{14} + \beta_{15} ) q^{92} -\beta_{6} q^{93} + ( -1 - \beta_{1} + 3 \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 4 \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 3 \beta_{15} ) q^{94} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{95} + ( 6 - 2 \beta_{1} + \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{14} ) q^{96} + ( -4 \beta_{1} + 4 \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{97} + ( -1 + \beta_{1} - 6 \beta_{2} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + \beta_{11} - 4 \beta_{14} - 3 \beta_{15} ) q^{98} + ( -2 + 4 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 4q^{2} - 2q^{3} + 18q^{4} - 4q^{5} - 6q^{6} - 2q^{7} - 12q^{8} + 10q^{9} + O(q^{10})$$ $$16q - 4q^{2} - 2q^{3} + 18q^{4} - 4q^{5} - 6q^{6} - 2q^{7} - 12q^{8} + 10q^{9} - 2q^{10} - 14q^{11} + 8q^{12} - 8q^{14} + 14q^{16} + 4q^{17} + 28q^{18} - 22q^{19} - 28q^{20} - 12q^{21} - 8q^{22} + 4q^{23} + 8q^{24} - 2q^{25} + 10q^{27} - 16q^{28} - 8q^{29} - 20q^{30} + 16q^{31} - 48q^{32} - 10q^{33} - 8q^{34} - 2q^{35} + 22q^{36} - 16q^{37} - 6q^{38} + 14q^{40} - 44q^{41} + 14q^{42} + 16q^{43} - 4q^{44} - 56q^{45} - 10q^{47} + 32q^{49} - 2q^{50} - 6q^{53} - 24q^{54} + 22q^{55} - 4q^{56} + 8q^{57} - 74q^{58} - 2q^{59} - 40q^{60} + 8q^{61} - 4q^{62} - 56q^{63} + 38q^{64} - 34q^{66} + 8q^{67} + 32q^{68} - 10q^{69} + 108q^{70} - 50q^{71} + 44q^{72} - 14q^{73} + 8q^{74} - 44q^{76} + 16q^{77} + 32q^{79} - 68q^{80} - 8q^{81} - 6q^{82} + 20q^{83} - 136q^{84} + 32q^{85} - 8q^{86} - 36q^{87} - 40q^{88} - 52q^{89} - 34q^{90} + 14q^{92} - 2q^{93} + 44q^{94} - 2q^{95} + 80q^{96} - 18q^{97} - 12q^{98} - 38q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} - 17 x^{14} + 80 x^{13} + 98 x^{12} - 628 x^{11} - 158 x^{10} + 2458 x^{9} - 510 x^{8} - 5030 x^{7} + 2318 x^{6} + 5112 x^{5} - 3154 x^{4} - 2086 x^{3} + 1542 x^{2} + 124 x - 147$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$234985 \nu^{15} - 649739 \nu^{14} - 4442990 \nu^{13} + 11737815 \nu^{12} + 32793880 \nu^{11} - 78785016 \nu^{10} - 121872261 \nu^{9} + 238801983 \nu^{8} + 252843528 \nu^{7} - 306742475 \nu^{6} - 322608389 \nu^{5} + 92048176 \nu^{4} + 252747823 \nu^{3} + 63836256 \nu^{2} - 79028469 \nu - 14255795$$$$)/3154099$$ $$\beta_{4}$$ $$=$$ $$($$$$-410788 \nu^{15} + 1274386 \nu^{14} + 8301681 \nu^{13} - 26138494 \nu^{12} - 65794620 \nu^{11} + 211314394 \nu^{10} + 256832189 \nu^{9} - 855882940 \nu^{8} - 500253866 \nu^{7} + 1822426814 \nu^{6} + 408312823 \nu^{5} - 1944612848 \nu^{4} - 20212581 \nu^{3} + 874475616 \nu^{2} - 75264514 \nu - 99677416$$$$)/3154099$$ $$\beta_{5}$$ $$=$$ $$($$$$-526980 \nu^{15} + 1004839 \nu^{14} + 11285484 \nu^{13} - 18208549 \nu^{12} - 97120171 \nu^{11} + 122723076 \nu^{10} + 427652755 \nu^{9} - 376249117 \nu^{8} - 1011084352 \nu^{7} + 508287520 \nu^{6} + 1227400581 \nu^{5} - 226361438 \nu^{4} - 654092986 \nu^{3} - 22175440 \nu^{2} + 99782059 \nu + 20992771$$$$)/3154099$$ $$\beta_{6}$$ $$=$$ $$($$$$-576101 \nu^{15} + 1321985 \nu^{14} + 12664367 \nu^{13} - 27267582 \nu^{12} - 111100193 \nu^{11} + 221666839 \nu^{10} + 491414968 \nu^{9} - 903595035 \nu^{8} - 1131337528 \nu^{7} + 1940652701 \nu^{6} + 1240159741 \nu^{5} - 2089826468 \nu^{4} - 467362734 \nu^{3} + 934560659 \nu^{2} - 16597867 \nu - 92150633$$$$)/3154099$$ $$\beta_{7}$$ $$=$$ $$($$$$784358 \nu^{15} - 1348140 \nu^{14} - 18214112 \nu^{13} + 27248288 \nu^{12} + 170196011 \nu^{11} - 214828824 \nu^{10} - 810806967 \nu^{9} + 837601748 \nu^{8} + 2051999077 \nu^{7} - 1691698840 \nu^{6} - 2604857321 \nu^{5} + 1682466163 \nu^{4} + 1382284810 \nu^{3} - 686414940 \nu^{2} - 176712133 \nu + 59706798$$$$)/3154099$$ $$\beta_{8}$$ $$=$$ $$($$$$-802740 \nu^{15} + 2267916 \nu^{14} + 16615333 \nu^{13} - 46241574 \nu^{12} - 136526765 \nu^{11} + 371532436 \nu^{10} + 564859057 \nu^{9} - 1497749457 \nu^{8} - 1221377987 \nu^{7} + 3191604442 \nu^{6} + 1265211546 \nu^{5} - 3450561986 \nu^{4} - 425159394 \nu^{3} + 1595613048 \nu^{2} - 63863150 \nu - 178878045$$$$)/3154099$$ $$\beta_{9}$$ $$=$$ $$($$$$-806190 \nu^{15} + 2331414 \nu^{14} + 15709912 \nu^{13} - 44758234 \nu^{12} - 119944681 \nu^{11} + 331635318 \nu^{10} + 455521285 \nu^{9} - 1197491878 \nu^{8} - 900316104 \nu^{7} + 2200020862 \nu^{6} + 873342061 \nu^{5} - 1969063272 \nu^{4} - 311347262 \nu^{3} + 732065680 \nu^{2} - 13992420 \nu - 64556158$$$$)/3154099$$ $$\beta_{10}$$ $$=$$ $$($$$$850612 \nu^{15} - 2605957 \nu^{14} - 15560111 \nu^{13} + 48167216 \nu^{12} + 108207470 \nu^{11} - 337529523 \nu^{10} - 355883087 \nu^{9} + 1118150931 \nu^{8} + 556539484 \nu^{7} - 1788644715 \nu^{6} - 356875457 \nu^{5} + 1285967036 \nu^{4} + 39160268 \nu^{3} - 358367395 \nu^{2} + 12269881 \nu + 34193170$$$$)/3154099$$ $$\beta_{11}$$ $$=$$ $$($$$$878874 \nu^{15} - 2418510 \nu^{14} - 17786322 \nu^{13} + 47599181 \nu^{12} + 142637281 \nu^{11} - 365724170 \nu^{10} - 576236155 \nu^{9} + 1394361242 \nu^{8} + 1225080435 \nu^{7} - 2780449711 \nu^{6} - 1283514326 \nu^{5} + 2802225260 \nu^{4} + 502687904 \nu^{3} - 1221951532 \nu^{2} + 3930381 \nu + 131774368$$$$)/3154099$$ $$\beta_{12}$$ $$=$$ $$($$$$-1064679 \nu^{15} + 2903608 \nu^{14} + 21450107 \nu^{13} - 56574959 \nu^{12} - 170882518 \nu^{11} + 428082420 \nu^{10} + 683297136 \nu^{9} - 1594225024 \nu^{8} - 1428069111 \nu^{7} + 3067103065 \nu^{6} + 1447767821 \nu^{5} - 2928194275 \nu^{4} - 515646724 \nu^{3} + 1167096052 \nu^{2} - 25654616 \nu - 96915048$$$$)/3154099$$ $$\beta_{13}$$ $$=$$ $$($$$$1168224 \nu^{15} - 3355748 \nu^{14} - 22484283 \nu^{13} + 63618073 \nu^{12} + 168369826 \nu^{11} - 462749260 \nu^{10} - 619522204 \nu^{9} + 1625012820 \nu^{8} + 1160124005 \nu^{7} - 2862215213 \nu^{6} - 1019455339 \nu^{5} + 2413508402 \nu^{4} + 276990748 \nu^{3} - 838973088 \nu^{2} + 55054740 \nu + 71906220$$$$)/3154099$$ $$\beta_{14}$$ $$=$$ $$($$$$-1756897 \nu^{15} + 5326013 \nu^{14} + 33209620 \nu^{13} - 101451904 \nu^{12} - 242221543 \nu^{11} + 743728799 \nu^{10} + 856967212 \nu^{9} - 2646688184 \nu^{8} - 1509972365 \nu^{7} + 4775361305 \nu^{6} + 1193774521 \nu^{5} - 4218689050 \nu^{4} - 228817785 \nu^{3} + 1600624687 \nu^{2} - 89762660 \nu - 153367298$$$$)/3154099$$ $$\beta_{15}$$ $$=$$ $$($$$$1970964 \nu^{15} - 5623664 \nu^{14} - 39099616 \nu^{13} + 109859647 \nu^{12} + 304896591 \nu^{11} - 834281696 \nu^{10} - 1184381261 \nu^{9} + 3122762277 \nu^{8} + 2381501992 \nu^{7} - 6053819655 \nu^{6} - 2284666885 \nu^{5} + 5864070388 \nu^{4} + 705304241 \nu^{3} - 2434586136 \nu^{2} + 103147395 \nu + 247630166$$$$)/3154099$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} - \beta_{13} + \beta_{8} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{15} + \beta_{14} + \beta_{12} + \beta_{10} + 8 \beta_{2} + 14$$ $$\nu^{5}$$ $$=$$ $$9 \beta_{15} + \beta_{14} - 7 \beta_{13} + \beta_{9} + 8 \beta_{8} + \beta_{6} + 2 \beta_{2} + 28 \beta_{1} + 11$$ $$\nu^{6}$$ $$=$$ $$11 \beta_{15} + 11 \beta_{14} + 8 \beta_{12} - \beta_{11} + 11 \beta_{10} + 2 \beta_{9} + \beta_{5} + \beta_{4} + \beta_{3} + 56 \beta_{2} + 3 \beta_{1} + 78$$ $$\nu^{7}$$ $$=$$ $$67 \beta_{15} + 13 \beta_{14} - 42 \beta_{13} - \beta_{12} + 3 \beta_{10} + 17 \beta_{9} + 54 \beta_{8} + \beta_{7} + 12 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + 30 \beta_{2} + 167 \beta_{1} + 94$$ $$\nu^{8}$$ $$=$$ $$96 \beta_{15} + 94 \beta_{14} + 51 \beta_{12} - 15 \beta_{11} + 95 \beta_{10} + 34 \beta_{9} - \beta_{8} + \beta_{7} + 3 \beta_{6} + 14 \beta_{5} + 14 \beta_{4} + 14 \beta_{3} + 386 \beta_{2} + 44 \beta_{1} + 476$$ $$\nu^{9}$$ $$=$$ $$476 \beta_{15} + 127 \beta_{14} - 241 \beta_{13} - 16 \beta_{12} - 5 \beta_{11} + 54 \beta_{10} + 188 \beta_{9} + 347 \beta_{8} + 18 \beta_{7} + 107 \beta_{6} + 36 \beta_{5} + 5 \beta_{4} + 28 \beta_{3} + 316 \beta_{2} + 1034 \beta_{1} + 736$$ $$\nu^{10}$$ $$=$$ $$775 \beta_{15} + 736 \beta_{14} + 6 \beta_{13} + 301 \beta_{12} - 157 \beta_{11} + 754 \beta_{10} + 392 \beta_{9} - 11 \beta_{8} + 25 \beta_{7} + 55 \beta_{6} + 151 \beta_{5} + 141 \beta_{4} + 138 \beta_{3} + 2671 \beta_{2} + 459 \beta_{1} + 3060$$ $$\nu^{11}$$ $$=$$ $$3345 \beta_{15} + 1112 \beta_{14} - 1352 \beta_{13} - 174 \beta_{12} - 97 \beta_{11} + 642 \beta_{10} + 1756 \beta_{9} + 2191 \beta_{8} + 216 \beta_{7} + 858 \beta_{6} + 434 \beta_{5} + 105 \beta_{4} + 278 \beta_{3} + 2884 \beta_{2} + 6574 \beta_{1} + 5537$$ $$\nu^{12}$$ $$=$$ $$6031 \beta_{15} + 5537 \beta_{14} + 131 \beta_{13} + 1697 \beta_{12} - 1410 \beta_{11} + 5750 \beta_{10} + 3839 \beta_{9} - 58 \beta_{8} + 363 \beta_{7} + 657 \beta_{6} + 1473 \beta_{5} + 1261 \beta_{4} + 1189 \beta_{3} + 18608 \beta_{2} + 4187 \beta_{1} + 20321$$ $$\nu^{13}$$ $$=$$ $$23516 \beta_{15} + 9190 \beta_{14} - 7419 \beta_{13} - 1610 \beta_{12} - 1213 \beta_{11} + 6380 \beta_{10} + 15093 \beta_{9} + 13770 \beta_{8} + 2176 \beta_{7} + 6534 \beta_{6} + 4410 \beta_{5} + 1410 \beta_{4} + 2430 \beta_{3} + 24428 \beta_{2} + 42681 \beta_{1} + 40818$$ $$\nu^{14}$$ $$=$$ $$45979 \beta_{15} + 40805 \beta_{14} + 1821 \beta_{13} + 9164 \beta_{12} - 11647 \beta_{11} + 42913 \beta_{10} + 34436 \beta_{9} + 61 \beta_{8} + 4142 \beta_{7} + 6496 \beta_{6} + 13524 \beta_{5} + 10689 \beta_{4} + 9587 \beta_{3} + 130468 \beta_{2} + 35646 \beta_{1} + 137920$$ $$\nu^{15}$$ $$=$$ $$165995 \beta_{15} + 73275 \beta_{14} - 39425 \beta_{13} - 13668 \beta_{12} - 12464 \beta_{11} + 57471 \beta_{10} + 123834 \beta_{9} + 86752 \beta_{8} + 19944 \beta_{7} + 48292 \beta_{6} + 40854 \beta_{5} + 15518 \beta_{4} + 19994 \beta_{3} + 197948 \beta_{2} + 282032 \beta_{1} + 297726$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.71381 2.67507 2.44691 1.88981 1.48947 1.44935 1.01245 0.624841 0.475674 −0.327248 −0.937316 −1.40506 −1.44015 −1.99449 −2.21678 −2.45635
−2.71381 0.265781 5.36475 0.954130 −0.721280 −4.28452 −9.13130 −2.92936 −2.58933
1.2 −2.67507 −1.67043 5.15599 −2.86564 4.46851 5.21058 −8.44248 −0.209666 7.66579
1.3 −2.44691 2.35527 3.98738 −2.67528 −5.76313 −0.955539 −4.86294 2.54728 6.54617
1.4 −1.88981 −0.327950 1.57140 1.75790 0.619765 0.652358 0.809976 −2.89245 −3.32211
1.5 −1.48947 2.99747 0.218531 0.692562 −4.46466 −1.15260 2.65345 5.98485 −1.03155
1.6 −1.44935 0.0382575 0.100611 3.85977 −0.0554484 −1.66255 2.75288 −2.99854 −5.59415
1.7 −1.01245 −1.05136 −0.974950 −3.24438 1.06445 1.79572 3.01198 −1.89463 3.28477
1.8 −0.624841 −2.78388 −1.60957 −0.619834 1.73948 −2.35216 2.25541 4.74999 0.387298
1.9 −0.475674 0.345654 −1.77373 −0.161594 −0.164419 4.07996 1.79507 −2.88052 0.0768661
1.10 0.327248 −1.21221 −1.89291 2.01769 −0.396693 0.699230 −1.27395 −1.53055 0.660284
1.11 0.937316 1.68575 −1.12144 −0.290760 1.58008 3.85549 −2.92577 −0.158249 −0.272534
1.12 1.40506 1.52886 −0.0258025 1.01618 2.14814 −2.90958 −2.84638 −0.662589 1.42779
1.13 1.44015 −2.79139 0.0740342 −2.61757 −4.02002 −4.99660 −2.77368 4.79185 −3.76969
1.14 1.99449 −1.58843 1.97798 2.66789 −3.16810 1.73927 −0.0439183 −0.476895 5.32107
1.15 2.21678 −2.59178 2.91410 −1.00571 −5.74540 1.82096 2.02635 3.71732 −2.22943
1.16 2.45635 2.80039 4.03363 −3.48535 6.87871 −3.54003 4.99531 4.84216 −8.56123
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5239.2.a.k 16
13.b even 2 1 5239.2.a.l 16
13.d odd 4 2 403.2.c.b 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.c.b 32 13.d odd 4 2
5239.2.a.k 16 1.a even 1 1 trivial
5239.2.a.l 16 13.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$13$$ $$-1$$
$$31$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5239))$$:

 $$T_{2}^{16} + \cdots$$ $$T_{5}^{16} + \cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 15 T^{2} + 40 T^{3} + 102 T^{4} + 228 T^{5} + 486 T^{6} + 958 T^{7} + 1818 T^{8} + 3266 T^{9} + 5702 T^{10} + 9500 T^{11} + 15446 T^{12} + 24110 T^{13} + 36798 T^{14} + 54136 T^{15} + 77909 T^{16} + 108272 T^{17} + 147192 T^{18} + 192880 T^{19} + 247136 T^{20} + 304000 T^{21} + 364928 T^{22} + 418048 T^{23} + 465408 T^{24} + 490496 T^{25} + 497664 T^{26} + 466944 T^{27} + 417792 T^{28} + 327680 T^{29} + 245760 T^{30} + 131072 T^{31} + 65536 T^{32}$$
$3$ $$1 + 2 T + 21 T^{2} + 34 T^{3} + 218 T^{4} + 300 T^{5} + 1555 T^{6} + 1884 T^{7} + 8731 T^{8} + 9520 T^{9} + 41144 T^{10} + 41066 T^{11} + 167989 T^{12} + 155486 T^{13} + 603142 T^{14} + 522428 T^{15} + 1919026 T^{16} + 1567284 T^{17} + 5428278 T^{18} + 4198122 T^{19} + 13607109 T^{20} + 9979038 T^{21} + 29993976 T^{22} + 20820240 T^{23} + 57284091 T^{24} + 37082772 T^{25} + 91821195 T^{26} + 53144100 T^{27} + 115854138 T^{28} + 54206982 T^{29} + 100442349 T^{30} + 28697814 T^{31} + 43046721 T^{32}$$
$5$ $$1 + 4 T + 49 T^{2} + 170 T^{3} + 1169 T^{4} + 3540 T^{5} + 18017 T^{6} + 48152 T^{7} + 201957 T^{8} + 482166 T^{9} + 1765550 T^{10} + 3812784 T^{11} + 12648819 T^{12} + 25046692 T^{13} + 77121452 T^{14} + 141984952 T^{15} + 410664657 T^{16} + 709924760 T^{17} + 1928036300 T^{18} + 3130836500 T^{19} + 7905511875 T^{20} + 11914950000 T^{21} + 27586718750 T^{22} + 37669218750 T^{23} + 78889453125 T^{24} + 94046875000 T^{25} + 175947265625 T^{26} + 172851562500 T^{27} + 285400390625 T^{28} + 207519531250 T^{29} + 299072265625 T^{30} + 122070312500 T^{31} + 152587890625 T^{32}$$
$7$ $$1 + 2 T + 42 T^{2} + 70 T^{3} + 850 T^{4} + 1110 T^{5} + 10960 T^{6} + 9916 T^{7} + 101646 T^{8} + 41362 T^{9} + 719683 T^{10} - 249098 T^{11} + 3929080 T^{12} - 6823952 T^{13} + 16827506 T^{14} - 74650080 T^{15} + 82389777 T^{16} - 522550560 T^{17} + 824547794 T^{18} - 2340615536 T^{19} + 9433721080 T^{20} - 4186590086 T^{21} + 84669985267 T^{22} + 34063385566 T^{23} + 585968962446 T^{24} + 400146367012 T^{25} + 3095928729040 T^{26} + 2194832684730 T^{27} + 11765094120850 T^{28} + 6782230728490 T^{29} + 28485369059658 T^{30} + 9495123019886 T^{31} + 33232930569601 T^{32}$$
$11$ $$1 + 14 T + 193 T^{2} + 1766 T^{3} + 15042 T^{4} + 106000 T^{5} + 696396 T^{6} + 4067690 T^{7} + 22319681 T^{8} + 112387838 T^{9} + 535177240 T^{10} + 2376412498 T^{11} + 10027317573 T^{12} + 39793437236 T^{13} + 150520146935 T^{14} + 537782186286 T^{15} + 1834017527270 T^{16} + 5915604049146 T^{17} + 18212937779135 T^{18} + 52965064961116 T^{19} + 146809956586293 T^{20} + 382723609215398 T^{21} + 948099126471640 T^{22} + 2190121017426298 T^{23} + 4784421843436961 T^{24} + 9591400243203790 T^{25} + 18062718742437996 T^{26} + 30243037084766000 T^{27} + 47208239642637282 T^{28} + 60967109646182146 T^{29} + 73291717881565513 T^{30} + 58481474371819114 T^{31} + 45949729863572161 T^{32}$$
$13$ 
$17$ $$1 - 4 T + 167 T^{2} - 638 T^{3} + 14230 T^{4} - 51436 T^{5} + 811354 T^{6} - 2759804 T^{7} + 34419993 T^{8} - 109711108 T^{9} + 1147272314 T^{10} - 3412018000 T^{11} + 30998579977 T^{12} - 85579508576 T^{13} + 691428574449 T^{14} - 1760474326102 T^{15} + 12855128851858 T^{16} - 29928063543734 T^{17} + 199822858015761 T^{18} - 420452125633888 T^{19} + 2589032398259017 T^{20} - 4844577641426000 T^{21} + 27692364640964666 T^{22} - 45018710470079684 T^{23} + 240105522288917913 T^{24} - 327279295907926588 T^{25} + 1635684715104897946 T^{26} - 1762809258479410988 T^{27} + 8290714435779499030 T^{28} - 6319120784993987806 T^{29} + 28119097035419955143 T^{30} - 11449692206039263172 T^{31} + 48661191875666868481 T^{32}$$
$19$ $$1 + 22 T + 423 T^{2} + 5510 T^{3} + 64524 T^{4} + 622948 T^{5} + 5538440 T^{6} + 43415552 T^{7} + 318418086 T^{8} + 2126188226 T^{9} + 13422249200 T^{10} + 78519921788 T^{11} + 437210858146 T^{12} + 2278943534750 T^{13} + 11352944126987 T^{14} + 53226592711684 T^{15} + 238957361874987 T^{16} + 1011305261521996 T^{17} + 4098412829842307 T^{18} + 15631273704850250 T^{19} + 56977756244444866 T^{20} + 194423099819345012 T^{21} + 631461538615545200 T^{22} + 1900539567015945014 T^{23} + 5407873636975559526 T^{24} + 14009664522684459008 T^{25} + 33956542604855370440 T^{26} + 72567373800127729612 T^{27} +$$$$14\!\cdots\!64$$$$T^{28} +$$$$23\!\cdots\!90$$$$T^{29} +$$$$33\!\cdots\!83$$$$T^{30} +$$$$33\!\cdots\!78$$$$T^{31} +$$$$28\!\cdots\!81$$$$T^{32}$$
$23$ $$1 - 4 T + 182 T^{2} - 876 T^{3} + 16882 T^{4} - 85596 T^{5} + 1066996 T^{6} - 5179642 T^{7} + 50683499 T^{8} - 225557226 T^{9} + 1897105692 T^{10} - 7709902006 T^{11} + 58547822885 T^{12} - 219399866216 T^{13} + 1565541834150 T^{14} - 5489823491962 T^{15} + 37688475152030 T^{16} - 126265940315126 T^{17} + 828171630265350 T^{18} - 2669438172250072 T^{19} + 16384081303961285 T^{20} - 49623573807004058 T^{21} + 280839727642180188 T^{22} - 767982982839530022 T^{23} + 3969074744178578219 T^{24} - 9329325973725536246 T^{25} + 44201921758918628404 T^{26} - 81556704038400495492 T^{27} +$$$$36\!\cdots\!22$$$$T^{28} -$$$$44\!\cdots\!08$$$$T^{29} +$$$$21\!\cdots\!38$$$$T^{30} -$$$$10\!\cdots\!28$$$$T^{31} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$1 + 8 T + 220 T^{2} + 1110 T^{3} + 21310 T^{4} + 68526 T^{5} + 1342857 T^{6} + 2179216 T^{7} + 63203332 T^{8} - 1461056 T^{9} + 2417825971 T^{10} - 4095947054 T^{11} + 80860738337 T^{12} - 256714762740 T^{13} + 2493206148736 T^{14} - 10114188632182 T^{15} + 73571341464604 T^{16} - 293311470333278 T^{17} + 2096786371086976 T^{18} - 6261016348465860 T^{19} + 57191263871731697 T^{20} - 84012580320705046 T^{21} + 1438179273670269691 T^{22} - 25203035280522304 T^{23} + 31617240120183186052 T^{24} + 31614204624949338704 T^{25} +$$$$56\!\cdots\!57$$$$T^{26} +$$$$83\!\cdots\!54$$$$T^{27} +$$$$75\!\cdots\!10$$$$T^{28} +$$$$11\!\cdots\!90$$$$T^{29} +$$$$65\!\cdots\!20$$$$T^{30} +$$$$69\!\cdots\!92$$$$T^{31} +$$$$25\!\cdots\!21$$$$T^{32}$$
$31$ $$( 1 - T )^{16}$$
$37$ $$1 + 16 T + 281 T^{2} + 3018 T^{3} + 34927 T^{4} + 313190 T^{5} + 2884151 T^{6} + 22212378 T^{7} + 173346155 T^{8} + 1184293678 T^{9} + 8159404947 T^{10} + 50203096966 T^{11} + 314774277836 T^{12} + 1809673031534 T^{13} + 10920687079539 T^{14} + 62095847832320 T^{15} + 384596080229138 T^{16} + 2297546369795840 T^{17} + 14950420611888891 T^{18} + 91665368066291702 T^{19} + 589937675323395596 T^{20} + 3481281397277134462 T^{21} + 20934800754243145323 T^{22} +$$$$11\!\cdots\!74$$$$T^{23} +$$$$60\!\cdots\!55$$$$T^{24} +$$$$28\!\cdots\!06$$$$T^{25} +$$$$13\!\cdots\!99$$$$T^{26} +$$$$55\!\cdots\!70$$$$T^{27} +$$$$22\!\cdots\!87$$$$T^{28} +$$$$73\!\cdots\!46$$$$T^{29} +$$$$25\!\cdots\!09$$$$T^{30} +$$$$53\!\cdots\!88$$$$T^{31} +$$$$12\!\cdots\!41$$$$T^{32}$$
$41$ $$1 + 44 T + 1267 T^{2} + 27010 T^{3} + 474658 T^{4} + 7134028 T^{5} + 94746078 T^{6} + 1131504052 T^{7} + 12335577335 T^{8} + 123970933278 T^{9} + 1158128963871 T^{10} + 10116089856576 T^{11} + 83030611047783 T^{12} + 642594586767924 T^{13} + 4702649197331611 T^{14} + 32598947411850872 T^{15} + 214314375261374857 T^{16} + 1336556843885885752 T^{17} + 7905153300714438091 T^{18} + 44288261514632090004 T^{19} +$$$$23\!\cdots\!63$$$$T^{20} +$$$$11\!\cdots\!76$$$$T^{21} +$$$$55\!\cdots\!11$$$$T^{22} +$$$$24\!\cdots\!18$$$$T^{23} +$$$$98\!\cdots\!35$$$$T^{24} +$$$$37\!\cdots\!72$$$$T^{25} +$$$$12\!\cdots\!78$$$$T^{26} +$$$$39\!\cdots\!48$$$$T^{27} +$$$$10\!\cdots\!98$$$$T^{28} +$$$$24\!\cdots\!10$$$$T^{29} +$$$$48\!\cdots\!87$$$$T^{30} +$$$$68\!\cdots\!44$$$$T^{31} +$$$$63\!\cdots\!41$$$$T^{32}$$
$43$ $$1 - 16 T + 491 T^{2} - 6320 T^{3} + 113133 T^{4} - 1240932 T^{5} + 16691615 T^{6} - 161301798 T^{7} + 1792302122 T^{8} - 15573445800 T^{9} + 149872010422 T^{10} - 1185239906190 T^{11} + 10146060020032 T^{12} - 73496852836876 T^{13} + 568551289690556 T^{14} - 3780355123235520 T^{15} + 26672834634296236 T^{16} - 162555270299127360 T^{17} + 1051251334637838044 T^{18} - 5843514278501500132 T^{19} + 34687360142545421632 T^{20} -$$$$17\!\cdots\!70$$$$T^{21} +$$$$94\!\cdots\!78$$$$T^{22} -$$$$42\!\cdots\!00$$$$T^{23} +$$$$20\!\cdots\!22$$$$T^{24} -$$$$81\!\cdots\!14$$$$T^{25} +$$$$36\!\cdots\!35$$$$T^{26} -$$$$11\!\cdots\!24$$$$T^{27} +$$$$45\!\cdots\!33$$$$T^{28} -$$$$10\!\cdots\!60$$$$T^{29} +$$$$36\!\cdots\!59$$$$T^{30} -$$$$50\!\cdots\!12$$$$T^{31} +$$$$13\!\cdots\!01$$$$T^{32}$$
$47$ $$1 + 10 T + 319 T^{2} + 2158 T^{3} + 45197 T^{4} + 227486 T^{5} + 4372173 T^{6} + 17808894 T^{7} + 345920114 T^{8} + 1191969258 T^{9} + 23616948612 T^{10} + 70926148726 T^{11} + 1429263970430 T^{12} + 3855128819978 T^{13} + 77579226337384 T^{14} + 192114798287706 T^{15} + 3812710321024804 T^{16} + 9029395519522182 T^{17} + 171372510979281256 T^{18} + 400251039476575894 T^{19} + 6974352240491832830 T^{20} + 16266558076047511082 T^{21} +$$$$25\!\cdots\!48$$$$T^{22} +$$$$60\!\cdots\!54$$$$T^{23} +$$$$82\!\cdots\!54$$$$T^{24} +$$$$19\!\cdots\!98$$$$T^{25} +$$$$22\!\cdots\!77$$$$T^{26} +$$$$56\!\cdots\!58$$$$T^{27} +$$$$52\!\cdots\!77$$$$T^{28} +$$$$11\!\cdots\!66$$$$T^{29} +$$$$81\!\cdots\!11$$$$T^{30} +$$$$12\!\cdots\!30$$$$T^{31} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 + 6 T + 469 T^{2} + 3382 T^{3} + 113080 T^{4} + 910192 T^{5} + 18630434 T^{6} + 157418214 T^{7} + 2337521583 T^{8} + 19768857434 T^{9} + 234972166028 T^{10} + 1922908429030 T^{11} + 19415354023723 T^{12} + 150287264202844 T^{13} + 1336877045345107 T^{14} + 9629706833693278 T^{15} + 77256050069348266 T^{16} + 510374462185743734 T^{17} + 3755287620374405563 T^{18} + 22374317032726806188 T^{19} +$$$$15\!\cdots\!63$$$$T^{20} +$$$$80\!\cdots\!90$$$$T^{21} +$$$$52\!\cdots\!12$$$$T^{22} +$$$$23\!\cdots\!58$$$$T^{23} +$$$$14\!\cdots\!63$$$$T^{24} +$$$$51\!\cdots\!62$$$$T^{25} +$$$$32\!\cdots\!66$$$$T^{26} +$$$$84\!\cdots\!24$$$$T^{27} +$$$$55\!\cdots\!80$$$$T^{28} +$$$$88\!\cdots\!86$$$$T^{29} +$$$$64\!\cdots\!61$$$$T^{30} +$$$$43\!\cdots\!42$$$$T^{31} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$1 + 2 T + 311 T^{2} + 280 T^{3} + 56871 T^{4} + 27892 T^{5} + 7683267 T^{6} + 3616286 T^{7} + 836116241 T^{8} + 479405556 T^{9} + 76215312886 T^{10} + 55605533344 T^{11} + 6016831986079 T^{12} + 5203987075418 T^{13} + 419411701854944 T^{14} + 386178217640108 T^{15} + 26118253198678209 T^{16} + 22784514840766372 T^{17} + 1459972134157060064 T^{18} + 1068789661562273422 T^{19} + 72908125251666217519 T^{20} + 39753746946480325856 T^{21} +$$$$32\!\cdots\!26$$$$T^{22} +$$$$11\!\cdots\!64$$$$T^{23} +$$$$12\!\cdots\!61$$$$T^{24} +$$$$31\!\cdots\!54$$$$T^{25} +$$$$39\!\cdots\!67$$$$T^{26} +$$$$84\!\cdots\!28$$$$T^{27} +$$$$10\!\cdots\!51$$$$T^{28} +$$$$29\!\cdots\!20$$$$T^{29} +$$$$19\!\cdots\!71$$$$T^{30} +$$$$73\!\cdots\!98$$$$T^{31} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$1 - 8 T + 605 T^{2} - 4160 T^{3} + 179090 T^{4} - 1077090 T^{5} + 34658063 T^{6} - 184200124 T^{7} + 4937753413 T^{8} - 23384070246 T^{9} + 552964363604 T^{10} - 2353709635954 T^{11} + 50716482075547 T^{12} - 195982769600702 T^{13} + 3911640281431720 T^{14} - 13871953720143360 T^{15} + 257587502471292710 T^{16} - 846189176928744960 T^{17} + 14555213487207430120 T^{18} - 44484365026736940662 T^{19} +$$$$70\!\cdots\!27$$$$T^{20} -$$$$19\!\cdots\!54$$$$T^{21} +$$$$28\!\cdots\!44$$$$T^{22} -$$$$73\!\cdots\!66$$$$T^{23} +$$$$94\!\cdots\!53$$$$T^{24} -$$$$21\!\cdots\!84$$$$T^{25} +$$$$24\!\cdots\!63$$$$T^{26} -$$$$46\!\cdots\!90$$$$T^{27} +$$$$47\!\cdots\!90$$$$T^{28} -$$$$67\!\cdots\!60$$$$T^{29} +$$$$59\!\cdots\!05$$$$T^{30} -$$$$48\!\cdots\!08$$$$T^{31} +$$$$36\!\cdots\!61$$$$T^{32}$$
$67$ $$1 - 8 T + 540 T^{2} - 4000 T^{3} + 150116 T^{4} - 1042956 T^{5} + 28529992 T^{6} - 187196214 T^{7} + 4144141138 T^{8} - 25724621186 T^{9} + 487341620837 T^{10} - 2857205859716 T^{11} + 47936582041667 T^{12} - 264447903444180 T^{13} + 4018219941545291 T^{14} - 20744677924238584 T^{15} + 289809615272188020 T^{16} - 1389893420923985128 T^{17} + 18037789317596811299 T^{18} - 79536144783581909340 T^{19} +$$$$96\!\cdots\!07$$$$T^{20} -$$$$38\!\cdots\!12$$$$T^{21} +$$$$44\!\cdots\!53$$$$T^{22} -$$$$15\!\cdots\!78$$$$T^{23} +$$$$16\!\cdots\!58$$$$T^{24} -$$$$50\!\cdots\!58$$$$T^{25} +$$$$52\!\cdots\!08$$$$T^{26} -$$$$12\!\cdots\!48$$$$T^{27} +$$$$12\!\cdots\!76$$$$T^{28} -$$$$21\!\cdots\!00$$$$T^{29} +$$$$19\!\cdots\!60$$$$T^{30} -$$$$19\!\cdots\!44$$$$T^{31} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$1 + 50 T + 2023 T^{2} + 57796 T^{3} + 1435378 T^{4} + 29942746 T^{5} + 562520724 T^{6} + 9392394082 T^{7} + 144049911404 T^{8} + 2015332844396 T^{9} + 26207157566060 T^{10} + 315243738983424 T^{11} + 3550204210177108 T^{12} + 37273943292315142 T^{13} + 367928198923001699 T^{14} + 3399544170131210334 T^{15} + 29588800057516388083 T^{16} +$$$$24\!\cdots\!14$$$$T^{17} +$$$$18\!\cdots\!59$$$$T^{18} +$$$$13\!\cdots\!62$$$$T^{19} +$$$$90\!\cdots\!48$$$$T^{20} +$$$$56\!\cdots\!24$$$$T^{21} +$$$$33\!\cdots\!60$$$$T^{22} +$$$$18\!\cdots\!36$$$$T^{23} +$$$$93\!\cdots\!44$$$$T^{24} +$$$$43\!\cdots\!42$$$$T^{25} +$$$$18\!\cdots\!24$$$$T^{26} +$$$$69\!\cdots\!66$$$$T^{27} +$$$$23\!\cdots\!98$$$$T^{28} +$$$$67\!\cdots\!56$$$$T^{29} +$$$$16\!\cdots\!63$$$$T^{30} +$$$$29\!\cdots\!50$$$$T^{31} +$$$$41\!\cdots\!21$$$$T^{32}$$
$73$ $$1 + 14 T + 728 T^{2} + 10292 T^{3} + 272102 T^{4} + 3665304 T^{5} + 68341783 T^{6} + 849758052 T^{7} + 12736535884 T^{8} + 144527743928 T^{9} + 1853079469817 T^{10} + 19174355415656 T^{11} + 217117284321741 T^{12} + 2053154612783592 T^{13} + 20894461478312908 T^{14} + 180776600638000586 T^{15} + 1670393972000490008 T^{16} + 13196691846574042778 T^{17} +$$$$11\!\cdots\!32$$$$T^{18} +$$$$79\!\cdots\!64$$$$T^{19} +$$$$61\!\cdots\!81$$$$T^{20} +$$$$39\!\cdots\!08$$$$T^{21} +$$$$28\!\cdots\!13$$$$T^{22} +$$$$15\!\cdots\!16$$$$T^{23} +$$$$10\!\cdots\!04$$$$T^{24} +$$$$50\!\cdots\!76$$$$T^{25} +$$$$29\!\cdots\!67$$$$T^{26} +$$$$11\!\cdots\!08$$$$T^{27} +$$$$62\!\cdots\!42$$$$T^{28} +$$$$17\!\cdots\!36$$$$T^{29} +$$$$88\!\cdots\!52$$$$T^{30} +$$$$12\!\cdots\!98$$$$T^{31} +$$$$65\!\cdots\!61$$$$T^{32}$$
$79$ $$1 - 32 T + 1315 T^{2} - 30182 T^{3} + 734463 T^{4} - 13343174 T^{5} + 243579549 T^{6} - 3692680710 T^{7} + 55185674243 T^{8} - 721646400906 T^{9} + 9239431689267 T^{10} - 106654598620084 T^{11} + 1203548318055452 T^{12} - 12472310267258742 T^{13} + 126429719218540843 T^{14} - 1189380926010068494 T^{15} + 10954443490430363522 T^{16} - 93961093154795411026 T^{17} +$$$$78\!\cdots\!63$$$$T^{18} -$$$$61\!\cdots\!38$$$$T^{19} +$$$$46\!\cdots\!12$$$$T^{20} -$$$$32\!\cdots\!16$$$$T^{21} +$$$$22\!\cdots\!07$$$$T^{22} -$$$$13\!\cdots\!54$$$$T^{23} +$$$$83\!\cdots\!23$$$$T^{24} -$$$$44\!\cdots\!90$$$$T^{25} +$$$$23\!\cdots\!49$$$$T^{26} -$$$$99\!\cdots\!46$$$$T^{27} +$$$$43\!\cdots\!83$$$$T^{28} -$$$$14\!\cdots\!98$$$$T^{29} +$$$$48\!\cdots\!15$$$$T^{30} -$$$$93\!\cdots\!68$$$$T^{31} +$$$$23\!\cdots\!21$$$$T^{32}$$
$83$ $$1 - 20 T + 817 T^{2} - 12646 T^{3} + 315387 T^{4} - 4128658 T^{5} + 80006467 T^{6} - 924494530 T^{7} + 15177213168 T^{8} - 158382621132 T^{9} + 2295866758492 T^{10} - 21922133767156 T^{11} + 286991147048010 T^{12} - 2525615695552308 T^{13} + 30248822567620700 T^{14} - 246118235375152034 T^{15} + 2715969577494055212 T^{16} - 20427813536137618822 T^{17} +$$$$20\!\cdots\!00$$$$T^{18} -$$$$14\!\cdots\!96$$$$T^{19} +$$$$13\!\cdots\!10$$$$T^{20} -$$$$86\!\cdots\!08$$$$T^{21} +$$$$75\!\cdots\!48$$$$T^{22} -$$$$42\!\cdots\!64$$$$T^{23} +$$$$34\!\cdots\!88$$$$T^{24} -$$$$17\!\cdots\!90$$$$T^{25} +$$$$12\!\cdots\!83$$$$T^{26} -$$$$53\!\cdots\!86$$$$T^{27} +$$$$33\!\cdots\!07$$$$T^{28} -$$$$11\!\cdots\!98$$$$T^{29} +$$$$60\!\cdots\!93$$$$T^{30} -$$$$12\!\cdots\!40$$$$T^{31} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$1 + 52 T + 2233 T^{2} + 65480 T^{3} + 1673018 T^{4} + 34959414 T^{5} + 655417961 T^{6} + 10652368896 T^{7} + 158010711225 T^{8} + 2086163588344 T^{9} + 25456627659406 T^{10} + 281146798817394 T^{11} + 2919262491032063 T^{12} + 28071678826415070 T^{13} + 263476173129907260 T^{14} + 2398723875918036890 T^{15} + 22562710182142469814 T^{16} +$$$$21\!\cdots\!10$$$$T^{17} +$$$$20\!\cdots\!60$$$$T^{18} +$$$$19\!\cdots\!30$$$$T^{19} +$$$$18\!\cdots\!83$$$$T^{20} +$$$$15\!\cdots\!06$$$$T^{21} +$$$$12\!\cdots\!66$$$$T^{22} +$$$$92\!\cdots\!76$$$$T^{23} +$$$$62\!\cdots\!25$$$$T^{24} +$$$$37\!\cdots\!64$$$$T^{25} +$$$$20\!\cdots\!61$$$$T^{26} +$$$$97\!\cdots\!46$$$$T^{27} +$$$$41\!\cdots\!78$$$$T^{28} +$$$$14\!\cdots\!20$$$$T^{29} +$$$$43\!\cdots\!53$$$$T^{30} +$$$$90\!\cdots\!48$$$$T^{31} +$$$$15\!\cdots\!61$$$$T^{32}$$
$97$ $$1 + 18 T + 867 T^{2} + 14750 T^{3} + 390563 T^{4} + 6189082 T^{5} + 119716322 T^{6} + 1751108066 T^{7} + 27705709782 T^{8} + 372138728000 T^{9} + 5100351015684 T^{10} + 62793193346864 T^{11} + 769145311830846 T^{12} + 8682674022988166 T^{13} + 96645487447424545 T^{14} + 1001500160242145840 T^{15} + 10209699886837918733 T^{16} + 97145515543488146480 T^{17} +$$$$90\!\cdots\!05$$$$T^{18} +$$$$79\!\cdots\!18$$$$T^{19} +$$$$68\!\cdots\!26$$$$T^{20} +$$$$53\!\cdots\!48$$$$T^{21} +$$$$42\!\cdots\!36$$$$T^{22} +$$$$30\!\cdots\!00$$$$T^{23} +$$$$21\!\cdots\!02$$$$T^{24} +$$$$13\!\cdots\!22$$$$T^{25} +$$$$88\!\cdots\!78$$$$T^{26} +$$$$44\!\cdots\!46$$$$T^{27} +$$$$27\!\cdots\!83$$$$T^{28} +$$$$99\!\cdots\!50$$$$T^{29} +$$$$56\!\cdots\!23$$$$T^{30} +$$$$11\!\cdots\!74$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$