# Properties

 Label 350.2.h.d Level 350 Weight 2 Character orbit 350.h Analytic conductor 2.795 Analytic rank 0 Dimension 20 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 350.h (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$5$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$5^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{14} q^{2} + \beta_{1} q^{3} + \beta_{6} q^{4} + ( \beta_{10} + \beta_{13} + \beta_{15} + \beta_{17} ) q^{5} -\beta_{5} q^{6} + q^{7} + ( 1 - \beta_{4} + \beta_{6} - \beta_{14} ) q^{8} + ( -1 - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} - \beta_{17} ) q^{9} +O(q^{10})$$ $$q + \beta_{14} q^{2} + \beta_{1} q^{3} + \beta_{6} q^{4} + ( \beta_{10} + \beta_{13} + \beta_{15} + \beta_{17} ) q^{5} -\beta_{5} q^{6} + q^{7} + ( 1 - \beta_{4} + \beta_{6} - \beta_{14} ) q^{8} + ( -1 - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} - \beta_{17} ) q^{9} + \beta_{13} q^{10} + ( -1 + \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} + 2 \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{16} - \beta_{19} ) q^{11} + \beta_{9} q^{12} + ( \beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{10} + \beta_{11} - \beta_{15} + \beta_{16} ) q^{13} + \beta_{14} q^{14} + ( -\beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{10} - \beta_{11} + \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{15} -\beta_{4} q^{16} + ( 1 - \beta_{1} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{15} - \beta_{16} + \beta_{18} ) q^{17} + ( -2 + \beta_{2} - \beta_{3} ) q^{18} + ( -2 + 3 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{18} - \beta_{19} ) q^{19} -\beta_{10} q^{20} + \beta_{1} q^{21} + ( -\beta_{2} + \beta_{8} - \beta_{11} ) q^{22} + ( -2 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + \beta_{13} + 2 \beta_{14} - \beta_{15} + \beta_{16} - \beta_{19} ) q^{23} + \beta_{7} q^{24} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{15} - 2 \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{25} + ( 1 - \beta_{2} - \beta_{5} - \beta_{13} - \beta_{15} ) q^{26} + ( 1 - 2 \beta_{1} - 3 \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} - \beta_{12} + 2 \beta_{15} - 2 \beta_{16} + \beta_{18} + \beta_{19} ) q^{27} + \beta_{6} q^{28} + ( -2 + \beta_{1} + \beta_{3} + 3 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{14} - \beta_{15} ) q^{29} + ( -1 + \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{16} ) q^{30} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{14} - \beta_{15} + \beta_{16} - \beta_{19} ) q^{31} - q^{32} + ( 2 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 3 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} + \beta_{16} - 2 \beta_{17} + \beta_{19} ) q^{33} + ( 1 - 3 \beta_{4} + 2 \beta_{6} + \beta_{8} + \beta_{11} + \beta_{13} - 2 \beta_{14} + \beta_{17} + \beta_{19} ) q^{34} + ( \beta_{10} + \beta_{13} + \beta_{15} + \beta_{17} ) q^{35} + ( \beta_{1} - \beta_{3} + \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{36} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} + 3 \beta_{15} - \beta_{16} + \beta_{17} ) q^{37} + ( \beta_{4} - \beta_{9} + \beta_{10} + \beta_{15} - \beta_{19} ) q^{38} + ( -5 + 3 \beta_{1} + 4 \beta_{4} - 3 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{13} + 3 \beta_{14} - 5 \beta_{15} + 2 \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{39} + \beta_{15} q^{40} + ( -1 - \beta_{1} + 3 \beta_{4} - \beta_{6} - \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} - \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{41} -\beta_{5} q^{42} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{9} - \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{17} + \beta_{18} ) q^{43} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{16} + \beta_{17} ) q^{44} + ( 6 - 3 \beta_{1} + \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} - \beta_{16} + \beta_{18} + 2 \beta_{19} ) q^{45} + ( -1 - 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} - \beta_{10} - \beta_{14} - \beta_{16} - \beta_{18} ) q^{46} + ( -1 - \beta_{1} + \beta_{3} + 2 \beta_{4} - 3 \beta_{6} + \beta_{10} + \beta_{15} + \beta_{17} + \beta_{18} ) q^{47} + ( -\beta_{1} - \beta_{5} + \beta_{7} - \beta_{9} ) q^{48} + q^{49} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{11} + \beta_{12} - 2 \beta_{14} - \beta_{18} + \beta_{19} ) q^{50} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} + \beta_{13} + \beta_{14} - \beta_{17} - \beta_{18} ) q^{51} + ( -\beta_{1} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{16} ) q^{52} + ( 1 - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{13} - \beta_{14} + \beta_{15} - 3 \beta_{17} + \beta_{18} + \beta_{19} ) q^{53} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{15} - \beta_{16} - \beta_{17} + \beta_{19} ) q^{54} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 5 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{16} - \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{55} + ( 1 - \beta_{4} + \beta_{6} - \beta_{14} ) q^{56} + ( 6 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} - 3 \beta_{7} + \beta_{12} ) q^{57} + ( 1 + \beta_{1} + \beta_{2} + \beta_{5} + \beta_{9} - \beta_{10} - \beta_{14} - \beta_{16} + \beta_{18} ) q^{58} + ( -2 - \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} + 4 \beta_{14} - \beta_{15} - \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{59} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{10} + \beta_{12} - 2 \beta_{14} + \beta_{16} ) q^{60} + ( -2 \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} - 6 \beta_{14} - 2 \beta_{16} + 2 \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{61} + ( 1 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{62} + ( -1 - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} - \beta_{17} ) q^{63} -\beta_{14} q^{64} + ( 1 + 4 \beta_{1} - 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{14} - \beta_{15} ) q^{65} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} - 3 \beta_{13} + \beta_{14} - \beta_{15} - 2 \beta_{17} ) q^{66} + ( 1 - 3 \beta_{1} + \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{67} + ( -\beta_{2} + \beta_{3} - \beta_{5} - \beta_{8} - \beta_{9} - \beta_{14} + \beta_{16} + \beta_{17} ) q^{68} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + \beta_{13} - 3 \beta_{14} + 2 \beta_{17} - \beta_{18} ) q^{69} + \beta_{13} q^{70} + ( 1 + 2 \beta_{2} + \beta_{3} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + \beta_{13} + \beta_{15} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{71} + ( -1 - \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{18} - \beta_{19} ) q^{72} + ( 2 \beta_{1} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{15} + 2 \beta_{16} - \beta_{18} ) q^{73} + ( 1 - \beta_{1} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} - \beta_{17} - \beta_{18} ) q^{74} + ( 3 - 4 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} + \beta_{17} ) q^{75} + ( 1 - \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{16} - \beta_{17} ) q^{76} + ( -1 + \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} + 2 \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{16} - \beta_{19} ) q^{77} + ( -2 + \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} + \beta_{10} - \beta_{14} + 2 \beta_{15} - \beta_{16} + 3 \beta_{17} - \beta_{19} ) q^{78} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 7 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + \beta_{14} + \beta_{15} + 3 \beta_{16} + \beta_{17} - 2 \beta_{19} ) q^{79} -\beta_{17} q^{80} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} + 3 \beta_{14} + 3 \beta_{15} - 2 \beta_{16} + \beta_{17} + \beta_{19} ) q^{81} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{14} + \beta_{15} - \beta_{19} ) q^{82} + ( 2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} + \beta_{16} - 2 \beta_{17} - \beta_{19} ) q^{83} + \beta_{9} q^{84} + ( -3 + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 3 \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} - \beta_{16} - \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{85} + ( 1 + \beta_{1} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} + \beta_{8} + 3 \beta_{9} + \beta_{11} - \beta_{12} + \beta_{15} - 2 \beta_{16} + 2 \beta_{18} + \beta_{19} ) q^{86} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 6 \beta_{4} + \beta_{5} + 6 \beta_{6} + \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} - 7 \beta_{14} + 2 \beta_{15} + 2 \beta_{17} + \beta_{19} ) q^{87} + ( 1 - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} + \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{88} + ( 1 + \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - 3 \beta_{14} + \beta_{15} + \beta_{17} - 2 \beta_{18} ) q^{89} + ( 1 - \beta_{2} + 2 \beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} + 3 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} + \beta_{18} + \beta_{19} ) q^{90} + ( \beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{10} + \beta_{11} - \beta_{15} + \beta_{16} ) q^{91} + ( -\beta_{1} + 2 \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{14} + \beta_{16} + \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{92} + ( -2 + 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{93} + ( \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{94} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{15} + \beta_{16} - \beta_{18} - 2 \beta_{19} ) q^{95} -\beta_{1} q^{96} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{11} - 3 \beta_{14} + 3 \beta_{15} - 3 \beta_{16} + 2 \beta_{17} + \beta_{18} + \beta_{19} ) q^{97} + \beta_{14} q^{98} + ( 4 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} - 5 \beta_{5} + 7 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{12} - 4 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} + 2 \beta_{17} - 2 \beta_{19} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 5q^{2} + 3q^{3} - 5q^{4} - 5q^{5} - 3q^{6} + 20q^{7} + 5q^{8} - 6q^{9} + O(q^{10})$$ $$20q + 5q^{2} + 3q^{3} - 5q^{4} - 5q^{5} - 3q^{6} + 20q^{7} + 5q^{8} - 6q^{9} - 9q^{11} - 2q^{12} + 5q^{13} + 5q^{14} - 5q^{16} - 12q^{17} - 34q^{18} + 2q^{19} + 5q^{20} + 3q^{21} - 6q^{22} - 5q^{23} + 2q^{24} - 35q^{25} + 20q^{26} - 6q^{27} - 5q^{28} - 22q^{29} - 25q^{30} - 7q^{31} - 20q^{32} + 25q^{33} - 18q^{34} - 5q^{35} - 6q^{36} - 3q^{37} + 8q^{38} - 22q^{39} + 19q^{41} - 3q^{42} + 2q^{43} + 6q^{44} + 45q^{45} - 10q^{46} - 14q^{47} - 2q^{48} + 20q^{49} + 10q^{50} + 38q^{51} + 5q^{52} - q^{53} - 19q^{54} - 20q^{55} + 5q^{56} + 116q^{57} + 22q^{58} + 17q^{59} - 5q^{60} - 38q^{61} + 7q^{62} - 6q^{63} - 5q^{64} + 15q^{65} - 16q^{67} - 12q^{68} + 35q^{69} + q^{71} + 11q^{72} + 19q^{73} + 18q^{74} + 35q^{75} + 12q^{76} - 9q^{77} - 18q^{78} - 64q^{79} - 40q^{81} + 26q^{82} + 57q^{83} - 2q^{84} - 40q^{85} - 2q^{86} - 78q^{87} + 9q^{88} - 6q^{89} + 10q^{90} + 5q^{91} + 10q^{92} - 22q^{93} + 14q^{94} + 60q^{95} - 3q^{96} - 18q^{97} + 5q^{98} + 112q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 3 x^{19} + 15 x^{18} - 30 x^{17} + 145 x^{16} - 194 x^{15} + 1187 x^{14} - 1490 x^{13} + 10170 x^{12} - 13920 x^{11} + 42087 x^{10} - 591 x^{9} + 65635 x^{8} + 120715 x^{7} + 257180 x^{6} + 306283 x^{5} + 246171 x^{4} + 99850 x^{3} + 26260 x^{2} + 4200 x + 400$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$42\!\cdots\!71$$$$\nu^{19} +$$$$45\!\cdots\!15$$$$\nu^{18} -$$$$16\!\cdots\!25$$$$\nu^{17} +$$$$63\!\cdots\!20$$$$\nu^{16} -$$$$16\!\cdots\!45$$$$\nu^{15} +$$$$57\!\cdots\!84$$$$\nu^{14} -$$$$12\!\cdots\!55$$$$\nu^{13} +$$$$45\!\cdots\!80$$$$\nu^{12} -$$$$98\!\cdots\!70$$$$\nu^{11} +$$$$39\!\cdots\!70$$$$\nu^{10} -$$$$68\!\cdots\!37$$$$\nu^{9} +$$$$14\!\cdots\!25$$$$\nu^{8} -$$$$48\!\cdots\!55$$$$\nu^{7} +$$$$15\!\cdots\!05$$$$\nu^{6} +$$$$25\!\cdots\!20$$$$\nu^{5} +$$$$62\!\cdots\!47$$$$\nu^{4} +$$$$74\!\cdots\!15$$$$\nu^{3} +$$$$55\!\cdots\!70$$$$\nu^{2} +$$$$14\!\cdots\!00$$$$\nu +$$$$24\!\cdots\!00$$$$)/$$$$40\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$45\!\cdots\!31$$$$\nu^{19} +$$$$46\!\cdots\!45$$$$\nu^{18} -$$$$17\!\cdots\!65$$$$\nu^{17} +$$$$64\!\cdots\!40$$$$\nu^{16} -$$$$17\!\cdots\!75$$$$\nu^{15} +$$$$57\!\cdots\!84$$$$\nu^{14} -$$$$12\!\cdots\!35$$$$\nu^{13} +$$$$46\!\cdots\!20$$$$\nu^{12} -$$$$10\!\cdots\!40$$$$\nu^{11} +$$$$40\!\cdots\!50$$$$\nu^{10} -$$$$69\!\cdots\!17$$$$\nu^{9} +$$$$14\!\cdots\!95$$$$\nu^{8} -$$$$47\!\cdots\!65$$$$\nu^{7} +$$$$14\!\cdots\!35$$$$\nu^{6} +$$$$25\!\cdots\!50$$$$\nu^{5} +$$$$61\!\cdots\!57$$$$\nu^{4} +$$$$74\!\cdots\!15$$$$\nu^{3} +$$$$55\!\cdots\!70$$$$\nu^{2} +$$$$14\!\cdots\!00$$$$\nu +$$$$41\!\cdots\!00$$$$)/$$$$40\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$96\!\cdots\!31$$$$\nu^{19} -$$$$48\!\cdots\!64$$$$\nu^{18} +$$$$21\!\cdots\!20$$$$\nu^{17} -$$$$61\!\cdots\!85$$$$\nu^{16} +$$$$21\!\cdots\!87$$$$\nu^{15} -$$$$49\!\cdots\!51$$$$\nu^{14} +$$$$16\!\cdots\!89$$$$\nu^{13} -$$$$39\!\cdots\!25$$$$\nu^{12} +$$$$13\!\cdots\!20$$$$\nu^{11} -$$$$34\!\cdots\!52$$$$\nu^{10} +$$$$76\!\cdots\!19$$$$\nu^{9} -$$$$96\!\cdots\!76$$$$\nu^{8} +$$$$10\!\cdots\!20$$$$\nu^{7} -$$$$19\!\cdots\!20$$$$\nu^{6} +$$$$63\!\cdots\!73$$$$\nu^{5} -$$$$12\!\cdots\!85$$$$\nu^{4} -$$$$16\!\cdots\!70$$$$\nu^{3} -$$$$18\!\cdots\!75$$$$\nu^{2} -$$$$93\!\cdots\!00$$$$\nu -$$$$28\!\cdots\!00$$$$)/$$$$80\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$13\!\cdots\!77$$$$\nu^{19} +$$$$41\!\cdots\!19$$$$\nu^{18} -$$$$20\!\cdots\!29$$$$\nu^{17} +$$$$40\!\cdots\!42$$$$\nu^{16} -$$$$19\!\cdots\!41$$$$\nu^{15} +$$$$26\!\cdots\!52$$$$\nu^{14} -$$$$16\!\cdots\!39$$$$\nu^{13} +$$$$20\!\cdots\!34$$$$\nu^{12} -$$$$14\!\cdots\!42$$$$\nu^{11} +$$$$18\!\cdots\!86$$$$\nu^{10} -$$$$57\!\cdots\!23$$$$\nu^{9} -$$$$50\!\cdots\!49$$$$\nu^{8} -$$$$93\!\cdots\!41$$$$\nu^{7} -$$$$16\!\cdots\!77$$$$\nu^{6} -$$$$36\!\cdots\!14$$$$\nu^{5} -$$$$43\!\cdots\!45$$$$\nu^{4} -$$$$35\!\cdots\!65$$$$\nu^{3} -$$$$13\!\cdots\!50$$$$\nu^{2} -$$$$34\!\cdots\!00$$$$\nu -$$$$53\!\cdots\!00$$$$)/$$$$80\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$10\!\cdots\!43$$$$\nu^{19} -$$$$48\!\cdots\!83$$$$\nu^{18} +$$$$21\!\cdots\!20$$$$\nu^{17} -$$$$57\!\cdots\!69$$$$\nu^{16} +$$$$20\!\cdots\!43$$$$\nu^{15} -$$$$45\!\cdots\!72$$$$\nu^{14} +$$$$16\!\cdots\!33$$$$\nu^{13} -$$$$35\!\cdots\!25$$$$\nu^{12} +$$$$13\!\cdots\!14$$$$\nu^{11} -$$$$31\!\cdots\!33$$$$\nu^{10} +$$$$71\!\cdots\!66$$$$\nu^{9} -$$$$75\!\cdots\!27$$$$\nu^{8} +$$$$85\!\cdots\!65$$$$\nu^{7} +$$$$19\!\cdots\!54$$$$\nu^{6} +$$$$93\!\cdots\!87$$$$\nu^{5} -$$$$57\!\cdots\!61$$$$\nu^{4} -$$$$14\!\cdots\!50$$$$\nu^{3} -$$$$18\!\cdots\!85$$$$\nu^{2} -$$$$48\!\cdots\!25$$$$\nu -$$$$91\!\cdots\!00$$$$)/$$$$40\!\cdots\!50$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$12\!\cdots\!87$$$$\nu^{19} +$$$$36\!\cdots\!37$$$$\nu^{18} -$$$$18\!\cdots\!06$$$$\nu^{17} +$$$$36\!\cdots\!53$$$$\nu^{16} -$$$$17\!\cdots\!12$$$$\nu^{15} +$$$$22\!\cdots\!02$$$$\nu^{14} -$$$$14\!\cdots\!87$$$$\nu^{13} +$$$$17\!\cdots\!36$$$$\nu^{12} -$$$$12\!\cdots\!48$$$$\nu^{11} +$$$$16\!\cdots\!37$$$$\nu^{10} -$$$$50\!\cdots\!18$$$$\nu^{9} -$$$$27\!\cdots\!12$$$$\nu^{8} -$$$$78\!\cdots\!29$$$$\nu^{7} -$$$$15\!\cdots\!58$$$$\nu^{6} -$$$$32\!\cdots\!28$$$$\nu^{5} -$$$$38\!\cdots\!40$$$$\nu^{4} -$$$$30\!\cdots\!55$$$$\nu^{3} -$$$$11\!\cdots\!50$$$$\nu^{2} -$$$$22\!\cdots\!00$$$$\nu -$$$$26\!\cdots\!00$$$$)/$$$$36\!\cdots\!50$$ $$\beta_{8}$$ $$=$$ $$($$$$70\!\cdots\!63$$$$\nu^{19} +$$$$12\!\cdots\!40$$$$\nu^{18} -$$$$68\!\cdots\!10$$$$\nu^{17} +$$$$32\!\cdots\!15$$$$\nu^{16} -$$$$17\!\cdots\!25$$$$\nu^{15} +$$$$38\!\cdots\!13$$$$\nu^{14} +$$$$38\!\cdots\!55$$$$\nu^{13} +$$$$31\!\cdots\!45$$$$\nu^{12} +$$$$69\!\cdots\!10$$$$\nu^{11} +$$$$26\!\cdots\!50$$$$\nu^{10} -$$$$29\!\cdots\!79$$$$\nu^{9} +$$$$15\!\cdots\!40$$$$\nu^{8} -$$$$10\!\cdots\!00$$$$\nu^{7} +$$$$31\!\cdots\!10$$$$\nu^{6} +$$$$50\!\cdots\!75$$$$\nu^{5} +$$$$94\!\cdots\!69$$$$\nu^{4} +$$$$91\!\cdots\!30$$$$\nu^{3} +$$$$57\!\cdots\!65$$$$\nu^{2} +$$$$11\!\cdots\!50$$$$\nu +$$$$17\!\cdots\!00$$$$)/$$$$20\!\cdots\!50$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$16\!\cdots\!54$$$$\nu^{19} +$$$$53\!\cdots\!75$$$$\nu^{18} -$$$$25\!\cdots\!79$$$$\nu^{17} +$$$$55\!\cdots\!08$$$$\nu^{16} -$$$$25\!\cdots\!30$$$$\nu^{15} +$$$$37\!\cdots\!92$$$$\nu^{14} -$$$$20\!\cdots\!55$$$$\nu^{13} +$$$$29\!\cdots\!04$$$$\nu^{12} -$$$$17\!\cdots\!73$$$$\nu^{11} +$$$$26\!\cdots\!25$$$$\nu^{10} -$$$$74\!\cdots\!14$$$$\nu^{9} +$$$$16\!\cdots\!60$$$$\nu^{8} -$$$$10\!\cdots\!91$$$$\nu^{7} -$$$$17\!\cdots\!53$$$$\nu^{6} -$$$$37\!\cdots\!30$$$$\nu^{5} -$$$$40\!\cdots\!03$$$$\nu^{4} -$$$$29\!\cdots\!35$$$$\nu^{3} -$$$$76\!\cdots\!05$$$$\nu^{2} -$$$$13\!\cdots\!00$$$$\nu -$$$$42\!\cdots\!00$$$$)/$$$$40\!\cdots\!50$$ $$\beta_{10}$$ $$=$$ $$($$$$12\!\cdots\!24$$$$\nu^{19} -$$$$41\!\cdots\!70$$$$\nu^{18} +$$$$20\!\cdots\!70$$$$\nu^{17} -$$$$45\!\cdots\!95$$$$\nu^{16} +$$$$20\!\cdots\!10$$$$\nu^{15} -$$$$31\!\cdots\!31$$$$\nu^{14} +$$$$16\!\cdots\!05$$$$\nu^{13} -$$$$24\!\cdots\!65$$$$\nu^{12} +$$$$13\!\cdots\!45$$$$\nu^{11} -$$$$22\!\cdots\!35$$$$\nu^{10} +$$$$62\!\cdots\!88$$$$\nu^{9} -$$$$23\!\cdots\!15$$$$\nu^{8} +$$$$96\!\cdots\!50$$$$\nu^{7} +$$$$12\!\cdots\!20$$$$\nu^{6} +$$$$29\!\cdots\!15$$$$\nu^{5} +$$$$31\!\cdots\!17$$$$\nu^{4} +$$$$24\!\cdots\!40$$$$\nu^{3} +$$$$90\!\cdots\!70$$$$\nu^{2} +$$$$31\!\cdots\!00$$$$\nu +$$$$39\!\cdots\!50$$$$)/$$$$20\!\cdots\!50$$ $$\beta_{11}$$ $$=$$ $$($$$$50\!\cdots\!17$$$$\nu^{19} -$$$$17\!\cdots\!75$$$$\nu^{18} +$$$$82\!\cdots\!45$$$$\nu^{17} -$$$$18\!\cdots\!20$$$$\nu^{16} +$$$$79\!\cdots\!65$$$$\nu^{15} -$$$$12\!\cdots\!18$$$$\nu^{14} +$$$$64\!\cdots\!05$$$$\nu^{13} -$$$$99\!\cdots\!80$$$$\nu^{12} +$$$$54\!\cdots\!20$$$$\nu^{11} -$$$$90\!\cdots\!40$$$$\nu^{10} +$$$$24\!\cdots\!99$$$$\nu^{9} -$$$$84\!\cdots\!05$$$$\nu^{8} +$$$$32\!\cdots\!65$$$$\nu^{7} +$$$$48\!\cdots\!45$$$$\nu^{6} +$$$$10\!\cdots\!60$$$$\nu^{5} +$$$$10\!\cdots\!31$$$$\nu^{4} +$$$$60\!\cdots\!45$$$$\nu^{3} +$$$$66\!\cdots\!10$$$$\nu^{2} -$$$$68\!\cdots\!00$$$$\nu -$$$$17\!\cdots\!00$$$$)/$$$$40\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$14\!\cdots\!44$$$$\nu^{19} +$$$$50\!\cdots\!50$$$$\nu^{18} -$$$$23\!\cdots\!25$$$$\nu^{17} +$$$$54\!\cdots\!55$$$$\nu^{16} -$$$$22\!\cdots\!50$$$$\nu^{15} +$$$$38\!\cdots\!21$$$$\nu^{14} -$$$$18\!\cdots\!85$$$$\nu^{13} +$$$$30\!\cdots\!20$$$$\nu^{12} -$$$$15\!\cdots\!55$$$$\nu^{11} +$$$$27\!\cdots\!50$$$$\nu^{10} -$$$$71\!\cdots\!38$$$$\nu^{9} +$$$$34\!\cdots\!60$$$$\nu^{8} -$$$$99\!\cdots\!70$$$$\nu^{7} -$$$$11\!\cdots\!30$$$$\nu^{6} -$$$$28\!\cdots\!25$$$$\nu^{5} -$$$$25\!\cdots\!12$$$$\nu^{4} -$$$$14\!\cdots\!65$$$$\nu^{3} +$$$$96\!\cdots\!05$$$$\nu^{2} +$$$$11\!\cdots\!75$$$$\nu +$$$$34\!\cdots\!25$$$$)/$$$$10\!\cdots\!75$$ $$\beta_{13}$$ $$=$$ $$($$$$64\!\cdots\!13$$$$\nu^{19} -$$$$22\!\cdots\!30$$$$\nu^{18} +$$$$10\!\cdots\!40$$$$\nu^{17} -$$$$24\!\cdots\!25$$$$\nu^{16} +$$$$10\!\cdots\!75$$$$\nu^{15} -$$$$17\!\cdots\!07$$$$\nu^{14} +$$$$84\!\cdots\!25$$$$\nu^{13} -$$$$13\!\cdots\!05$$$$\nu^{12} +$$$$71\!\cdots\!00$$$$\nu^{11} -$$$$12\!\cdots\!50$$$$\nu^{10} +$$$$32\!\cdots\!41$$$$\nu^{9} -$$$$16\!\cdots\!40$$$$\nu^{8} +$$$$44\!\cdots\!50$$$$\nu^{7} +$$$$55\!\cdots\!50$$$$\nu^{6} +$$$$12\!\cdots\!75$$$$\nu^{5} +$$$$11\!\cdots\!89$$$$\nu^{4} +$$$$68\!\cdots\!30$$$$\nu^{3} +$$$$29\!\cdots\!15$$$$\nu^{2} +$$$$36\!\cdots\!00$$$$\nu +$$$$19\!\cdots\!00$$$$)/$$$$40\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$26\!\cdots\!15$$$$\nu^{19} +$$$$83\!\cdots\!99$$$$\nu^{18} -$$$$41\!\cdots\!63$$$$\nu^{17} +$$$$85\!\cdots\!08$$$$\nu^{16} -$$$$39\!\cdots\!59$$$$\nu^{15} +$$$$56\!\cdots\!92$$$$\nu^{14} -$$$$32\!\cdots\!09$$$$\nu^{13} +$$$$43\!\cdots\!28$$$$\nu^{12} -$$$$27\!\cdots\!18$$$$\nu^{11} +$$$$40\!\cdots\!84$$$$\nu^{10} -$$$$11\!\cdots\!77$$$$\nu^{9} +$$$$13\!\cdots\!11$$$$\nu^{8} -$$$$17\!\cdots\!27$$$$\nu^{7} -$$$$30\!\cdots\!43$$$$\nu^{6} -$$$$66\!\cdots\!46$$$$\nu^{5} -$$$$75\!\cdots\!17$$$$\nu^{4} -$$$$57\!\cdots\!75$$$$\nu^{3} -$$$$19\!\cdots\!20$$$$\nu^{2} -$$$$43\!\cdots\!00$$$$\nu -$$$$43\!\cdots\!00$$$$)/$$$$16\!\cdots\!00$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$19\!\cdots\!09$$$$\nu^{19} +$$$$58\!\cdots\!85$$$$\nu^{18} -$$$$29\!\cdots\!85$$$$\nu^{17} +$$$$58\!\cdots\!60$$$$\nu^{16} -$$$$28\!\cdots\!65$$$$\nu^{15} +$$$$37\!\cdots\!16$$$$\nu^{14} -$$$$22\!\cdots\!95$$$$\nu^{13} +$$$$28\!\cdots\!80$$$$\nu^{12} -$$$$19\!\cdots\!10$$$$\nu^{11} +$$$$27\!\cdots\!40$$$$\nu^{10} -$$$$81\!\cdots\!03$$$$\nu^{9} +$$$$87\!\cdots\!25$$$$\nu^{8} -$$$$12\!\cdots\!85$$$$\nu^{7} -$$$$23\!\cdots\!85$$$$\nu^{6} -$$$$49\!\cdots\!10$$$$\nu^{5} -$$$$58\!\cdots\!67$$$$\nu^{4} -$$$$46\!\cdots\!65$$$$\nu^{3} -$$$$17\!\cdots\!20$$$$\nu^{2} -$$$$42\!\cdots\!00$$$$\nu -$$$$56\!\cdots\!00$$$$)/$$$$80\!\cdots\!00$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!67$$$$\nu^{19} +$$$$38\!\cdots\!10$$$$\nu^{18} -$$$$18\!\cdots\!90$$$$\nu^{17} +$$$$39\!\cdots\!65$$$$\nu^{16} -$$$$17\!\cdots\!25$$$$\nu^{15} +$$$$26\!\cdots\!73$$$$\nu^{14} -$$$$14\!\cdots\!15$$$$\nu^{13} +$$$$20\!\cdots\!25$$$$\nu^{12} -$$$$12\!\cdots\!40$$$$\nu^{11} +$$$$19\!\cdots\!00$$$$\nu^{10} -$$$$53\!\cdots\!29$$$$\nu^{9} +$$$$11\!\cdots\!20$$$$\nu^{8} -$$$$79\!\cdots\!70$$$$\nu^{7} -$$$$12\!\cdots\!40$$$$\nu^{6} -$$$$27\!\cdots\!75$$$$\nu^{5} -$$$$30\!\cdots\!11$$$$\nu^{4} -$$$$22\!\cdots\!20$$$$\nu^{3} -$$$$68\!\cdots\!35$$$$\nu^{2} -$$$$15\!\cdots\!50$$$$\nu -$$$$94\!\cdots\!00$$$$)/$$$$40\!\cdots\!00$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$66\!\cdots\!46$$$$\nu^{19} +$$$$22\!\cdots\!80$$$$\nu^{18} -$$$$10\!\cdots\!15$$$$\nu^{17} +$$$$23\!\cdots\!90$$$$\nu^{16} -$$$$10\!\cdots\!80$$$$\nu^{15} +$$$$16\!\cdots\!99$$$$\nu^{14} -$$$$83\!\cdots\!45$$$$\nu^{13} +$$$$12\!\cdots\!20$$$$\nu^{12} -$$$$71\!\cdots\!15$$$$\nu^{11} +$$$$11\!\cdots\!30$$$$\nu^{10} -$$$$31\!\cdots\!77$$$$\nu^{9} +$$$$10\!\cdots\!35$$$$\nu^{8} -$$$$45\!\cdots\!40$$$$\nu^{7} -$$$$65\!\cdots\!15$$$$\nu^{6} -$$$$14\!\cdots\!70$$$$\nu^{5} -$$$$15\!\cdots\!93$$$$\nu^{4} -$$$$10\!\cdots\!85$$$$\nu^{3} -$$$$21\!\cdots\!55$$$$\nu^{2} -$$$$28\!\cdots\!25$$$$\nu +$$$$17\!\cdots\!00$$$$)/$$$$20\!\cdots\!50$$ $$\beta_{18}$$ $$=$$ $$($$$$35\!\cdots\!79$$$$\nu^{19} -$$$$11\!\cdots\!15$$$$\nu^{18} +$$$$55\!\cdots\!55$$$$\nu^{17} -$$$$11\!\cdots\!40$$$$\nu^{16} +$$$$54\!\cdots\!95$$$$\nu^{15} -$$$$80\!\cdots\!76$$$$\nu^{14} +$$$$43\!\cdots\!25$$$$\nu^{13} -$$$$62\!\cdots\!00$$$$\nu^{12} +$$$$37\!\cdots\!90$$$$\nu^{11} -$$$$57\!\cdots\!20$$$$\nu^{10} +$$$$16\!\cdots\!73$$$$\nu^{9} -$$$$34\!\cdots\!95$$$$\nu^{8} +$$$$24\!\cdots\!15$$$$\nu^{7} +$$$$38\!\cdots\!15$$$$\nu^{6} +$$$$83\!\cdots\!30$$$$\nu^{5} +$$$$92\!\cdots\!57$$$$\nu^{4} +$$$$69\!\cdots\!15$$$$\nu^{3} +$$$$22\!\cdots\!20$$$$\nu^{2} +$$$$54\!\cdots\!00$$$$\nu +$$$$85\!\cdots\!00$$$$)/$$$$80\!\cdots\!00$$ $$\beta_{19}$$ $$=$$ $$($$$$-$$$$65\!\cdots\!83$$$$\nu^{19} +$$$$20\!\cdots\!45$$$$\nu^{18} -$$$$10\!\cdots\!25$$$$\nu^{17} +$$$$20\!\cdots\!10$$$$\nu^{16} -$$$$97\!\cdots\!35$$$$\nu^{15} +$$$$13\!\cdots\!82$$$$\nu^{14} -$$$$79\!\cdots\!65$$$$\nu^{13} +$$$$10\!\cdots\!90$$$$\nu^{12} -$$$$67\!\cdots\!10$$$$\nu^{11} +$$$$99\!\cdots\!60$$$$\nu^{10} -$$$$28\!\cdots\!01$$$$\nu^{9} +$$$$35\!\cdots\!25$$$$\nu^{8} -$$$$42\!\cdots\!65$$$$\nu^{7} -$$$$74\!\cdots\!85$$$$\nu^{6} -$$$$15\!\cdots\!40$$$$\nu^{5} -$$$$17\!\cdots\!69$$$$\nu^{4} -$$$$13\!\cdots\!05$$$$\nu^{3} -$$$$42\!\cdots\!90$$$$\nu^{2} -$$$$73\!\cdots\!00$$$$\nu -$$$$64\!\cdots\!00$$$$)/$$$$73\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{17} - \beta_{15} + \beta_{14} - \beta_{13} - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} - 3 \beta_{4} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{19} + \beta_{18} - 2 \beta_{16} + 2 \beta_{15} - \beta_{12} - 7 \beta_{9} + \beta_{8} + 6 \beta_{7} - 7 \beta_{5} - 3 \beta_{4} - 8 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$-8 \beta_{19} + \beta_{17} - 2 \beta_{16} + 3 \beta_{15} + 39 \beta_{14} + 2 \beta_{13} - 10 \beta_{12} - 10 \beta_{11} - 8 \beta_{10} - 7 \beta_{9} + \beta_{8} + 8 \beta_{7} - 29 \beta_{6} - 9 \beta_{5} + 36 \beta_{4} + \beta_{2} - 10 \beta_{1} - 38$$ $$\nu^{5}$$ $$=$$ $$\beta_{19} - 11 \beta_{18} - 12 \beta_{17} - \beta_{16} + 2 \beta_{15} + 6 \beta_{14} + 14 \beta_{13} + 9 \beta_{12} + 11 \beta_{10} + 12 \beta_{9} + \beta_{8} - 66 \beta_{7} - 11 \beta_{6} + 11 \beta_{5} + 4 \beta_{4} - 12 \beta_{3} + 15 \beta_{2} - 3 \beta_{1} - 25$$ $$\nu^{6}$$ $$=$$ $$80 \beta_{19} + 63 \beta_{18} - 26 \beta_{17} - 51 \beta_{16} + 47 \beta_{15} - 64 \beta_{14} + 20 \beta_{13} + 96 \beta_{11} + 3 \beta_{10} + 71 \beta_{9} + 68 \beta_{8} - 51 \beta_{7} + 297 \beta_{6} + 80 \beta_{5} - 115 \beta_{4} - \beta_{3} + 96 \beta_{2} - 19 \beta_{1} + 34$$ $$\nu^{7}$$ $$=$$ $$-20 \beta_{18} + 22 \beta_{17} + 175 \beta_{16} - 278 \beta_{15} + 58 \beta_{14} + 62 \beta_{13} - 65 \beta_{12} + 112 \beta_{11} - 33 \beta_{10} + 532 \beta_{9} + 2 \beta_{8} + 120 \beta_{7} + 47 \beta_{6} - 80 \beta_{5} + 261 \beta_{4} + 105 \beta_{3} - 65 \beta_{2} + 70 \beta_{1} - 18$$ $$\nu^{8}$$ $$=$$ $$-122 \beta_{19} - 717 \beta_{18} + 855 \beta_{17} + 1119 \beta_{16} - 739 \beta_{15} - 2710 \beta_{14} + 710 \beta_{13} + 927 \beta_{12} + 1135 \beta_{11} + 1495 \beta_{10} + 1004 \beta_{9} - 192 \beta_{8} - 57 \beta_{7} + 230 \beta_{6} + 149 \beta_{5} + 531 \beta_{4} - 500 \beta_{3} + 1506 \beta_{1} + 38$$ $$\nu^{9}$$ $$=$$ $$41 \beta_{19} - 20 \beta_{18} + 628 \beta_{17} + 869 \beta_{16} - 1826 \beta_{15} - 2303 \beta_{14} - 909 \beta_{13} + 1935 \beta_{12} + 1935 \beta_{11} - 279 \beta_{10} + 524 \beta_{9} - 1257 \beta_{8} + 239 \beta_{7} + 1753 \beta_{6} + 5083 \beta_{5} - 257 \beta_{4} + 280 \beta_{3} + 388 \beta_{2} + 2350 \beta_{1} + 1326$$ $$\nu^{10}$$ $$=$$ $$-917 \beta_{19} + 1172 \beta_{18} + 2504 \beta_{17} + 1332 \beta_{16} + 41 \beta_{15} - 4042 \beta_{14} - 2048 \beta_{13} + 2307 \beta_{12} - 1172 \beta_{10} - 2504 \beta_{9} - 1332 \beta_{8} + 5357 \beta_{7} + 1172 \beta_{6} - 1967 \beta_{5} - 1793 \beta_{4} + 6494 \beta_{3} - 9015 \beta_{2} + 496 \beta_{1} + 26490$$ $$\nu^{11}$$ $$=$$ $$-13040 \beta_{19} - 546 \beta_{18} + 8692 \beta_{17} + 5967 \beta_{16} + 11411 \beta_{15} + 1083 \beta_{14} - 10875 \beta_{13} - 20477 \beta_{11} + 1619 \beta_{10} - 22607 \beta_{9} - 4521 \beta_{8} + 11732 \beta_{7} - 20999 \beta_{6} - 13040 \beta_{5} + 7870 \beta_{4} + 537 \beta_{3} - 20477 \beta_{2} + 30428 \beta_{1} + 5707$$ $$\nu^{12}$$ $$=$$ $$-5765 \beta_{19} + 10875 \beta_{18} - 62454 \beta_{17} - 50555 \beta_{16} + 15651 \beta_{15} + 71879 \beta_{14} - 98144 \beta_{13} - 24780 \beta_{12} - 112974 \beta_{11} - 74549 \beta_{10} - 117404 \beta_{9} - 37639 \beta_{8} - 37090 \beta_{7} - 82789 \beta_{6} + 1400 \beta_{5} - 132862 \beta_{4} - 10910 \beta_{3} - 24780 \beta_{2} - 39645 \beta_{1} - 26764$$ $$\nu^{13}$$ $$=$$ $$74729 \beta_{19} + 129714 \beta_{18} - 97400 \beta_{17} - 327713 \beta_{16} + 320318 \beta_{15} + 235655 \beta_{14} - 89455 \beta_{13} - 213339 \beta_{12} - 220670 \beta_{11} - 147210 \beta_{10} - 432708 \beta_{9} + 114374 \beta_{8} + 146639 \beta_{7} - 5865 \beta_{6} - 335308 \beta_{5} - 312987 \beta_{4} + 5830 \beta_{3} - 556412 \beta_{1} + 108509$$ $$\nu^{14}$$ $$=$$ $$-253682 \beta_{19} + 72730 \beta_{18} - 201766 \beta_{17} - 559093 \beta_{16} + 658492 \beta_{15} + 2155756 \beta_{14} + 216903 \beta_{13} - 867400 \beta_{12} - 867400 \beta_{11} - 472602 \beta_{10} - 349118 \beta_{9} + 295399 \beta_{8} + 130412 \beta_{7} - 1333621 \beta_{6} - 764696 \beta_{5} + 1168634 \beta_{4} - 123270 \beta_{3} + 263694 \beta_{2} - 1007610 \beta_{1} - 1983627$$ $$\nu^{15}$$ $$=$$ $$441589 \beta_{19} - 648694 \beta_{18} - 1230493 \beta_{17} - 581799 \beta_{16} + 448678 \beta_{15} + 1003219 \beta_{14} + 1538961 \beta_{13} - 266189 \beta_{12} + 648694 \beta_{10} + 1230493 \beta_{9} + 581799 \beta_{8} - 3497289 \beta_{7} - 648694 \beta_{6} + 364624 \beta_{5} - 20169 \beta_{4} - 1282978 \beta_{3} + 2204950 \beta_{2} - 1314547 \beta_{1} - 4086135$$ $$\nu^{16}$$ $$=$$ $$5120440 \beta_{19} + 2000607 \beta_{18} - 1994864 \beta_{17} - 1429014 \beta_{16} - 253357 \beta_{15} - 2866476 \beta_{14} + 4816395 \beta_{13} + 8573274 \beta_{11} + 1696562 \beta_{10} + 8110794 \beta_{9} + 4230037 \beta_{8} - 3294399 \beta_{7} + 15001008 \beta_{6} + 5120440 \beta_{5} - 2609295 \beta_{4} - 865869 \beta_{3} + 8573274 \beta_{2} - 3187886 \beta_{1} - 3377014$$ $$\nu^{17}$$ $$=$$ $$1865385 \beta_{19} - 4816395 \beta_{18} + 7387453 \beta_{17} + 20439520 \beta_{16} - 25316767 \beta_{15} - 8700223 \beta_{14} + 17339053 \beta_{13} + 4368680 \beta_{12} + 27060448 \beta_{11} + 9979353 \beta_{10} + 46932363 \beta_{9} + 2252248 \beta_{8} + 5603630 \beta_{7} + 14283143 \beta_{6} + 4347970 \beta_{5} + 31162704 \beta_{4} + 5582920 \beta_{3} + 4368680 \beta_{2} + 14856600 \beta_{1} - 2564147$$ $$\nu^{18}$$ $$=$$ $$-7855878 \beta_{19} - 48167313 \beta_{18} + 63064055 \beta_{17} + 107843376 \beta_{16} - 92965066 \beta_{15} - 187242405 \beta_{14} + 52487530 \beta_{13} + 85130898 \beta_{12} + 114884240 \beta_{11} + 100694985 \beta_{10} + 117301516 \beta_{9} - 22712478 \beta_{8} - 8912053 \beta_{7} + 49960565 \beta_{6} + 54237461 \beta_{5} + 63211724 \beta_{4} - 15523945 \beta_{3} + 154364649 \beta_{1} + 27248087$$ $$\nu^{19}$$ $$=$$ $$1871189 \beta_{19} - 23258565 \beta_{18} + 105477727 \beta_{17} + 181609146 \beta_{16} - 229270514 \beta_{15} - 398714212 \beta_{14} - 69082576 \beta_{13} + 232979060 \beta_{12} + 232979060 \beta_{11} + 62577574 \beta_{10} + 114774551 \beta_{9} - 122740118 \beta_{8} + 49948996 \beta_{7} + 233714297 \beta_{6} + 338621527 \beta_{5} - 61025183 \beta_{4} + 51820185 \beta_{3} - 58869028 \beta_{2} + 344045065 \beta_{1} + 327794279$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
71.1
 −2.31300 − 1.68049i −0.649647 − 0.471996i −0.210620 − 0.153024i 1.88692 + 1.37093i 2.59536 + 1.88564i −0.924929 − 2.84664i −0.329549 − 1.01425i −0.0727237 − 0.223820i 0.671167 + 2.06564i 0.847017 + 2.60685i −0.924929 + 2.84664i −0.329549 + 1.01425i −0.0727237 + 0.223820i 0.671167 − 2.06564i 0.847017 − 2.60685i −2.31300 + 1.68049i −0.649647 + 0.471996i −0.210620 + 0.153024i 1.88692 − 1.37093i 2.59536 − 1.88564i
−0.309017 + 0.951057i −2.31300 1.68049i −0.809017 0.587785i −0.771064 2.09892i 2.31300 1.68049i 1.00000 0.809017 0.587785i 1.59885 + 4.92077i 2.23446 0.0847237i
71.2 −0.309017 + 0.951057i −0.649647 0.471996i −0.809017 0.587785i 0.192568 + 2.22776i 0.649647 0.471996i 1.00000 0.809017 0.587785i −0.727790 2.23991i −2.17823 0.505273i
71.3 −0.309017 + 0.951057i −0.210620 0.153024i −0.809017 0.587785i −2.22785 + 0.191483i 0.210620 0.153024i 1.00000 0.809017 0.587785i −0.906107 2.78871i 0.506334 2.17799i
71.4 −0.309017 + 0.951057i 1.88692 + 1.37093i −0.809017 0.587785i 1.06221 1.96767i −1.88692 + 1.37093i 1.00000 0.809017 0.587785i 0.753978 + 2.32051i 1.54312 + 1.61826i
71.5 −0.309017 + 0.951057i 2.59536 + 1.88564i −0.809017 0.587785i −0.0648780 + 2.23513i −2.59536 + 1.88564i 1.00000 0.809017 0.587785i 2.25320 + 6.93464i −2.10568 0.752395i
141.1 0.809017 + 0.587785i −0.924929 2.84664i 0.309017 + 0.951057i −1.56551 1.59661i 0.924929 2.84664i 1.00000 −0.309017 + 0.951057i −4.82080 + 3.50252i −0.328061 2.21187i
141.2 0.809017 + 0.587785i −0.329549 1.01425i 0.309017 + 0.951057i −1.58296 + 1.57932i 0.329549 1.01425i 1.00000 −0.309017 + 0.951057i 1.50696 1.09487i −2.20894 + 0.347253i
141.3 0.809017 + 0.587785i −0.0727237 0.223820i 0.309017 + 0.951057i 1.97315 1.05198i 0.0727237 0.223820i 1.00000 −0.309017 + 0.951057i 2.38224 1.73080i 2.21465 + 0.308716i
141.4 0.809017 + 0.587785i 0.671167 + 2.06564i 0.309017 + 0.951057i −0.284771 + 2.21786i −0.671167 + 2.06564i 1.00000 −0.309017 + 0.951057i −1.38935 + 1.00942i −1.53401 + 1.62690i
141.5 0.809017 + 0.587785i 0.847017 + 2.60685i 0.309017 + 0.951057i 0.769108 2.09964i −0.847017 + 2.60685i 1.00000 −0.309017 + 0.951057i −3.65118 + 2.65274i 1.85636 1.24657i
211.1 0.809017 0.587785i −0.924929 + 2.84664i 0.309017 0.951057i −1.56551 + 1.59661i 0.924929 + 2.84664i 1.00000 −0.309017 0.951057i −4.82080 3.50252i −0.328061 + 2.21187i
211.2 0.809017 0.587785i −0.329549 + 1.01425i 0.309017 0.951057i −1.58296 1.57932i 0.329549 + 1.01425i 1.00000 −0.309017 0.951057i 1.50696 + 1.09487i −2.20894 0.347253i
211.3 0.809017 0.587785i −0.0727237 + 0.223820i 0.309017 0.951057i 1.97315 + 1.05198i 0.0727237 + 0.223820i 1.00000 −0.309017 0.951057i 2.38224 + 1.73080i 2.21465 0.308716i
211.4 0.809017 0.587785i 0.671167 2.06564i 0.309017 0.951057i −0.284771 2.21786i −0.671167 2.06564i 1.00000 −0.309017 0.951057i −1.38935 1.00942i −1.53401 1.62690i
211.5 0.809017 0.587785i 0.847017 2.60685i 0.309017 0.951057i 0.769108 + 2.09964i −0.847017 2.60685i 1.00000 −0.309017 0.951057i −3.65118 2.65274i 1.85636 + 1.24657i
281.1 −0.309017 0.951057i −2.31300 + 1.68049i −0.809017 + 0.587785i −0.771064 + 2.09892i 2.31300 + 1.68049i 1.00000 0.809017 + 0.587785i 1.59885 4.92077i 2.23446 + 0.0847237i
281.2 −0.309017 0.951057i −0.649647 + 0.471996i −0.809017 + 0.587785i 0.192568 2.22776i 0.649647 + 0.471996i 1.00000 0.809017 + 0.587785i −0.727790 + 2.23991i −2.17823 + 0.505273i
281.3 −0.309017 0.951057i −0.210620 + 0.153024i −0.809017 + 0.587785i −2.22785 0.191483i 0.210620 + 0.153024i 1.00000 0.809017 + 0.587785i −0.906107 + 2.78871i 0.506334 + 2.17799i
281.4 −0.309017 0.951057i 1.88692 1.37093i −0.809017 + 0.587785i 1.06221 + 1.96767i −1.88692 1.37093i 1.00000 0.809017 + 0.587785i 0.753978 2.32051i 1.54312 1.61826i
281.5 −0.309017 0.951057i 2.59536 1.88564i −0.809017 + 0.587785i −0.0648780 2.23513i −2.59536 1.88564i 1.00000 0.809017 + 0.587785i 2.25320 6.93464i −2.10568 + 0.752395i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 281.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.h.d 20
25.d even 5 1 inner 350.2.h.d 20
25.d even 5 1 8750.2.a.w 10
25.e even 10 1 8750.2.a.x 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.h.d 20 1.a even 1 1 trivial
350.2.h.d 20 25.d even 5 1 inner
8750.2.a.w 10 25.d even 5 1
8750.2.a.x 10 25.e even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{20} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(350, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}$$
$3$ $$1 - 3 T + 9 T^{3} - 5 T^{4} - 11 T^{5} - 28 T^{6} + 181 T^{7} - 201 T^{8} - 198 T^{9} + 342 T^{10} + 432 T^{11} + 391 T^{12} - 6101 T^{13} + 12092 T^{14} - 1175 T^{15} - 16887 T^{16} - 11609 T^{17} + 41866 T^{18} + 124287 T^{19} - 435782 T^{20} + 372861 T^{21} + 376794 T^{22} - 313443 T^{23} - 1367847 T^{24} - 285525 T^{25} + 8815068 T^{26} - 13342887 T^{27} + 2565351 T^{28} + 8503056 T^{29} + 20194758 T^{30} - 35075106 T^{31} - 106819641 T^{32} + 288572463 T^{33} - 133923132 T^{34} - 157837977 T^{35} - 215233605 T^{36} + 1162261467 T^{37} - 3486784401 T^{39} + 3486784401 T^{40}$$
$5$ $$1 + 5 T + 30 T^{2} + 110 T^{3} + 390 T^{4} + 1125 T^{5} + 3005 T^{6} + 7340 T^{7} + 16455 T^{8} + 37775 T^{9} + 80575 T^{10} + 188875 T^{11} + 411375 T^{12} + 917500 T^{13} + 1878125 T^{14} + 3515625 T^{15} + 6093750 T^{16} + 8593750 T^{17} + 11718750 T^{18} + 9765625 T^{19} + 9765625 T^{20}$$
$7$ $$( 1 - T )^{20}$$
$11$ $$1 + 9 T + T^{2} - 270 T^{3} - 1106 T^{4} + 1437 T^{5} + 28254 T^{6} + 81600 T^{7} - 185184 T^{8} - 1956121 T^{9} - 3939441 T^{10} + 14683187 T^{11} + 99595113 T^{12} + 123974460 T^{13} - 845289048 T^{14} - 3927803039 T^{15} - 1004769233 T^{16} + 41106907430 T^{17} + 112816425513 T^{18} - 173691256523 T^{19} - 1636513680309 T^{20} - 1910603821753 T^{21} + 13650787487073 T^{22} + 54713293789330 T^{23} - 14710826340353 T^{24} - 632576607233989 T^{25} - 1497481111163928 T^{26} + 2415911501652660 T^{27} + 21349096975748553 T^{28} + 34622186883171217 T^{29} - 102178953907588041 T^{30} - 558104150427259931 T^{31} - 581186720514701664 T^{32} + 2817053310944769600 T^{33} + 10729451798060891214 T^{34} + 6002705619450290487 T^{35} - 50820401229110810066 T^{36} -$$$$13\!\cdots\!70$$$$T^{37} + 5559917313492231481 T^{38} +$$$$55\!\cdots\!19$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$
$13$ $$1 - 5 T + 180 T^{4} - 1119 T^{5} + 5205 T^{6} - 8500 T^{7} + 26300 T^{8} - 238895 T^{9} + 685648 T^{10} - 3004655 T^{11} + 11645825 T^{12} - 25367850 T^{13} + 167751875 T^{14} - 586109075 T^{15} + 1407193925 T^{16} - 9517504150 T^{17} + 31286861100 T^{18} - 80311296305 T^{19} + 353659054271 T^{20} - 1044046851965 T^{21} + 5287479525900 T^{22} - 20909956617550 T^{23} + 40190865691925 T^{24} - 217618196783975 T^{25} + 809706260016875 T^{26} - 1591794966978450 T^{27} + 9499857223889825 T^{28} - 31862862063581315 T^{29} + 94522399219283152 T^{30} - 428138157333469115 T^{31} + 612739638721250300 T^{32} - 2574438406034150500 T^{33} + 20494044087564799245 T^{34} - 57277014282767557083 T^{35} +$$$$11\!\cdots\!80$$$$T^{36} -$$$$73\!\cdots\!85$$$$T^{39} +$$$$19\!\cdots\!01$$$$T^{40}$$
$17$ $$1 + 12 T + 6 T^{2} - 466 T^{3} - 2255 T^{4} - 2604 T^{5} + 41979 T^{6} + 475288 T^{7} + 1352471 T^{8} - 6403588 T^{9} - 54774001 T^{10} - 170639242 T^{11} + 107637136 T^{12} + 5518940082 T^{13} + 25925948559 T^{14} + 1565494782 T^{15} - 419550373786 T^{16} - 2073571073444 T^{17} - 3941818822220 T^{18} + 22861596658438 T^{19} + 188979842286129 T^{20} + 388647143193446 T^{21} - 1139185639621580 T^{22} - 10187454683830372 T^{23} - 35041266768980506 T^{24} + 2222778724686174 T^{25} + 625789372233313071 T^{26} + 2264634549614391186 T^{27} + 750850552379928976 T^{28} - 20235745355837695274 T^{29} -$$$$11\!\cdots\!49$$$$T^{30} -$$$$21\!\cdots\!04$$$$T^{31} +$$$$78\!\cdots\!31$$$$T^{32} +$$$$47\!\cdots\!56$$$$T^{33} +$$$$70\!\cdots\!91$$$$T^{34} -$$$$74\!\cdots\!72$$$$T^{35} -$$$$10\!\cdots\!55$$$$T^{36} -$$$$38\!\cdots\!82$$$$T^{37} +$$$$84\!\cdots\!54$$$$T^{38} +$$$$28\!\cdots\!36$$$$T^{39} +$$$$40\!\cdots\!01$$$$T^{40}$$
$19$ $$1 - 2 T - 35 T^{2} - 22 T^{3} + 475 T^{4} + 682 T^{5} + 13747 T^{6} + 12118 T^{7} - 305918 T^{8} - 809522 T^{9} - 2291715 T^{10} - 21808678 T^{11} + 161227877 T^{12} + 967062682 T^{13} + 279777607 T^{14} - 5638086218 T^{15} - 48823661508 T^{16} - 380542806978 T^{17} + 242658755757 T^{18} + 4750008910922 T^{19} + 14190850058469 T^{20} + 90250169307518 T^{21} + 87599810828277 T^{22} - 2610143113062102 T^{23} - 6362748391384068 T^{24} - 13960459646303582 T^{25} + 13162384005386767 T^{26} + 864430001281343998 T^{27} + 2738223812996093957 T^{28} - 7037392095423526162 T^{29} - 14050656508996418715 T^{30} - 94301427363804041318 T^{31} -$$$$67\!\cdots\!98$$$$T^{32} +$$$$50\!\cdots\!62$$$$T^{33} +$$$$10\!\cdots\!87$$$$T^{34} +$$$$10\!\cdots\!18$$$$T^{35} +$$$$13\!\cdots\!75$$$$T^{36} -$$$$12\!\cdots\!58$$$$T^{37} -$$$$36\!\cdots\!35$$$$T^{38} -$$$$39\!\cdots\!58$$$$T^{39} +$$$$37\!\cdots\!01$$$$T^{40}$$
$23$ $$1 + 5 T - 35 T^{2} - 120 T^{3} + 1180 T^{4} + 4415 T^{5} - 24560 T^{6} - 152080 T^{7} + 462030 T^{8} + 3027805 T^{9} - 18239153 T^{10} - 62982805 T^{11} + 423352235 T^{12} + 329737570 T^{13} - 11032544730 T^{14} - 9735553715 T^{15} + 254133694985 T^{16} + 440207234730 T^{17} - 6047843923755 T^{18} - 348343474205 T^{19} + 189940842334619 T^{20} - 8011899906715 T^{21} - 3199309435666395 T^{22} + 5356001424959910 T^{23} + 71117027338297385 T^{24} - 62661363004664245 T^{25} - 1633212567037814970 T^{26} + 1122698869167943790 T^{27} + 33153130643763453035 T^{28} -$$$$11\!\cdots\!15$$$$T^{29} -$$$$75\!\cdots\!97$$$$T^{30} +$$$$28\!\cdots\!35$$$$T^{31} +$$$$10\!\cdots\!30$$$$T^{32} -$$$$76\!\cdots\!40$$$$T^{33} -$$$$28\!\cdots\!40$$$$T^{34} +$$$$11\!\cdots\!05$$$$T^{35} +$$$$72\!\cdots\!80$$$$T^{36} -$$$$16\!\cdots\!60$$$$T^{37} -$$$$11\!\cdots\!15$$$$T^{38} +$$$$37\!\cdots\!35$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$
$29$ $$1 + 22 T + 162 T^{2} + 364 T^{3} + 1221 T^{4} + 31882 T^{5} + 241485 T^{6} + 885670 T^{7} + 3609905 T^{8} + 27103550 T^{9} + 174298901 T^{10} + 735666692 T^{11} + 2745459462 T^{12} + 17332002484 T^{13} + 71322017141 T^{14} - 56024592128 T^{15} - 538624284330 T^{16} + 6226688616470 T^{17} + 24729523205740 T^{18} - 193015275164740 T^{19} - 1931410756140335 T^{20} - 5597442979777460 T^{21} + 20797529016027340 T^{22} + 151862708667086830 T^{23} - 380958722445206730 T^{24} - 1149128756801635072 T^{25} + 42423999096228545261 T^{26} +$$$$29\!\cdots\!56$$$$T^{27} +$$$$13\!\cdots\!82$$$$T^{28} +$$$$10\!\cdots\!48$$$$T^{29} +$$$$73\!\cdots\!01$$$$T^{30} +$$$$33\!\cdots\!50$$$$T^{31} +$$$$12\!\cdots\!05$$$$T^{32} +$$$$90\!\cdots\!30$$$$T^{33} +$$$$71\!\cdots\!85$$$$T^{34} +$$$$27\!\cdots\!18$$$$T^{35} +$$$$30\!\cdots\!41$$$$T^{36} +$$$$26\!\cdots\!76$$$$T^{37} +$$$$34\!\cdots\!82$$$$T^{38} +$$$$13\!\cdots\!18$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$
$31$ $$1 + 7 T - 79 T^{2} - 579 T^{3} + 3032 T^{4} + 28139 T^{5} + 4936 T^{6} - 743317 T^{7} - 4912332 T^{8} + 9560721 T^{9} + 212484051 T^{10} + 81948331 T^{11} - 2914483577 T^{12} - 1389679507 T^{13} - 55278648174 T^{14} - 279870156423 T^{15} + 759056753083 T^{16} + 13090764459319 T^{17} + 147368113191249 T^{18} - 221840826023487 T^{19} - 7945817595531147 T^{20} - 6877065606728097 T^{21} + 141620756776790289 T^{22} + 389986964007572329 T^{23} + 701004851663965243 T^{24} - 8012444968627686873 T^{25} - 49060003735128928494 T^{26} - 38233716014055723277 T^{27} -$$$$24\!\cdots\!57$$$$T^{28} +$$$$21\!\cdots\!01$$$$T^{29} +$$$$17\!\cdots\!51$$$$T^{30} +$$$$24\!\cdots\!51$$$$T^{31} -$$$$38\!\cdots\!52$$$$T^{32} -$$$$18\!\cdots\!47$$$$T^{33} +$$$$37\!\cdots\!56$$$$T^{34} +$$$$66\!\cdots\!89$$$$T^{35} +$$$$22\!\cdots\!92$$$$T^{36} -$$$$13\!\cdots\!69$$$$T^{37} -$$$$55\!\cdots\!39$$$$T^{38} +$$$$15\!\cdots\!97$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$
$37$ $$1 + 3 T - 75 T^{2} - 431 T^{3} + 1910 T^{4} + 1531 T^{5} - 52233 T^{6} + 918601 T^{7} + 7816136 T^{8} - 5198737 T^{9} - 134683907 T^{10} - 1561396851 T^{11} - 16230592826 T^{12} - 14283194053 T^{13} + 439735380987 T^{14} + 1622954356537 T^{15} + 13721182465584 T^{16} + 141107352922883 T^{17} - 5174944573003 T^{18} - 4875052887814471 T^{19} - 34960991903139092 T^{20} - 180376956849135427 T^{21} - 7084499120441107 T^{22} + 7147510747602792599 T^{23} + 25715705050881375024 T^{24} +$$$$11\!\cdots\!09$$$$T^{25} +$$$$11\!\cdots\!83$$$$T^{26} -$$$$13\!\cdots\!49$$$$T^{27} -$$$$57\!\cdots\!46$$$$T^{28} -$$$$20\!\cdots\!27$$$$T^{29} -$$$$64\!\cdots\!43$$$$T^{30} -$$$$92\!\cdots\!81$$$$T^{31} +$$$$51\!\cdots\!16$$$$T^{32} +$$$$22\!\cdots\!97$$$$T^{33} -$$$$47\!\cdots\!37$$$$T^{34} +$$$$51\!\cdots\!83$$$$T^{35} +$$$$23\!\cdots\!10$$$$T^{36} -$$$$19\!\cdots\!27$$$$T^{37} -$$$$12\!\cdots\!75$$$$T^{38} +$$$$18\!\cdots\!19$$$$T^{39} +$$$$23\!\cdots\!01$$$$T^{40}$$
$41$ $$1 - 19 T + 12 T^{2} + 1913 T^{3} - 9016 T^{4} - 84640 T^{5} + 727901 T^{6} + 733147 T^{7} - 19334902 T^{8} + 149819 T^{9} + 159905008 T^{10} + 1763738609 T^{11} + 15953599463 T^{12} - 202733667043 T^{13} - 1098060502629 T^{14} + 11161254620488 T^{15} + 56842341836837 T^{16} - 697945434779521 T^{17} + 859311312315416 T^{18} + 12981930863804423 T^{19} - 102522143793719309 T^{20} + 532259165415981343 T^{21} + 1444502316002214296 T^{22} - 48103097310439366841 T^{23} +$$$$16\!\cdots\!57$$$$T^{24} +$$$$12\!\cdots\!88$$$$T^{25} -$$$$52\!\cdots\!89$$$$T^{26} -$$$$39\!\cdots\!83$$$$T^{27} +$$$$12\!\cdots\!23$$$$T^{28} +$$$$57\!\cdots\!49$$$$T^{29} +$$$$21\!\cdots\!08$$$$T^{30} +$$$$82\!\cdots\!79$$$$T^{31} -$$$$43\!\cdots\!62$$$$T^{32} +$$$$67\!\cdots\!87$$$$T^{33} +$$$$27\!\cdots\!61$$$$T^{34} -$$$$13\!\cdots\!40$$$$T^{35} -$$$$57\!\cdots\!56$$$$T^{36} +$$$$50\!\cdots\!53$$$$T^{37} +$$$$12\!\cdots\!52$$$$T^{38} -$$$$83\!\cdots\!59$$$$T^{39} +$$$$18\!\cdots\!01$$$$T^{40}$$
$43$ $$( 1 - T + 131 T^{2} - 222 T^{3} + 13120 T^{4} - 17529 T^{5} + 947878 T^{6} - 1296581 T^{7} + 53628143 T^{8} - 65230381 T^{9} + 2588452654 T^{10} - 2804906383 T^{11} + 99158436407 T^{12} - 103087265567 T^{13} + 3240606254278 T^{14} - 2576910997347 T^{15} + 82936283202880 T^{16} - 60343731665754 T^{17} + 1531154236365731 T^{18} - 502592611936843 T^{19} + 21611482313284249 T^{20} )^{2}$$
$47$ $$1 + 14 T - 17 T^{2} - 332 T^{3} + 11875 T^{4} + 85608 T^{5} - 375631 T^{6} + 491530 T^{7} + 67558562 T^{8} + 169238914 T^{9} - 1464748913 T^{10} + 15388021112 T^{11} + 192696067221 T^{12} - 64267686812 T^{13} - 257501703091 T^{14} + 62244145699390 T^{15} + 272409586441108 T^{16} + 72500860159662 T^{17} + 11516861002196303 T^{18} + 93024868043107652 T^{19} + 301659113965774101 T^{20} + 4372168798026059644 T^{21} + 25440745953851633327 T^{22} + 7527256804356587826 T^{23} +$$$$13\!\cdots\!48$$$$T^{24} +$$$$14\!\cdots\!30$$$$T^{25} -$$$$27\!\cdots\!39$$$$T^{26} -$$$$32\!\cdots\!56$$$$T^{27} +$$$$45\!\cdots\!81$$$$T^{28} +$$$$17\!\cdots\!04$$$$T^{29} -$$$$77\!\cdots\!37$$$$T^{30} +$$$$41\!\cdots\!42$$$$T^{31} +$$$$78\!\cdots\!42$$$$T^{32} +$$$$26\!\cdots\!10$$$$T^{33} -$$$$96\!\cdots\!39$$$$T^{34} +$$$$10\!\cdots\!44$$$$T^{35} +$$$$67\!\cdots\!75$$$$T^{36} -$$$$88\!\cdots\!84$$$$T^{37} -$$$$21\!\cdots\!13$$$$T^{38} +$$$$82\!\cdots\!62$$$$T^{39} +$$$$27\!\cdots\!01$$$$T^{40}$$
$53$ $$1 + T + 222 T^{2} + 576 T^{3} + 29866 T^{4} + 121159 T^{5} + 2973080 T^{6} + 16226713 T^{7} + 233637046 T^{8} + 1673331728 T^{9} + 15995234209 T^{10} + 136618682878 T^{11} + 1032730160982 T^{12} + 9346217467105 T^{13} + 66824236403364 T^{14} + 556826155800381 T^{15} + 4361081323456902 T^{16} + 29933837836232390 T^{17} + 270757625857435954 T^{18} + 1566157358977656593 T^{19} + 15195332378604934484 T^{20} + 83006340025815799429 T^{21} +$$$$76\!\cdots\!86$$$$T^{22} +$$$$44\!\cdots\!30$$$$T^{23} +$$$$34\!\cdots\!62$$$$T^{24} +$$$$23\!\cdots\!33$$$$T^{25} +$$$$14\!\cdots\!56$$$$T^{26} +$$$$10\!\cdots\!85$$$$T^{27} +$$$$64\!\cdots\!02$$$$T^{28} +$$$$45\!\cdots\!74$$$$T^{29} +$$$$27\!\cdots\!41$$$$T^{30} +$$$$15\!\cdots\!16$$$$T^{31} +$$$$11\!\cdots\!86$$$$T^{32} +$$$$42\!\cdots\!49$$$$T^{33} +$$$$41\!\cdots\!20$$$$T^{34} +$$$$88\!\cdots\!63$$$$T^{35} +$$$$11\!\cdots\!86$$$$T^{36} +$$$$11\!\cdots\!88$$$$T^{37} +$$$$24\!\cdots\!58$$$$T^{38} +$$$$57\!\cdots\!17$$$$T^{39} +$$$$30\!\cdots\!01$$$$T^{40}$$
$59$ $$1 - 17 T - 24 T^{2} + 1080 T^{3} + 3041 T^{4} + 91298 T^{5} - 1705592 T^{6} - 6176752 T^{7} + 59242869 T^{8} + 435071881 T^{9} + 9725236734 T^{10} - 84667840483 T^{11} - 544988908505 T^{12} + 396013967022 T^{13} + 15351394464494 T^{14} + 534770086845542 T^{15} - 1784279235241195 T^{16} - 18902754975455470 T^{17} - 90766731594831180 T^{18} - 113158777818941145 T^{19} + 18511300884404128250 T^{20} - 6676367891317527555 T^{21} -$$$$31\!\cdots\!80$$$$T^{22} -$$$$38\!\cdots\!30$$$$T^{23} -$$$$21\!\cdots\!95$$$$T^{24} +$$$$38\!\cdots\!58$$$$T^{25} +$$$$64\!\cdots\!54$$$$T^{26} +$$$$98\!\cdots\!18$$$$T^{27} -$$$$80\!\cdots\!05$$$$T^{28} -$$$$73\!\cdots\!37$$$$T^{29} +$$$$49\!\cdots\!34$$$$T^{30} +$$$$13\!\cdots\!79$$$$T^{31} +$$$$10\!\cdots\!89$$$$T^{32} -$$$$64\!\cdots\!08$$$$T^{33} -$$$$10\!\cdots\!12$$$$T^{34} +$$$$33\!\cdots\!02$$$$T^{35} +$$$$65\!\cdots\!81$$$$T^{36} +$$$$13\!\cdots\!20$$$$T^{37} -$$$$18\!\cdots\!04$$$$T^{38} -$$$$75\!\cdots\!63$$$$T^{39} +$$$$26\!\cdots\!01$$$$T^{40}$$
$61$ $$1 + 38 T + 613 T^{2} + 3660 T^{3} - 42463 T^{4} - 1139168 T^{5} - 10432970 T^{6} - 11592012 T^{7} + 876621133 T^{8} + 11435430830 T^{9} + 57987789192 T^{10} - 257674721658 T^{11} - 7267129029675 T^{12} - 60744458842252 T^{13} - 147149444018752 T^{14} + 2537573290658796 T^{15} + 36705196328155185 T^{16} + 228237801345912040 T^{17} + 134377432547021165 T^{18} - 12634742916802888610 T^{19} -$$$$14\!\cdots\!30$$$$T^{20} -$$$$77\!\cdots\!10$$$$T^{21} +$$$$50\!\cdots\!65$$$$T^{22} +$$$$51\!\cdots\!40$$$$T^{23} +$$$$50\!\cdots\!85$$$$T^{24} +$$$$21\!\cdots\!96$$$$T^{25} -$$$$75\!\cdots\!72$$$$T^{26} -$$$$19\!\cdots\!92$$$$T^{27} -$$$$13\!\cdots\!75$$$$T^{28} -$$$$30\!\cdots\!78$$$$T^{29} +$$$$41\!\cdots\!92$$$$T^{30} +$$$$49\!\cdots\!30$$$$T^{31} +$$$$23\!\cdots\!93$$$$T^{32} -$$$$18\!\cdots\!72$$$$T^{33} -$$$$10\!\cdots\!70$$$$T^{34} -$$$$68\!\cdots\!68$$$$T^{35} -$$$$15\!\cdots\!43$$$$T^{36} +$$$$82\!\cdots\!60$$$$T^{37} +$$$$83\!\cdots\!53$$$$T^{38} +$$$$31\!\cdots\!58$$$$T^{39} +$$$$50\!\cdots\!01$$$$T^{40}$$
$67$ $$1 + 16 T - 58 T^{2} - 2978 T^{3} - 18445 T^{4} + 126976 T^{5} + 2658477 T^{6} + 13876972 T^{7} - 93385065 T^{8} - 1939016994 T^{9} - 8366709641 T^{10} + 75540871444 T^{11} + 1126536292624 T^{12} + 3842182564850 T^{13} - 44052154844443 T^{14} - 587678167507398 T^{15} - 1161816367846488 T^{16} + 27430169634446196 T^{17} + 203676466527797368 T^{18} - 490172917501845106 T^{19} - 13679143434364915411 T^{20} - 32841585472623622102 T^{21} +$$$$91\!\cdots\!52$$$$T^{22} +$$$$82\!\cdots\!48$$$$T^{23} -$$$$23\!\cdots\!48$$$$T^{24} -$$$$79\!\cdots\!86$$$$T^{25} -$$$$39\!\cdots\!67$$$$T^{26} +$$$$23\!\cdots\!50$$$$T^{27} +$$$$45\!\cdots\!84$$$$T^{28} +$$$$20\!\cdots\!68$$$$T^{29} -$$$$15\!\cdots\!09$$$$T^{30} -$$$$23\!\cdots\!02$$$$T^{31} -$$$$76\!\cdots\!65$$$$T^{32} +$$$$76\!\cdots\!64$$$$T^{33} +$$$$97\!\cdots\!33$$$$T^{34} +$$$$31\!\cdots\!68$$$$T^{35} -$$$$30\!\cdots\!45$$$$T^{36} -$$$$32\!\cdots\!06$$$$T^{37} -$$$$42\!\cdots\!22$$$$T^{38} +$$$$79\!\cdots\!48$$$$T^{39} +$$$$33\!\cdots\!01$$$$T^{40}$$
$71$ $$1 - T - 374 T^{2} + 385 T^{3} + 59424 T^{4} - 61198 T^{5} - 4274911 T^{6} + 3121075 T^{7} - 5229794 T^{8} + 450375939 T^{9} + 21565718974 T^{10} - 89365774783 T^{11} - 422299011827 T^{12} + 5204430801845 T^{13} - 172828405013793 T^{14} + 177634632473756 T^{15} + 14915998344657807 T^{16} - 43484081312487565 T^{17} - 30084719421630102 T^{18} + 1740254535967054397 T^{19} - 48390764100475009279 T^{20} +$$$$12\!\cdots\!87$$$$T^{21} -$$$$15\!\cdots\!82$$$$T^{22} -$$$$15\!\cdots\!15$$$$T^{23} +$$$$37\!\cdots\!67$$$$T^{24} +$$$$32\!\cdots\!56$$$$T^{25} -$$$$22\!\cdots\!53$$$$T^{26} +$$$$47\!\cdots\!95$$$$T^{27} -$$$$27\!\cdots\!47$$$$T^{28} -$$$$40\!\cdots\!73$$$$T^{29} +$$$$70\!\cdots\!74$$$$T^{30} +$$$$10\!\cdots\!69$$$$T^{31} -$$$$85\!\cdots\!54$$$$T^{32} +$$$$36\!\cdots\!25$$$$T^{33} -$$$$35\!\cdots\!91$$$$T^{34} -$$$$35\!\cdots\!98$$$$T^{35} +$$$$24\!\cdots\!04$$$$T^{36} +$$$$11\!\cdots\!35$$$$T^{37} -$$$$78\!\cdots\!14$$$$T^{38} -$$$$14\!\cdots\!31$$$$T^{39} +$$$$10\!\cdots\!01$$$$T^{40}$$
$73$ $$1 - 19 T - 37 T^{2} + 1911 T^{3} + 14984 T^{4} - 343095 T^{5} + 44886 T^{6} + 20424421 T^{7} + 38131320 T^{8} - 2107575143 T^{9} + 6644560253 T^{10} + 31127895089 T^{11} + 163541353759 T^{12} - 996108478075 T^{13} + 8406248222548 T^{14} - 631161430088471 T^{15} + 5329956394145463 T^{16} + 17871030714682627 T^{17} - 359037128943857193 T^{18} - 2008361662490082115 T^{19} + 51740292756835167037 T^{20} -$$$$14\!\cdots\!95$$$$T^{21} -$$$$19\!\cdots\!97$$$$T^{22} +$$$$69\!\cdots\!59$$$$T^{23} +$$$$15\!\cdots\!83$$$$T^{24} -$$$$13\!\cdots\!03$$$$T^{25} +$$$$12\!\cdots\!72$$$$T^{26} -$$$$11\!\cdots\!75$$$$T^{27} +$$$$13\!\cdots\!79$$$$T^{28} +$$$$18\!\cdots\!57$$$$T^{29} +$$$$28\!\cdots\!97$$$$T^{30} -$$$$66\!\cdots\!11$$$$T^{31} +$$$$87\!\cdots\!20$$$$T^{32} +$$$$34\!\cdots\!93$$$$T^{33} +$$$$54\!\cdots\!74$$$$T^{34} -$$$$30\!\cdots\!15$$$$T^{35} +$$$$97\!\cdots\!24$$$$T^{36} +$$$$90\!\cdots\!83$$$$T^{37} -$$$$12\!\cdots\!53$$$$T^{38} -$$$$48\!\cdots\!03$$$$T^{39} +$$$$18\!\cdots\!01$$$$T^{40}$$
$79$ $$1 + 64 T + 1813 T^{2} + 28300 T^{3} + 220923 T^{4} - 272960 T^{5} - 26038614 T^{6} - 272781482 T^{7} - 1191792493 T^{8} - 1015821708 T^{9} + 10837883272 T^{10} + 850456415848 T^{11} + 23123314759077 T^{12} + 258076348414252 T^{13} + 716331205356126 T^{14} - 13571832494284164 T^{15} - 144937325341815863 T^{16} - 90759777606546922 T^{17} + 6424026371015525703 T^{18} + 21997663317142103344 T^{19} -$$$$14\!\cdots\!34$$$$T^{20} +$$$$17\!\cdots\!76$$$$T^{21} +$$$$40\!\cdots\!23$$$$T^{22} -$$$$44\!\cdots\!58$$$$T^{23} -$$$$56\!\cdots\!03$$$$T^{24} -$$$$41\!\cdots\!36$$$$T^{25} +$$$$17\!\cdots\!46$$$$T^{26} +$$$$49\!\cdots\!68$$$$T^{27} +$$$$35\!\cdots\!97$$$$T^{28} +$$$$10\!\cdots\!12$$$$T^{29} +$$$$10\!\cdots\!72$$$$T^{30} -$$$$75\!\cdots\!32$$$$T^{31} -$$$$70\!\cdots\!13$$$$T^{32} -$$$$12\!\cdots\!98$$$$T^{33} -$$$$96\!\cdots\!34$$$$T^{34} -$$$$79\!\cdots\!40$$$$T^{35} +$$$$50\!\cdots\!83$$$$T^{36} +$$$$51\!\cdots\!00$$$$T^{37} +$$$$26\!\cdots\!93$$$$T^{38} +$$$$72\!\cdots\!16$$$$T^{39} +$$$$89\!\cdots\!01$$$$T^{40}$$
$83$ $$1 - 57 T + 1531 T^{2} - 25602 T^{3} + 283378 T^{4} - 1661127 T^{5} - 8691182 T^{6} + 349426040 T^{7} - 4444505934 T^{8} + 28616053427 T^{9} + 47050780495 T^{10} - 3583740129277 T^{11} + 45464734870983 T^{12} - 281124959027648 T^{13} - 395656472000900 T^{14} + 30726600590053373 T^{15} - 378959258485740125 T^{16} + 2393776145680167358 T^{17} + 572766742440286311 T^{18} -$$$$20\!\cdots\!17$$$$T^{19} +$$$$26\!\cdots\!27$$$$T^{20} -$$$$17\!\cdots\!11$$$$T^{21} +$$$$39\!\cdots\!79$$$$T^{22} +$$$$13\!\cdots\!46$$$$T^{23} -$$$$17\!\cdots\!25$$$$T^{24} +$$$$12\!\cdots\!39$$$$T^{25} -$$$$12\!\cdots\!00$$$$T^{26} -$$$$76\!\cdots\!96$$$$T^{27} +$$$$10\!\cdots\!03$$$$T^{28} -$$$$66\!\cdots\!31$$$$T^{29} +$$$$73\!\cdots\!55$$$$T^{30} +$$$$36\!\cdots\!09$$$$T^{31} -$$$$47\!\cdots\!74$$$$T^{32} +$$$$31\!\cdots\!20$$$$T^{33} -$$$$63\!\cdots\!78$$$$T^{34} -$$$$10\!\cdots\!89$$$$T^{35} +$$$$14\!\cdots\!18$$$$T^{36} -$$$$10\!\cdots\!46$$$$T^{37} +$$$$53\!\cdots\!79$$$$T^{38} -$$$$16\!\cdots\!79$$$$T^{39} +$$$$24\!\cdots\!01$$$$T^{40}$$
$89$ $$1 + 6 T - 225 T^{2} - 2414 T^{3} + 10475 T^{4} + 275574 T^{5} + 2441593 T^{6} + 4668416 T^{7} - 252701912 T^{8} - 3560779949 T^{9} - 8899915733 T^{10} + 174253060095 T^{11} + 1975857342718 T^{12} + 12822124485500 T^{13} + 98844654551923 T^{14} - 556869091622998 T^{15} - 19095987187955715 T^{16} - 193768608185247050 T^{17} - 611163344428279505 T^{18} + 12505565957375472620 T^{19} +$$$$22\!\cdots\!40$$$$T^{20} +$$$$11\!\cdots\!80$$$$T^{21} -$$$$48\!\cdots\!05$$$$T^{22} -$$$$13\!\cdots\!50$$$$T^{23} -$$$$11\!\cdots\!15$$$$T^{24} -$$$$31\!\cdots\!02$$$$T^{25} +$$$$49\!\cdots\!03$$$$T^{26} +$$$$56\!\cdots\!00$$$$T^{27} +$$$$77\!\cdots\!58$$$$T^{28} +$$$$61\!\cdots\!55$$$$T^{29} -$$$$27\!\cdots\!33$$$$T^{30} -$$$$98\!\cdots\!61$$$$T^{31} -$$$$62\!\cdots\!52$$$$T^{32} +$$$$10\!\cdots\!04$$$$T^{33} +$$$$47\!\cdots\!13$$$$T^{34} +$$$$47\!\cdots\!26$$$$T^{35} +$$$$16\!\cdots\!75$$$$T^{36} -$$$$33\!\cdots\!06$$$$T^{37} -$$$$27\!\cdots\!25$$$$T^{38} +$$$$65\!\cdots\!54$$$$T^{39} +$$$$97\!\cdots\!01$$$$T^{40}$$
$97$ $$1 + 18 T - 181 T^{2} - 5224 T^{3} - 12425 T^{4} + 240780 T^{5} + 3449805 T^{6} + 82687874 T^{7} + 326931282 T^{8} - 11449359402 T^{9} - 99823613357 T^{10} - 119817410092 T^{11} + 1031495320729 T^{12} + 126529834617016 T^{13} + 1496866997685305 T^{14} - 6508878292176770 T^{15} - 128528710934989580 T^{16} - 589248127879190526 T^{17} - 8372017197617105093 T^{18} + 50791694907309315928 T^{19} +$$$$20\!\cdots\!29$$$$T^{20} +$$$$49\!\cdots\!16$$$$T^{21} -$$$$78\!\cdots\!37$$$$T^{22} -$$$$53\!\cdots\!98$$$$T^{23} -$$$$11\!\cdots\!80$$$$T^{24} -$$$$55\!\cdots\!90$$$$T^{25} +$$$$12\!\cdots\!45$$$$T^{26} +$$$$10\!\cdots\!08$$$$T^{27} +$$$$80\!\cdots\!69$$$$T^{28} -$$$$91\!\cdots\!64$$$$T^{29} -$$$$73\!\cdots\!93$$$$T^{30} -$$$$81\!\cdots\!06$$$$T^{31} +$$$$22\!\cdots\!62$$$$T^{32} +$$$$55\!\cdots\!98$$$$T^{33} +$$$$22\!\cdots\!45$$$$T^{34} +$$$$15\!\cdots\!40$$$$T^{35} -$$$$76\!\cdots\!25$$$$T^{36} -$$$$31\!\cdots\!88$$$$T^{37} -$$$$10\!\cdots\!09$$$$T^{38} +$$$$10\!\cdots\!94$$$$T^{39} +$$$$54\!\cdots\!01$$$$T^{40}$$