# Properties

 Label 231.2.j.g Level 231 Weight 2 Character orbit 231.j Analytic conductor 1.845 Analytic rank 0 Dimension 20 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 231.j (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$5$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{2} -\beta_{4} q^{3} + ( \beta_{3} - \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{19} ) q^{4} + \beta_{13} q^{5} -\beta_{1} q^{6} + \beta_{9} q^{7} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{10} - \beta_{14} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{8} -\beta_{6} q^{9} +O(q^{10})$$ $$q + \beta_{7} q^{2} -\beta_{4} q^{3} + ( \beta_{3} - \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{19} ) q^{4} + \beta_{13} q^{5} -\beta_{1} q^{6} + \beta_{9} q^{7} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{10} - \beta_{14} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{8} -\beta_{6} q^{9} + ( \beta_{3} - \beta_{7} - \beta_{8} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{18} ) q^{10} + ( -\beta_{1} + \beta_{2} - \beta_{8} - \beta_{9} + \beta_{12} + \beta_{15} ) q^{11} + ( 2 - \beta_{3} ) q^{12} + ( 1 + \beta_{2} + \beta_{5} - \beta_{9} + \beta_{19} ) q^{13} + ( -\beta_{1} + \beta_{2} + \beta_{5} + \beta_{7} ) q^{14} -\beta_{15} q^{15} + ( -4 - 3 \beta_{1} + \beta_{2} + 5 \beta_{4} + \beta_{5} + 4 \beta_{6} + 5 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{16} + ( 1 - \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{18} ) q^{17} + \beta_{2} q^{18} + ( \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} + \beta_{14} + \beta_{17} ) q^{19} + ( -3 - \beta_{2} - \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + 5 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} - 2 \beta_{15} - \beta_{19} ) q^{20} - q^{21} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{10} + \beta_{11} - \beta_{13} - \beta_{15} + \beta_{16} + \beta_{18} - \beta_{19} ) q^{22} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - 3 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} - 2 \beta_{16} - \beta_{17} ) q^{23} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} ) q^{24} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{14} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{25} + ( 1 + 2 \beta_{1} - \beta_{2} - 5 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} + \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} + \beta_{18} - \beta_{19} ) q^{26} + ( -1 + \beta_{4} + \beta_{6} + \beta_{9} ) q^{27} + ( 2 - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{9} + \beta_{18} ) q^{28} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{8} - 3 \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{29} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} + \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{16} ) q^{30} + ( -1 + \beta_{2} + \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} + \beta_{15} + \beta_{16} - \beta_{17} - \beta_{19} ) q^{31} + ( 3 + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - 2 \beta_{11} + \beta_{13} + 2 \beta_{14} + \beta_{15} - 2 \beta_{16} - \beta_{17} - 2 \beta_{18} ) q^{32} + ( 1 - \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} + \beta_{10} - \beta_{16} + \beta_{18} ) q^{33} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{4} - 4 \beta_{5} - \beta_{6} + \beta_{8} + 3 \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{16} + \beta_{17} - \beta_{18} ) q^{34} -\beta_{10} q^{35} + ( -2 \beta_{4} + \beta_{6} + \beta_{10} + \beta_{18} - \beta_{19} ) q^{36} + ( -1 - \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - 3 \beta_{9} - \beta_{11} + 2 \beta_{12} + 2 \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{18} + \beta_{19} ) q^{37} + ( -1 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{10} + 4 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} + 4 \beta_{17} + \beta_{18} + \beta_{19} ) q^{38} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{13} - \beta_{19} ) q^{39} + ( -6 \beta_{1} + 6 \beta_{2} - \beta_{3} + 3 \beta_{4} + 6 \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} + 3 \beta_{15} - \beta_{16} - \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{40} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{8} ) q^{41} -\beta_{7} q^{42} + ( \beta_{1} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{18} ) q^{43} + ( 1 - 2 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} + \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} - 2 \beta_{18} ) q^{44} + ( 1 - \beta_{11} ) q^{45} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{12} + \beta_{13} + \beta_{14} + 3 \beta_{15} - \beta_{16} - \beta_{17} - \beta_{19} ) q^{46} + ( 2 \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} - 2 \beta_{8} - 3 \beta_{10} + 2 \beta_{11} - \beta_{13} + \beta_{15} - \beta_{18} + \beta_{19} ) q^{47} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{8} - 5 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} + \beta_{19} ) q^{48} + ( -1 + \beta_{4} + \beta_{6} + \beta_{9} ) q^{49} + ( -4 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 4 \beta_{4} - 4 \beta_{6} - \beta_{7} - 3 \beta_{9} - 2 \beta_{10} + \beta_{15} + \beta_{19} ) q^{50} + ( \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{19} ) q^{51} + ( -6 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - \beta_{10} + 2 \beta_{12} + \beta_{13} - \beta_{15} + 2 \beta_{16} - \beta_{18} + \beta_{19} ) q^{52} + ( 1 + 2 \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{53} + \beta_{5} q^{54} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{19} ) q^{55} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{56} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{5} + \beta_{6} + 2 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{16} - \beta_{17} ) q^{57} + ( 3 + 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{10} - \beta_{11} + \beta_{13} - 2 \beta_{14} - \beta_{15} - 2 \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{58} + ( 1 + \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} - 2 \beta_{17} - \beta_{18} - \beta_{19} ) q^{59} + ( -4 - \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{9} - 3 \beta_{10} + 2 \beta_{15} - 2 \beta_{18} + \beta_{19} ) q^{60} + ( -2 + 3 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{7} - \beta_{9} - \beta_{10} - \beta_{13} + 2 \beta_{15} - 2 \beta_{18} - \beta_{19} ) q^{61} + ( -2 - \beta_{1} + 3 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{9} + 2 \beta_{11} - 2 \beta_{13} - 2 \beta_{15} + \beta_{18} ) q^{62} + \beta_{4} q^{63} + ( -3 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{17} + \beta_{19} ) q^{64} + ( 1 + 5 \beta_{1} - 4 \beta_{2} + \beta_{3} - 2 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + 4 \beta_{11} - 2 \beta_{12} - \beta_{13} - 2 \beta_{14} - 2 \beta_{15} + 4 \beta_{16} + 2 \beta_{17} - 2 \beta_{18} ) q^{65} + ( 2 - 3 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} + \beta_{17} ) q^{66} + ( 5 - \beta_{3} - \beta_{5} + \beta_{7} + \beta_{10} - \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{67} + ( 2 + 3 \beta_{2} + 3 \beta_{5} + 8 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{11} - \beta_{15} + \beta_{19} ) q^{68} + ( -1 - \beta_{1} + \beta_{4} + \beta_{7} - \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{17} ) q^{69} + ( -1 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{5} + 3 \beta_{11} - \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{70} + ( 1 - 5 \beta_{1} + 4 \beta_{2} - \beta_{3} + \beta_{5} + 5 \beta_{6} + 3 \beta_{7} + 5 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} - 2 \beta_{17} + \beta_{18} - \beta_{19} ) q^{71} + ( -1 - \beta_{1} + \beta_{9} - \beta_{10} + \beta_{14} + \beta_{16} - \beta_{18} ) q^{72} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} - 2 \beta_{15} + \beta_{18} - \beta_{19} ) q^{73} + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{3} - 5 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + \beta_{8} + \beta_{10} + 2 \beta_{14} + 2 \beta_{17} + \beta_{18} - \beta_{19} ) q^{74} + ( -2 \beta_{1} - \beta_{5} - 3 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} + \beta_{19} ) q^{75} + ( 5 - 2 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 6 \beta_{9} - \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} + 2 \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{76} + ( 1 - \beta_{4} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{14} + \beta_{18} - \beta_{19} ) q^{77} + ( -4 - 2 \beta_{1} + 5 \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{8} + 5 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{13} - \beta_{18} ) q^{78} + ( 1 + \beta_{6} - 3 \beta_{7} + \beta_{10} - \beta_{11} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{79} + ( 11 + \beta_{1} - \beta_{2} - 3 \beta_{3} - 7 \beta_{4} + \beta_{5} - 10 \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{10} - \beta_{13} + \beta_{15} + \beta_{18} - \beta_{19} ) q^{80} -\beta_{9} q^{81} + ( -4 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} + 2 \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{82} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{15} + 2 \beta_{18} ) q^{83} + ( -\beta_{3} + \beta_{6} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{19} ) q^{84} + ( -3 - \beta_{1} + 2 \beta_{2} + 4 \beta_{4} + \beta_{5} + 4 \beta_{6} - 3 \beta_{11} - \beta_{12} + \beta_{14} - \beta_{16} + \beta_{17} - 3 \beta_{18} + 3 \beta_{19} ) q^{85} + ( -2 + \beta_{2} + \beta_{5} + 4 \beta_{6} - 2 \beta_{8} - 3 \beta_{10} + 2 \beta_{11} + 2 \beta_{15} ) q^{86} + ( 2 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} + \beta_{18} ) q^{87} + ( 13 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 7 \beta_{6} + 2 \beta_{7} - \beta_{8} - 8 \beta_{9} - 3 \beta_{10} - \beta_{11} + \beta_{13} + \beta_{15} - 2 \beta_{16} - \beta_{17} + 2 \beta_{19} ) q^{88} + ( -1 - 3 \beta_{1} + \beta_{3} - \beta_{4} + 3 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{13} + \beta_{18} ) q^{89} + ( -1 - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} + \beta_{15} + \beta_{16} - \beta_{17} - \beta_{19} ) q^{90} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{8} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{15} ) q^{91} + ( -3 - 3 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} + 6 \beta_{9} + 3 \beta_{11} + \beta_{12} - 3 \beta_{13} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{92} + ( -3 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{15} + \beta_{16} + 2 \beta_{17} + \beta_{19} ) q^{93} + ( -\beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{7} - \beta_{9} + 3 \beta_{10} - \beta_{13} - 2 \beta_{14} - 2 \beta_{15} - 2 \beta_{16} + 2 \beta_{18} + \beta_{19} ) q^{94} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - 5 \beta_{4} + \beta_{5} - 7 \beta_{6} - 2 \beta_{8} - 5 \beta_{9} - 2 \beta_{10} - 5 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} + 4 \beta_{14} + \beta_{15} - 2 \beta_{16} - 2 \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{95} + ( -2 + 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{10} + \beta_{12} - \beta_{14} + \beta_{16} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{96} + ( -1 + \beta_{2} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{9} + \beta_{19} ) q^{97} + \beta_{5} q^{98} + ( 1 - \beta_{3} - \beta_{4} + \beta_{8} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{17} - \beta_{19} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 5q^{3} - 14q^{4} - 5q^{5} + 5q^{7} - 9q^{8} - 5q^{9} + O(q^{10})$$ $$20q - 5q^{3} - 14q^{4} - 5q^{5} + 5q^{7} - 9q^{8} - 5q^{9} + 12q^{10} - q^{11} + 36q^{12} + 13q^{13} - 24q^{16} - q^{17} + 10q^{19} - 46q^{20} - 20q^{21} + 26q^{22} + 6q^{24} - 8q^{25} - 53q^{26} - 5q^{27} + 4q^{28} + 3q^{29} - 3q^{30} - 13q^{31} + 82q^{32} + 9q^{33} + 42q^{34} + 5q^{35} - 14q^{36} - 32q^{37} + 16q^{38} + 13q^{39} + 20q^{40} - 3q^{41} + 12q^{43} + 25q^{44} + 10q^{45} - 13q^{46} + 20q^{47} - 14q^{48} - 5q^{49} - 83q^{50} + 9q^{51} - 80q^{52} + 3q^{53} - 28q^{55} - 6q^{56} - 10q^{57} + 2q^{58} - 9q^{59} - 46q^{60} - 15q^{61} - 37q^{62} + 5q^{63} - 49q^{64} + 58q^{65} - 4q^{66} + 76q^{67} + 51q^{68} + 3q^{70} + 37q^{71} + 6q^{72} + 27q^{73} - 32q^{74} - 23q^{75} + 4q^{76} + 6q^{77} + 2q^{78} + 5q^{79} + 137q^{80} - 5q^{81} - 55q^{82} - 42q^{83} + 14q^{84} - 48q^{85} + 3q^{86} + 28q^{87} + 151q^{88} - 18q^{89} - 3q^{90} + 7q^{91} + 39q^{92} - 13q^{93} - 35q^{94} - 96q^{95} - 48q^{96} - 27q^{97} - q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 12 x^{18} - 3 x^{17} + 94 x^{16} - 10 x^{15} + 662 x^{14} - 153 x^{13} + 4638 x^{12} - 1174 x^{11} + 15808 x^{10} - 3393 x^{9} + 26062 x^{8} - 15494 x^{7} + 11660 x^{6} - 33295 x^{5} + 67756 x^{4} - 43696 x^{3} + 13084 x^{2} - 1155 x + 121$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$46\!\cdots\!50$$$$\nu^{19} -$$$$13\!\cdots\!96$$$$\nu^{18} -$$$$60\!\cdots\!30$$$$\nu^{17} -$$$$15\!\cdots\!69$$$$\nu^{16} -$$$$45\!\cdots\!19$$$$\nu^{15} -$$$$12\!\cdots\!18$$$$\nu^{14} -$$$$34\!\cdots\!39$$$$\nu^{13} -$$$$86\!\cdots\!29$$$$\nu^{12} -$$$$23\!\cdots\!29$$$$\nu^{11} -$$$$59\!\cdots\!21$$$$\nu^{10} -$$$$83\!\cdots\!87$$$$\nu^{9} -$$$$20\!\cdots\!82$$$$\nu^{8} -$$$$16\!\cdots\!59$$$$\nu^{7} -$$$$30\!\cdots\!49$$$$\nu^{6} -$$$$15\!\cdots\!62$$$$\nu^{5} +$$$$14\!\cdots\!80$$$$\nu^{4} -$$$$30\!\cdots\!18$$$$\nu^{3} -$$$$60\!\cdots\!57$$$$\nu^{2} +$$$$15\!\cdots\!72$$$$\nu +$$$$53\!\cdots\!27$$$$)/$$$$18\!\cdots\!60$$ $$\beta_{3}$$ $$=$$ $$($$$$15\!\cdots\!62$$$$\nu^{19} +$$$$30\!\cdots\!22$$$$\nu^{18} +$$$$18\!\cdots\!69$$$$\nu^{17} -$$$$20\!\cdots\!56$$$$\nu^{16} +$$$$14\!\cdots\!57$$$$\nu^{15} +$$$$20\!\cdots\!07$$$$\nu^{14} +$$$$10\!\cdots\!38$$$$\nu^{13} +$$$$27\!\cdots\!56$$$$\nu^{12} +$$$$70\!\cdots\!24$$$$\nu^{11} +$$$$25\!\cdots\!52$$$$\nu^{10} +$$$$24\!\cdots\!77$$$$\nu^{9} +$$$$23\!\cdots\!21$$$$\nu^{8} +$$$$40\!\cdots\!30$$$$\nu^{7} -$$$$71\!\cdots\!17$$$$\nu^{6} +$$$$17\!\cdots\!86$$$$\nu^{5} -$$$$35\!\cdots\!36$$$$\nu^{4} +$$$$91\!\cdots\!89$$$$\nu^{3} -$$$$42\!\cdots\!65$$$$\nu^{2} +$$$$35\!\cdots\!19$$$$\nu +$$$$73\!\cdots\!95$$$$)/$$$$18\!\cdots\!60$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$48\!\cdots\!57$$$$\nu^{19} -$$$$50\!\cdots\!50$$$$\nu^{18} -$$$$59\!\cdots\!40$$$$\nu^{17} +$$$$77\!\cdots\!41$$$$\nu^{16} -$$$$47\!\cdots\!17$$$$\nu^{15} -$$$$22\!\cdots\!39$$$$\nu^{14} -$$$$33\!\cdots\!32$$$$\nu^{13} +$$$$35\!\cdots\!92$$$$\nu^{12} -$$$$23\!\cdots\!85$$$$\nu^{11} +$$$$31\!\cdots\!99$$$$\nu^{10} -$$$$82\!\cdots\!87$$$$\nu^{9} +$$$$72\!\cdots\!44$$$$\nu^{8} -$$$$14\!\cdots\!36$$$$\nu^{7} +$$$$56\!\cdots\!09$$$$\nu^{6} -$$$$89\!\cdots\!59$$$$\nu^{5} +$$$$15\!\cdots\!33$$$$\nu^{4} -$$$$32\!\cdots\!12$$$$\nu^{3} +$$$$20\!\cdots\!74$$$$\nu^{2} -$$$$12\!\cdots\!15$$$$\nu +$$$$22\!\cdots\!27$$$$)/$$$$20\!\cdots\!60$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$49\!\cdots\!48$$$$\nu^{19} -$$$$12\!\cdots\!79$$$$\nu^{18} -$$$$59\!\cdots\!87$$$$\nu^{17} -$$$$80\!\cdots\!48$$$$\nu^{16} -$$$$46\!\cdots\!65$$$$\nu^{15} -$$$$72\!\cdots\!49$$$$\nu^{14} -$$$$32\!\cdots\!53$$$$\nu^{13} -$$$$10\!\cdots\!03$$$$\nu^{12} -$$$$22\!\cdots\!49$$$$\nu^{11} -$$$$24\!\cdots\!00$$$$\nu^{10} -$$$$77\!\cdots\!99$$$$\nu^{9} -$$$$39\!\cdots\!85$$$$\nu^{8} -$$$$12\!\cdots\!08$$$$\nu^{7} +$$$$41\!\cdots\!48$$$$\nu^{6} -$$$$47\!\cdots\!98$$$$\nu^{5} +$$$$14\!\cdots\!81$$$$\nu^{4} -$$$$29\!\cdots\!98$$$$\nu^{3} +$$$$13\!\cdots\!43$$$$\nu^{2} -$$$$11\!\cdots\!86$$$$\nu -$$$$68\!\cdots\!52$$$$)/$$$$18\!\cdots\!60$$ $$\beta_{6}$$ $$=$$ $$($$$$62\!\cdots\!32$$$$\nu^{19} -$$$$54\!\cdots\!28$$$$\nu^{18} +$$$$73\!\cdots\!15$$$$\nu^{17} -$$$$83\!\cdots\!53$$$$\nu^{16} +$$$$58\!\cdots\!80$$$$\nu^{15} -$$$$56\!\cdots\!35$$$$\nu^{14} +$$$$40\!\cdots\!45$$$$\nu^{13} -$$$$45\!\cdots\!79$$$$\nu^{12} +$$$$28\!\cdots\!83$$$$\nu^{11} -$$$$32\!\cdots\!07$$$$\nu^{10} +$$$$97\!\cdots\!56$$$$\nu^{9} -$$$$10\!\cdots\!65$$$$\nu^{8} +$$$$15\!\cdots\!49$$$$\nu^{7} -$$$$23\!\cdots\!96$$$$\nu^{6} +$$$$11\!\cdots\!48$$$$\nu^{5} -$$$$25\!\cdots\!18$$$$\nu^{4} +$$$$58\!\cdots\!83$$$$\nu^{3} -$$$$59\!\cdots\!50$$$$\nu^{2} +$$$$22\!\cdots\!61$$$$\nu -$$$$19\!\cdots\!06$$$$)/$$$$20\!\cdots\!60$$ $$\beta_{7}$$ $$=$$ $$($$$$56\!\cdots\!45$$$$\nu^{19} +$$$$15\!\cdots\!62$$$$\nu^{18} +$$$$68\!\cdots\!62$$$$\nu^{17} +$$$$13\!\cdots\!34$$$$\nu^{16} +$$$$53\!\cdots\!74$$$$\nu^{15} +$$$$86\!\cdots\!07$$$$\nu^{14} +$$$$37\!\cdots\!97$$$$\nu^{13} +$$$$14\!\cdots\!53$$$$\nu^{12} +$$$$26\!\cdots\!66$$$$\nu^{11} +$$$$43\!\cdots\!94$$$$\nu^{10} +$$$$89\!\cdots\!12$$$$\nu^{9} +$$$$50\!\cdots\!92$$$$\nu^{8} +$$$$14\!\cdots\!11$$$$\nu^{7} -$$$$46\!\cdots\!00$$$$\nu^{6} +$$$$58\!\cdots\!83$$$$\nu^{5} -$$$$17\!\cdots\!89$$$$\nu^{4} +$$$$34\!\cdots\!84$$$$\nu^{3} -$$$$15\!\cdots\!31$$$$\nu^{2} +$$$$31\!\cdots\!15$$$$\nu -$$$$29\!\cdots\!56$$$$)/$$$$18\!\cdots\!60$$ $$\beta_{8}$$ $$=$$ $$($$$$26\!\cdots\!84$$$$\nu^{19} -$$$$21\!\cdots\!82$$$$\nu^{18} +$$$$31\!\cdots\!69$$$$\nu^{17} -$$$$33\!\cdots\!03$$$$\nu^{16} +$$$$24\!\cdots\!18$$$$\nu^{15} -$$$$22\!\cdots\!11$$$$\nu^{14} +$$$$17\!\cdots\!49$$$$\nu^{13} -$$$$17\!\cdots\!97$$$$\nu^{12} +$$$$12\!\cdots\!63$$$$\nu^{11} -$$$$12\!\cdots\!71$$$$\nu^{10} +$$$$41\!\cdots\!76$$$$\nu^{9} -$$$$41\!\cdots\!11$$$$\nu^{8} +$$$$67\!\cdots\!35$$$$\nu^{7} -$$$$95\!\cdots\!02$$$$\nu^{6} +$$$$48\!\cdots\!62$$$$\nu^{5} -$$$$10\!\cdots\!74$$$$\nu^{4} +$$$$24\!\cdots\!59$$$$\nu^{3} -$$$$24\!\cdots\!92$$$$\nu^{2} +$$$$91\!\cdots\!97$$$$\nu -$$$$78\!\cdots\!74$$$$)/$$$$20\!\cdots\!60$$ $$\beta_{9}$$ $$=$$ $$($$$$26\!\cdots\!96$$$$\nu^{19} +$$$$62\!\cdots\!95$$$$\nu^{18} +$$$$32\!\cdots\!34$$$$\nu^{17} -$$$$52\!\cdots\!06$$$$\nu^{16} +$$$$25\!\cdots\!98$$$$\nu^{15} +$$$$31\!\cdots\!54$$$$\nu^{14} +$$$$17\!\cdots\!29$$$$\nu^{13} +$$$$48\!\cdots\!79$$$$\nu^{12} +$$$$12\!\cdots\!31$$$$\nu^{11} -$$$$25\!\cdots\!78$$$$\nu^{10} +$$$$42\!\cdots\!02$$$$\nu^{9} +$$$$76\!\cdots\!04$$$$\nu^{8} +$$$$70\!\cdots\!64$$$$\nu^{7} -$$$$24\!\cdots\!03$$$$\nu^{6} +$$$$26\!\cdots\!60$$$$\nu^{5} -$$$$82\!\cdots\!07$$$$\nu^{4} +$$$$16\!\cdots\!97$$$$\nu^{3} -$$$$78\!\cdots\!92$$$$\nu^{2} +$$$$17\!\cdots\!23$$$$\nu +$$$$41\!\cdots\!85$$$$)/$$$$20\!\cdots\!60$$ $$\beta_{10}$$ $$=$$ $$($$$$29\!\cdots\!75$$$$\nu^{19} +$$$$23\!\cdots\!79$$$$\nu^{18} +$$$$36\!\cdots\!54$$$$\nu^{17} +$$$$18\!\cdots\!93$$$$\nu^{16} +$$$$27\!\cdots\!13$$$$\nu^{15} +$$$$18\!\cdots\!09$$$$\nu^{14} +$$$$19\!\cdots\!39$$$$\nu^{13} +$$$$10\!\cdots\!31$$$$\nu^{12} +$$$$13\!\cdots\!92$$$$\nu^{11} +$$$$70\!\cdots\!93$$$$\nu^{10} +$$$$46\!\cdots\!19$$$$\nu^{9} +$$$$24\!\cdots\!74$$$$\nu^{8} +$$$$77\!\cdots\!92$$$$\nu^{7} +$$$$84\!\cdots\!80$$$$\nu^{6} +$$$$87\!\cdots\!81$$$$\nu^{5} -$$$$89\!\cdots\!48$$$$\nu^{4} +$$$$12\!\cdots\!03$$$$\nu^{3} +$$$$22\!\cdots\!58$$$$\nu^{2} -$$$$18\!\cdots\!70$$$$\nu +$$$$14\!\cdots\!88$$$$)/$$$$20\!\cdots\!60$$ $$\beta_{11}$$ $$=$$ $$($$$$30\!\cdots\!21$$$$\nu^{19} +$$$$77\!\cdots\!28$$$$\nu^{18} +$$$$35\!\cdots\!64$$$$\nu^{17} -$$$$14\!\cdots\!09$$$$\nu^{16} +$$$$27\!\cdots\!25$$$$\nu^{15} +$$$$37\!\cdots\!73$$$$\nu^{14} +$$$$19\!\cdots\!06$$$$\nu^{13} +$$$$10\!\cdots\!66$$$$\nu^{12} +$$$$13\!\cdots\!73$$$$\nu^{11} -$$$$22\!\cdots\!05$$$$\nu^{10} +$$$$45\!\cdots\!83$$$$\nu^{9} -$$$$14\!\cdots\!80$$$$\nu^{8} +$$$$70\!\cdots\!36$$$$\nu^{7} -$$$$34\!\cdots\!71$$$$\nu^{6} +$$$$12\!\cdots\!31$$$$\nu^{5} -$$$$10\!\cdots\!97$$$$\nu^{4} +$$$$17\!\cdots\!06$$$$\nu^{3} -$$$$74\!\cdots\!46$$$$\nu^{2} +$$$$63\!\cdots\!67$$$$\nu -$$$$89\!\cdots\!91$$$$)/$$$$18\!\cdots\!60$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$39\!\cdots\!52$$$$\nu^{19} -$$$$25\!\cdots\!75$$$$\nu^{18} -$$$$47\!\cdots\!43$$$$\nu^{17} -$$$$18\!\cdots\!25$$$$\nu^{16} -$$$$37\!\cdots\!34$$$$\nu^{15} -$$$$19\!\cdots\!87$$$$\nu^{14} -$$$$26\!\cdots\!48$$$$\nu^{13} -$$$$10\!\cdots\!02$$$$\nu^{12} -$$$$18\!\cdots\!16$$$$\nu^{11} -$$$$70\!\cdots\!91$$$$\nu^{10} -$$$$62\!\cdots\!08$$$$\nu^{9} -$$$$25\!\cdots\!95$$$$\nu^{8} -$$$$10\!\cdots\!13$$$$\nu^{7} -$$$$26\!\cdots\!17$$$$\nu^{6} -$$$$20\!\cdots\!30$$$$\nu^{5} +$$$$11\!\cdots\!45$$$$\nu^{4} -$$$$18\!\cdots\!44$$$$\nu^{3} +$$$$24\!\cdots\!36$$$$\nu^{2} +$$$$18\!\cdots\!96$$$$\nu -$$$$74\!\cdots\!47$$$$)/$$$$20\!\cdots\!60$$ $$\beta_{13}$$ $$=$$ $$($$$$94\!\cdots\!92$$$$\nu^{19} -$$$$62\!\cdots\!32$$$$\nu^{18} +$$$$11\!\cdots\!08$$$$\nu^{17} -$$$$37\!\cdots\!85$$$$\nu^{16} +$$$$86\!\cdots\!69$$$$\nu^{15} -$$$$16\!\cdots\!18$$$$\nu^{14} +$$$$60\!\cdots\!27$$$$\nu^{13} -$$$$19\!\cdots\!45$$$$\nu^{12} +$$$$42\!\cdots\!45$$$$\nu^{11} -$$$$14\!\cdots\!45$$$$\nu^{10} +$$$$14\!\cdots\!63$$$$\nu^{9} -$$$$46\!\cdots\!68$$$$\nu^{8} +$$$$21\!\cdots\!11$$$$\nu^{7} -$$$$17\!\cdots\!67$$$$\nu^{6} +$$$$62\!\cdots\!74$$$$\nu^{5} -$$$$32\!\cdots\!38$$$$\nu^{4} +$$$$64\!\cdots\!56$$$$\nu^{3} -$$$$39\!\cdots\!87$$$$\nu^{2} +$$$$70\!\cdots\!34$$$$\nu -$$$$43\!\cdots\!83$$$$)/$$$$40\!\cdots\!12$$ $$\beta_{14}$$ $$=$$ $$($$$$50\!\cdots\!16$$$$\nu^{19} +$$$$13\!\cdots\!36$$$$\nu^{18} +$$$$60\!\cdots\!71$$$$\nu^{17} +$$$$55\!\cdots\!22$$$$\nu^{16} +$$$$46\!\cdots\!75$$$$\nu^{15} +$$$$70\!\cdots\!79$$$$\nu^{14} +$$$$32\!\cdots\!42$$$$\nu^{13} +$$$$81\!\cdots\!14$$$$\nu^{12} +$$$$22\!\cdots\!24$$$$\nu^{11} +$$$$61\!\cdots\!98$$$$\nu^{10} +$$$$77\!\cdots\!23$$$$\nu^{9} +$$$$26\!\cdots\!17$$$$\nu^{8} +$$$$12\!\cdots\!36$$$$\nu^{7} -$$$$48\!\cdots\!25$$$$\nu^{6} +$$$$32\!\cdots\!16$$$$\nu^{5} -$$$$15\!\cdots\!50$$$$\nu^{4} +$$$$29\!\cdots\!53$$$$\nu^{3} -$$$$12\!\cdots\!53$$$$\nu^{2} +$$$$11\!\cdots\!87$$$$\nu +$$$$45\!\cdots\!99$$$$)/$$$$20\!\cdots\!60$$ $$\beta_{15}$$ $$=$$ $$($$$$54\!\cdots\!50$$$$\nu^{19} +$$$$26\!\cdots\!47$$$$\nu^{18} +$$$$66\!\cdots\!95$$$$\nu^{17} +$$$$15\!\cdots\!28$$$$\nu^{16} +$$$$51\!\cdots\!43$$$$\nu^{15} +$$$$19\!\cdots\!71$$$$\nu^{14} +$$$$36\!\cdots\!53$$$$\nu^{13} +$$$$91\!\cdots\!83$$$$\nu^{12} +$$$$25\!\cdots\!93$$$$\nu^{11} +$$$$57\!\cdots\!42$$$$\nu^{10} +$$$$87\!\cdots\!39$$$$\nu^{9} +$$$$22\!\cdots\!19$$$$\nu^{8} +$$$$14\!\cdots\!98$$$$\nu^{7} -$$$$16\!\cdots\!82$$$$\nu^{6} +$$$$48\!\cdots\!94$$$$\nu^{5} -$$$$16\!\cdots\!45$$$$\nu^{4} +$$$$28\!\cdots\!96$$$$\nu^{3} -$$$$10\!\cdots\!01$$$$\nu^{2} +$$$$20\!\cdots\!16$$$$\nu +$$$$33\!\cdots\!56$$$$)/$$$$20\!\cdots\!60$$ $$\beta_{16}$$ $$=$$ $$($$$$64\!\cdots\!20$$$$\nu^{19} +$$$$50\!\cdots\!70$$$$\nu^{18} +$$$$77\!\cdots\!03$$$$\nu^{17} -$$$$13\!\cdots\!59$$$$\nu^{16} +$$$$60\!\cdots\!62$$$$\nu^{15} -$$$$19\!\cdots\!17$$$$\nu^{14} +$$$$42\!\cdots\!81$$$$\nu^{13} -$$$$66\!\cdots\!21$$$$\nu^{12} +$$$$29\!\cdots\!21$$$$\nu^{11} -$$$$53\!\cdots\!45$$$$\nu^{10} +$$$$10\!\cdots\!96$$$$\nu^{9} -$$$$14\!\cdots\!21$$$$\nu^{8} +$$$$16\!\cdots\!17$$$$\nu^{7} -$$$$89\!\cdots\!24$$$$\nu^{6} +$$$$66\!\cdots\!26$$$$\nu^{5} -$$$$21\!\cdots\!82$$$$\nu^{4} +$$$$41\!\cdots\!67$$$$\nu^{3} -$$$$25\!\cdots\!42$$$$\nu^{2} +$$$$72\!\cdots\!99$$$$\nu -$$$$66\!\cdots\!86$$$$)/$$$$20\!\cdots\!60$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$64\!\cdots\!30$$$$\nu^{19} -$$$$57\!\cdots\!12$$$$\nu^{18} -$$$$78\!\cdots\!29$$$$\nu^{17} +$$$$12\!\cdots\!27$$$$\nu^{16} -$$$$61\!\cdots\!32$$$$\nu^{15} +$$$$13\!\cdots\!41$$$$\nu^{14} -$$$$43\!\cdots\!01$$$$\nu^{13} +$$$$62\!\cdots\!01$$$$\nu^{12} -$$$$30\!\cdots\!75$$$$\nu^{11} +$$$$50\!\cdots\!01$$$$\nu^{10} -$$$$10\!\cdots\!86$$$$\nu^{9} +$$$$13\!\cdots\!47$$$$\nu^{8} -$$$$16\!\cdots\!79$$$$\nu^{7} +$$$$89\!\cdots\!56$$$$\nu^{6} -$$$$66\!\cdots\!92$$$$\nu^{5} +$$$$21\!\cdots\!02$$$$\nu^{4} -$$$$41\!\cdots\!47$$$$\nu^{3} +$$$$25\!\cdots\!54$$$$\nu^{2} -$$$$73\!\cdots\!41$$$$\nu +$$$$62\!\cdots\!90$$$$)/$$$$20\!\cdots\!60$$ $$\beta_{18}$$ $$=$$ $$($$$$11\!\cdots\!15$$$$\nu^{19} +$$$$10\!\cdots\!47$$$$\nu^{18} +$$$$13\!\cdots\!24$$$$\nu^{17} -$$$$32\!\cdots\!55$$$$\nu^{16} +$$$$10\!\cdots\!25$$$$\nu^{15} -$$$$10\!\cdots\!27$$$$\nu^{14} +$$$$73\!\cdots\!51$$$$\nu^{13} -$$$$16\!\cdots\!07$$$$\nu^{12} +$$$$51\!\cdots\!44$$$$\nu^{11} -$$$$12\!\cdots\!09$$$$\nu^{10} +$$$$17\!\cdots\!45$$$$\nu^{9} -$$$$36\!\cdots\!20$$$$\nu^{8} +$$$$28\!\cdots\!04$$$$\nu^{7} -$$$$17\!\cdots\!58$$$$\nu^{6} +$$$$11\!\cdots\!27$$$$\nu^{5} -$$$$36\!\cdots\!78$$$$\nu^{4} +$$$$74\!\cdots\!57$$$$\nu^{3} -$$$$46\!\cdots\!66$$$$\nu^{2} +$$$$11\!\cdots\!64$$$$\nu -$$$$52\!\cdots\!28$$$$)/$$$$20\!\cdots\!60$$ $$\beta_{19}$$ $$=$$ $$($$$$16\!\cdots\!50$$$$\nu^{19} +$$$$20\!\cdots\!98$$$$\nu^{18} +$$$$20\!\cdots\!53$$$$\nu^{17} -$$$$24\!\cdots\!79$$$$\nu^{16} +$$$$15\!\cdots\!86$$$$\nu^{15} +$$$$26\!\cdots\!03$$$$\nu^{14} +$$$$11\!\cdots\!63$$$$\nu^{13} -$$$$11\!\cdots\!23$$$$\nu^{12} +$$$$77\!\cdots\!59$$$$\nu^{11} -$$$$10\!\cdots\!19$$$$\nu^{10} +$$$$26\!\cdots\!68$$$$\nu^{9} -$$$$24\!\cdots\!87$$$$\nu^{8} +$$$$43\!\cdots\!89$$$$\nu^{7} -$$$$20\!\cdots\!10$$$$\nu^{6} +$$$$17\!\cdots\!92$$$$\nu^{5} -$$$$54\!\cdots\!76$$$$\nu^{4} +$$$$10\!\cdots\!91$$$$\nu^{3} -$$$$60\!\cdots\!14$$$$\nu^{2} +$$$$16\!\cdots\!15$$$$\nu -$$$$14\!\cdots\!54$$$$)/$$$$20\!\cdots\!60$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{19} + \beta_{18} + \beta_{10} + \beta_{6} - 4 \beta_{4}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{19} - \beta_{17} - \beta_{12} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - 6 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$\beta_{17} - \beta_{16} + \beta_{15} + \beta_{11} + 7 \beta_{8} + \beta_{7} - 23 \beta_{6} + \beta_{5} + \beta_{2} - 1$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{18} + 9 \beta_{17} + 18 \beta_{16} - 9 \beta_{15} - 10 \beta_{14} - 9 \beta_{13} - 8 \beta_{12} + 18 \beta_{11} - 8 \beta_{10} - 11 \beta_{9} - 2 \beta_{8} + \beta_{7} - 9 \beta_{6} - 48 \beta_{5} - 11 \beta_{4} + 11 \beta_{3} - 18 \beta_{2} + 16 \beta_{1} - 11$$ $$\nu^{6}$$ $$=$$ $$-\beta_{19} - 49 \beta_{18} - 22 \beta_{17} - 10 \beta_{16} + \beta_{15} + 12 \beta_{14} + 10 \beta_{13} + 11 \beta_{12} - 22 \beta_{11} + 10 \beta_{10} + 168 \beta_{9} - 3 \beta_{7} + 157 \beta_{6} + 11 \beta_{5} + 170 \beta_{4} - \beta_{3} + 8 \beta_{2} - 33 \beta_{1} - 160$$ $$\nu^{7}$$ $$=$$ $$71 \beta_{19} - 71 \beta_{18} + 69 \beta_{17} - 13 \beta_{16} + \beta_{15} + 69 \beta_{14} - \beta_{13} - 13 \beta_{12} + 2 \beta_{11} - 73 \beta_{10} + 26 \beta_{8} + 197 \beta_{7} - 94 \beta_{6} + 266 \beta_{5} + 39 \beta_{4} - 26 \beta_{3} + 276 \beta_{2} - 207 \beta_{1} + 34$$ $$\nu^{8}$$ $$=$$ $$348 \beta_{19} + 18 \beta_{18} + 113 \beta_{17} + 97 \beta_{16} - 85 \beta_{15} - 194 \beta_{14} - 17 \beta_{13} - 81 \beta_{12} + 114 \beta_{11} - 332 \beta_{10} - 1174 \beta_{9} - 348 \beta_{8} - 16 \beta_{7} - 320 \beta_{6} - 242 \beta_{5} + 12 \beta_{4} + 366 \beta_{3} - 82 \beta_{2} + 242 \beta_{1} + 80$$ $$\nu^{9}$$ $$=$$ $$-255 \beta_{19} - 635 \beta_{17} - 383 \beta_{16} + 488 \beta_{15} + 509 \beta_{14} + 509 \beta_{13} + 1018 \beta_{12} - 1039 \beta_{11} + 676 \beta_{10} + 930 \beta_{9} - 540 \beta_{8} - 1747 \beta_{7} + 1318 \beta_{6} + 570 \beta_{5} + 1079 \beta_{2} - 1018 \beta_{1} - 400$$ $$\nu^{10}$$ $$=$$ $$-232 \beta_{18} - 178 \beta_{17} - 356 \beta_{16} + 178 \beta_{15} + 980 \beta_{14} - 419 \beta_{13} - 624 \beta_{12} + 346 \beta_{11} - 1221 \beta_{10} + 42 \beta_{9} + 232 \beta_{8} + 1331 \beta_{7} + 178 \beta_{6} + 2469 \beta_{5} + 42 \beta_{4} - 2727 \beta_{3} + 356 \beta_{2} - 885 \beta_{1} + 7048$$ $$\nu^{11}$$ $$=$$ $$2265 \beta_{19} + 4052 \beta_{18} + 2224 \beta_{17} - 2604 \beta_{16} - 834 \beta_{15} - 4828 \beta_{14} - 907 \beta_{13} - 1112 \beta_{12} + 2224 \beta_{11} + 1173 \beta_{10} - 11321 \beta_{9} - 1303 \beta_{7} - 8758 \beta_{6} - 1112 \beta_{5} - 3448 \beta_{4} + 2265 \beta_{3} - 2415 \beta_{2} + 12186 \beta_{1} + 6052$$ $$\nu^{12}$$ $$=$$ $$-18008 \beta_{19} + 18008 \beta_{18} - 1726 \beta_{17} + 6464 \beta_{16} - 834 \beta_{15} - 1726 \beta_{14} + 834 \beta_{13} + 6464 \beta_{12} - 5179 \beta_{11} + 23187 \beta_{10} - 2557 \beta_{8} - 17679 \beta_{7} + 11655 \beta_{6} - 19405 \beta_{5} - 59581 \beta_{4} + 2557 \beta_{3} - 7909 \beta_{2} + 6183 \beta_{1} + 5068$$ $$\nu^{13}$$ $$=$$ $$-30262 \beta_{19} - 19232 \beta_{18} - 36553 \beta_{17} - 9462 \beta_{16} + 7343 \beta_{15} + 18924 \beta_{14} + 6451 \beta_{13} - 17629 \beta_{12} - 15913 \beta_{11} + 3171 \beta_{10} + 94278 \beta_{9} + 30262 \beta_{8} + 27091 \beta_{7} + 67994 \beta_{6} + 10783 \beta_{5} + 64823 \beta_{4} - 49494 \beta_{3} - 78458 \beta_{2} - 10783 \beta_{1} - 3011$$ $$\nu^{14}$$ $$=$$ $$25683 \beta_{19} + 67097 \beta_{17} - 35809 \beta_{16} + 15779 \beta_{15} - 15644 \beta_{14} - 15644 \beta_{13} - 31288 \beta_{12} + 62711 \beta_{11} - 27748 \beta_{10} - 27687 \beta_{9} + 130655 \beta_{8} + 108271 \beta_{7} - 376853 \beta_{6} + 80300 \beta_{5} + 64656 \beta_{2} + 31288 \beta_{1} - 19380$$ $$\nu^{15}$$ $$=$$ $$159501 \beta_{18} + 197887 \beta_{17} + 395774 \beta_{16} - 197887 \beta_{15} - 277088 \beta_{14} - 168664 \beta_{13} - 118686 \beta_{12} + 346521 \beta_{11} - 89463 \beta_{10} - 524470 \beta_{9} - 159501 \beta_{8} - 34051 \beta_{7} - 197887 \beta_{6} - 928965 \beta_{5} - 524470 \beta_{4} + 385359 \beta_{3} - 395774 \beta_{2} + 350624 \beta_{1} - 314427$$ $$\nu^{16}$$ $$=$$ $$-242805 \beta_{19} - 952108 \beta_{18} - 813282 \beta_{17} - 270246 \beta_{16} + 178806 \beta_{15} + 543036 \beta_{14} + 356663 \beta_{13} + 406641 \beta_{12} - 813282 \beta_{11} + 334245 \beta_{10} + 3389971 \beta_{9} - 377660 \beta_{7} + 3089740 \beta_{6} + 406641 \beta_{5} + 3066295 \beta_{4} - 242805 \beta_{3} + 28981 \beta_{2} - 1184903 \beta_{1} - 2796049$$ $$\nu^{17}$$ $$=$$ $$1687959 \beta_{19} - 1687959 \beta_{18} + 1450189 \beta_{17} - 657013 \beta_{16} + 178806 \beta_{15} + 1450189 \beta_{14} - 178806 \beta_{13} - 657013 \beta_{12} + 444029 \beta_{11} - 2131988 \beta_{10} + 1304919 \beta_{8} + 4014407 \beta_{7} - 3776479 \beta_{6} + 5464596 \beta_{5} + 2004623 \beta_{4} - 1304919 \beta_{3} + 5011973 \beta_{2} - 3561784 \beta_{1} + 2301504$$ $$\nu^{18}$$ $$=$$ $$6965155 \beta_{19} + 2201571 \beta_{18} + 4359143 \beta_{17} + 3199137 \beta_{16} - 2807347 \beta_{15} - 6398274 \beta_{14} - 1535964 \beta_{13} - 2039131 \beta_{12} + 4735101 \beta_{11} - 5805149 \beta_{10} - 25499168 \beta_{9} - 6965155 \beta_{8} - 1160006 \beta_{7} - 9158330 \beta_{6} - 9526490 \beta_{5} - 3353181 \beta_{4} + 9166726 \beta_{3} - 2381505 \beta_{2} + 9526490 \beta_{1} + 1663173$$ $$\nu^{19}$$ $$=$$ $$-10583887 \beta_{19} - 16083999 \beta_{17} - 5250919 \beta_{16} + 8363279 \beta_{15} + 10667459 \beta_{14} + 10667459 \beta_{13} + 21334918 \beta_{12} - 23639098 \beta_{11} + 19674950 \beta_{10} + 33316132 \beta_{9} - 12645097 \beta_{8} - 35720110 \beta_{7} + 49296455 \beta_{6} + 5718483 \beta_{5} + 16385942 \beta_{2} - 21334918 \beta_{1} - 20344493$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/231\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{9}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 0.739775 + 2.27679i 0.648215 + 1.99500i 0.0347586 + 0.106976i −0.557915 − 1.71709i −0.864833 − 2.66168i 0.739775 − 2.27679i 0.648215 − 1.99500i 0.0347586 − 0.106976i −0.557915 + 1.71709i −0.864833 + 2.66168i 2.13576 − 1.55172i 0.705143 − 0.512316i 0.383242 − 0.278442i −1.07866 + 0.783695i −2.14548 + 1.55878i 2.13576 + 1.55172i 0.705143 + 0.512316i 0.383242 + 0.278442i −1.07866 − 0.783695i −2.14548 − 1.55878i
−0.739775 + 2.27679i −0.809017 + 0.587785i −3.01849 2.19306i −1.21700 3.74554i −0.739775 2.27679i 0.809017 + 0.587785i 3.35264 2.43583i 0.309017 0.951057i 9.42812
64.2 −0.648215 + 1.99500i −0.809017 + 0.587785i −1.94181 1.41081i 0.976330 + 3.00483i −0.648215 1.99500i 0.809017 + 0.587785i 0.679172 0.493447i 0.309017 0.951057i −6.62751
64.3 −0.0347586 + 0.106976i −0.809017 + 0.587785i 1.60780 + 1.16813i 0.327964 + 1.00937i −0.0347586 0.106976i 0.809017 + 0.587785i −0.362845 + 0.263622i 0.309017 0.951057i −0.119378
64.4 0.557915 1.71709i −0.809017 + 0.587785i −1.01908 0.740407i 0.858186 + 2.64122i 0.557915 + 1.71709i 0.809017 + 0.587785i 1.08138 0.785667i 0.309017 0.951057i 5.01400
64.5 0.864833 2.66168i −0.809017 + 0.587785i −4.71859 3.42825i −0.518429 1.59556i 0.864833 + 2.66168i 0.809017 + 0.587785i −8.67739 + 6.30449i 0.309017 0.951057i −4.69523
148.1 −0.739775 2.27679i −0.809017 0.587785i −3.01849 + 2.19306i −1.21700 + 3.74554i −0.739775 + 2.27679i 0.809017 0.587785i 3.35264 + 2.43583i 0.309017 + 0.951057i 9.42812
148.2 −0.648215 1.99500i −0.809017 0.587785i −1.94181 + 1.41081i 0.976330 3.00483i −0.648215 + 1.99500i 0.809017 0.587785i 0.679172 + 0.493447i 0.309017 + 0.951057i −6.62751
148.3 −0.0347586 0.106976i −0.809017 0.587785i 1.60780 1.16813i 0.327964 1.00937i −0.0347586 + 0.106976i 0.809017 0.587785i −0.362845 0.263622i 0.309017 + 0.951057i −0.119378
148.4 0.557915 + 1.71709i −0.809017 0.587785i −1.01908 + 0.740407i 0.858186 2.64122i 0.557915 1.71709i 0.809017 0.587785i 1.08138 + 0.785667i 0.309017 + 0.951057i 5.01400
148.5 0.864833 + 2.66168i −0.809017 0.587785i −4.71859 + 3.42825i −0.518429 + 1.59556i 0.864833 2.66168i 0.809017 0.587785i −8.67739 6.30449i 0.309017 + 0.951057i −4.69523
169.1 −2.13576 1.55172i 0.309017 0.951057i 1.53559 + 4.72606i −2.49476 + 1.81255i −2.13576 + 1.55172i −0.309017 0.951057i 2.42230 7.45506i −0.809017 0.587785i 8.14075
169.2 −0.705143 0.512316i 0.309017 0.951057i −0.383276 1.17960i 3.28814 2.38897i −0.705143 + 0.512316i −0.309017 0.951057i −0.872746 + 2.68604i −0.809017 0.587785i −3.54252
169.3 −0.383242 0.278442i 0.309017 0.951057i −0.548689 1.68869i −3.04094 + 2.20937i −0.383242 + 0.278442i −0.309017 0.951057i −0.552692 + 1.70101i −0.809017 0.587785i 1.78060
169.4 1.07866 + 0.783695i 0.309017 0.951057i −0.0686960 0.211425i 0.706442 0.513260i 1.07866 0.783695i −0.309017 0.951057i 0.915619 2.81798i −0.809017 0.587785i 1.16425
169.5 2.14548 + 1.55878i 0.309017 0.951057i 1.55524 + 4.78653i −1.38593 + 1.00694i 2.14548 1.55878i −0.309017 0.951057i −2.48543 + 7.64935i −0.809017 0.587785i −4.54308
190.1 −2.13576 + 1.55172i 0.309017 + 0.951057i 1.53559 4.72606i −2.49476 1.81255i −2.13576 1.55172i −0.309017 + 0.951057i 2.42230 + 7.45506i −0.809017 + 0.587785i 8.14075
190.2 −0.705143 + 0.512316i 0.309017 + 0.951057i −0.383276 + 1.17960i 3.28814 + 2.38897i −0.705143 0.512316i −0.309017 + 0.951057i −0.872746 2.68604i −0.809017 + 0.587785i −3.54252
190.3 −0.383242 + 0.278442i 0.309017 + 0.951057i −0.548689 + 1.68869i −3.04094 2.20937i −0.383242 0.278442i −0.309017 + 0.951057i −0.552692 1.70101i −0.809017 + 0.587785i 1.78060
190.4 1.07866 0.783695i 0.309017 + 0.951057i −0.0686960 + 0.211425i 0.706442 + 0.513260i 1.07866 + 0.783695i −0.309017 + 0.951057i 0.915619 + 2.81798i −0.809017 + 0.587785i 1.16425
190.5 2.14548 1.55878i 0.309017 + 0.951057i 1.55524 4.78653i −1.38593 1.00694i 2.14548 + 1.55878i −0.309017 + 0.951057i −2.48543 7.64935i −0.809017 + 0.587785i −4.54308
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 190.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.j.g 20
3.b odd 2 1 693.2.m.j 20
11.c even 5 1 inner 231.2.j.g 20
11.c even 5 1 2541.2.a.bq 10
11.d odd 10 1 2541.2.a.br 10
33.f even 10 1 7623.2.a.cy 10
33.h odd 10 1 693.2.m.j 20
33.h odd 10 1 7623.2.a.cx 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.g 20 1.a even 1 1 trivial
231.2.j.g 20 11.c even 5 1 inner
693.2.m.j 20 3.b odd 2 1
693.2.m.j 20 33.h odd 10 1
2541.2.a.bq 10 11.c even 5 1
2541.2.a.br 10 11.d odd 10 1
7623.2.a.cx 10 33.h odd 10 1
7623.2.a.cy 10 33.f even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{20} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(231, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T^{2} + 3 T^{3} + 6 T^{4} - 8 T^{5} + 14 T^{6} + 5 T^{7} + 10 T^{8} + 2 T^{9} + 176 T^{10} + 67 T^{11} + 170 T^{12} + 190 T^{13} + 510 T^{14} - 495 T^{15} + 710 T^{16} + 492 T^{17} + 1850 T^{18} + 927 T^{19} + 8347 T^{20} + 1854 T^{21} + 7400 T^{22} + 3936 T^{23} + 11360 T^{24} - 15840 T^{25} + 32640 T^{26} + 24320 T^{27} + 43520 T^{28} + 34304 T^{29} + 180224 T^{30} + 4096 T^{31} + 40960 T^{32} + 40960 T^{33} + 229376 T^{34} - 262144 T^{35} + 393216 T^{36} + 393216 T^{37} + 524288 T^{38} + 1048576 T^{40}$$
$3$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{5}$$
$5$ $$1 + 5 T + 4 T^{2} - 32 T^{3} - 105 T^{4} - 10 T^{5} + 535 T^{6} + 956 T^{7} - 452 T^{8} - 4251 T^{9} - 8164 T^{10} - 5041 T^{11} + 36064 T^{12} + 140288 T^{13} + 128201 T^{14} - 579422 T^{15} - 1696447 T^{16} - 345724 T^{17} + 4805888 T^{18} + 5640991 T^{19} - 3186474 T^{20} + 28204955 T^{21} + 120147200 T^{22} - 43215500 T^{23} - 1060279375 T^{24} - 1810693750 T^{25} + 2003140625 T^{26} + 10960000000 T^{27} + 14087500000 T^{28} - 9845703125 T^{29} - 79726562500 T^{30} - 207568359375 T^{31} - 110351562500 T^{32} + 1166992187500 T^{33} + 3265380859375 T^{34} - 305175781250 T^{35} - 16021728515625 T^{36} - 24414062500000 T^{37} + 15258789062500 T^{38} + 95367431640625 T^{39} + 95367431640625 T^{40}$$
$7$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}$$
$11$ $$1 + T - T^{2} + 25 T^{3} + 246 T^{4} + 17 T^{5} - 807 T^{6} + 1153 T^{7} + 25424 T^{8} - 22607 T^{9} - 174843 T^{10} - 248677 T^{11} + 3076304 T^{12} + 1534643 T^{13} - 11815287 T^{14} + 2737867 T^{15} + 435804006 T^{16} + 487179275 T^{17} - 214358881 T^{18} + 2357947691 T^{19} + 25937424601 T^{20}$$
$13$ $$1 - 13 T + 66 T^{2} - 139 T^{3} + 51 T^{4} - 2859 T^{5} + 33752 T^{6} - 143455 T^{7} + 202546 T^{8} + 225355 T^{9} + 2636810 T^{10} - 30800939 T^{11} + 94014139 T^{12} + 15525383 T^{13} - 616206028 T^{14} - 489756397 T^{15} + 8014753706 T^{16} + 3731069987 T^{17} - 135009335494 T^{18} + 334766284429 T^{19} - 343304150125 T^{20} + 4351961697577 T^{21} - 22816577698486 T^{22} + 8197160761439 T^{23} + 228909380597066 T^{24} - 181843121911321 T^{25} - 2974308801804652 T^{26} + 974194759107011 T^{27} + 76690221390664219 T^{28} - 326628538313311247 T^{29} + 363506649892361690 T^{30} + 403872305598208135 T^{31} + 4718933949218036626 T^{32} - 43448948416191654115 T^{33} +$$$$13\!\cdots\!28$$$$T^{34} -$$$$14\!\cdots\!63$$$$T^{35} + 33936247068342171891 T^{36} -$$$$12\!\cdots\!87$$$$T^{37} +$$$$74\!\cdots\!14$$$$T^{38} -$$$$19\!\cdots\!01$$$$T^{39} +$$$$19\!\cdots\!01$$$$T^{40}$$
$17$ $$1 + T - 34 T^{2} - 41 T^{3} + 811 T^{4} - 3837 T^{5} - 18522 T^{6} + 139403 T^{7} + 389000 T^{8} - 3251903 T^{9} + 2685136 T^{10} + 59936211 T^{11} - 176272043 T^{12} - 1070308079 T^{13} + 4384181682 T^{14} + 6978997953 T^{15} - 68502447326 T^{16} - 33742794795 T^{17} + 1218398812778 T^{18} - 458231992717 T^{19} - 15300717915965 T^{20} - 7789943876189 T^{21} + 352117256892842 T^{22} - 165778350827835 T^{23} - 5721392903114846 T^{24} + 9909179096552721 T^{25} + 105823487857811058 T^{26} - 439188796838039167 T^{27} - 1229631015597521963 T^{28} + 7107707987766132867 T^{29} + 5413217797876026064 T^{30} -$$$$11\!\cdots\!99$$$$T^{31} +$$$$22\!\cdots\!00$$$$T^{32} +$$$$13\!\cdots\!11$$$$T^{33} -$$$$31\!\cdots\!38$$$$T^{34} -$$$$10\!\cdots\!41$$$$T^{35} +$$$$39\!\cdots\!91$$$$T^{36} -$$$$33\!\cdots\!57$$$$T^{37} -$$$$47\!\cdots\!06$$$$T^{38} +$$$$23\!\cdots\!53$$$$T^{39} +$$$$40\!\cdots\!01$$$$T^{40}$$
$19$ $$1 - 10 T - 2 T^{2} + 248 T^{3} + 82 T^{4} - 2032 T^{5} - 23994 T^{6} + 41732 T^{7} + 518146 T^{8} - 926242 T^{9} + 540084 T^{10} - 30787074 T^{11} + 50342382 T^{12} + 703180340 T^{13} - 2471883182 T^{14} - 3864155488 T^{15} - 21312045690 T^{16} + 241124991192 T^{17} + 917942443426 T^{18} - 4413260805162 T^{19} - 5762405945050 T^{20} - 83851955298078 T^{21} + 331377222076786 T^{22} + 1653876314585928 T^{23} - 2777407106366490 T^{24} - 9568031539681312 T^{25} - 116291922026273342 T^{26} + 628553033346411260 T^{27} + 854993018331103662 T^{28} - 9934610030411708646 T^{29} + 3311290788778195284 T^{30} -$$$$10\!\cdots\!98$$$$T^{31} +$$$$11\!\cdots\!06$$$$T^{32} +$$$$17\!\cdots\!88$$$$T^{33} -$$$$19\!\cdots\!74$$$$T^{34} -$$$$30\!\cdots\!68$$$$T^{35} +$$$$23\!\cdots\!42$$$$T^{36} +$$$$13\!\cdots\!72$$$$T^{37} -$$$$20\!\cdots\!82$$$$T^{38} -$$$$19\!\cdots\!90$$$$T^{39} +$$$$37\!\cdots\!01$$$$T^{40}$$
$23$ $$( 1 + 124 T^{2} + 38 T^{3} + 8330 T^{4} + 3634 T^{5} + 381506 T^{6} + 180342 T^{7} + 12965000 T^{8} + 5936040 T^{9} + 338330610 T^{10} + 136528920 T^{11} + 6858485000 T^{12} + 2194221114 T^{13} + 106761020546 T^{14} + 23389670462 T^{15} + 1233138955370 T^{16} + 129383366986 T^{17} + 9710562174844 T^{18} + 41426511213649 T^{20} )^{2}$$
$29$ $$1 - 3 T - 4 T^{2} + 59 T^{3} + 609 T^{4} + 7469 T^{5} - 58340 T^{6} + 18675 T^{7} + 564354 T^{8} + 1565053 T^{9} - 1348916 T^{10} - 228651093 T^{11} + 603011151 T^{12} - 2322397715 T^{13} - 7665582012 T^{14} + 36331461243 T^{15} + 746249961816 T^{16} - 576665139899 T^{17} - 30471253369428 T^{18} + 122834837258411 T^{19} + 929104458856239 T^{20} + 3562210280493919 T^{21} - 25626324083688948 T^{22} - 14064286096996711 T^{23} + 527808419243182296 T^{24} + 745200014942898207 T^{25} - 4559666949775701852 T^{26} - 40061073324054233935 T^{27} +$$$$30\!\cdots\!11$$$$T^{28} -$$$$33\!\cdots\!17$$$$T^{29} -$$$$56\!\cdots\!16$$$$T^{30} +$$$$19\!\cdots\!37$$$$T^{31} +$$$$19\!\cdots\!14$$$$T^{32} +$$$$19\!\cdots\!75$$$$T^{33} -$$$$17\!\cdots\!40$$$$T^{34} +$$$$64\!\cdots\!81$$$$T^{35} +$$$$15\!\cdots\!89$$$$T^{36} +$$$$42\!\cdots\!31$$$$T^{37} -$$$$84\!\cdots\!44$$$$T^{38} -$$$$18\!\cdots\!07$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$
$31$ $$1 + 13 T + 65 T^{2} + 146 T^{3} + 295 T^{4} + 2802 T^{5} + 48827 T^{6} + 252198 T^{7} + 448847 T^{8} + 5903225 T^{9} + 65860972 T^{10} + 114046885 T^{11} - 1198096047 T^{12} - 10361131234 T^{13} - 7489384555 T^{14} + 470405899010 T^{15} + 3347900754057 T^{16} + 6862384130018 T^{17} + 25638402034775 T^{18} + 239884734794937 T^{19} + 1493178408224822 T^{20} + 7436426778643047 T^{21} + 24638504355418775 T^{22} + 204437285617366238 T^{23} + 3091856652287474697 T^{24} + 13467321514048040510 T^{25} - 6646856360987046955 T^{26} -$$$$28\!\cdots\!74$$$$T^{27} -$$$$10\!\cdots\!27$$$$T^{28} +$$$$30\!\cdots\!35$$$$T^{29} +$$$$53\!\cdots\!72$$$$T^{30} +$$$$14\!\cdots\!75$$$$T^{31} +$$$$35\!\cdots\!67$$$$T^{32} +$$$$61\!\cdots\!18$$$$T^{33} +$$$$36\!\cdots\!67$$$$T^{34} +$$$$65\!\cdots\!02$$$$T^{35} +$$$$21\!\cdots\!95$$$$T^{36} +$$$$32\!\cdots\!06$$$$T^{37} +$$$$45\!\cdots\!65$$$$T^{38} +$$$$28\!\cdots\!23$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$
$37$ $$1 + 32 T + 496 T^{2} + 5340 T^{3} + 50712 T^{4} + 444144 T^{5} + 3184776 T^{6} + 17020156 T^{7} + 60430936 T^{8} - 66824016 T^{9} - 4566815524 T^{10} - 57414614544 T^{11} - 482483421256 T^{12} - 3283683956236 T^{13} - 19092317847912 T^{14} - 82953054964976 T^{15} - 130679609958264 T^{16} + 1790702451469140 T^{17} + 24993747942875904 T^{18} + 219578098429763712 T^{19} + 1506082500954951158 T^{20} + 8124389641901257344 T^{21} + 34216440933797112576 T^{22} + 90704451274266348420 T^{23} -$$$$24\!\cdots\!04$$$$T^{24} -$$$$57\!\cdots\!32$$$$T^{25} -$$$$48\!\cdots\!08$$$$T^{26} -$$$$31\!\cdots\!88$$$$T^{27} -$$$$16\!\cdots\!76$$$$T^{28} -$$$$74\!\cdots\!88$$$$T^{29} -$$$$21\!\cdots\!76$$$$T^{30} -$$$$11\!\cdots\!08$$$$T^{31} +$$$$39\!\cdots\!16$$$$T^{32} +$$$$41\!\cdots\!32$$$$T^{33} +$$$$28\!\cdots\!64$$$$T^{34} +$$$$14\!\cdots\!92$$$$T^{35} +$$$$62\!\cdots\!92$$$$T^{36} +$$$$24\!\cdots\!80$$$$T^{37} +$$$$83\!\cdots\!84$$$$T^{38} +$$$$19\!\cdots\!36$$$$T^{39} +$$$$23\!\cdots\!01$$$$T^{40}$$
$41$ $$1 + 3 T - 151 T^{2} + 283 T^{3} + 15260 T^{4} - 53807 T^{5} - 908059 T^{6} + 5105899 T^{7} + 41696524 T^{8} - 332390297 T^{9} - 1424132099 T^{10} + 18190782425 T^{11} + 13662607978 T^{12} - 883993384739 T^{13} + 2395186168445 T^{14} + 33386146780263 T^{15} - 238454984830174 T^{16} - 934346443615571 T^{17} + 13814720376087777 T^{18} + 13438189208809197 T^{19} - 619586511510882716 T^{20} + 550965757561177077 T^{21} + 23222544952203553137 T^{22} - 64396091240428768891 T^{23} -$$$$67\!\cdots\!14$$$$T^{24} +$$$$38\!\cdots\!63$$$$T^{25} +$$$$11\!\cdots\!45$$$$T^{26} -$$$$17\!\cdots\!59$$$$T^{27} +$$$$10\!\cdots\!38$$$$T^{28} +$$$$59\!\cdots\!25$$$$T^{29} -$$$$19\!\cdots\!99$$$$T^{30} -$$$$18\!\cdots\!77$$$$T^{31} +$$$$94\!\cdots\!44$$$$T^{32} +$$$$47\!\cdots\!79$$$$T^{33} -$$$$34\!\cdots\!99$$$$T^{34} -$$$$83\!\cdots\!07$$$$T^{35} +$$$$97\!\cdots\!60$$$$T^{36} +$$$$73\!\cdots\!23$$$$T^{37} -$$$$16\!\cdots\!71$$$$T^{38} +$$$$13\!\cdots\!83$$$$T^{39} +$$$$18\!\cdots\!01$$$$T^{40}$$
$43$ $$( 1 - 6 T + 312 T^{2} - 1828 T^{3} + 47319 T^{4} - 259736 T^{5} + 4576717 T^{6} - 22838020 T^{7} + 311402464 T^{8} - 1377461130 T^{9} + 15540959670 T^{10} - 59230828590 T^{11} + 575783155936 T^{12} - 1815782456140 T^{13} + 15646884656317 T^{14} - 38183384951048 T^{15} + 299120578115631 T^{16} - 496884421103596 T^{17} + 3646718486611512 T^{18} - 3015555671621058 T^{19} + 21611482313284249 T^{20} )^{2}$$
$47$ $$1 - 20 T + 141 T^{2} - 502 T^{3} + 4037 T^{4} + 6218 T^{5} - 801001 T^{6} + 7197116 T^{7} - 33989974 T^{8} + 317744608 T^{9} - 1914149671 T^{10} - 9965865086 T^{11} + 157004029439 T^{12} - 867584843046 T^{13} + 9623990203403 T^{14} - 81333165828100 T^{15} + 137676757256356 T^{16} + 1184018410587624 T^{17} - 6791369921638187 T^{18} + 141773649130018030 T^{19} - 1649383233650553737 T^{20} + 6663361509110847410 T^{21} - 15002136156898755083 T^{22} +$$$$12\!\cdots\!52$$$$T^{23} +$$$$67\!\cdots\!36$$$$T^{24} -$$$$18\!\cdots\!00$$$$T^{25} +$$$$10\!\cdots\!87$$$$T^{26} -$$$$43\!\cdots\!98$$$$T^{27} +$$$$37\!\cdots\!79$$$$T^{28} -$$$$11\!\cdots\!62$$$$T^{29} -$$$$10\!\cdots\!79$$$$T^{30} +$$$$78\!\cdots\!24$$$$T^{31} -$$$$39\!\cdots\!34$$$$T^{32} +$$$$39\!\cdots\!32$$$$T^{33} -$$$$20\!\cdots\!69$$$$T^{34} +$$$$75\!\cdots\!74$$$$T^{35} +$$$$22\!\cdots\!77$$$$T^{36} -$$$$13\!\cdots\!74$$$$T^{37} +$$$$17\!\cdots\!49$$$$T^{38} -$$$$11\!\cdots\!60$$$$T^{39} +$$$$27\!\cdots\!01$$$$T^{40}$$
$53$ $$1 - 3 T - 126 T^{2} + 1842 T^{3} - 51 T^{4} - 176024 T^{5} + 1687499 T^{6} + 693786 T^{7} - 129427770 T^{8} + 1030515651 T^{9} + 22821416 T^{10} - 58177807579 T^{11} + 439171427598 T^{12} - 484180552566 T^{13} - 15615841075935 T^{14} + 125664083280776 T^{15} - 350599946515409 T^{16} - 1684175373629766 T^{17} + 16682975693935818 T^{18} - 76739832358344117 T^{19} + 165299081934488990 T^{20} - 4067211114992238201 T^{21} + 46862478724265712762 T^{22} -$$$$25\!\cdots\!82$$$$T^{23} -$$$$27\!\cdots\!29$$$$T^{24} +$$$$52\!\cdots\!68$$$$T^{25} -$$$$34\!\cdots\!15$$$$T^{26} -$$$$56\!\cdots\!42$$$$T^{27} +$$$$27\!\cdots\!78$$$$T^{28} -$$$$19\!\cdots\!07$$$$T^{29} +$$$$39\!\cdots\!84$$$$T^{30} +$$$$95\!\cdots\!47$$$$T^{31} -$$$$63\!\cdots\!70$$$$T^{32} +$$$$18\!\cdots\!78$$$$T^{33} +$$$$23\!\cdots\!31$$$$T^{34} -$$$$12\!\cdots\!68$$$$T^{35} -$$$$19\!\cdots\!71$$$$T^{36} +$$$$37\!\cdots\!46$$$$T^{37} -$$$$13\!\cdots\!14$$$$T^{38} -$$$$17\!\cdots\!51$$$$T^{39} +$$$$30\!\cdots\!01$$$$T^{40}$$
$59$ $$1 + 9 T - 11 T^{2} - 2009 T^{3} - 16566 T^{4} - 39875 T^{5} + 1481949 T^{6} + 16040319 T^{7} + 91259286 T^{8} - 482457503 T^{9} - 9939821477 T^{10} - 75582121635 T^{11} - 12007856396 T^{12} + 4148742606411 T^{13} + 36852203890145 T^{14} + 106193218742209 T^{15} - 1378906768809232 T^{16} - 14879013797272333 T^{17} - 71002723235317991 T^{18} + 551840607836296325 T^{19} + 6375429074335094320 T^{20} + 32558595862341483175 T^{21} -$$$$24\!\cdots\!71$$$$T^{22} -$$$$30\!\cdots\!07$$$$T^{23} -$$$$16\!\cdots\!52$$$$T^{24} +$$$$75\!\cdots\!91$$$$T^{25} +$$$$15\!\cdots\!45$$$$T^{26} +$$$$10\!\cdots\!09$$$$T^{27} -$$$$17\!\cdots\!16$$$$T^{28} -$$$$65\!\cdots\!65$$$$T^{29} -$$$$50\!\cdots\!77$$$$T^{30} -$$$$14\!\cdots\!77$$$$T^{31} +$$$$16\!\cdots\!66$$$$T^{32} +$$$$16\!\cdots\!01$$$$T^{33} +$$$$91\!\cdots\!89$$$$T^{34} -$$$$14\!\cdots\!25$$$$T^{35} -$$$$35\!\cdots\!06$$$$T^{36} -$$$$25\!\cdots\!71$$$$T^{37} -$$$$82\!\cdots\!31$$$$T^{38} +$$$$39\!\cdots\!51$$$$T^{39} +$$$$26\!\cdots\!01$$$$T^{40}$$
$61$ $$1 + 15 T - 90 T^{2} - 3165 T^{3} - 19585 T^{4} + 92683 T^{5} + 2977530 T^{6} + 30435495 T^{7} + 58060720 T^{8} - 2575904645 T^{9} - 32871179972 T^{10} - 119504505635 T^{11} + 1285823633985 T^{12} + 22572191661785 T^{13} + 150060548339790 T^{14} - 142916662288619 T^{15} - 12943544265704350 T^{16} - 115634960295586275 T^{17} - 240942316112795150 T^{18} + 4864605484258100025 T^{19} + 61052179162441039695 T^{20} +$$$$29\!\cdots\!25$$$$T^{21} -$$$$89\!\cdots\!50$$$$T^{22} -$$$$26\!\cdots\!75$$$$T^{23} -$$$$17\!\cdots\!50$$$$T^{24} -$$$$12\!\cdots\!19$$$$T^{25} +$$$$77\!\cdots\!90$$$$T^{26} +$$$$70\!\cdots\!85$$$$T^{27} +$$$$24\!\cdots\!85$$$$T^{28} -$$$$13\!\cdots\!35$$$$T^{29} -$$$$23\!\cdots\!72$$$$T^{30} -$$$$11\!\cdots\!45$$$$T^{31} +$$$$15\!\cdots\!20$$$$T^{32} +$$$$49\!\cdots\!95$$$$T^{33} +$$$$29\!\cdots\!30$$$$T^{34} +$$$$55\!\cdots\!83$$$$T^{35} -$$$$71\!\cdots\!85$$$$T^{36} -$$$$70\!\cdots\!65$$$$T^{37} -$$$$12\!\cdots\!90$$$$T^{38} +$$$$12\!\cdots\!15$$$$T^{39} +$$$$50\!\cdots\!01$$$$T^{40}$$
$67$ $$( 1 - 38 T + 939 T^{2} - 16920 T^{3} + 253022 T^{4} - 3231648 T^{5} + 37034650 T^{6} - 384823180 T^{7} + 3720320353 T^{8} - 33446868814 T^{9} + 283134556406 T^{10} - 2240940210538 T^{11} + 16700518064617 T^{12} - 115740574086340 T^{13} + 746289713342650 T^{14} - 4363129101786336 T^{15} + 22887960773164718 T^{16} - 102547240362065160 T^{17} + 381297549225685899 T^{18} - 1033848307059207986 T^{19} + 1822837804551761449 T^{20} )^{2}$$
$71$ $$1 - 37 T + 515 T^{2} - 3274 T^{3} + 13215 T^{4} - 211688 T^{5} + 4684917 T^{6} - 60912592 T^{7} + 467488647 T^{8} - 2980461565 T^{9} + 33268020472 T^{10} - 428318804705 T^{11} + 4339324633273 T^{12} - 35873672109364 T^{13} + 288523918504695 T^{14} - 2785853372108440 T^{15} + 26650571131092837 T^{16} - 220452230480408382 T^{17} + 1885088405364310445 T^{18} - 17288624106824488873 T^{19} +$$$$15\!\cdots\!82$$$$T^{20} -$$$$12\!\cdots\!83$$$$T^{21} +$$$$95\!\cdots\!45$$$$T^{22} -$$$$78\!\cdots\!02$$$$T^{23} +$$$$67\!\cdots\!97$$$$T^{24} -$$$$50\!\cdots\!40$$$$T^{25} +$$$$36\!\cdots\!95$$$$T^{26} -$$$$32\!\cdots\!24$$$$T^{27} +$$$$28\!\cdots\!53$$$$T^{28} -$$$$19\!\cdots\!55$$$$T^{29} +$$$$10\!\cdots\!72$$$$T^{30} -$$$$68\!\cdots\!15$$$$T^{31} +$$$$76\!\cdots\!27$$$$T^{32} -$$$$70\!\cdots\!12$$$$T^{33} +$$$$38\!\cdots\!77$$$$T^{34} -$$$$12\!\cdots\!88$$$$T^{35} +$$$$55\!\cdots\!15$$$$T^{36} -$$$$96\!\cdots\!34$$$$T^{37} +$$$$10\!\cdots\!15$$$$T^{38} -$$$$55\!\cdots\!47$$$$T^{39} +$$$$10\!\cdots\!01$$$$T^{40}$$
$73$ $$1 - 27 T + 117 T^{2} + 3825 T^{3} - 60386 T^{4} + 206645 T^{5} + 5076633 T^{6} - 88912135 T^{7} + 412271510 T^{8} + 5307403509 T^{9} - 94198639093 T^{10} + 498629925791 T^{11} + 2922251864844 T^{12} - 76647599821181 T^{13} + 554955524573989 T^{14} + 735710433225271 T^{15} - 49117193322293732 T^{16} + 443373919368750715 T^{17} - 1123922652740353551 T^{18} - 21964116604269227841 T^{19} +$$$$31\!\cdots\!04$$$$T^{20} -$$$$16\!\cdots\!93$$$$T^{21} -$$$$59\!\cdots\!79$$$$T^{22} +$$$$17\!\cdots\!55$$$$T^{23} -$$$$13\!\cdots\!12$$$$T^{24} +$$$$15\!\cdots\!03$$$$T^{25} +$$$$83\!\cdots\!21$$$$T^{26} -$$$$84\!\cdots\!57$$$$T^{27} +$$$$23\!\cdots\!64$$$$T^{28} +$$$$29\!\cdots\!83$$$$T^{29} -$$$$40\!\cdots\!57$$$$T^{30} +$$$$16\!\cdots\!93$$$$T^{31} +$$$$94\!\cdots\!10$$$$T^{32} -$$$$14\!\cdots\!55$$$$T^{33} +$$$$61\!\cdots\!97$$$$T^{34} +$$$$18\!\cdots\!65$$$$T^{35} -$$$$39\!\cdots\!46$$$$T^{36} +$$$$18\!\cdots\!25$$$$T^{37} +$$$$40\!\cdots\!73$$$$T^{38} -$$$$68\!\cdots\!99$$$$T^{39} +$$$$18\!\cdots\!01$$$$T^{40}$$
$79$ $$1 - 5 T - 226 T^{2} + 1188 T^{3} + 28293 T^{4} - 182558 T^{5} - 2094253 T^{6} + 20887148 T^{7} + 66066202 T^{8} - 1685198093 T^{9} + 5203173736 T^{10} + 81171391019 T^{11} - 899452563978 T^{12} + 259180666124 T^{13} + 50685690960293 T^{14} - 453111325450158 T^{15} + 2532348314805923 T^{16} + 41547309775010276 T^{17} - 818375975431692878 T^{18} - 1321261704586397645 T^{19} + 85108949992387444270 T^{20} -$$$$10\!\cdots\!55$$$$T^{21} -$$$$51\!\cdots\!98$$$$T^{22} +$$$$20\!\cdots\!64$$$$T^{23} +$$$$98\!\cdots\!63$$$$T^{24} -$$$$13\!\cdots\!42$$$$T^{25} +$$$$12\!\cdots\!53$$$$T^{26} +$$$$49\!\cdots\!16$$$$T^{27} -$$$$13\!\cdots\!58$$$$T^{28} +$$$$97\!\cdots\!61$$$$T^{29} +$$$$49\!\cdots\!36$$$$T^{30} -$$$$12\!\cdots\!47$$$$T^{31} +$$$$39\!\cdots\!82$$$$T^{32} +$$$$97\!\cdots\!72$$$$T^{33} -$$$$77\!\cdots\!93$$$$T^{34} -$$$$53\!\cdots\!42$$$$T^{35} +$$$$65\!\cdots\!53$$$$T^{36} +$$$$21\!\cdots\!92$$$$T^{37} -$$$$32\!\cdots\!86$$$$T^{38} -$$$$56\!\cdots\!95$$$$T^{39} +$$$$89\!\cdots\!01$$$$T^{40}$$
$83$ $$1 + 42 T + 661 T^{2} + 3786 T^{3} - 19549 T^{4} - 561934 T^{5} - 5968953 T^{6} - 20177190 T^{7} + 432561126 T^{8} + 6143013970 T^{9} + 28916165545 T^{10} + 66576463426 T^{11} + 144701828969 T^{12} - 5596879899702 T^{13} - 23882904399413 T^{14} - 248862974645262 T^{15} - 21903508692713044 T^{16} - 235757216137931078 T^{17} + 483849794965596541 T^{18} + 23802972835548797594 T^{19} +$$$$24\!\cdots\!45$$$$T^{20} +$$$$19\!\cdots\!02$$$$T^{21} +$$$$33\!\cdots\!49$$$$T^{22} -$$$$13\!\cdots\!86$$$$T^{23} -$$$$10\!\cdots\!24$$$$T^{24} -$$$$98\!\cdots\!66$$$$T^{25} -$$$$78\!\cdots\!97$$$$T^{26} -$$$$15\!\cdots\!54$$$$T^{27} +$$$$32\!\cdots\!29$$$$T^{28} +$$$$12\!\cdots\!78$$$$T^{29} +$$$$44\!\cdots\!05$$$$T^{30} +$$$$79\!\cdots\!90$$$$T^{31} +$$$$46\!\cdots\!86$$$$T^{32} -$$$$17\!\cdots\!70$$$$T^{33} -$$$$43\!\cdots\!37$$$$T^{34} -$$$$34\!\cdots\!38$$$$T^{35} -$$$$99\!\cdots\!69$$$$T^{36} +$$$$15\!\cdots\!78$$$$T^{37} +$$$$23\!\cdots\!49$$$$T^{38} +$$$$12\!\cdots\!74$$$$T^{39} +$$$$24\!\cdots\!01$$$$T^{40}$$
$89$ $$( 1 + 9 T + 640 T^{2} + 4050 T^{3} + 178355 T^{4} + 698876 T^{5} + 29313265 T^{6} + 51732294 T^{7} + 3369419368 T^{8} + 1036172651 T^{9} + 317703552166 T^{10} + 92219365939 T^{11} + 26689170813928 T^{12} + 36469663568886 T^{13} + 1839179937126865 T^{14} + 3902565131479324 T^{15} + 88639098149349155 T^{16} + 179136906326892450 T^{17} + 2519416835649331840 T^{18} + 3153207633367366881 T^{19} + 31181719929966183601 T^{20} )^{2}$$
$97$ $$1 + 27 T - 146 T^{2} - 10565 T^{3} - 42323 T^{4} + 1720741 T^{5} + 15722390 T^{6} - 118678851 T^{7} - 2079209414 T^{8} - 3980410181 T^{9} + 86169204810 T^{10} + 1330688850015 T^{11} + 13895090921091 T^{12} - 18497220903723 T^{13} - 2497628718197930 T^{14} - 21619068182342995 T^{15} + 116376235825961792 T^{16} + 3142408121720289719 T^{17} + 13315354121489765710 T^{18} -$$$$14\!\cdots\!49$$$$T^{19} -$$$$24\!\cdots\!21$$$$T^{20} -$$$$14\!\cdots\!53$$$$T^{21} +$$$$12\!\cdots\!90$$$$T^{22} +$$$$28\!\cdots\!87$$$$T^{23} +$$$$10\!\cdots\!52$$$$T^{24} -$$$$18\!\cdots\!15$$$$T^{25} -$$$$20\!\cdots\!70$$$$T^{26} -$$$$14\!\cdots\!99$$$$T^{27} +$$$$10\!\cdots\!51$$$$T^{28} +$$$$10\!\cdots\!55$$$$T^{29} +$$$$63\!\cdots\!90$$$$T^{30} -$$$$28\!\cdots\!93$$$$T^{31} -$$$$14\!\cdots\!74$$$$T^{32} -$$$$79\!\cdots\!27$$$$T^{33} +$$$$10\!\cdots\!10$$$$T^{34} +$$$$10\!\cdots\!13$$$$T^{35} -$$$$25\!\cdots\!83$$$$T^{36} -$$$$62\!\cdots\!05$$$$T^{37} -$$$$84\!\cdots\!94$$$$T^{38} +$$$$15\!\cdots\!91$$$$T^{39} +$$$$54\!\cdots\!01$$$$T^{40}$$