# Properties

 Label 966.2.q.e Level $966$ Weight $2$ Character orbit 966.q Analytic conductor $7.714$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.q (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.71354883526$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$2$$ over $$\Q(\zeta_{11})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 9 x^{19} + 42 x^{18} - 126 x^{17} + 277 x^{16} - 538 x^{15} + 1046 x^{14} - 1170 x^{13} - 619 x^{12} + 2218 x^{11} + 1660 x^{10} - 3225 x^{9} + 2719 x^{8} + 8761 x^{7} + 14705 x^{6} + 21691 x^{5} + 21163 x^{4} + 14656 x^{3} + 7846 x^{2} + 2622 x + 529$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 9 x^{19} + 42 x^{18} - 126 x^{17} + 277 x^{16} - 538 x^{15} + 1046 x^{14} - 1170 x^{13} - 619 x^{12} + 2218 x^{11} + 1660 x^{10} - 3225 x^{9} + 2719 x^{8} + 8761 x^{7} + 14705 x^{6} + 21691 x^{5} + 21163 x^{4} + 14656 x^{3} + 7846 x^{2} + 2622 x + 529$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$22\!\cdots\!16$$$$\nu^{19} -$$$$25\!\cdots\!51$$$$\nu^{18} +$$$$14\!\cdots\!15$$$$\nu^{17} -$$$$53\!\cdots\!15$$$$\nu^{16} +$$$$14\!\cdots\!70$$$$\nu^{15} -$$$$30\!\cdots\!74$$$$\nu^{14} +$$$$61\!\cdots\!18$$$$\nu^{13} -$$$$99\!\cdots\!06$$$$\nu^{12} +$$$$83\!\cdots\!58$$$$\nu^{11} +$$$$35\!\cdots\!44$$$$\nu^{10} -$$$$72\!\cdots\!80$$$$\nu^{9} -$$$$12\!\cdots\!29$$$$\nu^{8} +$$$$29\!\cdots\!28$$$$\nu^{7} -$$$$44\!\cdots\!23$$$$\nu^{6} -$$$$42\!\cdots\!35$$$$\nu^{5} -$$$$76\!\cdots\!93$$$$\nu^{4} -$$$$24\!\cdots\!67$$$$\nu^{3} -$$$$25\!\cdots\!31$$$$\nu^{2} -$$$$13\!\cdots\!06$$$$\nu -$$$$56\!\cdots\!30$$$$)/$$$$58\!\cdots\!49$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$23\!\cdots\!45$$$$\nu^{19} +$$$$25\!\cdots\!60$$$$\nu^{18} -$$$$13\!\cdots\!65$$$$\nu^{17} +$$$$47\!\cdots\!64$$$$\nu^{16} -$$$$12\!\cdots\!96$$$$\nu^{15} +$$$$24\!\cdots\!88$$$$\nu^{14} -$$$$47\!\cdots\!65$$$$\nu^{13} +$$$$70\!\cdots\!62$$$$\nu^{12} -$$$$31\!\cdots\!94$$$$\nu^{11} -$$$$93\!\cdots\!22$$$$\nu^{10} +$$$$10\!\cdots\!19$$$$\nu^{9} +$$$$85\!\cdots\!72$$$$\nu^{8} -$$$$23\!\cdots\!27$$$$\nu^{7} -$$$$54\!\cdots\!25$$$$\nu^{6} +$$$$43\!\cdots\!47$$$$\nu^{5} -$$$$84\!\cdots\!98$$$$\nu^{4} +$$$$15\!\cdots\!67$$$$\nu^{3} +$$$$11\!\cdots\!88$$$$\nu^{2} -$$$$24\!\cdots\!24$$$$\nu +$$$$21\!\cdots\!76$$$$)/$$$$58\!\cdots\!49$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$23\!\cdots\!45$$$$\nu^{19} +$$$$25\!\cdots\!60$$$$\nu^{18} -$$$$13\!\cdots\!65$$$$\nu^{17} +$$$$47\!\cdots\!64$$$$\nu^{16} -$$$$12\!\cdots\!96$$$$\nu^{15} +$$$$24\!\cdots\!88$$$$\nu^{14} -$$$$47\!\cdots\!65$$$$\nu^{13} +$$$$70\!\cdots\!62$$$$\nu^{12} -$$$$31\!\cdots\!94$$$$\nu^{11} -$$$$93\!\cdots\!22$$$$\nu^{10} +$$$$10\!\cdots\!19$$$$\nu^{9} +$$$$85\!\cdots\!72$$$$\nu^{8} -$$$$23\!\cdots\!27$$$$\nu^{7} -$$$$54\!\cdots\!25$$$$\nu^{6} +$$$$43\!\cdots\!47$$$$\nu^{5} -$$$$84\!\cdots\!98$$$$\nu^{4} +$$$$15\!\cdots\!67$$$$\nu^{3} +$$$$11\!\cdots\!88$$$$\nu^{2} +$$$$33\!\cdots\!25$$$$\nu +$$$$21\!\cdots\!76$$$$)/$$$$58\!\cdots\!49$$ $$\beta_{4}$$ $$=$$ $$($$$$23\!\cdots\!28$$$$\nu^{19} -$$$$21\!\cdots\!34$$$$\nu^{18} +$$$$10\!\cdots\!67$$$$\nu^{17} -$$$$33\!\cdots\!34$$$$\nu^{16} +$$$$75\!\cdots\!24$$$$\nu^{15} -$$$$14\!\cdots\!78$$$$\nu^{14} +$$$$28\!\cdots\!93$$$$\nu^{13} -$$$$34\!\cdots\!94$$$$\nu^{12} -$$$$79\!\cdots\!10$$$$\nu^{11} +$$$$63\!\cdots\!40$$$$\nu^{10} -$$$$14\!\cdots\!00$$$$\nu^{9} -$$$$66\!\cdots\!27$$$$\nu^{8} +$$$$14\!\cdots\!39$$$$\nu^{7} +$$$$11\!\cdots\!06$$$$\nu^{6} +$$$$28\!\cdots\!08$$$$\nu^{5} +$$$$55\!\cdots\!36$$$$\nu^{4} +$$$$47\!\cdots\!25$$$$\nu^{3} +$$$$30\!\cdots\!54$$$$\nu^{2} +$$$$22\!\cdots\!02$$$$\nu +$$$$23\!\cdots\!62$$$$)/$$$$58\!\cdots\!49$$ $$\beta_{5}$$ $$=$$ $$($$$$40\!\cdots\!44$$$$\nu^{19} -$$$$34\!\cdots\!51$$$$\nu^{18} +$$$$14\!\cdots\!88$$$$\nu^{17} -$$$$37\!\cdots\!79$$$$\nu^{16} +$$$$64\!\cdots\!24$$$$\nu^{15} -$$$$96\!\cdots\!76$$$$\nu^{14} +$$$$17\!\cdots\!36$$$$\nu^{13} +$$$$39\!\cdots\!85$$$$\nu^{12} -$$$$95\!\cdots\!98$$$$\nu^{11} +$$$$12\!\cdots\!86$$$$\nu^{10} +$$$$16\!\cdots\!62$$$$\nu^{9} -$$$$23\!\cdots\!19$$$$\nu^{8} +$$$$24\!\cdots\!64$$$$\nu^{7} +$$$$58\!\cdots\!11$$$$\nu^{6} +$$$$64\!\cdots\!45$$$$\nu^{5} +$$$$83\!\cdots\!57$$$$\nu^{4} +$$$$93\!\cdots\!70$$$$\nu^{3} +$$$$44\!\cdots\!97$$$$\nu^{2} +$$$$19\!\cdots\!36$$$$\nu +$$$$72\!\cdots\!43$$$$)/$$$$58\!\cdots\!49$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$46\!\cdots\!83$$$$\nu^{19} +$$$$48\!\cdots\!48$$$$\nu^{18} -$$$$25\!\cdots\!12$$$$\nu^{17} +$$$$87\!\cdots\!38$$$$\nu^{16} -$$$$21\!\cdots\!54$$$$\nu^{15} +$$$$45\!\cdots\!96$$$$\nu^{14} -$$$$87\!\cdots\!83$$$$\nu^{13} +$$$$12\!\cdots\!33$$$$\nu^{12} -$$$$59\!\cdots\!49$$$$\nu^{11} -$$$$13\!\cdots\!83$$$$\nu^{10} +$$$$11\!\cdots\!75$$$$\nu^{9} +$$$$16\!\cdots\!68$$$$\nu^{8} -$$$$38\!\cdots\!71$$$$\nu^{7} -$$$$13\!\cdots\!76$$$$\nu^{6} -$$$$15\!\cdots\!06$$$$\nu^{5} -$$$$44\!\cdots\!30$$$$\nu^{4} +$$$$13\!\cdots\!03$$$$\nu^{3} +$$$$89\!\cdots\!31$$$$\nu^{2} +$$$$17\!\cdots\!54$$$$\nu +$$$$41\!\cdots\!40$$$$)/$$$$58\!\cdots\!49$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$47\!\cdots\!93$$$$\nu^{19} +$$$$52\!\cdots\!22$$$$\nu^{18} -$$$$28\!\cdots\!67$$$$\nu^{17} +$$$$10\!\cdots\!16$$$$\nu^{16} -$$$$26\!\cdots\!87$$$$\nu^{15} +$$$$57\!\cdots\!96$$$$\nu^{14} -$$$$11\!\cdots\!74$$$$\nu^{13} +$$$$18\!\cdots\!67$$$$\nu^{12} -$$$$13\!\cdots\!08$$$$\nu^{11} -$$$$79\!\cdots\!61$$$$\nu^{10} +$$$$10\!\cdots\!89$$$$\nu^{9} +$$$$22\!\cdots\!91$$$$\nu^{8} -$$$$45\!\cdots\!73$$$$\nu^{7} -$$$$36\!\cdots\!46$$$$\nu^{6} -$$$$89\!\cdots\!61$$$$\nu^{5} +$$$$16\!\cdots\!42$$$$\nu^{4} +$$$$40\!\cdots\!07$$$$\nu^{3} +$$$$46\!\cdots\!26$$$$\nu^{2} +$$$$26\!\cdots\!49$$$$\nu +$$$$11\!\cdots\!19$$$$)/$$$$58\!\cdots\!49$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$48\!\cdots\!91$$$$\nu^{19} +$$$$31\!\cdots\!53$$$$\nu^{18} -$$$$91\!\cdots\!74$$$$\nu^{17} +$$$$59\!\cdots\!51$$$$\nu^{16} +$$$$39\!\cdots\!50$$$$\nu^{15} -$$$$13\!\cdots\!73$$$$\nu^{14} +$$$$27\!\cdots\!47$$$$\nu^{13} -$$$$92\!\cdots\!69$$$$\nu^{12} +$$$$21\!\cdots\!66$$$$\nu^{11} -$$$$73\!\cdots\!18$$$$\nu^{10} -$$$$44\!\cdots\!01$$$$\nu^{9} +$$$$15\!\cdots\!60$$$$\nu^{8} +$$$$39\!\cdots\!07$$$$\nu^{7} -$$$$95\!\cdots\!96$$$$\nu^{6} -$$$$16\!\cdots\!43$$$$\nu^{5} -$$$$19\!\cdots\!92$$$$\nu^{4} -$$$$24\!\cdots\!97$$$$\nu^{3} -$$$$15\!\cdots\!91$$$$\nu^{2} -$$$$68\!\cdots\!59$$$$\nu -$$$$12\!\cdots\!09$$$$)/$$$$58\!\cdots\!49$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$49\!\cdots\!74$$$$\nu^{19} +$$$$45\!\cdots\!37$$$$\nu^{18} -$$$$21\!\cdots\!11$$$$\nu^{17} +$$$$66\!\cdots\!54$$$$\nu^{16} -$$$$15\!\cdots\!82$$$$\nu^{15} +$$$$30\!\cdots\!25$$$$\nu^{14} -$$$$61\!\cdots\!19$$$$\nu^{13} +$$$$76\!\cdots\!60$$$$\nu^{12} -$$$$27\!\cdots\!33$$$$\nu^{11} -$$$$73\!\cdots\!75$$$$\nu^{10} -$$$$83\!\cdots\!72$$$$\nu^{9} +$$$$12\!\cdots\!98$$$$\nu^{8} -$$$$16\!\cdots\!72$$$$\nu^{7} -$$$$34\!\cdots\!34$$$$\nu^{6} -$$$$75\!\cdots\!94$$$$\nu^{5} -$$$$11\!\cdots\!19$$$$\nu^{4} -$$$$11\!\cdots\!36$$$$\nu^{3} -$$$$81\!\cdots\!09$$$$\nu^{2} -$$$$43\!\cdots\!86$$$$\nu -$$$$14\!\cdots\!26$$$$)/$$$$58\!\cdots\!49$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$54\!\cdots\!21$$$$\nu^{19} +$$$$50\!\cdots\!54$$$$\nu^{18} -$$$$23\!\cdots\!98$$$$\nu^{17} +$$$$73\!\cdots\!26$$$$\nu^{16} -$$$$16\!\cdots\!21$$$$\nu^{15} +$$$$31\!\cdots\!04$$$$\nu^{14} -$$$$60\!\cdots\!42$$$$\nu^{13} +$$$$70\!\cdots\!96$$$$\nu^{12} +$$$$32\!\cdots\!21$$$$\nu^{11} -$$$$15\!\cdots\!66$$$$\nu^{10} -$$$$46\!\cdots\!09$$$$\nu^{9} +$$$$22\!\cdots\!89$$$$\nu^{8} -$$$$22\!\cdots\!72$$$$\nu^{7} -$$$$45\!\cdots\!46$$$$\nu^{6} -$$$$59\!\cdots\!77$$$$\nu^{5} -$$$$94\!\cdots\!87$$$$\nu^{4} -$$$$82\!\cdots\!61$$$$\nu^{3} -$$$$37\!\cdots\!32$$$$\nu^{2} -$$$$24\!\cdots\!11$$$$\nu -$$$$61\!\cdots\!61$$$$)/$$$$58\!\cdots\!49$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$62\!\cdots\!20$$$$\nu^{19} +$$$$52\!\cdots\!08$$$$\nu^{18} -$$$$22\!\cdots\!33$$$$\nu^{17} +$$$$60\!\cdots\!42$$$$\nu^{16} -$$$$11\!\cdots\!73$$$$\nu^{15} +$$$$22\!\cdots\!16$$$$\nu^{14} -$$$$44\!\cdots\!73$$$$\nu^{13} +$$$$34\!\cdots\!31$$$$\nu^{12} +$$$$79\!\cdots\!30$$$$\nu^{11} -$$$$86\!\cdots\!11$$$$\nu^{10} -$$$$25\!\cdots\!88$$$$\nu^{9} +$$$$12\!\cdots\!23$$$$\nu^{8} +$$$$91\!\cdots\!11$$$$\nu^{7} -$$$$66\!\cdots\!11$$$$\nu^{6} -$$$$14\!\cdots\!75$$$$\nu^{5} -$$$$17\!\cdots\!15$$$$\nu^{4} -$$$$17\!\cdots\!81$$$$\nu^{3} -$$$$12\!\cdots\!36$$$$\nu^{2} -$$$$56\!\cdots\!83$$$$\nu -$$$$12\!\cdots\!32$$$$)/$$$$58\!\cdots\!49$$ $$\beta_{12}$$ $$=$$ $$($$$$70\!\cdots\!54$$$$\nu^{19} -$$$$62\!\cdots\!76$$$$\nu^{18} +$$$$28\!\cdots\!61$$$$\nu^{17} -$$$$80\!\cdots\!81$$$$\nu^{16} +$$$$16\!\cdots\!62$$$$\nu^{15} -$$$$28\!\cdots\!59$$$$\nu^{14} +$$$$52\!\cdots\!63$$$$\nu^{13} -$$$$40\!\cdots\!31$$$$\nu^{12} -$$$$11\!\cdots\!44$$$$\nu^{11} +$$$$22\!\cdots\!06$$$$\nu^{10} +$$$$14\!\cdots\!89$$$$\nu^{9} -$$$$38\!\cdots\!53$$$$\nu^{8} +$$$$25\!\cdots\!05$$$$\nu^{7} +$$$$76\!\cdots\!49$$$$\nu^{6} +$$$$88\!\cdots\!65$$$$\nu^{5} +$$$$14\!\cdots\!32$$$$\nu^{4} +$$$$14\!\cdots\!09$$$$\nu^{3} +$$$$70\!\cdots\!37$$$$\nu^{2} +$$$$46\!\cdots\!92$$$$\nu +$$$$10\!\cdots\!17$$$$)/$$$$58\!\cdots\!49$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$91\!\cdots\!85$$$$\nu^{19} +$$$$86\!\cdots\!61$$$$\nu^{18} -$$$$42\!\cdots\!98$$$$\nu^{17} +$$$$13\!\cdots\!26$$$$\nu^{16} -$$$$31\!\cdots\!62$$$$\nu^{15} +$$$$64\!\cdots\!96$$$$\nu^{14} -$$$$12\!\cdots\!57$$$$\nu^{13} +$$$$16\!\cdots\!75$$$$\nu^{12} -$$$$27\!\cdots\!13$$$$\nu^{11} -$$$$18\!\cdots\!69$$$$\nu^{10} -$$$$69\!\cdots\!66$$$$\nu^{9} +$$$$32\!\cdots\!06$$$$\nu^{8} -$$$$38\!\cdots\!27$$$$\nu^{7} -$$$$61\!\cdots\!04$$$$\nu^{6} -$$$$10\!\cdots\!05$$$$\nu^{5} -$$$$14\!\cdots\!66$$$$\nu^{4} -$$$$11\!\cdots\!34$$$$\nu^{3} -$$$$63\!\cdots\!27$$$$\nu^{2} -$$$$24\!\cdots\!65$$$$\nu -$$$$25\!\cdots\!97$$$$)/$$$$58\!\cdots\!49$$ $$\beta_{14}$$ $$=$$ $$($$$$10\!\cdots\!52$$$$\nu^{19} -$$$$82\!\cdots\!00$$$$\nu^{18} +$$$$33\!\cdots\!87$$$$\nu^{17} -$$$$81\!\cdots\!33$$$$\nu^{16} +$$$$12\!\cdots\!78$$$$\nu^{15} -$$$$17\!\cdots\!38$$$$\nu^{14} +$$$$30\!\cdots\!81$$$$\nu^{13} +$$$$32\!\cdots\!01$$$$\nu^{12} -$$$$27\!\cdots\!62$$$$\nu^{11} +$$$$29\!\cdots\!95$$$$\nu^{10} +$$$$35\!\cdots\!04$$$$\nu^{9} -$$$$23\!\cdots\!31$$$$\nu^{8} -$$$$17\!\cdots\!40$$$$\nu^{7} +$$$$14\!\cdots\!74$$$$\nu^{6} +$$$$21\!\cdots\!34$$$$\nu^{5} +$$$$32\!\cdots\!84$$$$\nu^{4} +$$$$34\!\cdots\!34$$$$\nu^{3} +$$$$25\!\cdots\!64$$$$\nu^{2} +$$$$12\!\cdots\!02$$$$\nu +$$$$45\!\cdots\!18$$$$)/$$$$58\!\cdots\!49$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$10\!\cdots\!70$$$$\nu^{19} +$$$$93\!\cdots\!14$$$$\nu^{18} -$$$$42\!\cdots\!89$$$$\nu^{17} +$$$$11\!\cdots\!05$$$$\nu^{16} -$$$$24\!\cdots\!75$$$$\nu^{15} +$$$$43\!\cdots\!90$$$$\nu^{14} -$$$$80\!\cdots\!46$$$$\nu^{13} +$$$$63\!\cdots\!82$$$$\nu^{12} +$$$$16\!\cdots\!36$$$$\nu^{11} -$$$$32\!\cdots\!18$$$$\nu^{10} -$$$$21\!\cdots\!44$$$$\nu^{9} +$$$$41\!\cdots\!30$$$$\nu^{8} -$$$$16\!\cdots\!01$$$$\nu^{7} -$$$$12\!\cdots\!98$$$$\nu^{6} -$$$$15\!\cdots\!27$$$$\nu^{5} -$$$$22\!\cdots\!35$$$$\nu^{4} -$$$$21\!\cdots\!17$$$$\nu^{3} -$$$$13\!\cdots\!53$$$$\nu^{2} -$$$$58\!\cdots\!89$$$$\nu -$$$$14\!\cdots\!34$$$$)/$$$$58\!\cdots\!49$$ $$\beta_{16}$$ $$=$$ $$($$$$14\!\cdots\!89$$$$\nu^{19} -$$$$13\!\cdots\!38$$$$\nu^{18} +$$$$65\!\cdots\!02$$$$\nu^{17} -$$$$20\!\cdots\!52$$$$\nu^{16} +$$$$45\!\cdots\!38$$$$\nu^{15} -$$$$87\!\cdots\!56$$$$\nu^{14} +$$$$16\!\cdots\!19$$$$\nu^{13} -$$$$19\!\cdots\!17$$$$\nu^{12} -$$$$81\!\cdots\!45$$$$\nu^{11} +$$$$43\!\cdots\!41$$$$\nu^{10} +$$$$66\!\cdots\!06$$$$\nu^{9} -$$$$58\!\cdots\!21$$$$\nu^{8} +$$$$57\!\cdots\!59$$$$\nu^{7} +$$$$12\!\cdots\!97$$$$\nu^{6} +$$$$14\!\cdots\!21$$$$\nu^{5} +$$$$22\!\cdots\!24$$$$\nu^{4} +$$$$17\!\cdots\!63$$$$\nu^{3} +$$$$87\!\cdots\!23$$$$\nu^{2} +$$$$37\!\cdots\!44$$$$\nu +$$$$57\!\cdots\!63$$$$)/$$$$58\!\cdots\!49$$ $$\beta_{17}$$ $$=$$ $$($$$$15\!\cdots\!30$$$$\nu^{19} -$$$$15\!\cdots\!36$$$$\nu^{18} +$$$$73\!\cdots\!21$$$$\nu^{17} -$$$$23\!\cdots\!61$$$$\nu^{16} +$$$$54\!\cdots\!98$$$$\nu^{15} -$$$$10\!\cdots\!95$$$$\nu^{14} +$$$$21\!\cdots\!41$$$$\nu^{13} -$$$$28\!\cdots\!43$$$$\nu^{12} +$$$$22\!\cdots\!63$$$$\nu^{11} +$$$$35\!\cdots\!56$$$$\nu^{10} +$$$$11\!\cdots\!88$$$$\nu^{9} -$$$$64\!\cdots\!62$$$$\nu^{8} +$$$$70\!\cdots\!26$$$$\nu^{7} +$$$$11\!\cdots\!84$$$$\nu^{6} +$$$$17\!\cdots\!22$$$$\nu^{5} +$$$$23\!\cdots\!95$$$$\nu^{4} +$$$$18\!\cdots\!48$$$$\nu^{3} +$$$$97\!\cdots\!67$$$$\nu^{2} +$$$$37\!\cdots\!36$$$$\nu +$$$$29\!\cdots\!01$$$$)/$$$$58\!\cdots\!49$$ $$\beta_{18}$$ $$=$$ $$($$$$-$$$$55\!\cdots\!48$$$$\nu^{19} +$$$$46\!\cdots\!86$$$$\nu^{18} -$$$$19\!\cdots\!77$$$$\nu^{17} +$$$$48\!\cdots\!59$$$$\nu^{16} -$$$$81\!\cdots\!14$$$$\nu^{15} +$$$$11\!\cdots\!67$$$$\nu^{14} -$$$$19\!\cdots\!31$$$$\nu^{13} -$$$$11\!\cdots\!09$$$$\nu^{12} +$$$$15\!\cdots\!50$$$$\nu^{11} -$$$$19\!\cdots\!64$$$$\nu^{10} -$$$$16\!\cdots\!77$$$$\nu^{9} +$$$$22\!\cdots\!56$$$$\nu^{8} -$$$$45\!\cdots\!22$$$$\nu^{7} -$$$$76\!\cdots\!39$$$$\nu^{6} -$$$$95\!\cdots\!90$$$$\nu^{5} -$$$$14\!\cdots\!47$$$$\nu^{4} -$$$$15\!\cdots\!77$$$$\nu^{3} -$$$$89\!\cdots\!31$$$$\nu^{2} -$$$$42\!\cdots\!22$$$$\nu -$$$$12\!\cdots\!39$$$$)/$$$$18\!\cdots\!07$$ $$\beta_{19}$$ $$=$$ $$($$$$-$$$$28\!\cdots\!08$$$$\nu^{19} +$$$$26\!\cdots\!64$$$$\nu^{18} -$$$$12\!\cdots\!42$$$$\nu^{17} +$$$$37\!\cdots\!87$$$$\nu^{16} -$$$$82\!\cdots\!33$$$$\nu^{15} +$$$$15\!\cdots\!10$$$$\nu^{14} -$$$$29\!\cdots\!15$$$$\nu^{13} +$$$$32\!\cdots\!72$$$$\nu^{12} +$$$$25\!\cdots\!20$$$$\nu^{11} -$$$$87\!\cdots\!62$$$$\nu^{10} -$$$$26\!\cdots\!51$$$$\nu^{9} +$$$$12\!\cdots\!45$$$$\nu^{8} -$$$$10\!\cdots\!64$$$$\nu^{7} -$$$$26\!\cdots\!32$$$$\nu^{6} -$$$$31\!\cdots\!21$$$$\nu^{5} -$$$$51\!\cdots\!29$$$$\nu^{4} -$$$$44\!\cdots\!02$$$$\nu^{3} -$$$$23\!\cdots\!31$$$$\nu^{2} -$$$$11\!\cdots\!93$$$$\nu -$$$$23\!\cdots\!18$$$$)/$$$$58\!\cdots\!49$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{3} - \beta_{2}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{17} + \beta_{14} + \beta_{11} + 5 \beta_{10} - \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{18} - 4 \beta_{16} + \beta_{15} + 7 \beta_{14} + 2 \beta_{13} + 6 \beta_{11} + 15 \beta_{10} + 6 \beta_{9} - 8 \beta_{8} - 4 \beta_{7} - 15 \beta_{6} + 8 \beta_{5} + 6 \beta_{4} - 2 \beta_{3} + 8 \beta_{1} + 7$$ $$\nu^{4}$$ $$=$$ $$7 \beta_{19} + 2 \beta_{18} + 9 \beta_{17} - 6 \beta_{16} + 3 \beta_{15} + 9 \beta_{14} + 9 \beta_{12} + 9 \beta_{11} + 8 \beta_{10} + 9 \beta_{9} - 11 \beta_{8} + 11 \beta_{7} - 36 \beta_{6} + 6 \beta_{5} - 11 \beta_{4} - 5 \beta_{3} + \beta_{2} + 10 \beta_{1} + 2$$ $$\nu^{5}$$ $$=$$ $$35 \beta_{19} - 9 \beta_{18} + 5 \beta_{17} - 27 \beta_{16} - 65 \beta_{15} - 11 \beta_{14} - 20 \beta_{13} + 47 \beta_{12} - 3 \beta_{11} - 18 \beta_{10} + 6 \beta_{9} + 11 \beta_{8} + 5 \beta_{7} - 52 \beta_{6} - 26 \beta_{5} - 23 \beta_{4} - \beta_{3} - 13 \beta_{1} + 6$$ $$\nu^{6}$$ $$=$$ $$70 \beta_{19} - 46 \beta_{18} - 65 \beta_{17} - 154 \beta_{16} - 312 \beta_{15} - 41 \beta_{14} - 31 \beta_{13} + 75 \beta_{12} - 46 \beta_{11} + 16 \beta_{10} + 40 \beta_{9} + 70 \beta_{8} - 154 \beta_{7} - 70 \beta_{6} - 46 \beta_{5} + 115 \beta_{4} + 55 \beta_{3} - 24 \beta_{2} + 5 \beta_{1} + 91$$ $$\nu^{7}$$ $$=$$ $$-53 \beta_{18} - 330 \beta_{17} - 117 \beta_{16} - 330 \beta_{15} + 64 \beta_{14} + 83 \beta_{13} - 45 \beta_{12} - 83 \beta_{11} + 328 \beta_{10} + 25 \beta_{9} + 53 \beta_{8} - 493 \beta_{7} - 25 \beta_{6} + 166 \beta_{5} + 373 \beta_{4} + 162 \beta_{3} - 45 \beta_{2} + 341 \beta_{1} + 4$$ $$\nu^{8}$$ $$=$$ $$-646 \beta_{19} + 671 \beta_{18} - 780 \beta_{17} + 988 \beta_{16} + 1466 \beta_{15} + 1044 \beta_{14} + 551 \beta_{13} - 348 \beta_{12} + 373 \beta_{11} + 1538 \beta_{10} + 224 \beta_{9} - 1019 \beta_{8} - 373 \beta_{7} - 133 \beta_{6} + 892 \beta_{5} + 133 \beta_{4} - 646 \beta_{3} + 622 \beta_{2} + 1690 \beta_{1} - 342$$ $$\nu^{9}$$ $$=$$ $$-2337 \beta_{19} + 3434 \beta_{18} + 804 \beta_{17} + 3278 \beta_{16} + 6903 \beta_{15} + 3426 \beta_{14} + 700 \beta_{13} + 1880 \beta_{11} + 848 \beta_{10} + 2622 \beta_{9} - 4126 \beta_{8} + 4048 \beta_{7} - 778 \beta_{6} + 86 \beta_{5} - 3477 \beta_{4} - 4126 \beta_{3} + 3434 \beta_{2} + 3426 \beta_{1} + 86$$ $$\nu^{10}$$ $$=$$ $$-1844 \beta_{19} + 5208 \beta_{18} + 8197 \beta_{17} + 3393 \beta_{16} + 7560 \beta_{15} + 42 \beta_{14} - 5970 \beta_{13} + 4255 \beta_{12} - 17985 \beta_{10} + 6902 \beta_{9} + 17985 \beta_{7} + 5237 \beta_{6} - 11773 \beta_{5} - 14254 \beta_{4} - 4126 \beta_{3} + 5970 \beta_{2} - 6057 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$9017 \beta_{19} - 9017 \beta_{18} + 6186 \beta_{17} - 14930 \beta_{16} - 41932 \beta_{15} - 27002 \beta_{14} - 30857 \beta_{13} + 9017 \beta_{12} - 18718 \beta_{11} - 57859 \beta_{10} + 4584 \beta_{9} + 39874 \beta_{8} + 8842 \beta_{7} + 39699 \beta_{6} - 33188 \beta_{5} + 4584 \beta_{4} + 28803 \beta_{3} - 8284 \beta_{2} - 47521 \beta_{1} + 1982$$ $$\nu^{12}$$ $$=$$ $$14008 \beta_{19} - 48118 \beta_{18} - 68930 \beta_{17} - 68930 \beta_{16} - 158194 \beta_{15} - 48118 \beta_{14} - 44283 \beta_{13} - 44283 \beta_{12} - 43986 \beta_{11} + 21088 \beta_{10} - 7560 \beta_{9} + 102277 \beta_{8} - 144186 \beta_{7} + 88269 \beta_{6} + 50731 \beta_{5} + 195985 \beta_{4} + 122264 \beta_{3} - 63973 \beta_{2} - 58291 \beta_{1} + 21088$$ $$\nu^{13}$$ $$=$$ $$-98908 \beta_{19} - 244346 \beta_{17} - 3245 \beta_{16} + 98908 \beta_{15} + 173330 \beta_{14} + 107463 \beta_{13} - 284254 \beta_{12} + 74422 \beta_{11} + 590422 \beta_{10} + 3245 \beta_{9} - 92621 \beta_{8} - 417676 \beta_{7} - 65063 \beta_{6} + 565374 \beta_{5} + 565374 \beta_{4} + 74422 \beta_{3} - 74422 \beta_{2} + 280793 \beta_{1} - 9359$$ $$\nu^{14}$$ $$=$$ $$-460076 \beta_{19} + 521894 \beta_{18} + 592649 \beta_{16} + 1900647 \beta_{15} + 1004853 \beta_{14} + 521894 \beta_{13} - 500311 \beta_{12} + 661599 \beta_{11} + 1430397 \beta_{10} + 243079 \beta_{9} - 1183493 \beta_{8} + 278815 \beta_{7} - 970321 \beta_{6} + 1392383 \beta_{5} - 30520 \beta_{4} - 1022205 \beta_{3} + 460076 \beta_{2} + 1183493 \beta_{1} - 257078$$ $$\nu^{15}$$ $$=$$ $$-213172 \beta_{19} + 1214013 \beta_{18} + 2487134 \beta_{17} + 771418 \beta_{16} + 3799098 \beta_{15} + 1000841 \beta_{14} + 1000841 \beta_{12} + 1000841 \beta_{11} - 1857843 \beta_{10} + 1000841 \beta_{9} - 1941255 \beta_{8} + 4522515 \beta_{7} - 2239628 \beta_{6} - 771418 \beta_{5} - 4522515 \beta_{4} - 3612080 \beta_{3} + 1900647 \beta_{2} + 40608 \beta_{1} - 545879$$ $$\nu^{16}$$ $$=$$ $$4686154 \beta_{19} - 3386541 \beta_{18} + 6746559 \beta_{17} - 5924164 \beta_{16} - 9813145 \beta_{15} - 8145872 \beta_{14} - 5523356 \beta_{13} + 6762143 \beta_{12} - 4698505 \beta_{11} - 19651762 \beta_{10} - 418291 \beta_{9} + 8145872 \beta_{8} + 8147590 \beta_{7} + 3092058 \beta_{6} - 13610120 \beta_{5} - 10221861 \beta_{4} + 60826 \beta_{3} - 12905203 \beta_{1} - 418291$$ $$\nu^{17}$$ $$=$$ $$16559704 \beta_{19} - 25515100 \beta_{18} - 7526168 \beta_{17} - 27924648 \beta_{16} - 67499365 \beta_{15} - 34945001 \beta_{14} - 13252655 \beta_{13} + 7129803 \beta_{12} - 25515100 \beta_{11} - 35057653 \beta_{10} - 15824182 \beta_{9} + 44748571 \beta_{8} - 27924648 \beta_{7} + 30092600 \beta_{6} - 25515100 \beta_{5} + 18796421 \beta_{4} + 32486126 \beta_{3} - 19233471 \beta_{2} - 37618768 \beta_{1} - 437050$$ $$\nu^{18}$$ $$=$$ $$-44764593 \beta_{18} - 86288805 \beta_{17} - 22943573 \beta_{16} - 86288805 \beta_{15} - 21821020 \beta_{14} + 30496175 \beta_{13} - 48889021 \beta_{12} - 30496175 \beta_{11} + 92497875 \beta_{10} - 53316936 \beta_{9} + 44764593 \beta_{8} - 162936345 \beta_{7} + 53316936 \beta_{6} + 63003212 \beta_{5} + 141386896 \beta_{4} + 71832594 \beta_{3} - 48889021 \beta_{2} + 23883759 \beta_{1} - 8829382$$ $$\nu^{19}$$ $$=$$ $$-163489500 \beta_{19} + 110172564 \beta_{18} - 141421132 \beta_{17} + 231063521 \beta_{16} + 458616410 \beta_{15} + 251559460 \beta_{14} + 246749456 \beta_{13} - 194703832 \beta_{12} + 141386896 \beta_{11} + 526344742 \beta_{10} - 43567450 \beta_{9} - 304876396 \beta_{8} - 141386896 \beta_{7} - 108368497 \beta_{6} + 362855242 \beta_{5} + 108368497 \beta_{4} - 163489500 \beta_{3} + 70112925 \beta_{2} + 415048960 \beta_{1} - 67574021$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/966\mathbb{Z}\right)^\times$$.

 $$n$$ $$323$$ $$829$$ $$925$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{17}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
85.1
 −0.748652 − 0.219824i 2.58991 + 0.760465i −0.308405 + 0.355918i 1.16609 − 1.34574i 2.33806 − 1.50258i −0.922645 + 0.592948i −0.0895295 + 0.622691i 0.130037 − 0.904424i 2.33806 + 1.50258i −0.922645 − 0.592948i 1.31816 − 2.88636i −0.973019 + 2.13061i −0.0895295 − 0.622691i 0.130037 + 0.904424i −0.308405 − 0.355918i 1.16609 + 1.34574i −0.748652 + 0.219824i 2.58991 − 0.760465i 1.31816 + 2.88636i −0.973019 − 2.13061i
0.654861 0.755750i 0.841254 0.540641i −0.142315 0.989821i −1.38365 3.02977i 0.142315 0.989821i −0.959493 + 0.281733i −0.841254 0.540641i 0.415415 0.909632i −3.19585 0.938386i
85.2 0.654861 0.755750i 0.841254 0.540641i −0.142315 0.989821i −0.182739 0.400142i 0.142315 0.989821i −0.959493 + 0.281733i −0.841254 0.540641i 0.415415 0.909632i −0.422076 0.123933i
127.1 −0.841254 + 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i −0.180741 + 0.0530703i −0.415415 0.909632i −0.654861 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i 0.123357 0.142361i
127.2 −0.841254 + 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i 3.96506 1.16425i −0.415415 0.909632i −0.654861 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i −2.70618 + 3.12310i
169.1 0.142315 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −1.18479 1.36732i 0.959493 0.281733i 0.841254 + 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i −1.52202 + 0.978140i
169.2 0.142315 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i 2.13961 + 2.46924i 0.959493 0.281733i 0.841254 + 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i 2.74860 1.76642i
211.1 −0.415415 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i −0.117330 0.0754032i 0.654861 + 0.755750i −0.142315 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i −0.0198486 + 0.138050i
211.2 −0.415415 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i 2.06640 + 1.32799i 0.654861 + 0.755750i −0.142315 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i 0.349573 2.43133i
463.1 0.142315 + 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i −1.18479 + 1.36732i 0.959493 + 0.281733i 0.841254 0.540641i −0.415415 0.909632i −0.654861 0.755750i −1.52202 0.978140i
463.2 0.142315 + 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i 2.13961 2.46924i 0.959493 + 0.281733i 0.841254 0.540641i −0.415415 0.909632i −0.654861 0.755750i 2.74860 + 1.76642i
547.1 0.959493 + 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i −0.172611 + 1.20054i −0.841254 + 0.540641i 0.415415 + 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i −0.503850 + 1.10328i
547.2 0.959493 + 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i 0.0508009 0.353328i −0.841254 + 0.540641i 0.415415 + 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i 0.148287 0.324704i
673.1 −0.415415 + 0.909632i −0.959493 0.281733i −0.654861 0.755750i −0.117330 + 0.0754032i 0.654861 0.755750i −0.142315 + 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i −0.0198486 0.138050i
673.2 −0.415415 + 0.909632i −0.959493 0.281733i −0.654861 0.755750i 2.06640 1.32799i 0.654861 0.755750i −0.142315 + 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i 0.349573 + 2.43133i
715.1 −0.841254 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i −0.180741 0.0530703i −0.415415 + 0.909632i −0.654861 + 0.755750i 0.142315 0.989821i −0.959493 + 0.281733i 0.123357 + 0.142361i
715.2 −0.841254 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i 3.96506 + 1.16425i −0.415415 + 0.909632i −0.654861 + 0.755750i 0.142315 0.989821i −0.959493 + 0.281733i −2.70618 3.12310i
841.1 0.654861 + 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i −1.38365 + 3.02977i 0.142315 + 0.989821i −0.959493 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i −3.19585 + 0.938386i
841.2 0.654861 + 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i −0.182739 + 0.400142i 0.142315 + 0.989821i −0.959493 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.422076 + 0.123933i
883.1 0.959493 0.281733i −0.654861 0.755750i 0.841254 0.540641i −0.172611 1.20054i −0.841254 0.540641i 0.415415 0.909632i 0.654861 0.755750i −0.142315 + 0.989821i −0.503850 1.10328i
883.2 0.959493 0.281733i −0.654861 0.755750i 0.841254 0.540641i 0.0508009 + 0.353328i −0.841254 0.540641i 0.415415 0.909632i 0.654861 0.755750i −0.142315 + 0.989821i 0.148287 + 0.324704i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 883.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.q.e 20
23.c even 11 1 inner 966.2.q.e 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.q.e 20 1.a even 1 1 trivial
966.2.q.e 20 23.c even 11 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{20} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(966, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}$$
$3$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}$$
$5$ $$1 + 23 T + 231 T^{2} + 1286 T^{3} + 4696 T^{4} + 13728 T^{5} + 32077 T^{6} + 47037 T^{7} + 59510 T^{8} + 1770 T^{9} + 22923 T^{10} - 20879 T^{11} + 8459 T^{12} - 3409 T^{13} + 3389 T^{14} - 1683 T^{15} + 527 T^{16} - 144 T^{17} + 44 T^{18} - 10 T^{19} + T^{20}$$
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}$$
$11$ $$279841 + 2068390 T + 11681378 T^{2} + 30550808 T^{3} + 84459956 T^{4} + 46342631 T^{5} + 49410492 T^{6} + 38029381 T^{7} + 21513251 T^{8} + 6532585 T^{9} + 3166581 T^{10} + 105262 T^{11} - 87215 T^{12} - 64313 T^{13} + 3801 T^{14} + 4405 T^{15} + 620 T^{16} - 157 T^{17} - 40 T^{18} + 3 T^{19} + T^{20}$$
$13$ $$1024 - 4096 T + 185600 T^{2} + 59136 T^{3} - 2407744 T^{4} + 3015136 T^{5} + 17850464 T^{6} - 13308240 T^{7} + 9403684 T^{8} + 1987706 T^{9} + 1789259 T^{10} + 669431 T^{11} + 146710 T^{12} + 36971 T^{13} - 398 T^{14} - 377 T^{15} + 754 T^{16} - 55 T^{17} + 29 T^{18} - 6 T^{19} + T^{20}$$
$17$ $$529 + 21896 T + 168569 T^{2} - 7653274 T^{3} + 47579376 T^{4} - 59213214 T^{5} + 54316888 T^{6} - 33290680 T^{7} + 21753359 T^{8} - 10919440 T^{9} + 5099072 T^{10} - 1991278 T^{11} + 724662 T^{12} - 245502 T^{13} + 75378 T^{14} - 20553 T^{15} + 5262 T^{16} - 1075 T^{17} + 147 T^{18} - 14 T^{19} + T^{20}$$
$19$ $$12499910809 - 27172601120 T + 22304839975 T^{2} - 2888821008 T^{3} - 2999979970 T^{4} + 1561925772 T^{5} + 235302457 T^{6} - 103131845 T^{7} + 623898 T^{8} - 8321414 T^{9} + 776544 T^{10} - 856829 T^{11} + 394273 T^{12} + 284170 T^{13} + 129721 T^{14} + 33649 T^{15} + 7464 T^{16} + 1147 T^{17} + 154 T^{18} + 13 T^{19} + T^{20}$$
$23$ $$41426511213649 + 3602305322926 T - 4072171234612 T^{2} - 211099177714 T^{3} + 367129004720 T^{4} + 17950960627 T^{5} - 21182564495 T^{6} - 366482207 T^{7} + 1189268705 T^{8} + 17753240 T^{9} - 51933563 T^{10} + 771880 T^{11} + 2248145 T^{12} - 30121 T^{13} - 75695 T^{14} + 2789 T^{15} + 2480 T^{16} - 62 T^{17} - 52 T^{18} + 2 T^{19} + T^{20}$$
$29$ $$41729946256384 + 61352970233344 T - 11153368697088 T^{2} - 9838217765888 T^{3} + 27423922623616 T^{4} - 21186510161312 T^{5} + 10691891893200 T^{6} - 3504906714624 T^{7} + 818671484628 T^{8} - 139161280408 T^{9} + 18941233761 T^{10} - 2220233552 T^{11} + 249881986 T^{12} - 27148907 T^{13} + 3070934 T^{14} - 329592 T^{15} + 37289 T^{16} - 4019 T^{17} + 366 T^{18} - 26 T^{19} + T^{20}$$
$31$ $$6404133748321 - 13121320194137 T + 14461653268852 T^{2} - 9504791908684 T^{3} + 4651662553297 T^{4} - 1895526773024 T^{5} + 580092440193 T^{6} - 141247836739 T^{7} + 32994433365 T^{8} - 3610588122 T^{9} + 945439682 T^{10} - 142505984 T^{11} + 22440874 T^{12} - 3256474 T^{13} + 569247 T^{14} - 123436 T^{15} + 14116 T^{16} - 549 T^{17} + 67 T^{18} - 15 T^{19} + T^{20}$$
$37$ $$744266818681 - 2971633933442 T + 7954765902312 T^{2} - 2402101588804 T^{3} + 10115742367381 T^{4} - 930317586296 T^{5} + 469330154954 T^{6} + 179000465200 T^{7} - 3594541701 T^{8} - 3736716493 T^{9} - 197543358 T^{10} + 3195102 T^{11} + 5019071 T^{12} + 1289953 T^{13} + 324465 T^{14} + 60444 T^{15} + 11538 T^{16} + 2116 T^{17} + 311 T^{18} + 26 T^{19} + T^{20}$$
$41$ $$55504332715129 - 104748923083198 T + 139758764248112 T^{2} - 107669364047030 T^{3} + 65302004499195 T^{4} - 26793099234417 T^{5} + 10151758877206 T^{6} - 2285132266894 T^{7} + 273931822524 T^{8} - 30684017687 T^{9} + 12160613816 T^{10} - 4032185896 T^{11} + 841508351 T^{12} - 130454605 T^{13} + 16881214 T^{14} - 1830093 T^{15} + 159873 T^{16} - 11766 T^{17} + 784 T^{18} - 40 T^{19} + T^{20}$$
$43$ $$41787170080162816 + 4312337108931072 T + 14731349819109888 T^{2} + 2735270468658688 T^{3} + 1249104173794240 T^{4} + 102008223751104 T^{5} + 11443107946096 T^{6} - 3965585600656 T^{7} + 284086101304 T^{8} + 53714032688 T^{9} + 2792983929 T^{10} - 1098153585 T^{11} + 22309543 T^{12} + 6157279 T^{13} + 1095804 T^{14} - 146289 T^{15} - 5106 T^{16} + 1813 T^{17} - 60 T^{18} - 8 T^{19} + T^{20}$$
$47$ $$( 2073248 - 1847936 T - 1369216 T^{2} + 760436 T^{3} + 281944 T^{4} - 78379 T^{5} - 11029 T^{6} + 3417 T^{7} - 107 T^{8} - 18 T^{9} + T^{10} )^{2}$$
$53$ $$1938817024 + 1262485504 T - 5148704768 T^{2} - 7461437440 T^{3} + 26021249024 T^{4} - 39537969152 T^{5} + 43559556864 T^{6} - 25332639040 T^{7} + 10509561456 T^{8} - 2625282236 T^{9} + 388966181 T^{10} + 8061367 T^{11} - 13025366 T^{12} + 2492221 T^{13} - 10374 T^{14} - 31911 T^{15} + 10033 T^{16} - 523 T^{17} + 57 T^{18} + 5 T^{19} + T^{20}$$
$59$ $$6045905961419776 - 3319601742094336 T + 970315669278208 T^{2} - 59528593919744 T^{3} + 66099889727104 T^{4} - 27906821726784 T^{5} + 1748001148624 T^{6} - 333946992728 T^{7} + 298976617412 T^{8} - 38343433280 T^{9} + 1571242441 T^{10} - 1236127326 T^{11} + 316614210 T^{12} - 33510242 T^{13} + 5137669 T^{14} - 758307 T^{15} + 66094 T^{16} - 5756 T^{17} + 418 T^{18} - 13 T^{19} + T^{20}$$
$61$ $$76535691699119104 - 400685086758151680 T + 1213520461487114240 T^{2} - 196898525299952768 T^{3} + 32582794390038080 T^{4} + 2583401289509376 T^{5} - 814293398719744 T^{6} + 124462684299208 T^{7} + 935250426740 T^{8} - 2345015027232 T^{9} + 388967521789 T^{10} - 29787843783 T^{11} + 751860268 T^{12} + 14318633 T^{13} + 13929594 T^{14} - 3594651 T^{15} + 435537 T^{16} - 33571 T^{17} + 1761 T^{18} - 59 T^{19} + T^{20}$$
$67$ $$682426593903616 - 65317435400192 T + 512518814596096 T^{2} - 116209182132224 T^{3} + 159417620567872 T^{4} - 40445549344992 T^{5} + 26388974406336 T^{6} - 6274816409248 T^{7} + 1041222821496 T^{8} + 25965845462 T^{9} - 1991737485 T^{10} - 946654459 T^{11} + 134523631 T^{12} + 41724017 T^{13} + 5260666 T^{14} + 130020 T^{15} - 48744 T^{16} - 4772 T^{17} + 66 T^{18} + 26 T^{19} + T^{20}$$
$71$ $$9548694366891241 - 6506681769509752 T + 3991162438081357 T^{2} - 1490560015902544 T^{3} + 414523631006124 T^{4} - 80300161197916 T^{5} + 8875304059231 T^{6} - 128593342303 T^{7} - 16207515368 T^{8} - 30357054088 T^{9} + 9341546202 T^{10} - 1126142011 T^{11} + 37804767 T^{12} + 5979428 T^{13} - 722669 T^{14} + 33231 T^{15} - 1138 T^{16} + 63 T^{17} + 88 T^{18} - 13 T^{19} + T^{20}$$
$73$ $$53223169182966784 - 58450602461614592 T + 52983355484946944 T^{2} - 19011150434053888 T^{3} + 3936908892635264 T^{4} - 230883850421408 T^{5} - 66794857418352 T^{6} - 25695919661104 T^{7} + 5027437201880 T^{8} + 1650820197696 T^{9} + 295297675441 T^{10} + 25005653385 T^{11} + 688878793 T^{12} - 153827971 T^{13} - 17679867 T^{14} - 741194 T^{15} + 38310 T^{16} + 6424 T^{17} + 611 T^{18} + 34 T^{19} + T^{20}$$
$79$ $$3072419271230464 + 6868085892397568 T + 7820649934415872 T^{2} + 5917967692711808 T^{3} + 3300978533721664 T^{4} + 1405655151016000 T^{5} + 462356730588704 T^{6} + 118127339126504 T^{7} + 23557130936068 T^{8} + 3681134374668 T^{9} + 448443956653 T^{10} + 41607199935 T^{11} + 3182304401 T^{12} + 225667425 T^{13} + 16341170 T^{14} + 1048140 T^{15} + 60564 T^{16} + 4556 T^{17} + 401 T^{18} + 27 T^{19} + T^{20}$$
$83$ $$24050157577216 - 39380910931968 T + 32885753063168 T^{2} - 24516471601664 T^{3} + 14596196839616 T^{4} - 5689505702880 T^{5} + 2155621656160 T^{6} - 531799515248 T^{7} + 125956492012 T^{8} - 17349860190 T^{9} + 3296722681 T^{10} - 788033740 T^{11} + 273097886 T^{12} - 61977445 T^{13} + 9428478 T^{14} - 851973 T^{15} + 53686 T^{16} - 4861 T^{17} + 617 T^{18} - 41 T^{19} + T^{20}$$
$89$ $$2037500161310329 - 3591960326206470 T + 6096056693656239 T^{2} - 5883929588844769 T^{3} + 3575611414516225 T^{4} - 1397621497714362 T^{5} + 375519691320612 T^{6} - 73564524298331 T^{7} + 11750702511355 T^{8} - 1203935290845 T^{9} + 32770126212 T^{10} + 7080073130 T^{11} - 912999870 T^{12} + 51616433 T^{13} - 214628 T^{14} - 469791 T^{15} + 102638 T^{16} - 12507 T^{17} + 963 T^{18} - 43 T^{19} + T^{20}$$
$97$ $$834755060368384 + 1393645771222528 T + 1592003300743424 T^{2} - 1171059800893184 T^{3} + 573679015637568 T^{4} - 168069668245600 T^{5} + 37760079762352 T^{6} - 5758680483800 T^{7} + 647033141936 T^{8} - 30117226290 T^{9} - 525560553 T^{10} + 427797217 T^{11} + 8824742 T^{12} - 889181 T^{13} + 381871 T^{14} + 33440 T^{15} + 3364 T^{16} - 1175 T^{17} - 93 T^{18} + 8 T^{19} + T^{20}$$