# Properties

 Label 483.2.q.c Level $483$ Weight $2$ Character orbit 483.q Analytic conductor $3.857$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$483 = 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 483.q (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.85677441763$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$2$$ over $$\Q(\zeta_{11})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 8 x^{19} + 40 x^{18} - 117 x^{17} + 295 x^{16} - 575 x^{15} + 1777 x^{14} - 1560 x^{13} + 4383 x^{12} - 6446 x^{11} + 7261 x^{10} + 7700 x^{9} + 7852 x^{8} - 39430 x^{7} - 101709 x^{6} + 156742 x^{5} + 999838 x^{4} + 2029154 x^{3} + 3616480 x^{2} + 4299390 x + 2374681$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{2} - \beta_{6} ) q^{2} + \beta_{12} q^{3} + ( -\beta_{5} + \beta_{11} ) q^{4} + ( \beta_{6} - \beta_{14} ) q^{5} + ( -\beta_{5} - \beta_{13} ) q^{6} -\beta_{13} q^{7} + ( -\beta_{2} - \beta_{3} - \beta_{12} - \beta_{15} ) q^{8} + \beta_{8} q^{9} +O(q^{10})$$ $$q + ( -\beta_{2} - \beta_{6} ) q^{2} + \beta_{12} q^{3} + ( -\beta_{5} + \beta_{11} ) q^{4} + ( \beta_{6} - \beta_{14} ) q^{5} + ( -\beta_{5} - \beta_{13} ) q^{6} -\beta_{13} q^{7} + ( -\beta_{2} - \beta_{3} - \beta_{12} - \beta_{15} ) q^{8} + \beta_{8} q^{9} + ( 1 + \beta_{9} + \beta_{10} ) q^{10} + ( \beta_{1} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} - \beta_{13} + \beta_{19} ) q^{11} + ( 1 + \beta_{3} ) q^{12} + ( \beta_{2} - \beta_{4} - \beta_{6} + \beta_{8} + \beta_{14} ) q^{13} + ( 1 - \beta_{15} ) q^{14} + ( -1 + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{11} - \beta_{12} + \beta_{15} - \beta_{18} ) q^{15} + ( -1 + \beta_{2} - \beta_{3} + 3 \beta_{5} - 3 \beta_{8} - \beta_{11} - \beta_{12} + 3 \beta_{13} + \beta_{15} ) q^{16} + ( \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{9} + \beta_{11} - \beta_{16} + \beta_{19} ) q^{17} + ( 1 + \beta_{11} ) q^{18} + ( 1 + \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{12} - 2 \beta_{13} - 2 \beta_{15} + \beta_{17} + \beta_{19} ) q^{19} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} + \beta_{14} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{20} + \beta_{11} q^{21} + ( \beta_{3} + \beta_{5} + \beta_{9} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{22} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{8} + \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - \beta_{15} + \beta_{16} + \beta_{18} + \beta_{19} ) q^{23} + ( -\beta_{2} + \beta_{6} - \beta_{8} - \beta_{13} ) q^{24} + ( 1 - \beta_{2} - \beta_{6} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{15} + \beta_{17} ) q^{25} + ( 1 - \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{13} - \beta_{15} + \beta_{18} ) q^{26} + ( -1 + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{15} ) q^{27} + ( -\beta_{2} + \beta_{8} ) q^{28} + ( 3 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} - \beta_{18} - 2 \beta_{19} ) q^{29} + ( 1 - \beta_{1} + \beta_{3} + \beta_{12} - \beta_{15} + \beta_{16} ) q^{30} + ( -\beta_{2} + \beta_{5} + \beta_{7} + \beta_{9} - \beta_{11} - \beta_{13} - \beta_{15} - \beta_{18} ) q^{31} + ( -3 + \beta_{2} - 4 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} - 4 \beta_{11} + \beta_{13} + 3 \beta_{15} ) q^{32} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{14} + \beta_{17} - \beta_{18} ) q^{33} + ( -2 - \beta_{2} - 2 \beta_{3} - \beta_{4} - 3 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{34} + ( 1 - \beta_{1} + \beta_{3} ) q^{35} + ( -\beta_{6} + \beta_{12} ) q^{36} + ( 2 \beta_{3} + 2 \beta_{6} + 2 \beta_{8} + \beta_{10} + 2 \beta_{11} + 2 \beta_{13} + \beta_{16} ) q^{37} + ( 1 - 2 \beta_{3} - \beta_{4} - \beta_{5} - 4 \beta_{6} - \beta_{7} + 3 \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{38} + ( -\beta_{6} + \beta_{8} + \beta_{13} + \beta_{18} + \beta_{19} ) q^{39} + ( 2 - 2 \beta_{2} + 2 \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} - 2 \beta_{15} + \beta_{18} - \beta_{19} ) q^{40} + ( -1 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} + \beta_{15} + \beta_{17} ) q^{41} + ( -\beta_{2} + \beta_{12} ) q^{42} + ( -3 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{7} - 2 \beta_{8} - \beta_{10} - 3 \beta_{11} + 4 \beta_{13} + 2 \beta_{14} + \beta_{15} + \beta_{17} - \beta_{19} ) q^{43} + ( -1 - \beta_{1} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} + \beta_{15} - \beta_{17} - \beta_{18} ) q^{44} + ( \beta_{11} - \beta_{15} + \beta_{17} ) q^{45} + ( 1 + \beta_{1} - 3 \beta_{2} + \beta_{3} - 3 \beta_{5} - 3 \beta_{6} + \beta_{7} + 3 \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} - 3 \beta_{13} + \beta_{14} - 2 \beta_{15} - \beta_{16} - \beta_{17} ) q^{46} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{15} - \beta_{17} + \beta_{18} ) q^{47} + ( -2 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{12} - 2 \beta_{13} - 3 \beta_{15} ) q^{48} + \beta_{12} q^{49} + ( -2 + \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{11} + \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} ) q^{50} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{12} - \beta_{13} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{51} + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{18} - \beta_{19} ) q^{52} + ( 3 - \beta_{1} + 5 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 3 \beta_{12} - 4 \beta_{13} - 2 \beta_{14} - 5 \beta_{15} + \beta_{16} + \beta_{17} ) q^{53} + ( \beta_{3} + \beta_{12} ) q^{54} + ( -6 + 4 \beta_{2} - 2 \beta_{3} - \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - \beta_{7} - 6 \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} + 3 \beta_{13} + \beta_{14} + 6 \beta_{15} ) q^{55} + ( 2 - \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{8} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{56} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + \beta_{7} - 3 \beta_{8} + 2 \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} + 2 \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{57} + ( -1 - \beta_{1} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + \beta_{7} - 3 \beta_{8} + 2 \beta_{10} - \beta_{11} - \beta_{12} + 3 \beta_{13} + 2 \beta_{16} + 2 \beta_{17} - \beta_{19} ) q^{58} + ( 3 - 2 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + 6 \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{16} + \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{59} + ( -\beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{14} ) q^{60} + ( 2 + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{61} + ( -2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{11} + \beta_{13} + 2 \beta_{16} + \beta_{17} ) q^{62} + \beta_{3} q^{63} + ( \beta_{2} + \beta_{3} - \beta_{5} - \beta_{8} - 4 \beta_{12} + 4 \beta_{13} ) q^{64} + ( -1 + \beta_{2} - \beta_{3} + 5 \beta_{5} + \beta_{6} - 2 \beta_{9} + \beta_{10} - 5 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{18} ) q^{65} + ( -\beta_{1} + \beta_{3} - \beta_{4} - \beta_{8} - \beta_{14} - \beta_{15} ) q^{66} + ( \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} - 3 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} - \beta_{12} + 3 \beta_{13} + \beta_{15} + \beta_{16} - 2 \beta_{19} ) q^{67} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} + 3 \beta_{15} - \beta_{16} - 2 \beta_{17} + \beta_{18} + \beta_{19} ) q^{68} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{14} + \beta_{16} - \beta_{18} - \beta_{19} ) q^{69} + ( \beta_{6} - \beta_{8} - \beta_{14} - \beta_{19} ) q^{70} + ( -2 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} + \beta_{4} + 5 \beta_{6} - 4 \beta_{8} + 2 \beta_{10} - 4 \beta_{11} - 3 \beta_{12} + 6 \beta_{13} - 2 \beta_{18} - \beta_{19} ) q^{71} + ( 1 - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{8} + 2 \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{72} + ( 6 - 2 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} - 4 \beta_{5} - 3 \beta_{6} + \beta_{7} + 6 \beta_{8} + 3 \beta_{9} + \beta_{10} + 4 \beta_{11} + 3 \beta_{12} - 4 \beta_{13} - 8 \beta_{15} + \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{73} + ( 1 + \beta_{1} - 4 \beta_{2} + \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{74} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{12} - \beta_{13} - \beta_{15} - \beta_{16} ) q^{75} + ( -1 + \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{11} - \beta_{12} + 4 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} - \beta_{16} - 2 \beta_{17} ) q^{76} + ( -\beta_{2} - \beta_{4} - \beta_{6} + \beta_{8} - \beta_{10} - \beta_{13} - \beta_{15} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{77} + ( -2 + \beta_{1} - \beta_{10} + 2 \beta_{15} - \beta_{16} - \beta_{17} ) q^{78} + ( -5 + 2 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - 3 \beta_{9} - 3 \beta_{10} + 3 \beta_{13} + 2 \beta_{14} + 4 \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{79} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} - 3 \beta_{11} - 3 \beta_{12} + \beta_{13} + 5 \beta_{15} - 2 \beta_{16} - 2 \beta_{17} ) q^{80} -\beta_{15} q^{81} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 4 \beta_{6} - \beta_{7} + 7 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{12} - 5 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} - 3 \beta_{16} - \beta_{17} + \beta_{18} + 3 \beta_{19} ) q^{82} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + 3 \beta_{8} - \beta_{9} + 2 \beta_{11} - \beta_{12} - \beta_{14} - 2 \beta_{15} + \beta_{17} ) q^{83} + ( -1 + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{11} - \beta_{12} + \beta_{15} ) q^{84} + ( -5 + \beta_{1} - 3 \beta_{3} + 7 \beta_{5} + \beta_{6} - \beta_{7} - 8 \beta_{8} - 5 \beta_{12} + 6 \beta_{13} + 3 \beta_{15} - \beta_{16} - \beta_{18} - \beta_{19} ) q^{85} + ( -5 + \beta_{1} + 4 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{18} + 2 \beta_{19} ) q^{86} + ( 2 - \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{87} + ( -1 - \beta_{1} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - 3 \beta_{11} + 2 \beta_{13} - \beta_{15} - 2 \beta_{17} - \beta_{19} ) q^{88} + ( 4 - 3 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{6} + 2 \beta_{7} - \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} + 3 \beta_{12} - 5 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} + 2 \beta_{17} - 3 \beta_{18} - 2 \beta_{19} ) q^{89} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{90} + ( -2 + \beta_{1} - \beta_{10} ) q^{91} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} - 2 \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} - \beta_{17} ) q^{92} + ( -\beta_{1} - \beta_{2} + \beta_{10} + \beta_{11} - \beta_{13} - \beta_{15} + \beta_{17} ) q^{93} + ( \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + 4 \beta_{6} - \beta_{7} - 3 \beta_{8} - 2 \beta_{12} + 3 \beta_{13} - \beta_{16} + \beta_{18} - \beta_{19} ) q^{94} + ( -3 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} - 6 \beta_{8} + \beta_{9} + 2 \beta_{11} - 5 \beta_{15} + 3 \beta_{17} - \beta_{18} - 3 \beta_{19} ) q^{95} + ( 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{6} + \beta_{8} - 2 \beta_{12} - \beta_{15} ) q^{96} + ( -2 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{5} + 4 \beta_{6} + 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{12} + \beta_{13} + 4 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} - 2 \beta_{17} + 2 \beta_{18} ) q^{97} + ( -\beta_{5} - \beta_{13} ) q^{98} + ( \beta_{1} - \beta_{3} - \beta_{7} - \beta_{9} + \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 4q^{2} - 2q^{3} - 4q^{4} - q^{5} - 4q^{6} - 2q^{7} - 2q^{9} + O(q^{10})$$ $$20q - 4q^{2} - 2q^{3} - 4q^{4} - q^{5} - 4q^{6} - 2q^{7} - 2q^{9} + 9q^{10} + 3q^{11} + 18q^{12} - 2q^{13} + 18q^{14} - q^{15} + 8q^{16} + 8q^{17} + 18q^{18} + 6q^{19} - 2q^{20} - 2q^{21} + 6q^{22} + 11q^{23} + 9q^{25} + 7q^{26} - 2q^{27} - 4q^{28} + 23q^{29} + 9q^{30} + q^{31} - 28q^{32} + 14q^{33} - 28q^{34} + 10q^{35} - 4q^{36} - 9q^{37} + 34q^{38} - 2q^{39} - 15q^{41} - 4q^{42} - 23q^{43} - 16q^{44} - 12q^{45} + 11q^{46} - 66q^{47} - 36q^{48} - 2q^{49} - 26q^{50} - 14q^{51} + 7q^{52} + 9q^{53} - 4q^{54} - 62q^{55} + 22q^{56} - 27q^{57} - 20q^{58} + 49q^{59} - 2q^{60} + 46q^{61} - 9q^{62} - 2q^{63} + 16q^{64} + 11q^{65} - 16q^{66} + 14q^{67} + 38q^{68} + 11q^{69} - 2q^{70} + 36q^{71} - q^{73} + 4q^{74} - 2q^{75} + 34q^{76} - 8q^{77} - 15q^{78} - 22q^{79} + 15q^{80} - 2q^{81} - 30q^{82} + 8q^{83} - 4q^{84} - 32q^{85} - 68q^{86} + q^{87} - 11q^{88} - 2q^{89} - 2q^{90} - 24q^{91} + 11q^{92} - 32q^{93} + 33q^{94} - 107q^{95} + 16q^{96} + 18q^{97} - 4q^{98} + 14q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 8 x^{19} + 40 x^{18} - 117 x^{17} + 295 x^{16} - 575 x^{15} + 1777 x^{14} - 1560 x^{13} + 4383 x^{12} - 6446 x^{11} + 7261 x^{10} + 7700 x^{9} + 7852 x^{8} - 39430 x^{7} - 101709 x^{6} + 156742 x^{5} + 999838 x^{4} + 2029154 x^{3} + 3616480 x^{2} + 4299390 x + 2374681$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$27\!\cdots\!67$$$$\nu^{19} +$$$$99\!\cdots\!87$$$$\nu^{18} -$$$$81\!\cdots\!67$$$$\nu^{17} +$$$$41\!\cdots\!26$$$$\nu^{16} -$$$$13\!\cdots\!64$$$$\nu^{15} +$$$$37\!\cdots\!83$$$$\nu^{14} -$$$$86\!\cdots\!38$$$$\nu^{13} +$$$$21\!\cdots\!49$$$$\nu^{12} -$$$$31\!\cdots\!66$$$$\nu^{11} +$$$$53\!\cdots\!00$$$$\nu^{10} -$$$$83\!\cdots\!06$$$$\nu^{9} +$$$$69\!\cdots\!13$$$$\nu^{8} +$$$$12\!\cdots\!98$$$$\nu^{7} -$$$$29\!\cdots\!07$$$$\nu^{6} +$$$$40\!\cdots\!84$$$$\nu^{5} -$$$$26\!\cdots\!53$$$$\nu^{4} +$$$$37\!\cdots\!25$$$$\nu^{3} +$$$$38\!\cdots\!93$$$$\nu^{2} +$$$$89\!\cdots\!63$$$$\nu +$$$$93\!\cdots\!23$$$$)/$$$$16\!\cdots\!33$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$32\!\cdots\!48$$$$\nu^{19} +$$$$30\!\cdots\!71$$$$\nu^{18} -$$$$16\!\cdots\!62$$$$\nu^{17} +$$$$53\!\cdots\!57$$$$\nu^{16} -$$$$13\!\cdots\!99$$$$\nu^{15} +$$$$26\!\cdots\!53$$$$\nu^{14} -$$$$66\!\cdots\!47$$$$\nu^{13} +$$$$84\!\cdots\!10$$$$\nu^{12} -$$$$11\!\cdots\!00$$$$\nu^{11} +$$$$15\!\cdots\!23$$$$\nu^{10} -$$$$17\!\cdots\!97$$$$\nu^{9} -$$$$50\!\cdots\!68$$$$\nu^{8} +$$$$97\!\cdots\!77$$$$\nu^{7} +$$$$10\!\cdots\!89$$$$\nu^{6} -$$$$19\!\cdots\!01$$$$\nu^{5} -$$$$62\!\cdots\!53$$$$\nu^{4} -$$$$30\!\cdots\!03$$$$\nu^{3} +$$$$38\!\cdots\!04$$$$\nu^{2} -$$$$72\!\cdots\!03$$$$\nu -$$$$33\!\cdots\!96$$$$)/$$$$16\!\cdots\!33$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$64\!\cdots\!61$$$$\nu^{19} +$$$$50\!\cdots\!26$$$$\nu^{18} -$$$$23\!\cdots\!23$$$$\nu^{17} +$$$$59\!\cdots\!17$$$$\nu^{16} -$$$$11\!\cdots\!59$$$$\nu^{15} +$$$$12\!\cdots\!53$$$$\nu^{14} -$$$$50\!\cdots\!90$$$$\nu^{13} -$$$$46\!\cdots\!52$$$$\nu^{12} +$$$$80\!\cdots\!55$$$$\nu^{11} -$$$$83\!\cdots\!93$$$$\nu^{10} +$$$$37\!\cdots\!04$$$$\nu^{9} -$$$$18\!\cdots\!19$$$$\nu^{8} +$$$$33\!\cdots\!53$$$$\nu^{7} +$$$$51\!\cdots\!99$$$$\nu^{6} +$$$$17\!\cdots\!11$$$$\nu^{5} -$$$$27\!\cdots\!63$$$$\nu^{4} -$$$$15\!\cdots\!88$$$$\nu^{3} -$$$$66\!\cdots\!11$$$$\nu^{2} -$$$$15\!\cdots\!72$$$$\nu -$$$$11\!\cdots\!64$$$$)/$$$$24\!\cdots\!99$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$52\!\cdots\!69$$$$\nu^{19} +$$$$39\!\cdots\!41$$$$\nu^{18} -$$$$16\!\cdots\!71$$$$\nu^{17} +$$$$27\!\cdots\!15$$$$\nu^{16} +$$$$14\!\cdots\!84$$$$\nu^{15} -$$$$28\!\cdots\!49$$$$\nu^{14} +$$$$71\!\cdots\!87$$$$\nu^{13} -$$$$32\!\cdots\!09$$$$\nu^{12} +$$$$75\!\cdots\!76$$$$\nu^{11} -$$$$15\!\cdots\!99$$$$\nu^{10} +$$$$30\!\cdots\!78$$$$\nu^{9} -$$$$68\!\cdots\!44$$$$\nu^{8} +$$$$88\!\cdots\!19$$$$\nu^{7} -$$$$65\!\cdots\!87$$$$\nu^{6} +$$$$13\!\cdots\!32$$$$\nu^{5} -$$$$21\!\cdots\!60$$$$\nu^{4} -$$$$64\!\cdots\!66$$$$\nu^{3} -$$$$49\!\cdots\!72$$$$\nu^{2} -$$$$11\!\cdots\!28$$$$\nu -$$$$13\!\cdots\!60$$$$)/$$$$16\!\cdots\!33$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$57\!\cdots\!21$$$$\nu^{19} +$$$$62\!\cdots\!41$$$$\nu^{18} -$$$$39\!\cdots\!84$$$$\nu^{17} +$$$$16\!\cdots\!83$$$$\nu^{16} -$$$$54\!\cdots\!37$$$$\nu^{15} +$$$$14\!\cdots\!76$$$$\nu^{14} -$$$$38\!\cdots\!79$$$$\nu^{13} +$$$$80\!\cdots\!82$$$$\nu^{12} -$$$$15\!\cdots\!39$$$$\nu^{11} +$$$$29\!\cdots\!36$$$$\nu^{10} -$$$$47\!\cdots\!77$$$$\nu^{9} +$$$$61\!\cdots\!98$$$$\nu^{8} -$$$$73\!\cdots\!89$$$$\nu^{7} +$$$$10\!\cdots\!42$$$$\nu^{6} -$$$$34\!\cdots\!52$$$$\nu^{5} -$$$$10\!\cdots\!53$$$$\nu^{4} -$$$$30\!\cdots\!94$$$$\nu^{3} -$$$$13\!\cdots\!61$$$$\nu^{2} -$$$$32\!\cdots\!99$$$$\nu +$$$$13\!\cdots\!18$$$$)/$$$$16\!\cdots\!33$$ $$\beta_{7}$$ $$=$$ $$($$$$99\!\cdots\!65$$$$\nu^{19} -$$$$91\!\cdots\!08$$$$\nu^{18} +$$$$48\!\cdots\!82$$$$\nu^{17} -$$$$15\!\cdots\!23$$$$\nu^{16} +$$$$35\!\cdots\!63$$$$\nu^{15} -$$$$51\!\cdots\!89$$$$\nu^{14} +$$$$10\!\cdots\!41$$$$\nu^{13} +$$$$22\!\cdots\!23$$$$\nu^{12} -$$$$30\!\cdots\!18$$$$\nu^{11} +$$$$11\!\cdots\!55$$$$\nu^{10} -$$$$28\!\cdots\!41$$$$\nu^{9} +$$$$79\!\cdots\!58$$$$\nu^{8} -$$$$16\!\cdots\!38$$$$\nu^{7} +$$$$17\!\cdots\!54$$$$\nu^{6} -$$$$24\!\cdots\!58$$$$\nu^{5} +$$$$41\!\cdots\!78$$$$\nu^{4} +$$$$44\!\cdots\!66$$$$\nu^{3} +$$$$58\!\cdots\!12$$$$\nu^{2} +$$$$28\!\cdots\!24$$$$\nu +$$$$25\!\cdots\!14$$$$)/$$$$24\!\cdots\!99$$ $$\beta_{8}$$ $$=$$ $$($$$$72\!\cdots\!98$$$$\nu^{19} -$$$$64\!\cdots\!39$$$$\nu^{18} +$$$$35\!\cdots\!56$$$$\nu^{17} -$$$$11\!\cdots\!60$$$$\nu^{16} +$$$$31\!\cdots\!51$$$$\nu^{15} -$$$$65\!\cdots\!71$$$$\nu^{14} +$$$$16\!\cdots\!09$$$$\nu^{13} -$$$$18\!\cdots\!27$$$$\nu^{12} +$$$$30\!\cdots\!93$$$$\nu^{11} -$$$$25\!\cdots\!02$$$$\nu^{10} -$$$$23\!\cdots\!07$$$$\nu^{9} +$$$$24\!\cdots\!47$$$$\nu^{8} -$$$$47\!\cdots\!90$$$$\nu^{7} +$$$$84\!\cdots\!06$$$$\nu^{6} -$$$$18\!\cdots\!00$$$$\nu^{5} +$$$$27\!\cdots\!02$$$$\nu^{4} +$$$$44\!\cdots\!98$$$$\nu^{3} +$$$$11\!\cdots\!70$$$$\nu^{2} +$$$$22\!\cdots\!36$$$$\nu +$$$$12\!\cdots\!12$$$$)/$$$$16\!\cdots\!33$$ $$\beta_{9}$$ $$=$$ $$($$$$86\!\cdots\!19$$$$\nu^{19} -$$$$22\!\cdots\!73$$$$\nu^{18} -$$$$14\!\cdots\!55$$$$\nu^{17} +$$$$18\!\cdots\!50$$$$\nu^{16} -$$$$84\!\cdots\!96$$$$\nu^{15} +$$$$27\!\cdots\!94$$$$\nu^{14} -$$$$65\!\cdots\!83$$$$\nu^{13} +$$$$19\!\cdots\!34$$$$\nu^{12} -$$$$36\!\cdots\!42$$$$\nu^{11} +$$$$69\!\cdots\!58$$$$\nu^{10} -$$$$13\!\cdots\!57$$$$\nu^{9} +$$$$22\!\cdots\!48$$$$\nu^{8} -$$$$21\!\cdots\!66$$$$\nu^{7} +$$$$13\!\cdots\!81$$$$\nu^{6} -$$$$39\!\cdots\!51$$$$\nu^{5} +$$$$77\!\cdots\!75$$$$\nu^{4} +$$$$24\!\cdots\!05$$$$\nu^{3} +$$$$33\!\cdots\!68$$$$\nu^{2} +$$$$57\!\cdots\!36$$$$\nu +$$$$68\!\cdots\!56$$$$)/$$$$16\!\cdots\!33$$ $$\beta_{10}$$ $$=$$ $$($$$$13\!\cdots\!54$$$$\nu^{19} -$$$$98\!\cdots\!78$$$$\nu^{18} +$$$$47\!\cdots\!17$$$$\nu^{17} -$$$$13\!\cdots\!55$$$$\nu^{16} +$$$$35\!\cdots\!60$$$$\nu^{15} -$$$$74\!\cdots\!34$$$$\nu^{14} +$$$$25\!\cdots\!90$$$$\nu^{13} -$$$$21\!\cdots\!22$$$$\nu^{12} +$$$$77\!\cdots\!10$$$$\nu^{11} -$$$$11\!\cdots\!48$$$$\nu^{10} +$$$$14\!\cdots\!10$$$$\nu^{9} +$$$$47\!\cdots\!14$$$$\nu^{8} +$$$$29\!\cdots\!55$$$$\nu^{7} -$$$$57\!\cdots\!56$$$$\nu^{6} -$$$$19\!\cdots\!70$$$$\nu^{5} +$$$$20\!\cdots\!00$$$$\nu^{4} +$$$$14\!\cdots\!13$$$$\nu^{3} +$$$$33\!\cdots\!91$$$$\nu^{2} +$$$$62\!\cdots\!94$$$$\nu +$$$$72\!\cdots\!74$$$$)/$$$$24\!\cdots\!99$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$14\!\cdots\!57$$$$\nu^{19} +$$$$14\!\cdots\!41$$$$\nu^{18} -$$$$85\!\cdots\!14$$$$\nu^{17} +$$$$31\!\cdots\!71$$$$\nu^{16} -$$$$93\!\cdots\!83$$$$\nu^{15} +$$$$22\!\cdots\!15$$$$\nu^{14} -$$$$59\!\cdots\!11$$$$\nu^{13} +$$$$11\!\cdots\!32$$$$\nu^{12} -$$$$20\!\cdots\!61$$$$\nu^{11} +$$$$38\!\cdots\!70$$$$\nu^{10} -$$$$58\!\cdots\!49$$$$\nu^{9} +$$$$56\!\cdots\!67$$$$\nu^{8} -$$$$49\!\cdots\!88$$$$\nu^{7} +$$$$96\!\cdots\!08$$$$\nu^{6} +$$$$47\!\cdots\!53$$$$\nu^{5} -$$$$48\!\cdots\!87$$$$\nu^{4} -$$$$68\!\cdots\!99$$$$\nu^{3} -$$$$11\!\cdots\!97$$$$\nu^{2} -$$$$15\!\cdots\!85$$$$\nu -$$$$37\!\cdots\!15$$$$)/$$$$16\!\cdots\!33$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$18\!\cdots\!15$$$$\nu^{19} +$$$$16\!\cdots\!36$$$$\nu^{18} -$$$$82\!\cdots\!23$$$$\nu^{17} +$$$$25\!\cdots\!13$$$$\nu^{16} -$$$$62\!\cdots\!23$$$$\nu^{15} +$$$$11\!\cdots\!58$$$$\nu^{14} -$$$$31\!\cdots\!64$$$$\nu^{13} +$$$$23\!\cdots\!45$$$$\nu^{12} -$$$$37\!\cdots\!80$$$$\nu^{11} +$$$$38\!\cdots\!30$$$$\nu^{10} +$$$$41\!\cdots\!94$$$$\nu^{9} -$$$$52\!\cdots\!12$$$$\nu^{8} +$$$$66\!\cdots\!32$$$$\nu^{7} -$$$$13\!\cdots\!80$$$$\nu^{6} +$$$$27\!\cdots\!99$$$$\nu^{5} -$$$$57\!\cdots\!39$$$$\nu^{4} -$$$$17\!\cdots\!68$$$$\nu^{3} -$$$$27\!\cdots\!42$$$$\nu^{2} -$$$$43\!\cdots\!08$$$$\nu -$$$$43\!\cdots\!97$$$$)/$$$$16\!\cdots\!33$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$20\!\cdots\!94$$$$\nu^{19} +$$$$19\!\cdots\!06$$$$\nu^{18} -$$$$11\!\cdots\!85$$$$\nu^{17} +$$$$39\!\cdots\!57$$$$\nu^{16} -$$$$11\!\cdots\!39$$$$\nu^{15} +$$$$25\!\cdots\!07$$$$\nu^{14} -$$$$65\!\cdots\!84$$$$\nu^{13} +$$$$11\!\cdots\!85$$$$\nu^{12} -$$$$20\!\cdots\!12$$$$\nu^{11} +$$$$34\!\cdots\!38$$$$\nu^{10} -$$$$51\!\cdots\!96$$$$\nu^{9} +$$$$38\!\cdots\!94$$$$\nu^{8} -$$$$25\!\cdots\!11$$$$\nu^{7} +$$$$95\!\cdots\!14$$$$\nu^{6} +$$$$44\!\cdots\!31$$$$\nu^{5} -$$$$44\!\cdots\!87$$$$\nu^{4} -$$$$12\!\cdots\!81$$$$\nu^{3} -$$$$16\!\cdots\!94$$$$\nu^{2} -$$$$40\!\cdots\!91$$$$\nu -$$$$27\!\cdots\!67$$$$)/$$$$16\!\cdots\!33$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$20\!\cdots\!37$$$$\nu^{19} +$$$$20\!\cdots\!54$$$$\nu^{18} -$$$$12\!\cdots\!94$$$$\nu^{17} +$$$$45\!\cdots\!68$$$$\nu^{16} -$$$$13\!\cdots\!10$$$$\nu^{15} +$$$$33\!\cdots\!09$$$$\nu^{14} -$$$$85\!\cdots\!55$$$$\nu^{13} +$$$$15\!\cdots\!63$$$$\nu^{12} -$$$$26\!\cdots\!16$$$$\nu^{11} +$$$$45\!\cdots\!57$$$$\nu^{10} -$$$$60\!\cdots\!89$$$$\nu^{9} +$$$$31\!\cdots\!82$$$$\nu^{8} +$$$$16\!\cdots\!16$$$$\nu^{7} -$$$$46\!\cdots\!52$$$$\nu^{6} +$$$$46\!\cdots\!71$$$$\nu^{5} -$$$$10\!\cdots\!79$$$$\nu^{4} -$$$$80\!\cdots\!78$$$$\nu^{3} -$$$$19\!\cdots\!61$$$$\nu^{2} -$$$$27\!\cdots\!25$$$$\nu +$$$$10\!\cdots\!02$$$$)/$$$$16\!\cdots\!33$$ $$\beta_{15}$$ $$=$$ $$($$$$22\!\cdots\!47$$$$\nu^{19} -$$$$20\!\cdots\!28$$$$\nu^{18} +$$$$11\!\cdots\!10$$$$\nu^{17} -$$$$39\!\cdots\!45$$$$\nu^{16} +$$$$11\!\cdots\!97$$$$\nu^{15} -$$$$25\!\cdots\!32$$$$\nu^{14} +$$$$66\!\cdots\!65$$$$\nu^{13} -$$$$10\!\cdots\!57$$$$\nu^{12} +$$$$19\!\cdots\!61$$$$\nu^{11} -$$$$30\!\cdots\!52$$$$\nu^{10} +$$$$40\!\cdots\!56$$$$\nu^{9} -$$$$92\!\cdots\!97$$$$\nu^{8} -$$$$85\!\cdots\!88$$$$\nu^{7} -$$$$29\!\cdots\!64$$$$\nu^{6} -$$$$22\!\cdots\!54$$$$\nu^{5} +$$$$65\!\cdots\!22$$$$\nu^{4} +$$$$13\!\cdots\!28$$$$\nu^{3} +$$$$28\!\cdots\!70$$$$\nu^{2} +$$$$54\!\cdots\!38$$$$\nu +$$$$41\!\cdots\!45$$$$)/$$$$16\!\cdots\!33$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$41\!\cdots\!56$$$$\nu^{19} +$$$$37\!\cdots\!90$$$$\nu^{18} -$$$$20\!\cdots\!38$$$$\nu^{17} +$$$$67\!\cdots\!96$$$$\nu^{16} -$$$$18\!\cdots\!21$$$$\nu^{15} +$$$$39\!\cdots\!38$$$$\nu^{14} -$$$$10\!\cdots\!11$$$$\nu^{13} +$$$$14\!\cdots\!80$$$$\nu^{12} -$$$$24\!\cdots\!70$$$$\nu^{11} +$$$$35\!\cdots\!67$$$$\nu^{10} -$$$$39\!\cdots\!91$$$$\nu^{9} -$$$$39\!\cdots\!96$$$$\nu^{8} +$$$$88\!\cdots\!38$$$$\nu^{7} +$$$$51\!\cdots\!07$$$$\nu^{6} +$$$$44\!\cdots\!44$$$$\nu^{5} -$$$$12\!\cdots\!74$$$$\nu^{4} -$$$$26\!\cdots\!04$$$$\nu^{3} -$$$$39\!\cdots\!66$$$$\nu^{2} -$$$$82\!\cdots\!55$$$$\nu -$$$$79\!\cdots\!21$$$$)/$$$$24\!\cdots\!99$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$42\!\cdots\!55$$$$\nu^{19} +$$$$38\!\cdots\!02$$$$\nu^{18} -$$$$21\!\cdots\!06$$$$\nu^{17} +$$$$74\!\cdots\!04$$$$\nu^{16} -$$$$21\!\cdots\!20$$$$\nu^{15} +$$$$48\!\cdots\!66$$$$\nu^{14} -$$$$13\!\cdots\!36$$$$\nu^{13} +$$$$21\!\cdots\!90$$$$\nu^{12} -$$$$42\!\cdots\!44$$$$\nu^{11} +$$$$71\!\cdots\!16$$$$\nu^{10} -$$$$10\!\cdots\!01$$$$\nu^{9} +$$$$56\!\cdots\!72$$$$\nu^{8} -$$$$56\!\cdots\!94$$$$\nu^{7} +$$$$15\!\cdots\!80$$$$\nu^{6} +$$$$37\!\cdots\!79$$$$\nu^{5} -$$$$11\!\cdots\!01$$$$\nu^{4} -$$$$28\!\cdots\!43$$$$\nu^{3} -$$$$56\!\cdots\!25$$$$\nu^{2} -$$$$89\!\cdots\!45$$$$\nu -$$$$52\!\cdots\!51$$$$)/$$$$24\!\cdots\!99$$ $$\beta_{18}$$ $$=$$ $$($$$$-$$$$49\!\cdots\!62$$$$\nu^{19} +$$$$44\!\cdots\!75$$$$\nu^{18} -$$$$23\!\cdots\!77$$$$\nu^{17} +$$$$78\!\cdots\!27$$$$\nu^{16} -$$$$20\!\cdots\!71$$$$\nu^{15} +$$$$43\!\cdots\!38$$$$\nu^{14} -$$$$11\!\cdots\!35$$$$\nu^{13} +$$$$17\!\cdots\!30$$$$\nu^{12} -$$$$31\!\cdots\!42$$$$\nu^{11} +$$$$55\!\cdots\!86$$$$\nu^{10} -$$$$81\!\cdots\!82$$$$\nu^{9} +$$$$14\!\cdots\!69$$$$\nu^{8} -$$$$23\!\cdots\!82$$$$\nu^{7} +$$$$24\!\cdots\!45$$$$\nu^{6} +$$$$19\!\cdots\!17$$$$\nu^{5} -$$$$12\!\cdots\!73$$$$\nu^{4} -$$$$36\!\cdots\!46$$$$\nu^{3} -$$$$48\!\cdots\!87$$$$\nu^{2} -$$$$89\!\cdots\!79$$$$\nu -$$$$72\!\cdots\!42$$$$)/$$$$24\!\cdots\!99$$ $$\beta_{19}$$ $$=$$ $$($$$$-$$$$49\!\cdots\!86$$$$\nu^{19} +$$$$48\!\cdots\!30$$$$\nu^{18} -$$$$28\!\cdots\!51$$$$\nu^{17} +$$$$10\!\cdots\!42$$$$\nu^{16} -$$$$31\!\cdots\!28$$$$\nu^{15} +$$$$75\!\cdots\!16$$$$\nu^{14} -$$$$19\!\cdots\!94$$$$\nu^{13} +$$$$34\!\cdots\!90$$$$\nu^{12} -$$$$64\!\cdots\!14$$$$\nu^{11} +$$$$10\!\cdots\!72$$$$\nu^{10} -$$$$16\!\cdots\!64$$$$\nu^{9} +$$$$14\!\cdots\!42$$$$\nu^{8} -$$$$13\!\cdots\!22$$$$\nu^{7} +$$$$20\!\cdots\!91$$$$\nu^{6} +$$$$21\!\cdots\!50$$$$\nu^{5} -$$$$11\!\cdots\!46$$$$\nu^{4} -$$$$29\!\cdots\!46$$$$\nu^{3} -$$$$47\!\cdots\!06$$$$\nu^{2} -$$$$10\!\cdots\!34$$$$\nu -$$$$39\!\cdots\!62$$$$)/$$$$16\!\cdots\!33$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{16} - \beta_{15} - \beta_{13} + \beta_{12} - \beta_{5} - \beta_{4} + 5 \beta_{3} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{19} - 2 \beta_{18} + 3 \beta_{17} - \beta_{15} - 2 \beta_{14} + 2 \beta_{13} + 7 \beta_{12} - \beta_{11} + 3 \beta_{10} + 2 \beta_{9} - 5 \beta_{8} + 3 \beta_{7} + 4 \beta_{6} - 5 \beta_{4} + 6 \beta_{3} - 4 \beta_{2} - 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-16 \beta_{19} - 19 \beta_{18} + 19 \beta_{17} + 16 \beta_{16} - 2 \beta_{15} - 3 \beta_{14} + 14 \beta_{13} + 25 \beta_{12} + 17 \beta_{10} + 17 \beta_{9} - 11 \beta_{8} + 20 \beta_{7} - 6 \beta_{6} + 12 \beta_{5} + 3 \beta_{4} + 8 \beta_{3} - 45 \beta_{2} - 20 \beta_{1} + 5$$ $$\nu^{5}$$ $$=$$ $$-59 \beta_{18} + 42 \beta_{17} + 59 \beta_{16} - 57 \beta_{15} + 36 \beta_{14} - 66 \beta_{13} + 66 \beta_{12} + 27 \beta_{11} + 42 \beta_{10} + 57 \beta_{9} + 177 \beta_{8} + 37 \beta_{7} - 204 \beta_{6} + 20 \beta_{5} + 37 \beta_{4} + 37 \beta_{3} - 177 \beta_{2} - 57 \beta_{1} + 84$$ $$\nu^{6}$$ $$=$$ $$234 \beta_{19} + 27 \beta_{17} + 71 \beta_{16} - 298 \beta_{15} + 170 \beta_{14} - 705 \beta_{13} + 86 \beta_{12} + 62 \beta_{11} + 71 \beta_{10} + 86 \beta_{9} + 1071 \beta_{8} - 170 \beta_{7} - 875 \beta_{6} - 125 \beta_{5} + 86 \beta_{4} + 148 \beta_{3} - 381 \beta_{2} - 27 \beta_{1} + 298$$ $$\nu^{7}$$ $$=$$ $$946 \beta_{19} + 946 \beta_{18} - 35 \beta_{17} - 244 \beta_{16} + 122 \beta_{15} + 366 \beta_{14} - 2109 \beta_{13} - 591 \beta_{12} - 514 \beta_{11} - 366 \beta_{9} + 2005 \beta_{8} - 1312 \beta_{7} - 946 \beta_{6} - 1059 \beta_{5} - 35 \beta_{4} - 122 \beta_{3} + 514 \beta_{2} + 244 \beta_{1} - 591$$ $$\nu^{8}$$ $$=$$ $$1258 \beta_{19} + 4679 \beta_{18} + 167 \beta_{17} - 713 \beta_{16} + 8585 \beta_{15} - 167 \beta_{14} + 2371 \beta_{13} - 6559 \beta_{12} - 3876 \beta_{11} - 1258 \beta_{10} - 3629 \beta_{9} - 6414 \beta_{8} - 3629 \beta_{7} + 8458 \beta_{6} - 2253 \beta_{5} - 713 \beta_{4} - 6559 \beta_{3} + 8752 \beta_{2} - 9549$$ $$\nu^{9}$$ $$=$$ $$-3325 \beta_{19} + 11070 \beta_{18} - 2348 \beta_{17} + 2906 \beta_{16} + 46259 \beta_{15} - 2906 \beta_{14} + 41513 \beta_{13} - 35187 \beta_{12} - 7284 \beta_{11} - 11070 \beta_{10} - 15694 \beta_{9} - 52490 \beta_{8} - 2348 \beta_{7} + 48571 \beta_{6} + 7284 \beta_{5} - 42340 \beta_{3} + 38093 \beta_{2} - 3325 \beta_{1} - 38188$$ $$\nu^{10}$$ $$=$$ $$-10478 \beta_{19} + 9998 \beta_{18} - 44227 \beta_{17} + 12049 \beta_{16} + 101459 \beta_{15} + 143351 \beta_{13} - 119150 \beta_{12} + 30124 \beta_{11} - 60233 \beta_{10} - 44227 \beta_{9} - 129628 \beta_{8} + 12049 \beta_{7} + 101333 \beta_{6} + 57232 \beta_{5} + 10478 \beta_{4} - 132873 \beta_{3} + 90855 \beta_{2} + 9998 \beta_{1} - 44227$$ $$\nu^{11}$$ $$=$$ $$40344 \beta_{19} - 3725 \beta_{18} - 276261 \beta_{17} - 82394 \beta_{16} - 138853 \beta_{15} + 40344 \beta_{14} + 111905 \beta_{13} - 204811 \beta_{12} + 221247 \beta_{11} - 211908 \beta_{10} - 82394 \beta_{9} + 32635 \beta_{8} - 32635 \beta_{6} + 122417 \beta_{5} + 3725 \beta_{4} - 82394 \beta_{3} + 111905 \beta_{2} + 211908 \beta_{1} + 247584$$ $$\nu^{12}$$ $$=$$ $$158777 \beta_{19} + 10512 \beta_{18} - 835558 \beta_{17} - 835558 \beta_{16} - 1743371 \beta_{15} + 10512 \beta_{14} - 746572 \beta_{13} + 473990 \beta_{12} + 473990 \beta_{11} - 334325 \beta_{10} + 757084 \beta_{8} - 158777 \beta_{7} - 357772 \beta_{6} - 188483 \beta_{5} - 334325 \beta_{4} + 1316101 \beta_{3} - 10512 \beta_{2} + 1073803 \beta_{1} + 1316101$$ $$\nu^{13}$$ $$=$$ $$-1286101 \beta_{19} - 728012 \beta_{18} - 2844417 \beta_{16} - 5660088 \beta_{15} - 1299036 \beta_{14} - 1491315 \beta_{13} + 5660088 \beta_{12} - 702058 \beta_{11} + 1299036 \beta_{10} + 1286101 \beta_{9} - 1299036 \beta_{8} + 728012 \beta_{7} + 2001094 \beta_{6} - 1491315 \beta_{5} - 1915914 \beta_{4} + 7262566 \beta_{3} - 1123540 \beta_{2} + 1915914 \beta_{1} + 2422576$$ $$\nu^{14}$$ $$=$$ $$-12421513 \beta_{19} - 10385128 \beta_{18} + 12421513 \beta_{17} - 9310711 \beta_{15} - 7380965 \beta_{14} + 7684182 \beta_{13} + 27876859 \beta_{12} - 5647797 \beta_{11} + 12447467 \beta_{10} + 10385128 \beta_{9} - 17603825 \beta_{8} + 12447467 \beta_{7} + 10491767 \beta_{6} - 3518392 \beta_{4} + 21688844 \beta_{3} - 14307879 \beta_{2} - 7380965 \beta_{1} + 2198653$$ $$\nu^{15}$$ $$=$$ $$-41887018 \beta_{19} - 61912028 \beta_{18} + 61912028 \beta_{17} + 41887018 \beta_{16} - 23729214 \beta_{15} - 13568980 \beta_{14} + 18157804 \beta_{13} + 98772521 \beta_{12} + 54227846 \beta_{10} + 54227846 \beta_{9} - 4183627 \beta_{8} + 61446543 \beta_{7} - 43316523 \beta_{6} + 21185098 \beta_{5} + 13568980 \beta_{4} + 51272371 \beta_{3} - 108720413 \beta_{2} - 61446543 \beta_{1} + 33042748$$ $$\nu^{16}$$ $$=$$ $$-184832898 \beta_{18} + 159456397 \beta_{17} + 184832898 \beta_{16} - 187860192 \beta_{15} + 51834910 \beta_{14} - 260741535 \beta_{13} + 260741535 \beta_{12} + 79842443 \beta_{11} + 159456397 \beta_{10} + 187860192 \beta_{9} + 494513837 \beta_{8} + 108946693 \beta_{7} - 584791977 \beta_{6} + 33674795 \beta_{5} + 108946693 \beta_{4} + 154185397 \beta_{3} - 494513837 \beta_{2} - 187860192 \beta_{1} + 267702635$$ $$\nu^{17}$$ $$=$$ $$617830033 \beta_{19} + 206032295 \beta_{17} + 311251239 \beta_{16} - 716403632 \beta_{15} + 418605200 \beta_{14} - 2022568957 \beta_{13} + 294416330 \beta_{12} + 122214018 \beta_{11} + 311251239 \beta_{10} + 294416330 \beta_{9} + 2785864285 \beta_{8} - 418605200 \beta_{7} - 2441174157 \beta_{6} - 412491513 \beta_{5} + 294416330 \beta_{4} + 416630348 \beta_{3} - 1125513043 \beta_{2} - 206032295 \beta_{1} + 716403632$$ $$\nu^{18}$$ $$=$$ $$2752425396 \beta_{19} + 2752425396 \beta_{18} + 83818277 \beta_{17} - 304339042 \beta_{16} + 1171879029 \beta_{15} + 1142347952 \beta_{14} - 5605852118 \beta_{13} - 2298968873 \beta_{12} - 1596270520 \beta_{11} - 1142347952 \beta_{9} + 5427476594 \beta_{8} - 3515720724 \beta_{7} - 2752425396 \beta_{6} - 2675051198 \beta_{5} + 83818277 \beta_{4} - 1171879029 \beta_{3} + 1596270520 \beta_{2} + 304339042 \beta_{1} - 2298968873$$ $$\nu^{19}$$ $$=$$ $$4830873504 \beta_{19} + 13952446346 \beta_{18} - 398359311 \beta_{17} - 1431428886 \beta_{16} + 25882340842 \beta_{15} + 398359311 \beta_{14} + 6190771922 \beta_{13} - 21651999752 \beta_{12} - 10260369542 \beta_{11} - 4830873504 \beta_{10} - 11021645426 \beta_{9} - 16052782153 \beta_{8} - 11021645426 \beta_{7} + 22232251405 \beta_{6} - 5051416075 \beta_{5} - 1431428886 \beta_{4} - 21651999752 \beta_{3} + 25483981531 \beta_{2} - 27461484220$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/483\mathbb{Z}\right)^\times$$.

 $$n$$ $$323$$ $$346$$ $$442$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{11}$$

## 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
 2.10485 − 0.618040i −1.22309 + 0.359132i −1.29470 − 1.49416i 1.49752 + 1.72823i −1.02673 − 0.659842i 3.28340 + 2.11011i 0.216617 + 1.50661i −0.318425 − 2.21469i 1.58077 + 3.46140i −0.820216 − 1.79602i 1.58077 − 3.46140i −0.820216 + 1.79602i −1.02673 + 0.659842i 3.28340 − 2.11011i −1.29470 + 1.49416i 1.49752 − 1.72823i 2.10485 + 0.618040i −1.22309 − 0.359132i 0.216617 − 1.50661i −0.318425 + 2.21469i
0.273100 + 0.0801894i −0.654861 + 0.755750i −1.61435 1.03748i −0.0390977 + 0.271930i −0.239446 + 0.153882i 0.415415 + 0.909632i −0.730471 0.843008i −0.142315 0.989821i −0.0324835 + 0.0711290i
64.2 0.273100 + 0.0801894i −0.654861 + 0.755750i −1.61435 1.03748i 0.454513 3.16121i −0.239446 + 0.153882i 0.415415 + 0.909632i −0.730471 0.843008i −0.142315 0.989821i 0.377623 0.826879i
85.1 −0.544078 + 0.627899i 0.841254 0.540641i 0.186393 + 1.29639i −1.36538 2.98976i −0.118239 + 0.822373i −0.959493 + 0.281733i −2.31329 1.48666i 0.415415 0.909632i 2.62014 + 0.769343i
85.2 −0.544078 + 0.627899i 0.841254 0.540641i 0.186393 + 1.29639i 0.405886 + 0.888766i −0.118239 + 0.822373i −0.959493 + 0.281733i −2.31329 1.48666i 0.415415 0.909632i −0.778889 0.228702i
127.1 −1.61435 + 1.03748i −0.142315 + 0.989821i 0.698939 1.53046i −2.78540 + 0.817866i −0.797176 1.74557i −0.654861 0.755750i −0.0867074 0.603063i −0.959493 0.281733i 3.64809 4.21012i
127.2 −1.61435 + 1.03748i −0.142315 + 0.989821i 0.698939 1.53046i 2.13054 0.625582i −0.797176 1.74557i −0.654861 0.755750i −0.0867074 0.603063i −0.959493 0.281733i −2.79041 + 3.22030i
169.1 0.186393 1.29639i 0.415415 + 0.909632i 0.273100 + 0.0801894i −0.810370 0.935217i 1.25667 0.368991i 0.841254 + 0.540641i 1.24302 2.72183i −0.654861 + 0.755750i −1.36345 + 0.876238i
169.2 0.186393 1.29639i 0.415415 + 0.909632i 0.273100 + 0.0801894i 1.65162 + 1.90608i 1.25667 0.368991i 0.841254 + 0.540641i 1.24302 2.72183i −0.654861 + 0.755750i 2.77887 1.78587i
190.1 0.698939 1.53046i −0.959493 0.281733i −0.544078 0.627899i −2.50227 + 1.60811i −1.10181 + 1.27155i −0.142315 + 0.989821i 1.88745 0.554206i 0.841254 + 0.540641i 0.712219 + 4.95359i
190.2 0.698939 1.53046i −0.959493 0.281733i −0.544078 0.627899i 2.35995 1.51665i −1.10181 + 1.27155i −0.142315 + 0.989821i 1.88745 0.554206i 0.841254 + 0.540641i −0.671712 4.67186i
211.1 0.698939 + 1.53046i −0.959493 + 0.281733i −0.544078 + 0.627899i −2.50227 1.60811i −1.10181 1.27155i −0.142315 0.989821i 1.88745 + 0.554206i 0.841254 0.540641i 0.712219 4.95359i
211.2 0.698939 + 1.53046i −0.959493 + 0.281733i −0.544078 + 0.627899i 2.35995 + 1.51665i −1.10181 1.27155i −0.142315 0.989821i 1.88745 + 0.554206i 0.841254 0.540641i −0.671712 + 4.67186i
232.1 −1.61435 1.03748i −0.142315 0.989821i 0.698939 + 1.53046i −2.78540 0.817866i −0.797176 + 1.74557i −0.654861 + 0.755750i −0.0867074 + 0.603063i −0.959493 + 0.281733i 3.64809 + 4.21012i
232.2 −1.61435 1.03748i −0.142315 0.989821i 0.698939 + 1.53046i 2.13054 + 0.625582i −0.797176 + 1.74557i −0.654861 + 0.755750i −0.0867074 + 0.603063i −0.959493 + 0.281733i −2.79041 3.22030i
358.1 −0.544078 0.627899i 0.841254 + 0.540641i 0.186393 1.29639i −1.36538 + 2.98976i −0.118239 0.822373i −0.959493 0.281733i −2.31329 + 1.48666i 0.415415 + 0.909632i 2.62014 0.769343i
358.2 −0.544078 0.627899i 0.841254 + 0.540641i 0.186393 1.29639i 0.405886 0.888766i −0.118239 0.822373i −0.959493 0.281733i −2.31329 + 1.48666i 0.415415 + 0.909632i −0.778889 + 0.228702i
400.1 0.273100 0.0801894i −0.654861 0.755750i −1.61435 + 1.03748i −0.0390977 0.271930i −0.239446 0.153882i 0.415415 0.909632i −0.730471 + 0.843008i −0.142315 + 0.989821i −0.0324835 0.0711290i
400.2 0.273100 0.0801894i −0.654861 0.755750i −1.61435 + 1.03748i 0.454513 + 3.16121i −0.239446 0.153882i 0.415415 0.909632i −0.730471 + 0.843008i −0.142315 + 0.989821i 0.377623 + 0.826879i
463.1 0.186393 + 1.29639i 0.415415 0.909632i 0.273100 0.0801894i −0.810370 + 0.935217i 1.25667 + 0.368991i 0.841254 0.540641i 1.24302 + 2.72183i −0.654861 0.755750i −1.36345 0.876238i
463.2 0.186393 + 1.29639i 0.415415 0.909632i 0.273100 0.0801894i 1.65162 1.90608i 1.25667 + 0.368991i 0.841254 0.540641i 1.24302 + 2.72183i −0.654861 0.755750i 2.77887 + 1.78587i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 463.2 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 483.2.q.c 20
23.c even 11 1 inner 483.2.q.c 20

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(483, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 5 T + 3 T^{2} + 7 T^{3} + 20 T^{4} + 10 T^{5} + 16 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2}$$
$3$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}$$
$5$ $$223729 + 145684 T + 3022701 T^{2} - 952391 T^{3} + 2071883 T^{4} + 473330 T^{5} + 971366 T^{6} - 1039555 T^{7} + 97471 T^{8} + 141240 T^{9} - 21174 T^{10} - 11725 T^{11} + 3609 T^{12} + 474 T^{13} + 309 T^{14} + 56 T^{15} + 12 T^{16} + T^{17} + T^{18} + T^{19} + T^{20}$$
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}$$
$11$ $$2778449521 + 9633831337 T + 16151136745 T^{2} + 16106902539 T^{3} + 10354859191 T^{4} + 4382886590 T^{5} + 1304755119 T^{6} + 292047590 T^{7} + 47681363 T^{8} - 702757 T^{9} - 376848 T^{10} - 491480 T^{11} + 70238 T^{12} + 10576 T^{13} - 2102 T^{14} + 2910 T^{15} + 293 T^{16} + 15 T^{17} + 35 T^{18} - 3 T^{19} + T^{20}$$
$13$ $$210511081 + 666006627 T + 1021488776 T^{2} + 1071178757 T^{3} + 835774588 T^{4} + 468801047 T^{5} + 209998360 T^{6} + 88622017 T^{7} + 33712679 T^{8} + 7496115 T^{9} + 1270600 T^{10} + 186192 T^{11} + 43585 T^{12} - 8419 T^{13} + 2990 T^{14} - 738 T^{15} + 137 T^{16} - 69 T^{17} + 15 T^{18} + 2 T^{19} + T^{20}$$
$17$ $$14014061161 + 52448110145 T + 142434369480 T^{2} + 187207017039 T^{3} + 142164495061 T^{4} + 68109215103 T^{5} + 21541051760 T^{6} + 4611104323 T^{7} + 708366091 T^{8} + 81506480 T^{9} + 4553792 T^{10} - 307747 T^{11} + 95582 T^{12} - 14811 T^{13} - 6227 T^{14} + 3448 T^{15} + 150 T^{16} - 114 T^{17} + 68 T^{18} - 8 T^{19} + T^{20}$$
$19$ $$5727311041 + 10282430051 T + 67795446382 T^{2} - 22448271618 T^{3} + 190229777005 T^{4} - 95237640350 T^{5} + 20538413638 T^{6} + 7658020484 T^{7} - 1494102353 T^{8} - 283877767 T^{9} + 68284667 T^{10} + 6134781 T^{11} - 418494 T^{12} - 286655 T^{13} + 3101 T^{14} + 6660 T^{15} + 1021 T^{16} - 350 T^{17} + 38 T^{18} - 6 T^{19} + T^{20}$$
$23$ $$41426511213649 - 19812679276093 T + 6108256851918 T^{2} - 1011233157759 T^{3} - 9622332785 T^{4} + 50409438376 T^{5} - 15532574705 T^{6} + 2111278675 T^{7} + 104934556 T^{8} - 124996421 T^{9} + 37583943 T^{10} - 5434627 T^{11} + 198364 T^{12} + 173525 T^{13} - 55505 T^{14} + 7832 T^{15} - 65 T^{16} - 297 T^{17} + 78 T^{18} - 11 T^{19} + T^{20}$$
$29$ $$13615855969 + 7568318820 T + 22578888597 T^{2} - 22815471579 T^{3} + 25429594831 T^{4} - 40139177875 T^{5} + 51363741811 T^{6} - 15078603343 T^{7} + 1800176944 T^{8} + 130291376 T^{9} - 40671235 T^{10} + 18055655 T^{11} - 3790014 T^{12} + 350946 T^{13} + 220299 T^{14} - 79942 T^{15} + 17150 T^{16} - 2470 T^{17} + 273 T^{18} - 23 T^{19} + T^{20}$$
$31$ $$1745041 + 35507159 T + 424911969 T^{2} + 1918623544 T^{3} + 7090922773 T^{4} + 3577474638 T^{5} + 1320284972 T^{6} + 707753968 T^{7} + 329843351 T^{8} + 79301816 T^{9} + 21183570 T^{10} + 6957082 T^{11} + 1697130 T^{12} + 331840 T^{13} + 32933 T^{14} - 13221 T^{15} - 3245 T^{16} + 149 T^{17} + 58 T^{18} - T^{19} + T^{20}$$
$37$ $$6032784241 + 14402300517 T + 6491321122 T^{2} - 10651774716 T^{3} - 1843964980 T^{4} + 20861820760 T^{5} + 22953395938 T^{6} + 11230715045 T^{7} + 3373268261 T^{8} + 542009171 T^{9} + 119476996 T^{10} - 3983480 T^{11} + 3107778 T^{12} - 1513970 T^{13} + 40918 T^{14} - 27279 T^{15} + 5956 T^{16} + 586 T^{17} + 169 T^{18} + 9 T^{19} + T^{20}$$
$41$ $$7334352656401 - 5498455073898 T + 6487746537590 T^{2} - 13046184277701 T^{3} + 10418392776044 T^{4} - 3743750574398 T^{5} + 839792046593 T^{6} - 136839989573 T^{7} + 2625131066 T^{8} + 477581148 T^{9} + 994460347 T^{10} - 103662372 T^{11} + 54438798 T^{12} + 1808710 T^{13} + 633802 T^{14} + 131055 T^{15} + 22297 T^{16} + 1283 T^{17} + 189 T^{18} + 15 T^{19} + T^{20}$$
$43$ $$744809433692809 + 1053379945237024 T + 1166217120390548 T^{2} + 961356031498506 T^{3} + 613500817160671 T^{4} + 293849335610337 T^{5} + 102811417983580 T^{6} + 25295999654492 T^{7} + 4414396609915 T^{8} + 569224654823 T^{9} + 56366416943 T^{10} + 4693917426 T^{11} + 466814185 T^{12} + 59971358 T^{13} + 6384962 T^{14} + 471670 T^{15} + 25839 T^{16} + 2114 T^{17} + 293 T^{18} + 23 T^{19} + T^{20}$$
$47$ $$( -22732832 - 15895088 T + 535016 T^{2} + 2433772 T^{3} + 387348 T^{4} - 78089 T^{5} - 22999 T^{6} - 506 T^{7} + 315 T^{8} + 33 T^{9} + T^{10} )^{2}$$
$53$ $$7927879291201 - 9691328706848 T + 13808569186524 T^{2} - 23634487245835 T^{3} + 9871783181751 T^{4} + 681746180252 T^{5} + 6550182032371 T^{6} - 1270598036108 T^{7} + 1511262766269 T^{8} + 219002145054 T^{9} - 3910490275 T^{10} - 1132713890 T^{11} + 522675091 T^{12} - 2662153 T^{13} - 757243 T^{14} + 372807 T^{15} - 3305 T^{16} + 515 T^{17} + 183 T^{18} - 9 T^{19} + T^{20}$$
$59$ $$12638970647689 - 48601547767730 T + 153555776734759 T^{2} - 292119411333711 T^{3} + 427081840983072 T^{4} - 422917682840388 T^{5} + 271469560326821 T^{6} - 109846330178461 T^{7} + 29418690036016 T^{8} - 5511730814975 T^{9} + 751526400043 T^{10} - 75905988906 T^{11} + 5707658915 T^{12} - 336231700 T^{13} + 21068592 T^{14} - 2128686 T^{15} + 242858 T^{16} - 21184 T^{17} + 1285 T^{18} - 49 T^{19} + T^{20}$$
$61$ $$571516344169 - 2002165042644 T + 2911045047553 T^{2} - 1846355188807 T^{3} + 974780048143 T^{4} - 746830969369 T^{5} + 558978136505 T^{6} - 258384394064 T^{7} + 113026891315 T^{8} - 39309793028 T^{9} + 12598729969 T^{10} - 3398485684 T^{11} + 762257701 T^{12} - 149014900 T^{13} + 23764006 T^{14} - 2860716 T^{15} + 264264 T^{16} - 19664 T^{17} + 1145 T^{18} - 46 T^{19} + T^{20}$$
$67$ $$4285227066241 - 726686742397 T - 192978918737 T^{2} - 1389237427413 T^{3} + 970133207010 T^{4} - 264163388115 T^{5} + 115361686836 T^{6} - 77756122477 T^{7} + 45919737468 T^{8} - 13977028648 T^{9} + 2023508708 T^{10} + 75647337 T^{11} - 60374221 T^{12} + 10549871 T^{13} + 8990 T^{14} - 147214 T^{15} + 30265 T^{16} - 3349 T^{17} + 317 T^{18} - 14 T^{19} + T^{20}$$
$71$ $$52922322201529 + 562237705965485 T + 2038595793475230 T^{2} - 946296075459526 T^{3} + 703859091628143 T^{4} - 375394813373624 T^{5} + 181056493420726 T^{6} - 64465015013102 T^{7} + 16627519681311 T^{8} - 3221212583930 T^{9} + 474735676416 T^{10} - 52125821411 T^{11} + 3946347531 T^{12} - 139880189 T^{13} - 9147003 T^{14} + 1737339 T^{15} - 102901 T^{16} - 478 T^{17} + 526 T^{18} - 36 T^{19} + T^{20}$$
$73$ $$2186895646828878129 + 354306532199502051 T - 107092896700427208 T^{2} - 11057812781143734 T^{3} + 3252192819860493 T^{4} + 23298722954919 T^{5} - 39583990809927 T^{6} + 1250602384923 T^{7} + 1498853463849 T^{8} + 247227112731 T^{9} + 68724652546 T^{10} + 7326200935 T^{11} + 963766015 T^{12} + 91338454 T^{13} + 9905392 T^{14} + 451885 T^{15} + 78079 T^{16} + 952 T^{17} + 361 T^{18} + T^{19} + T^{20}$$
$79$ $$1322647498704841 + 1684701976237544 T + 3791696313856652 T^{2} - 328833730919881 T^{3} + 532343579142835 T^{4} - 53024044437485 T^{5} - 8335194074961 T^{6} - 4335689904282 T^{7} + 199180237095 T^{8} + 64848940266 T^{9} + 44333112517 T^{10} - 1084213680 T^{11} - 286221790 T^{12} - 57394942 T^{13} + 8844251 T^{14} - 57750 T^{15} - 26604 T^{16} - 1364 T^{17} + 268 T^{18} + 22 T^{19} + T^{20}$$
$83$ $$211498849 + 13481215570 T + 173290067710 T^{2} - 2277848931684 T^{3} + 5914549581908 T^{4} + 487304022655 T^{5} + 908531102481 T^{6} + 36815774610 T^{7} + 85039231394 T^{8} + 11488177899 T^{9} + 5354389074 T^{10} + 712903521 T^{11} + 181161835 T^{12} + 18914605 T^{13} + 3430996 T^{14} + 251338 T^{15} + 22858 T^{16} + 223 T^{17} + 16 T^{18} - 8 T^{19} + T^{20}$$
$89$ $$3480098656354321 - 15154989883020417 T + 25553005569601022 T^{2} - 21154443569745917 T^{3} + 10131658278794998 T^{4} - 3154566750911093 T^{5} + 711072249692058 T^{6} - 124891147793056 T^{7} + 17788042701563 T^{8} - 2007553443181 T^{9} + 193155259353 T^{10} - 13316684788 T^{11} + 1107986490 T^{12} - 57066939 T^{13} + 9046387 T^{14} - 250328 T^{15} + 60005 T^{16} + 852 T^{17} + 248 T^{18} + 2 T^{19} + T^{20}$$
$97$ $$662287188273481 - 16914836685336604 T + 122774597498714641 T^{2} - 83950131758399724 T^{3} + 33787489392721909 T^{4} - 8579302765171806 T^{5} + 1817489728152148 T^{6} - 212848050096772 T^{7} + 39806544688586 T^{8} - 2853889949702 T^{9} + 956171985506 T^{10} - 54799051604 T^{11} + 3973796847 T^{12} - 224114786 T^{13} + 10074842 T^{14} - 1319948 T^{15} + 120278 T^{16} - 5990 T^{17} + 318 T^{18} - 18 T^{19} + T^{20}$$