# Properties

 Label 273.2.l.c Level $273$ Weight $2$ Character orbit 273.l Analytic conductor $2.180$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.l (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 18 x^{18} - 4 x^{17} + 211 x^{16} - 59 x^{15} + 1458 x^{14} - 526 x^{13} + 7324 x^{12} - 2645 x^{11} + 23428 x^{10} - 8506 x^{9} + 54235 x^{8} - 18801 x^{7} + 74141 x^{6} - 25533 x^{5} + 68867 x^{4} - 16327 x^{3} + 16588 x^{2} + 2997 x + 1369$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 18 x^{18} - 4 x^{17} + 211 x^{16} - 59 x^{15} + 1458 x^{14} - 526 x^{13} + 7324 x^{12} - 2645 x^{11} + 23428 x^{10} - 8506 x^{9} + 54235 x^{8} - 18801 x^{7} + 74141 x^{6} - 25533 x^{5} + 68867 x^{4} - 16327 x^{3} + 16588 x^{2} + 2997 x + 1369$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$40\!\cdots\!97$$$$\nu^{19} +$$$$29\!\cdots\!91$$$$\nu^{18} -$$$$71\!\cdots\!53$$$$\nu^{17} +$$$$64\!\cdots\!94$$$$\nu^{16} -$$$$84\!\cdots\!57$$$$\nu^{15} +$$$$77\!\cdots\!67$$$$\nu^{14} -$$$$58\!\cdots\!55$$$$\nu^{13} +$$$$54\!\cdots\!85$$$$\nu^{12} -$$$$29\!\cdots\!73$$$$\nu^{11} +$$$$25\!\cdots\!36$$$$\nu^{10} -$$$$91\!\cdots\!84$$$$\nu^{9} +$$$$71\!\cdots\!80$$$$\nu^{8} -$$$$20\!\cdots\!20$$$$\nu^{7} +$$$$14\!\cdots\!41$$$$\nu^{6} -$$$$26\!\cdots\!19$$$$\nu^{5} +$$$$10\!\cdots\!39$$$$\nu^{4} -$$$$23\!\cdots\!58$$$$\nu^{3} -$$$$37\!\cdots\!70$$$$\nu^{2} -$$$$40\!\cdots\!95$$$$\nu -$$$$54\!\cdots\!82$$$$)/$$$$13\!\cdots\!02$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$30\!\cdots\!58$$$$\nu^{19} +$$$$13\!\cdots\!99$$$$\nu^{18} -$$$$61\!\cdots\!96$$$$\nu^{17} +$$$$26\!\cdots\!25$$$$\nu^{16} -$$$$81\!\cdots\!68$$$$\nu^{15} +$$$$32\!\cdots\!41$$$$\nu^{14} -$$$$67\!\cdots\!33$$$$\nu^{13} +$$$$23\!\cdots\!47$$$$\nu^{12} -$$$$39\!\cdots\!23$$$$\nu^{11} +$$$$11\!\cdots\!21$$$$\nu^{10} -$$$$15\!\cdots\!78$$$$\nu^{9} +$$$$39\!\cdots\!13$$$$\nu^{8} -$$$$44\!\cdots\!92$$$$\nu^{7} +$$$$93\!\cdots\!98$$$$\nu^{6} -$$$$86\!\cdots\!48$$$$\nu^{5} +$$$$13\!\cdots\!79$$$$\nu^{4} -$$$$10\!\cdots\!07$$$$\nu^{3} +$$$$11\!\cdots\!53$$$$\nu^{2} -$$$$75\!\cdots\!06$$$$\nu +$$$$31\!\cdots\!39$$$$)/$$$$66\!\cdots\!26$$ $$\beta_{4}$$ $$=$$ $$($$$$95\!\cdots\!54$$$$\nu^{19} +$$$$40\!\cdots\!97$$$$\nu^{18} +$$$$16\!\cdots\!81$$$$\nu^{17} +$$$$33\!\cdots\!37$$$$\nu^{16} +$$$$19\!\cdots\!00$$$$\nu^{15} +$$$$28\!\cdots\!71$$$$\nu^{14} +$$$$13\!\cdots\!65$$$$\nu^{13} +$$$$80\!\cdots\!51$$$$\nu^{12} +$$$$64\!\cdots\!11$$$$\nu^{11} +$$$$40\!\cdots\!43$$$$\nu^{10} +$$$$19\!\cdots\!76$$$$\nu^{9} +$$$$10\!\cdots\!60$$$$\nu^{8} +$$$$44\!\cdots\!10$$$$\nu^{7} +$$$$29\!\cdots\!66$$$$\nu^{6} +$$$$55\!\cdots\!73$$$$\nu^{5} +$$$$25\!\cdots\!37$$$$\nu^{4} +$$$$54\!\cdots\!79$$$$\nu^{3} +$$$$80\!\cdots\!00$$$$\nu^{2} +$$$$22\!\cdots\!20$$$$\nu +$$$$32\!\cdots\!33$$$$)/$$$$13\!\cdots\!02$$ $$\beta_{5}$$ $$=$$ $$($$$$69\!\cdots\!90$$$$\nu^{19} -$$$$87\!\cdots\!59$$$$\nu^{18} +$$$$11\!\cdots\!77$$$$\nu^{17} -$$$$18\!\cdots\!13$$$$\nu^{16} +$$$$14\!\cdots\!03$$$$\nu^{15} -$$$$21\!\cdots\!10$$$$\nu^{14} +$$$$96\!\cdots\!00$$$$\nu^{13} -$$$$14\!\cdots\!76$$$$\nu^{12} +$$$$49\!\cdots\!92$$$$\nu^{11} -$$$$68\!\cdots\!46$$$$\nu^{10} +$$$$15\!\cdots\!33$$$$\nu^{9} -$$$$19\!\cdots\!59$$$$\nu^{8} +$$$$35\!\cdots\!95$$$$\nu^{7} -$$$$34\!\cdots\!72$$$$\nu^{6} +$$$$46\!\cdots\!33$$$$\nu^{5} -$$$$28\!\cdots\!13$$$$\nu^{4} +$$$$42\!\cdots\!78$$$$\nu^{3} -$$$$23\!\cdots\!37$$$$\nu^{2} +$$$$10\!\cdots\!15$$$$\nu +$$$$76\!\cdots\!03$$$$)/$$$$59\!\cdots\!66$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$22\!\cdots\!28$$$$\nu^{19} +$$$$11\!\cdots\!73$$$$\nu^{18} -$$$$40\!\cdots\!59$$$$\nu^{17} +$$$$32\!\cdots\!12$$$$\nu^{16} -$$$$48\!\cdots\!35$$$$\nu^{15} +$$$$42\!\cdots\!01$$$$\nu^{14} -$$$$34\!\cdots\!01$$$$\nu^{13} +$$$$34\!\cdots\!99$$$$\nu^{12} -$$$$17\!\cdots\!59$$$$\nu^{11} +$$$$18\!\cdots\!13$$$$\nu^{10} -$$$$57\!\cdots\!40$$$$\nu^{9} +$$$$65\!\cdots\!70$$$$\nu^{8} -$$$$12\!\cdots\!81$$$$\nu^{7} +$$$$16\!\cdots\!77$$$$\nu^{6} -$$$$17\!\cdots\!51$$$$\nu^{5} +$$$$26\!\cdots\!66$$$$\nu^{4} -$$$$13\!\cdots\!32$$$$\nu^{3} +$$$$24\!\cdots\!88$$$$\nu^{2} -$$$$46\!\cdots\!95$$$$\nu +$$$$69\!\cdots\!02$$$$)/$$$$17\!\cdots\!98$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$87\!\cdots\!09$$$$\nu^{19} +$$$$35\!\cdots\!98$$$$\nu^{18} -$$$$15\!\cdots\!73$$$$\nu^{17} +$$$$97\!\cdots\!33$$$$\nu^{16} -$$$$18\!\cdots\!30$$$$\nu^{15} +$$$$12\!\cdots\!31$$$$\nu^{14} -$$$$12\!\cdots\!95$$$$\nu^{13} +$$$$94\!\cdots\!39$$$$\nu^{12} -$$$$64\!\cdots\!29$$$$\nu^{11} +$$$$47\!\cdots\!12$$$$\nu^{10} -$$$$20\!\cdots\!61$$$$\nu^{9} +$$$$14\!\cdots\!66$$$$\nu^{8} -$$$$47\!\cdots\!95$$$$\nu^{7} +$$$$32\!\cdots\!79$$$$\nu^{6} -$$$$64\!\cdots\!27$$$$\nu^{5} +$$$$43\!\cdots\!98$$$$\nu^{4} -$$$$59\!\cdots\!34$$$$\nu^{3} +$$$$34\!\cdots\!66$$$$\nu^{2} -$$$$11\!\cdots\!92$$$$\nu +$$$$33\!\cdots\!41$$$$)/$$$$51\!\cdots\!74$$ $$\beta_{8}$$ $$=$$ $$($$$$76\!\cdots\!29$$$$\nu^{19} -$$$$38\!\cdots\!32$$$$\nu^{18} +$$$$13\!\cdots\!19$$$$\nu^{17} -$$$$92\!\cdots\!65$$$$\nu^{16} +$$$$16\!\cdots\!12$$$$\nu^{15} -$$$$11\!\cdots\!72$$$$\nu^{14} +$$$$11\!\cdots\!38$$$$\nu^{13} -$$$$82\!\cdots\!86$$$$\nu^{12} +$$$$59\!\cdots\!40$$$$\nu^{11} -$$$$39\!\cdots\!25$$$$\nu^{10} +$$$$19\!\cdots\!52$$$$\nu^{9} -$$$$11\!\cdots\!73$$$$\nu^{8} +$$$$47\!\cdots\!20$$$$\nu^{7} -$$$$24\!\cdots\!94$$$$\nu^{6} +$$$$72\!\cdots\!79$$$$\nu^{5} -$$$$27\!\cdots\!60$$$$\nu^{4} +$$$$76\!\cdots\!16$$$$\nu^{3} -$$$$22\!\cdots\!77$$$$\nu^{2} +$$$$30\!\cdots\!31$$$$\nu +$$$$39\!\cdots\!22$$$$)/$$$$31\!\cdots\!06$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$54\!\cdots\!61$$$$\nu^{19} +$$$$51\!\cdots\!59$$$$\nu^{18} -$$$$10\!\cdots\!01$$$$\nu^{17} +$$$$11\!\cdots\!76$$$$\nu^{16} -$$$$12\!\cdots\!58$$$$\nu^{15} +$$$$14\!\cdots\!12$$$$\nu^{14} -$$$$88\!\cdots\!44$$$$\nu^{13} +$$$$10\!\cdots\!84$$$$\nu^{12} -$$$$47\!\cdots\!92$$$$\nu^{11} +$$$$52\!\cdots\!97$$$$\nu^{10} -$$$$16\!\cdots\!41$$$$\nu^{9} +$$$$16\!\cdots\!29$$$$\nu^{8} -$$$$41\!\cdots\!21$$$$\nu^{7} +$$$$38\!\cdots\!67$$$$\nu^{6} -$$$$66\!\cdots\!31$$$$\nu^{5} +$$$$52\!\cdots\!53$$$$\nu^{4} -$$$$70\!\cdots\!71$$$$\nu^{3} +$$$$44\!\cdots\!47$$$$\nu^{2} -$$$$27\!\cdots\!62$$$$\nu +$$$$41\!\cdots\!48$$$$)/$$$$22\!\cdots\!42$$ $$\beta_{10}$$ $$=$$ $$($$$$25\!\cdots\!27$$$$\nu^{19} -$$$$66\!\cdots\!74$$$$\nu^{18} +$$$$46\!\cdots\!87$$$$\nu^{17} -$$$$24\!\cdots\!90$$$$\nu^{16} +$$$$53\!\cdots\!07$$$$\nu^{15} -$$$$32\!\cdots\!85$$$$\nu^{14} +$$$$36\!\cdots\!57$$$$\nu^{13} -$$$$26\!\cdots\!23$$$$\nu^{12} +$$$$18\!\cdots\!37$$$$\nu^{11} -$$$$13\!\cdots\!70$$$$\nu^{10} +$$$$58\!\cdots\!01$$$$\nu^{9} -$$$$44\!\cdots\!14$$$$\nu^{8} +$$$$13\!\cdots\!24$$$$\nu^{7} -$$$$97\!\cdots\!16$$$$\nu^{6} +$$$$18\!\cdots\!47$$$$\nu^{5} -$$$$12\!\cdots\!73$$$$\nu^{4} +$$$$16\!\cdots\!01$$$$\nu^{3} -$$$$65\!\cdots\!66$$$$\nu^{2} +$$$$32\!\cdots\!53$$$$\nu +$$$$49\!\cdots\!78$$$$)/$$$$85\!\cdots\!38$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$10\!\cdots\!62$$$$\nu^{19} -$$$$73\!\cdots\!70$$$$\nu^{18} -$$$$18\!\cdots\!47$$$$\nu^{17} -$$$$93\!\cdots\!94$$$$\nu^{16} -$$$$21\!\cdots\!51$$$$\nu^{15} -$$$$97\!\cdots\!61$$$$\nu^{14} -$$$$14\!\cdots\!53$$$$\nu^{13} -$$$$56\!\cdots\!93$$$$\nu^{12} -$$$$74\!\cdots\!55$$$$\nu^{11} -$$$$27\!\cdots\!63$$$$\nu^{10} -$$$$23\!\cdots\!51$$$$\nu^{9} -$$$$87\!\cdots\!50$$$$\nu^{8} -$$$$55\!\cdots\!23$$$$\nu^{7} -$$$$19\!\cdots\!12$$$$\nu^{6} -$$$$77\!\cdots\!13$$$$\nu^{5} -$$$$22\!\cdots\!08$$$$\nu^{4} -$$$$70\!\cdots\!71$$$$\nu^{3} -$$$$19\!\cdots\!20$$$$\nu^{2} -$$$$14\!\cdots\!33$$$$\nu -$$$$95\!\cdots\!36$$$$)/$$$$31\!\cdots\!06$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$81\!\cdots\!78$$$$\nu^{19} -$$$$11\!\cdots\!43$$$$\nu^{18} -$$$$14\!\cdots\!44$$$$\nu^{17} +$$$$72\!\cdots\!91$$$$\nu^{16} -$$$$17\!\cdots\!19$$$$\nu^{15} +$$$$18\!\cdots\!82$$$$\nu^{14} -$$$$12\!\cdots\!10$$$$\nu^{13} +$$$$21\!\cdots\!92$$$$\nu^{12} -$$$$63\!\cdots\!88$$$$\nu^{11} +$$$$11\!\cdots\!68$$$$\nu^{10} -$$$$20\!\cdots\!91$$$$\nu^{9} +$$$$34\!\cdots\!86$$$$\nu^{8} -$$$$48\!\cdots\!81$$$$\nu^{7} +$$$$80\!\cdots\!62$$$$\nu^{6} -$$$$69\!\cdots\!22$$$$\nu^{5} +$$$$12\!\cdots\!99$$$$\nu^{4} -$$$$62\!\cdots\!74$$$$\nu^{3} +$$$$11\!\cdots\!14$$$$\nu^{2} -$$$$22\!\cdots\!61$$$$\nu +$$$$33\!\cdots\!27$$$$)/$$$$22\!\cdots\!42$$ $$\beta_{13}$$ $$=$$ $$($$$$79\!\cdots\!49$$$$\nu^{19} +$$$$13\!\cdots\!35$$$$\nu^{18} +$$$$14\!\cdots\!60$$$$\nu^{17} -$$$$12\!\cdots\!87$$$$\nu^{16} +$$$$16\!\cdots\!79$$$$\nu^{15} -$$$$26\!\cdots\!85$$$$\nu^{14} +$$$$11\!\cdots\!39$$$$\nu^{13} -$$$$30\!\cdots\!99$$$$\nu^{12} +$$$$57\!\cdots\!75$$$$\nu^{11} -$$$$16\!\cdots\!96$$$$\nu^{10} +$$$$18\!\cdots\!58$$$$\nu^{9} -$$$$60\!\cdots\!95$$$$\nu^{8} +$$$$40\!\cdots\!75$$$$\nu^{7} -$$$$14\!\cdots\!49$$$$\nu^{6} +$$$$53\!\cdots\!24$$$$\nu^{5} -$$$$25\!\cdots\!10$$$$\nu^{4} +$$$$47\!\cdots\!50$$$$\nu^{3} -$$$$20\!\cdots\!47$$$$\nu^{2} +$$$$14\!\cdots\!20$$$$\nu -$$$$49\!\cdots\!74$$$$)/$$$$17\!\cdots\!98$$ $$\beta_{14}$$ $$=$$ $$($$$$96\!\cdots\!47$$$$\nu^{19} -$$$$36\!\cdots\!26$$$$\nu^{18} +$$$$16\!\cdots\!48$$$$\nu^{17} -$$$$10\!\cdots\!12$$$$\nu^{16} +$$$$19\!\cdots\!11$$$$\nu^{15} -$$$$13\!\cdots\!20$$$$\nu^{14} +$$$$13\!\cdots\!76$$$$\nu^{13} -$$$$99\!\cdots\!62$$$$\nu^{12} +$$$$66\!\cdots\!68$$$$\nu^{11} -$$$$48\!\cdots\!59$$$$\nu^{10} +$$$$20\!\cdots\!80$$$$\nu^{9} -$$$$15\!\cdots\!26$$$$\nu^{8} +$$$$47\!\cdots\!61$$$$\nu^{7} -$$$$32\!\cdots\!37$$$$\nu^{6} +$$$$61\!\cdots\!86$$$$\nu^{5} -$$$$39\!\cdots\!61$$$$\nu^{4} +$$$$59\!\cdots\!53$$$$\nu^{3} -$$$$22\!\cdots\!26$$$$\nu^{2} +$$$$55\!\cdots\!60$$$$\nu +$$$$53\!\cdots\!65$$$$)/$$$$17\!\cdots\!98$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!66$$$$\nu^{19} +$$$$49\!\cdots\!06$$$$\nu^{18} -$$$$20\!\cdots\!13$$$$\nu^{17} +$$$$13\!\cdots\!14$$$$\nu^{16} -$$$$23\!\cdots\!90$$$$\nu^{15} +$$$$17\!\cdots\!50$$$$\nu^{14} -$$$$16\!\cdots\!40$$$$\nu^{13} +$$$$13\!\cdots\!06$$$$\nu^{12} -$$$$82\!\cdots\!98$$$$\nu^{11} +$$$$67\!\cdots\!40$$$$\nu^{10} -$$$$25\!\cdots\!18$$$$\nu^{9} +$$$$21\!\cdots\!21$$$$\nu^{8} -$$$$59\!\cdots\!86$$$$\nu^{7} +$$$$49\!\cdots\!56$$$$\nu^{6} -$$$$79\!\cdots\!75$$$$\nu^{5} +$$$$66\!\cdots\!86$$$$\nu^{4} -$$$$73\!\cdots\!96$$$$\nu^{3} +$$$$48\!\cdots\!73$$$$\nu^{2} -$$$$18\!\cdots\!80$$$$\nu +$$$$17\!\cdots\!44$$$$)/$$$$17\!\cdots\!98$$ $$\beta_{16}$$ $$=$$ $$($$$$17\!\cdots\!18$$$$\nu^{19} -$$$$70\!\cdots\!96$$$$\nu^{18} +$$$$31\!\cdots\!46$$$$\nu^{17} -$$$$19\!\cdots\!66$$$$\nu^{16} +$$$$36\!\cdots\!60$$$$\nu^{15} -$$$$24\!\cdots\!62$$$$\nu^{14} +$$$$25\!\cdots\!90$$$$\nu^{13} -$$$$18\!\cdots\!78$$$$\nu^{12} +$$$$12\!\cdots\!58$$$$\nu^{11} -$$$$94\!\cdots\!24$$$$\nu^{10} +$$$$40\!\cdots\!22$$$$\nu^{9} -$$$$29\!\cdots\!32$$$$\nu^{8} +$$$$94\!\cdots\!90$$$$\nu^{7} -$$$$65\!\cdots\!58$$$$\nu^{6} +$$$$12\!\cdots\!54$$$$\nu^{5} -$$$$86\!\cdots\!96$$$$\nu^{4} +$$$$11\!\cdots\!68$$$$\nu^{3} -$$$$66\!\cdots\!95$$$$\nu^{2} +$$$$23\!\cdots\!84$$$$\nu +$$$$35\!\cdots\!66$$$$)/$$$$25\!\cdots\!37$$ $$\beta_{17}$$ $$=$$ $$($$$$16\!\cdots\!22$$$$\nu^{19} -$$$$10\!\cdots\!96$$$$\nu^{18} +$$$$30\!\cdots\!16$$$$\nu^{17} -$$$$84\!\cdots\!36$$$$\nu^{16} +$$$$36\!\cdots\!00$$$$\nu^{15} -$$$$12\!\cdots\!09$$$$\nu^{14} +$$$$25\!\cdots\!37$$$$\nu^{13} -$$$$10\!\cdots\!07$$$$\nu^{12} +$$$$12\!\cdots\!11$$$$\nu^{11} -$$$$52\!\cdots\!63$$$$\nu^{10} +$$$$42\!\cdots\!21$$$$\nu^{9} -$$$$16\!\cdots\!57$$$$\nu^{8} +$$$$98\!\cdots\!13$$$$\nu^{7} -$$$$37\!\cdots\!27$$$$\nu^{6} +$$$$13\!\cdots\!52$$$$\nu^{5} -$$$$51\!\cdots\!14$$$$\nu^{4} +$$$$12\!\cdots\!32$$$$\nu^{3} -$$$$34\!\cdots\!21$$$$\nu^{2} +$$$$30\!\cdots\!05$$$$\nu +$$$$53\!\cdots\!99$$$$)/$$$$22\!\cdots\!42$$ $$\beta_{18}$$ $$=$$ $$($$$$-$$$$17\!\cdots\!04$$$$\nu^{19} -$$$$51\!\cdots\!27$$$$\nu^{18} -$$$$30\!\cdots\!68$$$$\nu^{17} -$$$$21\!\cdots\!16$$$$\nu^{16} -$$$$35\!\cdots\!61$$$$\nu^{15} -$$$$31\!\cdots\!19$$$$\nu^{14} -$$$$24\!\cdots\!45$$$$\nu^{13} +$$$$19\!\cdots\!49$$$$\nu^{12} -$$$$12\!\cdots\!89$$$$\nu^{11} +$$$$10\!\cdots\!43$$$$\nu^{10} -$$$$37\!\cdots\!02$$$$\nu^{9} +$$$$41\!\cdots\!09$$$$\nu^{8} -$$$$84\!\cdots\!69$$$$\nu^{7} +$$$$10\!\cdots\!85$$$$\nu^{6} -$$$$10\!\cdots\!48$$$$\nu^{5} +$$$$18\!\cdots\!72$$$$\nu^{4} -$$$$94\!\cdots\!78$$$$\nu^{3} +$$$$10\!\cdots\!93$$$$\nu^{2} -$$$$16\!\cdots\!41$$$$\nu -$$$$43\!\cdots\!26$$$$)/$$$$22\!\cdots\!42$$ $$\beta_{19}$$ $$=$$ $$($$$$14\!\cdots\!24$$$$\nu^{19} -$$$$42\!\cdots\!75$$$$\nu^{18} +$$$$26\!\cdots\!83$$$$\nu^{17} -$$$$13\!\cdots\!77$$$$\nu^{16} +$$$$31\!\cdots\!31$$$$\nu^{15} -$$$$17\!\cdots\!05$$$$\nu^{14} +$$$$21\!\cdots\!63$$$$\nu^{13} -$$$$14\!\cdots\!09$$$$\nu^{12} +$$$$11\!\cdots\!07$$$$\nu^{11} -$$$$71\!\cdots\!23$$$$\nu^{10} +$$$$36\!\cdots\!30$$$$\nu^{9} -$$$$23\!\cdots\!34$$$$\nu^{8} +$$$$85\!\cdots\!64$$$$\nu^{7} -$$$$52\!\cdots\!65$$$$\nu^{6} +$$$$12\!\cdots\!69$$$$\nu^{5} -$$$$72\!\cdots\!53$$$$\nu^{4} +$$$$11\!\cdots\!84$$$$\nu^{3} -$$$$57\!\cdots\!44$$$$\nu^{2} +$$$$36\!\cdots\!20$$$$\nu -$$$$55\!\cdots\!06$$$$)/$$$$17\!\cdots\!98$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{16} + 4 \beta_{7} - 4$$ $$\nu^{3}$$ $$=$$ $$-\beta_{15} - \beta_{14} - \beta_{13} + 5 \beta_{4}$$ $$\nu^{4}$$ $$=$$ $$\beta_{18} - 7 \beta_{16} + \beta_{14} + \beta_{13} - \beta_{11} - 23 \beta_{7} - \beta_{6} - 7 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{18} - \beta_{17} + 2 \beta_{16} + 9 \beta_{13} - \beta_{12} - \beta_{11} - 2 \beta_{8} + \beta_{7} + 9 \beta_{6} - 29 \beta_{4} - 29 \beta_{1} - 1$$ $$\nu^{6}$$ $$=$$ $$-\beta_{19} - 3 \beta_{17} + \beta_{16} - 10 \beta_{15} - 10 \beta_{13} - 13 \beta_{12} - 2 \beta_{11} - 2 \beta_{9} - 14 \beta_{8} - \beta_{7} + 10 \beta_{6} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 45 \beta_{2} + 149$$ $$\nu^{7}$$ $$=$$ $$-3 \beta_{19} - 15 \beta_{18} - 26 \beta_{16} + 67 \beta_{15} + 68 \beta_{14} + \beta_{13} - 12 \beta_{12} + 27 \beta_{11} - 16 \beta_{9} + 12 \beta_{8} - 17 \beta_{7} - 68 \beta_{6} + 3 \beta_{3} - 26 \beta_{2} + 181 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$30 \beta_{19} - 106 \beta_{18} + 42 \beta_{17} + 289 \beta_{16} + 79 \beta_{15} - 79 \beta_{14} + 137 \beta_{12} + 122 \beta_{11} - 19 \beta_{10} + 15 \beta_{9} + 121 \beta_{8} + 997 \beta_{7} - 30 \beta_{4} + 15 \beta_{3} + 15 \beta_{2} - 30 \beta_{1} - 1012$$ $$\nu^{9}$$ $$=$$ $$-50 \beta_{19} - 60 \beta_{18} + 79 \beta_{17} + 50 \beta_{16} - 490 \beta_{15} - 474 \beta_{14} - 490 \beta_{13} + 163 \beta_{12} - 160 \beta_{11} + 129 \beta_{9} + 53 \beta_{8} - 50 \beta_{7} + 16 \beta_{6} + 3 \beta_{5} + 1175 \beta_{4} - 100 \beta_{3} + 200 \beta_{2} + 294$$ $$\nu^{10}$$ $$=$$ $$-163 \beta_{19} + 853 \beta_{18} - 2039 \beta_{16} + 3 \beta_{15} + 587 \beta_{14} + 584 \beta_{13} - 176 \beta_{12} - 677 \beta_{11} + 229 \beta_{10} + 94 \beta_{9} + 176 \beta_{8} - 6699 \beta_{7} - 587 \beta_{6} - 229 \beta_{5} + 163 \beta_{3} - 2039 \beta_{2} + 308 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$1136 \beta_{19} + 1690 \beta_{18} - 567 \beta_{17} + 1575 \beta_{16} + 179 \beta_{15} - 179 \beta_{14} + 3300 \beta_{13} - 199 \beta_{12} - 767 \beta_{11} - 69 \beta_{10} + 568 \beta_{9} - 1122 \beta_{8} + 2436 \beta_{7} + 3300 \beta_{6} - 7803 \beta_{4} + 568 \beta_{3} + 568 \beta_{2} - 7803 \beta_{1} - 3004$$ $$\nu^{12}$$ $$=$$ $$-1560 \beta_{19} - 110 \beta_{18} - 3680 \beta_{17} + 1560 \beta_{16} - 4287 \beta_{15} - 64 \beta_{14} - 4287 \beta_{13} - 7885 \beta_{12} - 3230 \beta_{11} - 2120 \beta_{9} - 9555 \beta_{8} - 1560 \beta_{7} + 4223 \beta_{6} + 2271 \beta_{5} + 2720 \beta_{4} - 3120 \beta_{3} + 12324 \beta_{2} + 49354$$ $$\nu^{13}$$ $$=$$ $$-5501 \beta_{19} - 10744 \beta_{18} - 17337 \beta_{16} + 22872 \beta_{15} + 24606 \beta_{14} + 1734 \beta_{13} - 7453 \beta_{12} + 18197 \beta_{11} + 1000 \beta_{10} - 14808 \beta_{9} + 7453 \beta_{8} - 16482 \beta_{7} - 24606 \beta_{6} - 1000 \beta_{5} + 5501 \beta_{3} - 17337 \beta_{2} + 52573 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$27908 \beta_{19} - 46312 \beta_{18} + 30151 \beta_{17} + 81854 \beta_{16} + 30325 \beta_{15} - 30325 \beta_{14} + 874 \beta_{13} + 74954 \beta_{12} + 61000 \beta_{11} - 20309 \beta_{10} + 13954 \beta_{9} + 60266 \beta_{8} + 335458 \beta_{7} + 874 \beta_{6} - 22302 \beta_{4} + 13954 \beta_{3} + 13954 \beta_{2} - 22302 \beta_{1} - 349412$$ $$\nu^{15}$$ $$=$$ $$-48951 \beta_{19} - 10016 \beta_{18} + 24025 \beta_{17} + 48951 \beta_{16} - 174053 \beta_{15} - 158491 \beta_{14} - 174053 \beta_{13} + 43533 \beta_{12} - 107918 \beta_{11} + 72976 \beta_{9} - 15434 \beta_{8} - 48951 \beta_{7} + 15562 \beta_{6} + 11711 \beta_{5} + 357821 \beta_{4} - 97902 \beta_{3} + 86733 \beta_{2} + 236300$$ $$\nu^{16}$$ $$=$$ $$-119629 \beta_{19} + 337294 \beta_{18} - 668559 \beta_{16} + 9811 \beta_{15} + 227269 \beta_{14} + 217458 \beta_{13} - 123480 \beta_{12} - 213814 \beta_{11} + 170878 \beta_{10} - 1616 \beta_{9} + 123480 \beta_{8} - 2248024 \beta_{7} - 227269 \beta_{6} - 170878 \beta_{5} + 119629 \beta_{3} - 668559 \beta_{2} + 175656 \beta_{1}$$ $$\nu^{17}$$ $$=$$ $$827974 \beta_{19} + 660926 \beta_{18} - 141583 \beta_{17} + 626550 \beta_{16} + 133291 \beta_{15} - 133291 \beta_{14} + 1099831 \beta_{13} + 213628 \beta_{12} - 200359 \beta_{11} - 121245 \beta_{10} + 413987 \beta_{9} - 246939 \beta_{8} + 1541668 \beta_{7} + 1099831 \beta_{6} - 2454193 \beta_{4} + 413987 \beta_{3} + 413987 \beta_{2} - 2454193 \beta_{1} - 1955655$$ $$\nu^{18}$$ $$=$$ $$-995799 \beta_{19} - 12046 \beta_{18} - 1828222 \beta_{17} + 995799 \beta_{16} - 1659313 \beta_{15} - 98915 \beta_{14} - 1659313 \beta_{13} - 3370902 \beta_{12} - 2003644 \beta_{11} - 832423 \beta_{9} - 4378747 \beta_{8} - 995799 \beta_{7} + 1560398 \beta_{6} + 1383544 \beta_{5} + 1352440 \beta_{4} - 1991598 \beta_{3} + 3712233 \beta_{2} + 17778807$$ $$\nu^{19}$$ $$=$$ $$-3387188 \beta_{19} - 4486483 \beta_{18} - 7877160 \beta_{16} + 7647734 \beta_{15} + 8754494 \beta_{14} + 1106760 \beta_{13} - 3564042 \beta_{12} + 8050525 \beta_{11} + 1159175 \beta_{10} - 7533580 \beta_{9} + 3564042 \beta_{8} - 9012635 \beta_{7} - 8754494 \beta_{6} - 1159175 \beta_{5} + 3387188 \beta_{3} - 7877160 \beta_{2} + 16936871 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{7}$$ $$-1 + \beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 −1.35774 − 2.35168i −1.27537 − 2.20901i −0.828334 − 1.43472i −0.707433 − 1.22531i −0.130586 − 0.226181i 0.328258 + 0.568560i 0.613260 + 1.06220i 0.904928 + 1.56738i 1.14017 + 1.97483i 1.31285 + 2.27393i −1.35774 + 2.35168i −1.27537 + 2.20901i −0.828334 + 1.43472i −0.707433 + 1.22531i −0.130586 + 0.226181i 0.328258 − 0.568560i 0.613260 − 1.06220i 0.904928 − 1.56738i 1.14017 − 1.97483i 1.31285 − 2.27393i
−2.71549 −0.500000 + 0.866025i 5.37387 1.94413 3.36734i 1.35774 2.35168i 1.94064 1.79830i −9.16172 −0.500000 0.866025i −5.27927 + 9.14396i
16.2 −2.55074 −0.500000 + 0.866025i 4.50630 −1.40932 + 2.44101i 1.27537 2.20901i −2.27422 + 1.35201i −6.39292 −0.500000 0.866025i 3.59481 6.22639i
16.3 −1.65667 −0.500000 + 0.866025i 0.744548 1.05011 1.81885i 0.828334 1.43472i −1.49221 + 2.18479i 2.07987 −0.500000 0.866025i −1.73968 + 3.01322i
16.4 −1.41487 −0.500000 + 0.866025i 0.00184802 −1.42962 + 2.47618i 0.707433 1.22531i 2.56093 + 0.664549i 2.82712 −0.500000 0.866025i 2.02273 3.50347i
16.5 −0.261171 −0.500000 + 0.866025i −1.93179 0.708533 1.22721i 0.130586 0.226181i 2.55312 0.693954i 1.02687 −0.500000 0.866025i −0.185048 + 0.320513i
16.6 0.656517 −0.500000 + 0.866025i −1.56899 −0.0109774 + 0.0190133i −0.328258 + 0.568560i −1.55912 + 2.13756i −2.34310 −0.500000 0.866025i −0.00720682 + 0.0124826i
16.7 1.22652 −0.500000 + 0.866025i −0.495647 −2.10660 + 3.64874i −0.613260 + 1.06220i 0.113533 2.64331i −3.06096 −0.500000 0.866025i −2.58379 + 4.47526i
16.8 1.80986 −0.500000 + 0.866025i 1.27558 1.98776 3.44291i −0.904928 + 1.56738i 1.70815 + 2.02045i −1.31110 −0.500000 0.866025i 3.59756 6.23116i
16.9 2.28034 −0.500000 + 0.866025i 3.19995 −1.46862 + 2.54373i −1.14017 + 1.97483i 0.102378 + 2.64377i 2.73629 −0.500000 0.866025i −3.34896 + 5.80057i
16.10 2.62571 −0.500000 + 0.866025i 4.89433 0.734607 1.27238i −1.31285 + 2.27393i −2.15321 1.53742i 7.59966 −0.500000 0.866025i 1.92886 3.34089i
256.1 −2.71549 −0.500000 0.866025i 5.37387 1.94413 + 3.36734i 1.35774 + 2.35168i 1.94064 + 1.79830i −9.16172 −0.500000 + 0.866025i −5.27927 9.14396i
256.2 −2.55074 −0.500000 0.866025i 4.50630 −1.40932 2.44101i 1.27537 + 2.20901i −2.27422 1.35201i −6.39292 −0.500000 + 0.866025i 3.59481 + 6.22639i
256.3 −1.65667 −0.500000 0.866025i 0.744548 1.05011 + 1.81885i 0.828334 + 1.43472i −1.49221 2.18479i 2.07987 −0.500000 + 0.866025i −1.73968 3.01322i
256.4 −1.41487 −0.500000 0.866025i 0.00184802 −1.42962 2.47618i 0.707433 + 1.22531i 2.56093 0.664549i 2.82712 −0.500000 + 0.866025i 2.02273 + 3.50347i
256.5 −0.261171 −0.500000 0.866025i −1.93179 0.708533 + 1.22721i 0.130586 + 0.226181i 2.55312 + 0.693954i 1.02687 −0.500000 + 0.866025i −0.185048 0.320513i
256.6 0.656517 −0.500000 0.866025i −1.56899 −0.0109774 0.0190133i −0.328258 0.568560i −1.55912 2.13756i −2.34310 −0.500000 + 0.866025i −0.00720682 0.0124826i
256.7 1.22652 −0.500000 0.866025i −0.495647 −2.10660 3.64874i −0.613260 1.06220i 0.113533 + 2.64331i −3.06096 −0.500000 + 0.866025i −2.58379 4.47526i
256.8 1.80986 −0.500000 0.866025i 1.27558 1.98776 + 3.44291i −0.904928 1.56738i 1.70815 2.02045i −1.31110 −0.500000 + 0.866025i 3.59756 + 6.23116i
256.9 2.28034 −0.500000 0.866025i 3.19995 −1.46862 2.54373i −1.14017 1.97483i 0.102378 2.64377i 2.73629 −0.500000 + 0.866025i −3.34896 5.80057i
256.10 2.62571 −0.500000 0.866025i 4.89433 0.734607 + 1.27238i −1.31285 2.27393i −2.15321 + 1.53742i 7.59966 −0.500000 + 0.866025i 1.92886 + 3.34089i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 256.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.l.c yes 20
3.b odd 2 1 819.2.s.f 20
7.c even 3 1 273.2.j.c 20
13.c even 3 1 273.2.j.c 20
21.h odd 6 1 819.2.n.f 20
39.i odd 6 1 819.2.n.f 20
91.h even 3 1 inner 273.2.l.c yes 20
273.s odd 6 1 819.2.s.f 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.j.c 20 7.c even 3 1
273.2.j.c 20 13.c even 3 1
273.2.l.c yes 20 1.a even 1 1 trivial
273.2.l.c yes 20 91.h even 3 1 inner
819.2.n.f 20 21.h odd 6 1
819.2.n.f 20 39.i odd 6 1
819.2.s.f 20 3.b odd 2 1
819.2.s.f 20 273.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -37 - 81 T + 271 T^{2} + 76 T^{3} - 290 T^{4} - 23 T^{5} + 113 T^{6} + 2 T^{7} - 18 T^{8} + T^{10} )^{2}$$
$3$ $$( 1 + T + T^{2} )^{10}$$
$5$ $$21904 + 973840 T + 44388788 T^{2} - 48038324 T^{3} + 65723253 T^{4} - 31950698 T^{5} + 27505483 T^{6} - 8096156 T^{7} + 7642531 T^{8} - 1358388 T^{9} + 1409381 T^{10} - 113648 T^{11} + 193385 T^{12} - 6876 T^{13} + 17091 T^{14} - 250 T^{15} + 1097 T^{16} - 6 T^{17} + 41 T^{18} + T^{20}$$
$7$ $$282475249 - 121060821 T + 34588806 T^{2} - 27176919 T^{3} + 15059072 T^{4} - 6201783 T^{5} + 2941225 T^{6} - 1252293 T^{7} + 424585 T^{8} - 192360 T^{9} + 96809 T^{10} - 27480 T^{11} + 8665 T^{12} - 3651 T^{13} + 1225 T^{14} - 369 T^{15} + 128 T^{16} - 33 T^{17} + 6 T^{18} - 3 T^{19} + T^{20}$$
$11$ $$4096 - 23552 T + 126464 T^{2} - 268096 T^{3} + 718224 T^{4} - 705040 T^{5} + 2675556 T^{6} - 1806188 T^{7} + 2985669 T^{8} + 260933 T^{9} + 1078899 T^{10} + 296280 T^{11} + 354700 T^{12} + 120888 T^{13} + 51637 T^{14} + 10230 T^{15} + 2603 T^{16} + 366 T^{17} + 81 T^{18} + 8 T^{19} + T^{20}$$
$13$ $$137858491849 + 53022496865 T - 13867422257 T^{2} - 7216079455 T^{3} - 501988136 T^{4} - 18193357 T^{5} + 50610092 T^{6} + 22411597 T^{7} - 5659810 T^{8} - 186862 T^{9} + 958780 T^{10} - 14374 T^{11} - 33490 T^{12} + 10201 T^{13} + 1772 T^{14} - 49 T^{15} - 104 T^{16} - 115 T^{17} - 17 T^{18} + 5 T^{19} + T^{20}$$
$17$ $$( 18804 - 151812 T + 256257 T^{2} + 29774 T^{3} - 51153 T^{4} - 2054 T^{5} + 3551 T^{6} + 46 T^{7} - 102 T^{8} + T^{10} )^{2}$$
$19$ $$4253126656 + 12043047424 T + 25800035200 T^{2} + 24322288088 T^{3} + 19209560865 T^{4} + 9427944816 T^{5} + 4636707789 T^{6} + 1566781200 T^{7} + 650348999 T^{8} + 166642797 T^{9} + 58904464 T^{10} + 9747195 T^{11} + 3397766 T^{12} + 333445 T^{13} + 151553 T^{14} + 2983 T^{15} + 4498 T^{16} - 195 T^{17} + 114 T^{18} - 7 T^{19} + T^{20}$$
$23$ $$( 156324 - 127008 T - 285489 T^{2} + 219619 T^{3} + 62616 T^{4} - 33475 T^{5} - 1108 T^{6} + 1294 T^{7} - 52 T^{8} - 14 T^{9} + T^{10} )^{2}$$
$29$ $$1882384 + 9411920 T + 78062684 T^{2} - 81602444 T^{3} + 600779901 T^{4} - 355617598 T^{5} + 2374013675 T^{6} - 2718924106 T^{7} + 4031830689 T^{8} - 1557031481 T^{9} + 679142868 T^{10} - 34869883 T^{11} + 27512288 T^{12} + 14171 T^{13} + 864015 T^{14} + 33949 T^{15} + 15174 T^{16} + 701 T^{17} + 190 T^{18} + 9 T^{19} + T^{20}$$
$31$ $$64085935104 + 261921185280 T + 756837512352 T^{2} + 1040630058000 T^{3} + 1050662929545 T^{4} + 648705932469 T^{5} + 317476826179 T^{6} + 100060555692 T^{7} + 30939258292 T^{8} + 6587017103 T^{9} + 1944756458 T^{10} + 299751613 T^{11} + 67670703 T^{12} + 7484372 T^{13} + 1545541 T^{14} + 135902 T^{15} + 22639 T^{16} + 1331 T^{17} + 204 T^{18} + 9 T^{19} + T^{20}$$
$37$ $$( 17337 + 142380 T - 225882 T^{2} - 67031 T^{3} + 53457 T^{4} + 13616 T^{5} - 3127 T^{6} - 895 T^{7} + 30 T^{8} + 18 T^{9} + T^{10} )^{2}$$
$41$ $$5541909136 - 10704153872 T + 25485188004 T^{2} - 14755145932 T^{3} + 29377645777 T^{4} - 20447210680 T^{5} + 21334149520 T^{6} - 9865119752 T^{7} + 6470904248 T^{8} - 2459144385 T^{9} + 1283790892 T^{10} - 351554495 T^{11} + 105341557 T^{12} - 11339752 T^{13} + 2097731 T^{14} - 81478 T^{15} + 31185 T^{16} - 490 T^{17} + 205 T^{18} + T^{19} + T^{20}$$
$43$ $$2988746904541489 - 2154091832745125 T + 2223003658681029 T^{2} - 19113604128220 T^{3} + 294015164693034 T^{4} - 11297692784674 T^{5} + 21155753310012 T^{6} - 25133212091 T^{7} + 808237643635 T^{8} + 16014493349 T^{9} + 22459986660 T^{10} + 820971472 T^{11} + 421614783 T^{12} + 18255660 T^{13} + 5853149 T^{14} + 266750 T^{15} + 54400 T^{16} + 2161 T^{17} + 338 T^{18} + 11 T^{19} + T^{20}$$
$47$ $$6617497104 - 23440914288 T + 99577379052 T^{2} - 49385771004 T^{3} + 262931491617 T^{4} - 344381757756 T^{5} + 384290459824 T^{6} - 221666708762 T^{7} + 96961200829 T^{8} - 28612314884 T^{9} + 6879192767 T^{10} - 1209493030 T^{11} + 197895994 T^{12} - 25515846 T^{13} + 3567182 T^{14} - 355205 T^{15} + 39596 T^{16} - 2695 T^{17} + 270 T^{18} - 13 T^{19} + T^{20}$$
$53$ $$406192078224 + 156600120384 T + 651928741380 T^{2} + 643759769256 T^{3} + 1056685335177 T^{4} + 691260531051 T^{5} + 423339028669 T^{6} + 133442030762 T^{7} + 45782556337 T^{8} + 8076589660 T^{9} + 2879582113 T^{10} + 390112848 T^{11} + 100356265 T^{12} + 10073857 T^{13} + 2269606 T^{14} + 186567 T^{15} + 30776 T^{16} + 1404 T^{17} + 210 T^{18} + 6 T^{19} + T^{20}$$
$59$ $$( -3044508 + 39196512 T + 34989129 T^{2} + 6064769 T^{3} - 1286289 T^{4} - 293945 T^{5} + 23549 T^{6} + 4195 T^{7} - 282 T^{8} - 15 T^{9} + T^{10} )^{2}$$
$61$ $$26713003629441424 - 3137029595035952 T + 5115373199622084 T^{2} - 110865667640428 T^{3} + 619602062045449 T^{4} - 3816665807265 T^{5} + 39061180241750 T^{6} + 1757697774293 T^{7} + 1622355414146 T^{8} + 66379960997 T^{9} + 38288279414 T^{10} + 1268385622 T^{11} + 646952226 T^{12} + 14859386 T^{13} + 7327895 T^{14} + 98304 T^{15} + 60739 T^{16} + 390 T^{17} + 308 T^{18} + T^{20}$$
$67$ $$83534872576 - 94503911424 T + 565236002816 T^{2} + 1247393931264 T^{3} + 2176021424128 T^{4} + 1851205412352 T^{5} + 1224430126448 T^{6} + 513942036644 T^{7} + 191992924437 T^{8} + 56341426033 T^{9} + 16496649389 T^{10} + 3847097210 T^{11} + 789656270 T^{12} + 118331096 T^{13} + 14779805 T^{14} + 1342542 T^{15} + 111139 T^{16} + 7028 T^{17} + 495 T^{18} + 22 T^{19} + T^{20}$$
$71$ $$1342437380496 + 6077068992720 T + 22070393593596 T^{2} + 24306684386772 T^{3} + 22219676855109 T^{4} + 8557985307498 T^{5} + 4649305132075 T^{6} + 1671849814326 T^{7} + 666628282159 T^{8} + 165400854559 T^{9} + 37481526306 T^{10} + 5791477539 T^{11} + 899163028 T^{12} + 103125071 T^{13} + 13291201 T^{14} + 1121877 T^{15} + 104554 T^{16} + 4635 T^{17} + 374 T^{18} + 11 T^{19} + T^{20}$$
$73$ $$21167631081 + 66343896000 T + 1127008030167 T^{2} - 3664201438884 T^{3} + 38509113632298 T^{4} - 18076808346426 T^{5} + 14566865747797 T^{6} + 1980543913936 T^{7} + 1313999536463 T^{8} + 112074099462 T^{9} + 47751920411 T^{10} + 3696946376 T^{11} + 1167480453 T^{12} + 51800986 T^{13} + 13755978 T^{14} + 239688 T^{15} + 118412 T^{16} + 764 T^{17} + 398 T^{18} + T^{20}$$
$79$ $$22605069576830976 + 818903770988544 T + 9150160623501312 T^{2} + 6945314972958720 T^{3} + 4200044509827072 T^{4} + 1542811898634240 T^{5} + 453990955138048 T^{6} + 101617375094784 T^{7} + 19752699208192 T^{8} + 3227304038912 T^{9} + 484898817984 T^{10} + 62971048064 T^{11} + 7453068320 T^{12} + 737743008 T^{13} + 65685932 T^{14} + 4842680 T^{15} + 338973 T^{16} + 19388 T^{17} + 1045 T^{18} + 36 T^{19} + T^{20}$$
$83$ $$( 548326464 - 1237459440 T + 7340772 T^{2} + 66428716 T^{3} - 1726265 T^{4} - 1184759 T^{5} + 43352 T^{6} + 8460 T^{7} - 378 T^{8} - 20 T^{9} + T^{10} )^{2}$$
$89$ $$( -451948 - 1131840 T - 923977 T^{2} - 205977 T^{3} + 80863 T^{4} + 43244 T^{5} + 3653 T^{6} - 1007 T^{7} - 166 T^{8} + 2 T^{9} + T^{10} )^{2}$$
$97$ $$479579795289 + 4067227825353 T + 27986565300003 T^{2} + 63845221136712 T^{3} + 125108940206676 T^{4} - 56930956920732 T^{5} + 38607494021365 T^{6} - 4908707369839 T^{7} + 1940150029491 T^{8} - 176830516984 T^{9} + 66531678192 T^{10} - 4452113328 T^{11} + 1216431363 T^{12} - 73596997 T^{13} + 16035305 T^{14} - 822352 T^{15} + 121468 T^{16} - 5300 T^{17} + 623 T^{18} - 21 T^{19} + T^{20}$$