# Properties

 Label 429.2.n.b Level $429$ Weight $2$ Character orbit 429.n Analytic conductor $3.426$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$5$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - x^{19} + 4 x^{18} + 4 x^{17} + 37 x^{16} - 74 x^{15} + 398 x^{14} - 224 x^{13} + 978 x^{12} + 115 x^{11} + 1963 x^{10} + 323 x^{9} + 3007 x^{8} + 1828 x^{7} + 6736 x^{6} + 4512 x^{5} + 5197 x^{4} + 2152 x^{3} + 1259 x^{2} + 495 x + 121$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} -\beta_{9} q^{3} + ( -\beta_{3} - \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{4} + ( \beta_{6} + \beta_{8} + \beta_{12} + \beta_{16} - \beta_{18} ) q^{5} -\beta_{1} q^{6} + ( -\beta_{11} - \beta_{14} + \beta_{17} - \beta_{19} ) q^{7} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{12} - \beta_{13} + \beta_{16} - \beta_{18} ) q^{8} -\beta_{10} q^{9} +O(q^{10})$$ $$q + \beta_{5} q^{2} -\beta_{9} q^{3} + ( -\beta_{3} - \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{4} + ( \beta_{6} + \beta_{8} + \beta_{12} + \beta_{16} - \beta_{18} ) q^{5} -\beta_{1} q^{6} + ( -\beta_{11} - \beta_{14} + \beta_{17} - \beta_{19} ) q^{7} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{12} - \beta_{13} + \beta_{16} - \beta_{18} ) q^{8} -\beta_{10} q^{9} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{11} - \beta_{12} + \beta_{14} + \beta_{18} + \beta_{19} ) q^{10} + ( \beta_{1} - \beta_{7} + \beta_{8} + \beta_{13} + \beta_{18} + \beta_{19} ) q^{11} + ( -\beta_{3} - \beta_{8} - \beta_{10} - \beta_{12} + \beta_{18} ) q^{12} + \beta_{10} q^{13} + ( \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{9} - \beta_{12} + \beta_{13} - \beta_{15} + \beta_{16} ) q^{14} + ( -1 + \beta_{4} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{13} ) q^{15} + ( -2 + 3 \beta_{1} - 2 \beta_{3} - \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} - 3 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{16} + ( -\beta_{1} + \beta_{3} + \beta_{7} - \beta_{11} + \beta_{12} - \beta_{14} + \beta_{17} ) q^{17} + ( -\beta_{3} - \beta_{5} - \beta_{7} ) q^{18} + ( -1 - \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{12} - \beta_{13} + \beta_{15} - 2 \beta_{16} + \beta_{17} + \beta_{18} ) q^{19} + ( -3 \beta_{5} - \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} - \beta_{11} + 2 \beta_{13} - 2 \beta_{15} - \beta_{16} + \beta_{17} + \beta_{18} ) q^{20} -\beta_{14} q^{21} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{12} - 2 \beta_{13} - \beta_{16} ) q^{22} + ( 1 - \beta_{1} - \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{10} - \beta_{12} - \beta_{14} + \beta_{18} ) q^{23} + ( \beta_{2} - \beta_{5} - \beta_{7} + \beta_{10} + \beta_{13} - \beta_{15} - \beta_{16} + \beta_{18} ) q^{24} + ( -\beta_{1} - \beta_{2} + \beta_{6} + \beta_{7} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{18} - \beta_{19} ) q^{25} + ( \beta_{3} + \beta_{5} + \beta_{7} ) q^{26} + \beta_{8} q^{27} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{11} + \beta_{16} - \beta_{18} ) q^{28} + ( 1 + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} - \beta_{17} + \beta_{19} ) q^{29} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{12} - \beta_{13} + \beta_{14} + \beta_{19} ) q^{30} + ( -3 \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{16} - \beta_{18} - \beta_{19} ) q^{31} + ( 1 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} + 4 \beta_{8} + 4 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} - 2 \beta_{15} - \beta_{18} - \beta_{19} ) q^{32} + ( -1 + \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{15} ) q^{33} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{8} - 3 \beta_{10} - 2 \beta_{12} + 2 \beta_{18} ) q^{34} + ( -2 \beta_{2} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{13} + \beta_{15} - \beta_{17} - \beta_{18} ) q^{35} + ( 1 - \beta_{1} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{36} + ( -1 - \beta_{1} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} - \beta_{17} + \beta_{19} ) q^{37} + ( -\beta_{1} - 2 \beta_{6} - 2 \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{16} + \beta_{18} ) q^{38} -\beta_{8} q^{39} + ( 4 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} - \beta_{6} + 5 \beta_{7} - 4 \beta_{8} + 7 \beta_{9} - 7 \beta_{10} + \beta_{11} + \beta_{12} - 3 \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} - \beta_{18} ) q^{40} + ( -2 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{10} - 2 \beta_{14} + \beta_{17} - 2 \beta_{19} ) q^{41} + ( \beta_{2} + \beta_{5} + \beta_{7} - \beta_{10} + \beta_{18} ) q^{42} + ( 2 - 2 \beta_{3} - \beta_{6} + \beta_{11} - \beta_{12} - \beta_{15} + \beta_{18} + \beta_{19} ) q^{43} + ( -2 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} + 2 \beta_{18} ) q^{44} + ( -1 - \beta_{2} + \beta_{4} + \beta_{6} + \beta_{15} ) q^{45} + ( 2 \beta_{2} - 2 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{13} - 2 \beta_{15} + \beta_{16} - \beta_{18} - 2 \beta_{19} ) q^{46} + ( 1 - \beta_{1} + 2 \beta_{2} - 3 \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} + 3 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{47} + ( 1 - 2 \beta_{1} + \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{8} + 3 \beta_{9} - 3 \beta_{10} + \beta_{12} - 3 \beta_{13} + \beta_{15} - \beta_{17} + \beta_{19} ) q^{48} + ( -\beta_{1} - \beta_{6} + \beta_{8} + \beta_{11} - 2 \beta_{13} - \beta_{16} - \beta_{18} ) q^{49} + ( -2 + 3 \beta_{1} - \beta_{3} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{11} - 2 \beta_{12} + 3 \beta_{13} - \beta_{14} + \beta_{17} + 3 \beta_{18} ) q^{50} + ( \beta_{1} - \beta_{5} - \beta_{7} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{51} + ( -1 + \beta_{1} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} ) q^{52} + ( -\beta_{5} + 2 \beta_{7} + \beta_{10} + \beta_{13} - \beta_{15} + \beta_{16} + 2 \beta_{19} ) q^{53} -\beta_{3} q^{54} + ( -2 + 3 \beta_{1} + \beta_{3} + 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + 3 \beta_{10} + \beta_{12} + \beta_{15} + \beta_{16} - 3 \beta_{18} - \beta_{19} ) q^{55} + ( 1 - \beta_{3} - 2 \beta_{6} - 5 \beta_{8} - 5 \beta_{10} + \beta_{11} - 3 \beta_{12} - \beta_{15} + 3 \beta_{18} + \beta_{19} ) q^{56} + ( -2 \beta_{2} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{18} + \beta_{19} ) q^{57} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} - 3 \beta_{16} + \beta_{18} + \beta_{19} ) q^{58} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + 4 \beta_{9} - 4 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{18} ) q^{59} + ( -3 + 3 \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} - 3 \beta_{9} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{16} + \beta_{17} + 3 \beta_{18} ) q^{60} + ( 1 + \beta_{1} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{11} - 3 \beta_{12} + \beta_{13} + \beta_{16} ) q^{61} + ( 4 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - 4 \beta_{8} + 8 \beta_{9} - 8 \beta_{10} + \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} + 2 \beta_{16} - 2 \beta_{18} ) q^{62} -\beta_{17} q^{63} + ( -2 \beta_{2} + 5 \beta_{5} + 6 \beta_{7} - 4 \beta_{8} + 4 \beta_{9} - 8 \beta_{10} - \beta_{11} - 3 \beta_{13} + 3 \beta_{15} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{64} + ( 1 + \beta_{2} - \beta_{4} - \beta_{6} - \beta_{15} ) q^{65} + ( -1 - \beta_{1} + \beta_{2} - \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} - \beta_{12} + \beta_{13} - 2 \beta_{15} ) q^{66} + ( 2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{14} ) q^{67} + ( 3 \beta_{2} - 5 \beta_{5} - 4 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} + 5 \beta_{10} - \beta_{11} + 2 \beta_{13} - 2 \beta_{15} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{68} + ( 1 - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{10} + \beta_{12} - \beta_{13} + 2 \beta_{16} - \beta_{17} - 2 \beta_{18} ) q^{69} + ( -3 + 2 \beta_{1} - \beta_{3} - 3 \beta_{5} - 3 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} + 3 \beta_{13} - 2 \beta_{14} - \beta_{15} + 2 \beta_{17} - 2 \beta_{19} ) q^{70} + ( -2 - \beta_{1} - 3 \beta_{6} + \beta_{8} - 2 \beta_{9} + 2 \beta_{11} + \beta_{12} - 4 \beta_{13} + \beta_{14} - 3 \beta_{16} - \beta_{17} - \beta_{18} ) q^{71} + ( \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{12} + \beta_{13} - \beta_{16} + 2 \beta_{18} ) q^{72} + ( 2 - \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{14} + \beta_{16} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{73} + ( -3 - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{9} + 3 \beta_{10} + \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} - 2 \beta_{16} + \beta_{18} + 2 \beta_{19} ) q^{74} + ( -\beta_{5} + \beta_{11} - \beta_{16} - \beta_{17} ) q^{75} + ( 1 + \beta_{2} - \beta_{4} - 2 \beta_{6} + 4 \beta_{8} + 4 \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{18} - 2 \beta_{19} ) q^{76} + ( 1 - \beta_{5} - \beta_{7} + 2 \beta_{8} + 4 \beta_{9} - \beta_{10} + \beta_{13} - \beta_{17} - 2 \beta_{19} ) q^{77} + \beta_{3} q^{78} + ( -3 \beta_{2} - \beta_{5} - 2 \beta_{8} + 2 \beta_{9} - \beta_{13} + \beta_{15} - 2 \beta_{16} - \beta_{18} + \beta_{19} ) q^{79} + ( -4 + 11 \beta_{1} - \beta_{2} - 6 \beta_{3} + \beta_{4} - 6 \beta_{5} + \beta_{6} - 11 \beta_{7} - 9 \beta_{9} + 4 \beta_{10} - 7 \beta_{12} + 7 \beta_{13} - 2 \beta_{15} - \beta_{16} + \beta_{17} + 3 \beta_{18} ) q^{80} + ( -1 + \beta_{8} - \beta_{9} + \beta_{10} ) q^{81} + ( -3 + 2 \beta_{1} + \beta_{4} + 4 \beta_{6} - 3 \beta_{9} + 2 \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} + 4 \beta_{16} - 2 \beta_{17} - 3 \beta_{18} ) q^{82} + ( 2 - 3 \beta_{1} + 3 \beta_{3} + \beta_{4} + \beta_{6} + 3 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{11} + 3 \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{16} - 2 \beta_{17} - 2 \beta_{18} ) q^{83} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{14} - \beta_{16} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{84} + ( -2 \beta_{1} - \beta_{2} + 4 \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} + 2 \beta_{7} + 4 \beta_{9} + 3 \beta_{12} - 3 \beta_{13} - \beta_{14} + 3 \beta_{15} + \beta_{16} - 2 \beta_{17} - 4 \beta_{18} - \beta_{19} ) q^{85} + ( \beta_{2} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 6 \beta_{10} + 2 \beta_{13} - 2 \beta_{15} - \beta_{16} - \beta_{19} ) q^{86} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{14} - \beta_{18} + \beta_{19} ) q^{87} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{4} - 5 \beta_{5} + \beta_{6} - 2 \beta_{7} + 7 \beta_{8} + 6 \beta_{10} + \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} - \beta_{17} - 5 \beta_{18} ) q^{88} + ( -3 - 3 \beta_{1} - 3 \beta_{3} - 3 \beta_{5} + \beta_{6} - 3 \beta_{8} - 3 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} - 3 \beta_{14} + \beta_{15} - 2 \beta_{18} - 3 \beta_{19} ) q^{89} + ( -2 \beta_{5} - \beta_{7} - \beta_{11} + \beta_{13} - \beta_{15} - \beta_{16} + \beta_{17} + \beta_{18} ) q^{90} + \beta_{17} q^{91} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} - \beta_{18} ) q^{92} + ( -1 + 3 \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{16} - \beta_{18} ) q^{93} + ( \beta_{1} - \beta_{3} + 5 \beta_{4} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + 5 \beta_{11} - 3 \beta_{12} + \beta_{13} + 2 \beta_{14} + 2 \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{94} + ( 2 + 3 \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} - 3 \beta_{17} + 3 \beta_{19} ) q^{95} + ( -4 + 4 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} - 4 \beta_{7} - 5 \beta_{9} + 4 \beta_{10} - 3 \beta_{12} + 3 \beta_{13} - \beta_{14} - 2 \beta_{15} - \beta_{16} + 3 \beta_{18} - \beta_{19} ) q^{96} + ( -2 \beta_{2} - 5 \beta_{5} - 2 \beta_{7} + 2 \beta_{10} + \beta_{11} - 2 \beta_{13} + 2 \beta_{15} - \beta_{17} - 2 \beta_{18} ) q^{97} + ( -1 - \beta_{1} - \beta_{2} - 5 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{11} - \beta_{14} + \beta_{15} - \beta_{19} ) q^{98} + ( -1 - \beta_{1} - \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + q^{2} + 5q^{3} + 3q^{4} + 4q^{5} - q^{6} - 3q^{7} - 7q^{8} - 5q^{9} + O(q^{10})$$ $$20q + q^{2} + 5q^{3} + 3q^{4} + 4q^{5} - q^{6} - 3q^{7} - 7q^{8} - 5q^{9} - 2q^{10} + 14q^{11} - 18q^{12} + 5q^{13} - q^{14} - 4q^{15} - 35q^{16} + 2q^{17} - 4q^{18} - 2q^{19} + 45q^{20} - 2q^{21} + 11q^{22} + 6q^{23} + 2q^{24} - 7q^{25} + 4q^{26} + 5q^{27} + 12q^{28} + 26q^{29} - 3q^{30} + 20q^{31} + 42q^{32} + q^{33} - 24q^{34} - 18q^{35} + 3q^{36} - 6q^{37} - 3q^{38} - 5q^{39} - 26q^{41} - 9q^{42} + 28q^{43} - 38q^{44} - 16q^{45} - 17q^{46} + 8q^{47} - 20q^{48} + 2q^{49} - 29q^{50} + 3q^{51} - 3q^{52} + q^{53} - 6q^{54} - 36q^{56} - 8q^{57} + 22q^{58} - 21q^{59} - 45q^{60} + 26q^{61} - 10q^{62} - 3q^{63} - 87q^{64} + 16q^{65} + 14q^{66} + 56q^{67} + 65q^{68} + 4q^{69} - 24q^{70} - 28q^{71} - 2q^{72} + 45q^{73} - 29q^{74} - 3q^{75} + 60q^{76} + 4q^{77} + 6q^{78} - 15q^{79} - 7q^{80} - 5q^{81} - 46q^{82} + 36q^{83} + 8q^{84} + 39q^{86} + 24q^{87} + 73q^{88} - 126q^{89} - 2q^{90} + 3q^{91} + 2q^{92} - 20q^{93} - 3q^{94} + 47q^{95} - 47q^{96} + 18q^{97} - 54q^{98} - 16q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - x^{19} + 4 x^{18} + 4 x^{17} + 37 x^{16} - 74 x^{15} + 398 x^{14} - 224 x^{13} + 978 x^{12} + 115 x^{11} + 1963 x^{10} + 323 x^{9} + 3007 x^{8} + 1828 x^{7} + 6736 x^{6} + 4512 x^{5} + 5197 x^{4} + 2152 x^{3} + 1259 x^{2} + 495 x + 121$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$10\!\cdots\!45$$$$\nu^{19} +$$$$18\!\cdots\!24$$$$\nu^{18} +$$$$12\!\cdots\!49$$$$\nu^{17} +$$$$15\!\cdots\!93$$$$\nu^{16} +$$$$52\!\cdots\!90$$$$\nu^{15} +$$$$24\!\cdots\!63$$$$\nu^{14} +$$$$20\!\cdots\!00$$$$\nu^{13} +$$$$90\!\cdots\!62$$$$\nu^{12} +$$$$44\!\cdots\!43$$$$\nu^{11} +$$$$25\!\cdots\!14$$$$\nu^{10} +$$$$29\!\cdots\!32$$$$\nu^{9} +$$$$49\!\cdots\!28$$$$\nu^{8} +$$$$43\!\cdots\!45$$$$\nu^{7} +$$$$93\!\cdots\!57$$$$\nu^{6} +$$$$12\!\cdots\!00$$$$\nu^{5} +$$$$21\!\cdots\!12$$$$\nu^{4} +$$$$16\!\cdots\!62$$$$\nu^{3} +$$$$12\!\cdots\!94$$$$\nu^{2} +$$$$67\!\cdots\!63$$$$\nu +$$$$21\!\cdots\!64$$$$)/$$$$14\!\cdots\!69$$ $$\beta_{3}$$ $$=$$ $$($$$$11\!\cdots\!86$$$$\nu^{19} -$$$$20\!\cdots\!25$$$$\nu^{18} +$$$$49\!\cdots\!89$$$$\nu^{17} +$$$$23\!\cdots\!82$$$$\nu^{16} +$$$$36\!\cdots\!52$$$$\nu^{15} -$$$$11\!\cdots\!84$$$$\nu^{14} +$$$$50\!\cdots\!72$$$$\nu^{13} -$$$$54\!\cdots\!69$$$$\nu^{12} +$$$$10\!\cdots\!52$$$$\nu^{11} -$$$$33\!\cdots\!81$$$$\nu^{10} +$$$$14\!\cdots\!58$$$$\nu^{9} -$$$$84\!\cdots\!91$$$$\nu^{8} +$$$$23\!\cdots\!79$$$$\nu^{7} +$$$$31\!\cdots\!16$$$$\nu^{6} +$$$$52\!\cdots\!95$$$$\nu^{5} -$$$$25\!\cdots\!86$$$$\nu^{4} -$$$$19\!\cdots\!26$$$$\nu^{3} -$$$$18\!\cdots\!91$$$$\nu^{2} -$$$$69\!\cdots\!14$$$$\nu +$$$$45\!\cdots\!91$$$$)/$$$$13\!\cdots\!79$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$13\!\cdots\!34$$$$\nu^{19} +$$$$21\!\cdots\!93$$$$\nu^{18} -$$$$66\!\cdots\!18$$$$\nu^{17} -$$$$12\!\cdots\!07$$$$\nu^{16} -$$$$46\!\cdots\!06$$$$\nu^{15} +$$$$12\!\cdots\!90$$$$\nu^{14} -$$$$60\!\cdots\!37$$$$\nu^{13} +$$$$68\!\cdots\!18$$$$\nu^{12} -$$$$16\!\cdots\!69$$$$\nu^{11} +$$$$73\!\cdots\!77$$$$\nu^{10} -$$$$24\!\cdots\!63$$$$\nu^{9} +$$$$72\!\cdots\!03$$$$\nu^{8} -$$$$32\!\cdots\!07$$$$\nu^{7} -$$$$66\!\cdots\!68$$$$\nu^{6} -$$$$67\!\cdots\!95$$$$\nu^{5} -$$$$15\!\cdots\!78$$$$\nu^{4} -$$$$32\!\cdots\!94$$$$\nu^{3} +$$$$12\!\cdots\!64$$$$\nu^{2} +$$$$34\!\cdots\!57$$$$\nu +$$$$42\!\cdots\!51$$$$)/$$$$14\!\cdots\!69$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$12\!\cdots\!30$$$$\nu^{19} +$$$$38\!\cdots\!05$$$$\nu^{18} -$$$$39\!\cdots\!17$$$$\nu^{17} -$$$$86\!\cdots\!91$$$$\nu^{16} -$$$$49\!\cdots\!40$$$$\nu^{15} +$$$$62\!\cdots\!31$$$$\nu^{14} -$$$$42\!\cdots\!98$$$$\nu^{13} -$$$$78\!\cdots\!16$$$$\nu^{12} -$$$$95\!\cdots\!71$$$$\nu^{11} -$$$$98\!\cdots\!16$$$$\nu^{10} -$$$$23\!\cdots\!70$$$$\nu^{9} -$$$$18\!\cdots\!50$$$$\nu^{8} -$$$$38\!\cdots\!76$$$$\nu^{7} -$$$$44\!\cdots\!44$$$$\nu^{6} -$$$$97\!\cdots\!95$$$$\nu^{5} -$$$$10\!\cdots\!18$$$$\nu^{4} -$$$$93\!\cdots\!75$$$$\nu^{3} -$$$$50\!\cdots\!63$$$$\nu^{2} -$$$$28\!\cdots\!22$$$$\nu -$$$$10\!\cdots\!15$$$$)/$$$$13\!\cdots\!79$$ $$\beta_{6}$$ $$=$$ $$($$$$15\!\cdots\!03$$$$\nu^{19} -$$$$44\!\cdots\!77$$$$\nu^{18} +$$$$79\!\cdots\!28$$$$\nu^{17} -$$$$26\!\cdots\!63$$$$\nu^{16} +$$$$37\!\cdots\!53$$$$\nu^{15} -$$$$21\!\cdots\!77$$$$\nu^{14} +$$$$78\!\cdots\!36$$$$\nu^{13} -$$$$13\!\cdots\!69$$$$\nu^{12} +$$$$15\!\cdots\!88$$$$\nu^{11} -$$$$16\!\cdots\!23$$$$\nu^{10} +$$$$12\!\cdots\!95$$$$\nu^{9} -$$$$37\!\cdots\!15$$$$\nu^{8} +$$$$20\!\cdots\!02$$$$\nu^{7} -$$$$30\!\cdots\!29$$$$\nu^{6} +$$$$19\!\cdots\!64$$$$\nu^{5} -$$$$99\!\cdots\!75$$$$\nu^{4} -$$$$86\!\cdots\!76$$$$\nu^{3} -$$$$49\!\cdots\!03$$$$\nu^{2} -$$$$14\!\cdots\!02$$$$\nu +$$$$42\!\cdots\!08$$$$)/$$$$13\!\cdots\!79$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$18\!\cdots\!89$$$$\nu^{19} +$$$$39\!\cdots\!34$$$$\nu^{18} -$$$$94\!\cdots\!77$$$$\nu^{17} +$$$$59\!\cdots\!00$$$$\nu^{16} -$$$$58\!\cdots\!88$$$$\nu^{15} +$$$$20\!\cdots\!13$$$$\nu^{14} -$$$$88\!\cdots\!07$$$$\nu^{13} +$$$$12\!\cdots\!09$$$$\nu^{12} -$$$$22\!\cdots\!93$$$$\nu^{11} +$$$$16\!\cdots\!33$$$$\nu^{10} -$$$$31\!\cdots\!65$$$$\nu^{9} +$$$$28\!\cdots\!66$$$$\nu^{8} -$$$$46\!\cdots\!72$$$$\nu^{7} +$$$$18\!\cdots\!72$$$$\nu^{6} -$$$$85\!\cdots\!16$$$$\nu^{5} +$$$$44\!\cdots\!45$$$$\nu^{4} -$$$$12\!\cdots\!23$$$$\nu^{3} +$$$$32\!\cdots\!51$$$$\nu^{2} +$$$$15\!\cdots\!72$$$$\nu +$$$$22\!\cdots\!69$$$$)/$$$$13\!\cdots\!79$$ $$\beta_{8}$$ $$=$$ $$($$$$20\!\cdots\!79$$$$\nu^{19} -$$$$86\!\cdots\!00$$$$\nu^{18} +$$$$40\!\cdots\!42$$$$\nu^{17} +$$$$18\!\cdots\!63$$$$\nu^{16} +$$$$76\!\cdots\!23$$$$\nu^{15} -$$$$90\!\cdots\!78$$$$\nu^{14} +$$$$60\!\cdots\!99$$$$\nu^{13} +$$$$50\!\cdots\!81$$$$\nu^{12} +$$$$69\!\cdots\!63$$$$\nu^{11} +$$$$26\!\cdots\!08$$$$\nu^{10} +$$$$23\!\cdots\!14$$$$\nu^{9} +$$$$41\!\cdots\!32$$$$\nu^{8} +$$$$31\!\cdots\!27$$$$\nu^{7} +$$$$89\!\cdots\!04$$$$\nu^{6} +$$$$11\!\cdots\!52$$$$\nu^{5} +$$$$18\!\cdots\!24$$$$\nu^{4} +$$$$59\!\cdots\!68$$$$\nu^{3} +$$$$59\!\cdots\!61$$$$\nu^{2} -$$$$94\!\cdots\!00$$$$\nu +$$$$79\!\cdots\!82$$$$)/$$$$14\!\cdots\!69$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$29\!\cdots\!05$$$$\nu^{19} +$$$$50\!\cdots\!68$$$$\nu^{18} -$$$$14\!\cdots\!74$$$$\nu^{17} -$$$$24\!\cdots\!65$$$$\nu^{16} -$$$$10\!\cdots\!86$$$$\nu^{15} +$$$$29\!\cdots\!06$$$$\nu^{14} -$$$$13\!\cdots\!50$$$$\nu^{13} +$$$$15\!\cdots\!83$$$$\nu^{12} -$$$$35\!\cdots\!54$$$$\nu^{11} +$$$$20\!\cdots\!57$$$$\nu^{10} -$$$$60\!\cdots\!11$$$$\nu^{9} +$$$$35\!\cdots\!32$$$$\nu^{8} -$$$$89\!\cdots\!10$$$$\nu^{7} +$$$$14\!\cdots\!19$$$$\nu^{6} -$$$$17\!\cdots\!64$$$$\nu^{5} +$$$$12\!\cdots\!16$$$$\nu^{4} -$$$$80\!\cdots\!36$$$$\nu^{3} +$$$$56\!\cdots\!04$$$$\nu^{2} -$$$$14\!\cdots\!62$$$$\nu +$$$$62\!\cdots\!29$$$$)/$$$$14\!\cdots\!69$$ $$\beta_{10}$$ $$=$$ $$($$$$40\!\cdots\!81$$$$\nu^{19} -$$$$53\!\cdots\!27$$$$\nu^{18} +$$$$18\!\cdots\!99$$$$\nu^{17} +$$$$10\!\cdots\!45$$$$\nu^{16} +$$$$14\!\cdots\!95$$$$\nu^{15} -$$$$34\!\cdots\!66$$$$\nu^{14} +$$$$17\!\cdots\!62$$$$\nu^{13} -$$$$14\!\cdots\!36$$$$\nu^{12} +$$$$46\!\cdots\!77$$$$\nu^{11} -$$$$68\!\cdots\!57$$$$\nu^{10} +$$$$84\!\cdots\!94$$$$\nu^{9} -$$$$31\!\cdots\!75$$$$\nu^{8} +$$$$13\!\cdots\!68$$$$\nu^{7} +$$$$48\!\cdots\!99$$$$\nu^{6} +$$$$27\!\cdots\!40$$$$\nu^{5} +$$$$12\!\cdots\!27$$$$\nu^{4} +$$$$21\!\cdots\!03$$$$\nu^{3} +$$$$90\!\cdots\!98$$$$\nu^{2} +$$$$53\!\cdots\!80$$$$\nu +$$$$21\!\cdots\!49$$$$)/$$$$14\!\cdots\!69$$ $$\beta_{11}$$ $$=$$ $$($$$$43\!\cdots\!35$$$$\nu^{19} -$$$$12\!\cdots\!65$$$$\nu^{18} +$$$$29\!\cdots\!76$$$$\nu^{17} -$$$$20\!\cdots\!01$$$$\nu^{16} +$$$$14\!\cdots\!93$$$$\nu^{15} -$$$$59\!\cdots\!17$$$$\nu^{14} +$$$$24\!\cdots\!54$$$$\nu^{13} -$$$$45\!\cdots\!15$$$$\nu^{12} +$$$$79\!\cdots\!69$$$$\nu^{11} -$$$$87\!\cdots\!73$$$$\nu^{10} +$$$$11\!\cdots\!79$$$$\nu^{9} -$$$$14\!\cdots\!07$$$$\nu^{8} +$$$$17\!\cdots\!47$$$$\nu^{7} -$$$$17\!\cdots\!01$$$$\nu^{6} +$$$$26\!\cdots\!28$$$$\nu^{5} -$$$$28\!\cdots\!07$$$$\nu^{4} +$$$$11\!\cdots\!34$$$$\nu^{3} -$$$$21\!\cdots\!71$$$$\nu^{2} -$$$$25\!\cdots\!47$$$$\nu -$$$$48\!\cdots\!89$$$$)/$$$$14\!\cdots\!69$$ $$\beta_{12}$$ $$=$$ $$($$$$66\!\cdots\!18$$$$\nu^{19} -$$$$10\!\cdots\!07$$$$\nu^{18} +$$$$30\!\cdots\!95$$$$\nu^{17} +$$$$10\!\cdots\!31$$$$\nu^{16} +$$$$23\!\cdots\!92$$$$\nu^{15} -$$$$64\!\cdots\!63$$$$\nu^{14} +$$$$29\!\cdots\!33$$$$\nu^{13} -$$$$30\!\cdots\!82$$$$\nu^{12} +$$$$75\!\cdots\!15$$$$\nu^{11} -$$$$34\!\cdots\!18$$$$\nu^{10} +$$$$13\!\cdots\!94$$$$\nu^{9} -$$$$64\!\cdots\!36$$$$\nu^{8} +$$$$19\!\cdots\!14$$$$\nu^{7} -$$$$56\!\cdots\!78$$$$\nu^{6} +$$$$38\!\cdots\!75$$$$\nu^{5} +$$$$18\!\cdots\!80$$$$\nu^{4} +$$$$18\!\cdots\!53$$$$\nu^{3} -$$$$10\!\cdots\!78$$$$\nu^{2} -$$$$28\!\cdots\!98$$$$\nu -$$$$30\!\cdots\!10$$$$)/$$$$14\!\cdots\!69$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$81\!\cdots\!57$$$$\nu^{19} +$$$$14\!\cdots\!98$$$$\nu^{18} -$$$$40\!\cdots\!73$$$$\nu^{17} -$$$$73\!\cdots\!09$$$$\nu^{16} -$$$$28\!\cdots\!29$$$$\nu^{15} +$$$$81\!\cdots\!64$$$$\nu^{14} -$$$$37\!\cdots\!56$$$$\nu^{13} +$$$$43\!\cdots\!02$$$$\nu^{12} -$$$$10\!\cdots\!47$$$$\nu^{11} +$$$$49\!\cdots\!73$$$$\nu^{10} -$$$$16\!\cdots\!48$$$$\nu^{9} +$$$$77\!\cdots\!61$$$$\nu^{8} -$$$$25\!\cdots\!76$$$$\nu^{7} +$$$$91\!\cdots\!72$$$$\nu^{6} -$$$$49\!\cdots\!33$$$$\nu^{5} -$$$$22\!\cdots\!04$$$$\nu^{4} -$$$$28\!\cdots\!11$$$$\nu^{3} -$$$$37\!\cdots\!72$$$$\nu^{2} -$$$$96\!\cdots\!41$$$$\nu -$$$$32\!\cdots\!44$$$$)/$$$$14\!\cdots\!69$$ $$\beta_{14}$$ $$=$$ $$($$$$80\!\cdots\!07$$$$\nu^{19} -$$$$18\!\cdots\!77$$$$\nu^{18} +$$$$26\!\cdots\!63$$$$\nu^{17} +$$$$53\!\cdots\!69$$$$\nu^{16} +$$$$32\!\cdots\!58$$$$\nu^{15} -$$$$37\!\cdots\!35$$$$\nu^{14} +$$$$27\!\cdots\!74$$$$\nu^{13} +$$$$51\!\cdots\!42$$$$\nu^{12} +$$$$67\!\cdots\!33$$$$\nu^{11} +$$$$57\!\cdots\!83$$$$\nu^{10} +$$$$16\!\cdots\!90$$$$\nu^{9} +$$$$12\!\cdots\!49$$$$\nu^{8} +$$$$24\!\cdots\!33$$$$\nu^{7} +$$$$29\!\cdots\!97$$$$\nu^{6} +$$$$63\!\cdots\!18$$$$\nu^{5} +$$$$72\!\cdots\!75$$$$\nu^{4} +$$$$63\!\cdots\!04$$$$\nu^{3} +$$$$34\!\cdots\!72$$$$\nu^{2} +$$$$10\!\cdots\!50$$$$\nu +$$$$61\!\cdots\!13$$$$)/$$$$13\!\cdots\!79$$ $$\beta_{15}$$ $$=$$ $$($$$$92\!\cdots\!19$$$$\nu^{19} -$$$$60\!\cdots\!73$$$$\nu^{18} +$$$$32\!\cdots\!34$$$$\nu^{17} +$$$$51\!\cdots\!60$$$$\nu^{16} +$$$$34\!\cdots\!37$$$$\nu^{15} -$$$$57\!\cdots\!70$$$$\nu^{14} +$$$$33\!\cdots\!55$$$$\nu^{13} -$$$$69\!\cdots\!74$$$$\nu^{12} +$$$$77\!\cdots\!31$$$$\nu^{11} +$$$$45\!\cdots\!37$$$$\nu^{10} +$$$$16\!\cdots\!55$$$$\nu^{9} +$$$$89\!\cdots\!64$$$$\nu^{8} +$$$$25\!\cdots\!23$$$$\nu^{7} +$$$$25\!\cdots\!86$$$$\nu^{6} +$$$$63\!\cdots\!31$$$$\nu^{5} +$$$$60\!\cdots\!76$$$$\nu^{4} +$$$$52\!\cdots\!44$$$$\nu^{3} +$$$$28\!\cdots\!69$$$$\nu^{2} +$$$$84\!\cdots\!93$$$$\nu +$$$$42\!\cdots\!18$$$$)/$$$$13\!\cdots\!79$$ $$\beta_{16}$$ $$=$$ $$($$$$10\!\cdots\!52$$$$\nu^{19} -$$$$91\!\cdots\!57$$$$\nu^{18} +$$$$38\!\cdots\!54$$$$\nu^{17} +$$$$51\!\cdots\!27$$$$\nu^{16} +$$$$39\!\cdots\!73$$$$\nu^{15} -$$$$75\!\cdots\!17$$$$\nu^{14} +$$$$40\!\cdots\!69$$$$\nu^{13} -$$$$15\!\cdots\!94$$$$\nu^{12} +$$$$93\!\cdots\!50$$$$\nu^{11} +$$$$28\!\cdots\!20$$$$\nu^{10} +$$$$19\!\cdots\!14$$$$\nu^{9} +$$$$50\!\cdots\!19$$$$\nu^{8} +$$$$29\!\cdots\!86$$$$\nu^{7} +$$$$22\!\cdots\!99$$$$\nu^{6} +$$$$71\!\cdots\!87$$$$\nu^{5} +$$$$51\!\cdots\!48$$$$\nu^{4} +$$$$50\!\cdots\!63$$$$\nu^{3} +$$$$18\!\cdots\!76$$$$\nu^{2} +$$$$10\!\cdots\!74$$$$\nu +$$$$44\!\cdots\!72$$$$)/$$$$14\!\cdots\!69$$ $$\beta_{17}$$ $$=$$ $$($$$$11\!\cdots\!59$$$$\nu^{19} -$$$$10\!\cdots\!01$$$$\nu^{18} +$$$$47\!\cdots\!24$$$$\nu^{17} +$$$$39\!\cdots\!29$$$$\nu^{16} +$$$$43\!\cdots\!18$$$$\nu^{15} -$$$$81\!\cdots\!78$$$$\nu^{14} +$$$$45\!\cdots\!78$$$$\nu^{13} -$$$$26\!\cdots\!23$$$$\nu^{12} +$$$$12\!\cdots\!83$$$$\nu^{11} -$$$$74\!\cdots\!42$$$$\nu^{10} +$$$$26\!\cdots\!28$$$$\nu^{9} +$$$$97\!\cdots\!40$$$$\nu^{8} +$$$$40\!\cdots\!03$$$$\nu^{7} +$$$$16\!\cdots\!50$$$$\nu^{6} +$$$$86\!\cdots\!29$$$$\nu^{5} +$$$$49\!\cdots\!93$$$$\nu^{4} +$$$$76\!\cdots\!68$$$$\nu^{3} +$$$$18\!\cdots\!36$$$$\nu^{2} +$$$$19\!\cdots\!93$$$$\nu +$$$$24\!\cdots\!71$$$$)/$$$$14\!\cdots\!69$$ $$\beta_{18}$$ $$=$$ $$($$$$15\!\cdots\!99$$$$\nu^{19} -$$$$19\!\cdots\!42$$$$\nu^{18} +$$$$62\!\cdots\!96$$$$\nu^{17} +$$$$45\!\cdots\!71$$$$\nu^{16} +$$$$53\!\cdots\!61$$$$\nu^{15} -$$$$12\!\cdots\!93$$$$\nu^{14} +$$$$62\!\cdots\!82$$$$\nu^{13} -$$$$49\!\cdots\!55$$$$\nu^{12} +$$$$14\!\cdots\!53$$$$\nu^{11} -$$$$19\!\cdots\!78$$$$\nu^{10} +$$$$27\!\cdots\!00$$$$\nu^{9} -$$$$34\!\cdots\!54$$$$\nu^{8} +$$$$40\!\cdots\!22$$$$\nu^{7} +$$$$15\!\cdots\!66$$$$\nu^{6} +$$$$89\!\cdots\!50$$$$\nu^{5} +$$$$36\!\cdots\!00$$$$\nu^{4} +$$$$48\!\cdots\!71$$$$\nu^{3} +$$$$58\!\cdots\!52$$$$\nu^{2} +$$$$44\!\cdots\!43$$$$\nu +$$$$31\!\cdots\!30$$$$)/$$$$14\!\cdots\!69$$ $$\beta_{19}$$ $$=$$ $$($$$$-$$$$21\!\cdots\!12$$$$\nu^{19} +$$$$31\!\cdots\!24$$$$\nu^{18} -$$$$95\!\cdots\!43$$$$\nu^{17} -$$$$44\!\cdots\!53$$$$\nu^{16} -$$$$73\!\cdots\!78$$$$\nu^{15} +$$$$19\!\cdots\!38$$$$\nu^{14} -$$$$91\!\cdots\!52$$$$\nu^{13} +$$$$88\!\cdots\!11$$$$\nu^{12} -$$$$22\!\cdots\!85$$$$\nu^{11} +$$$$67\!\cdots\!48$$$$\nu^{10} -$$$$39\!\cdots\!78$$$$\nu^{9} +$$$$10\!\cdots\!04$$$$\nu^{8} -$$$$59\!\cdots\!95$$$$\nu^{7} -$$$$11\!\cdots\!97$$$$\nu^{6} -$$$$12\!\cdots\!89$$$$\nu^{5} -$$$$34\!\cdots\!44$$$$\nu^{4} -$$$$66\!\cdots\!51$$$$\nu^{3} -$$$$75\!\cdots\!01$$$$\nu^{2} -$$$$61\!\cdots\!75$$$$\nu -$$$$44\!\cdots\!12$$$$)/$$$$14\!\cdots\!69$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{13} + \beta_{12} - \beta_{10} + 3 \beta_{9} + \beta_{7} - \beta_{1} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{18} - \beta_{16} - \beta_{15} + \beta_{13} - \beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} - 5 \beta_{7} - 5 \beta_{5} + \beta_{4} - 4 \beta_{3} + \beta_{2} + \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{19} + \beta_{18} - \beta_{17} + \beta_{16} + 7 \beta_{15} - 7 \beta_{13} + \beta_{11} - 17 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} + 2 \beta_{7} + 9 \beta_{5}$$ $$\nu^{5}$$ $$=$$ $$-\beta_{19} - 9 \beta_{18} - 2 \beta_{15} - \beta_{14} + 9 \beta_{12} - \beta_{11} + 4 \beta_{10} + 4 \beta_{8} - 11 \beta_{5} - 9 \beta_{4} + 21 \beta_{3} + 9 \beta_{2} - 11 \beta_{1} + 9$$ $$\nu^{6}$$ $$=$$ $$59 \beta_{18} + 9 \beta_{17} - 10 \beta_{16} - 9 \beta_{14} + 49 \beta_{13} - 70 \beta_{12} - 60 \beta_{9} - 54 \beta_{8} - 26 \beta_{7} - 10 \beta_{6} + 2 \beta_{4} - 26 \beta_{3} + 71 \beta_{1} - 60$$ $$\nu^{7}$$ $$=$$ $$10 \beta_{19} - 97 \beta_{18} - 2 \beta_{17} + 69 \beta_{16} + 28 \beta_{15} + 10 \beta_{14} - 102 \beta_{13} + 102 \beta_{12} - 56 \beta_{10} + 125 \beta_{9} + 228 \beta_{7} + 66 \beta_{6} + 100 \beta_{5} - 69 \beta_{4} + 100 \beta_{3} - 66 \beta_{2} - 228 \beta_{1} + 56$$ $$\nu^{8}$$ $$=$$ $$-69 \beta_{19} + 38 \beta_{18} + 69 \beta_{17} - 38 \beta_{16} - 353 \beta_{15} - 3 \beta_{14} + 530 \beta_{13} - 353 \beta_{12} - 3 \beta_{11} + 810 \beta_{10} - 810 \beta_{9} + 359 \beta_{8} - 550 \beta_{7} + 43 \beta_{6} - 550 \beta_{5} + 81 \beta_{4} - 299 \beta_{3} + 38 \beta_{2} + 251 \beta_{1} - 359$$ $$\nu^{9}$$ $$=$$ $$38 \beta_{19} - 176 \beta_{18} - 81 \beta_{17} + 462 \beta_{16} + 603 \beta_{15} - 603 \beta_{13} + 81 \beta_{11} - 1106 \beta_{10} + 575 \beta_{9} - 575 \beta_{8} + 848 \beta_{7} + 1697 \beta_{5} - 518 \beta_{2}$$ $$\nu^{10}$$ $$=$$ $$-462 \beta_{19} - 2596 \beta_{18} - 1409 \beta_{15} + 56 \beta_{14} + 2596 \beta_{12} - 462 \beta_{11} + 3403 \beta_{10} + 3403 \beta_{8} - 163 \beta_{6} - 2189 \beta_{5} - 628 \beta_{4} + 2078 \beta_{3} + 628 \beta_{2} - 2189 \beta_{1} + 2504$$ $$\nu^{11}$$ $$=$$ $$8119 \beta_{18} + 628 \beta_{17} - 3221 \beta_{16} - 628 \beta_{14} + 4898 \beta_{13} - 7496 \beta_{12} - 163 \beta_{11} - 5271 \beta_{9} - 4158 \beta_{8} - 6948 \beta_{7} - 3221 \beta_{6} + 3895 \beta_{4} - 6948 \beta_{3} + 12883 \beta_{1} - 5271$$ $$\nu^{12}$$ $$=$$ $$3221 \beta_{19} - 15746 \beta_{18} - 3895 \beta_{17} + 4740 \beta_{16} + 11006 \beta_{15} + 3221 \beta_{14} - 30377 \beta_{13} + 30377 \beta_{12} - 25771 \beta_{10} + 43749 \beta_{9} + 33233 \beta_{7} + 4852 \beta_{6} + 18247 \beta_{5} - 4740 \beta_{4} + 18247 \beta_{3} - 4852 \beta_{2} - 33233 \beta_{1} + 25771$$ $$\nu^{13}$$ $$=$$ $$-4740 \beta_{19} + 29420 \beta_{18} + 4740 \beta_{17} - 29420 \beta_{16} - 39604 \beta_{15} + 112 \beta_{14} + 61917 \beta_{13} - 39604 \beta_{12} + 112 \beta_{11} + 78635 \beta_{10} - 78635 \beta_{9} + 32914 \beta_{8} - 98829 \beta_{7} - 6716 \beta_{6} - 98829 \beta_{5} + 22704 \beta_{4} - 42960 \beta_{3} + 29420 \beta_{2} + 55869 \beta_{1} - 32914$$ $$\nu^{14}$$ $$=$$ $$29420 \beta_{19} + 47661 \beta_{18} - 22704 \beta_{17} + 37740 \beta_{16} + 146101 \beta_{15} - 146101 \beta_{13} + 22704 \beta_{11} - 327639 \beta_{10} + 195922 \beta_{9} - 195922 \beta_{8} + 148583 \beta_{7} + 259600 \beta_{5} - 43953 \beta_{2}$$ $$\nu^{15}$$ $$=$$ $$-37740 \beta_{19} - 318589 \beta_{18} - 185820 \beta_{15} + 6213 \beta_{14} + 318589 \beta_{12} - 37740 \beta_{11} + 384253 \beta_{10} + 384253 \beta_{8} + 60590 \beta_{6} - 444439 \beta_{5} - 162592 \beta_{4} + 318470 \beta_{3} + 162592 \beta_{2} - 444439 \beta_{1} + 261753$$ $$\nu^{16}$$ $$=$$ $$1406682 \beta_{18} + 162592 \beta_{17} - 295739 \beta_{16} - 162592 \beta_{14} + 1110943 \beta_{13} - 1772351 \beta_{12} + 60590 \beta_{11} - 1495412 \beta_{9} - 979759 \beta_{8} - 1193741 \beta_{7} - 295739 \beta_{6} + 385770 \beta_{4} - 1193741 \beta_{3} + 2031585 \beta_{1} - 1495412$$ $$\nu^{17}$$ $$=$$ $$295739 \beta_{19} - 3218715 \beta_{18} - 385770 \beta_{17} + 1699794 \beta_{16} + 1518921 \beta_{15} + 295739 \beta_{14} - 4069423 \beta_{13} + 4069423 \beta_{12} - 3165633 \beta_{10} + 5249299 \beta_{9} + 5913434 \beta_{7} + 1183504 \beta_{6} + 3513149 \beta_{5} - 1699794 \beta_{4} + 3513149 \beta_{3} - 1183504 \beta_{2} - 5913434 \beta_{1} + 3165633$$ $$\nu^{18}$$ $$=$$ $$-1699794 \beta_{19} + 3271689 \beta_{18} + 1699794 \beta_{17} - 3271689 \beta_{16} - 8500993 \beta_{15} - 516290 \beta_{14} + 13623905 \beta_{13} - 8500993 \beta_{12} - 516290 \beta_{11} + 18830848 \beta_{10} - 18830848 \beta_{9} + 7372980 \beta_{8} - 15915024 \beta_{7} - 941738 \beta_{6} - 15915024 \beta_{5} + 2329951 \beta_{4} - 6403297 \beta_{3} + 3271689 \beta_{2} + 9511727 \beta_{1} - 7372980$$ $$\nu^{19}$$ $$=$$ $$3271689 \beta_{19} + 3524367 \beta_{18} - 2329951 \beta_{17} + 8742759 \beta_{16} + 20333160 \beta_{15} - 20333160 \beta_{13} + 2329951 \beta_{11} - 42305436 \beta_{10} + 25732709 \beta_{9} - 25732709 \beta_{8} + 27664402 \beta_{7} + 45965207 \beta_{5} - 12993748 \beta_{2}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/429\mathbb{Z}\right)^\times$$.

 $$n$$ $$67$$ $$79$$ $$287$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{8} - \beta_{9} + \beta_{10}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
157.1
 −2.26411 − 1.64497i −0.972448 − 0.706525i −0.297979 − 0.216494i 1.21086 + 0.879744i 2.01466 + 1.46373i −0.375422 − 1.15543i −0.262109 − 0.806688i 0.169480 + 0.521606i 0.566002 + 1.74197i 0.711066 + 2.18844i −2.26411 + 1.64497i −0.972448 + 0.706525i −0.297979 + 0.216494i 1.21086 − 0.879744i 2.01466 − 1.46373i −0.375422 + 1.15543i −0.262109 + 0.806688i 0.169480 − 0.521606i 0.566002 − 1.74197i 0.711066 − 2.18844i
−2.26411 + 1.64497i −0.309017 0.951057i 1.80222 5.54667i 2.68510 + 1.95084i 2.26411 + 1.64497i −0.449002 + 1.38189i 3.31406 + 10.1996i −0.809017 + 0.587785i −9.28845
157.2 −0.972448 + 0.706525i −0.309017 0.951057i −0.171556 + 0.527996i −1.69471 1.23128i 0.972448 + 0.706525i −0.640431 + 1.97104i −0.949097 2.92102i −0.809017 + 0.587785i 2.51795
157.3 −0.297979 + 0.216494i −0.309017 0.951057i −0.576112 + 1.77309i −0.859409 0.624397i 0.297979 + 0.216494i 1.08879 3.35095i −0.439831 1.35366i −0.809017 + 0.587785i 0.391264
157.4 1.21086 0.879744i −0.309017 0.951057i 0.0742075 0.228387i 2.08500 + 1.51484i −1.21086 0.879744i −0.780732 + 2.40285i 0.813950 + 2.50508i −0.809017 + 0.587785i 3.85732
157.5 2.01466 1.46373i −0.309017 0.951057i 1.29829 3.99572i 1.02009 + 0.741137i −2.01466 1.46373i 0.590393 1.81704i −1.69400 5.21361i −0.809017 + 0.587785i 3.13995
196.1 −0.375422 + 1.15543i 0.809017 0.587785i 0.423959 + 0.308024i 0.339998 + 1.04640i 0.375422 + 1.15543i 0.237146 + 0.172297i −2.48080 + 1.80240i 0.309017 0.951057i −1.33669
196.2 −0.262109 + 0.806688i 0.809017 0.587785i 1.03599 + 0.752690i −1.08398 3.33616i 0.262109 + 0.806688i −2.82268 2.05079i −2.25115 + 1.63555i 0.309017 0.951057i 2.97536
196.3 0.169480 0.521606i 0.809017 0.587785i 1.37468 + 0.998767i −0.266070 0.818878i −0.169480 0.521606i 2.33406 + 1.69579i 1.64135 1.19251i 0.309017 0.951057i −0.472225
196.4 0.566002 1.74197i 0.809017 0.587785i −1.09608 0.796352i 0.725100 + 2.23163i −0.566002 1.74197i −1.01039 0.734092i 0.956015 0.694585i 0.309017 0.951057i 4.29785
196.5 0.711066 2.18844i 0.809017 0.587785i −2.66560 1.93667i −0.951111 2.92722i −0.711066 2.18844i −0.0471527 0.0342585i −2.41051 + 1.75134i 0.309017 0.951057i −7.08233
235.1 −2.26411 1.64497i −0.309017 + 0.951057i 1.80222 + 5.54667i 2.68510 1.95084i 2.26411 1.64497i −0.449002 1.38189i 3.31406 10.1996i −0.809017 0.587785i −9.28845
235.2 −0.972448 0.706525i −0.309017 + 0.951057i −0.171556 0.527996i −1.69471 + 1.23128i 0.972448 0.706525i −0.640431 1.97104i −0.949097 + 2.92102i −0.809017 0.587785i 2.51795
235.3 −0.297979 0.216494i −0.309017 + 0.951057i −0.576112 1.77309i −0.859409 + 0.624397i 0.297979 0.216494i 1.08879 + 3.35095i −0.439831 + 1.35366i −0.809017 0.587785i 0.391264
235.4 1.21086 + 0.879744i −0.309017 + 0.951057i 0.0742075 + 0.228387i 2.08500 1.51484i −1.21086 + 0.879744i −0.780732 2.40285i 0.813950 2.50508i −0.809017 0.587785i 3.85732
235.5 2.01466 + 1.46373i −0.309017 + 0.951057i 1.29829 + 3.99572i 1.02009 0.741137i −2.01466 + 1.46373i 0.590393 + 1.81704i −1.69400 + 5.21361i −0.809017 0.587785i 3.13995
313.1 −0.375422 1.15543i 0.809017 + 0.587785i 0.423959 0.308024i 0.339998 1.04640i 0.375422 1.15543i 0.237146 0.172297i −2.48080 1.80240i 0.309017 + 0.951057i −1.33669
313.2 −0.262109 0.806688i 0.809017 + 0.587785i 1.03599 0.752690i −1.08398 + 3.33616i 0.262109 0.806688i −2.82268 + 2.05079i −2.25115 1.63555i 0.309017 + 0.951057i 2.97536
313.3 0.169480 + 0.521606i 0.809017 + 0.587785i 1.37468 0.998767i −0.266070 + 0.818878i −0.169480 + 0.521606i 2.33406 1.69579i 1.64135 + 1.19251i 0.309017 + 0.951057i −0.472225
313.4 0.566002 + 1.74197i 0.809017 + 0.587785i −1.09608 + 0.796352i 0.725100 2.23163i −0.566002 + 1.74197i −1.01039 + 0.734092i 0.956015 + 0.694585i 0.309017 + 0.951057i 4.29785
313.5 0.711066 + 2.18844i 0.809017 + 0.587785i −2.66560 + 1.93667i −0.951111 + 2.92722i −0.711066 + 2.18844i −0.0471527 + 0.0342585i −2.41051 1.75134i 0.309017 + 0.951057i −7.08233
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 313.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.n.b 20
11.c even 5 1 inner 429.2.n.b 20
11.c even 5 1 4719.2.a.bn 10
11.d odd 10 1 4719.2.a.bi 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.n.b 20 1.a even 1 1 trivial
429.2.n.b 20 11.c even 5 1 inner
4719.2.a.bi 10 11.d odd 10 1
4719.2.a.bn 10 11.c even 5 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{20} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$121 + 495 T + 1259 T^{2} + 2152 T^{3} + 5197 T^{4} + 4512 T^{5} + 6736 T^{6} + 1828 T^{7} + 3007 T^{8} + 323 T^{9} + 1963 T^{10} + 115 T^{11} + 978 T^{12} - 224 T^{13} + 398 T^{14} - 74 T^{15} + 37 T^{16} + 4 T^{17} + 4 T^{18} - T^{19} + T^{20}$$
$3$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}$$
$5$ $$331776 + 55296 T + 483840 T^{2} - 86784 T^{3} + 363712 T^{4} - 40160 T^{5} + 235336 T^{6} - 145836 T^{7} + 274477 T^{8} - 74453 T^{9} + 50068 T^{10} - 13843 T^{11} + 12155 T^{12} - 5091 T^{13} + 2842 T^{14} - 878 T^{15} + 390 T^{16} - 97 T^{17} + 24 T^{18} - 4 T^{19} + T^{20}$$
$7$ $$121 + 2893 T + 23230 T^{2} - 120793 T^{3} + 195659 T^{4} + 392987 T^{5} + 633787 T^{6} + 522131 T^{7} + 420453 T^{8} + 207537 T^{9} + 129945 T^{10} + 39861 T^{11} + 21802 T^{12} + 3968 T^{13} + 2516 T^{14} + 516 T^{15} + 291 T^{16} + 62 T^{17} + 21 T^{18} + 3 T^{19} + T^{20}$$
$11$ $$25937424601 - 33011267674 T + 27437936768 T^{2} - 18356915082 T^{3} + 10250251946 T^{4} - 5022375435 T^{5} + 2211508409 T^{6} - 882969428 T^{7} + 323865817 T^{8} - 109615506 T^{9} + 34315243 T^{10} - 9965046 T^{11} + 2676577 T^{12} - 663388 T^{13} + 151049 T^{14} - 31185 T^{15} + 5786 T^{16} - 942 T^{17} + 128 T^{18} - 14 T^{19} + T^{20}$$
$13$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{5}$$
$17$ $$9554281 + 17736158 T + 30583237 T^{2} + 16811337 T^{3} + 7778446 T^{4} + 1547373 T^{5} + 27222904 T^{6} + 5870682 T^{7} + 9785995 T^{8} + 5247707 T^{9} + 2896411 T^{10} + 678236 T^{11} + 516028 T^{12} + 70025 T^{13} + 24562 T^{14} + 3132 T^{15} + 1055 T^{16} - 88 T^{17} + 31 T^{18} - 2 T^{19} + T^{20}$$
$19$ $$32364721 - 60815410 T + 71750180 T^{2} + 102733141 T^{3} + 82716436 T^{4} + 263836092 T^{5} + 1030084531 T^{6} + 891235402 T^{7} + 421986015 T^{8} + 128762855 T^{9} + 43575772 T^{10} + 11343649 T^{11} + 2314289 T^{12} + 262442 T^{13} + 119706 T^{14} - 1594 T^{15} + 4054 T^{16} - 84 T^{17} + 101 T^{18} + 2 T^{19} + T^{20}$$
$23$ $$( 3824 - 24816 T + 3812 T^{2} + 22344 T^{3} - 9365 T^{4} - 2974 T^{5} + 1571 T^{6} + 139 T^{7} - 82 T^{8} - 3 T^{9} + T^{10} )^{2}$$
$29$ $$8008281121 - 27642436188 T + 46870821814 T^{2} - 48922362322 T^{3} + 39222352274 T^{4} - 26546214380 T^{5} + 15483966803 T^{6} - 6632846398 T^{7} + 2565451345 T^{8} - 658888753 T^{9} + 179577518 T^{10} - 33103364 T^{11} + 9459053 T^{12} - 2119698 T^{13} + 541484 T^{14} - 106655 T^{15} + 18702 T^{16} - 2761 T^{17} + 341 T^{18} - 26 T^{19} + T^{20}$$
$31$ $$41229241 - 220362299 T + 3439601024 T^{2} + 656934107 T^{3} + 5783251714 T^{4} + 1565348962 T^{5} + 4189056167 T^{6} + 1264663415 T^{7} + 1070664272 T^{8} + 152536119 T^{9} + 287464673 T^{10} - 119701609 T^{11} + 19746025 T^{12} - 1308206 T^{13} + 275541 T^{14} - 58553 T^{15} + 6933 T^{16} - 791 T^{17} + 175 T^{18} - 20 T^{19} + T^{20}$$
$37$ $$1085964073216 + 362332610816 T + 882464790464 T^{2} - 56143138752 T^{3} + 187072905264 T^{4} - 70732195664 T^{5} + 32734603660 T^{6} - 7708493356 T^{7} + 3815694833 T^{8} - 595511694 T^{9} + 139006608 T^{10} - 17608379 T^{11} + 6516138 T^{12} - 89102 T^{13} + 201859 T^{14} - 13036 T^{15} + 5980 T^{16} - 214 T^{17} + 3 T^{18} + 6 T^{19} + T^{20}$$
$41$ $$9398786273536 + 19609921129216 T + 28002375676224 T^{2} + 24848805748480 T^{3} + 15083850526048 T^{4} + 6621833787024 T^{5} + 2353711780112 T^{6} + 695938251516 T^{7} + 175730834473 T^{8} + 36782515350 T^{9} + 6487879328 T^{10} + 918243724 T^{11} + 124193074 T^{12} + 16680776 T^{13} + 2288654 T^{14} + 278274 T^{15} + 37314 T^{16} + 4139 T^{17} + 402 T^{18} + 26 T^{19} + T^{20}$$
$43$ $$( 450224 - 591040 T - 107444 T^{2} + 199228 T^{3} + 19273 T^{4} - 19510 T^{5} - 757 T^{6} + 840 T^{7} - 17 T^{8} - 14 T^{9} + T^{10} )^{2}$$
$47$ $$283802574366481 + 317095505010700 T + 234981186096887 T^{2} + 133130797231330 T^{3} + 74977490146041 T^{4} + 27202341921674 T^{5} + 8481165704186 T^{6} + 1671374290681 T^{7} + 373719044044 T^{8} + 32327378754 T^{9} + 16769326379 T^{10} - 1721476481 T^{11} + 478675275 T^{12} - 40938458 T^{13} + 3579144 T^{14} - 160097 T^{15} + 28114 T^{16} - 725 T^{17} + 59 T^{18} - 8 T^{19} + T^{20}$$
$53$ $$2152584874561 - 12344311012286 T + 111597954663559 T^{2} - 167129582678096 T^{3} + 122892640280178 T^{4} - 52774314785816 T^{5} + 14364931379752 T^{6} - 2652192077594 T^{7} + 412933169526 T^{8} - 65796439764 T^{9} + 9736714526 T^{10} - 1072204669 T^{11} + 148543280 T^{12} - 20336719 T^{13} + 2348657 T^{14} - 180208 T^{15} + 36451 T^{16} - 732 T^{17} + 285 T^{18} - T^{19} + T^{20}$$
$59$ $$14896967596281 + 38587751380890 T + 57617791782165 T^{2} + 50883178040763 T^{3} + 29453046145399 T^{4} + 6868784300831 T^{5} + 1379362701529 T^{6} + 242488935494 T^{7} + 98478173717 T^{8} + 33065285835 T^{9} + 8984003529 T^{10} + 1649281411 T^{11} + 239108572 T^{12} + 24192869 T^{13} + 2169939 T^{14} + 176086 T^{15} + 23786 T^{16} + 2955 T^{17} + 328 T^{18} + 21 T^{19} + T^{20}$$
$61$ $$51631683299361 - 91879909317504 T + 132091007527251 T^{2} - 115325460250857 T^{3} + 80097805523308 T^{4} - 42376567778195 T^{5} + 18494063306382 T^{6} - 6805412285700 T^{7} + 2127737836603 T^{8} - 516885224184 T^{9} + 96689118662 T^{10} - 14379917451 T^{11} + 1862691175 T^{12} - 194287254 T^{13} + 15540233 T^{14} - 837116 T^{15} + 33058 T^{16} - 1948 T^{17} + 312 T^{18} - 26 T^{19} + T^{20}$$
$67$ $$( 207765424 - 50135296 T - 64607020 T^{2} + 10975016 T^{3} + 2387489 T^{4} - 422444 T^{5} - 25338 T^{6} + 5901 T^{7} - 16 T^{8} - 28 T^{9} + T^{10} )^{2}$$
$71$ $$1742115822072921 + 17536905746742699 T + 71291211701032485 T^{2} + 61000486811621034 T^{3} + 31991817609346159 T^{4} + 11616507373047700 T^{5} + 3163007013984825 T^{6} + 658150824819061 T^{7} + 107085336109000 T^{8} + 13544026124861 T^{9} + 1363623214588 T^{10} + 112137691872 T^{11} + 8243142996 T^{12} + 564046702 T^{13} + 41500024 T^{14} + 2874580 T^{15} + 203064 T^{16} + 12218 T^{17} + 713 T^{18} + 28 T^{19} + T^{20}$$
$73$ $$2430617787279616 - 2748631655818752 T + 1583081414803520 T^{2} - 591456014938048 T^{3} + 210553612587168 T^{4} - 62865883593648 T^{5} + 15420762891984 T^{6} - 3133111564224 T^{7} + 725612285713 T^{8} - 161017100877 T^{9} + 36811712428 T^{10} - 7332101450 T^{11} + 1318177274 T^{12} - 204156419 T^{13} + 27426139 T^{14} - 3081663 T^{15} + 286409 T^{16} - 20979 T^{17} + 1174 T^{18} - 45 T^{19} + T^{20}$$
$79$ $$3035957760000 - 49114771200000 T + 308365630272000 T^{2} - 127371542880000 T^{3} + 19926722180800 T^{4} + 4511631332000 T^{5} + 4072274290120 T^{6} + 628993394220 T^{7} + 257587890021 T^{8} + 25938579599 T^{9} + 8412399966 T^{10} + 513934467 T^{11} + 156837335 T^{12} + 14034569 T^{13} + 3280887 T^{14} + 304423 T^{15} + 48304 T^{16} + 3352 T^{17} + 402 T^{18} + 15 T^{19} + T^{20}$$
$83$ $$24104358725161 - 54181401523535 T + 168079907164962 T^{2} - 174671054772071 T^{3} + 229496861301132 T^{4} - 90899345413986 T^{5} + 61174312802991 T^{6} + 1592221132259 T^{7} + 2505618164106 T^{8} - 190966977005 T^{9} + 17856096781 T^{10} - 4523330861 T^{11} + 1565202097 T^{12} - 247403360 T^{13} + 22976653 T^{14} - 894251 T^{15} - 1241 T^{16} - 1409 T^{17} + 503 T^{18} - 36 T^{19} + T^{20}$$
$89$ $$( 994744784 + 1393546064 T + 477251404 T^{2} + 32661252 T^{3} - 11528303 T^{4} - 2394134 T^{5} - 119199 T^{6} + 9525 T^{7} + 1412 T^{8} + 63 T^{9} + T^{10} )^{2}$$
$97$ $$39577830238034176 + 11665302764819968 T + 11528025196810816 T^{2} + 1015825511688384 T^{3} + 904164226473840 T^{4} - 39827340292400 T^{5} + 52647832452692 T^{6} + 2236881199232 T^{7} + 3264217289553 T^{8} + 293263984668 T^{9} + 63325887890 T^{10} + 4881449377 T^{11} + 691904498 T^{12} + 9209622 T^{13} + 1690818 T^{14} - 240540 T^{15} + 32288 T^{16} + 1612 T^{17} + 74 T^{18} - 18 T^{19} + T^{20}$$