Properties

 Label 430.2.k.c Level 430 Weight 2 Character orbit 430.k Analytic conductor 3.434 Analytic rank 0 Dimension 18 CM no Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ = $$430 = 2 \cdot 5 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 430.k (of order $$7$$, degree $$6$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.43356728692$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$3$$ over $$\Q(\zeta_{7})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 5 x^{17} + 20 x^{16} - 61 x^{15} + 142 x^{14} - 195 x^{13} + 244 x^{12} + 320 x^{11} + 64 x^{10} - 562 x^{9} + 4114 x^{8} - 1933 x^{7} + 2941 x^{6} - 2555 x^{5} + 2807 x^{4} - 2366 x^{3} + 1225 x^{2} - 343 x + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$82\!\cdots\!53$$$$\nu^{17} +$$$$15\!\cdots\!30$$$$\nu^{16} -$$$$35\!\cdots\!88$$$$\nu^{15} +$$$$28\!\cdots\!38$$$$\nu^{14} -$$$$11\!\cdots\!13$$$$\nu^{13} +$$$$36\!\cdots\!73$$$$\nu^{12} -$$$$44\!\cdots\!58$$$$\nu^{11} +$$$$10\!\cdots\!06$$$$\nu^{10} +$$$$17\!\cdots\!21$$$$\nu^{9} +$$$$28\!\cdots\!32$$$$\nu^{8} +$$$$83\!\cdots\!25$$$$\nu^{7} +$$$$15\!\cdots\!74$$$$\nu^{6} -$$$$74\!\cdots\!57$$$$\nu^{5} +$$$$71\!\cdots\!03$$$$\nu^{4} -$$$$73\!\cdots\!93$$$$\nu^{3} +$$$$77\!\cdots\!07$$$$\nu^{2} -$$$$67\!\cdots\!23$$$$\nu +$$$$17\!\cdots\!38$$$$)/$$$$14\!\cdots\!08$$ $$\beta_{3}$$ $$=$$ $$($$$$19\!\cdots\!05$$$$\nu^{17} -$$$$59\!\cdots\!74$$$$\nu^{16} +$$$$21\!\cdots\!66$$$$\nu^{15} -$$$$49\!\cdots\!92$$$$\nu^{14} +$$$$68\!\cdots\!85$$$$\nu^{13} +$$$$84\!\cdots\!95$$$$\nu^{12} -$$$$10\!\cdots\!18$$$$\nu^{11} +$$$$12\!\cdots\!28$$$$\nu^{10} +$$$$15\!\cdots\!47$$$$\nu^{9} -$$$$69\!\cdots\!58$$$$\nu^{8} +$$$$55\!\cdots\!33$$$$\nu^{7} +$$$$10\!\cdots\!42$$$$\nu^{6} +$$$$16\!\cdots\!85$$$$\nu^{5} +$$$$18\!\cdots\!49$$$$\nu^{4} -$$$$36\!\cdots\!73$$$$\nu^{3} -$$$$42\!\cdots\!11$$$$\nu^{2} -$$$$46\!\cdots\!31$$$$\nu +$$$$12\!\cdots\!36$$$$)/$$$$14\!\cdots\!08$$ $$\beta_{4}$$ $$=$$ $$($$$$27\!\cdots\!98$$$$\nu^{17} -$$$$17\!\cdots\!64$$$$\nu^{16} +$$$$71\!\cdots\!01$$$$\nu^{15} -$$$$23\!\cdots\!63$$$$\nu^{14} +$$$$57\!\cdots\!83$$$$\nu^{13} -$$$$93\!\cdots\!11$$$$\nu^{12} +$$$$11\!\cdots\!86$$$$\nu^{11} +$$$$33\!\cdots\!21$$$$\nu^{10} -$$$$13\!\cdots\!69$$$$\nu^{9} -$$$$26\!\cdots\!69$$$$\nu^{8} +$$$$13\!\cdots\!82$$$$\nu^{7} -$$$$19\!\cdots\!94$$$$\nu^{6} +$$$$68\!\cdots\!81$$$$\nu^{5} -$$$$16\!\cdots\!11$$$$\nu^{4} +$$$$11\!\cdots\!12$$$$\nu^{3} -$$$$14\!\cdots\!63$$$$\nu^{2} +$$$$71\!\cdots\!66$$$$\nu -$$$$18\!\cdots\!83$$$$)/$$$$14\!\cdots\!08$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$10\!\cdots\!44$$$$\nu^{17} +$$$$16\!\cdots\!29$$$$\nu^{16} -$$$$47\!\cdots\!05$$$$\nu^{15} +$$$$49\!\cdots\!87$$$$\nu^{14} +$$$$36\!\cdots\!01$$$$\nu^{13} -$$$$20\!\cdots\!24$$$$\nu^{12} +$$$$21\!\cdots\!65$$$$\nu^{11} -$$$$92\!\cdots\!41$$$$\nu^{10} -$$$$15\!\cdots\!59$$$$\nu^{9} -$$$$39\!\cdots\!82$$$$\nu^{8} -$$$$26\!\cdots\!06$$$$\nu^{7} -$$$$11\!\cdots\!13$$$$\nu^{6} -$$$$36\!\cdots\!87$$$$\nu^{5} -$$$$66\!\cdots\!76$$$$\nu^{4} +$$$$12\!\cdots\!91$$$$\nu^{3} -$$$$44\!\cdots\!22$$$$\nu^{2} +$$$$33\!\cdots\!49$$$$\nu -$$$$78\!\cdots\!18$$$$)/$$$$14\!\cdots\!08$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$58\!\cdots\!99$$$$\nu^{17} +$$$$27\!\cdots\!64$$$$\nu^{16} -$$$$10\!\cdots\!31$$$$\nu^{15} +$$$$32\!\cdots\!27$$$$\nu^{14} -$$$$71\!\cdots\!56$$$$\nu^{13} +$$$$86\!\cdots\!12$$$$\nu^{12} -$$$$10\!\cdots\!92$$$$\nu^{11} -$$$$23\!\cdots\!89$$$$\nu^{10} -$$$$90\!\cdots\!64$$$$\nu^{9} +$$$$33\!\cdots\!95$$$$\nu^{8} -$$$$23\!\cdots\!93$$$$\nu^{7} +$$$$35\!\cdots\!68$$$$\nu^{6} -$$$$12\!\cdots\!46$$$$\nu^{5} +$$$$95\!\cdots\!68$$$$\nu^{4} -$$$$10\!\cdots\!95$$$$\nu^{3} +$$$$81\!\cdots\!36$$$$\nu^{2} -$$$$28\!\cdots\!53$$$$\nu -$$$$30\!\cdots\!09$$$$)/$$$$73\!\cdots\!54$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!40$$$$\nu^{17} +$$$$62\!\cdots\!36$$$$\nu^{16} -$$$$24\!\cdots\!61$$$$\nu^{15} +$$$$77\!\cdots\!55$$$$\nu^{14} -$$$$18\!\cdots\!37$$$$\nu^{13} +$$$$25\!\cdots\!01$$$$\nu^{12} -$$$$29\!\cdots\!22$$$$\nu^{11} -$$$$36\!\cdots\!05$$$$\nu^{10} +$$$$11\!\cdots\!11$$$$\nu^{9} +$$$$93\!\cdots\!33$$$$\nu^{8} -$$$$49\!\cdots\!96$$$$\nu^{7} +$$$$34\!\cdots\!02$$$$\nu^{6} -$$$$20\!\cdots\!75$$$$\nu^{5} +$$$$44\!\cdots\!13$$$$\nu^{4} -$$$$27\!\cdots\!22$$$$\nu^{3} +$$$$31\!\cdots\!93$$$$\nu^{2} -$$$$13\!\cdots\!04$$$$\nu +$$$$19\!\cdots\!91$$$$)/$$$$14\!\cdots\!08$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$15\!\cdots\!34$$$$\nu^{17} +$$$$66\!\cdots\!89$$$$\nu^{16} -$$$$26\!\cdots\!77$$$$\nu^{15} +$$$$75\!\cdots\!11$$$$\nu^{14} -$$$$16\!\cdots\!13$$$$\nu^{13} +$$$$18\!\cdots\!86$$$$\nu^{12} -$$$$22\!\cdots\!79$$$$\nu^{11} -$$$$67\!\cdots\!49$$$$\nu^{10} -$$$$44\!\cdots\!81$$$$\nu^{9} +$$$$67\!\cdots\!26$$$$\nu^{8} -$$$$58\!\cdots\!68$$$$\nu^{7} -$$$$72\!\cdots\!29$$$$\nu^{6} -$$$$38\!\cdots\!05$$$$\nu^{5} +$$$$13\!\cdots\!78$$$$\nu^{4} -$$$$24\!\cdots\!43$$$$\nu^{3} +$$$$15\!\cdots\!76$$$$\nu^{2} -$$$$35\!\cdots\!77$$$$\nu -$$$$64\!\cdots\!78$$$$)/$$$$14\!\cdots\!08$$ $$\beta_{9}$$ $$=$$ $$($$$$19\!\cdots\!61$$$$\nu^{17} -$$$$91\!\cdots\!15$$$$\nu^{16} +$$$$35\!\cdots\!43$$$$\nu^{15} -$$$$10\!\cdots\!87$$$$\nu^{14} +$$$$23\!\cdots\!60$$$$\nu^{13} -$$$$29\!\cdots\!09$$$$\nu^{12} +$$$$35\!\cdots\!89$$$$\nu^{11} +$$$$76\!\cdots\!69$$$$\nu^{10} +$$$$33\!\cdots\!14$$$$\nu^{9} -$$$$10\!\cdots\!84$$$$\nu^{8} +$$$$76\!\cdots\!87$$$$\nu^{7} -$$$$12\!\cdots\!41$$$$\nu^{6} +$$$$45\!\cdots\!40$$$$\nu^{5} -$$$$34\!\cdots\!35$$$$\nu^{4} +$$$$39\!\cdots\!60$$$$\nu^{3} -$$$$30\!\cdots\!21$$$$\nu^{2} +$$$$10\!\cdots\!20$$$$\nu -$$$$15\!\cdots\!50$$$$)/$$$$14\!\cdots\!08$$ $$\beta_{10}$$ $$=$$ $$($$$$22\!\cdots\!40$$$$\nu^{17} -$$$$99\!\cdots\!19$$$$\nu^{16} +$$$$38\!\cdots\!81$$$$\nu^{15} -$$$$11\!\cdots\!75$$$$\nu^{14} +$$$$24\!\cdots\!31$$$$\nu^{13} -$$$$26\!\cdots\!74$$$$\nu^{12} +$$$$33\!\cdots\!01$$$$\nu^{11} +$$$$99\!\cdots\!37$$$$\nu^{10} +$$$$71\!\cdots\!03$$$$\nu^{9} -$$$$99\!\cdots\!20$$$$\nu^{8} +$$$$86\!\cdots\!02$$$$\nu^{7} +$$$$13\!\cdots\!23$$$$\nu^{6} +$$$$59\!\cdots\!95$$$$\nu^{5} -$$$$19\!\cdots\!38$$$$\nu^{4} +$$$$43\!\cdots\!99$$$$\nu^{3} -$$$$21\!\cdots\!04$$$$\nu^{2} +$$$$50\!\cdots\!17$$$$\nu +$$$$10\!\cdots\!72$$$$)/$$$$14\!\cdots\!08$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$29\!\cdots\!71$$$$\nu^{17} +$$$$17\!\cdots\!34$$$$\nu^{16} -$$$$68\!\cdots\!08$$$$\nu^{15} +$$$$21\!\cdots\!78$$$$\nu^{14} -$$$$52\!\cdots\!17$$$$\nu^{13} +$$$$80\!\cdots\!17$$$$\nu^{12} -$$$$95\!\cdots\!62$$$$\nu^{11} -$$$$62\!\cdots\!38$$$$\nu^{10} +$$$$87\!\cdots\!93$$$$\nu^{9} +$$$$25\!\cdots\!48$$$$\nu^{8} -$$$$12\!\cdots\!19$$$$\nu^{7} +$$$$14\!\cdots\!62$$$$\nu^{6} -$$$$59\!\cdots\!65$$$$\nu^{5} +$$$$14\!\cdots\!55$$$$\nu^{4} -$$$$96\!\cdots\!73$$$$\nu^{3} +$$$$10\!\cdots\!15$$$$\nu^{2} -$$$$55\!\cdots\!43$$$$\nu +$$$$14\!\cdots\!42$$$$)/$$$$14\!\cdots\!08$$ $$\beta_{12}$$ $$=$$ $$($$$$31\!\cdots\!50$$$$\nu^{17} -$$$$13\!\cdots\!89$$$$\nu^{16} +$$$$54\!\cdots\!85$$$$\nu^{15} -$$$$15\!\cdots\!07$$$$\nu^{14} +$$$$34\!\cdots\!13$$$$\nu^{13} -$$$$38\!\cdots\!90$$$$\nu^{12} +$$$$48\!\cdots\!91$$$$\nu^{11} +$$$$13\!\cdots\!89$$$$\nu^{10} +$$$$96\!\cdots\!69$$$$\nu^{9} -$$$$14\!\cdots\!86$$$$\nu^{8} +$$$$11\!\cdots\!16$$$$\nu^{7} +$$$$15\!\cdots\!37$$$$\nu^{6} +$$$$81\!\cdots\!09$$$$\nu^{5} -$$$$35\!\cdots\!10$$$$\nu^{4} +$$$$54\!\cdots\!15$$$$\nu^{3} -$$$$36\!\cdots\!40$$$$\nu^{2} +$$$$86\!\cdots\!29$$$$\nu -$$$$30\!\cdots\!30$$$$)/$$$$14\!\cdots\!08$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$32\!\cdots\!24$$$$\nu^{17} +$$$$16\!\cdots\!97$$$$\nu^{16} -$$$$66\!\cdots\!89$$$$\nu^{15} +$$$$20\!\cdots\!03$$$$\nu^{14} -$$$$48\!\cdots\!95$$$$\nu^{13} +$$$$69\!\cdots\!36$$$$\nu^{12} -$$$$86\!\cdots\!07$$$$\nu^{11} -$$$$93\!\cdots\!81$$$$\nu^{10} +$$$$44\!\cdots\!25$$$$\nu^{9} +$$$$18\!\cdots\!78$$$$\nu^{8} -$$$$13\!\cdots\!38$$$$\nu^{7} +$$$$85\!\cdots\!67$$$$\nu^{6} -$$$$97\!\cdots\!79$$$$\nu^{5} +$$$$89\!\cdots\!92$$$$\nu^{4} -$$$$10\!\cdots\!09$$$$\nu^{3} +$$$$87\!\cdots\!18$$$$\nu^{2} -$$$$44\!\cdots\!11$$$$\nu +$$$$12\!\cdots\!70$$$$)/$$$$14\!\cdots\!08$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$34\!\cdots\!35$$$$\nu^{17} +$$$$17\!\cdots\!53$$$$\nu^{16} -$$$$70\!\cdots\!32$$$$\nu^{15} +$$$$21\!\cdots\!50$$$$\nu^{14} -$$$$50\!\cdots\!23$$$$\nu^{13} +$$$$70\!\cdots\!30$$$$\nu^{12} -$$$$84\!\cdots\!89$$$$\nu^{11} -$$$$10\!\cdots\!62$$$$\nu^{10} +$$$$11\!\cdots\!83$$$$\nu^{9} +$$$$22\!\cdots\!35$$$$\nu^{8} -$$$$14\!\cdots\!85$$$$\nu^{7} +$$$$83\!\cdots\!69$$$$\nu^{6} -$$$$81\!\cdots\!49$$$$\nu^{5} +$$$$90\!\cdots\!24$$$$\nu^{4} -$$$$99\!\cdots\!02$$$$\nu^{3} +$$$$78\!\cdots\!98$$$$\nu^{2} -$$$$40\!\cdots\!28$$$$\nu +$$$$53\!\cdots\!59$$$$)/$$$$14\!\cdots\!08$$ $$\beta_{15}$$ $$=$$ $$($$$$37\!\cdots\!67$$$$\nu^{17} -$$$$18\!\cdots\!37$$$$\nu^{16} +$$$$73\!\cdots\!76$$$$\nu^{15} -$$$$22\!\cdots\!86$$$$\nu^{14} +$$$$51\!\cdots\!51$$$$\nu^{13} -$$$$67\!\cdots\!82$$$$\nu^{12} +$$$$82\!\cdots\!37$$$$\nu^{11} +$$$$13\!\cdots\!26$$$$\nu^{10} +$$$$27\!\cdots\!09$$$$\nu^{9} -$$$$22\!\cdots\!23$$$$\nu^{8} +$$$$15\!\cdots\!69$$$$\nu^{7} -$$$$59\!\cdots\!29$$$$\nu^{6} +$$$$92\!\cdots\!53$$$$\nu^{5} -$$$$89\!\cdots\!04$$$$\nu^{4} +$$$$89\!\cdots\!58$$$$\nu^{3} -$$$$78\!\cdots\!10$$$$\nu^{2} +$$$$31\!\cdots\!12$$$$\nu -$$$$57\!\cdots\!15$$$$)/$$$$14\!\cdots\!08$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$52\!\cdots\!74$$$$\nu^{17} +$$$$24\!\cdots\!83$$$$\nu^{16} -$$$$97\!\cdots\!35$$$$\nu^{15} +$$$$29\!\cdots\!77$$$$\nu^{14} -$$$$65\!\cdots\!67$$$$\nu^{13} +$$$$80\!\cdots\!42$$$$\nu^{12} -$$$$96\!\cdots\!85$$$$\nu^{11} -$$$$20\!\cdots\!27$$$$\nu^{10} -$$$$66\!\cdots\!71$$$$\nu^{9} +$$$$32\!\cdots\!46$$$$\nu^{8} -$$$$20\!\cdots\!16$$$$\nu^{7} +$$$$45\!\cdots\!37$$$$\nu^{6} -$$$$10\!\cdots\!99$$$$\nu^{5} +$$$$10\!\cdots\!74$$$$\nu^{4} -$$$$95\!\cdots\!85$$$$\nu^{3} +$$$$86\!\cdots\!56$$$$\nu^{2} -$$$$22\!\cdots\!83$$$$\nu +$$$$53\!\cdots\!74$$$$)/$$$$14\!\cdots\!08$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$12\!\cdots\!18$$$$\nu^{17} +$$$$56\!\cdots\!98$$$$\nu^{16} -$$$$22\!\cdots\!59$$$$\nu^{15} +$$$$66\!\cdots\!37$$$$\nu^{14} -$$$$14\!\cdots\!49$$$$\nu^{13} +$$$$18\!\cdots\!03$$$$\nu^{12} -$$$$22\!\cdots\!12$$$$\nu^{11} -$$$$47\!\cdots\!55$$$$\nu^{10} -$$$$23\!\cdots\!45$$$$\nu^{9} +$$$$63\!\cdots\!13$$$$\nu^{8} -$$$$47\!\cdots\!14$$$$\nu^{7} +$$$$64\!\cdots\!36$$$$\nu^{6} -$$$$31\!\cdots\!11$$$$\nu^{5} +$$$$20\!\cdots\!43$$$$\nu^{4} -$$$$25\!\cdots\!62$$$$\nu^{3} +$$$$18\!\cdots\!99$$$$\nu^{2} -$$$$73\!\cdots\!12$$$$\nu +$$$$92\!\cdots\!83$$$$)/$$$$14\!\cdots\!08$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{16} - \beta_{15} + \beta_{14} - \beta_{13} + \beta_{11} + \beta_{10} - \beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$-\beta_{16} + \beta_{14} + 2 \beta_{12} - 2 \beta_{10} - \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} - 7 \beta_{4} - 2 \beta_{3} - 7 \beta_{2} - 8 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{17} + \beta_{16} + 9 \beta_{15} + 8 \beta_{13} - 8 \beta_{11} + 3 \beta_{9} - 10 \beta_{8} + 19 \beta_{7} + 9 \beta_{6} - \beta_{5} - 9 \beta_{4} - 10 \beta_{2} - 7 \beta_{1} + 9$$ $$\nu^{5}$$ $$=$$ $$10 \beta_{17} + 5 \beta_{15} - 7 \beta_{14} + 10 \beta_{13} + 4 \beta_{12} - 7 \beta_{11} + 18 \beta_{10} + 13 \beta_{9} - 53 \beta_{8} + 4 \beta_{7} + 53 \beta_{6} + 2 \beta_{5} + 8 \beta_{4} - 5 \beta_{3} - 13 \beta_{2} + 8 \beta_{1} - 18$$ $$\nu^{6}$$ $$=$$ $$53 \beta_{17} - 8 \beta_{16} + 3 \beta_{15} - 53 \beta_{14} + 5 \beta_{13} + 143 \beta_{12} + 8 \beta_{11} - 74 \beta_{10} + 87 \beta_{9} - 79 \beta_{8} + 3 \beta_{7} + 87 \beta_{6} - 74 \beta_{3} - 41 \beta_{2} - 33 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$79 \beta_{17} + 79 \beta_{16} + 202 \beta_{15} - 120 \beta_{14} + 120 \beta_{13} + 202 \beta_{12} - 33 \beta_{11} - 219 \beta_{10} + 423 \beta_{9} + 48 \beta_{8} + 219 \beta_{7} + 136 \beta_{6} - 33 \beta_{5} - 48 \beta_{4} + 157$$ $$\nu^{8}$$ $$=$$ $$-48 \beta_{17} + 423 \beta_{16} + 537 \beta_{15} - 88 \beta_{14} + 423 \beta_{13} - 538 \beta_{12} + 538 \beta_{10} + 697 \beta_{9} + 429 \beta_{6} + 48 \beta_{5} + 697 \beta_{4} + 601 \beta_{3} + 429 \beta_{2} + 389 \beta_{1} - 601$$ $$\nu^{9}$$ $$=$$ $$737 \beta_{16} - 182 \beta_{15} - 737 \beta_{14} + 308 \beta_{13} + 389 \beta_{11} + 182 \beta_{10} - 224 \beta_{9} - 224 \beta_{8} - 1965 \beta_{7} - 308 \beta_{5} + 3466 \beta_{4} + 599 \beta_{3} + 1317 \beta_{2} - 224 \beta_{1} - 1965$$ $$\nu^{10}$$ $$=$$ $$224 \beta_{17} + 1317 \beta_{16} - 3690 \beta_{14} + 4028 \beta_{12} - 224 \beta_{11} - 8414 \beta_{10} + 6151 \beta_{8} - 855 \beta_{7} - 4100 \beta_{6} - 3690 \beta_{5} + 6245 \beta_{4} + 855 \beta_{3} + 6245 \beta_{2} + 2145 \beta_{1} + 4028$$ $$\nu^{11}$$ $$=$$ $$-6151 \beta_{17} + 4100 \beta_{16} + 1643 \beta_{15} - 2145 \beta_{13} - 15533 \beta_{12} + 2145 \beta_{11} + 474 \beta_{9} + 28821 \beta_{8} - 3789 \beta_{7} - 12293 \beta_{6} - 4100 \beta_{5} + 12293 \beta_{4} + 15533 \beta_{3} + 28821 \beta_{2} + 29295 \beta_{1} + 1643$$ $$\nu^{12}$$ $$=$$ $$-28821 \beta_{17} - 36579 \beta_{15} + 29295 \beta_{14} - 28821 \beta_{13} - 76402 \beta_{12} + 29295 \beta_{11} + 66641 \beta_{10} - 54131 \beta_{9} + 53293 \beta_{8} - 76402 \beta_{7} - 53293 \beta_{6} + 17002 \beta_{5} + 37575 \beta_{4} + 36579 \beta_{3} + 54131 \beta_{2} + 37575 \beta_{1} - 66641$$ $$\nu^{13}$$ $$=$$ $$-53293 \beta_{17} - 37575 \beta_{16} - 134492 \beta_{15} + 53293 \beta_{14} - 91706 \beta_{13} - 34893 \beta_{12} + 37575 \beta_{11} - 13549 \beta_{10} - 241942 \beta_{9} + 112448 \beta_{8} - 134492 \beta_{7} - 241942 \beta_{6} - 13549 \beta_{3} + 6593 \beta_{2} - 122901 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-112448 \beta_{17} - 112448 \beta_{16} - 102208 \beta_{15} + 119041 \beta_{14} - 119041 \beta_{13} - 102208 \beta_{12} - 122901 \beta_{11} - 224116 \beta_{10} - 458294 \beta_{9} + 336477 \beta_{8} + 224116 \beta_{7} - 474733 \beta_{6} - 122901 \beta_{5} - 336477 \beta_{4} + 532296$$ $$\nu^{15}$$ $$=$$ $$-336477 \beta_{17} - 458294 \beta_{16} - 430535 \beta_{15} + 811210 \beta_{14} - 458294 \beta_{13} - 1265079 \beta_{12} + 1265079 \beta_{10} - 1016020 \beta_{9} - 134309 \beta_{6} + 336477 \beta_{5} - 1016020 \beta_{4} + 104770 \beta_{3} - 134309 \beta_{2} + 894493 \beta_{1} - 104770$$ $$\nu^{16}$$ $$=$$ $$-2044822 \beta_{16} - 2686846 \beta_{15} + 2044822 \beta_{14} - 1910513 \beta_{13} + 894493 \beta_{11} + 2686846 \beta_{10} - 2970492 \beta_{9} - 2970492 \beta_{8} - 1668184 \beta_{7} + 1910513 \beta_{5} - 3966718 \beta_{4} - 2611383 \beta_{3} - 4150650 \beta_{2} - 2970492 \beta_{1} - 1668184$$ $$\nu^{17}$$ $$=$$ $$2970492 \beta_{17} - 4150650 \beta_{16} + 996226 \beta_{14} + 10750085 \beta_{12} - 2970492 \beta_{11} - 6923340 \beta_{10} - 9106201 \beta_{8} + 9987154 \beta_{7} + 1722282 \beta_{6} + 996226 \beta_{5} - 17370932 \beta_{4} - 9987154 \beta_{3} - 17370932 \beta_{2} - 15648650 \beta_{1} + 10750085$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/430\mathbb{Z}\right)^\times$$.

 $$n$$ $$87$$ $$261$$ $$\chi(n)$$ $$1$$ $$-\beta_{3}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −0.583263 + 0.731388i 0.219976 − 0.275841i 1.14077 − 1.43048i −1.43931 + 0.693135i 0.403095 − 0.194120i 2.65970 − 1.28085i −1.43931 − 0.693135i 0.403095 + 0.194120i 2.65970 + 1.28085i −0.635189 − 2.78294i 0.168441 + 0.737990i 0.565778 + 2.47884i −0.635189 + 2.78294i 0.168441 − 0.737990i 0.565778 − 2.47884i −0.583263 − 0.731388i 0.219976 + 0.275841i 1.14077 + 1.43048i
−0.623490 0.781831i −0.583263 + 0.731388i −0.222521 + 0.974928i −0.900969 + 0.433884i 0.935481 2.01178 0.900969 0.433884i 0.472829 + 2.07160i 0.900969 + 0.433884i
11.2 −0.623490 0.781831i 0.219976 0.275841i −0.222521 + 0.974928i −0.900969 + 0.433884i −0.352814 −0.963199 0.900969 0.433884i 0.639864 + 2.80343i 0.900969 + 0.433884i
11.3 −0.623490 0.781831i 1.14077 1.43048i −0.222521 + 0.974928i −0.900969 + 0.433884i −1.82965 4.80227 0.900969 0.433884i −0.0773496 0.338891i 0.900969 + 0.433884i
21.1 0.900969 + 0.433884i −1.43931 + 0.693135i 0.623490 + 0.781831i −0.222521 0.974928i −1.59751 −1.15014 0.222521 + 0.974928i −0.279294 + 0.350223i 0.222521 0.974928i
21.2 0.900969 + 0.433884i 0.403095 0.194120i 0.623490 + 0.781831i −0.222521 0.974928i 0.447401 2.78084 0.222521 + 0.974928i −1.74567 + 2.18900i 0.222521 0.974928i
21.3 0.900969 + 0.433884i 2.65970 1.28085i 0.623490 + 0.781831i −0.222521 0.974928i 2.95205 −1.54255 0.222521 + 0.974928i 3.56299 4.46785i 0.222521 0.974928i
41.1 0.900969 0.433884i −1.43931 0.693135i 0.623490 0.781831i −0.222521 + 0.974928i −1.59751 −1.15014 0.222521 0.974928i −0.279294 0.350223i 0.222521 + 0.974928i
41.2 0.900969 0.433884i 0.403095 + 0.194120i 0.623490 0.781831i −0.222521 + 0.974928i 0.447401 2.78084 0.222521 0.974928i −1.74567 2.18900i 0.222521 + 0.974928i
41.3 0.900969 0.433884i 2.65970 + 1.28085i 0.623490 0.781831i −0.222521 + 0.974928i 2.95205 −1.54255 0.222521 0.974928i 3.56299 + 4.46785i 0.222521 + 0.974928i
121.1 0.222521 0.974928i −0.635189 2.78294i −0.900969 0.433884i 0.623490 0.781831i −2.85451 −1.93646 −0.623490 + 0.781831i −4.63840 + 2.23374i −0.623490 0.781831i
121.2 0.222521 0.974928i 0.168441 + 0.737990i −0.900969 0.433884i 0.623490 0.781831i 0.756969 −2.52891 −0.623490 + 0.781831i 2.18665 1.05304i −0.623490 0.781831i
121.3 0.222521 0.974928i 0.565778 + 2.47884i −0.900969 0.433884i 0.623490 0.781831i 2.54259 2.52637 −0.623490 + 0.781831i −3.12162 + 1.50329i −0.623490 0.781831i
231.1 0.222521 + 0.974928i −0.635189 + 2.78294i −0.900969 + 0.433884i 0.623490 + 0.781831i −2.85451 −1.93646 −0.623490 0.781831i −4.63840 2.23374i −0.623490 + 0.781831i
231.2 0.222521 + 0.974928i 0.168441 0.737990i −0.900969 + 0.433884i 0.623490 + 0.781831i 0.756969 −2.52891 −0.623490 0.781831i 2.18665 + 1.05304i −0.623490 + 0.781831i
231.3 0.222521 + 0.974928i 0.565778 2.47884i −0.900969 + 0.433884i 0.623490 + 0.781831i 2.54259 2.52637 −0.623490 0.781831i −3.12162 1.50329i −0.623490 + 0.781831i
391.1 −0.623490 + 0.781831i −0.583263 0.731388i −0.222521 0.974928i −0.900969 0.433884i 0.935481 2.01178 0.900969 + 0.433884i 0.472829 2.07160i 0.900969 0.433884i
391.2 −0.623490 + 0.781831i 0.219976 + 0.275841i −0.222521 0.974928i −0.900969 0.433884i −0.352814 −0.963199 0.900969 + 0.433884i 0.639864 2.80343i 0.900969 0.433884i
391.3 −0.623490 + 0.781831i 1.14077 + 1.43048i −0.222521 0.974928i −0.900969 0.433884i −1.82965 4.80227 0.900969 + 0.433884i −0.0773496 + 0.338891i 0.900969 0.433884i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 391.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.e even 7 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.k.c 18
43.e even 7 1 inner 430.2.k.c 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.k.c 18 1.a even 1 1 trivial
430.2.k.c 18 43.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{18} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(430, [\chi])$$.

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{3}$$
$3$ $$1 - 5 T + 11 T^{2} - 22 T^{3} + 64 T^{4} - 177 T^{5} + 364 T^{6} - 721 T^{7} + 1705 T^{8} - 3727 T^{9} + 6910 T^{10} - 13168 T^{11} + 27619 T^{12} - 53717 T^{13} + 95903 T^{14} - 176450 T^{15} + 331870 T^{16} - 581677 T^{17} + 983731 T^{18} - 1745031 T^{19} + 2986830 T^{20} - 4764150 T^{21} + 7768143 T^{22} - 13053231 T^{23} + 20134251 T^{24} - 28798416 T^{25} + 45336510 T^{26} - 73358541 T^{27} + 100678545 T^{28} - 127722987 T^{29} + 193444524 T^{30} - 282195171 T^{31} + 306110016 T^{32} - 315675954 T^{33} + 473513931 T^{34} - 645700815 T^{35} + 387420489 T^{36}$$
$5$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{3}$$
$7$ $$( 1 - 4 T + 43 T^{2} - 165 T^{3} + 954 T^{4} - 3272 T^{5} + 13473 T^{6} - 40583 T^{7} + 131977 T^{8} - 340920 T^{9} + 923839 T^{10} - 1988567 T^{11} + 4621239 T^{12} - 7856072 T^{13} + 16033878 T^{14} - 19412085 T^{15} + 35412349 T^{16} - 23059204 T^{17} + 40353607 T^{18} )^{2}$$
$11$ $$1 - 5 T + 29 T^{2} - 2 T^{3} - 64 T^{4} + 1743 T^{5} - 258 T^{6} + 563 T^{7} + 93155 T^{8} - 93657 T^{9} + 443272 T^{10} + 1889454 T^{11} - 1463949 T^{12} + 6857689 T^{13} + 40881021 T^{14} - 138737432 T^{15} + 197538356 T^{16} - 688470581 T^{17} - 1743464851 T^{18} - 7573176391 T^{19} + 23902141076 T^{20} - 184659521992 T^{21} + 598539028461 T^{22} + 1104437671139 T^{23} - 2593474954389 T^{24} + 36820113194634 T^{25} + 95019289898632 T^{26} - 220838306895987 T^{27} + 2416200788706155 T^{28} + 160630470553993 T^{29} - 809714521194018 T^{30} + 60173087266871733 T^{31} - 24303989349327424 T^{32} - 8354496338831302 T^{33} + 1332542166043592669 T^{34} - 2527235142496468855 T^{35} + 5559917313492231481 T^{36}$$
$13$ $$1 - 22 T + 189 T^{2} - 776 T^{3} + 2016 T^{4} - 11560 T^{5} + 76130 T^{6} - 262662 T^{7} + 880343 T^{8} - 5919980 T^{9} + 28592867 T^{10} - 84734756 T^{11} + 312372760 T^{12} - 1531317100 T^{13} + 5382091707 T^{14} - 17490109252 T^{15} + 77239495484 T^{16} - 309727210712 T^{17} + 1082670643211 T^{18} - 4026453739256 T^{19} + 13053474736796 T^{20} - 38425770026644 T^{21} + 153717921243627 T^{22} - 568567320010300 T^{23} + 1507763649322840 T^{24} - 5316980277356852 T^{25} + 23324080013367107 T^{26} - 62778424198172540 T^{27} + 121362758289824207 T^{28} - 470732433418546494 T^{29} + 1773683220374478530 T^{30} - 3501236232206444680 T^{31} + 7937750793569766624 T^{32} - 39720252978934427432 T^{33} +$$$$12\!\cdots\!49$$$$T^{34} -$$$$19\!\cdots\!26$$$$T^{35} +$$$$11\!\cdots\!29$$$$T^{36}$$
$17$ $$1 - T + 17 T^{2} - 71 T^{3} + 163 T^{4} - 1386 T^{5} + 5859 T^{6} - 25092 T^{7} + 79275 T^{8} - 574216 T^{9} + 380351 T^{10} - 8959160 T^{11} + 10289590 T^{12} - 149888072 T^{13} + 943088259 T^{14} - 2036970771 T^{15} + 18923873831 T^{16} - 49782589785 T^{17} + 210671203633 T^{18} - 846304026345 T^{19} + 5468999537159 T^{20} - 10007637397923 T^{21} + 78767674479939 T^{22} - 212819628245704 T^{23} + 248365688606710 T^{24} - 3676289825594680 T^{25} + 2653236318441791 T^{26} - 68095056090601352 T^{27} + 159817916458094475 T^{28} - 859950422151127236 T^{29} + 3413583687929169699 T^{30} - 13727745153607628682 T^{31} + 27445585729182351427 T^{32} -$$$$20\!\cdots\!03$$$$T^{33} +$$$$82\!\cdots\!77$$$$T^{34} -$$$$82\!\cdots\!77$$$$T^{35} +$$$$14\!\cdots\!09$$$$T^{36}$$
$19$ $$1 + 7 T + 80 T^{2} + 528 T^{3} + 3734 T^{4} + 20920 T^{5} + 120996 T^{6} + 605461 T^{7} + 2904657 T^{8} + 13467676 T^{9} + 54757744 T^{10} + 216429928 T^{11} + 724355904 T^{12} + 2023912444 T^{13} + 3282662694 T^{14} - 10380996494 T^{15} - 115691998692 T^{16} - 856017011464 T^{17} - 3500283086684 T^{18} - 16264323217816 T^{19} - 41764811527812 T^{20} - 71203254952346 T^{21} + 427799884944774 T^{22} + 5011407578675956 T^{23} + 34077961661231424 T^{24} + 193460596113004792 T^{25} + 929981597206939504 T^{26} + 4345853362873491604 T^{27} + 17808644523185479257 T^{28} + 70530308642774573959 T^{29} +$$$$26\!\cdots\!56$$$$T^{30} +$$$$87\!\cdots\!80$$$$T^{31} +$$$$29\!\cdots\!14$$$$T^{32} +$$$$80\!\cdots\!72$$$$T^{33} +$$$$23\!\cdots\!80$$$$T^{34} +$$$$38\!\cdots\!73$$$$T^{35} +$$$$10\!\cdots\!41$$$$T^{36}$$
$23$ $$1 - 8 T + 13 T^{2} + 20 T^{3} + 794 T^{4} - 8894 T^{5} + 44620 T^{6} - 76030 T^{7} - 348697 T^{8} - 785090 T^{9} + 27523733 T^{10} - 101765222 T^{11} - 302476000 T^{12} + 4014732254 T^{13} - 12572137619 T^{14} - 14995597814 T^{15} + 64743711756 T^{16} + 2079662840830 T^{17} - 17083611139087 T^{18} + 47832245339090 T^{19} + 34249423518924 T^{20} - 182451438602938 T^{21} - 3518199563438579 T^{22} + 25840193839907122 T^{23} - 44777303561164000 T^{24} - 346492817485204234 T^{25} + 2155410649841173973 T^{26} - 1414066942987986670 T^{27} - 14445300180665765353 T^{28} - 72442125894195869810 T^{29} +$$$$97\!\cdots\!20$$$$T^{30} -$$$$44\!\cdots\!02$$$$T^{31} +$$$$92\!\cdots\!46$$$$T^{32} +$$$$53\!\cdots\!40$$$$T^{33} +$$$$79\!\cdots\!93$$$$T^{34} -$$$$11\!\cdots\!24$$$$T^{35} +$$$$32\!\cdots\!69$$$$T^{36}$$
$29$ $$1 + 6 T - 71 T^{2} - 679 T^{3} + 1326 T^{4} + 34252 T^{5} + 88405 T^{6} - 867289 T^{7} - 8027938 T^{8} - 681278 T^{9} + 325349830 T^{10} + 1194610606 T^{11} - 6989192779 T^{12} - 61491095320 T^{13} - 35442154336 T^{14} + 1740460758039 T^{15} + 9086251390251 T^{16} - 21170258850237 T^{17} - 364928839121169 T^{18} - 613937506656873 T^{19} + 7641537419201091 T^{20} + 42448097427813171 T^{21} - 25067562360920416 T^{22} - 1261253018281722680 T^{23} - 4157334859913999059 T^{24} + 20606885190919533254 T^{25} +$$$$16\!\cdots\!30$$$$T^{26} - 9883399396148080582 T^{27} -$$$$33\!\cdots\!38$$$$T^{28} -$$$$10\!\cdots\!81$$$$T^{29} +$$$$31\!\cdots\!05$$$$T^{30} +$$$$35\!\cdots\!28$$$$T^{31} +$$$$39\!\cdots\!06$$$$T^{32} -$$$$58\!\cdots\!71$$$$T^{33} -$$$$17\!\cdots\!91$$$$T^{34} +$$$$43\!\cdots\!54$$$$T^{35} +$$$$21\!\cdots\!61$$$$T^{36}$$
$31$ $$1 + 5 T - 66 T^{2} - 503 T^{3} + 337 T^{4} + 12028 T^{5} + 33978 T^{6} + 406763 T^{7} + 2995682 T^{8} - 18538475 T^{9} - 206059741 T^{10} - 275557932 T^{11} + 1910953762 T^{12} + 11736742065 T^{13} + 105697996694 T^{14} + 712002834894 T^{15} - 316523157013 T^{16} - 20711586044207 T^{17} - 106673227770053 T^{18} - 642059167370417 T^{19} - 304178753889493 T^{20} + 21211276454327154 T^{21} + 97614319604839574 T^{22} + 336012960826936815 T^{23} + 1695978497995797922 T^{24} - 7581319048341178452 T^{25} -$$$$17\!\cdots\!81$$$$T^{26} -$$$$49\!\cdots\!25$$$$T^{27} +$$$$24\!\cdots\!82$$$$T^{28} +$$$$10\!\cdots\!53$$$$T^{29} +$$$$26\!\cdots\!58$$$$T^{30} +$$$$29\!\cdots\!48$$$$T^{31} +$$$$25\!\cdots\!77$$$$T^{32} -$$$$11\!\cdots\!53$$$$T^{33} -$$$$48\!\cdots\!46$$$$T^{34} +$$$$11\!\cdots\!55$$$$T^{35} +$$$$69\!\cdots\!41$$$$T^{36}$$
$37$ $$( 1 + 9 T + 218 T^{2} + 1614 T^{3} + 23401 T^{4} + 151257 T^{5} + 1644298 T^{6} + 9291632 T^{7} + 82504230 T^{8} + 404531792 T^{9} + 3052656510 T^{10} + 12720244208 T^{11} + 83288626594 T^{12} + 283479970377 T^{13} + 1622717937757 T^{14} + 4141082424126 T^{15} + 20695149214994 T^{16} + 31612315085289 T^{17} + 129961739795077 T^{18} )^{2}$$
$41$ $$1 - 16 T - 52 T^{2} + 2313 T^{3} - 7721 T^{4} - 117110 T^{5} + 860282 T^{6} + 1872521 T^{7} - 38916852 T^{8} + 82707819 T^{9} + 1144968899 T^{10} - 8680504064 T^{11} - 18775033826 T^{12} + 451896528557 T^{13} - 387529241846 T^{14} - 14990466569601 T^{15} + 45356087778625 T^{16} + 236151025913079 T^{17} - 2271300748262055 T^{18} + 9682192062436239 T^{19} + 76243583555868625 T^{20} - 1033157946443470521 T^{21} - 1095065017967994806 T^{22} + 52355015043702031957 T^{23} - 89183367801801056066 T^{24} -$$$$16\!\cdots\!84$$$$T^{25} +$$$$91\!\cdots\!79$$$$T^{26} +$$$$27\!\cdots\!59$$$$T^{27} -$$$$52\!\cdots\!52$$$$T^{28} +$$$$10\!\cdots\!61$$$$T^{29} +$$$$19\!\cdots\!42$$$$T^{30} -$$$$10\!\cdots\!10$$$$T^{31} -$$$$29\!\cdots\!81$$$$T^{32} +$$$$35\!\cdots\!13$$$$T^{33} -$$$$33\!\cdots\!32$$$$T^{34} -$$$$41\!\cdots\!96$$$$T^{35} +$$$$10\!\cdots\!21$$$$T^{36}$$
$43$ $$1 + 13 T + 135 T^{2} + 173 T^{3} - 5485 T^{4} - 91815 T^{5} - 487924 T^{6} - 685681 T^{7} + 23032937 T^{8} + 203701481 T^{9} + 990416291 T^{10} - 1267824169 T^{11} - 38793373468 T^{12} - 313897213815 T^{13} - 806341309855 T^{14} + 1093595807477 T^{15} + 36695512499445 T^{16} + 151946603608813 T^{17} + 502592611936843 T^{18}$$
$47$ $$1 + 36 T + 612 T^{2} + 6578 T^{3} + 56686 T^{4} + 524814 T^{5} + 5385180 T^{6} + 49774124 T^{7} + 394711081 T^{8} + 3091549542 T^{9} + 26602209040 T^{10} + 226026429292 T^{11} + 1707793138576 T^{12} + 12049342225310 T^{13} + 89717535794078 T^{14} + 702257335176408 T^{15} + 5175913711157672 T^{16} + 34826239488864972 T^{17} + 232179354339637252 T^{18} + 1636833255976653684 T^{19} + 11433593387947297448 T^{20} + 72910463310020207784 T^{21} +$$$$43\!\cdots\!18$$$$T^{22} +$$$$27\!\cdots\!70$$$$T^{23} +$$$$18\!\cdots\!04$$$$T^{24} +$$$$11\!\cdots\!96$$$$T^{25} +$$$$63\!\cdots\!40$$$$T^{26} +$$$$34\!\cdots\!14$$$$T^{27} +$$$$20\!\cdots\!69$$$$T^{28} +$$$$12\!\cdots\!72$$$$T^{29} +$$$$62\!\cdots\!80$$$$T^{30} +$$$$28\!\cdots\!78$$$$T^{31} +$$$$14\!\cdots\!34$$$$T^{32} +$$$$79\!\cdots\!54$$$$T^{33} +$$$$34\!\cdots\!52$$$$T^{34} +$$$$95\!\cdots\!32$$$$T^{35} +$$$$12\!\cdots\!89$$$$T^{36}$$
$53$ $$1 + 18 T + 219 T^{2} + 1873 T^{3} + 21028 T^{4} + 191940 T^{5} + 1927031 T^{6} + 16691841 T^{7} + 157628752 T^{8} + 1143000974 T^{9} + 9987462820 T^{10} + 79427444768 T^{11} + 685958917601 T^{12} + 4783166371000 T^{13} + 40313855954708 T^{14} + 280316920456111 T^{15} + 2208536845954869 T^{16} + 15550871535373675 T^{17} + 128962261045140095 T^{18} + 824196191374804775 T^{19} + 6203780000287227021 T^{20} + 41732742166744437347 T^{21} +$$$$31\!\cdots\!48$$$$T^{22} +$$$$20\!\cdots\!00$$$$T^{23} +$$$$15\!\cdots\!29$$$$T^{24} +$$$$93\!\cdots\!16$$$$T^{25} +$$$$62\!\cdots\!20$$$$T^{26} +$$$$37\!\cdots\!42$$$$T^{27} +$$$$27\!\cdots\!48$$$$T^{28} +$$$$15\!\cdots\!77$$$$T^{29} +$$$$94\!\cdots\!71$$$$T^{30} +$$$$49\!\cdots\!20$$$$T^{31} +$$$$29\!\cdots\!32$$$$T^{32} +$$$$13\!\cdots\!61$$$$T^{33} +$$$$84\!\cdots\!99$$$$T^{34} +$$$$36\!\cdots\!34$$$$T^{35} +$$$$10\!\cdots\!89$$$$T^{36}$$
$59$ $$1 + 4 T + 182 T^{2} + 889 T^{3} + 19535 T^{4} + 92775 T^{5} + 1403700 T^{6} + 3337697 T^{7} + 60900249 T^{8} - 100866924 T^{9} + 279379214 T^{10} - 22396990778 T^{11} - 136590509711 T^{12} - 1169980179938 T^{13} - 4940983425545 T^{14} + 7131058308663 T^{15} + 145810789722975 T^{16} + 5887152771171894 T^{17} + 24165780228834877 T^{18} + 347342013499141746 T^{19} + 507567359025675975 T^{20} + 1464569624374898277 T^{21} - 59871679862345386745 T^{22} -$$$$83\!\cdots\!62$$$$T^{23} -$$$$57\!\cdots\!51$$$$T^{24} -$$$$55\!\cdots\!82$$$$T^{25} +$$$$41\!\cdots\!94$$$$T^{26} -$$$$87\!\cdots\!36$$$$T^{27} +$$$$31\!\cdots\!49$$$$T^{28} +$$$$10\!\cdots\!23$$$$T^{29} +$$$$24\!\cdots\!00$$$$T^{30} +$$$$97\!\cdots\!25$$$$T^{31} +$$$$12\!\cdots\!35$$$$T^{32} +$$$$32\!\cdots\!11$$$$T^{33} +$$$$39\!\cdots\!62$$$$T^{34} +$$$$50\!\cdots\!76$$$$T^{35} +$$$$75\!\cdots\!21$$$$T^{36}$$
$61$ $$1 - 42 T + 933 T^{2} - 14861 T^{3} + 192423 T^{4} - 2141538 T^{5} + 21336371 T^{6} - 196037772 T^{7} + 1665767448 T^{8} - 12759457459 T^{9} + 83761862657 T^{10} - 415805126156 T^{11} + 730146937632 T^{12} + 16028186575331 T^{13} - 292933998347776 T^{14} + 3505813369714012 T^{15} - 35535339534952817 T^{16} + 322709127419838593 T^{17} - 2650994802911766219 T^{18} + 19685256772610154173 T^{19} -$$$$13\!\cdots\!57$$$$T^{20} +$$$$79\!\cdots\!72$$$$T^{21} -$$$$40\!\cdots\!16$$$$T^{22} +$$$$13\!\cdots\!31$$$$T^{23} +$$$$37\!\cdots\!52$$$$T^{24} -$$$$13\!\cdots\!76$$$$T^{25} +$$$$16\!\cdots\!17$$$$T^{26} -$$$$14\!\cdots\!19$$$$T^{27} +$$$$11\!\cdots\!48$$$$T^{28} -$$$$85\!\cdots\!92$$$$T^{29} +$$$$56\!\cdots\!91$$$$T^{30} -$$$$34\!\cdots\!78$$$$T^{31} +$$$$19\!\cdots\!43$$$$T^{32} -$$$$89\!\cdots\!61$$$$T^{33} +$$$$34\!\cdots\!13$$$$T^{34} -$$$$94\!\cdots\!82$$$$T^{35} +$$$$13\!\cdots\!81$$$$T^{36}$$
$67$ $$1 + 19 T - 121 T^{2} - 3509 T^{3} + 17241 T^{4} + 454616 T^{5} - 1635832 T^{6} - 44808601 T^{7} + 41507781 T^{8} + 3267974565 T^{9} + 6835163934 T^{10} - 240159436956 T^{11} - 1558153168075 T^{12} + 15530120367369 T^{13} + 158968204106237 T^{14} - 817261224750396 T^{15} - 11907700881595164 T^{16} + 24442230601190901 T^{17} + 850574669249442709 T^{18} + 1637629450279790367 T^{19} - 53453669257480691196 T^{20} -$$$$24\!\cdots\!48$$$$T^{21} +$$$$32\!\cdots\!77$$$$T^{22} +$$$$20\!\cdots\!83$$$$T^{23} -$$$$14\!\cdots\!75$$$$T^{24} -$$$$14\!\cdots\!88$$$$T^{25} +$$$$27\!\cdots\!94$$$$T^{26} +$$$$88\!\cdots\!55$$$$T^{27} +$$$$75\!\cdots\!69$$$$T^{28} -$$$$54\!\cdots\!83$$$$T^{29} -$$$$13\!\cdots\!52$$$$T^{30} +$$$$24\!\cdots\!92$$$$T^{31} +$$$$63\!\cdots\!89$$$$T^{32} -$$$$86\!\cdots\!87$$$$T^{33} -$$$$19\!\cdots\!01$$$$T^{34} +$$$$20\!\cdots\!13$$$$T^{35} +$$$$74\!\cdots\!09$$$$T^{36}$$
$71$ $$1 + 27 T + 192 T^{2} - 3064 T^{3} - 71457 T^{4} - 431170 T^{5} + 3006306 T^{6} + 66658940 T^{7} + 407727641 T^{8} - 36960119 T^{9} - 18346711662 T^{10} - 203539733210 T^{11} - 1941959274995 T^{12} - 10817050171315 T^{13} + 59547414301229 T^{14} + 1594397690229843 T^{15} + 9439982388216595 T^{16} - 39777894586305512 T^{17} - 893741411732820099 T^{18} - 2824230515627691352 T^{19} + 47586951218999855395 T^{20} +$$$$57\!\cdots\!73$$$$T^{21} +$$$$15\!\cdots\!49$$$$T^{22} -$$$$19\!\cdots\!65$$$$T^{23} -$$$$24\!\cdots\!95$$$$T^{24} -$$$$18\!\cdots\!10$$$$T^{25} -$$$$11\!\cdots\!82$$$$T^{26} -$$$$16\!\cdots\!89$$$$T^{27} +$$$$13\!\cdots\!41$$$$T^{28} +$$$$15\!\cdots\!40$$$$T^{29} +$$$$49\!\cdots\!46$$$$T^{30} -$$$$50\!\cdots\!70$$$$T^{31} -$$$$59\!\cdots\!17$$$$T^{32} -$$$$17\!\cdots\!64$$$$T^{33} +$$$$80\!\cdots\!32$$$$T^{34} +$$$$79\!\cdots\!57$$$$T^{35} +$$$$21\!\cdots\!61$$$$T^{36}$$
$73$ $$1 - 28 T + 206 T^{2} + 1439 T^{3} - 30467 T^{4} + 115902 T^{5} + 1193696 T^{6} - 12456075 T^{7} - 52758690 T^{8} + 1181272439 T^{9} + 858716819 T^{10} - 81754299384 T^{11} - 22479646170 T^{12} + 7830326968241 T^{13} - 27470831061776 T^{14} - 692066597072037 T^{15} + 6983730927869749 T^{16} + 20546869285875433 T^{17} - 650284685443406899 T^{18} + 1499921457868906609 T^{19} + 37216302114617892421 T^{20} -$$$$26\!\cdots\!29$$$$T^{21} -$$$$78\!\cdots\!16$$$$T^{22} +$$$$16\!\cdots\!13$$$$T^{23} -$$$$34\!\cdots\!30$$$$T^{24} -$$$$90\!\cdots\!48$$$$T^{25} +$$$$69\!\cdots\!39$$$$T^{26} +$$$$69\!\cdots\!07$$$$T^{27} -$$$$22\!\cdots\!10$$$$T^{28} -$$$$39\!\cdots\!75$$$$T^{29} +$$$$27\!\cdots\!16$$$$T^{30} +$$$$19\!\cdots\!66$$$$T^{31} -$$$$37\!\cdots\!03$$$$T^{32} +$$$$12\!\cdots\!23$$$$T^{33} +$$$$13\!\cdots\!66$$$$T^{34} -$$$$13\!\cdots\!84$$$$T^{35} +$$$$34\!\cdots\!69$$$$T^{36}$$
$79$ $$( 1 + 13 T + 473 T^{2} + 3826 T^{3} + 90460 T^{4} + 498762 T^{5} + 11312861 T^{6} + 50578492 T^{7} + 1144441533 T^{8} + 4524834638 T^{9} + 90410881107 T^{10} + 315660368572 T^{11} + 5577681674579 T^{12} + 19426820299722 T^{13} + 278350521853540 T^{14} + 930052604823346 T^{15} + 9083448950453207 T^{16} + 19722414528785293 T^{17} + 119851595982618319 T^{18} )^{2}$$
$83$ $$1 - 62 T + 1955 T^{2} - 43474 T^{3} + 791403 T^{4} - 12660991 T^{5} + 182312666 T^{6} - 2380818115 T^{7} + 28454173700 T^{8} - 313999851042 T^{9} + 3203343358059 T^{10} - 30029118019905 T^{11} + 256415447656186 T^{12} - 1966785367099865 T^{13} + 13121718252161262 T^{14} - 70231361214705904 T^{15} + 228113352519056074 T^{16} + 528279145154473206 T^{17} - 13256029529997156100 T^{18} + 43847169047821276098 T^{19} +$$$$15\!\cdots\!86$$$$T^{20} -$$$$40\!\cdots\!48$$$$T^{21} +$$$$62\!\cdots\!02$$$$T^{22} -$$$$77\!\cdots\!95$$$$T^{23} +$$$$83\!\cdots\!34$$$$T^{24} -$$$$81\!\cdots\!35$$$$T^{25} +$$$$72\!\cdots\!19$$$$T^{26} -$$$$58\!\cdots\!26$$$$T^{27} +$$$$44\!\cdots\!00$$$$T^{28} -$$$$30\!\cdots\!05$$$$T^{29} +$$$$19\!\cdots\!26$$$$T^{30} -$$$$11\!\cdots\!33$$$$T^{31} +$$$$58\!\cdots\!87$$$$T^{32} -$$$$26\!\cdots\!18$$$$T^{33} +$$$$99\!\cdots\!55$$$$T^{34} -$$$$26\!\cdots\!26$$$$T^{35} +$$$$34\!\cdots\!09$$$$T^{36}$$
$89$ $$1 + 12 T + 116 T^{2} + 1499 T^{3} + 3935 T^{4} - 42611 T^{5} - 1895934 T^{6} - 26145279 T^{7} - 209446975 T^{8} - 2196428168 T^{9} - 7262368198 T^{10} + 44027191578 T^{11} + 1289869174743 T^{12} + 22210109524986 T^{13} + 120472292193179 T^{14} + 920900716991509 T^{15} + 1767591597163055 T^{16} - 97723090524153866 T^{17} - 862701798538908603 T^{18} - 8697355056649694074 T^{19} + 14001093041128558655 T^{20} +$$$$64\!\cdots\!21$$$$T^{21} +$$$$75\!\cdots\!39$$$$T^{22} +$$$$12\!\cdots\!14$$$$T^{23} +$$$$64\!\cdots\!23$$$$T^{24} +$$$$19\!\cdots\!62$$$$T^{25} -$$$$28\!\cdots\!38$$$$T^{26} -$$$$76\!\cdots\!12$$$$T^{27} -$$$$65\!\cdots\!75$$$$T^{28} -$$$$72\!\cdots\!31$$$$T^{29} -$$$$46\!\cdots\!14$$$$T^{30} -$$$$93\!\cdots\!59$$$$T^{31} +$$$$76\!\cdots\!35$$$$T^{32} +$$$$26\!\cdots\!51$$$$T^{33} +$$$$17\!\cdots\!76$$$$T^{34} +$$$$16\!\cdots\!48$$$$T^{35} +$$$$12\!\cdots\!81$$$$T^{36}$$
$97$ $$1 - 71 T + 2368 T^{2} - 50969 T^{3} + 836880 T^{4} - 11828308 T^{5} + 155370763 T^{6} - 1975485248 T^{7} + 25025905391 T^{8} - 318238582892 T^{9} + 3957848635408 T^{10} - 46785223241176 T^{11} + 523567041637077 T^{12} - 5637158848736682 T^{13} + 59668959182508043 T^{14} - 630662658429013549 T^{15} + 6659652238475527590 T^{16} - 69228635653545298441 T^{17} +$$$$69\!\cdots\!39$$$$T^{18} -$$$$67\!\cdots\!77$$$$T^{19} +$$$$62\!\cdots\!10$$$$T^{20} -$$$$57\!\cdots\!77$$$$T^{21} +$$$$52\!\cdots\!83$$$$T^{22} -$$$$48\!\cdots\!74$$$$T^{23} +$$$$43\!\cdots\!33$$$$T^{24} -$$$$37\!\cdots\!88$$$$T^{25} +$$$$31\!\cdots\!88$$$$T^{26} -$$$$24\!\cdots\!64$$$$T^{27} +$$$$18\!\cdots\!59$$$$T^{28} -$$$$14\!\cdots\!44$$$$T^{29} +$$$$10\!\cdots\!83$$$$T^{30} -$$$$79\!\cdots\!16$$$$T^{31} +$$$$54\!\cdots\!20$$$$T^{32} -$$$$32\!\cdots\!17$$$$T^{33} +$$$$14\!\cdots\!28$$$$T^{34} -$$$$42\!\cdots\!27$$$$T^{35} +$$$$57\!\cdots\!89$$$$T^{36}$$