Properties

Label 690.2.m.f
Level $690$
Weight $2$
Character orbit 690.m
Analytic conductor $5.510$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.m (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 6 x^{19} + 36 x^{18} - 172 x^{17} + 691 x^{16} - 2342 x^{15} + 7870 x^{14} - 23240 x^{13} + 63837 x^{12} - 154607 x^{11} + 334467 x^{10} - 664202 x^{9} + 1099560 x^{8} - 1331000 x^{7} + 1218096 x^{6} + 395648 x^{5} - 588544 x^{4} + 665984 x^{3} + 2323200 x^{2} + 991232 x + 123904\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 6 x^{19} + 36 x^{18} - 172 x^{17} + 691 x^{16} - 2342 x^{15} + 7870 x^{14} - 23240 x^{13} + 63837 x^{12} - 154607 x^{11} + 334467 x^{10} - 664202 x^{9} + 1099560 x^{8} - 1331000 x^{7} + 1218096 x^{6} + 395648 x^{5} - 588544 x^{4} + 665984 x^{3} + 2323200 x^{2} + 991232 x + 123904\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(16\!\cdots\!51\)\( \nu^{19} - \)\(10\!\cdots\!58\)\( \nu^{18} + \)\(63\!\cdots\!92\)\( \nu^{17} - \)\(30\!\cdots\!72\)\( \nu^{16} + \)\(12\!\cdots\!61\)\( \nu^{15} - \)\(43\!\cdots\!06\)\( \nu^{14} + \)\(14\!\cdots\!34\)\( \nu^{13} - \)\(43\!\cdots\!52\)\( \nu^{12} + \)\(12\!\cdots\!75\)\( \nu^{11} - \)\(29\!\cdots\!45\)\( \nu^{10} + \)\(65\!\cdots\!37\)\( \nu^{9} - \)\(13\!\cdots\!66\)\( \nu^{8} + \)\(22\!\cdots\!48\)\( \nu^{7} - \)\(30\!\cdots\!00\)\( \nu^{6} + \)\(30\!\cdots\!96\)\( \nu^{5} - \)\(43\!\cdots\!52\)\( \nu^{4} - \)\(48\!\cdots\!76\)\( \nu^{3} + \)\(62\!\cdots\!60\)\( \nu^{2} + \)\(52\!\cdots\!68\)\( \nu + \)\(15\!\cdots\!32\)\(\)\()/ \)\(18\!\cdots\!64\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(24\!\cdots\!55\)\( \nu^{19} - \)\(15\!\cdots\!16\)\( \nu^{18} + \)\(93\!\cdots\!96\)\( \nu^{17} - \)\(45\!\cdots\!88\)\( \nu^{16} + \)\(18\!\cdots\!25\)\( \nu^{15} - \)\(63\!\cdots\!68\)\( \nu^{14} + \)\(21\!\cdots\!18\)\( \nu^{13} - \)\(63\!\cdots\!60\)\( \nu^{12} + \)\(17\!\cdots\!83\)\( \nu^{11} - \)\(43\!\cdots\!99\)\( \nu^{10} + \)\(97\!\cdots\!15\)\( \nu^{9} - \)\(19\!\cdots\!48\)\( \nu^{8} + \)\(34\!\cdots\!36\)\( \nu^{7} - \)\(45\!\cdots\!92\)\( \nu^{6} + \)\(48\!\cdots\!16\)\( \nu^{5} - \)\(11\!\cdots\!40\)\( \nu^{4} + \)\(29\!\cdots\!68\)\( \nu^{3} + \)\(14\!\cdots\!88\)\( \nu^{2} + \)\(54\!\cdots\!72\)\( \nu + \)\(94\!\cdots\!04\)\(\)\()/ \)\(18\!\cdots\!64\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(14\!\cdots\!19\)\( \nu^{19} - \)\(86\!\cdots\!34\)\( \nu^{18} + \)\(51\!\cdots\!40\)\( \nu^{17} - \)\(24\!\cdots\!84\)\( \nu^{16} + \)\(98\!\cdots\!25\)\( \nu^{15} - \)\(33\!\cdots\!70\)\( \nu^{14} + \)\(11\!\cdots\!06\)\( \nu^{13} - \)\(33\!\cdots\!28\)\( \nu^{12} + \)\(90\!\cdots\!87\)\( \nu^{11} - \)\(21\!\cdots\!81\)\( \nu^{10} + \)\(47\!\cdots\!25\)\( \nu^{9} - \)\(93\!\cdots\!90\)\( \nu^{8} + \)\(15\!\cdots\!60\)\( \nu^{7} - \)\(18\!\cdots\!24\)\( \nu^{6} + \)\(16\!\cdots\!48\)\( \nu^{5} + \)\(65\!\cdots\!08\)\( \nu^{4} - \)\(11\!\cdots\!16\)\( \nu^{3} + \)\(58\!\cdots\!88\)\( \nu^{2} + \)\(32\!\cdots\!24\)\( \nu + \)\(78\!\cdots\!48\)\(\)\()/ \)\(79\!\cdots\!68\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(72\!\cdots\!99\)\( \nu^{19} + \)\(42\!\cdots\!52\)\( \nu^{18} - \)\(25\!\cdots\!04\)\( \nu^{17} + \)\(11\!\cdots\!04\)\( \nu^{16} - \)\(47\!\cdots\!13\)\( \nu^{15} + \)\(15\!\cdots\!32\)\( \nu^{14} - \)\(52\!\cdots\!70\)\( \nu^{13} + \)\(15\!\cdots\!48\)\( \nu^{12} - \)\(41\!\cdots\!79\)\( \nu^{11} + \)\(98\!\cdots\!59\)\( \nu^{10} - \)\(20\!\cdots\!87\)\( \nu^{9} + \)\(39\!\cdots\!64\)\( \nu^{8} - \)\(61\!\cdots\!84\)\( \nu^{7} + \)\(62\!\cdots\!20\)\( \nu^{6} - \)\(34\!\cdots\!12\)\( \nu^{5} - \)\(92\!\cdots\!92\)\( \nu^{4} + \)\(92\!\cdots\!60\)\( \nu^{3} - \)\(47\!\cdots\!76\)\( \nu^{2} - \)\(18\!\cdots\!88\)\( \nu - \)\(77\!\cdots\!28\)\(\)\()/ \)\(36\!\cdots\!28\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(63\!\cdots\!63\)\( \nu^{19} - \)\(41\!\cdots\!94\)\( \nu^{18} + \)\(24\!\cdots\!30\)\( \nu^{17} - \)\(12\!\cdots\!08\)\( \nu^{16} + \)\(49\!\cdots\!33\)\( \nu^{15} - \)\(17\!\cdots\!34\)\( \nu^{14} + \)\(58\!\cdots\!52\)\( \nu^{13} - \)\(17\!\cdots\!68\)\( \nu^{12} + \)\(48\!\cdots\!91\)\( \nu^{11} - \)\(12\!\cdots\!21\)\( \nu^{10} + \)\(26\!\cdots\!03\)\( \nu^{9} - \)\(54\!\cdots\!72\)\( \nu^{8} + \)\(95\!\cdots\!82\)\( \nu^{7} - \)\(12\!\cdots\!00\)\( \nu^{6} + \)\(13\!\cdots\!96\)\( \nu^{5} - \)\(31\!\cdots\!80\)\( \nu^{4} - \)\(32\!\cdots\!80\)\( \nu^{3} + \)\(64\!\cdots\!72\)\( \nu^{2} + \)\(10\!\cdots\!92\)\( \nu + \)\(42\!\cdots\!48\)\(\)\()/ \)\(18\!\cdots\!64\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(14\!\cdots\!03\)\( \nu^{19} + \)\(91\!\cdots\!56\)\( \nu^{18} - \)\(54\!\cdots\!12\)\( \nu^{17} + \)\(26\!\cdots\!68\)\( \nu^{16} - \)\(10\!\cdots\!25\)\( \nu^{15} + \)\(36\!\cdots\!04\)\( \nu^{14} - \)\(12\!\cdots\!46\)\( \nu^{13} + \)\(36\!\cdots\!68\)\( \nu^{12} - \)\(99\!\cdots\!99\)\( \nu^{11} + \)\(24\!\cdots\!67\)\( \nu^{10} - \)\(53\!\cdots\!43\)\( \nu^{9} + \)\(10\!\cdots\!48\)\( \nu^{8} - \)\(17\!\cdots\!64\)\( \nu^{7} + \)\(22\!\cdots\!76\)\( \nu^{6} - \)\(22\!\cdots\!12\)\( \nu^{5} - \)\(79\!\cdots\!36\)\( \nu^{4} + \)\(64\!\cdots\!36\)\( \nu^{3} - \)\(80\!\cdots\!04\)\( \nu^{2} - \)\(32\!\cdots\!44\)\( \nu - \)\(10\!\cdots\!64\)\(\)\()/ \)\(36\!\cdots\!28\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(15\!\cdots\!77\)\( \nu^{19} - \)\(96\!\cdots\!72\)\( \nu^{18} + \)\(58\!\cdots\!04\)\( \nu^{17} - \)\(28\!\cdots\!36\)\( \nu^{16} + \)\(11\!\cdots\!83\)\( \nu^{15} - \)\(39\!\cdots\!84\)\( \nu^{14} + \)\(13\!\cdots\!26\)\( \nu^{13} - \)\(39\!\cdots\!16\)\( \nu^{12} + \)\(11\!\cdots\!69\)\( \nu^{11} - \)\(27\!\cdots\!05\)\( \nu^{10} + \)\(60\!\cdots\!57\)\( \nu^{9} - \)\(12\!\cdots\!84\)\( \nu^{8} + \)\(20\!\cdots\!16\)\( \nu^{7} - \)\(27\!\cdots\!72\)\( \nu^{6} + \)\(27\!\cdots\!76\)\( \nu^{5} - \)\(36\!\cdots\!36\)\( \nu^{4} - \)\(66\!\cdots\!08\)\( \nu^{3} + \)\(96\!\cdots\!32\)\( \nu^{2} + \)\(32\!\cdots\!24\)\( \nu + \)\(43\!\cdots\!20\)\(\)\()/ \)\(36\!\cdots\!28\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(96\!\cdots\!03\)\( \nu^{19} - \)\(58\!\cdots\!92\)\( \nu^{18} + \)\(35\!\cdots\!24\)\( \nu^{17} - \)\(17\!\cdots\!08\)\( \nu^{16} + \)\(68\!\cdots\!41\)\( \nu^{15} - \)\(23\!\cdots\!84\)\( \nu^{14} + \)\(79\!\cdots\!90\)\( \nu^{13} - \)\(23\!\cdots\!00\)\( \nu^{12} + \)\(65\!\cdots\!59\)\( \nu^{11} - \)\(16\!\cdots\!91\)\( \nu^{10} + \)\(35\!\cdots\!39\)\( \nu^{9} - \)\(71\!\cdots\!20\)\( \nu^{8} + \)\(12\!\cdots\!16\)\( \nu^{7} - \)\(15\!\cdots\!52\)\( \nu^{6} + \)\(16\!\cdots\!84\)\( \nu^{5} - \)\(16\!\cdots\!20\)\( \nu^{4} - \)\(12\!\cdots\!60\)\( \nu^{3} + \)\(67\!\cdots\!24\)\( \nu^{2} + \)\(20\!\cdots\!72\)\( \nu + \)\(91\!\cdots\!16\)\(\)\()/ \)\(18\!\cdots\!64\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(25\!\cdots\!41\)\( \nu^{19} - \)\(15\!\cdots\!48\)\( \nu^{18} + \)\(92\!\cdots\!92\)\( \nu^{17} - \)\(44\!\cdots\!36\)\( \nu^{16} + \)\(17\!\cdots\!75\)\( \nu^{15} - \)\(61\!\cdots\!44\)\( \nu^{14} + \)\(20\!\cdots\!82\)\( \nu^{13} - \)\(61\!\cdots\!08\)\( \nu^{12} + \)\(16\!\cdots\!21\)\( \nu^{11} - \)\(41\!\cdots\!37\)\( \nu^{10} + \)\(89\!\cdots\!37\)\( \nu^{9} - \)\(17\!\cdots\!56\)\( \nu^{8} + \)\(30\!\cdots\!92\)\( \nu^{7} - \)\(37\!\cdots\!96\)\( \nu^{6} + \)\(36\!\cdots\!36\)\( \nu^{5} + \)\(38\!\cdots\!76\)\( \nu^{4} - \)\(13\!\cdots\!00\)\( \nu^{3} + \)\(17\!\cdots\!96\)\( \nu^{2} + \)\(56\!\cdots\!80\)\( \nu + \)\(14\!\cdots\!76\)\(\)\()/ \)\(36\!\cdots\!28\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(25\!\cdots\!61\)\( \nu^{19} - \)\(15\!\cdots\!54\)\( \nu^{18} + \)\(93\!\cdots\!64\)\( \nu^{17} - \)\(44\!\cdots\!04\)\( \nu^{16} + \)\(18\!\cdots\!51\)\( \nu^{15} - \)\(61\!\cdots\!90\)\( \nu^{14} + \)\(20\!\cdots\!38\)\( \nu^{13} - \)\(61\!\cdots\!36\)\( \nu^{12} + \)\(17\!\cdots\!57\)\( \nu^{11} - \)\(41\!\cdots\!47\)\( \nu^{10} + \)\(90\!\cdots\!91\)\( \nu^{9} - \)\(18\!\cdots\!18\)\( \nu^{8} + \)\(30\!\cdots\!12\)\( \nu^{7} - \)\(38\!\cdots\!60\)\( \nu^{6} + \)\(36\!\cdots\!72\)\( \nu^{5} + \)\(53\!\cdots\!96\)\( \nu^{4} - \)\(18\!\cdots\!24\)\( \nu^{3} + \)\(22\!\cdots\!72\)\( \nu^{2} + \)\(52\!\cdots\!36\)\( \nu + \)\(13\!\cdots\!24\)\(\)\()/ \)\(36\!\cdots\!28\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(25\!\cdots\!39\)\( \nu^{19} + \)\(16\!\cdots\!64\)\( \nu^{18} - \)\(97\!\cdots\!04\)\( \nu^{17} + \)\(47\!\cdots\!20\)\( \nu^{16} - \)\(19\!\cdots\!81\)\( \nu^{15} + \)\(66\!\cdots\!96\)\( \nu^{14} - \)\(22\!\cdots\!26\)\( \nu^{13} + \)\(66\!\cdots\!48\)\( \nu^{12} - \)\(18\!\cdots\!03\)\( \nu^{11} + \)\(45\!\cdots\!27\)\( \nu^{10} - \)\(99\!\cdots\!19\)\( \nu^{9} + \)\(19\!\cdots\!36\)\( \nu^{8} - \)\(33\!\cdots\!04\)\( \nu^{7} + \)\(43\!\cdots\!04\)\( \nu^{6} - \)\(42\!\cdots\!96\)\( \nu^{5} + \)\(16\!\cdots\!92\)\( \nu^{4} + \)\(18\!\cdots\!80\)\( \nu^{3} - \)\(23\!\cdots\!64\)\( \nu^{2} - \)\(52\!\cdots\!80\)\( \nu - \)\(89\!\cdots\!76\)\(\)\()/ \)\(36\!\cdots\!28\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(27\!\cdots\!29\)\( \nu^{19} + \)\(17\!\cdots\!84\)\( \nu^{18} - \)\(10\!\cdots\!60\)\( \nu^{17} + \)\(51\!\cdots\!88\)\( \nu^{16} - \)\(20\!\cdots\!83\)\( \nu^{15} + \)\(71\!\cdots\!88\)\( \nu^{14} - \)\(24\!\cdots\!74\)\( \nu^{13} + \)\(72\!\cdots\!24\)\( \nu^{12} - \)\(20\!\cdots\!77\)\( \nu^{11} + \)\(49\!\cdots\!65\)\( \nu^{10} - \)\(10\!\cdots\!41\)\( \nu^{9} + \)\(22\!\cdots\!08\)\( \nu^{8} - \)\(37\!\cdots\!52\)\( \nu^{7} + \)\(49\!\cdots\!64\)\( \nu^{6} - \)\(50\!\cdots\!80\)\( \nu^{5} + \)\(62\!\cdots\!36\)\( \nu^{4} + \)\(13\!\cdots\!04\)\( \nu^{3} - \)\(21\!\cdots\!08\)\( \nu^{2} - \)\(58\!\cdots\!64\)\( \nu - \)\(11\!\cdots\!64\)\(\)\()/ \)\(36\!\cdots\!28\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(32\!\cdots\!95\)\( \nu^{19} - \)\(20\!\cdots\!80\)\( \nu^{18} + \)\(12\!\cdots\!72\)\( \nu^{17} - \)\(58\!\cdots\!92\)\( \nu^{16} + \)\(23\!\cdots\!13\)\( \nu^{15} - \)\(82\!\cdots\!52\)\( \nu^{14} + \)\(27\!\cdots\!10\)\( \nu^{13} - \)\(82\!\cdots\!92\)\( \nu^{12} + \)\(22\!\cdots\!35\)\( \nu^{11} - \)\(56\!\cdots\!63\)\( \nu^{10} + \)\(12\!\cdots\!39\)\( \nu^{9} - \)\(24\!\cdots\!64\)\( \nu^{8} + \)\(42\!\cdots\!20\)\( \nu^{7} - \)\(54\!\cdots\!20\)\( \nu^{6} + \)\(54\!\cdots\!92\)\( \nu^{5} - \)\(23\!\cdots\!68\)\( \nu^{4} - \)\(18\!\cdots\!36\)\( \nu^{3} + \)\(28\!\cdots\!24\)\( \nu^{2} + \)\(66\!\cdots\!28\)\( \nu + \)\(15\!\cdots\!64\)\(\)\()/ \)\(36\!\cdots\!28\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(59\!\cdots\!91\)\( \nu^{19} + \)\(37\!\cdots\!88\)\( \nu^{18} - \)\(22\!\cdots\!52\)\( \nu^{17} + \)\(10\!\cdots\!36\)\( \nu^{16} - \)\(44\!\cdots\!73\)\( \nu^{15} + \)\(15\!\cdots\!64\)\( \nu^{14} - \)\(51\!\cdots\!62\)\( \nu^{13} + \)\(15\!\cdots\!88\)\( \nu^{12} - \)\(42\!\cdots\!51\)\( \nu^{11} + \)\(10\!\cdots\!83\)\( \nu^{10} - \)\(23\!\cdots\!79\)\( \nu^{9} + \)\(46\!\cdots\!92\)\( \nu^{8} - \)\(80\!\cdots\!72\)\( \nu^{7} + \)\(10\!\cdots\!32\)\( \nu^{6} - \)\(10\!\cdots\!88\)\( \nu^{5} + \)\(16\!\cdots\!92\)\( \nu^{4} + \)\(20\!\cdots\!72\)\( \nu^{3} - \)\(43\!\cdots\!80\)\( \nu^{2} - \)\(11\!\cdots\!28\)\( \nu - \)\(23\!\cdots\!32\)\(\)\()/ \)\(36\!\cdots\!28\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(11\!\cdots\!83\)\( \nu^{19} - \)\(69\!\cdots\!08\)\( \nu^{18} + \)\(41\!\cdots\!48\)\( \nu^{17} - \)\(20\!\cdots\!52\)\( \nu^{16} + \)\(82\!\cdots\!69\)\( \nu^{15} - \)\(28\!\cdots\!52\)\( \nu^{14} + \)\(95\!\cdots\!26\)\( \nu^{13} - \)\(28\!\cdots\!64\)\( \nu^{12} + \)\(79\!\cdots\!19\)\( \nu^{11} - \)\(19\!\cdots\!07\)\( \nu^{10} + \)\(42\!\cdots\!83\)\( \nu^{9} - \)\(86\!\cdots\!92\)\( \nu^{8} + \)\(14\!\cdots\!56\)\( \nu^{7} - \)\(19\!\cdots\!12\)\( \nu^{6} + \)\(19\!\cdots\!64\)\( \nu^{5} - \)\(20\!\cdots\!24\)\( \nu^{4} - \)\(54\!\cdots\!16\)\( \nu^{3} + \)\(89\!\cdots\!64\)\( \nu^{2} + \)\(22\!\cdots\!04\)\( \nu + \)\(47\!\cdots\!92\)\(\)\()/ \)\(36\!\cdots\!28\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(11\!\cdots\!03\)\( \nu^{19} + \)\(70\!\cdots\!86\)\( \nu^{18} - \)\(42\!\cdots\!52\)\( \nu^{17} + \)\(20\!\cdots\!72\)\( \nu^{16} - \)\(83\!\cdots\!57\)\( \nu^{15} + \)\(28\!\cdots\!38\)\( \nu^{14} - \)\(96\!\cdots\!10\)\( \nu^{13} + \)\(28\!\cdots\!96\)\( \nu^{12} - \)\(79\!\cdots\!15\)\( \nu^{11} + \)\(19\!\cdots\!49\)\( \nu^{10} - \)\(42\!\cdots\!21\)\( \nu^{9} + \)\(86\!\cdots\!70\)\( \nu^{8} - \)\(14\!\cdots\!00\)\( \nu^{7} + \)\(18\!\cdots\!12\)\( \nu^{6} - \)\(18\!\cdots\!64\)\( \nu^{5} + \)\(59\!\cdots\!64\)\( \nu^{4} + \)\(65\!\cdots\!92\)\( \nu^{3} - \)\(96\!\cdots\!52\)\( \nu^{2} - \)\(23\!\cdots\!40\)\( \nu - \)\(52\!\cdots\!00\)\(\)\()/ \)\(36\!\cdots\!28\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(12\!\cdots\!11\)\( \nu^{19} + \)\(75\!\cdots\!62\)\( \nu^{18} - \)\(45\!\cdots\!76\)\( \nu^{17} + \)\(21\!\cdots\!00\)\( \nu^{16} - \)\(88\!\cdots\!61\)\( \nu^{15} + \)\(30\!\cdots\!26\)\( \nu^{14} - \)\(10\!\cdots\!90\)\( \nu^{13} + \)\(30\!\cdots\!40\)\( \nu^{12} - \)\(83\!\cdots\!47\)\( \nu^{11} + \)\(20\!\cdots\!25\)\( \nu^{10} - \)\(44\!\cdots\!25\)\( \nu^{9} + \)\(89\!\cdots\!18\)\( \nu^{8} - \)\(15\!\cdots\!56\)\( \nu^{7} + \)\(19\!\cdots\!12\)\( \nu^{6} - \)\(18\!\cdots\!20\)\( \nu^{5} - \)\(10\!\cdots\!56\)\( \nu^{4} + \)\(74\!\cdots\!60\)\( \nu^{3} - \)\(96\!\cdots\!28\)\( \nu^{2} - \)\(26\!\cdots\!36\)\( \nu - \)\(61\!\cdots\!20\)\(\)\()/ \)\(36\!\cdots\!28\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(69\!\cdots\!60\)\( \nu^{19} - \)\(43\!\cdots\!93\)\( \nu^{18} + \)\(25\!\cdots\!62\)\( \nu^{17} - \)\(12\!\cdots\!68\)\( \nu^{16} + \)\(50\!\cdots\!80\)\( \nu^{15} - \)\(17\!\cdots\!43\)\( \nu^{14} + \)\(58\!\cdots\!38\)\( \nu^{13} - \)\(17\!\cdots\!14\)\( \nu^{12} + \)\(47\!\cdots\!04\)\( \nu^{11} - \)\(11\!\cdots\!49\)\( \nu^{10} + \)\(25\!\cdots\!35\)\( \nu^{9} - \)\(51\!\cdots\!35\)\( \nu^{8} + \)\(86\!\cdots\!74\)\( \nu^{7} - \)\(10\!\cdots\!56\)\( \nu^{6} + \)\(10\!\cdots\!24\)\( \nu^{5} + \)\(79\!\cdots\!08\)\( \nu^{4} - \)\(41\!\cdots\!00\)\( \nu^{3} + \)\(52\!\cdots\!04\)\( \nu^{2} + \)\(15\!\cdots\!40\)\( \nu + \)\(38\!\cdots\!60\)\(\)\()/ \)\(18\!\cdots\!64\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{19} + \beta_{18} - \beta_{15} - 2 \beta_{13} - \beta_{10} - \beta_{9} - 5 \beta_{8} - \beta_{6} + \beta_{1} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{17} + \beta_{16} + \beta_{15} - 4 \beta_{14} + 2 \beta_{13} - 6 \beta_{12} + \beta_{11} + 3 \beta_{10} - 2 \beta_{8} + 3 \beta_{5} + 8 \beta_{3} + \beta_{2} - \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(-15 \beta_{19} - 3 \beta_{18} + 4 \beta_{17} + 12 \beta_{16} + 2 \beta_{15} + 22 \beta_{14} + 28 \beta_{13} + 12 \beta_{11} + 3 \beta_{10} + 16 \beta_{9} + 46 \beta_{8} + 30 \beta_{7} - 20 \beta_{6} - 30 \beta_{5} + 15 \beta_{4} + 12 \beta_{2} + 8 \beta_{1} - 9\)
\(\nu^{5}\)\(=\)\(3 \beta_{19} + 22 \beta_{18} - 21 \beta_{17} - 19 \beta_{16} - 77 \beta_{15} + 81 \beta_{14} - 81 \beta_{13} + 96 \beta_{12} + 30 \beta_{11} - 195 \beta_{10} + 19 \beta_{9} + 70 \beta_{8} + 123 \beta_{7} - 123 \beta_{6} - 207 \beta_{5} - 77 \beta_{3} - 3 \beta_{2} + 21 \beta_{1} - 133\)
\(\nu^{6}\)\(=\)\(71 \beta_{19} + 84 \beta_{18} - 139 \beta_{17} - 33 \beta_{16} - 407 \beta_{14} - 93 \beta_{13} - 56 \beta_{12} + 60 \beta_{11} - 71 \beta_{9} - 144 \beta_{8} - 160 \beta_{7} - 60 \beta_{6} + 144 \beta_{5} - 195 \beta_{4} + 33 \beta_{3} - 93\)
\(\nu^{7}\)\(=\)\(-344 \beta_{19} - 343 \beta_{18} + 321 \beta_{17} + 834 \beta_{16} + 834 \beta_{15} - 978 \beta_{14} + 1388 \beta_{13} - 842 \beta_{12} - 144 \beta_{11} + 2490 \beta_{10} - 85 \beta_{8} + 85 \beta_{6} + 1709 \beta_{5} + 76 \beta_{4} + 321 \beta_{3} + 76 \beta_{2} - 343 \beta_{1} + 1185\)
\(\nu^{8}\)\(=\)\(-1345 \beta_{19} - 1646 \beta_{18} + 464 \beta_{17} - 464 \beta_{15} + 3313 \beta_{14} + 325 \beta_{13} + 2214 \beta_{12} - 2333 \beta_{11} - 1349 \beta_{10} + 1238 \beta_{9} + 3151 \beta_{8} + 2694 \beta_{7} + 325 \beta_{6} - 4551 \beta_{5} + 1238 \beta_{4} - 1646 \beta_{3} - 1252 \beta_{2} - 301 \beta_{1} + 408\)
\(\nu^{9}\)\(=\)\(5122 \beta_{19} + 5039 \beta_{18} - 9782 \beta_{17} - 9782 \beta_{16} - 5039 \beta_{15} + 863 \beta_{14} - 11065 \beta_{13} + 11148 \beta_{12} - 3771 \beta_{11} - 19321 \beta_{10} - 3462 \beta_{9} - 3545 \beta_{8} - 12999 \beta_{7} + 3880 \beta_{6} - 7588 \beta_{5} - 4811 \beta_{4} - 5122 \beta_{3} - 3462 \beta_{2} + 4728 \beta_{1} - 14199\)
\(\nu^{10}\)\(=\)\(20067 \beta_{19} + 6414 \beta_{18} + 6414 \beta_{16} + 20067 \beta_{15} - 60255 \beta_{14} + 20067 \beta_{13} - 40188 \beta_{12} - 3877 \beta_{11} + 64523 \beta_{10} - 32337 \beta_{9} - 78176 \beta_{8} - 63158 \beta_{7} + 17530 \beta_{6} + 125051 \beta_{5} - 19649 \beta_{4} + 19649 \beta_{3} - 328 \beta_{2} - 13325 \beta_{1} + 43091\)
\(\nu^{11}\)\(=\)\(-75281 \beta_{19} - 120846 \beta_{18} + 120846 \beta_{17} + 75281 \beta_{16} + 73162 \beta_{15} + 157372 \beta_{13} - 132394 \beta_{12} - 57113 \beta_{11} + 276072 \beta_{10} + 29743 \beta_{9} + 7406 \beta_{8} + 55090 \beta_{7} + 110050 \beta_{6} + 155588 \beta_{5} + 73162 \beta_{4} + 64523 \beta_{3} + 8639 \beta_{2} - 75308 \beta_{1} + 298571\)
\(\nu^{12}\)\(=\)\(-89013 \beta_{19} - 89013 \beta_{17} - 251121 \beta_{16} - 276072 \beta_{15} + 823377 \beta_{14} - 376753 \beta_{13} + 689140 \beta_{12} - 1019988 \beta_{10} + 260673 \beta_{9} + 688780 \beta_{8} + 411256 \beta_{7} + 72534 \beta_{6} - 1314898 \beta_{5} + 251121 \beta_{4} - 285624 \beta_{3} - 24951 \beta_{2} + 244405 \beta_{1} - 599767\)
\(\nu^{13}\)\(=\)\(1542824 \beta_{19} + 1542824 \beta_{18} - 1088728 \beta_{17} - 1019988 \beta_{16} - 853137 \beta_{15} - 511864 \beta_{14} - 2014906 \beta_{13} + 858294 \beta_{12} + 645926 \beta_{11} - 3208738 \beta_{10} - 921877 \beta_{9} - 1905869 \beta_{8} - 1531852 \beta_{7} - 691443 \beta_{6} + 68740 \beta_{5} - 1088728 \beta_{4} - 312027 \beta_{3} - 68740 \beta_{2} + 609850 \beta_{1} - 2704593\)
\(\nu^{14}\)\(=\)\(-1239061 \beta_{18} + 3208738 \beta_{17} + 3807616 \beta_{16} + 4094376 \beta_{15} - 9431371 \beta_{14} + 6476989 \beta_{13} - 11434400 \beta_{12} + 452823 \beta_{11} + 15412851 \beta_{10} - 1837939 \beta_{9} - 7716050 \beta_{8} - 3641553 \beta_{7} + 1525821 \beta_{6} + 16651912 \beta_{5} - 1239061 \beta_{4} + 5463599 \beta_{3} + 1969677 \beta_{2} - 3495498 \beta_{1} + 10670432\)
\(\nu^{15}\)\(=\)\(-20106737 \beta_{19} - 15412851 \beta_{18} + 14027735 \beta_{17} + 11253417 \beta_{16} + 4480203 \beta_{15} + 24106032 \beta_{14} + 16864314 \beta_{13} + 439515 \beta_{11} + 17239039 \beta_{10} + 18721621 \beta_{9} + 40053335 \beta_{8} + 37049421 \beta_{7} - 93380 \beta_{6} - 30489890 \beta_{5} + 20106737 \beta_{4} + 4693886 \beta_{2} - 2774318 \beta_{1} + 14973336\)
\(\nu^{16}\)\(=\)\(17239039 \beta_{19} + 41649927 \beta_{18} - 52022757 \beta_{17} - 57587306 \beta_{16} - 72330500 \beta_{15} + 119362791 \beta_{14} - 119362791 \beta_{13} + 151466124 \beta_{12} + 21347839 \beta_{11} - 265446551 \beta_{10} + 24410888 \beta_{9} + 97781484 \beta_{8} + 60699858 \beta_{7} - 60699858 \beta_{6} - 252141528 \beta_{5} - 72330500 \beta_{3} - 17239039 \beta_{2} + 52022757 \beta_{1} - 197050067\)
\(\nu^{17}\)\(=\)\(215119523 \beta_{19} + 191441670 \beta_{18} - 148795495 \beta_{17} - 63231234 \beta_{16} - 433042192 \beta_{14} - 116115420 \beta_{13} - 152256010 \beta_{12} + 52884186 \beta_{11} - 215119523 \beta_{9} - 492495580 \beta_{8} - 430606767 \beta_{7} - 65958203 \beta_{6} + 492495580 \beta_{5} - 265446551 \beta_{4} + 63231234 \beta_{3} - 129189437\)
\(\nu^{18}\)\(=\)\(-546891998 \beta_{19} - 707249278 \beta_{18} + 800052670 \beta_{17} + 963399658 \beta_{16} + 963399658 \beta_{15} - 1158373948 \beta_{14} + 1708889874 \beta_{13} - 1774254534 \beta_{12} - 194974290 \beta_{11} + 3533210008 \beta_{10} - 441015098 \beta_{8} + 441015098 \beta_{6} + 2508942544 \beta_{5} + 239165097 \beta_{4} + 800052670 \beta_{3} + 239165097 \beta_{2} - 707249278 \beta_{1} + 2481503812\)
\(\nu^{19}\)\(=\)\(-2601376287 \beta_{19} - 1975124054 \beta_{18} + 882484415 \beta_{17} - 882484415 \beta_{15} + 6607755494 \beta_{14} - 66246942 \beta_{13} + 4080396742 \beta_{12} - 772747794 \beta_{11} - 3305382093 \beta_{10} + 2973829581 \beta_{9} + 7680478556 \beta_{8} + 5906758380 \beta_{7} - 66246942 \beta_{6} - 9581585075 \beta_{5} + 2973829581 \beta_{4} - 1975124054 \beta_{3} - 559380427 \beta_{2} + 626252233 \beta_{1} - 998705527\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/690\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.02072 + 1.17798i
−1.91971 2.21546i
2.64328 1.69874i
−0.626062 + 0.402346i
1.05064 2.30058i
−1.08726 + 2.38076i
2.64328 + 1.69874i
−0.626062 0.402346i
0.303318 2.10963i
−0.523366 + 3.64009i
−0.245245 0.0720105i
2.38367 + 0.699910i
0.303318 + 2.10963i
−0.523366 3.64009i
1.05064 + 2.30058i
−1.08726 2.38076i
−0.245245 + 0.0720105i
2.38367 0.699910i
1.02072 1.17798i
−1.91971 + 2.21546i
0.142315 0.989821i −0.415415 0.909632i −0.959493 0.281733i 0.654861 + 0.755750i −0.959493 + 0.281733i −2.22667 1.43099i −0.415415 + 0.909632i −0.654861 + 0.755750i 0.841254 0.540641i
31.2 0.142315 0.989821i −0.415415 0.909632i −0.959493 0.281733i 0.654861 + 0.755750i −0.959493 + 0.281733i 1.55070 + 0.996574i −0.415415 + 0.909632i −0.654861 + 0.755750i 0.841254 0.540641i
121.1 −0.415415 + 0.909632i 0.959493 + 0.281733i −0.654861 0.755750i −0.841254 + 0.540641i −0.654861 + 0.755750i 0.0123285 0.0857467i 0.959493 0.281733i 0.841254 + 0.540641i −0.142315 0.989821i
121.2 −0.415415 + 0.909632i 0.959493 + 0.281733i −0.654861 0.755750i −0.841254 + 0.540641i −0.654861 + 0.755750i 0.565404 3.93247i 0.959493 0.281733i 0.841254 + 0.540641i −0.142315 0.989821i
151.1 0.654861 + 0.755750i −0.841254 0.540641i −0.142315 + 0.989821i −0.415415 + 0.909632i −0.142315 0.989821i −3.76793 1.10637i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.959493 + 0.281733i
151.2 0.654861 + 0.755750i −0.841254 0.540641i −0.142315 + 0.989821i −0.415415 + 0.909632i −0.142315 0.989821i 1.17001 + 0.343546i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.959493 + 0.281733i
211.1 −0.415415 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i −0.841254 0.540641i −0.654861 0.755750i 0.0123285 + 0.0857467i 0.959493 + 0.281733i 0.841254 0.540641i −0.142315 + 0.989821i
211.2 −0.415415 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i −0.841254 0.540641i −0.654861 0.755750i 0.565404 + 3.93247i 0.959493 + 0.281733i 0.841254 0.540641i −0.142315 + 0.989821i
271.1 0.959493 + 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i 0.142315 0.989821i 0.841254 0.540641i −0.730522 1.59962i 0.654861 + 0.755750i −0.142315 0.989821i 0.415415 0.909632i
271.2 0.959493 + 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i 0.142315 0.989821i 0.841254 0.540641i 1.68256 + 3.68429i 0.654861 + 0.755750i −0.142315 0.989821i 0.415415 0.909632i
301.1 −0.841254 0.540641i 0.142315 + 0.989821i 0.415415 + 0.909632i 0.959493 + 0.281733i 0.415415 0.909632i −0.525067 + 0.605959i 0.142315 0.989821i −0.959493 + 0.281733i −0.654861 0.755750i
301.2 −0.841254 0.540641i 0.142315 + 0.989821i 0.415415 + 0.909632i 0.959493 + 0.281733i 0.415415 0.909632i 1.26919 1.46472i 0.142315 0.989821i −0.959493 + 0.281733i −0.654861 0.755750i
331.1 0.959493 0.281733i 0.654861 + 0.755750i 0.841254 0.540641i 0.142315 + 0.989821i 0.841254 + 0.540641i −0.730522 + 1.59962i 0.654861 0.755750i −0.142315 + 0.989821i 0.415415 + 0.909632i
331.2 0.959493 0.281733i 0.654861 + 0.755750i 0.841254 0.540641i 0.142315 + 0.989821i 0.841254 + 0.540641i 1.68256 3.68429i 0.654861 0.755750i −0.142315 + 0.989821i 0.415415 + 0.909632i
361.1 0.654861 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i −0.415415 0.909632i −0.142315 + 0.989821i −3.76793 + 1.10637i −0.841254 0.540641i 0.415415 0.909632i −0.959493 0.281733i
361.2 0.654861 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i −0.415415 0.909632i −0.142315 + 0.989821i 1.17001 0.343546i −0.841254 0.540641i 0.415415 0.909632i −0.959493 0.281733i
541.1 −0.841254 + 0.540641i 0.142315 0.989821i 0.415415 0.909632i 0.959493 0.281733i 0.415415 + 0.909632i −0.525067 0.605959i 0.142315 + 0.989821i −0.959493 0.281733i −0.654861 + 0.755750i
541.2 −0.841254 + 0.540641i 0.142315 0.989821i 0.415415 0.909632i 0.959493 0.281733i 0.415415 + 0.909632i 1.26919 + 1.46472i 0.142315 + 0.989821i −0.959493 0.281733i −0.654861 + 0.755750i
601.1 0.142315 + 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.654861 0.755750i −0.959493 0.281733i −2.22667 + 1.43099i −0.415415 0.909632i −0.654861 0.755750i 0.841254 + 0.540641i
601.2 0.142315 + 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.654861 0.755750i −0.959493 0.281733i 1.55070 0.996574i −0.415415 0.909632i −0.654861 0.755750i 0.841254 + 0.540641i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 601.2
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.m.f 20
23.c even 11 1 inner 690.2.m.f 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.m.f 20 1.a even 1 1 trivial
690.2.m.f 20 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{20} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$3$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$5$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$7$ \( 7921 - 27679 T + 1059283 T^{2} - 225140 T^{3} - 174878 T^{4} - 1404325 T^{5} + 1534992 T^{6} - 477373 T^{7} + 199834 T^{8} - 177870 T^{9} + 76330 T^{10} + 21912 T^{11} - 6516 T^{12} - 2760 T^{13} + 3060 T^{14} + 309 T^{15} + 132 T^{16} + 106 T^{17} + 12 T^{18} + 2 T^{19} + T^{20} \)
$11$ \( 7921 + 185654 T + 6212993 T^{2} + 1877265 T^{3} + 5536737 T^{4} + 19816594 T^{5} - 17461031 T^{6} - 5497224 T^{7} + 37890425 T^{8} - 39382486 T^{9} + 23778107 T^{10} - 8851557 T^{11} + 1969595 T^{12} - 240541 T^{13} + 21775 T^{14} - 9312 T^{15} + 4111 T^{16} - 1039 T^{17} + 171 T^{18} - 18 T^{19} + T^{20} \)
$13$ \( 1771561 - 15944049 T + 46221637 T^{2} - 49764759 T^{3} + 87743513 T^{4} - 122196448 T^{5} + 103208402 T^{6} - 60142929 T^{7} + 27361004 T^{8} - 11371547 T^{9} + 5277438 T^{10} - 2082537 T^{11} + 735364 T^{12} - 153556 T^{13} + 35792 T^{14} - 2122 T^{15} + 522 T^{16} - 63 T^{17} + 15 T^{18} - 2 T^{19} + T^{20} \)
$17$ \( 123904 - 991232 T + 4584448 T^{2} - 2617472 T^{3} + 1386176 T^{4} + 2461888 T^{5} + 9948576 T^{6} - 23667688 T^{7} - 22567072 T^{8} + 24876456 T^{9} + 30887297 T^{10} + 9949066 T^{11} + 2004522 T^{12} + 262995 T^{13} + 45659 T^{14} + 1231 T^{15} + 2106 T^{16} + 63 T^{17} + 81 T^{18} + 2 T^{19} + T^{20} \)
$19$ \( 16834609 - 267593557 T + 1085455305 T^{2} + 708131567 T^{3} - 399121162 T^{4} - 685371929 T^{5} + 350330585 T^{6} - 127993217 T^{7} + 113034174 T^{8} - 18583818 T^{9} + 16133052 T^{10} - 4552617 T^{11} + 1379706 T^{12} - 280472 T^{13} + 148295 T^{14} - 8405 T^{15} + 5096 T^{16} - 361 T^{17} + 93 T^{18} - 4 T^{19} + T^{20} \)
$23$ \( 41426511213649 + 1801152661463 T^{2} - 149812319668 T^{3} + 19688773237 T^{4} - 24355121912 T^{5} + 443547985 T^{6} - 933379238 T^{7} + 110282217 T^{8} - 9563400 T^{9} + 11025037 T^{10} - 415800 T^{11} + 208473 T^{12} - 76714 T^{13} + 1585 T^{14} - 3784 T^{15} + 133 T^{16} - 44 T^{17} + 23 T^{18} + T^{20} \)
$29$ \( 2135900406784 - 5774404481536 T + 6056685732864 T^{2} - 3026978229504 T^{3} + 786740583104 T^{4} - 186376546496 T^{5} + 64252918352 T^{6} - 12065979152 T^{7} + 2745312496 T^{8} - 942059096 T^{9} + 238248913 T^{10} - 18444987 T^{11} + 14644590 T^{12} - 230794 T^{13} + 546468 T^{14} - 597 T^{15} + 11727 T^{16} - 406 T^{17} + 138 T^{18} - 6 T^{19} + T^{20} \)
$31$ \( 1472103424 - 10311476736 T + 29346879232 T^{2} - 44341818752 T^{3} + 42051941184 T^{4} - 30103277600 T^{5} + 17446643280 T^{6} - 7071184120 T^{7} + 2078558460 T^{8} - 261936422 T^{9} + 14493821 T^{10} - 13525530 T^{11} + 9229709 T^{12} - 128075 T^{13} + 234963 T^{14} - 12704 T^{15} + 8364 T^{16} - 600 T^{17} + 152 T^{18} - 8 T^{19} + T^{20} \)
$37$ \( 412990025449 - 142889743121 T + 349251942250 T^{2} + 216920562159 T^{3} + 92317408271 T^{4} - 30696137957 T^{5} - 8981280623 T^{6} + 1346021532 T^{7} - 738793950 T^{8} + 686999225 T^{9} + 98630159 T^{10} - 185216042 T^{11} + 64247664 T^{12} - 11930126 T^{13} + 1552358 T^{14} - 176254 T^{15} + 20669 T^{16} - 2417 T^{17} + 252 T^{18} - 16 T^{19} + T^{20} \)
$41$ \( 2145911501449 - 2541557127354 T + 3882681759063 T^{2} - 2692732829544 T^{3} + 1473293773160 T^{4} - 472904753674 T^{5} + 85909411151 T^{6} + 14390912341 T^{7} - 5088712982 T^{8} + 359774514 T^{9} + 402663350 T^{10} + 60022259 T^{11} + 4021005 T^{12} - 7444 T^{13} + 574291 T^{14} + 104411 T^{15} + 3520 T^{16} - 1209 T^{17} - 58 T^{18} - 3 T^{19} + T^{20} \)
$43$ \( 402382160896 + 5155548078592 T + 37020631962368 T^{2} + 59636617424640 T^{3} + 50496431697216 T^{4} + 27761751059488 T^{5} + 10920357660208 T^{6} + 3182721631224 T^{7} + 683051893144 T^{8} + 103426335518 T^{9} + 9516588863 T^{10} + 143493515 T^{11} - 58901383 T^{12} - 8401666 T^{13} - 674034 T^{14} - 52135 T^{15} + 8153 T^{16} + 2012 T^{17} + 261 T^{18} + 16 T^{19} + T^{20} \)
$47$ \( ( 1104851 + 891407 T - 1016158 T^{2} - 625691 T^{3} + 257664 T^{4} + 101034 T^{5} - 2404 T^{6} - 2650 T^{7} - 100 T^{8} + 17 T^{9} + T^{10} )^{2} \)
$53$ \( 3622956481 + 365720516 T + 9230039473 T^{2} - 11022107047 T^{3} + 10911139577 T^{4} - 14933591546 T^{5} + 9393228719 T^{6} - 6420799645 T^{7} + 5663676822 T^{8} - 508318844 T^{9} - 43418517 T^{10} - 26579498 T^{11} + 20987968 T^{12} + 6960543 T^{13} + 1653933 T^{14} + 158856 T^{15} + 305 T^{16} - 1985 T^{17} - 115 T^{18} + 4 T^{19} + T^{20} \)
$59$ \( 1279000251450769 - 770544129050027 T + 1025036266678706 T^{2} - 317388874530374 T^{3} + 66007323283446 T^{4} + 4881481514970 T^{5} + 2146030875238 T^{6} - 1700700124102 T^{7} + 736929142525 T^{8} - 165647687535 T^{9} + 21121925584 T^{10} - 619450623 T^{11} - 179032290 T^{12} + 26176595 T^{13} - 914211 T^{14} - 59449 T^{15} + 15088 T^{16} - 2416 T^{17} + 299 T^{18} - 25 T^{19} + T^{20} \)
$61$ \( 36489718211584 - 300691990911488 T + 960510340838400 T^{2} - 1459821156368256 T^{3} + 1081105294511616 T^{4} - 306549797635488 T^{5} + 65401933073664 T^{6} - 4316924941208 T^{7} + 701448293876 T^{8} + 10886544856 T^{9} + 29165718253 T^{10} + 1285479059 T^{11} + 139416216 T^{12} + 27284809 T^{13} + 1842866 T^{14} + 392347 T^{15} + 66609 T^{16} + 5605 T^{17} + 537 T^{18} + 19 T^{19} + T^{20} \)
$67$ \( 2141141533696 + 1433179292160 T + 17289520762624 T^{2} + 655243985792 T^{3} + 23860267057088 T^{4} - 8650379868416 T^{5} + 4895361804256 T^{6} - 591248923232 T^{7} + 205492370732 T^{8} - 80622405512 T^{9} + 22585725097 T^{10} - 2800356146 T^{11} - 11251182 T^{12} + 52057199 T^{13} - 2308286 T^{14} - 464564 T^{15} + 22401 T^{16} + 2593 T^{17} - 40 T^{18} - 20 T^{19} + T^{20} \)
$71$ \( 2364115354624 + 3982768540672 T + 9396427676928 T^{2} + 8596979306240 T^{3} - 3755618983424 T^{4} - 3208507398656 T^{5} + 3437834565680 T^{6} - 181171916920 T^{7} - 58921594748 T^{8} + 45344029082 T^{9} + 14713014537 T^{10} + 1683916652 T^{11} + 244947001 T^{12} + 32385899 T^{13} + 2930324 T^{14} + 318836 T^{15} + 27643 T^{16} + 1839 T^{17} + 216 T^{18} + T^{19} + T^{20} \)
$73$ \( 9360858440704 - 6086305602560 T + 10232571318784 T^{2} - 8259298047232 T^{3} + 3803416826432 T^{4} - 379747286144 T^{5} + 598221115808 T^{6} - 48527309104 T^{7} + 43612386284 T^{8} - 5561659274 T^{9} + 1703202303 T^{10} - 422373481 T^{11} + 19419293 T^{12} - 10723264 T^{13} + 1548692 T^{14} + 205320 T^{15} + 32704 T^{16} - 2183 T^{17} - 233 T^{18} + T^{19} + T^{20} \)
$79$ \( 1669866496 + 7541206016 T + 20635894272 T^{2} + 92134201728 T^{3} + 418912052992 T^{4} + 425977717632 T^{5} - 144260275824 T^{6} - 88286349400 T^{7} + 197441094800 T^{8} - 85904128868 T^{9} + 27297886781 T^{10} - 6470664911 T^{11} + 1277957231 T^{12} - 194868229 T^{13} + 23881549 T^{14} - 2479318 T^{15} + 198372 T^{16} - 11086 T^{17} + 533 T^{18} - 28 T^{19} + T^{20} \)
$83$ \( 596379329536 - 6878206179840 T + 83053579099904 T^{2} - 199906308047104 T^{3} + 213056291473664 T^{4} - 122436256835104 T^{5} + 42656173340208 T^{6} - 10051545316640 T^{7} + 1818142064116 T^{8} - 276673889140 T^{9} + 38494873649 T^{10} - 4941109393 T^{11} + 593502880 T^{12} - 62853875 T^{13} + 6565159 T^{14} - 558423 T^{15} + 42485 T^{16} - 2836 T^{17} + 261 T^{18} - 9 T^{19} + T^{20} \)
$89$ \( 33243729682334161 - 25122129089885173 T - 2053802585647818 T^{2} - 8790185303601010 T^{3} + 7053301816331849 T^{4} + 3094765765529419 T^{5} + 838893310038343 T^{6} + 133613528844458 T^{7} + 17637143884177 T^{8} + 1224472478020 T^{9} + 126212470187 T^{10} + 4184949945 T^{11} + 1121248153 T^{12} + 19515481 T^{13} - 15668 T^{14} - 689472 T^{15} - 44943 T^{16} + 251 T^{17} + 459 T^{18} + 31 T^{19} + T^{20} \)
$97$ \( 61546877742490624 - 102407566510216704 T + 1621095585001728 T^{2} + 37974113385093248 T^{3} + 18198064412946880 T^{4} + 3920093761850464 T^{5} + 877068310550960 T^{6} + 189188170294728 T^{7} + 32620456648868 T^{8} + 4280007453194 T^{9} + 585306629271 T^{10} + 74048220153 T^{11} + 8794019292 T^{12} + 867329481 T^{13} + 77114673 T^{14} + 5799507 T^{15} + 378581 T^{16} + 22316 T^{17} + 1027 T^{18} + 35 T^{19} + T^{20} \)
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