Properties

Label 690.2.m.d
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: \(x^{20} - 5 x^{19} + 8 x^{18} - 32 x^{17} + 277 x^{16} - 1138 x^{15} + 2950 x^{14} - 6404 x^{13} + 24088 x^{12} - 93423 x^{11} + 318055 x^{10} - 798006 x^{9} + 1869818 x^{8} - 3381161 x^{7} + 6172602 x^{6} - 7296658 x^{5} + 10759748 x^{4} - 4308967 x^{3} + 7994447 x^{2} + 486652 x + 7921\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 5 x^{19} + 8 x^{18} - 32 x^{17} + 277 x^{16} - 1138 x^{15} + 2950 x^{14} - 6404 x^{13} + 24088 x^{12} - 93423 x^{11} + 318055 x^{10} - 798006 x^{9} + 1869818 x^{8} - 3381161 x^{7} + 6172602 x^{6} - 7296658 x^{5} + 10759748 x^{4} - 4308967 x^{3} + 7994447 x^{2} + 486652 x + 7921\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(18\!\cdots\!21\)\( \nu^{19} - \)\(78\!\cdots\!83\)\( \nu^{18} + \)\(47\!\cdots\!75\)\( \nu^{17} - \)\(35\!\cdots\!64\)\( \nu^{16} + \)\(45\!\cdots\!24\)\( \nu^{15} - \)\(16\!\cdots\!31\)\( \nu^{14} + \)\(31\!\cdots\!83\)\( \nu^{13} - \)\(46\!\cdots\!12\)\( \nu^{12} + \)\(28\!\cdots\!40\)\( \nu^{11} - \)\(12\!\cdots\!07\)\( \nu^{10} + \)\(40\!\cdots\!06\)\( \nu^{9} - \)\(79\!\cdots\!35\)\( \nu^{8} + \)\(15\!\cdots\!38\)\( \nu^{7} - \)\(18\!\cdots\!82\)\( \nu^{6} + \)\(30\!\cdots\!54\)\( \nu^{5} + \)\(99\!\cdots\!87\)\( \nu^{4} + \)\(63\!\cdots\!94\)\( \nu^{3} + \)\(14\!\cdots\!75\)\( \nu^{2} + \)\(84\!\cdots\!26\)\( \nu + \)\(75\!\cdots\!32\)\(\)\()/ \)\(27\!\cdots\!41\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(21\!\cdots\!92\)\( \nu^{19} + \)\(99\!\cdots\!57\)\( \nu^{18} - \)\(82\!\cdots\!02\)\( \nu^{17} + \)\(40\!\cdots\!89\)\( \nu^{16} - \)\(54\!\cdots\!95\)\( \nu^{15} + \)\(21\!\cdots\!92\)\( \nu^{14} - \)\(41\!\cdots\!18\)\( \nu^{13} + \)\(66\!\cdots\!06\)\( \nu^{12} - \)\(36\!\cdots\!50\)\( \nu^{11} + \)\(16\!\cdots\!87\)\( \nu^{10} - \)\(51\!\cdots\!25\)\( \nu^{9} + \)\(10\!\cdots\!81\)\( \nu^{8} - \)\(21\!\cdots\!51\)\( \nu^{7} + \)\(31\!\cdots\!09\)\( \nu^{6} - \)\(50\!\cdots\!28\)\( \nu^{5} + \)\(20\!\cdots\!56\)\( \nu^{4} - \)\(24\!\cdots\!32\)\( \nu^{3} - \)\(95\!\cdots\!28\)\( \nu^{2} - \)\(56\!\cdots\!45\)\( \nu + \)\(50\!\cdots\!92\)\(\)\()/ \)\(27\!\cdots\!41\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(82\!\cdots\!12\)\( \nu^{19} + \)\(35\!\cdots\!77\)\( \nu^{18} - \)\(25\!\cdots\!91\)\( \nu^{17} + \)\(15\!\cdots\!62\)\( \nu^{16} - \)\(20\!\cdots\!22\)\( \nu^{15} + \)\(76\!\cdots\!03\)\( \nu^{14} - \)\(14\!\cdots\!29\)\( \nu^{13} + \)\(22\!\cdots\!79\)\( \nu^{12} - \)\(13\!\cdots\!84\)\( \nu^{11} + \)\(58\!\cdots\!91\)\( \nu^{10} - \)\(18\!\cdots\!35\)\( \nu^{9} + \)\(37\!\cdots\!39\)\( \nu^{8} - \)\(75\!\cdots\!79\)\( \nu^{7} + \)\(10\!\cdots\!89\)\( \nu^{6} - \)\(16\!\cdots\!07\)\( \nu^{5} + \)\(12\!\cdots\!34\)\( \nu^{4} - \)\(56\!\cdots\!90\)\( \nu^{3} - \)\(56\!\cdots\!67\)\( \nu^{2} - \)\(33\!\cdots\!30\)\( \nu - \)\(15\!\cdots\!04\)\(\)\()/ \)\(92\!\cdots\!47\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(33\!\cdots\!00\)\( \nu^{19} - \)\(16\!\cdots\!58\)\( \nu^{18} + \)\(27\!\cdots\!11\)\( \nu^{17} - \)\(10\!\cdots\!88\)\( \nu^{16} + \)\(94\!\cdots\!48\)\( \nu^{15} - \)\(38\!\cdots\!18\)\( \nu^{14} + \)\(10\!\cdots\!55\)\( \nu^{13} - \)\(21\!\cdots\!12\)\( \nu^{12} + \)\(82\!\cdots\!76\)\( \nu^{11} - \)\(31\!\cdots\!33\)\( \nu^{10} + \)\(10\!\cdots\!39\)\( \nu^{9} - \)\(27\!\cdots\!58\)\( \nu^{8} + \)\(63\!\cdots\!09\)\( \nu^{7} - \)\(11\!\cdots\!95\)\( \nu^{6} + \)\(21\!\cdots\!99\)\( \nu^{5} - \)\(25\!\cdots\!94\)\( \nu^{4} + \)\(37\!\cdots\!49\)\( \nu^{3} - \)\(16\!\cdots\!06\)\( \nu^{2} + \)\(28\!\cdots\!58\)\( \nu - \)\(17\!\cdots\!95\)\(\)\()/ \)\(92\!\cdots\!47\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(10\!\cdots\!00\)\( \nu^{19} + \)\(51\!\cdots\!05\)\( \nu^{18} - \)\(81\!\cdots\!96\)\( \nu^{17} + \)\(33\!\cdots\!32\)\( \nu^{16} - \)\(28\!\cdots\!41\)\( \nu^{15} + \)\(11\!\cdots\!55\)\( \nu^{14} - \)\(30\!\cdots\!60\)\( \nu^{13} + \)\(66\!\cdots\!34\)\( \nu^{12} - \)\(25\!\cdots\!20\)\( \nu^{11} + \)\(97\!\cdots\!63\)\( \nu^{10} - \)\(32\!\cdots\!28\)\( \nu^{9} + \)\(82\!\cdots\!48\)\( \nu^{8} - \)\(19\!\cdots\!01\)\( \nu^{7} + \)\(34\!\cdots\!68\)\( \nu^{6} - \)\(64\!\cdots\!87\)\( \nu^{5} + \)\(75\!\cdots\!83\)\( \nu^{4} - \)\(11\!\cdots\!02\)\( \nu^{3} + \)\(44\!\cdots\!59\)\( \nu^{2} - \)\(86\!\cdots\!46\)\( \nu - \)\(52\!\cdots\!51\)\(\)\()/ \)\(27\!\cdots\!41\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(20\!\cdots\!79\)\( \nu^{19} - \)\(10\!\cdots\!89\)\( \nu^{18} + \)\(16\!\cdots\!07\)\( \nu^{17} - \)\(66\!\cdots\!80\)\( \nu^{16} + \)\(57\!\cdots\!18\)\( \nu^{15} - \)\(23\!\cdots\!07\)\( \nu^{14} + \)\(60\!\cdots\!49\)\( \nu^{13} - \)\(13\!\cdots\!30\)\( \nu^{12} + \)\(49\!\cdots\!97\)\( \nu^{11} - \)\(19\!\cdots\!97\)\( \nu^{10} + \)\(65\!\cdots\!61\)\( \nu^{9} - \)\(16\!\cdots\!85\)\( \nu^{8} + \)\(38\!\cdots\!95\)\( \nu^{7} - \)\(69\!\cdots\!19\)\( \nu^{6} + \)\(12\!\cdots\!28\)\( \nu^{5} - \)\(15\!\cdots\!79\)\( \nu^{4} + \)\(22\!\cdots\!88\)\( \nu^{3} - \)\(88\!\cdots\!19\)\( \nu^{2} + \)\(16\!\cdots\!03\)\( \nu + \)\(73\!\cdots\!70\)\(\)\()/ \)\(27\!\cdots\!41\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(94\!\cdots\!81\)\( \nu^{19} - \)\(47\!\cdots\!49\)\( \nu^{18} + \)\(78\!\cdots\!92\)\( \nu^{17} - \)\(30\!\cdots\!90\)\( \nu^{16} + \)\(26\!\cdots\!39\)\( \nu^{15} - \)\(10\!\cdots\!30\)\( \nu^{14} + \)\(28\!\cdots\!61\)\( \nu^{13} - \)\(62\!\cdots\!46\)\( \nu^{12} + \)\(23\!\cdots\!20\)\( \nu^{11} - \)\(89\!\cdots\!72\)\( \nu^{10} + \)\(30\!\cdots\!72\)\( \nu^{9} - \)\(77\!\cdots\!57\)\( \nu^{8} + \)\(18\!\cdots\!66\)\( \nu^{7} - \)\(32\!\cdots\!15\)\( \nu^{6} + \)\(60\!\cdots\!41\)\( \nu^{5} - \)\(72\!\cdots\!65\)\( \nu^{4} + \)\(10\!\cdots\!23\)\( \nu^{3} - \)\(45\!\cdots\!56\)\( \nu^{2} + \)\(76\!\cdots\!69\)\( \nu + \)\(17\!\cdots\!37\)\(\)\()/ \)\(92\!\cdots\!47\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(96\!\cdots\!14\)\( \nu^{19} + \)\(48\!\cdots\!59\)\( \nu^{18} - \)\(78\!\cdots\!01\)\( \nu^{17} + \)\(30\!\cdots\!11\)\( \nu^{16} - \)\(26\!\cdots\!81\)\( \nu^{15} + \)\(11\!\cdots\!34\)\( \nu^{14} - \)\(28\!\cdots\!73\)\( \nu^{13} + \)\(62\!\cdots\!50\)\( \nu^{12} - \)\(23\!\cdots\!65\)\( \nu^{11} + \)\(90\!\cdots\!39\)\( \nu^{10} - \)\(30\!\cdots\!38\)\( \nu^{9} + \)\(77\!\cdots\!76\)\( \nu^{8} - \)\(18\!\cdots\!50\)\( \nu^{7} + \)\(33\!\cdots\!09\)\( \nu^{6} - \)\(60\!\cdots\!76\)\( \nu^{5} + \)\(71\!\cdots\!18\)\( \nu^{4} - \)\(10\!\cdots\!79\)\( \nu^{3} + \)\(43\!\cdots\!44\)\( \nu^{2} - \)\(76\!\cdots\!11\)\( \nu - \)\(29\!\cdots\!83\)\(\)\()/ \)\(92\!\cdots\!47\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(34\!\cdots\!00\)\( \nu^{19} - \)\(17\!\cdots\!58\)\( \nu^{18} + \)\(28\!\cdots\!61\)\( \nu^{17} - \)\(11\!\cdots\!40\)\( \nu^{16} + \)\(94\!\cdots\!82\)\( \nu^{15} - \)\(39\!\cdots\!91\)\( \nu^{14} + \)\(10\!\cdots\!32\)\( \nu^{13} - \)\(22\!\cdots\!39\)\( \nu^{12} + \)\(83\!\cdots\!93\)\( \nu^{11} - \)\(32\!\cdots\!87\)\( \nu^{10} + \)\(10\!\cdots\!96\)\( \nu^{9} - \)\(27\!\cdots\!51\)\( \nu^{8} + \)\(64\!\cdots\!99\)\( \nu^{7} - \)\(11\!\cdots\!54\)\( \nu^{6} + \)\(21\!\cdots\!43\)\( \nu^{5} - \)\(25\!\cdots\!57\)\( \nu^{4} + \)\(37\!\cdots\!43\)\( \nu^{3} - \)\(16\!\cdots\!61\)\( \nu^{2} + \)\(27\!\cdots\!95\)\( \nu + \)\(95\!\cdots\!13\)\(\)\()/ \)\(27\!\cdots\!41\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(11\!\cdots\!26\)\( \nu^{19} - \)\(57\!\cdots\!36\)\( \nu^{18} + \)\(93\!\cdots\!64\)\( \nu^{17} - \)\(36\!\cdots\!59\)\( \nu^{16} + \)\(31\!\cdots\!96\)\( \nu^{15} - \)\(13\!\cdots\!50\)\( \nu^{14} + \)\(34\!\cdots\!38\)\( \nu^{13} - \)\(74\!\cdots\!98\)\( \nu^{12} + \)\(27\!\cdots\!53\)\( \nu^{11} - \)\(10\!\cdots\!79\)\( \nu^{10} + \)\(36\!\cdots\!77\)\( \nu^{9} - \)\(92\!\cdots\!52\)\( \nu^{8} + \)\(21\!\cdots\!57\)\( \nu^{7} - \)\(39\!\cdots\!67\)\( \nu^{6} + \)\(71\!\cdots\!55\)\( \nu^{5} - \)\(85\!\cdots\!82\)\( \nu^{4} + \)\(12\!\cdots\!75\)\( \nu^{3} - \)\(52\!\cdots\!88\)\( \nu^{2} + \)\(92\!\cdots\!41\)\( \nu + \)\(24\!\cdots\!06\)\(\)\()/ \)\(92\!\cdots\!47\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(36\!\cdots\!28\)\( \nu^{19} + \)\(18\!\cdots\!26\)\( \nu^{18} - \)\(29\!\cdots\!65\)\( \nu^{17} + \)\(11\!\cdots\!78\)\( \nu^{16} - \)\(10\!\cdots\!32\)\( \nu^{15} + \)\(42\!\cdots\!75\)\( \nu^{14} - \)\(10\!\cdots\!10\)\( \nu^{13} + \)\(23\!\cdots\!62\)\( \nu^{12} - \)\(89\!\cdots\!41\)\( \nu^{11} + \)\(34\!\cdots\!07\)\( \nu^{10} - \)\(11\!\cdots\!70\)\( \nu^{9} + \)\(29\!\cdots\!83\)\( \nu^{8} - \)\(69\!\cdots\!31\)\( \nu^{7} + \)\(12\!\cdots\!64\)\( \nu^{6} - \)\(23\!\cdots\!79\)\( \nu^{5} + \)\(27\!\cdots\!11\)\( \nu^{4} - \)\(40\!\cdots\!77\)\( \nu^{3} + \)\(17\!\cdots\!53\)\( \nu^{2} - \)\(30\!\cdots\!52\)\( \nu - \)\(95\!\cdots\!01\)\(\)\()/ \)\(27\!\cdots\!41\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(37\!\cdots\!77\)\( \nu^{19} - \)\(18\!\cdots\!88\)\( \nu^{18} + \)\(29\!\cdots\!36\)\( \nu^{17} - \)\(11\!\cdots\!47\)\( \nu^{16} + \)\(10\!\cdots\!89\)\( \nu^{15} - \)\(42\!\cdots\!70\)\( \nu^{14} + \)\(11\!\cdots\!86\)\( \nu^{13} - \)\(23\!\cdots\!86\)\( \nu^{12} + \)\(89\!\cdots\!46\)\( \nu^{11} - \)\(34\!\cdots\!02\)\( \nu^{10} + \)\(11\!\cdots\!10\)\( \nu^{9} - \)\(29\!\cdots\!51\)\( \nu^{8} + \)\(69\!\cdots\!71\)\( \nu^{7} - \)\(12\!\cdots\!41\)\( \nu^{6} + \)\(23\!\cdots\!78\)\( \nu^{5} - \)\(27\!\cdots\!21\)\( \nu^{4} + \)\(40\!\cdots\!88\)\( \nu^{3} - \)\(16\!\cdots\!21\)\( \nu^{2} + \)\(30\!\cdots\!02\)\( \nu + \)\(87\!\cdots\!30\)\(\)\()/ \)\(27\!\cdots\!41\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(22\!\cdots\!49\)\( \nu^{19} + \)\(11\!\cdots\!80\)\( \nu^{18} - \)\(18\!\cdots\!37\)\( \nu^{17} + \)\(72\!\cdots\!38\)\( \nu^{16} - \)\(62\!\cdots\!70\)\( \nu^{15} + \)\(25\!\cdots\!44\)\( \nu^{14} - \)\(67\!\cdots\!02\)\( \nu^{13} + \)\(14\!\cdots\!94\)\( \nu^{12} - \)\(54\!\cdots\!26\)\( \nu^{11} + \)\(21\!\cdots\!59\)\( \nu^{10} - \)\(72\!\cdots\!60\)\( \nu^{9} + \)\(18\!\cdots\!57\)\( \nu^{8} - \)\(42\!\cdots\!83\)\( \nu^{7} + \)\(77\!\cdots\!51\)\( \nu^{6} - \)\(14\!\cdots\!43\)\( \nu^{5} + \)\(16\!\cdots\!06\)\( \nu^{4} - \)\(24\!\cdots\!10\)\( \nu^{3} + \)\(10\!\cdots\!96\)\( \nu^{2} - \)\(18\!\cdots\!71\)\( \nu - \)\(29\!\cdots\!36\)\(\)\()/ \)\(92\!\cdots\!47\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(22\!\cdots\!73\)\( \nu^{19} - \)\(11\!\cdots\!20\)\( \nu^{18} + \)\(18\!\cdots\!23\)\( \nu^{17} - \)\(72\!\cdots\!40\)\( \nu^{16} + \)\(63\!\cdots\!30\)\( \nu^{15} - \)\(26\!\cdots\!18\)\( \nu^{14} + \)\(67\!\cdots\!65\)\( \nu^{13} - \)\(14\!\cdots\!66\)\( \nu^{12} + \)\(54\!\cdots\!34\)\( \nu^{11} - \)\(21\!\cdots\!81\)\( \nu^{10} + \)\(72\!\cdots\!72\)\( \nu^{9} - \)\(18\!\cdots\!81\)\( \nu^{8} + \)\(42\!\cdots\!49\)\( \nu^{7} - \)\(77\!\cdots\!91\)\( \nu^{6} + \)\(14\!\cdots\!33\)\( \nu^{5} - \)\(16\!\cdots\!76\)\( \nu^{4} + \)\(24\!\cdots\!38\)\( \nu^{3} - \)\(10\!\cdots\!49\)\( \nu^{2} + \)\(18\!\cdots\!15\)\( \nu + \)\(28\!\cdots\!40\)\(\)\()/ \)\(92\!\cdots\!47\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(94\!\cdots\!41\)\( \nu^{19} - \)\(47\!\cdots\!70\)\( \nu^{18} + \)\(75\!\cdots\!74\)\( \nu^{17} - \)\(30\!\cdots\!58\)\( \nu^{16} + \)\(26\!\cdots\!81\)\( \nu^{15} - \)\(10\!\cdots\!01\)\( \nu^{14} + \)\(27\!\cdots\!31\)\( \nu^{13} - \)\(60\!\cdots\!49\)\( \nu^{12} + \)\(22\!\cdots\!30\)\( \nu^{11} - \)\(88\!\cdots\!31\)\( \nu^{10} + \)\(30\!\cdots\!31\)\( \nu^{9} - \)\(75\!\cdots\!48\)\( \nu^{8} + \)\(17\!\cdots\!73\)\( \nu^{7} - \)\(31\!\cdots\!67\)\( \nu^{6} + \)\(58\!\cdots\!89\)\( \nu^{5} - \)\(69\!\cdots\!59\)\( \nu^{4} + \)\(10\!\cdots\!75\)\( \nu^{3} - \)\(42\!\cdots\!17\)\( \nu^{2} + \)\(76\!\cdots\!73\)\( \nu + \)\(26\!\cdots\!33\)\(\)\()/ \)\(27\!\cdots\!41\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(10\!\cdots\!06\)\( \nu^{19} - \)\(51\!\cdots\!03\)\( \nu^{18} + \)\(82\!\cdots\!63\)\( \nu^{17} - \)\(32\!\cdots\!38\)\( \nu^{16} + \)\(28\!\cdots\!51\)\( \nu^{15} - \)\(11\!\cdots\!60\)\( \nu^{14} + \)\(30\!\cdots\!77\)\( \nu^{13} - \)\(65\!\cdots\!97\)\( \nu^{12} + \)\(24\!\cdots\!74\)\( \nu^{11} - \)\(95\!\cdots\!21\)\( \nu^{10} + \)\(32\!\cdots\!77\)\( \nu^{9} - \)\(82\!\cdots\!40\)\( \nu^{8} + \)\(19\!\cdots\!80\)\( \nu^{7} - \)\(34\!\cdots\!18\)\( \nu^{6} + \)\(63\!\cdots\!19\)\( \nu^{5} - \)\(75\!\cdots\!55\)\( \nu^{4} + \)\(11\!\cdots\!55\)\( \nu^{3} - \)\(46\!\cdots\!85\)\( \nu^{2} + \)\(81\!\cdots\!67\)\( \nu + \)\(31\!\cdots\!04\)\(\)\()/ \)\(27\!\cdots\!41\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(10\!\cdots\!38\)\( \nu^{19} + \)\(52\!\cdots\!04\)\( \nu^{18} - \)\(84\!\cdots\!94\)\( \nu^{17} + \)\(33\!\cdots\!76\)\( \nu^{16} - \)\(28\!\cdots\!10\)\( \nu^{15} + \)\(11\!\cdots\!92\)\( \nu^{14} - \)\(30\!\cdots\!48\)\( \nu^{13} + \)\(66\!\cdots\!75\)\( \nu^{12} - \)\(25\!\cdots\!09\)\( \nu^{11} + \)\(97\!\cdots\!08\)\( \nu^{10} - \)\(33\!\cdots\!79\)\( \nu^{9} + \)\(83\!\cdots\!90\)\( \nu^{8} - \)\(19\!\cdots\!28\)\( \nu^{7} + \)\(35\!\cdots\!72\)\( \nu^{6} - \)\(64\!\cdots\!30\)\( \nu^{5} + \)\(76\!\cdots\!59\)\( \nu^{4} - \)\(11\!\cdots\!87\)\( \nu^{3} + \)\(47\!\cdots\!30\)\( \nu^{2} - \)\(82\!\cdots\!20\)\( \nu - \)\(26\!\cdots\!71\)\(\)\()/ \)\(27\!\cdots\!41\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(13\!\cdots\!17\)\( \nu^{19} + \)\(66\!\cdots\!24\)\( \nu^{18} - \)\(10\!\cdots\!94\)\( \nu^{17} + \)\(42\!\cdots\!83\)\( \nu^{16} - \)\(36\!\cdots\!36\)\( \nu^{15} + \)\(15\!\cdots\!26\)\( \nu^{14} - \)\(39\!\cdots\!97\)\( \nu^{13} + \)\(85\!\cdots\!24\)\( \nu^{12} - \)\(31\!\cdots\!67\)\( \nu^{11} + \)\(12\!\cdots\!42\)\( \nu^{10} - \)\(42\!\cdots\!88\)\( \nu^{9} + \)\(10\!\cdots\!99\)\( \nu^{8} - \)\(24\!\cdots\!33\)\( \nu^{7} + \)\(45\!\cdots\!53\)\( \nu^{6} - \)\(82\!\cdots\!91\)\( \nu^{5} + \)\(98\!\cdots\!15\)\( \nu^{4} - \)\(14\!\cdots\!73\)\( \nu^{3} + \)\(60\!\cdots\!44\)\( \nu^{2} - \)\(10\!\cdots\!19\)\( \nu - \)\(30\!\cdots\!77\)\(\)\()/ \)\(27\!\cdots\!41\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{19} - \beta_{14} - 7 \beta_{13} - 8 \beta_{12} + 6 \beta_{11} - 7 \beta_{10} - 6 \beta_{9} - 7 \beta_{8} - 8 \beta_{7} - 6 \beta_{6} - 7 \beta_{5} + \beta_{4} - 6\)
\(\nu^{3}\)\(=\)\(-\beta_{19} - \beta_{18} - 2 \beta_{17} + \beta_{16} + 9 \beta_{15} + \beta_{14} - 9 \beta_{13} + 2 \beta_{12} + 10 \beta_{11} - 11 \beta_{10} + 3 \beta_{9} - 6 \beta_{8} + 2 \beta_{7} - 10 \beta_{6} - 10 \beta_{5} - \beta_{4} + 9 \beta_{3} - \beta_{2} + 3 \beta_{1} - 5\)
\(\nu^{4}\)\(=\)\(2 \beta_{19} + 2 \beta_{18} + 17 \beta_{17} + 14 \beta_{16} - 4 \beta_{15} - 15 \beta_{14} - 35 \beta_{13} + \beta_{12} - 41 \beta_{11} - 15 \beta_{10} - 26 \beta_{9} - 21 \beta_{8} - 17 \beta_{7} - 13 \beta_{6} + 3 \beta_{5} + 14 \beta_{4} - 2 \beta_{3} + 2 \beta_{1} - 39\)
\(\nu^{5}\)\(=\)\(-116 \beta_{19} - 20 \beta_{18} + 8 \beta_{17} - 15 \beta_{16} - 30 \beta_{14} - 134 \beta_{13} - 38 \beta_{12} + 68 \beta_{11} - 175 \beta_{10} + 30 \beta_{9} - 190 \beta_{8} - 163 \beta_{7} - 15 \beta_{6} - 198 \beta_{5} + 20 \beta_{4} + 30 \beta_{3} + 22 \beta_{2} - 13 \beta_{1} - 45\)
\(\nu^{6}\)\(=\)\(125 \beta_{19} - 180 \beta_{18} - 38 \beta_{17} + 213 \beta_{16} + 83 \beta_{15} + 38 \beta_{14} + 153 \beta_{13} + 825 \beta_{12} - 243 \beta_{11} - 15 \beta_{10} + 456 \beta_{9} + 516 \beta_{8} + 348 \beta_{7} - 83 \beta_{6} + 303 \beta_{5} - 42 \beta_{4} + 213 \beta_{3} - 42 \beta_{2} + 168 \beta_{1} + 90\)
\(\nu^{7}\)\(=\)\(-330 \beta_{19} + 206 \beta_{18} + 1439 \beta_{17} + 366 \beta_{16} - 1076 \beta_{15} - 1076 \beta_{14} - 1433 \beta_{13} + 133 \beta_{12} - 1436 \beta_{11} - 600 \beta_{10} + 133 \beta_{9} - 1335 \beta_{8} - 1400 \beta_{7} + 1436 \beta_{6} + 1076 \beta_{5} + 363 \beta_{4} - 206 \beta_{3} + 330 \beta_{2} - 33 \beta_{1} - 1129\)
\(\nu^{8}\)\(=\)\(-3573 \beta_{19} - 2931 \beta_{18} - 2038 \beta_{17} - 577 \beta_{16} + 577 \beta_{15} + 1303 \beta_{14} + 2113 \beta_{13} + 3480 \beta_{12} + 2468 \beta_{11} - 3404 \beta_{10} + 7622 \beta_{9} - 577 \beta_{8} - 3404 \beta_{7} + 1275 \beta_{6} - 5976 \beta_{5} + 2270 \beta_{3} + 2270 \beta_{2} + 65 \beta_{1} - 157\)
\(\nu^{9}\)\(=\)\(15909 \beta_{19} - 4206 \beta_{18} + 5032 \beta_{17} + 13021 \beta_{16} - 4206 \beta_{15} + 9658 \beta_{13} + 26644 \beta_{12} - 20308 \beta_{11} + 4775 \beta_{10} + 10843 \beta_{9} + 36217 \beta_{8} + 16249 \beta_{7} + 10843 \beta_{6} + 42553 \beta_{5} - 5452 \beta_{4} + 5032 \beta_{3} - 2888 \beta_{2} + 5452 \beta_{1} + 4775\)
\(\nu^{10}\)\(=\)\(-47812 \beta_{19} + 8147 \beta_{18} + 47243 \beta_{17} - 18689 \beta_{16} - 39665 \beta_{15} - 18689 \beta_{14} - 6106 \beta_{13} - 4039 \beta_{12} - 53180 \beta_{11} + 10542 \beta_{10} + 40301 \beta_{9} - 69971 \beta_{8} - 68017 \beta_{7} + 119863 \beta_{6} + 21612 \beta_{5} + 12043 \beta_{4} - 12043 \beta_{3} + 39665 \beta_{2} - 11474 \beta_{1} - 38894\)
\(\nu^{11}\)\(=\)\(-162998 \beta_{18} - 204415 \beta_{17} + 41417 \beta_{15} + 162998 \beta_{14} + 359840 \beta_{13} + 196842 \beta_{12} + 67251 \beta_{11} - 25834 \beta_{10} + 223068 \beta_{9} + 386066 \beta_{8} - 35972 \beta_{6} - 35972 \beta_{5} - 73385 \beta_{4} + 78559 \beta_{3} + 47193 \beta_{2} + 122122\)
\(\nu^{12}\)\(=\)\(644714 \beta_{19} + 259040 \beta_{18} + 644714 \beta_{17} + 259040 \beta_{16} - 359840 \beta_{15} - 112236 \beta_{14} + 523730 \beta_{13} + 894161 \beta_{12} - 1208484 \beta_{11} + 894161 \beta_{10} + 276126 \beta_{9} + 1332280 \beta_{8} + 960880 \beta_{7} + 1332280 \beta_{6} + 2619336 \beta_{5} - 359840 \beta_{4} - 112236 \beta_{2} + 163539 \beta_{1} + 359840\)
\(\nu^{13}\)\(=\)\(-2684859 \beta_{19} - 2084170 \beta_{16} - 1040965 \beta_{15} + 522184 \beta_{14} + 2603639 \beta_{13} - 1327328 \beta_{12} - 803650 \beta_{11} + 1705591 \beta_{10} + 1198596 \beta_{9} - 2424999 \beta_{8} - 3411498 \beta_{7} + 3204328 \beta_{6} - 2424999 \beta_{5} + 600689 \beta_{4} - 1107684 \beta_{3} + 2084170 \beta_{2} - 1107684 \beta_{1} - 1280520\)
\(\nu^{14}\)\(=\)\(8094417 \beta_{19} - 3545157 \beta_{18} - 8662246 \beta_{17} + 1539425 \beta_{16} + 2889314 \beta_{15} + 7122821 \beta_{14} + 17195169 \beta_{13} + 6760119 \beta_{12} + 4428739 \beta_{11} + 3109103 \beta_{10} + 2830735 \beta_{9} + 27121641 \beta_{8} + 10925152 \beta_{7} - 4985314 \beta_{6} + 10072348 \beta_{5} - 7122821 \beta_{4} + 2889314 \beta_{3} - 3545157 \beta_{2} + 567829 \beta_{1} + 12533051\)
\(\nu^{15}\)\(=\)\(14502816 \beta_{19} + 27022373 \beta_{18} + 35757971 \beta_{17} - 5702188 \beta_{16} - 15422365 \beta_{15} - 8735598 \beta_{14} + 19162884 \beta_{13} + 14437786 \beta_{12} - 53923595 \beta_{11} + 66578470 \beta_{10} - 633457 \beta_{9} + 13460696 \beta_{8} + 39442655 \beta_{7} + 67384577 \beta_{6} + 88818204 \beta_{5} - 5702188 \beta_{4} - 14502816 \beta_{3} + 919549 \beta_{1} + 5133761\)
\(\nu^{16}\)\(=\)\(-91422781 \beta_{19} - 21199320 \beta_{18} - 106064565 \beta_{17} - 95153849 \beta_{16} + 57729430 \beta_{14} + 122397657 \beta_{13} - 175615895 \beta_{12} + 117886465 \beta_{11} + 11288856 \beta_{10} - 57729430 \beta_{9} - 83864993 \beta_{8} - 163328177 \beta_{7} - 71120314 \beta_{6} - 279681346 \beta_{5} + 21199320 \beta_{4} - 57729430 \beta_{3} + 48335135 \beta_{2} - 68018034 \beta_{1} - 13390884\)
\(\nu^{17}\)\(=\)\(566784182 \beta_{19} + 61209850 \beta_{18} - 199649430 \beta_{17} + 126798067 \beta_{16} + 213040314 \beta_{15} + 199649430 \beta_{14} + 464502282 \beta_{13} + 305016469 \beta_{12} + 169525899 \beta_{11} + 351095099 \beta_{10} - 42727832 \beta_{9} + 1123411792 \beta_{8} + 1010004609 \beta_{7} - 213040314 \beta_{6} + 819572282 \beta_{5} - 353743868 \beta_{4} + 126798067 \beta_{3} - 353743868 \beta_{2} + 113407183 \beta_{1} + 692774215\)
\(\nu^{18}\)\(=\)\(-293860150 \beta_{19} + 1270863889 \beta_{18} + 1290373476 \beta_{17} - 735502047 \beta_{16} - 631463966 \beta_{15} - 631463966 \beta_{14} - 363758600 \beta_{13} - 1359292200 \beta_{12} - 1568458403 \beta_{11} + 2392178780 \beta_{10} - 1359292200 \beta_{9} - 2271139003 \beta_{8} + 1290760 \beta_{7} + 1568458403 \beta_{6} + 631463966 \beta_{5} + 658909510 \beta_{4} - 1270863889 \beta_{3} + 293860150 \beta_{2} - 365049360 \beta_{1} - 1000275114\)
\(\nu^{19}\)\(=\)\(-2287059920 \beta_{19} - 1833231470 \beta_{18} - 7591068386 \beta_{17} - 2726258213 \beta_{16} + 2726258213 \beta_{15} + 2927750603 \beta_{14} + 1526471841 \beta_{13} - 9610161858 \beta_{12} + 12085389652 \beta_{11} - 5193827541 \beta_{10} - 4787101056 \beta_{9} - 2726258213 \beta_{8} - 5193827541 \beta_{7} - 11241900938 \beta_{6} - 16644879335 \beta_{5} - 640690683 \beta_{3} - 640690683 \beta_{2} - 2272429763 \beta_{1} + 2167162524\)

Character values

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

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{7}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.63205 1.88348i
−2.42025 + 2.79312i
−2.75487 1.77045i
2.31649 + 1.48872i
1.01145 + 2.21477i
−0.764548 1.67413i
−2.75487 + 1.77045i
2.31649 1.48872i
0.271421 + 1.88778i
−0.162761 1.13203i
3.40070 0.998536i
−0.0296794 + 0.00871465i
0.271421 1.88778i
−0.162761 + 1.13203i
1.01145 2.21477i
−0.764548 + 1.67413i
3.40070 + 0.998536i
−0.0296794 0.00871465i
1.63205 + 1.88348i
−2.42025 2.79312i
−0.142315 + 0.989821i −0.415415 0.909632i −0.959493 0.281733i 0.654861 + 0.755750i 0.959493 0.281733i −2.73875 1.76009i 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 −1.25705 0.807858i 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 −0.257432 + 1.79048i −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 0.455355 3.16706i −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 −0.783059 0.229927i 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 4.58950 + 1.34760i 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 −0.257432 1.79048i −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 0.455355 + 3.16706i −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 −1.00994 2.21146i −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 1.42213 + 3.11402i −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 −1.67997 + 1.93879i −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 2.25922 2.60728i −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 −1.00994 + 2.21146i −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 1.42213 3.11402i −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 −0.783059 + 0.229927i 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 4.58950 1.34760i 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 −1.67997 1.93879i −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 2.25922 + 2.60728i −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 −2.73875 + 1.76009i 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 −1.25705 + 0.807858i 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.d 20
23.c even 11 1 inner 690.2.m.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.m.d 20 1.a even 1 1 trivial
690.2.m.d 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

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$3$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$5$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$7$ \( 65545216 + 255056384 T + 434779136 T^{2} + 449152000 T^{3} + 335493312 T^{4} + 191237376 T^{5} + 86736848 T^{6} + 31813760 T^{7} + 10690240 T^{8} + 3028564 T^{9} + 882069 T^{10} + 167315 T^{11} + 42925 T^{12} + 3183 T^{13} + 1384 T^{14} - 461 T^{15} + 16 T^{16} - 19 T^{17} + 4 T^{18} - 2 T^{19} + T^{20} \)
$11$ \( 7745089 + 88900152 T + 373169808 T^{2} + 691226899 T^{3} + 694390686 T^{4} + 371553732 T^{5} + 99944306 T^{6} + 39656298 T^{7} + 28608525 T^{8} + 5598824 T^{9} + 3093740 T^{10} + 363335 T^{11} + 197651 T^{12} + 21850 T^{13} + 20172 T^{14} + 1189 T^{15} + 1490 T^{16} + 3 T^{17} + 59 T^{18} - 2 T^{19} + T^{20} \)
$13$ \( 2374681 - 5119202 T + 53874572 T^{2} + 143602492 T^{3} + 227626932 T^{4} + 96944936 T^{5} + 38303725 T^{6} + 23470986 T^{7} + 7931388 T^{8} + 2435972 T^{9} + 1234201 T^{10} + 327514 T^{11} + 90347 T^{12} + 34474 T^{13} + 8926 T^{14} + 1848 T^{15} + 900 T^{16} + 121 T^{17} + 36 T^{18} + 11 T^{19} + T^{20} \)
$17$ \( 13980121 - 90453888 T + 801964883 T^{2} - 640907374 T^{3} + 2287188222 T^{4} - 2080437536 T^{5} + 676852712 T^{6} + 85615338 T^{7} - 169670955 T^{8} + 41207496 T^{9} + 14967404 T^{10} - 5917846 T^{11} + 836766 T^{12} + 208680 T^{13} - 58122 T^{14} - 2223 T^{15} + 6380 T^{16} - 1855 T^{17} + 287 T^{18} - 24 T^{19} + T^{20} \)
$19$ \( 1814078464 + 9070392320 T + 20408382720 T^{2} + 23418103808 T^{3} + 16398857024 T^{4} + 3354077408 T^{5} - 1284191392 T^{6} + 136805504 T^{7} + 219087924 T^{8} - 109421510 T^{9} - 7786229 T^{10} - 2776829 T^{11} + 9716058 T^{12} - 4226893 T^{13} + 1118766 T^{14} - 213543 T^{15} + 32186 T^{16} - 3707 T^{17} + 341 T^{18} - 22 T^{19} + T^{20} \)
$23$ \( 41426511213649 + 39625358552186 T + 19029569423283 T^{2} + 6741554385060 T^{3} + 2156142723285 T^{4} + 639321950190 T^{5} + 171107939927 T^{6} + 42557489260 T^{7} + 10076879209 T^{8} + 2248235418 T^{9} + 476884299 T^{10} + 97749366 T^{11} + 19048921 T^{12} + 3497780 T^{13} + 611447 T^{14} + 99330 T^{15} + 14565 T^{16} + 1980 T^{17} + 243 T^{18} + 22 T^{19} + T^{20} \)
$29$ \( 40157569 + 2185643974 T + 43311855100 T^{2} + 84252324102 T^{3} + 80293096345 T^{4} + 36631398693 T^{5} + 26851181822 T^{6} - 9198639018 T^{7} + 4496199828 T^{8} - 597567619 T^{9} + 163480714 T^{10} - 14555068 T^{11} + 1698355 T^{12} + 736497 T^{13} - 182872 T^{14} + 11109 T^{15} + 2783 T^{16} - 780 T^{17} + 118 T^{18} - 14 T^{19} + T^{20} \)
$31$ \( 13256881 + 32040800 T + 79790183 T^{2} + 53062614 T^{3} + 347949294 T^{4} + 372342553 T^{5} + 642498560 T^{6} + 839698739 T^{7} + 707535301 T^{8} + 442022339 T^{9} + 201369917 T^{10} + 61082039 T^{11} + 12118837 T^{12} + 1513140 T^{13} + 137756 T^{14} + 17038 T^{15} + 4008 T^{16} + 237 T^{17} - 2 T^{18} + 8 T^{19} + T^{20} \)
$37$ \( 148035889 - 476289382 T + 1247251337 T^{2} - 1991275554 T^{3} + 2090619109 T^{4} - 1243746620 T^{5} + 418967889 T^{6} - 201746220 T^{7} + 267717024 T^{8} - 257047703 T^{9} + 166840961 T^{10} - 81033667 T^{11} + 30726227 T^{12} - 9165694 T^{13} + 2097565 T^{14} - 345977 T^{15} + 39254 T^{16} - 3368 T^{17} + 282 T^{18} - 20 T^{19} + T^{20} \)
$41$ \( 286557184 - 3562463744 T + 17069466624 T^{2} - 19593503104 T^{3} + 1575792320 T^{4} + 46279466624 T^{5} + 19210832000 T^{6} - 427119672 T^{7} + 4108742500 T^{8} + 688532136 T^{9} + 141271625 T^{10} + 8686845 T^{11} + 22093085 T^{12} + 7946963 T^{13} + 2049005 T^{14} + 369397 T^{15} + 47774 T^{16} + 4823 T^{17} + 372 T^{18} + 21 T^{19} + T^{20} \)
$43$ \( 3604441369 - 988689316 T + 1238767653 T^{2} + 810290165 T^{3} + 1865830290 T^{4} - 3031808706 T^{5} + 3839668560 T^{6} - 2712969524 T^{7} + 1490954675 T^{8} - 532007564 T^{9} + 128981535 T^{10} - 10622997 T^{11} - 4574998 T^{12} + 2034781 T^{13} - 337104 T^{14} - 21381 T^{15} + 22247 T^{16} - 4766 T^{17} + 541 T^{18} - 34 T^{19} + T^{20} \)
$47$ \( ( -10400203 + 92631 T + 3491422 T^{2} + 221639 T^{3} - 319594 T^{4} - 24628 T^{5} + 11460 T^{6} + 776 T^{7} - 174 T^{8} - 7 T^{9} + T^{10} )^{2} \)
$53$ \( 474092839936 - 320773370368 T - 333045195776 T^{2} - 123095364480 T^{3} + 174771056256 T^{4} + 233863949792 T^{5} + 170057754832 T^{6} + 82239591720 T^{7} + 29817024196 T^{8} + 8116643062 T^{9} + 1715703265 T^{10} + 267886080 T^{11} + 29519978 T^{12} + 1041414 T^{13} - 242666 T^{14} - 57068 T^{15} + 185 T^{16} + 297 T^{17} + 3 T^{18} + T^{20} \)
$59$ \( 6968743441 + 37061336840 T + 115169499566 T^{2} + 229292510546 T^{3} + 314832161974 T^{4} + 317294328439 T^{5} + 245460953309 T^{6} + 149880247187 T^{7} + 73271264792 T^{8} + 28808086725 T^{9} + 9100588366 T^{10} + 2306399895 T^{11} + 471393888 T^{12} + 79283254 T^{13} + 11391681 T^{14} + 1436709 T^{15} + 155015 T^{16} + 13147 T^{17} + 852 T^{18} + 40 T^{19} + T^{20} \)
$61$ \( 2318871110656 + 4061520755712 T + 21567893871360 T^{2} + 22682564913664 T^{3} + 43837724344128 T^{4} + 23255443492128 T^{5} + 6037136361280 T^{6} + 554579019248 T^{7} - 6827846436 T^{8} + 2440029130 T^{9} + 5193082027 T^{10} + 816533091 T^{11} + 108954920 T^{12} + 6927057 T^{13} + 1589024 T^{14} + 48555 T^{15} - 8657 T^{16} - 2753 T^{17} - 61 T^{18} + 19 T^{19} + T^{20} \)
$67$ \( 977272622041 + 4574846593827 T + 3563379887610 T^{2} - 6134875842435 T^{3} + 1997843389025 T^{4} - 1214834180099 T^{5} + 695787049932 T^{6} - 19250790002 T^{7} + 55040673654 T^{8} - 4388850829 T^{9} + 2005205830 T^{10} - 4553516 T^{11} + 85534778 T^{12} - 4269984 T^{13} + 931872 T^{14} - 209661 T^{15} + 49918 T^{16} - 4817 T^{17} + 356 T^{18} - 18 T^{19} + T^{20} \)
$71$ \( 1143633470464 - 6268082611712 T + 31079882914048 T^{2} - 43459229570816 T^{3} + 246438086540096 T^{4} - 2947978554784 T^{5} + 23085575365712 T^{6} + 8314308440824 T^{7} + 1681164028232 T^{8} + 221328188554 T^{9} + 27769046581 T^{10} + 9181622516 T^{11} + 3408294557 T^{12} + 831836769 T^{13} + 137268084 T^{14} + 16052180 T^{15} + 1354617 T^{16} + 81639 T^{17} + 3360 T^{18} + 85 T^{19} + T^{20} \)
$73$ \( 369268184187904 + 200434852824064 T + 173496412263936 T^{2} - 25742358113280 T^{3} - 5740562579136 T^{4} - 12174428932832 T^{5} + 3350155918944 T^{6} - 44705013664 T^{7} + 301939491904 T^{8} - 130499507886 T^{9} + 12496881211 T^{10} + 2159017927 T^{11} - 496100955 T^{12} + 42119288 T^{13} + 3104492 T^{14} - 1195624 T^{15} + 172348 T^{16} - 15127 T^{17} + 955 T^{18} - 39 T^{19} + T^{20} \)
$79$ \( 188504457241 - 218330702428 T + 5219727707231 T^{2} - 4718969434110 T^{3} + 82355268671944 T^{4} - 82661976739215 T^{5} + 32428933648502 T^{6} - 3855123256659 T^{7} + 466552195201 T^{8} - 156781282529 T^{9} + 15681167799 T^{10} - 3629313417 T^{11} + 576191628 T^{12} + 105733920 T^{13} + 5706931 T^{14} + 550842 T^{15} + 79440 T^{16} + 4640 T^{17} + 313 T^{18} + 28 T^{19} + T^{20} \)
$83$ \( 2376925225984 - 1345176180736 T + 2390023174400 T^{2} + 4023908902016 T^{3} - 1004077140480 T^{4} - 985382799648 T^{5} + 704795121168 T^{6} + 291316563688 T^{7} + 181461651232 T^{8} + 6322987404 T^{9} + 3379860341 T^{10} + 149147181 T^{11} + 136039366 T^{12} - 45644637 T^{13} + 11719399 T^{14} - 1716907 T^{15} + 193895 T^{16} - 16986 T^{17} + 1127 T^{18} - 49 T^{19} + T^{20} \)
$89$ \( 48271830245254144 - 64025904303779328 T + 42724961217155328 T^{2} - 6901305282319232 T^{3} - 2011131327385664 T^{4} + 866858217098336 T^{5} + 9979231619568 T^{6} - 11832855204312 T^{7} + 5509564254212 T^{8} - 297650370534 T^{9} + 41540830013 T^{10} + 1342470206 T^{11} + 76368817 T^{12} + 42587242 T^{13} + 57296 T^{14} + 202412 T^{15} + 24937 T^{16} - 1081 T^{17} + 313 T^{18} - 3 T^{19} + T^{20} \)
$97$ \( 14460297228952576 + 17395490796035072 T + 38712234676912640 T^{2} + 26652335889180288 T^{3} + 10503581892957760 T^{4} + 2894212700576896 T^{5} + 525968512823360 T^{6} + 50847559322008 T^{7} + 1035180875496 T^{8} - 585638547472 T^{9} + 14718269809 T^{10} + 2067336953 T^{11} + 1203594826 T^{12} - 215813601 T^{13} + 12118531 T^{14} + 224977 T^{15} - 26951 T^{16} - 4942 T^{17} + 769 T^{18} - 43 T^{19} + T^{20} \)
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