Properties

Label 170.2.n.a
Level $170$
Weight $2$
Character orbit 170.n
Analytic conductor $1.357$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.n (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.35745683436\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 16 x^{15} + 52 x^{14} + 992 x^{13} + 6181 x^{12} + 8952 x^{11} + 6244 x^{10} - 11448 x^{9} - 14520 x^{8} + 27936 x^{7} + 27880 x^{6} - 121104 x^{5} + 187460 x^{4} - 142208 x^{3} + 73856 x^{2} - 19456 x + 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 16 x^{15} + 52 x^{14} + 992 x^{13} + 6181 x^{12} + 8952 x^{11} + 6244 x^{10} - 11448 x^{9} - 14520 x^{8} + 27936 x^{7} + 27880 x^{6} - 121104 x^{5} + 187460 x^{4} - 142208 x^{3} + 73856 x^{2} - 19456 x + 2048\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(24\!\cdots\!67\)\( \nu^{19} + \)\(64\!\cdots\!72\)\( \nu^{18} + \)\(11\!\cdots\!32\)\( \nu^{17} + \)\(10\!\cdots\!52\)\( \nu^{16} + \)\(12\!\cdots\!32\)\( \nu^{15} - \)\(35\!\cdots\!00\)\( \nu^{14} + \)\(25\!\cdots\!04\)\( \nu^{13} + \)\(25\!\cdots\!68\)\( \nu^{12} + \)\(22\!\cdots\!75\)\( \nu^{11} + \)\(73\!\cdots\!24\)\( \nu^{10} + \)\(15\!\cdots\!92\)\( \nu^{9} + \)\(19\!\cdots\!76\)\( \nu^{8} + \)\(14\!\cdots\!76\)\( \nu^{7} + \)\(45\!\cdots\!88\)\( \nu^{6} + \)\(72\!\cdots\!88\)\( \nu^{5} - \)\(61\!\cdots\!20\)\( \nu^{4} + \)\(73\!\cdots\!40\)\( \nu^{3} - \)\(44\!\cdots\!36\)\( \nu^{2} + \)\(50\!\cdots\!36\)\( \nu - \)\(24\!\cdots\!36\)\(\)\()/ \)\(51\!\cdots\!40\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(47\!\cdots\!67\)\( \nu^{19} - \)\(23\!\cdots\!52\)\( \nu^{18} - \)\(34\!\cdots\!72\)\( \nu^{17} - \)\(39\!\cdots\!12\)\( \nu^{16} - \)\(14\!\cdots\!92\)\( \nu^{15} + \)\(82\!\cdots\!80\)\( \nu^{14} + \)\(14\!\cdots\!76\)\( \nu^{13} - \)\(53\!\cdots\!08\)\( \nu^{12} - \)\(53\!\cdots\!75\)\( \nu^{11} - \)\(22\!\cdots\!44\)\( \nu^{10} - \)\(48\!\cdots\!72\)\( \nu^{9} - \)\(65\!\cdots\!76\)\( \nu^{8} - \)\(31\!\cdots\!76\)\( \nu^{7} + \)\(28\!\cdots\!72\)\( \nu^{6} + \)\(25\!\cdots\!72\)\( \nu^{5} - \)\(10\!\cdots\!20\)\( \nu^{4} - \)\(30\!\cdots\!40\)\( \nu^{3} - \)\(13\!\cdots\!44\)\( \nu^{2} + \)\(63\!\cdots\!84\)\( \nu - \)\(14\!\cdots\!04\)\(\)\()/ \)\(27\!\cdots\!60\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(42\!\cdots\!91\)\( \nu^{19} - \)\(28\!\cdots\!16\)\( \nu^{18} - \)\(10\!\cdots\!56\)\( \nu^{17} + \)\(34\!\cdots\!24\)\( \nu^{16} + \)\(36\!\cdots\!24\)\( \nu^{15} + \)\(64\!\cdots\!80\)\( \nu^{14} + \)\(22\!\cdots\!88\)\( \nu^{13} - \)\(55\!\cdots\!64\)\( \nu^{12} - \)\(54\!\cdots\!55\)\( \nu^{11} - \)\(22\!\cdots\!32\)\( \nu^{10} - \)\(33\!\cdots\!16\)\( \nu^{9} - \)\(19\!\cdots\!88\)\( \nu^{8} + \)\(37\!\cdots\!72\)\( \nu^{7} + \)\(44\!\cdots\!16\)\( \nu^{6} - \)\(84\!\cdots\!44\)\( \nu^{5} - \)\(76\!\cdots\!80\)\( \nu^{4} + \)\(23\!\cdots\!60\)\( \nu^{3} - \)\(31\!\cdots\!72\)\( \nu^{2} + \)\(16\!\cdots\!92\)\( \nu - \)\(49\!\cdots\!32\)\(\)\()/ \)\(10\!\cdots\!80\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(15\!\cdots\!03\)\( \nu^{19} + \)\(15\!\cdots\!36\)\( \nu^{18} + \)\(17\!\cdots\!92\)\( \nu^{17} - \)\(19\!\cdots\!44\)\( \nu^{16} - \)\(18\!\cdots\!72\)\( \nu^{15} + \)\(24\!\cdots\!36\)\( \nu^{14} - \)\(82\!\cdots\!88\)\( \nu^{13} - \)\(15\!\cdots\!76\)\( \nu^{12} - \)\(92\!\cdots\!75\)\( \nu^{11} - \)\(12\!\cdots\!08\)\( \nu^{10} - \)\(70\!\cdots\!80\)\( \nu^{9} + \)\(19\!\cdots\!68\)\( \nu^{8} + \)\(19\!\cdots\!44\)\( \nu^{7} - \)\(51\!\cdots\!24\)\( \nu^{6} - \)\(48\!\cdots\!92\)\( \nu^{5} + \)\(18\!\cdots\!48\)\( \nu^{4} - \)\(29\!\cdots\!60\)\( \nu^{3} + \)\(22\!\cdots\!12\)\( \nu^{2} - \)\(10\!\cdots\!40\)\( \nu + \)\(17\!\cdots\!44\)\(\)\()/ \)\(15\!\cdots\!76\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(38\!\cdots\!43\)\( \nu^{19} + \)\(12\!\cdots\!98\)\( \nu^{18} + \)\(10\!\cdots\!08\)\( \nu^{17} - \)\(16\!\cdots\!52\)\( \nu^{16} - \)\(13\!\cdots\!32\)\( \nu^{15} - \)\(62\!\cdots\!80\)\( \nu^{14} + \)\(18\!\cdots\!76\)\( \nu^{13} + \)\(39\!\cdots\!92\)\( \nu^{12} + \)\(25\!\cdots\!15\)\( \nu^{11} + \)\(42\!\cdots\!26\)\( \nu^{10} + \)\(36\!\cdots\!88\)\( \nu^{9} - \)\(36\!\cdots\!36\)\( \nu^{8} - \)\(72\!\cdots\!56\)\( \nu^{7} + \)\(86\!\cdots\!32\)\( \nu^{6} + \)\(14\!\cdots\!72\)\( \nu^{5} - \)\(42\!\cdots\!40\)\( \nu^{4} + \)\(59\!\cdots\!20\)\( \nu^{3} - \)\(32\!\cdots\!24\)\( \nu^{2} + \)\(14\!\cdots\!64\)\( \nu - \)\(19\!\cdots\!44\)\(\)\()/ \)\(25\!\cdots\!20\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(11\!\cdots\!32\)\( \nu^{19} + \)\(24\!\cdots\!67\)\( \nu^{18} + \)\(64\!\cdots\!72\)\( \nu^{17} + \)\(11\!\cdots\!32\)\( \nu^{16} + \)\(10\!\cdots\!52\)\( \nu^{15} - \)\(18\!\cdots\!80\)\( \nu^{14} + \)\(60\!\cdots\!64\)\( \nu^{13} + \)\(11\!\cdots\!48\)\( \nu^{12} + \)\(73\!\cdots\!60\)\( \nu^{11} + \)\(10\!\cdots\!39\)\( \nu^{10} + \)\(81\!\cdots\!32\)\( \nu^{9} - \)\(11\!\cdots\!44\)\( \nu^{8} - \)\(15\!\cdots\!64\)\( \nu^{7} + \)\(34\!\cdots\!28\)\( \nu^{6} + \)\(33\!\cdots\!48\)\( \nu^{5} - \)\(14\!\cdots\!40\)\( \nu^{4} + \)\(22\!\cdots\!00\)\( \nu^{3} - \)\(16\!\cdots\!16\)\( \nu^{2} + \)\(87\!\cdots\!56\)\( \nu - \)\(17\!\cdots\!56\)\(\)\()/ \)\(51\!\cdots\!40\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(74\!\cdots\!99\)\( \nu^{19} + \)\(31\!\cdots\!44\)\( \nu^{18} + \)\(10\!\cdots\!84\)\( \nu^{17} + \)\(86\!\cdots\!64\)\( \nu^{16} - \)\(12\!\cdots\!16\)\( \nu^{15} - \)\(11\!\cdots\!40\)\( \nu^{14} + \)\(33\!\cdots\!08\)\( \nu^{13} + \)\(75\!\cdots\!16\)\( \nu^{12} + \)\(49\!\cdots\!55\)\( \nu^{11} + \)\(87\!\cdots\!68\)\( \nu^{10} + \)\(81\!\cdots\!64\)\( \nu^{9} - \)\(56\!\cdots\!48\)\( \nu^{8} - \)\(13\!\cdots\!68\)\( \nu^{7} + \)\(15\!\cdots\!16\)\( \nu^{6} + \)\(27\!\cdots\!76\)\( \nu^{5} - \)\(79\!\cdots\!20\)\( \nu^{4} + \)\(10\!\cdots\!20\)\( \nu^{3} - \)\(59\!\cdots\!32\)\( \nu^{2} + \)\(29\!\cdots\!52\)\( \nu - \)\(33\!\cdots\!32\)\(\)\()/ \)\(20\!\cdots\!60\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(11\!\cdots\!25\)\( \nu^{19} - \)\(70\!\cdots\!88\)\( \nu^{18} - \)\(14\!\cdots\!52\)\( \nu^{17} + \)\(33\!\cdots\!72\)\( \nu^{16} - \)\(61\!\cdots\!28\)\( \nu^{15} + \)\(17\!\cdots\!60\)\( \nu^{14} - \)\(47\!\cdots\!36\)\( \nu^{13} - \)\(11\!\cdots\!00\)\( \nu^{12} - \)\(76\!\cdots\!77\)\( \nu^{11} - \)\(14\!\cdots\!84\)\( \nu^{10} - \)\(14\!\cdots\!04\)\( \nu^{9} + \)\(73\!\cdots\!68\)\( \nu^{8} + \)\(23\!\cdots\!68\)\( \nu^{7} - \)\(20\!\cdots\!20\)\( \nu^{6} - \)\(50\!\cdots\!28\)\( \nu^{5} + \)\(11\!\cdots\!56\)\( \nu^{4} - \)\(12\!\cdots\!96\)\( \nu^{3} + \)\(48\!\cdots\!84\)\( \nu^{2} - \)\(17\!\cdots\!00\)\( \nu - \)\(17\!\cdots\!00\)\(\)\()/ \)\(20\!\cdots\!16\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(17\!\cdots\!31\)\( \nu^{19} + \)\(30\!\cdots\!06\)\( \nu^{18} - \)\(31\!\cdots\!72\)\( \nu^{17} - \)\(35\!\cdots\!84\)\( \nu^{16} + \)\(38\!\cdots\!88\)\( \nu^{15} - \)\(27\!\cdots\!52\)\( \nu^{14} + \)\(84\!\cdots\!40\)\( \nu^{13} + \)\(17\!\cdots\!28\)\( \nu^{12} + \)\(10\!\cdots\!63\)\( \nu^{11} + \)\(17\!\cdots\!62\)\( \nu^{10} + \)\(13\!\cdots\!80\)\( \nu^{9} - \)\(18\!\cdots\!28\)\( \nu^{8} - \)\(28\!\cdots\!56\)\( \nu^{7} + \)\(44\!\cdots\!28\)\( \nu^{6} + \)\(58\!\cdots\!28\)\( \nu^{5} - \)\(19\!\cdots\!40\)\( \nu^{4} + \)\(28\!\cdots\!64\)\( \nu^{3} - \)\(18\!\cdots\!28\)\( \nu^{2} + \)\(81\!\cdots\!12\)\( \nu - \)\(12\!\cdots\!56\)\(\)\()/ \)\(30\!\cdots\!52\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(48\!\cdots\!23\)\( \nu^{19} - \)\(24\!\cdots\!14\)\( \nu^{18} + \)\(87\!\cdots\!88\)\( \nu^{17} + \)\(54\!\cdots\!72\)\( \nu^{16} + \)\(27\!\cdots\!68\)\( \nu^{15} + \)\(77\!\cdots\!64\)\( \nu^{14} - \)\(24\!\cdots\!20\)\( \nu^{13} - \)\(48\!\cdots\!72\)\( \nu^{12} - \)\(30\!\cdots\!63\)\( \nu^{11} - \)\(44\!\cdots\!90\)\( \nu^{10} - \)\(26\!\cdots\!48\)\( \nu^{9} + \)\(65\!\cdots\!04\)\( \nu^{8} + \)\(83\!\cdots\!56\)\( \nu^{7} - \)\(13\!\cdots\!80\)\( \nu^{6} - \)\(15\!\cdots\!24\)\( \nu^{5} + \)\(59\!\cdots\!16\)\( \nu^{4} - \)\(83\!\cdots\!68\)\( \nu^{3} + \)\(54\!\cdots\!00\)\( \nu^{2} - \)\(23\!\cdots\!80\)\( \nu + \)\(48\!\cdots\!24\)\(\)\()/ \)\(68\!\cdots\!72\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(10\!\cdots\!29\)\( \nu^{19} - \)\(15\!\cdots\!04\)\( \nu^{18} + \)\(19\!\cdots\!12\)\( \nu^{17} - \)\(73\!\cdots\!52\)\( \nu^{16} - \)\(12\!\cdots\!52\)\( \nu^{15} + \)\(16\!\cdots\!32\)\( \nu^{14} - \)\(50\!\cdots\!76\)\( \nu^{13} - \)\(10\!\cdots\!68\)\( \nu^{12} - \)\(64\!\cdots\!97\)\( \nu^{11} - \)\(10\!\cdots\!84\)\( \nu^{10} - \)\(77\!\cdots\!28\)\( \nu^{9} + \)\(10\!\cdots\!84\)\( \nu^{8} + \)\(15\!\cdots\!24\)\( \nu^{7} - \)\(28\!\cdots\!20\)\( \nu^{6} - \)\(33\!\cdots\!80\)\( \nu^{5} + \)\(12\!\cdots\!44\)\( \nu^{4} - \)\(16\!\cdots\!72\)\( \nu^{3} + \)\(10\!\cdots\!16\)\( \nu^{2} - \)\(46\!\cdots\!60\)\( \nu + \)\(44\!\cdots\!56\)\(\)\()/ \)\(13\!\cdots\!44\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(96\!\cdots\!58\)\( \nu^{19} - \)\(29\!\cdots\!53\)\( \nu^{18} - \)\(11\!\cdots\!16\)\( \nu^{17} + \)\(27\!\cdots\!04\)\( \nu^{16} + \)\(71\!\cdots\!44\)\( \nu^{15} + \)\(15\!\cdots\!68\)\( \nu^{14} - \)\(45\!\cdots\!32\)\( \nu^{13} - \)\(96\!\cdots\!72\)\( \nu^{12} - \)\(62\!\cdots\!46\)\( \nu^{11} - \)\(10\!\cdots\!05\)\( \nu^{10} - \)\(93\!\cdots\!16\)\( \nu^{9} + \)\(84\!\cdots\!96\)\( \nu^{8} + \)\(17\!\cdots\!56\)\( \nu^{7} - \)\(20\!\cdots\!96\)\( \nu^{6} - \)\(33\!\cdots\!48\)\( \nu^{5} + \)\(10\!\cdots\!24\)\( \nu^{4} - \)\(14\!\cdots\!16\)\( \nu^{3} + \)\(97\!\cdots\!32\)\( \nu^{2} - \)\(49\!\cdots\!52\)\( \nu + \)\(87\!\cdots\!56\)\(\)\()/ \)\(10\!\cdots\!08\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(48\!\cdots\!91\)\( \nu^{19} - \)\(18\!\cdots\!21\)\( \nu^{18} - \)\(11\!\cdots\!96\)\( \nu^{17} - \)\(48\!\cdots\!76\)\( \nu^{16} + \)\(11\!\cdots\!44\)\( \nu^{15} + \)\(77\!\cdots\!80\)\( \nu^{14} - \)\(22\!\cdots\!12\)\( \nu^{13} - \)\(48\!\cdots\!64\)\( \nu^{12} - \)\(31\!\cdots\!15\)\( \nu^{11} - \)\(55\!\cdots\!17\)\( \nu^{10} - \)\(53\!\cdots\!56\)\( \nu^{9} + \)\(31\!\cdots\!52\)\( \nu^{8} + \)\(80\!\cdots\!12\)\( \nu^{7} - \)\(95\!\cdots\!84\)\( \nu^{6} - \)\(15\!\cdots\!24\)\( \nu^{5} + \)\(51\!\cdots\!60\)\( \nu^{4} - \)\(73\!\cdots\!40\)\( \nu^{3} + \)\(46\!\cdots\!08\)\( \nu^{2} - \)\(23\!\cdots\!48\)\( \nu + \)\(37\!\cdots\!48\)\(\)\()/ \)\(51\!\cdots\!40\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(42\!\cdots\!55\)\( \nu^{19} + \)\(73\!\cdots\!04\)\( \nu^{18} + \)\(18\!\cdots\!24\)\( \nu^{17} + \)\(26\!\cdots\!36\)\( \nu^{16} + \)\(90\!\cdots\!60\)\( \nu^{15} - \)\(68\!\cdots\!84\)\( \nu^{14} + \)\(21\!\cdots\!84\)\( \nu^{13} + \)\(42\!\cdots\!36\)\( \nu^{12} + \)\(27\!\cdots\!23\)\( \nu^{11} + \)\(43\!\cdots\!44\)\( \nu^{10} + \)\(34\!\cdots\!48\)\( \nu^{9} - \)\(41\!\cdots\!64\)\( \nu^{8} - \)\(66\!\cdots\!80\)\( \nu^{7} + \)\(10\!\cdots\!28\)\( \nu^{6} + \)\(13\!\cdots\!80\)\( \nu^{5} - \)\(50\!\cdots\!12\)\( \nu^{4} + \)\(72\!\cdots\!08\)\( \nu^{3} - \)\(47\!\cdots\!20\)\( \nu^{2} + \)\(21\!\cdots\!84\)\( \nu - \)\(27\!\cdots\!84\)\(\)\()/ \)\(41\!\cdots\!32\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(24\!\cdots\!51\)\( \nu^{19} - \)\(52\!\cdots\!70\)\( \nu^{18} + \)\(12\!\cdots\!84\)\( \nu^{17} + \)\(23\!\cdots\!72\)\( \nu^{16} + \)\(13\!\cdots\!44\)\( \nu^{15} + \)\(39\!\cdots\!04\)\( \nu^{14} - \)\(11\!\cdots\!44\)\( \nu^{13} - \)\(24\!\cdots\!24\)\( \nu^{12} - \)\(15\!\cdots\!31\)\( \nu^{11} - \)\(25\!\cdots\!98\)\( \nu^{10} - \)\(18\!\cdots\!40\)\( \nu^{9} + \)\(27\!\cdots\!40\)\( \nu^{8} + \)\(45\!\cdots\!56\)\( \nu^{7} - \)\(59\!\cdots\!88\)\( \nu^{6} - \)\(86\!\cdots\!92\)\( \nu^{5} + \)\(27\!\cdots\!84\)\( \nu^{4} - \)\(38\!\cdots\!36\)\( \nu^{3} + \)\(24\!\cdots\!08\)\( \nu^{2} - \)\(10\!\cdots\!76\)\( \nu + \)\(18\!\cdots\!16\)\(\)\()/ \)\(20\!\cdots\!16\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(10\!\cdots\!13\)\( \nu^{19} + \)\(82\!\cdots\!48\)\( \nu^{18} + \)\(93\!\cdots\!38\)\( \nu^{17} + \)\(74\!\cdots\!08\)\( \nu^{16} - \)\(55\!\cdots\!32\)\( \nu^{15} - \)\(16\!\cdots\!60\)\( \nu^{14} + \)\(53\!\cdots\!96\)\( \nu^{13} + \)\(10\!\cdots\!92\)\( \nu^{12} + \)\(65\!\cdots\!25\)\( \nu^{11} + \)\(10\!\cdots\!96\)\( \nu^{10} + \)\(79\!\cdots\!78\)\( \nu^{9} - \)\(10\!\cdots\!76\)\( \nu^{8} - \)\(15\!\cdots\!76\)\( \nu^{7} + \)\(27\!\cdots\!32\)\( \nu^{6} + \)\(28\!\cdots\!72\)\( \nu^{5} - \)\(12\!\cdots\!60\)\( \nu^{4} + \)\(19\!\cdots\!60\)\( \nu^{3} - \)\(14\!\cdots\!64\)\( \nu^{2} + \)\(72\!\cdots\!24\)\( \nu - \)\(14\!\cdots\!64\)\(\)\()/ \)\(85\!\cdots\!40\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(49\!\cdots\!77\)\( \nu^{19} - \)\(21\!\cdots\!32\)\( \nu^{18} - \)\(68\!\cdots\!42\)\( \nu^{17} + \)\(48\!\cdots\!88\)\( \nu^{16} + \)\(21\!\cdots\!08\)\( \nu^{15} + \)\(79\!\cdots\!60\)\( \nu^{14} - \)\(22\!\cdots\!84\)\( \nu^{13} - \)\(50\!\cdots\!68\)\( \nu^{12} - \)\(32\!\cdots\!05\)\( \nu^{11} - \)\(58\!\cdots\!44\)\( \nu^{10} - \)\(54\!\cdots\!02\)\( \nu^{9} + \)\(38\!\cdots\!84\)\( \nu^{8} + \)\(94\!\cdots\!44\)\( \nu^{7} - \)\(98\!\cdots\!68\)\( \nu^{6} - \)\(18\!\cdots\!88\)\( \nu^{5} + \)\(52\!\cdots\!80\)\( \nu^{4} - \)\(69\!\cdots\!00\)\( \nu^{3} + \)\(40\!\cdots\!96\)\( \nu^{2} - \)\(19\!\cdots\!96\)\( \nu + \)\(22\!\cdots\!96\)\(\)\()/ \)\(25\!\cdots\!20\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(13\!\cdots\!01\)\( \nu^{19} - \)\(23\!\cdots\!76\)\( \nu^{18} + \)\(33\!\cdots\!04\)\( \nu^{17} + \)\(12\!\cdots\!24\)\( \nu^{16} - \)\(37\!\cdots\!56\)\( \nu^{15} + \)\(22\!\cdots\!00\)\( \nu^{14} - \)\(68\!\cdots\!12\)\( \nu^{13} - \)\(13\!\cdots\!44\)\( \nu^{12} - \)\(88\!\cdots\!25\)\( \nu^{11} - \)\(13\!\cdots\!12\)\( \nu^{10} - \)\(10\!\cdots\!96\)\( \nu^{9} + \)\(14\!\cdots\!12\)\( \nu^{8} + \)\(23\!\cdots\!32\)\( \nu^{7} - \)\(36\!\cdots\!84\)\( \nu^{6} - \)\(46\!\cdots\!24\)\( \nu^{5} + \)\(16\!\cdots\!60\)\( \nu^{4} - \)\(23\!\cdots\!00\)\( \nu^{3} + \)\(15\!\cdots\!48\)\( \nu^{2} - \)\(66\!\cdots\!28\)\( \nu + \)\(10\!\cdots\!68\)\(\)\()/ \)\(51\!\cdots\!40\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{18} - \beta_{16} + \beta_{11} + 4 \beta_{8}\)
\(\nu^{3}\)\(=\)\(\beta_{19} + \beta_{18} - \beta_{16} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 7 \beta_{6} + \beta_{4}\)
\(\nu^{4}\)\(=\)\(9 \beta_{19} - \beta_{16} + \beta_{15} + 28 \beta_{10} - 10 \beta_{9} + \beta_{8} + \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 10 \beta_{4} - \beta_{2} - \beta_{1}\)
\(\nu^{5}\)\(=\)\(13 \beta_{19} - 13 \beta_{17} + 13 \beta_{15} - 11 \beta_{14} + 13 \beta_{12} - 11 \beta_{11} + 23 \beta_{10} - 13 \beta_{9} - 4 \beta_{8} + 23 \beta_{7} + \beta_{6} - 57 \beta_{5} + 13 \beta_{4} + \beta_{2} + 4\)
\(\nu^{6}\)\(=\)\(3 \beta_{19} - 83 \beta_{17} - 2 \beta_{16} + 94 \beta_{15} + \beta_{14} - 14 \beta_{13} + 94 \beta_{12} - 2 \beta_{11} + 17 \beta_{10} - \beta_{9} + 232 \beta_{7} - 15 \beta_{6} - 51 \beta_{5} + 14 \beta_{4} - 3 \beta_{3} + 51 \beta_{2} + 15 \beta_{1} - 17\)
\(\nu^{7}\)\(=\)\(-3 \beta_{19} + 3 \beta_{18} - 146 \beta_{17} - 104 \beta_{16} + 147 \beta_{15} + 147 \beta_{14} - 150 \beta_{13} + 150 \beta_{12} - 57 \beta_{10} + 104 \beta_{9} + 57 \beta_{8} + 308 \beta_{7} - 22 \beta_{5} - 146 \beta_{3} + 499 \beta_{2} - 22 \beta_{1} - 308\)
\(\nu^{8}\)\(=\)\(-68 \beta_{18} - 68 \beta_{17} + 21 \beta_{16} + 21 \beta_{15} + 899 \beta_{14} - 899 \beta_{13} + 166 \beta_{12} - 166 \beta_{11} + 49 \beta_{9} - 233 \beta_{8} + 233 \beta_{7} + 167 \beta_{6} + 167 \beta_{5} + 49 \beta_{4} - 793 \beta_{3} + 647 \beta_{2} - 647 \beta_{1} - 2094\)
\(\nu^{9}\)\(=\)\(50 \beta_{19} - 1567 \beta_{18} + 50 \beta_{17} + 1585 \beta_{16} - 949 \beta_{15} + 1663 \beta_{14} - 1585 \beta_{13} + 20 \beta_{12} - 1663 \beta_{11} + 590 \beta_{10} + 20 \beta_{9} - 3557 \beta_{8} - 590 \beta_{7} - 343 \beta_{6} + 949 \beta_{4} - 1567 \beta_{3} + 343 \beta_{2} - 4587 \beta_{1} - 3557\)
\(\nu^{10}\)\(=\)\(-1057 \beta_{19} - 7757 \beta_{18} + 8766 \beta_{16} - 764 \beta_{15} + 1898 \beta_{14} - 345 \beta_{13} + 764 \beta_{12} - 8766 \beta_{11} - 2925 \beta_{10} + 1898 \beta_{9} - 19876 \beta_{8} - 7403 \beta_{6} - 1647 \beta_{5} - 345 \beta_{4} - 1057 \beta_{3} - 1647 \beta_{2} - 7403 \beta_{1} - 2925\)
\(\nu^{11}\)\(=\)\(-16514 \beta_{19} - 16514 \beta_{18} - 513 \beta_{17} + 17990 \beta_{16} - 596 \beta_{15} + 596 \beta_{14} + 8592 \beta_{13} + 8592 \beta_{12} - 16713 \beta_{11} - 38800 \beta_{10} + 17990 \beta_{9} - 38800 \beta_{8} + 5311 \beta_{7} - 43637 \beta_{6} + 4672 \beta_{5} - 16713 \beta_{4} + 513 \beta_{3} - 4672 \beta_{1} + 5311\)
\(\nu^{12}\)\(=\)\(-77063 \beta_{19} - 14070 \beta_{18} + 14070 \beta_{17} + 21468 \beta_{16} - 21468 \beta_{15} + 9911 \beta_{14} + 9911 \beta_{13} - 5069 \beta_{12} - 5069 \beta_{11} - 194370 \beta_{10} + 86733 \beta_{9} - 35087 \beta_{8} - 35087 \beta_{7} - 80829 \beta_{6} + 80829 \beta_{5} - 86733 \beta_{4} + 15097 \beta_{2} + 15097 \beta_{1}\)
\(\nu^{13}\)\(=\)\(-172631 \beta_{19} + 3540 \beta_{18} + 172631 \beta_{17} + 11440 \beta_{16} - 191543 \beta_{15} + 77849 \beta_{14} + 11440 \beta_{13} - 174291 \beta_{12} + 77849 \beta_{11} - 412605 \beta_{10} + 174291 \beta_{9} + 43136 \beta_{8} - 412605 \beta_{7} - 59295 \beta_{6} + 425077 \beta_{5} - 191543 \beta_{4} - 3540 \beta_{3} - 59295 \beta_{2} - 43136\)
\(\nu^{14}\)\(=\)\(-172989 \beta_{19} + 773659 \beta_{17} + 117234 \beta_{16} - 867100 \beta_{15} - 69075 \beta_{14} + 241486 \beta_{13} - 867100 \beta_{12} + 117234 \beta_{11} - 409591 \beta_{10} + 69075 \beta_{9} - 1934776 \beta_{7} + 130337 \beta_{6} + 861781 \beta_{5} - 241486 \beta_{4} + 172989 \beta_{3} - 861781 \beta_{2} - 130337 \beta_{1} + 409591\)
\(\nu^{15}\)\(=\)\(6085 \beta_{19} - 6085 \beta_{18} + 1797956 \beta_{17} + 707214 \beta_{16} - 1807851 \beta_{15} - 1807851 \beta_{14} + 2019104 \beta_{13} - 2019104 \beta_{12} + 180224 \beta_{11} + 310095 \beta_{10} - 707214 \beta_{9} - 310095 \beta_{8} - 4334562 \beta_{7} + 720730 \beta_{5} - 180224 \beta_{4} + 1797956 \beta_{3} - 4208167 \beta_{2} + 720730 \beta_{1} + 4334562\)
\(\nu^{16}\)\(=\)\(2032620 \beta_{18} + 2032620 \beta_{17} - 891059 \beta_{16} - 891059 \beta_{15} - 8732441 \beta_{14} + 8732441 \beta_{13} - 2702720 \beta_{12} + 2702720 \beta_{11} - 1317975 \beta_{9} + 4694581 \beta_{8} - 4694581 \beta_{7} - 1054423 \beta_{6} - 1054423 \beta_{5} - 1317975 \beta_{4} + 7823869 \beta_{3} - 9072035 \beta_{2} + 9072035 \beta_{1} + 19468618\)
\(\nu^{17}\)\(=\)\(330322 \beta_{19} + 18695535 \beta_{18} + 330322 \beta_{17} - 18707565 \beta_{16} + 6436269 \beta_{15} - 21155071 \beta_{14} + 18707565 \beta_{13} - 2539356 \beta_{12} + 21155071 \beta_{11} - 1770186 \beta_{10} - 2539356 \beta_{9} + 45263765 \beta_{8} + 1770186 \beta_{7} + 8509319 \beta_{6} - 6436269 \beta_{4} + 18695535 \beta_{3} - 8509319 \beta_{2} + 42121419 \beta_{1} + 45263765\)
\(\nu^{18}\)\(=\)\(23228121 \beta_{19} + 79536549 \beta_{18} - 88408294 \beta_{16} + 14388480 \beta_{15} - 30086562 \beta_{14} + 11036645 \beta_{13} - 14388480 \beta_{12} + 88408294 \beta_{11} + 53087193 \beta_{10} - 30086562 \beta_{9} + 197262636 \beta_{8} + 94846311 \beta_{6} + 7784283 \beta_{5} + 11036645 \beta_{4} + 23228121 \beta_{3} + 7784283 \beta_{2} + 94846311 \beta_{1} + 53087193\)
\(\nu^{19}\)\(=\)\(194291250 \beta_{19} + 194291250 \beta_{18} - 7913799 \beta_{17} - 220871206 \beta_{16} + 33338924 \beta_{15} - 33338924 \beta_{14} - 58575976 \beta_{13} - 58575976 \beta_{12} + 193432777 \beta_{11} + 471300144 \beta_{10} - 220871206 \beta_{9} + 471300144 \beta_{8} - 3698703 \beta_{7} + 424838813 \beta_{6} - 98388604 \beta_{5} + 193432777 \beta_{4} + 7913799 \beta_{3} + 98388604 \beta_{1} - 3698703\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/170\mathbb{Z}\right)^\times\).

\(n\) \(71\) \(137\)
\(\chi(n)\) \(-\beta_{7}\) \(-1\)

Embeddings

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

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

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
2.99334 1.23988i
0.953222 0.394838i
0.236338 0.0978946i
−1.86202 + 0.771273i
−2.32088 + 0.961341i
2.99334 + 1.23988i
0.953222 + 0.394838i
0.236338 + 0.0978946i
−1.86202 0.771273i
−2.32088 0.961341i
−0.969032 + 2.33945i
−0.826884 + 1.99627i
0.254075 0.613391i
0.355063 0.857197i
1.18678 2.86514i
−0.969032 2.33945i
−0.826884 1.99627i
0.254075 + 0.613391i
0.355063 + 0.857197i
1.18678 + 2.86514i
−0.707107 0.707107i −1.23988 + 2.99334i 1.00000i −2.18859 0.458323i 2.99334 1.23988i 1.49921 0.620992i 0.707107 0.707107i −5.30145 5.30145i 1.22349 + 1.87165i
9.2 −0.707107 0.707107i −0.394838 + 0.953222i 1.00000i 1.21037 1.88016i 0.953222 0.394838i −0.363965 + 0.150759i 0.707107 0.707107i 1.36858 + 1.36858i −2.18534 + 0.473610i
9.3 −0.707107 0.707107i −0.0978946 + 0.236338i 1.00000i −0.496623 + 2.18022i 0.236338 0.0978946i −3.48449 + 1.44332i 0.707107 0.707107i 2.07505 + 2.07505i 1.89281 1.19048i
9.4 −0.707107 0.707107i 0.771273 1.86202i 1.00000i 1.71825 + 1.43095i −1.86202 + 0.771273i 2.47378 1.02468i 0.707107 0.707107i −0.750924 0.750924i −0.203147 2.22682i
9.5 −0.707107 0.707107i 0.961341 2.32088i 1.00000i −1.24340 1.85848i −2.32088 + 0.961341i −0.124542 + 0.0515871i 0.707107 0.707107i −2.34100 2.34100i −0.434925 + 2.19336i
19.1 −0.707107 + 0.707107i −1.23988 2.99334i 1.00000i −2.18859 + 0.458323i 2.99334 + 1.23988i 1.49921 + 0.620992i 0.707107 + 0.707107i −5.30145 + 5.30145i 1.22349 1.87165i
19.2 −0.707107 + 0.707107i −0.394838 0.953222i 1.00000i 1.21037 + 1.88016i 0.953222 + 0.394838i −0.363965 0.150759i 0.707107 + 0.707107i 1.36858 1.36858i −2.18534 0.473610i
19.3 −0.707107 + 0.707107i −0.0978946 0.236338i 1.00000i −0.496623 2.18022i 0.236338 + 0.0978946i −3.48449 1.44332i 0.707107 + 0.707107i 2.07505 2.07505i 1.89281 + 1.19048i
19.4 −0.707107 + 0.707107i 0.771273 + 1.86202i 1.00000i 1.71825 1.43095i −1.86202 0.771273i 2.47378 + 1.02468i 0.707107 + 0.707107i −0.750924 + 0.750924i −0.203147 + 2.22682i
19.5 −0.707107 + 0.707107i 0.961341 + 2.32088i 1.00000i −1.24340 + 1.85848i −2.32088 0.961341i −0.124542 0.0515871i 0.707107 + 0.707107i −2.34100 + 2.34100i −0.434925 2.19336i
49.1 0.707107 0.707107i −2.33945 + 0.969032i 1.00000i −0.309710 + 2.21452i −0.969032 + 2.33945i −1.26758 + 3.06021i −0.707107 0.707107i 2.41268 2.41268i 1.34690 + 1.78490i
49.2 0.707107 0.707107i −1.99627 + 0.826884i 1.00000i −1.27343 1.83804i −0.826884 + 1.99627i 1.32795 3.20595i −0.707107 0.707107i 1.18005 1.18005i −2.20014 0.399242i
49.3 0.707107 0.707107i 0.613391 0.254075i 1.00000i 0.384345 + 2.20279i 0.254075 0.613391i 1.97319 4.76369i −0.707107 0.707107i −1.80963 + 1.80963i 1.82938 + 1.28583i
49.4 0.707107 0.707107i 0.857197 0.355063i 1.00000i 2.23422 0.0908004i 0.355063 0.857197i −0.939960 + 2.26926i −0.707107 0.707107i −1.51260 + 1.51260i 1.51563 1.64404i
49.5 0.707107 0.707107i 2.86514 1.18678i 1.00000i −2.03543 + 0.925748i 1.18678 2.86514i −1.09360 + 2.64018i −0.707107 0.707107i 4.67924 4.67924i −0.784666 + 2.09387i
59.1 0.707107 + 0.707107i −2.33945 0.969032i 1.00000i −0.309710 2.21452i −0.969032 2.33945i −1.26758 3.06021i −0.707107 + 0.707107i 2.41268 + 2.41268i 1.34690 1.78490i
59.2 0.707107 + 0.707107i −1.99627 0.826884i 1.00000i −1.27343 + 1.83804i −0.826884 1.99627i 1.32795 + 3.20595i −0.707107 + 0.707107i 1.18005 + 1.18005i −2.20014 + 0.399242i
59.3 0.707107 + 0.707107i 0.613391 + 0.254075i 1.00000i 0.384345 2.20279i 0.254075 + 0.613391i 1.97319 + 4.76369i −0.707107 + 0.707107i −1.80963 1.80963i 1.82938 1.28583i
59.4 0.707107 + 0.707107i 0.857197 + 0.355063i 1.00000i 2.23422 + 0.0908004i 0.355063 + 0.857197i −0.939960 2.26926i −0.707107 + 0.707107i −1.51260 1.51260i 1.51563 + 1.64404i
59.5 0.707107 + 0.707107i 2.86514 + 1.18678i 1.00000i −2.03543 0.925748i 1.18678 + 2.86514i −1.09360 2.64018i −0.707107 + 0.707107i 4.67924 + 4.67924i −0.784666 2.09387i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.m even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.2.n.a 20
5.b even 2 1 170.2.n.b yes 20
5.c odd 4 1 850.2.l.h 20
5.c odd 4 1 850.2.l.i 20
17.d even 8 1 170.2.n.b yes 20
85.k odd 8 1 850.2.l.i 20
85.m even 8 1 inner 170.2.n.a 20
85.n odd 8 1 850.2.l.h 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.n.a 20 1.a even 1 1 trivial
170.2.n.a 20 85.m even 8 1 inner
170.2.n.b yes 20 5.b even 2 1
170.2.n.b yes 20 17.d even 8 1
850.2.l.h 20 5.c odd 4 1
850.2.l.h 20 85.n odd 8 1
850.2.l.i 20 5.c odd 4 1
850.2.l.i 20 85.k odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{20} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(170, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{5} \)
$3$ \( 1 - 8 T^{3} + 32 T^{5} + 28 T^{6} - 72 T^{7} - 234 T^{8} - 24 T^{9} + 1224 T^{10} + 728 T^{11} - 2488 T^{12} - 6728 T^{13} + 1256 T^{14} + 27240 T^{15} + 14713 T^{16} - 60616 T^{17} - 122316 T^{18} + 66920 T^{19} + 485264 T^{20} + 200760 T^{21} - 1100844 T^{22} - 1636632 T^{23} + 1191753 T^{24} + 6619320 T^{25} + 915624 T^{26} - 14714136 T^{27} - 16323768 T^{28} + 14329224 T^{29} + 72275976 T^{30} - 4251528 T^{31} - 124357194 T^{32} - 114791256 T^{33} + 133923132 T^{34} + 459165024 T^{35} - 1033121304 T^{37} + 3486784401 T^{40} \)
$5$ \( 1 + 4 T + 14 T^{2} + 28 T^{3} + 41 T^{4} + 32 T^{5} - 72 T^{6} - 416 T^{7} - 1550 T^{8} - 4944 T^{9} - 10916 T^{10} - 24720 T^{11} - 38750 T^{12} - 52000 T^{13} - 45000 T^{14} + 100000 T^{15} + 640625 T^{16} + 2187500 T^{17} + 5468750 T^{18} + 7812500 T^{19} + 9765625 T^{20} \)
$7$ \( 1 + 28 T^{2} + 72 T^{3} + 392 T^{4} + 2152 T^{5} + 5636 T^{6} + 31696 T^{7} + 93057 T^{8} + 321232 T^{9} + 1251096 T^{10} + 3073136 T^{11} + 11857952 T^{12} + 32094064 T^{13} + 87528552 T^{14} + 298482128 T^{15} + 656393822 T^{16} + 2152770768 T^{17} + 5629095584 T^{18} + 13238676192 T^{19} + 44114872624 T^{20} + 92670733344 T^{21} + 275825683616 T^{22} + 738400373424 T^{23} + 1576001566622 T^{24} + 5016589125296 T^{25} + 10297646614248 T^{26} + 26430841748752 T^{27} + 68358733547552 T^{28} + 124012122401552 T^{29} + 353403654122904 T^{30} + 635180624307376 T^{31} + 1288028663063457 T^{32} + 3070994073860272 T^{33} + 3822465238576964 T^{34} + 10216752369397336 T^{35} + 13027308783283592 T^{36} + 16749397007078904 T^{37} + 45595580741492572 T^{38} + 79792266297612001 T^{40} \)
$11$ \( 1 + 8 T + 32 T^{2} + 88 T^{3} + 192 T^{4} + 1336 T^{5} + 8272 T^{6} + 31784 T^{7} + 74909 T^{8} + 75696 T^{9} + 299120 T^{10} + 2946256 T^{11} + 14548096 T^{12} + 38626064 T^{13} + 40847888 T^{14} + 70277616 T^{15} + 1208780434 T^{16} + 7630937792 T^{17} + 25235168784 T^{18} + 46671777024 T^{19} + 75581355904 T^{20} + 513389547264 T^{21} + 3053455422864 T^{22} + 10156778201152 T^{23} + 17697754334194 T^{24} + 11318280334416 T^{25} + 72364525313168 T^{26} + 752712714224944 T^{27} + 3118513579240576 T^{28} + 6947117532294896 T^{29} + 7758402446651120 T^{30} + 21596952218570256 T^{31} + 235096531271793389 T^{32} + 1097269882782702904 T^{33} + 3141290623400569552 T^{34} + 5580803554339309736 T^{35} + 8822348133805854912 T^{36} + 44479338507937851848 T^{37} + \)\(17\!\cdots\!92\)\( T^{38} + \)\(48\!\cdots\!28\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \)
$13$ \( ( 1 + 12 T + 126 T^{2} + 944 T^{3} + 6370 T^{4} + 36804 T^{5} + 195036 T^{6} + 930636 T^{7} + 4101389 T^{8} + 16591780 T^{9} + 62248284 T^{10} + 215693140 T^{11} + 693134741 T^{12} + 2044607292 T^{13} + 5570423196 T^{14} + 13665067572 T^{15} + 30746773330 T^{16} + 59234600048 T^{17} + 102782070846 T^{18} + 127253992476 T^{19} + 137858491849 T^{20} )^{2} \)
$17$ \( 1 + 18 T^{2} - 16 T^{3} + 803 T^{4} - 1696 T^{5} + 11588 T^{6} - 48032 T^{7} + 238596 T^{8} - 1229744 T^{9} + 3632100 T^{10} - 20905648 T^{11} + 68954244 T^{12} - 235981216 T^{13} + 967841348 T^{14} - 2408077472 T^{15} + 19382467907 T^{16} - 6565418768 T^{17} + 125563633938 T^{18} + 2015993900449 T^{20} \)
$19$ \( 1 - 104 T^{3} - 440 T^{4} - 1768 T^{5} + 5408 T^{6} - 5432 T^{7} + 242710 T^{8} + 738648 T^{9} + 4507360 T^{10} + 7934112 T^{11} + 6690854 T^{12} - 462658112 T^{13} - 1445661280 T^{14} - 8971090888 T^{15} - 40777542279 T^{16} - 43430694488 T^{17} + 214418884704 T^{18} + 3590696280576 T^{19} + 25541206222212 T^{20} + 68223229330944 T^{21} + 77405217378144 T^{22} - 297891133493192 T^{23} - 5314170087341559 T^{24} - 22213309176685912 T^{25} - 68012408545187680 T^{26} - 413557011135896768 T^{27} + 113634540707127014 T^{28} + 2560240335200737248 T^{29} + 27634922807761915360 T^{30} + 86045296754651667912 T^{31} + \)\(53\!\cdots\!10\)\( T^{32} - \)\(22\!\cdots\!88\)\( T^{33} + \)\(43\!\cdots\!68\)\( T^{34} - \)\(26\!\cdots\!32\)\( T^{35} - \)\(12\!\cdots\!40\)\( T^{36} - \)\(56\!\cdots\!56\)\( T^{37} + \)\(37\!\cdots\!01\)\( T^{40} \)
$23$ \( 1 - 16 T + 56 T^{2} + 376 T^{3} - 2400 T^{4} - 7256 T^{5} + 72440 T^{6} + 121072 T^{7} - 2621531 T^{8} + 4314208 T^{9} + 70866208 T^{10} - 388785408 T^{11} - 779461504 T^{12} + 12408165568 T^{13} - 8863476320 T^{14} - 287308681888 T^{15} + 756303717514 T^{16} + 5606916061312 T^{17} - 30507258885872 T^{18} - 66054575985616 T^{19} + 960733835992000 T^{20} - 1519255247669168 T^{21} - 16138339950626288 T^{22} + 68219347717983104 T^{23} + 211644788612835274 T^{24} - 1849217223509055584 T^{25} - 1312112596661648480 T^{26} + 42247637876515608896 T^{27} - 61040398366850122624 T^{28} - \)\(70\!\cdots\!04\)\( T^{29} + \)\(29\!\cdots\!92\)\( T^{30} + \)\(41\!\cdots\!16\)\( T^{31} - \)\(57\!\cdots\!51\)\( T^{32} + \)\(61\!\cdots\!76\)\( T^{33} + \)\(83\!\cdots\!60\)\( T^{34} - \)\(19\!\cdots\!92\)\( T^{35} - \)\(14\!\cdots\!00\)\( T^{36} + \)\(53\!\cdots\!28\)\( T^{37} + \)\(18\!\cdots\!64\)\( T^{38} - \)\(11\!\cdots\!92\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$29$ \( 1 + 12 T + 38 T^{2} + 212 T^{3} + 2978 T^{4} + 17140 T^{5} + 92422 T^{6} + 571116 T^{7} + 2331938 T^{8} + 14922092 T^{9} + 130048102 T^{10} + 770925860 T^{11} + 3851687522 T^{12} + 19564235252 T^{13} + 133504284918 T^{14} + 966299457452 T^{15} + 4642708862081 T^{16} + 19643962031712 T^{17} + 123559244554608 T^{18} + 710977606960016 T^{19} + 3560114011124736 T^{20} + 20618350601840464 T^{21} + 103913324670425328 T^{22} + 479096589991423968 T^{23} + 3283699766681511761 T^{24} + 19819912150417132348 T^{25} + 79411462122654972678 T^{26} + \)\(33\!\cdots\!68\)\( T^{27} + \)\(19\!\cdots\!42\)\( T^{28} + \)\(11\!\cdots\!40\)\( T^{29} + \)\(54\!\cdots\!02\)\( T^{30} + \)\(18\!\cdots\!68\)\( T^{31} + \)\(82\!\cdots\!58\)\( T^{32} + \)\(58\!\cdots\!24\)\( T^{33} + \)\(27\!\cdots\!82\)\( T^{34} + \)\(14\!\cdots\!60\)\( T^{35} + \)\(74\!\cdots\!38\)\( T^{36} + \)\(15\!\cdots\!08\)\( T^{37} + \)\(79\!\cdots\!18\)\( T^{38} + \)\(73\!\cdots\!28\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$31$ \( 1 - 8 T + 4 T^{2} + 384 T^{3} - 2296 T^{4} + 6096 T^{5} + 40040 T^{6} - 153440 T^{7} + 1584090 T^{8} - 14476232 T^{9} + 89328068 T^{10} + 315411224 T^{11} - 3760050272 T^{12} + 22889881832 T^{13} + 17254141740 T^{14} - 332815622952 T^{15} + 2990654769257 T^{16} - 6114910723832 T^{17} + 38570348356996 T^{18} + 367744608569728 T^{19} - 1791803809774288 T^{20} + 11400082865661568 T^{21} + 37066104771073156 T^{22} - 182169305373679112 T^{23} + 2761932483158993897 T^{24} - 9528228724651873752 T^{25} + 15313114306745744940 T^{26} + \)\(62\!\cdots\!52\)\( T^{27} - \)\(32\!\cdots\!52\)\( T^{28} + \)\(83\!\cdots\!04\)\( T^{29} + \)\(73\!\cdots\!68\)\( T^{30} - \)\(36\!\cdots\!92\)\( T^{31} + \)\(12\!\cdots\!90\)\( T^{32} - \)\(37\!\cdots\!40\)\( T^{33} + \)\(30\!\cdots\!40\)\( T^{34} + \)\(14\!\cdots\!96\)\( T^{35} - \)\(16\!\cdots\!76\)\( T^{36} + \)\(86\!\cdots\!24\)\( T^{37} + \)\(27\!\cdots\!64\)\( T^{38} - \)\(17\!\cdots\!68\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \)
$37$ \( 1 + 8 T + 118 T^{2} + 960 T^{3} + 8114 T^{4} + 50464 T^{5} + 338982 T^{6} + 1440600 T^{7} + 4335441 T^{8} - 9800336 T^{9} - 346829856 T^{10} - 3402505696 T^{11} - 26710359232 T^{12} - 167815912224 T^{13} - 904985189600 T^{14} - 3602804409392 T^{15} - 5581747209506 T^{16} + 63717621771584 T^{17} + 1054915539916916 T^{18} + 8107624670209056 T^{19} + 59840742028993692 T^{20} + 299982112797735072 T^{21} + 1444179374146258004 T^{22} + 3227488695596044352 T^{23} - 10461092931914974466 T^{24} - \)\(24\!\cdots\!44\)\( T^{25} - \)\(23\!\cdots\!00\)\( T^{26} - \)\(15\!\cdots\!92\)\( T^{27} - \)\(93\!\cdots\!72\)\( T^{28} - \)\(44\!\cdots\!92\)\( T^{29} - \)\(16\!\cdots\!44\)\( T^{30} - \)\(17\!\cdots\!68\)\( T^{31} + \)\(28\!\cdots\!21\)\( T^{32} + \)\(35\!\cdots\!00\)\( T^{33} + \)\(30\!\cdots\!98\)\( T^{34} + \)\(16\!\cdots\!52\)\( T^{35} + \)\(10\!\cdots\!74\)\( T^{36} + \)\(43\!\cdots\!20\)\( T^{37} + \)\(19\!\cdots\!22\)\( T^{38} + \)\(49\!\cdots\!84\)\( T^{39} + \)\(23\!\cdots\!01\)\( T^{40} \)
$41$ \( 1 - 4 T + 30 T^{2} + 92 T^{3} - 334 T^{4} - 10116 T^{5} + 85854 T^{6} - 842916 T^{7} - 2808099 T^{8} + 21400784 T^{9} - 13842264 T^{10} - 1079831088 T^{11} + 13460410040 T^{12} + 2388153040 T^{13} - 195580560792 T^{14} + 2644046148048 T^{15} + 10500427183938 T^{16} - 192141210610008 T^{17} + 664554433081812 T^{18} - 1101475111191192 T^{19} - 30298126262993364 T^{20} - 45160479558838872 T^{21} + 1117116002010525972 T^{22} - 13242564376452361368 T^{23} + 29671697619711826818 T^{24} + \)\(30\!\cdots\!48\)\( T^{25} - \)\(92\!\cdots\!72\)\( T^{26} + \)\(46\!\cdots\!40\)\( T^{27} + \)\(10\!\cdots\!40\)\( T^{28} - \)\(35\!\cdots\!68\)\( T^{29} - \)\(18\!\cdots\!64\)\( T^{30} + \)\(11\!\cdots\!44\)\( T^{31} - \)\(63\!\cdots\!19\)\( T^{32} - \)\(77\!\cdots\!36\)\( T^{33} + \)\(32\!\cdots\!94\)\( T^{34} - \)\(15\!\cdots\!16\)\( T^{35} - \)\(21\!\cdots\!94\)\( T^{36} + \)\(24\!\cdots\!52\)\( T^{37} + \)\(32\!\cdots\!30\)\( T^{38} - \)\(17\!\cdots\!44\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$43$ \( 1 - 16 T + 128 T^{2} - 1488 T^{3} + 20694 T^{4} - 176960 T^{5} + 1289600 T^{6} - 12374208 T^{7} + 114689745 T^{8} - 816890480 T^{9} + 5773220224 T^{10} - 47396668080 T^{11} + 376357329952 T^{12} - 2558318885360 T^{13} + 17028142721920 T^{14} - 127218542416432 T^{15} + 963177088145694 T^{16} - 6226032087351344 T^{17} + 38642451157210368 T^{18} - 278587214299949552 T^{19} + 2002378843515302292 T^{20} - 11979250214897830736 T^{21} + 71449892189681970432 T^{22} - \)\(49\!\cdots\!08\)\( T^{23} + \)\(32\!\cdots\!94\)\( T^{24} - \)\(18\!\cdots\!76\)\( T^{25} + \)\(10\!\cdots\!80\)\( T^{26} - \)\(69\!\cdots\!20\)\( T^{27} + \)\(43\!\cdots\!52\)\( T^{28} - \)\(23\!\cdots\!40\)\( T^{29} + \)\(12\!\cdots\!76\)\( T^{30} - \)\(75\!\cdots\!60\)\( T^{31} + \)\(45\!\cdots\!45\)\( T^{32} - \)\(21\!\cdots\!44\)\( T^{33} + \)\(95\!\cdots\!00\)\( T^{34} - \)\(56\!\cdots\!20\)\( T^{35} + \)\(28\!\cdots\!94\)\( T^{36} - \)\(87\!\cdots\!84\)\( T^{37} + \)\(32\!\cdots\!72\)\( T^{38} - \)\(17\!\cdots\!12\)\( T^{39} + \)\(46\!\cdots\!01\)\( T^{40} \)
$47$ \( ( 1 - 20 T + 388 T^{2} - 4976 T^{3} + 62092 T^{4} - 635100 T^{5} + 6214238 T^{6} - 53431788 T^{7} + 439450571 T^{8} - 3289992044 T^{9} + 23554230924 T^{10} - 154629626068 T^{11} + 970746311339 T^{12} - 5547448525524 T^{13} + 30323499098078 T^{14} - 145657013945700 T^{15} + 669303038208268 T^{16} - 2520956647423888 T^{17} + 9238779224763268 T^{18} - 22382609462055340 T^{19} + 52599132235830049 T^{20} )^{2} \)
$53$ \( 1 - 44 T + 968 T^{2} - 14204 T^{3} + 153360 T^{4} - 1253852 T^{5} + 7593816 T^{6} - 29525844 T^{7} + 21121854 T^{8} + 436104500 T^{9} + 1235332696 T^{10} - 80090622156 T^{11} + 970719706210 T^{12} - 6135983063340 T^{13} + 7164913588248 T^{14} + 319649478179492 T^{15} - 4261370408835583 T^{16} + 31329479689545576 T^{17} - 136710820885404048 T^{18} + 162474207458147632 T^{19} + 1532805880285790428 T^{20} + 8611132995281824496 T^{21} - \)\(38\!\cdots\!32\)\( T^{22} + \)\(46\!\cdots\!52\)\( T^{23} - \)\(33\!\cdots\!23\)\( T^{24} + \)\(13\!\cdots\!56\)\( T^{25} + \)\(15\!\cdots\!92\)\( T^{26} - \)\(72\!\cdots\!80\)\( T^{27} + \)\(60\!\cdots\!10\)\( T^{28} - \)\(26\!\cdots\!48\)\( T^{29} + \)\(21\!\cdots\!04\)\( T^{30} + \)\(40\!\cdots\!00\)\( T^{31} + \)\(10\!\cdots\!14\)\( T^{32} - \)\(76\!\cdots\!12\)\( T^{33} + \)\(10\!\cdots\!04\)\( T^{34} - \)\(91\!\cdots\!64\)\( T^{35} + \)\(59\!\cdots\!60\)\( T^{36} - \)\(29\!\cdots\!52\)\( T^{37} + \)\(10\!\cdots\!52\)\( T^{38} - \)\(25\!\cdots\!48\)\( T^{39} + \)\(30\!\cdots\!01\)\( T^{40} \)
$59$ \( 1 - 16 T + 128 T^{2} - 640 T^{3} + 6032 T^{4} - 73144 T^{5} + 603008 T^{6} - 4225576 T^{7} + 33076134 T^{8} - 244311232 T^{9} + 1417281312 T^{10} - 16887395168 T^{11} + 211053947230 T^{12} - 1522774288880 T^{13} + 6460849018400 T^{14} - 56720710039568 T^{15} + 808791885028057 T^{16} - 6447237579834824 T^{17} + 35399211288359360 T^{18} - 220142009338344872 T^{19} + 1853897520463942116 T^{20} - 12988378550962347448 T^{21} + \)\(12\!\cdots\!60\)\( T^{22} - \)\(13\!\cdots\!96\)\( T^{23} + \)\(98\!\cdots\!77\)\( T^{24} - \)\(40\!\cdots\!32\)\( T^{25} + \)\(27\!\cdots\!00\)\( T^{26} - \)\(37\!\cdots\!20\)\( T^{27} + \)\(30\!\cdots\!30\)\( T^{28} - \)\(14\!\cdots\!52\)\( T^{29} + \)\(72\!\cdots\!12\)\( T^{30} - \)\(73\!\cdots\!88\)\( T^{31} + \)\(58\!\cdots\!54\)\( T^{32} - \)\(44\!\cdots\!04\)\( T^{33} + \)\(37\!\cdots\!88\)\( T^{34} - \)\(26\!\cdots\!56\)\( T^{35} + \)\(13\!\cdots\!12\)\( T^{36} - \)\(81\!\cdots\!60\)\( T^{37} + \)\(96\!\cdots\!88\)\( T^{38} - \)\(70\!\cdots\!24\)\( T^{39} + \)\(26\!\cdots\!01\)\( T^{40} \)
$61$ \( 1 - 8 T + 10 T^{2} + 320 T^{3} - 2126 T^{4} - 27296 T^{5} - 72510 T^{6} + 8840 T^{7} - 3520990 T^{8} - 69652728 T^{9} + 659172698 T^{10} + 1929507872 T^{11} + 77537561570 T^{12} + 128446570208 T^{13} + 1861119756730 T^{14} + 37614663351992 T^{15} - 71694040127423 T^{16} - 1493114370031936 T^{17} + 3398743476006728 T^{18} - 110814064642641248 T^{19} - 1163583925604399136 T^{20} - 6759657943201116128 T^{21} + 12646724474221034888 T^{22} - \)\(33\!\cdots\!16\)\( T^{23} - \)\(99\!\cdots\!43\)\( T^{24} + \)\(31\!\cdots\!92\)\( T^{25} + \)\(95\!\cdots\!30\)\( T^{26} + \)\(40\!\cdots\!68\)\( T^{27} + \)\(14\!\cdots\!70\)\( T^{28} + \)\(22\!\cdots\!52\)\( T^{29} + \)\(47\!\cdots\!98\)\( T^{30} - \)\(30\!\cdots\!08\)\( T^{31} - \)\(93\!\cdots\!90\)\( T^{32} + \)\(14\!\cdots\!40\)\( T^{33} - \)\(71\!\cdots\!10\)\( T^{34} - \)\(16\!\cdots\!96\)\( T^{35} - \)\(78\!\cdots\!86\)\( T^{36} + \)\(71\!\cdots\!20\)\( T^{37} + \)\(13\!\cdots\!10\)\( T^{38} - \)\(66\!\cdots\!28\)\( T^{39} + \)\(50\!\cdots\!01\)\( T^{40} \)
$67$ \( 1 - 532 T^{2} + 148646 T^{4} - 29200180 T^{6} + 4500891213 T^{8} - 575578725168 T^{10} + 63179028972648 T^{12} - 6089051557526448 T^{14} + 523435323978922754 T^{16} - 40558419543281652984 T^{18} + \)\(28\!\cdots\!68\)\( T^{20} - \)\(18\!\cdots\!76\)\( T^{22} + \)\(10\!\cdots\!34\)\( T^{24} - \)\(55\!\cdots\!12\)\( T^{26} + \)\(25\!\cdots\!68\)\( T^{28} - \)\(10\!\cdots\!32\)\( T^{30} + \)\(36\!\cdots\!93\)\( T^{32} - \)\(10\!\cdots\!20\)\( T^{34} + \)\(24\!\cdots\!26\)\( T^{36} - \)\(39\!\cdots\!88\)\( T^{38} + \)\(33\!\cdots\!01\)\( T^{40} \)
$71$ \( 1 - 8 T + 4 T^{2} + 1064 T^{3} - 7736 T^{4} + 33056 T^{5} + 411864 T^{6} - 3498384 T^{7} + 43320906 T^{8} - 175160784 T^{9} + 1711157444 T^{10} + 23184409480 T^{11} - 282606722400 T^{12} + 2190349172904 T^{13} + 5955655055116 T^{14} - 92105865793280 T^{15} + 1484388456856825 T^{16} - 5521262001586648 T^{17} + 13570667323190324 T^{18} + 605072991295826840 T^{19} - 4474880001515114064 T^{20} + 42960182382003705640 T^{21} + 68409733976202423284 T^{22} - \)\(19\!\cdots\!28\)\( T^{23} + \)\(37\!\cdots\!25\)\( T^{24} - \)\(16\!\cdots\!80\)\( T^{25} + \)\(76\!\cdots\!36\)\( T^{26} + \)\(19\!\cdots\!64\)\( T^{27} - \)\(18\!\cdots\!00\)\( T^{28} + \)\(10\!\cdots\!80\)\( T^{29} + \)\(55\!\cdots\!44\)\( T^{30} - \)\(40\!\cdots\!64\)\( T^{31} + \)\(71\!\cdots\!46\)\( T^{32} - \)\(40\!\cdots\!24\)\( T^{33} + \)\(34\!\cdots\!84\)\( T^{34} + \)\(19\!\cdots\!56\)\( T^{35} - \)\(32\!\cdots\!56\)\( T^{36} + \)\(31\!\cdots\!24\)\( T^{37} + \)\(84\!\cdots\!44\)\( T^{38} - \)\(11\!\cdots\!48\)\( T^{39} + \)\(10\!\cdots\!01\)\( T^{40} \)
$73$ \( 1 + 60 T + 1682 T^{2} + 29988 T^{3} + 398642 T^{4} + 4449180 T^{5} + 44616114 T^{6} + 400755436 T^{7} + 3146402174 T^{8} + 22577269012 T^{9} + 172416257546 T^{10} + 1548661495372 T^{11} + 15112755685578 T^{12} + 149613932704668 T^{13} + 1492299482027146 T^{14} + 14514912884993716 T^{15} + 131595396462860353 T^{16} + 1114942726481802512 T^{17} + 9296763449815475432 T^{18} + 79328198294011301400 T^{19} + \)\(68\!\cdots\!92\)\( T^{20} + \)\(57\!\cdots\!00\)\( T^{21} + \)\(49\!\cdots\!28\)\( T^{22} + \)\(43\!\cdots\!04\)\( T^{23} + \)\(37\!\cdots\!73\)\( T^{24} + \)\(30\!\cdots\!88\)\( T^{25} + \)\(22\!\cdots\!94\)\( T^{26} + \)\(16\!\cdots\!96\)\( T^{27} + \)\(12\!\cdots\!18\)\( T^{28} + \)\(91\!\cdots\!36\)\( T^{29} + \)\(74\!\cdots\!54\)\( T^{30} + \)\(70\!\cdots\!24\)\( T^{31} + \)\(72\!\cdots\!54\)\( T^{32} + \)\(67\!\cdots\!88\)\( T^{33} + \)\(54\!\cdots\!26\)\( T^{34} + \)\(39\!\cdots\!60\)\( T^{35} + \)\(25\!\cdots\!62\)\( T^{36} + \)\(14\!\cdots\!64\)\( T^{37} + \)\(58\!\cdots\!58\)\( T^{38} + \)\(15\!\cdots\!20\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$79$ \( 1 - 56 T + 1608 T^{2} - 32544 T^{3} + 531232 T^{4} - 7502656 T^{5} + 94723432 T^{6} - 1079876968 T^{7} + 11152309445 T^{8} - 104467318464 T^{9} + 885876705344 T^{10} - 6748309500960 T^{11} + 45237704790656 T^{12} - 254352077824672 T^{13} + 1043944393809024 T^{14} - 1173550884529024 T^{15} - 28330582069716470 T^{16} + 326793601144278544 T^{17} - 2173623899474329584 T^{18} + 10801150466444693600 T^{19} - 64674172953547878720 T^{20} + \)\(85\!\cdots\!00\)\( T^{21} - \)\(13\!\cdots\!44\)\( T^{22} + \)\(16\!\cdots\!16\)\( T^{23} - \)\(11\!\cdots\!70\)\( T^{24} - \)\(36\!\cdots\!76\)\( T^{25} + \)\(25\!\cdots\!04\)\( T^{26} - \)\(48\!\cdots\!48\)\( T^{27} + \)\(68\!\cdots\!16\)\( T^{28} - \)\(80\!\cdots\!40\)\( T^{29} + \)\(83\!\cdots\!44\)\( T^{30} - \)\(78\!\cdots\!56\)\( T^{31} + \)\(65\!\cdots\!45\)\( T^{32} - \)\(50\!\cdots\!52\)\( T^{33} + \)\(34\!\cdots\!92\)\( T^{34} - \)\(21\!\cdots\!44\)\( T^{35} + \)\(12\!\cdots\!72\)\( T^{36} - \)\(59\!\cdots\!96\)\( T^{37} + \)\(23\!\cdots\!88\)\( T^{38} - \)\(63\!\cdots\!64\)\( T^{39} + \)\(89\!\cdots\!01\)\( T^{40} \)
$83$ \( 1 - 544 T^{3} - 1354 T^{4} + 59152 T^{5} + 147968 T^{6} + 546160 T^{7} - 31611727 T^{8} + 119552016 T^{9} + 1652717184 T^{10} - 6405201136 T^{11} + 252699874336 T^{12} + 937926276176 T^{13} + 17620618729088 T^{14} - 121733300402992 T^{15} - 1091081305080034 T^{16} + 10833819307749168 T^{17} + 164063275302237440 T^{18} + 183331162121083376 T^{19} - 10465764546854435628 T^{20} + 15216486456049920208 T^{21} + \)\(11\!\cdots\!60\)\( T^{22} + \)\(61\!\cdots\!16\)\( T^{23} - \)\(51\!\cdots\!14\)\( T^{24} - \)\(47\!\cdots\!56\)\( T^{25} + \)\(57\!\cdots\!72\)\( T^{26} + \)\(25\!\cdots\!52\)\( T^{27} + \)\(56\!\cdots\!76\)\( T^{28} - \)\(11\!\cdots\!08\)\( T^{29} + \)\(25\!\cdots\!16\)\( T^{30} + \)\(15\!\cdots\!72\)\( T^{31} - \)\(33\!\cdots\!47\)\( T^{32} + \)\(48\!\cdots\!80\)\( T^{33} + \)\(10\!\cdots\!72\)\( T^{34} + \)\(36\!\cdots\!64\)\( T^{35} - \)\(68\!\cdots\!74\)\( T^{36} - \)\(22\!\cdots\!12\)\( T^{37} + \)\(24\!\cdots\!01\)\( T^{40} \)
$89$ \( 1 - 948 T^{2} + 453936 T^{4} - 146353496 T^{6} + 35701988702 T^{8} - 7013810546556 T^{10} + 1152109743195558 T^{12} - 162048052656574324 T^{14} + 19816119477140894897 T^{16} - \)\(21\!\cdots\!68\)\( T^{18} + \)\(20\!\cdots\!84\)\( T^{20} - \)\(16\!\cdots\!28\)\( T^{22} + \)\(12\!\cdots\!77\)\( T^{24} - \)\(80\!\cdots\!64\)\( T^{26} + \)\(45\!\cdots\!98\)\( T^{28} - \)\(21\!\cdots\!56\)\( T^{30} + \)\(88\!\cdots\!42\)\( T^{32} - \)\(28\!\cdots\!36\)\( T^{34} + \)\(70\!\cdots\!96\)\( T^{36} - \)\(11\!\cdots\!88\)\( T^{38} + \)\(97\!\cdots\!01\)\( T^{40} \)
$97$ \( 1 - 48 T + 1142 T^{2} - 16936 T^{3} + 160946 T^{4} - 603200 T^{5} - 12193658 T^{6} + 329836064 T^{7} - 4766618146 T^{8} + 48871055912 T^{9} - 342706004482 T^{10} + 607199068352 T^{11} + 28990974681866 T^{12} - 605921738623824 T^{13} + 7806367455647038 T^{14} - 74491245978555832 T^{15} + 484575109311053409 T^{16} - 549564314450980192 T^{17} - 41722742455227040440 T^{18} + \)\(77\!\cdots\!16\)\( T^{19} - \)\(90\!\cdots\!40\)\( T^{20} + \)\(75\!\cdots\!52\)\( T^{21} - \)\(39\!\cdots\!60\)\( T^{22} - \)\(50\!\cdots\!16\)\( T^{23} + \)\(42\!\cdots\!29\)\( T^{24} - \)\(63\!\cdots\!24\)\( T^{25} + \)\(65\!\cdots\!02\)\( T^{26} - \)\(48\!\cdots\!12\)\( T^{27} + \)\(22\!\cdots\!26\)\( T^{28} + \)\(46\!\cdots\!84\)\( T^{29} - \)\(25\!\cdots\!18\)\( T^{30} + \)\(34\!\cdots\!36\)\( T^{31} - \)\(33\!\cdots\!86\)\( T^{32} + \)\(22\!\cdots\!28\)\( T^{33} - \)\(79\!\cdots\!02\)\( T^{34} - \)\(38\!\cdots\!00\)\( T^{35} + \)\(98\!\cdots\!66\)\( T^{36} - \)\(10\!\cdots\!32\)\( T^{37} + \)\(66\!\cdots\!38\)\( T^{38} - \)\(26\!\cdots\!84\)\( T^{39} + \)\(54\!\cdots\!01\)\( T^{40} \)
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