Properties

Label 350.2.h.d
Level 350
Weight 2
Character orbit 350.h
Analytic conductor 2.795
Analytic rank 0
Dimension 20
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 350.h (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{14} q^{2} + \beta_{1} q^{3} + \beta_{6} q^{4} + ( \beta_{10} + \beta_{13} + \beta_{15} + \beta_{17} ) q^{5} -\beta_{5} q^{6} + q^{7} + ( 1 - \beta_{4} + \beta_{6} - \beta_{14} ) q^{8} + ( -1 - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} - \beta_{17} ) q^{9} +O(q^{10})\) \( q + \beta_{14} q^{2} + \beta_{1} q^{3} + \beta_{6} q^{4} + ( \beta_{10} + \beta_{13} + \beta_{15} + \beta_{17} ) q^{5} -\beta_{5} q^{6} + q^{7} + ( 1 - \beta_{4} + \beta_{6} - \beta_{14} ) q^{8} + ( -1 - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} - \beta_{17} ) q^{9} + \beta_{13} q^{10} + ( -1 + \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} + 2 \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{16} - \beta_{19} ) q^{11} + \beta_{9} q^{12} + ( \beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{10} + \beta_{11} - \beta_{15} + \beta_{16} ) q^{13} + \beta_{14} q^{14} + ( -\beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{10} - \beta_{11} + \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{15} -\beta_{4} q^{16} + ( 1 - \beta_{1} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{15} - \beta_{16} + \beta_{18} ) q^{17} + ( -2 + \beta_{2} - \beta_{3} ) q^{18} + ( -2 + 3 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{18} - \beta_{19} ) q^{19} -\beta_{10} q^{20} + \beta_{1} q^{21} + ( -\beta_{2} + \beta_{8} - \beta_{11} ) q^{22} + ( -2 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + \beta_{13} + 2 \beta_{14} - \beta_{15} + \beta_{16} - \beta_{19} ) q^{23} + \beta_{7} q^{24} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{15} - 2 \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{25} + ( 1 - \beta_{2} - \beta_{5} - \beta_{13} - \beta_{15} ) q^{26} + ( 1 - 2 \beta_{1} - 3 \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} - \beta_{12} + 2 \beta_{15} - 2 \beta_{16} + \beta_{18} + \beta_{19} ) q^{27} + \beta_{6} q^{28} + ( -2 + \beta_{1} + \beta_{3} + 3 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{14} - \beta_{15} ) q^{29} + ( -1 + \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{16} ) q^{30} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{14} - \beta_{15} + \beta_{16} - \beta_{19} ) q^{31} - q^{32} + ( 2 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 3 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} + \beta_{16} - 2 \beta_{17} + \beta_{19} ) q^{33} + ( 1 - 3 \beta_{4} + 2 \beta_{6} + \beta_{8} + \beta_{11} + \beta_{13} - 2 \beta_{14} + \beta_{17} + \beta_{19} ) q^{34} + ( \beta_{10} + \beta_{13} + \beta_{15} + \beta_{17} ) q^{35} + ( \beta_{1} - \beta_{3} + \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{36} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} + 3 \beta_{15} - \beta_{16} + \beta_{17} ) q^{37} + ( \beta_{4} - \beta_{9} + \beta_{10} + \beta_{15} - \beta_{19} ) q^{38} + ( -5 + 3 \beta_{1} + 4 \beta_{4} - 3 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{13} + 3 \beta_{14} - 5 \beta_{15} + 2 \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{39} + \beta_{15} q^{40} + ( -1 - \beta_{1} + 3 \beta_{4} - \beta_{6} - \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} - \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{41} -\beta_{5} q^{42} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{9} - \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{17} + \beta_{18} ) q^{43} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{16} + \beta_{17} ) q^{44} + ( 6 - 3 \beta_{1} + \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} - \beta_{16} + \beta_{18} + 2 \beta_{19} ) q^{45} + ( -1 - 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} - \beta_{10} - \beta_{14} - \beta_{16} - \beta_{18} ) q^{46} + ( -1 - \beta_{1} + \beta_{3} + 2 \beta_{4} - 3 \beta_{6} + \beta_{10} + \beta_{15} + \beta_{17} + \beta_{18} ) q^{47} + ( -\beta_{1} - \beta_{5} + \beta_{7} - \beta_{9} ) q^{48} + q^{49} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{11} + \beta_{12} - 2 \beta_{14} - \beta_{18} + \beta_{19} ) q^{50} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} + \beta_{13} + \beta_{14} - \beta_{17} - \beta_{18} ) q^{51} + ( -\beta_{1} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{16} ) q^{52} + ( 1 - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{13} - \beta_{14} + \beta_{15} - 3 \beta_{17} + \beta_{18} + \beta_{19} ) q^{53} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{15} - \beta_{16} - \beta_{17} + \beta_{19} ) q^{54} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 5 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{16} - \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{55} + ( 1 - \beta_{4} + \beta_{6} - \beta_{14} ) q^{56} + ( 6 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} - 3 \beta_{7} + \beta_{12} ) q^{57} + ( 1 + \beta_{1} + \beta_{2} + \beta_{5} + \beta_{9} - \beta_{10} - \beta_{14} - \beta_{16} + \beta_{18} ) q^{58} + ( -2 - \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} + 4 \beta_{14} - \beta_{15} - \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{59} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{10} + \beta_{12} - 2 \beta_{14} + \beta_{16} ) q^{60} + ( -2 \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} - 6 \beta_{14} - 2 \beta_{16} + 2 \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{61} + ( 1 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{62} + ( -1 - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} - \beta_{17} ) q^{63} -\beta_{14} q^{64} + ( 1 + 4 \beta_{1} - 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{14} - \beta_{15} ) q^{65} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} - 3 \beta_{13} + \beta_{14} - \beta_{15} - 2 \beta_{17} ) q^{66} + ( 1 - 3 \beta_{1} + \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{67} + ( -\beta_{2} + \beta_{3} - \beta_{5} - \beta_{8} - \beta_{9} - \beta_{14} + \beta_{16} + \beta_{17} ) q^{68} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + \beta_{13} - 3 \beta_{14} + 2 \beta_{17} - \beta_{18} ) q^{69} + \beta_{13} q^{70} + ( 1 + 2 \beta_{2} + \beta_{3} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + \beta_{13} + \beta_{15} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{71} + ( -1 - \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{18} - \beta_{19} ) q^{72} + ( 2 \beta_{1} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{15} + 2 \beta_{16} - \beta_{18} ) q^{73} + ( 1 - \beta_{1} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} - \beta_{17} - \beta_{18} ) q^{74} + ( 3 - 4 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} + \beta_{17} ) q^{75} + ( 1 - \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{16} - \beta_{17} ) q^{76} + ( -1 + \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} + 2 \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{16} - \beta_{19} ) q^{77} + ( -2 + \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} + \beta_{10} - \beta_{14} + 2 \beta_{15} - \beta_{16} + 3 \beta_{17} - \beta_{19} ) q^{78} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 7 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + \beta_{14} + \beta_{15} + 3 \beta_{16} + \beta_{17} - 2 \beta_{19} ) q^{79} -\beta_{17} q^{80} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} + 3 \beta_{14} + 3 \beta_{15} - 2 \beta_{16} + \beta_{17} + \beta_{19} ) q^{81} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{14} + \beta_{15} - \beta_{19} ) q^{82} + ( 2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} + \beta_{16} - 2 \beta_{17} - \beta_{19} ) q^{83} + \beta_{9} q^{84} + ( -3 + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 3 \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} - \beta_{16} - \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{85} + ( 1 + \beta_{1} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} + \beta_{8} + 3 \beta_{9} + \beta_{11} - \beta_{12} + \beta_{15} - 2 \beta_{16} + 2 \beta_{18} + \beta_{19} ) q^{86} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 6 \beta_{4} + \beta_{5} + 6 \beta_{6} + \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} - 7 \beta_{14} + 2 \beta_{15} + 2 \beta_{17} + \beta_{19} ) q^{87} + ( 1 - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} + \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{88} + ( 1 + \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - 3 \beta_{14} + \beta_{15} + \beta_{17} - 2 \beta_{18} ) q^{89} + ( 1 - \beta_{2} + 2 \beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} + 3 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} + \beta_{18} + \beta_{19} ) q^{90} + ( \beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{10} + \beta_{11} - \beta_{15} + \beta_{16} ) q^{91} + ( -\beta_{1} + 2 \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{14} + \beta_{16} + \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{92} + ( -2 + 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{93} + ( \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{94} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{15} + \beta_{16} - \beta_{18} - 2 \beta_{19} ) q^{95} -\beta_{1} q^{96} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{11} - 3 \beta_{14} + 3 \beta_{15} - 3 \beta_{16} + 2 \beta_{17} + \beta_{18} + \beta_{19} ) q^{97} + \beta_{14} q^{98} + ( 4 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} - 5 \beta_{5} + 7 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{12} - 4 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} + 2 \beta_{17} - 2 \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 5q^{2} + 3q^{3} - 5q^{4} - 5q^{5} - 3q^{6} + 20q^{7} + 5q^{8} - 6q^{9} + O(q^{10}) \) \( 20q + 5q^{2} + 3q^{3} - 5q^{4} - 5q^{5} - 3q^{6} + 20q^{7} + 5q^{8} - 6q^{9} - 9q^{11} - 2q^{12} + 5q^{13} + 5q^{14} - 5q^{16} - 12q^{17} - 34q^{18} + 2q^{19} + 5q^{20} + 3q^{21} - 6q^{22} - 5q^{23} + 2q^{24} - 35q^{25} + 20q^{26} - 6q^{27} - 5q^{28} - 22q^{29} - 25q^{30} - 7q^{31} - 20q^{32} + 25q^{33} - 18q^{34} - 5q^{35} - 6q^{36} - 3q^{37} + 8q^{38} - 22q^{39} + 19q^{41} - 3q^{42} + 2q^{43} + 6q^{44} + 45q^{45} - 10q^{46} - 14q^{47} - 2q^{48} + 20q^{49} + 10q^{50} + 38q^{51} + 5q^{52} - q^{53} - 19q^{54} - 20q^{55} + 5q^{56} + 116q^{57} + 22q^{58} + 17q^{59} - 5q^{60} - 38q^{61} + 7q^{62} - 6q^{63} - 5q^{64} + 15q^{65} - 16q^{67} - 12q^{68} + 35q^{69} + q^{71} + 11q^{72} + 19q^{73} + 18q^{74} + 35q^{75} + 12q^{76} - 9q^{77} - 18q^{78} - 64q^{79} - 40q^{81} + 26q^{82} + 57q^{83} - 2q^{84} - 40q^{85} - 2q^{86} - 78q^{87} + 9q^{88} - 6q^{89} + 10q^{90} + 5q^{91} + 10q^{92} - 22q^{93} + 14q^{94} + 60q^{95} - 3q^{96} - 18q^{97} + 5q^{98} + 112q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 3 x^{19} + 15 x^{18} - 30 x^{17} + 145 x^{16} - 194 x^{15} + 1187 x^{14} - 1490 x^{13} + 10170 x^{12} - 13920 x^{11} + 42087 x^{10} - 591 x^{9} + 65635 x^{8} + 120715 x^{7} + 257180 x^{6} + 306283 x^{5} + 246171 x^{4} + 99850 x^{3} + 26260 x^{2} + 4200 x + 400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(42\!\cdots\!71\)\( \nu^{19} + \)\(45\!\cdots\!15\)\( \nu^{18} - \)\(16\!\cdots\!25\)\( \nu^{17} + \)\(63\!\cdots\!20\)\( \nu^{16} - \)\(16\!\cdots\!45\)\( \nu^{15} + \)\(57\!\cdots\!84\)\( \nu^{14} - \)\(12\!\cdots\!55\)\( \nu^{13} + \)\(45\!\cdots\!80\)\( \nu^{12} - \)\(98\!\cdots\!70\)\( \nu^{11} + \)\(39\!\cdots\!70\)\( \nu^{10} - \)\(68\!\cdots\!37\)\( \nu^{9} + \)\(14\!\cdots\!25\)\( \nu^{8} - \)\(48\!\cdots\!55\)\( \nu^{7} + \)\(15\!\cdots\!05\)\( \nu^{6} + \)\(25\!\cdots\!20\)\( \nu^{5} + \)\(62\!\cdots\!47\)\( \nu^{4} + \)\(74\!\cdots\!15\)\( \nu^{3} + \)\(55\!\cdots\!70\)\( \nu^{2} + \)\(14\!\cdots\!00\)\( \nu + \)\(24\!\cdots\!00\)\(\)\()/ \)\(40\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(45\!\cdots\!31\)\( \nu^{19} + \)\(46\!\cdots\!45\)\( \nu^{18} - \)\(17\!\cdots\!65\)\( \nu^{17} + \)\(64\!\cdots\!40\)\( \nu^{16} - \)\(17\!\cdots\!75\)\( \nu^{15} + \)\(57\!\cdots\!84\)\( \nu^{14} - \)\(12\!\cdots\!35\)\( \nu^{13} + \)\(46\!\cdots\!20\)\( \nu^{12} - \)\(10\!\cdots\!40\)\( \nu^{11} + \)\(40\!\cdots\!50\)\( \nu^{10} - \)\(69\!\cdots\!17\)\( \nu^{9} + \)\(14\!\cdots\!95\)\( \nu^{8} - \)\(47\!\cdots\!65\)\( \nu^{7} + \)\(14\!\cdots\!35\)\( \nu^{6} + \)\(25\!\cdots\!50\)\( \nu^{5} + \)\(61\!\cdots\!57\)\( \nu^{4} + \)\(74\!\cdots\!15\)\( \nu^{3} + \)\(55\!\cdots\!70\)\( \nu^{2} + \)\(14\!\cdots\!00\)\( \nu + \)\(41\!\cdots\!00\)\(\)\()/ \)\(40\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(96\!\cdots\!31\)\( \nu^{19} - \)\(48\!\cdots\!64\)\( \nu^{18} + \)\(21\!\cdots\!20\)\( \nu^{17} - \)\(61\!\cdots\!85\)\( \nu^{16} + \)\(21\!\cdots\!87\)\( \nu^{15} - \)\(49\!\cdots\!51\)\( \nu^{14} + \)\(16\!\cdots\!89\)\( \nu^{13} - \)\(39\!\cdots\!25\)\( \nu^{12} + \)\(13\!\cdots\!20\)\( \nu^{11} - \)\(34\!\cdots\!52\)\( \nu^{10} + \)\(76\!\cdots\!19\)\( \nu^{9} - \)\(96\!\cdots\!76\)\( \nu^{8} + \)\(10\!\cdots\!20\)\( \nu^{7} - \)\(19\!\cdots\!20\)\( \nu^{6} + \)\(63\!\cdots\!73\)\( \nu^{5} - \)\(12\!\cdots\!85\)\( \nu^{4} - \)\(16\!\cdots\!70\)\( \nu^{3} - \)\(18\!\cdots\!75\)\( \nu^{2} - \)\(93\!\cdots\!00\)\( \nu - \)\(28\!\cdots\!00\)\(\)\()/ \)\(80\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(13\!\cdots\!77\)\( \nu^{19} + \)\(41\!\cdots\!19\)\( \nu^{18} - \)\(20\!\cdots\!29\)\( \nu^{17} + \)\(40\!\cdots\!42\)\( \nu^{16} - \)\(19\!\cdots\!41\)\( \nu^{15} + \)\(26\!\cdots\!52\)\( \nu^{14} - \)\(16\!\cdots\!39\)\( \nu^{13} + \)\(20\!\cdots\!34\)\( \nu^{12} - \)\(14\!\cdots\!42\)\( \nu^{11} + \)\(18\!\cdots\!86\)\( \nu^{10} - \)\(57\!\cdots\!23\)\( \nu^{9} - \)\(50\!\cdots\!49\)\( \nu^{8} - \)\(93\!\cdots\!41\)\( \nu^{7} - \)\(16\!\cdots\!77\)\( \nu^{6} - \)\(36\!\cdots\!14\)\( \nu^{5} - \)\(43\!\cdots\!45\)\( \nu^{4} - \)\(35\!\cdots\!65\)\( \nu^{3} - \)\(13\!\cdots\!50\)\( \nu^{2} - \)\(34\!\cdots\!00\)\( \nu - \)\(53\!\cdots\!00\)\(\)\()/ \)\(80\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(10\!\cdots\!43\)\( \nu^{19} - \)\(48\!\cdots\!83\)\( \nu^{18} + \)\(21\!\cdots\!20\)\( \nu^{17} - \)\(57\!\cdots\!69\)\( \nu^{16} + \)\(20\!\cdots\!43\)\( \nu^{15} - \)\(45\!\cdots\!72\)\( \nu^{14} + \)\(16\!\cdots\!33\)\( \nu^{13} - \)\(35\!\cdots\!25\)\( \nu^{12} + \)\(13\!\cdots\!14\)\( \nu^{11} - \)\(31\!\cdots\!33\)\( \nu^{10} + \)\(71\!\cdots\!66\)\( \nu^{9} - \)\(75\!\cdots\!27\)\( \nu^{8} + \)\(85\!\cdots\!65\)\( \nu^{7} + \)\(19\!\cdots\!54\)\( \nu^{6} + \)\(93\!\cdots\!87\)\( \nu^{5} - \)\(57\!\cdots\!61\)\( \nu^{4} - \)\(14\!\cdots\!50\)\( \nu^{3} - \)\(18\!\cdots\!85\)\( \nu^{2} - \)\(48\!\cdots\!25\)\( \nu - \)\(91\!\cdots\!00\)\(\)\()/ \)\(40\!\cdots\!50\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(12\!\cdots\!87\)\( \nu^{19} + \)\(36\!\cdots\!37\)\( \nu^{18} - \)\(18\!\cdots\!06\)\( \nu^{17} + \)\(36\!\cdots\!53\)\( \nu^{16} - \)\(17\!\cdots\!12\)\( \nu^{15} + \)\(22\!\cdots\!02\)\( \nu^{14} - \)\(14\!\cdots\!87\)\( \nu^{13} + \)\(17\!\cdots\!36\)\( \nu^{12} - \)\(12\!\cdots\!48\)\( \nu^{11} + \)\(16\!\cdots\!37\)\( \nu^{10} - \)\(50\!\cdots\!18\)\( \nu^{9} - \)\(27\!\cdots\!12\)\( \nu^{8} - \)\(78\!\cdots\!29\)\( \nu^{7} - \)\(15\!\cdots\!58\)\( \nu^{6} - \)\(32\!\cdots\!28\)\( \nu^{5} - \)\(38\!\cdots\!40\)\( \nu^{4} - \)\(30\!\cdots\!55\)\( \nu^{3} - \)\(11\!\cdots\!50\)\( \nu^{2} - \)\(22\!\cdots\!00\)\( \nu - \)\(26\!\cdots\!00\)\(\)\()/ \)\(36\!\cdots\!50\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(70\!\cdots\!63\)\( \nu^{19} + \)\(12\!\cdots\!40\)\( \nu^{18} - \)\(68\!\cdots\!10\)\( \nu^{17} + \)\(32\!\cdots\!15\)\( \nu^{16} - \)\(17\!\cdots\!25\)\( \nu^{15} + \)\(38\!\cdots\!13\)\( \nu^{14} + \)\(38\!\cdots\!55\)\( \nu^{13} + \)\(31\!\cdots\!45\)\( \nu^{12} + \)\(69\!\cdots\!10\)\( \nu^{11} + \)\(26\!\cdots\!50\)\( \nu^{10} - \)\(29\!\cdots\!79\)\( \nu^{9} + \)\(15\!\cdots\!40\)\( \nu^{8} - \)\(10\!\cdots\!00\)\( \nu^{7} + \)\(31\!\cdots\!10\)\( \nu^{6} + \)\(50\!\cdots\!75\)\( \nu^{5} + \)\(94\!\cdots\!69\)\( \nu^{4} + \)\(91\!\cdots\!30\)\( \nu^{3} + \)\(57\!\cdots\!65\)\( \nu^{2} + \)\(11\!\cdots\!50\)\( \nu + \)\(17\!\cdots\!00\)\(\)\()/ \)\(20\!\cdots\!50\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(16\!\cdots\!54\)\( \nu^{19} + \)\(53\!\cdots\!75\)\( \nu^{18} - \)\(25\!\cdots\!79\)\( \nu^{17} + \)\(55\!\cdots\!08\)\( \nu^{16} - \)\(25\!\cdots\!30\)\( \nu^{15} + \)\(37\!\cdots\!92\)\( \nu^{14} - \)\(20\!\cdots\!55\)\( \nu^{13} + \)\(29\!\cdots\!04\)\( \nu^{12} - \)\(17\!\cdots\!73\)\( \nu^{11} + \)\(26\!\cdots\!25\)\( \nu^{10} - \)\(74\!\cdots\!14\)\( \nu^{9} + \)\(16\!\cdots\!60\)\( \nu^{8} - \)\(10\!\cdots\!91\)\( \nu^{7} - \)\(17\!\cdots\!53\)\( \nu^{6} - \)\(37\!\cdots\!30\)\( \nu^{5} - \)\(40\!\cdots\!03\)\( \nu^{4} - \)\(29\!\cdots\!35\)\( \nu^{3} - \)\(76\!\cdots\!05\)\( \nu^{2} - \)\(13\!\cdots\!00\)\( \nu - \)\(42\!\cdots\!00\)\(\)\()/ \)\(40\!\cdots\!50\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(12\!\cdots\!24\)\( \nu^{19} - \)\(41\!\cdots\!70\)\( \nu^{18} + \)\(20\!\cdots\!70\)\( \nu^{17} - \)\(45\!\cdots\!95\)\( \nu^{16} + \)\(20\!\cdots\!10\)\( \nu^{15} - \)\(31\!\cdots\!31\)\( \nu^{14} + \)\(16\!\cdots\!05\)\( \nu^{13} - \)\(24\!\cdots\!65\)\( \nu^{12} + \)\(13\!\cdots\!45\)\( \nu^{11} - \)\(22\!\cdots\!35\)\( \nu^{10} + \)\(62\!\cdots\!88\)\( \nu^{9} - \)\(23\!\cdots\!15\)\( \nu^{8} + \)\(96\!\cdots\!50\)\( \nu^{7} + \)\(12\!\cdots\!20\)\( \nu^{6} + \)\(29\!\cdots\!15\)\( \nu^{5} + \)\(31\!\cdots\!17\)\( \nu^{4} + \)\(24\!\cdots\!40\)\( \nu^{3} + \)\(90\!\cdots\!70\)\( \nu^{2} + \)\(31\!\cdots\!00\)\( \nu + \)\(39\!\cdots\!50\)\(\)\()/ \)\(20\!\cdots\!50\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(50\!\cdots\!17\)\( \nu^{19} - \)\(17\!\cdots\!75\)\( \nu^{18} + \)\(82\!\cdots\!45\)\( \nu^{17} - \)\(18\!\cdots\!20\)\( \nu^{16} + \)\(79\!\cdots\!65\)\( \nu^{15} - \)\(12\!\cdots\!18\)\( \nu^{14} + \)\(64\!\cdots\!05\)\( \nu^{13} - \)\(99\!\cdots\!80\)\( \nu^{12} + \)\(54\!\cdots\!20\)\( \nu^{11} - \)\(90\!\cdots\!40\)\( \nu^{10} + \)\(24\!\cdots\!99\)\( \nu^{9} - \)\(84\!\cdots\!05\)\( \nu^{8} + \)\(32\!\cdots\!65\)\( \nu^{7} + \)\(48\!\cdots\!45\)\( \nu^{6} + \)\(10\!\cdots\!60\)\( \nu^{5} + \)\(10\!\cdots\!31\)\( \nu^{4} + \)\(60\!\cdots\!45\)\( \nu^{3} + \)\(66\!\cdots\!10\)\( \nu^{2} - \)\(68\!\cdots\!00\)\( \nu - \)\(17\!\cdots\!00\)\(\)\()/ \)\(40\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(14\!\cdots\!44\)\( \nu^{19} + \)\(50\!\cdots\!50\)\( \nu^{18} - \)\(23\!\cdots\!25\)\( \nu^{17} + \)\(54\!\cdots\!55\)\( \nu^{16} - \)\(22\!\cdots\!50\)\( \nu^{15} + \)\(38\!\cdots\!21\)\( \nu^{14} - \)\(18\!\cdots\!85\)\( \nu^{13} + \)\(30\!\cdots\!20\)\( \nu^{12} - \)\(15\!\cdots\!55\)\( \nu^{11} + \)\(27\!\cdots\!50\)\( \nu^{10} - \)\(71\!\cdots\!38\)\( \nu^{9} + \)\(34\!\cdots\!60\)\( \nu^{8} - \)\(99\!\cdots\!70\)\( \nu^{7} - \)\(11\!\cdots\!30\)\( \nu^{6} - \)\(28\!\cdots\!25\)\( \nu^{5} - \)\(25\!\cdots\!12\)\( \nu^{4} - \)\(14\!\cdots\!65\)\( \nu^{3} + \)\(96\!\cdots\!05\)\( \nu^{2} + \)\(11\!\cdots\!75\)\( \nu + \)\(34\!\cdots\!25\)\(\)\()/ \)\(10\!\cdots\!75\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(64\!\cdots\!13\)\( \nu^{19} - \)\(22\!\cdots\!30\)\( \nu^{18} + \)\(10\!\cdots\!40\)\( \nu^{17} - \)\(24\!\cdots\!25\)\( \nu^{16} + \)\(10\!\cdots\!75\)\( \nu^{15} - \)\(17\!\cdots\!07\)\( \nu^{14} + \)\(84\!\cdots\!25\)\( \nu^{13} - \)\(13\!\cdots\!05\)\( \nu^{12} + \)\(71\!\cdots\!00\)\( \nu^{11} - \)\(12\!\cdots\!50\)\( \nu^{10} + \)\(32\!\cdots\!41\)\( \nu^{9} - \)\(16\!\cdots\!40\)\( \nu^{8} + \)\(44\!\cdots\!50\)\( \nu^{7} + \)\(55\!\cdots\!50\)\( \nu^{6} + \)\(12\!\cdots\!75\)\( \nu^{5} + \)\(11\!\cdots\!89\)\( \nu^{4} + \)\(68\!\cdots\!30\)\( \nu^{3} + \)\(29\!\cdots\!15\)\( \nu^{2} + \)\(36\!\cdots\!00\)\( \nu + \)\(19\!\cdots\!00\)\(\)\()/ \)\(40\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(26\!\cdots\!15\)\( \nu^{19} + \)\(83\!\cdots\!99\)\( \nu^{18} - \)\(41\!\cdots\!63\)\( \nu^{17} + \)\(85\!\cdots\!08\)\( \nu^{16} - \)\(39\!\cdots\!59\)\( \nu^{15} + \)\(56\!\cdots\!92\)\( \nu^{14} - \)\(32\!\cdots\!09\)\( \nu^{13} + \)\(43\!\cdots\!28\)\( \nu^{12} - \)\(27\!\cdots\!18\)\( \nu^{11} + \)\(40\!\cdots\!84\)\( \nu^{10} - \)\(11\!\cdots\!77\)\( \nu^{9} + \)\(13\!\cdots\!11\)\( \nu^{8} - \)\(17\!\cdots\!27\)\( \nu^{7} - \)\(30\!\cdots\!43\)\( \nu^{6} - \)\(66\!\cdots\!46\)\( \nu^{5} - \)\(75\!\cdots\!17\)\( \nu^{4} - \)\(57\!\cdots\!75\)\( \nu^{3} - \)\(19\!\cdots\!20\)\( \nu^{2} - \)\(43\!\cdots\!00\)\( \nu - \)\(43\!\cdots\!00\)\(\)\()/ \)\(16\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(19\!\cdots\!09\)\( \nu^{19} + \)\(58\!\cdots\!85\)\( \nu^{18} - \)\(29\!\cdots\!85\)\( \nu^{17} + \)\(58\!\cdots\!60\)\( \nu^{16} - \)\(28\!\cdots\!65\)\( \nu^{15} + \)\(37\!\cdots\!16\)\( \nu^{14} - \)\(22\!\cdots\!95\)\( \nu^{13} + \)\(28\!\cdots\!80\)\( \nu^{12} - \)\(19\!\cdots\!10\)\( \nu^{11} + \)\(27\!\cdots\!40\)\( \nu^{10} - \)\(81\!\cdots\!03\)\( \nu^{9} + \)\(87\!\cdots\!25\)\( \nu^{8} - \)\(12\!\cdots\!85\)\( \nu^{7} - \)\(23\!\cdots\!85\)\( \nu^{6} - \)\(49\!\cdots\!10\)\( \nu^{5} - \)\(58\!\cdots\!67\)\( \nu^{4} - \)\(46\!\cdots\!65\)\( \nu^{3} - \)\(17\!\cdots\!20\)\( \nu^{2} - \)\(42\!\cdots\!00\)\( \nu - \)\(56\!\cdots\!00\)\(\)\()/ \)\(80\!\cdots\!00\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(11\!\cdots\!67\)\( \nu^{19} + \)\(38\!\cdots\!10\)\( \nu^{18} - \)\(18\!\cdots\!90\)\( \nu^{17} + \)\(39\!\cdots\!65\)\( \nu^{16} - \)\(17\!\cdots\!25\)\( \nu^{15} + \)\(26\!\cdots\!73\)\( \nu^{14} - \)\(14\!\cdots\!15\)\( \nu^{13} + \)\(20\!\cdots\!25\)\( \nu^{12} - \)\(12\!\cdots\!40\)\( \nu^{11} + \)\(19\!\cdots\!00\)\( \nu^{10} - \)\(53\!\cdots\!29\)\( \nu^{9} + \)\(11\!\cdots\!20\)\( \nu^{8} - \)\(79\!\cdots\!70\)\( \nu^{7} - \)\(12\!\cdots\!40\)\( \nu^{6} - \)\(27\!\cdots\!75\)\( \nu^{5} - \)\(30\!\cdots\!11\)\( \nu^{4} - \)\(22\!\cdots\!20\)\( \nu^{3} - \)\(68\!\cdots\!35\)\( \nu^{2} - \)\(15\!\cdots\!50\)\( \nu - \)\(94\!\cdots\!00\)\(\)\()/ \)\(40\!\cdots\!00\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(66\!\cdots\!46\)\( \nu^{19} + \)\(22\!\cdots\!80\)\( \nu^{18} - \)\(10\!\cdots\!15\)\( \nu^{17} + \)\(23\!\cdots\!90\)\( \nu^{16} - \)\(10\!\cdots\!80\)\( \nu^{15} + \)\(16\!\cdots\!99\)\( \nu^{14} - \)\(83\!\cdots\!45\)\( \nu^{13} + \)\(12\!\cdots\!20\)\( \nu^{12} - \)\(71\!\cdots\!15\)\( \nu^{11} + \)\(11\!\cdots\!30\)\( \nu^{10} - \)\(31\!\cdots\!77\)\( \nu^{9} + \)\(10\!\cdots\!35\)\( \nu^{8} - \)\(45\!\cdots\!40\)\( \nu^{7} - \)\(65\!\cdots\!15\)\( \nu^{6} - \)\(14\!\cdots\!70\)\( \nu^{5} - \)\(15\!\cdots\!93\)\( \nu^{4} - \)\(10\!\cdots\!85\)\( \nu^{3} - \)\(21\!\cdots\!55\)\( \nu^{2} - \)\(28\!\cdots\!25\)\( \nu + \)\(17\!\cdots\!00\)\(\)\()/ \)\(20\!\cdots\!50\)\( \)
\(\beta_{18}\)\(=\)\((\)\(\)\(35\!\cdots\!79\)\( \nu^{19} - \)\(11\!\cdots\!15\)\( \nu^{18} + \)\(55\!\cdots\!55\)\( \nu^{17} - \)\(11\!\cdots\!40\)\( \nu^{16} + \)\(54\!\cdots\!95\)\( \nu^{15} - \)\(80\!\cdots\!76\)\( \nu^{14} + \)\(43\!\cdots\!25\)\( \nu^{13} - \)\(62\!\cdots\!00\)\( \nu^{12} + \)\(37\!\cdots\!90\)\( \nu^{11} - \)\(57\!\cdots\!20\)\( \nu^{10} + \)\(16\!\cdots\!73\)\( \nu^{9} - \)\(34\!\cdots\!95\)\( \nu^{8} + \)\(24\!\cdots\!15\)\( \nu^{7} + \)\(38\!\cdots\!15\)\( \nu^{6} + \)\(83\!\cdots\!30\)\( \nu^{5} + \)\(92\!\cdots\!57\)\( \nu^{4} + \)\(69\!\cdots\!15\)\( \nu^{3} + \)\(22\!\cdots\!20\)\( \nu^{2} + \)\(54\!\cdots\!00\)\( \nu + \)\(85\!\cdots\!00\)\(\)\()/ \)\(80\!\cdots\!00\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(65\!\cdots\!83\)\( \nu^{19} + \)\(20\!\cdots\!45\)\( \nu^{18} - \)\(10\!\cdots\!25\)\( \nu^{17} + \)\(20\!\cdots\!10\)\( \nu^{16} - \)\(97\!\cdots\!35\)\( \nu^{15} + \)\(13\!\cdots\!82\)\( \nu^{14} - \)\(79\!\cdots\!65\)\( \nu^{13} + \)\(10\!\cdots\!90\)\( \nu^{12} - \)\(67\!\cdots\!10\)\( \nu^{11} + \)\(99\!\cdots\!60\)\( \nu^{10} - \)\(28\!\cdots\!01\)\( \nu^{9} + \)\(35\!\cdots\!25\)\( \nu^{8} - \)\(42\!\cdots\!65\)\( \nu^{7} - \)\(74\!\cdots\!85\)\( \nu^{6} - \)\(15\!\cdots\!40\)\( \nu^{5} - \)\(17\!\cdots\!69\)\( \nu^{4} - \)\(13\!\cdots\!05\)\( \nu^{3} - \)\(42\!\cdots\!90\)\( \nu^{2} - \)\(73\!\cdots\!00\)\( \nu - \)\(64\!\cdots\!00\)\(\)\()/ \)\(73\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{17} - \beta_{15} + \beta_{14} - \beta_{13} - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} - 3 \beta_{4} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{19} + \beta_{18} - 2 \beta_{16} + 2 \beta_{15} - \beta_{12} - 7 \beta_{9} + \beta_{8} + 6 \beta_{7} - 7 \beta_{5} - 3 \beta_{4} - 8 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-8 \beta_{19} + \beta_{17} - 2 \beta_{16} + 3 \beta_{15} + 39 \beta_{14} + 2 \beta_{13} - 10 \beta_{12} - 10 \beta_{11} - 8 \beta_{10} - 7 \beta_{9} + \beta_{8} + 8 \beta_{7} - 29 \beta_{6} - 9 \beta_{5} + 36 \beta_{4} + \beta_{2} - 10 \beta_{1} - 38\)
\(\nu^{5}\)\(=\)\(\beta_{19} - 11 \beta_{18} - 12 \beta_{17} - \beta_{16} + 2 \beta_{15} + 6 \beta_{14} + 14 \beta_{13} + 9 \beta_{12} + 11 \beta_{10} + 12 \beta_{9} + \beta_{8} - 66 \beta_{7} - 11 \beta_{6} + 11 \beta_{5} + 4 \beta_{4} - 12 \beta_{3} + 15 \beta_{2} - 3 \beta_{1} - 25\)
\(\nu^{6}\)\(=\)\(80 \beta_{19} + 63 \beta_{18} - 26 \beta_{17} - 51 \beta_{16} + 47 \beta_{15} - 64 \beta_{14} + 20 \beta_{13} + 96 \beta_{11} + 3 \beta_{10} + 71 \beta_{9} + 68 \beta_{8} - 51 \beta_{7} + 297 \beta_{6} + 80 \beta_{5} - 115 \beta_{4} - \beta_{3} + 96 \beta_{2} - 19 \beta_{1} + 34\)
\(\nu^{7}\)\(=\)\(-20 \beta_{18} + 22 \beta_{17} + 175 \beta_{16} - 278 \beta_{15} + 58 \beta_{14} + 62 \beta_{13} - 65 \beta_{12} + 112 \beta_{11} - 33 \beta_{10} + 532 \beta_{9} + 2 \beta_{8} + 120 \beta_{7} + 47 \beta_{6} - 80 \beta_{5} + 261 \beta_{4} + 105 \beta_{3} - 65 \beta_{2} + 70 \beta_{1} - 18\)
\(\nu^{8}\)\(=\)\(-122 \beta_{19} - 717 \beta_{18} + 855 \beta_{17} + 1119 \beta_{16} - 739 \beta_{15} - 2710 \beta_{14} + 710 \beta_{13} + 927 \beta_{12} + 1135 \beta_{11} + 1495 \beta_{10} + 1004 \beta_{9} - 192 \beta_{8} - 57 \beta_{7} + 230 \beta_{6} + 149 \beta_{5} + 531 \beta_{4} - 500 \beta_{3} + 1506 \beta_{1} + 38\)
\(\nu^{9}\)\(=\)\(41 \beta_{19} - 20 \beta_{18} + 628 \beta_{17} + 869 \beta_{16} - 1826 \beta_{15} - 2303 \beta_{14} - 909 \beta_{13} + 1935 \beta_{12} + 1935 \beta_{11} - 279 \beta_{10} + 524 \beta_{9} - 1257 \beta_{8} + 239 \beta_{7} + 1753 \beta_{6} + 5083 \beta_{5} - 257 \beta_{4} + 280 \beta_{3} + 388 \beta_{2} + 2350 \beta_{1} + 1326\)
\(\nu^{10}\)\(=\)\(-917 \beta_{19} + 1172 \beta_{18} + 2504 \beta_{17} + 1332 \beta_{16} + 41 \beta_{15} - 4042 \beta_{14} - 2048 \beta_{13} + 2307 \beta_{12} - 1172 \beta_{10} - 2504 \beta_{9} - 1332 \beta_{8} + 5357 \beta_{7} + 1172 \beta_{6} - 1967 \beta_{5} - 1793 \beta_{4} + 6494 \beta_{3} - 9015 \beta_{2} + 496 \beta_{1} + 26490\)
\(\nu^{11}\)\(=\)\(-13040 \beta_{19} - 546 \beta_{18} + 8692 \beta_{17} + 5967 \beta_{16} + 11411 \beta_{15} + 1083 \beta_{14} - 10875 \beta_{13} - 20477 \beta_{11} + 1619 \beta_{10} - 22607 \beta_{9} - 4521 \beta_{8} + 11732 \beta_{7} - 20999 \beta_{6} - 13040 \beta_{5} + 7870 \beta_{4} + 537 \beta_{3} - 20477 \beta_{2} + 30428 \beta_{1} + 5707\)
\(\nu^{12}\)\(=\)\(-5765 \beta_{19} + 10875 \beta_{18} - 62454 \beta_{17} - 50555 \beta_{16} + 15651 \beta_{15} + 71879 \beta_{14} - 98144 \beta_{13} - 24780 \beta_{12} - 112974 \beta_{11} - 74549 \beta_{10} - 117404 \beta_{9} - 37639 \beta_{8} - 37090 \beta_{7} - 82789 \beta_{6} + 1400 \beta_{5} - 132862 \beta_{4} - 10910 \beta_{3} - 24780 \beta_{2} - 39645 \beta_{1} - 26764\)
\(\nu^{13}\)\(=\)\(74729 \beta_{19} + 129714 \beta_{18} - 97400 \beta_{17} - 327713 \beta_{16} + 320318 \beta_{15} + 235655 \beta_{14} - 89455 \beta_{13} - 213339 \beta_{12} - 220670 \beta_{11} - 147210 \beta_{10} - 432708 \beta_{9} + 114374 \beta_{8} + 146639 \beta_{7} - 5865 \beta_{6} - 335308 \beta_{5} - 312987 \beta_{4} + 5830 \beta_{3} - 556412 \beta_{1} + 108509\)
\(\nu^{14}\)\(=\)\(-253682 \beta_{19} + 72730 \beta_{18} - 201766 \beta_{17} - 559093 \beta_{16} + 658492 \beta_{15} + 2155756 \beta_{14} + 216903 \beta_{13} - 867400 \beta_{12} - 867400 \beta_{11} - 472602 \beta_{10} - 349118 \beta_{9} + 295399 \beta_{8} + 130412 \beta_{7} - 1333621 \beta_{6} - 764696 \beta_{5} + 1168634 \beta_{4} - 123270 \beta_{3} + 263694 \beta_{2} - 1007610 \beta_{1} - 1983627\)
\(\nu^{15}\)\(=\)\(441589 \beta_{19} - 648694 \beta_{18} - 1230493 \beta_{17} - 581799 \beta_{16} + 448678 \beta_{15} + 1003219 \beta_{14} + 1538961 \beta_{13} - 266189 \beta_{12} + 648694 \beta_{10} + 1230493 \beta_{9} + 581799 \beta_{8} - 3497289 \beta_{7} - 648694 \beta_{6} + 364624 \beta_{5} - 20169 \beta_{4} - 1282978 \beta_{3} + 2204950 \beta_{2} - 1314547 \beta_{1} - 4086135\)
\(\nu^{16}\)\(=\)\(5120440 \beta_{19} + 2000607 \beta_{18} - 1994864 \beta_{17} - 1429014 \beta_{16} - 253357 \beta_{15} - 2866476 \beta_{14} + 4816395 \beta_{13} + 8573274 \beta_{11} + 1696562 \beta_{10} + 8110794 \beta_{9} + 4230037 \beta_{8} - 3294399 \beta_{7} + 15001008 \beta_{6} + 5120440 \beta_{5} - 2609295 \beta_{4} - 865869 \beta_{3} + 8573274 \beta_{2} - 3187886 \beta_{1} - 3377014\)
\(\nu^{17}\)\(=\)\(1865385 \beta_{19} - 4816395 \beta_{18} + 7387453 \beta_{17} + 20439520 \beta_{16} - 25316767 \beta_{15} - 8700223 \beta_{14} + 17339053 \beta_{13} + 4368680 \beta_{12} + 27060448 \beta_{11} + 9979353 \beta_{10} + 46932363 \beta_{9} + 2252248 \beta_{8} + 5603630 \beta_{7} + 14283143 \beta_{6} + 4347970 \beta_{5} + 31162704 \beta_{4} + 5582920 \beta_{3} + 4368680 \beta_{2} + 14856600 \beta_{1} - 2564147\)
\(\nu^{18}\)\(=\)\(-7855878 \beta_{19} - 48167313 \beta_{18} + 63064055 \beta_{17} + 107843376 \beta_{16} - 92965066 \beta_{15} - 187242405 \beta_{14} + 52487530 \beta_{13} + 85130898 \beta_{12} + 114884240 \beta_{11} + 100694985 \beta_{10} + 117301516 \beta_{9} - 22712478 \beta_{8} - 8912053 \beta_{7} + 49960565 \beta_{6} + 54237461 \beta_{5} + 63211724 \beta_{4} - 15523945 \beta_{3} + 154364649 \beta_{1} + 27248087\)
\(\nu^{19}\)\(=\)\(1871189 \beta_{19} - 23258565 \beta_{18} + 105477727 \beta_{17} + 181609146 \beta_{16} - 229270514 \beta_{15} - 398714212 \beta_{14} - 69082576 \beta_{13} + 232979060 \beta_{12} + 232979060 \beta_{11} + 62577574 \beta_{10} + 114774551 \beta_{9} - 122740118 \beta_{8} + 49948996 \beta_{7} + 233714297 \beta_{6} + 338621527 \beta_{5} - 61025183 \beta_{4} + 51820185 \beta_{3} - 58869028 \beta_{2} + 344045065 \beta_{1} + 327794279\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
−2.31300 1.68049i
−0.649647 0.471996i
−0.210620 0.153024i
1.88692 + 1.37093i
2.59536 + 1.88564i
−0.924929 2.84664i
−0.329549 1.01425i
−0.0727237 0.223820i
0.671167 + 2.06564i
0.847017 + 2.60685i
−0.924929 + 2.84664i
−0.329549 + 1.01425i
−0.0727237 + 0.223820i
0.671167 2.06564i
0.847017 2.60685i
−2.31300 + 1.68049i
−0.649647 + 0.471996i
−0.210620 + 0.153024i
1.88692 1.37093i
2.59536 1.88564i
−0.309017 + 0.951057i −2.31300 1.68049i −0.809017 0.587785i −0.771064 2.09892i 2.31300 1.68049i 1.00000 0.809017 0.587785i 1.59885 + 4.92077i 2.23446 0.0847237i
71.2 −0.309017 + 0.951057i −0.649647 0.471996i −0.809017 0.587785i 0.192568 + 2.22776i 0.649647 0.471996i 1.00000 0.809017 0.587785i −0.727790 2.23991i −2.17823 0.505273i
71.3 −0.309017 + 0.951057i −0.210620 0.153024i −0.809017 0.587785i −2.22785 + 0.191483i 0.210620 0.153024i 1.00000 0.809017 0.587785i −0.906107 2.78871i 0.506334 2.17799i
71.4 −0.309017 + 0.951057i 1.88692 + 1.37093i −0.809017 0.587785i 1.06221 1.96767i −1.88692 + 1.37093i 1.00000 0.809017 0.587785i 0.753978 + 2.32051i 1.54312 + 1.61826i
71.5 −0.309017 + 0.951057i 2.59536 + 1.88564i −0.809017 0.587785i −0.0648780 + 2.23513i −2.59536 + 1.88564i 1.00000 0.809017 0.587785i 2.25320 + 6.93464i −2.10568 0.752395i
141.1 0.809017 + 0.587785i −0.924929 2.84664i 0.309017 + 0.951057i −1.56551 1.59661i 0.924929 2.84664i 1.00000 −0.309017 + 0.951057i −4.82080 + 3.50252i −0.328061 2.21187i
141.2 0.809017 + 0.587785i −0.329549 1.01425i 0.309017 + 0.951057i −1.58296 + 1.57932i 0.329549 1.01425i 1.00000 −0.309017 + 0.951057i 1.50696 1.09487i −2.20894 + 0.347253i
141.3 0.809017 + 0.587785i −0.0727237 0.223820i 0.309017 + 0.951057i 1.97315 1.05198i 0.0727237 0.223820i 1.00000 −0.309017 + 0.951057i 2.38224 1.73080i 2.21465 + 0.308716i
141.4 0.809017 + 0.587785i 0.671167 + 2.06564i 0.309017 + 0.951057i −0.284771 + 2.21786i −0.671167 + 2.06564i 1.00000 −0.309017 + 0.951057i −1.38935 + 1.00942i −1.53401 + 1.62690i
141.5 0.809017 + 0.587785i 0.847017 + 2.60685i 0.309017 + 0.951057i 0.769108 2.09964i −0.847017 + 2.60685i 1.00000 −0.309017 + 0.951057i −3.65118 + 2.65274i 1.85636 1.24657i
211.1 0.809017 0.587785i −0.924929 + 2.84664i 0.309017 0.951057i −1.56551 + 1.59661i 0.924929 + 2.84664i 1.00000 −0.309017 0.951057i −4.82080 3.50252i −0.328061 + 2.21187i
211.2 0.809017 0.587785i −0.329549 + 1.01425i 0.309017 0.951057i −1.58296 1.57932i 0.329549 + 1.01425i 1.00000 −0.309017 0.951057i 1.50696 + 1.09487i −2.20894 0.347253i
211.3 0.809017 0.587785i −0.0727237 + 0.223820i 0.309017 0.951057i 1.97315 + 1.05198i 0.0727237 + 0.223820i 1.00000 −0.309017 0.951057i 2.38224 + 1.73080i 2.21465 0.308716i
211.4 0.809017 0.587785i 0.671167 2.06564i 0.309017 0.951057i −0.284771 2.21786i −0.671167 2.06564i 1.00000 −0.309017 0.951057i −1.38935 1.00942i −1.53401 1.62690i
211.5 0.809017 0.587785i 0.847017 2.60685i 0.309017 0.951057i 0.769108 + 2.09964i −0.847017 2.60685i 1.00000 −0.309017 0.951057i −3.65118 2.65274i 1.85636 + 1.24657i
281.1 −0.309017 0.951057i −2.31300 + 1.68049i −0.809017 + 0.587785i −0.771064 + 2.09892i 2.31300 + 1.68049i 1.00000 0.809017 + 0.587785i 1.59885 4.92077i 2.23446 + 0.0847237i
281.2 −0.309017 0.951057i −0.649647 + 0.471996i −0.809017 + 0.587785i 0.192568 2.22776i 0.649647 + 0.471996i 1.00000 0.809017 + 0.587785i −0.727790 + 2.23991i −2.17823 + 0.505273i
281.3 −0.309017 0.951057i −0.210620 + 0.153024i −0.809017 + 0.587785i −2.22785 0.191483i 0.210620 + 0.153024i 1.00000 0.809017 + 0.587785i −0.906107 + 2.78871i 0.506334 + 2.17799i
281.4 −0.309017 0.951057i 1.88692 1.37093i −0.809017 + 0.587785i 1.06221 + 1.96767i −1.88692 1.37093i 1.00000 0.809017 + 0.587785i 0.753978 2.32051i 1.54312 1.61826i
281.5 −0.309017 0.951057i 2.59536 1.88564i −0.809017 + 0.587785i −0.0648780 2.23513i −2.59536 1.88564i 1.00000 0.809017 + 0.587785i 2.25320 6.93464i −2.10568 + 0.752395i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 281.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.h.d 20
25.d even 5 1 inner 350.2.h.d 20
25.d even 5 1 8750.2.a.w 10
25.e even 10 1 8750.2.a.x 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.h.d 20 1.a even 1 1 trivial
350.2.h.d 20 25.d even 5 1 inner
8750.2.a.w 10 25.d even 5 1
8750.2.a.x 10 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{20} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5} \)
$3$ \( 1 - 3 T + 9 T^{3} - 5 T^{4} - 11 T^{5} - 28 T^{6} + 181 T^{7} - 201 T^{8} - 198 T^{9} + 342 T^{10} + 432 T^{11} + 391 T^{12} - 6101 T^{13} + 12092 T^{14} - 1175 T^{15} - 16887 T^{16} - 11609 T^{17} + 41866 T^{18} + 124287 T^{19} - 435782 T^{20} + 372861 T^{21} + 376794 T^{22} - 313443 T^{23} - 1367847 T^{24} - 285525 T^{25} + 8815068 T^{26} - 13342887 T^{27} + 2565351 T^{28} + 8503056 T^{29} + 20194758 T^{30} - 35075106 T^{31} - 106819641 T^{32} + 288572463 T^{33} - 133923132 T^{34} - 157837977 T^{35} - 215233605 T^{36} + 1162261467 T^{37} - 3486784401 T^{39} + 3486784401 T^{40} \)
$5$ \( 1 + 5 T + 30 T^{2} + 110 T^{3} + 390 T^{4} + 1125 T^{5} + 3005 T^{6} + 7340 T^{7} + 16455 T^{8} + 37775 T^{9} + 80575 T^{10} + 188875 T^{11} + 411375 T^{12} + 917500 T^{13} + 1878125 T^{14} + 3515625 T^{15} + 6093750 T^{16} + 8593750 T^{17} + 11718750 T^{18} + 9765625 T^{19} + 9765625 T^{20} \)
$7$ \( ( 1 - T )^{20} \)
$11$ \( 1 + 9 T + T^{2} - 270 T^{3} - 1106 T^{4} + 1437 T^{5} + 28254 T^{6} + 81600 T^{7} - 185184 T^{8} - 1956121 T^{9} - 3939441 T^{10} + 14683187 T^{11} + 99595113 T^{12} + 123974460 T^{13} - 845289048 T^{14} - 3927803039 T^{15} - 1004769233 T^{16} + 41106907430 T^{17} + 112816425513 T^{18} - 173691256523 T^{19} - 1636513680309 T^{20} - 1910603821753 T^{21} + 13650787487073 T^{22} + 54713293789330 T^{23} - 14710826340353 T^{24} - 632576607233989 T^{25} - 1497481111163928 T^{26} + 2415911501652660 T^{27} + 21349096975748553 T^{28} + 34622186883171217 T^{29} - 102178953907588041 T^{30} - 558104150427259931 T^{31} - 581186720514701664 T^{32} + 2817053310944769600 T^{33} + 10729451798060891214 T^{34} + 6002705619450290487 T^{35} - 50820401229110810066 T^{36} - \)\(13\!\cdots\!70\)\( T^{37} + 5559917313492231481 T^{38} + \)\(55\!\cdots\!19\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \)
$13$ \( 1 - 5 T + 180 T^{4} - 1119 T^{5} + 5205 T^{6} - 8500 T^{7} + 26300 T^{8} - 238895 T^{9} + 685648 T^{10} - 3004655 T^{11} + 11645825 T^{12} - 25367850 T^{13} + 167751875 T^{14} - 586109075 T^{15} + 1407193925 T^{16} - 9517504150 T^{17} + 31286861100 T^{18} - 80311296305 T^{19} + 353659054271 T^{20} - 1044046851965 T^{21} + 5287479525900 T^{22} - 20909956617550 T^{23} + 40190865691925 T^{24} - 217618196783975 T^{25} + 809706260016875 T^{26} - 1591794966978450 T^{27} + 9499857223889825 T^{28} - 31862862063581315 T^{29} + 94522399219283152 T^{30} - 428138157333469115 T^{31} + 612739638721250300 T^{32} - 2574438406034150500 T^{33} + 20494044087564799245 T^{34} - 57277014282767557083 T^{35} + \)\(11\!\cdots\!80\)\( T^{36} - \)\(73\!\cdots\!85\)\( T^{39} + \)\(19\!\cdots\!01\)\( T^{40} \)
$17$ \( 1 + 12 T + 6 T^{2} - 466 T^{3} - 2255 T^{4} - 2604 T^{5} + 41979 T^{6} + 475288 T^{7} + 1352471 T^{8} - 6403588 T^{9} - 54774001 T^{10} - 170639242 T^{11} + 107637136 T^{12} + 5518940082 T^{13} + 25925948559 T^{14} + 1565494782 T^{15} - 419550373786 T^{16} - 2073571073444 T^{17} - 3941818822220 T^{18} + 22861596658438 T^{19} + 188979842286129 T^{20} + 388647143193446 T^{21} - 1139185639621580 T^{22} - 10187454683830372 T^{23} - 35041266768980506 T^{24} + 2222778724686174 T^{25} + 625789372233313071 T^{26} + 2264634549614391186 T^{27} + 750850552379928976 T^{28} - 20235745355837695274 T^{29} - \)\(11\!\cdots\!49\)\( T^{30} - \)\(21\!\cdots\!04\)\( T^{31} + \)\(78\!\cdots\!31\)\( T^{32} + \)\(47\!\cdots\!56\)\( T^{33} + \)\(70\!\cdots\!91\)\( T^{34} - \)\(74\!\cdots\!72\)\( T^{35} - \)\(10\!\cdots\!55\)\( T^{36} - \)\(38\!\cdots\!82\)\( T^{37} + \)\(84\!\cdots\!54\)\( T^{38} + \)\(28\!\cdots\!36\)\( T^{39} + \)\(40\!\cdots\!01\)\( T^{40} \)
$19$ \( 1 - 2 T - 35 T^{2} - 22 T^{3} + 475 T^{4} + 682 T^{5} + 13747 T^{6} + 12118 T^{7} - 305918 T^{8} - 809522 T^{9} - 2291715 T^{10} - 21808678 T^{11} + 161227877 T^{12} + 967062682 T^{13} + 279777607 T^{14} - 5638086218 T^{15} - 48823661508 T^{16} - 380542806978 T^{17} + 242658755757 T^{18} + 4750008910922 T^{19} + 14190850058469 T^{20} + 90250169307518 T^{21} + 87599810828277 T^{22} - 2610143113062102 T^{23} - 6362748391384068 T^{24} - 13960459646303582 T^{25} + 13162384005386767 T^{26} + 864430001281343998 T^{27} + 2738223812996093957 T^{28} - 7037392095423526162 T^{29} - 14050656508996418715 T^{30} - 94301427363804041318 T^{31} - \)\(67\!\cdots\!98\)\( T^{32} + \)\(50\!\cdots\!62\)\( T^{33} + \)\(10\!\cdots\!87\)\( T^{34} + \)\(10\!\cdots\!18\)\( T^{35} + \)\(13\!\cdots\!75\)\( T^{36} - \)\(12\!\cdots\!58\)\( T^{37} - \)\(36\!\cdots\!35\)\( T^{38} - \)\(39\!\cdots\!58\)\( T^{39} + \)\(37\!\cdots\!01\)\( T^{40} \)
$23$ \( 1 + 5 T - 35 T^{2} - 120 T^{3} + 1180 T^{4} + 4415 T^{5} - 24560 T^{6} - 152080 T^{7} + 462030 T^{8} + 3027805 T^{9} - 18239153 T^{10} - 62982805 T^{11} + 423352235 T^{12} + 329737570 T^{13} - 11032544730 T^{14} - 9735553715 T^{15} + 254133694985 T^{16} + 440207234730 T^{17} - 6047843923755 T^{18} - 348343474205 T^{19} + 189940842334619 T^{20} - 8011899906715 T^{21} - 3199309435666395 T^{22} + 5356001424959910 T^{23} + 71117027338297385 T^{24} - 62661363004664245 T^{25} - 1633212567037814970 T^{26} + 1122698869167943790 T^{27} + 33153130643763453035 T^{28} - \)\(11\!\cdots\!15\)\( T^{29} - \)\(75\!\cdots\!97\)\( T^{30} + \)\(28\!\cdots\!35\)\( T^{31} + \)\(10\!\cdots\!30\)\( T^{32} - \)\(76\!\cdots\!40\)\( T^{33} - \)\(28\!\cdots\!40\)\( T^{34} + \)\(11\!\cdots\!05\)\( T^{35} + \)\(72\!\cdots\!80\)\( T^{36} - \)\(16\!\cdots\!60\)\( T^{37} - \)\(11\!\cdots\!15\)\( T^{38} + \)\(37\!\cdots\!35\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$29$ \( 1 + 22 T + 162 T^{2} + 364 T^{3} + 1221 T^{4} + 31882 T^{5} + 241485 T^{6} + 885670 T^{7} + 3609905 T^{8} + 27103550 T^{9} + 174298901 T^{10} + 735666692 T^{11} + 2745459462 T^{12} + 17332002484 T^{13} + 71322017141 T^{14} - 56024592128 T^{15} - 538624284330 T^{16} + 6226688616470 T^{17} + 24729523205740 T^{18} - 193015275164740 T^{19} - 1931410756140335 T^{20} - 5597442979777460 T^{21} + 20797529016027340 T^{22} + 151862708667086830 T^{23} - 380958722445206730 T^{24} - 1149128756801635072 T^{25} + 42423999096228545261 T^{26} + \)\(29\!\cdots\!56\)\( T^{27} + \)\(13\!\cdots\!82\)\( T^{28} + \)\(10\!\cdots\!48\)\( T^{29} + \)\(73\!\cdots\!01\)\( T^{30} + \)\(33\!\cdots\!50\)\( T^{31} + \)\(12\!\cdots\!05\)\( T^{32} + \)\(90\!\cdots\!30\)\( T^{33} + \)\(71\!\cdots\!85\)\( T^{34} + \)\(27\!\cdots\!18\)\( T^{35} + \)\(30\!\cdots\!41\)\( T^{36} + \)\(26\!\cdots\!76\)\( T^{37} + \)\(34\!\cdots\!82\)\( T^{38} + \)\(13\!\cdots\!18\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$31$ \( 1 + 7 T - 79 T^{2} - 579 T^{3} + 3032 T^{4} + 28139 T^{5} + 4936 T^{6} - 743317 T^{7} - 4912332 T^{8} + 9560721 T^{9} + 212484051 T^{10} + 81948331 T^{11} - 2914483577 T^{12} - 1389679507 T^{13} - 55278648174 T^{14} - 279870156423 T^{15} + 759056753083 T^{16} + 13090764459319 T^{17} + 147368113191249 T^{18} - 221840826023487 T^{19} - 7945817595531147 T^{20} - 6877065606728097 T^{21} + 141620756776790289 T^{22} + 389986964007572329 T^{23} + 701004851663965243 T^{24} - 8012444968627686873 T^{25} - 49060003735128928494 T^{26} - 38233716014055723277 T^{27} - \)\(24\!\cdots\!57\)\( T^{28} + \)\(21\!\cdots\!01\)\( T^{29} + \)\(17\!\cdots\!51\)\( T^{30} + \)\(24\!\cdots\!51\)\( T^{31} - \)\(38\!\cdots\!52\)\( T^{32} - \)\(18\!\cdots\!47\)\( T^{33} + \)\(37\!\cdots\!56\)\( T^{34} + \)\(66\!\cdots\!89\)\( T^{35} + \)\(22\!\cdots\!92\)\( T^{36} - \)\(13\!\cdots\!69\)\( T^{37} - \)\(55\!\cdots\!39\)\( T^{38} + \)\(15\!\cdots\!97\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \)
$37$ \( 1 + 3 T - 75 T^{2} - 431 T^{3} + 1910 T^{4} + 1531 T^{5} - 52233 T^{6} + 918601 T^{7} + 7816136 T^{8} - 5198737 T^{9} - 134683907 T^{10} - 1561396851 T^{11} - 16230592826 T^{12} - 14283194053 T^{13} + 439735380987 T^{14} + 1622954356537 T^{15} + 13721182465584 T^{16} + 141107352922883 T^{17} - 5174944573003 T^{18} - 4875052887814471 T^{19} - 34960991903139092 T^{20} - 180376956849135427 T^{21} - 7084499120441107 T^{22} + 7147510747602792599 T^{23} + 25715705050881375024 T^{24} + \)\(11\!\cdots\!09\)\( T^{25} + \)\(11\!\cdots\!83\)\( T^{26} - \)\(13\!\cdots\!49\)\( T^{27} - \)\(57\!\cdots\!46\)\( T^{28} - \)\(20\!\cdots\!27\)\( T^{29} - \)\(64\!\cdots\!43\)\( T^{30} - \)\(92\!\cdots\!81\)\( T^{31} + \)\(51\!\cdots\!16\)\( T^{32} + \)\(22\!\cdots\!97\)\( T^{33} - \)\(47\!\cdots\!37\)\( T^{34} + \)\(51\!\cdots\!83\)\( T^{35} + \)\(23\!\cdots\!10\)\( T^{36} - \)\(19\!\cdots\!27\)\( T^{37} - \)\(12\!\cdots\!75\)\( T^{38} + \)\(18\!\cdots\!19\)\( T^{39} + \)\(23\!\cdots\!01\)\( T^{40} \)
$41$ \( 1 - 19 T + 12 T^{2} + 1913 T^{3} - 9016 T^{4} - 84640 T^{5} + 727901 T^{6} + 733147 T^{7} - 19334902 T^{8} + 149819 T^{9} + 159905008 T^{10} + 1763738609 T^{11} + 15953599463 T^{12} - 202733667043 T^{13} - 1098060502629 T^{14} + 11161254620488 T^{15} + 56842341836837 T^{16} - 697945434779521 T^{17} + 859311312315416 T^{18} + 12981930863804423 T^{19} - 102522143793719309 T^{20} + 532259165415981343 T^{21} + 1444502316002214296 T^{22} - 48103097310439366841 T^{23} + \)\(16\!\cdots\!57\)\( T^{24} + \)\(12\!\cdots\!88\)\( T^{25} - \)\(52\!\cdots\!89\)\( T^{26} - \)\(39\!\cdots\!83\)\( T^{27} + \)\(12\!\cdots\!23\)\( T^{28} + \)\(57\!\cdots\!49\)\( T^{29} + \)\(21\!\cdots\!08\)\( T^{30} + \)\(82\!\cdots\!79\)\( T^{31} - \)\(43\!\cdots\!62\)\( T^{32} + \)\(67\!\cdots\!87\)\( T^{33} + \)\(27\!\cdots\!61\)\( T^{34} - \)\(13\!\cdots\!40\)\( T^{35} - \)\(57\!\cdots\!56\)\( T^{36} + \)\(50\!\cdots\!53\)\( T^{37} + \)\(12\!\cdots\!52\)\( T^{38} - \)\(83\!\cdots\!59\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$43$ \( ( 1 - T + 131 T^{2} - 222 T^{3} + 13120 T^{4} - 17529 T^{5} + 947878 T^{6} - 1296581 T^{7} + 53628143 T^{8} - 65230381 T^{9} + 2588452654 T^{10} - 2804906383 T^{11} + 99158436407 T^{12} - 103087265567 T^{13} + 3240606254278 T^{14} - 2576910997347 T^{15} + 82936283202880 T^{16} - 60343731665754 T^{17} + 1531154236365731 T^{18} - 502592611936843 T^{19} + 21611482313284249 T^{20} )^{2} \)
$47$ \( 1 + 14 T - 17 T^{2} - 332 T^{3} + 11875 T^{4} + 85608 T^{5} - 375631 T^{6} + 491530 T^{7} + 67558562 T^{8} + 169238914 T^{9} - 1464748913 T^{10} + 15388021112 T^{11} + 192696067221 T^{12} - 64267686812 T^{13} - 257501703091 T^{14} + 62244145699390 T^{15} + 272409586441108 T^{16} + 72500860159662 T^{17} + 11516861002196303 T^{18} + 93024868043107652 T^{19} + 301659113965774101 T^{20} + 4372168798026059644 T^{21} + 25440745953851633327 T^{22} + 7527256804356587826 T^{23} + \)\(13\!\cdots\!48\)\( T^{24} + \)\(14\!\cdots\!30\)\( T^{25} - \)\(27\!\cdots\!39\)\( T^{26} - \)\(32\!\cdots\!56\)\( T^{27} + \)\(45\!\cdots\!81\)\( T^{28} + \)\(17\!\cdots\!04\)\( T^{29} - \)\(77\!\cdots\!37\)\( T^{30} + \)\(41\!\cdots\!42\)\( T^{31} + \)\(78\!\cdots\!42\)\( T^{32} + \)\(26\!\cdots\!10\)\( T^{33} - \)\(96\!\cdots\!39\)\( T^{34} + \)\(10\!\cdots\!44\)\( T^{35} + \)\(67\!\cdots\!75\)\( T^{36} - \)\(88\!\cdots\!84\)\( T^{37} - \)\(21\!\cdots\!13\)\( T^{38} + \)\(82\!\cdots\!62\)\( T^{39} + \)\(27\!\cdots\!01\)\( T^{40} \)
$53$ \( 1 + T + 222 T^{2} + 576 T^{3} + 29866 T^{4} + 121159 T^{5} + 2973080 T^{6} + 16226713 T^{7} + 233637046 T^{8} + 1673331728 T^{9} + 15995234209 T^{10} + 136618682878 T^{11} + 1032730160982 T^{12} + 9346217467105 T^{13} + 66824236403364 T^{14} + 556826155800381 T^{15} + 4361081323456902 T^{16} + 29933837836232390 T^{17} + 270757625857435954 T^{18} + 1566157358977656593 T^{19} + 15195332378604934484 T^{20} + 83006340025815799429 T^{21} + \)\(76\!\cdots\!86\)\( T^{22} + \)\(44\!\cdots\!30\)\( T^{23} + \)\(34\!\cdots\!62\)\( T^{24} + \)\(23\!\cdots\!33\)\( T^{25} + \)\(14\!\cdots\!56\)\( T^{26} + \)\(10\!\cdots\!85\)\( T^{27} + \)\(64\!\cdots\!02\)\( T^{28} + \)\(45\!\cdots\!74\)\( T^{29} + \)\(27\!\cdots\!41\)\( T^{30} + \)\(15\!\cdots\!16\)\( T^{31} + \)\(11\!\cdots\!86\)\( T^{32} + \)\(42\!\cdots\!49\)\( T^{33} + \)\(41\!\cdots\!20\)\( T^{34} + \)\(88\!\cdots\!63\)\( T^{35} + \)\(11\!\cdots\!86\)\( T^{36} + \)\(11\!\cdots\!88\)\( T^{37} + \)\(24\!\cdots\!58\)\( T^{38} + \)\(57\!\cdots\!17\)\( T^{39} + \)\(30\!\cdots\!01\)\( T^{40} \)
$59$ \( 1 - 17 T - 24 T^{2} + 1080 T^{3} + 3041 T^{4} + 91298 T^{5} - 1705592 T^{6} - 6176752 T^{7} + 59242869 T^{8} + 435071881 T^{9} + 9725236734 T^{10} - 84667840483 T^{11} - 544988908505 T^{12} + 396013967022 T^{13} + 15351394464494 T^{14} + 534770086845542 T^{15} - 1784279235241195 T^{16} - 18902754975455470 T^{17} - 90766731594831180 T^{18} - 113158777818941145 T^{19} + 18511300884404128250 T^{20} - 6676367891317527555 T^{21} - \)\(31\!\cdots\!80\)\( T^{22} - \)\(38\!\cdots\!30\)\( T^{23} - \)\(21\!\cdots\!95\)\( T^{24} + \)\(38\!\cdots\!58\)\( T^{25} + \)\(64\!\cdots\!54\)\( T^{26} + \)\(98\!\cdots\!18\)\( T^{27} - \)\(80\!\cdots\!05\)\( T^{28} - \)\(73\!\cdots\!37\)\( T^{29} + \)\(49\!\cdots\!34\)\( T^{30} + \)\(13\!\cdots\!79\)\( T^{31} + \)\(10\!\cdots\!89\)\( T^{32} - \)\(64\!\cdots\!08\)\( T^{33} - \)\(10\!\cdots\!12\)\( T^{34} + \)\(33\!\cdots\!02\)\( T^{35} + \)\(65\!\cdots\!81\)\( T^{36} + \)\(13\!\cdots\!20\)\( T^{37} - \)\(18\!\cdots\!04\)\( T^{38} - \)\(75\!\cdots\!63\)\( T^{39} + \)\(26\!\cdots\!01\)\( T^{40} \)
$61$ \( 1 + 38 T + 613 T^{2} + 3660 T^{3} - 42463 T^{4} - 1139168 T^{5} - 10432970 T^{6} - 11592012 T^{7} + 876621133 T^{8} + 11435430830 T^{9} + 57987789192 T^{10} - 257674721658 T^{11} - 7267129029675 T^{12} - 60744458842252 T^{13} - 147149444018752 T^{14} + 2537573290658796 T^{15} + 36705196328155185 T^{16} + 228237801345912040 T^{17} + 134377432547021165 T^{18} - 12634742916802888610 T^{19} - \)\(14\!\cdots\!30\)\( T^{20} - \)\(77\!\cdots\!10\)\( T^{21} + \)\(50\!\cdots\!65\)\( T^{22} + \)\(51\!\cdots\!40\)\( T^{23} + \)\(50\!\cdots\!85\)\( T^{24} + \)\(21\!\cdots\!96\)\( T^{25} - \)\(75\!\cdots\!72\)\( T^{26} - \)\(19\!\cdots\!92\)\( T^{27} - \)\(13\!\cdots\!75\)\( T^{28} - \)\(30\!\cdots\!78\)\( T^{29} + \)\(41\!\cdots\!92\)\( T^{30} + \)\(49\!\cdots\!30\)\( T^{31} + \)\(23\!\cdots\!93\)\( T^{32} - \)\(18\!\cdots\!72\)\( T^{33} - \)\(10\!\cdots\!70\)\( T^{34} - \)\(68\!\cdots\!68\)\( T^{35} - \)\(15\!\cdots\!43\)\( T^{36} + \)\(82\!\cdots\!60\)\( T^{37} + \)\(83\!\cdots\!53\)\( T^{38} + \)\(31\!\cdots\!58\)\( T^{39} + \)\(50\!\cdots\!01\)\( T^{40} \)
$67$ \( 1 + 16 T - 58 T^{2} - 2978 T^{3} - 18445 T^{4} + 126976 T^{5} + 2658477 T^{6} + 13876972 T^{7} - 93385065 T^{8} - 1939016994 T^{9} - 8366709641 T^{10} + 75540871444 T^{11} + 1126536292624 T^{12} + 3842182564850 T^{13} - 44052154844443 T^{14} - 587678167507398 T^{15} - 1161816367846488 T^{16} + 27430169634446196 T^{17} + 203676466527797368 T^{18} - 490172917501845106 T^{19} - 13679143434364915411 T^{20} - 32841585472623622102 T^{21} + \)\(91\!\cdots\!52\)\( T^{22} + \)\(82\!\cdots\!48\)\( T^{23} - \)\(23\!\cdots\!48\)\( T^{24} - \)\(79\!\cdots\!86\)\( T^{25} - \)\(39\!\cdots\!67\)\( T^{26} + \)\(23\!\cdots\!50\)\( T^{27} + \)\(45\!\cdots\!84\)\( T^{28} + \)\(20\!\cdots\!68\)\( T^{29} - \)\(15\!\cdots\!09\)\( T^{30} - \)\(23\!\cdots\!02\)\( T^{31} - \)\(76\!\cdots\!65\)\( T^{32} + \)\(76\!\cdots\!64\)\( T^{33} + \)\(97\!\cdots\!33\)\( T^{34} + \)\(31\!\cdots\!68\)\( T^{35} - \)\(30\!\cdots\!45\)\( T^{36} - \)\(32\!\cdots\!06\)\( T^{37} - \)\(42\!\cdots\!22\)\( T^{38} + \)\(79\!\cdots\!48\)\( T^{39} + \)\(33\!\cdots\!01\)\( T^{40} \)
$71$ \( 1 - T - 374 T^{2} + 385 T^{3} + 59424 T^{4} - 61198 T^{5} - 4274911 T^{6} + 3121075 T^{7} - 5229794 T^{8} + 450375939 T^{9} + 21565718974 T^{10} - 89365774783 T^{11} - 422299011827 T^{12} + 5204430801845 T^{13} - 172828405013793 T^{14} + 177634632473756 T^{15} + 14915998344657807 T^{16} - 43484081312487565 T^{17} - 30084719421630102 T^{18} + 1740254535967054397 T^{19} - 48390764100475009279 T^{20} + \)\(12\!\cdots\!87\)\( T^{21} - \)\(15\!\cdots\!82\)\( T^{22} - \)\(15\!\cdots\!15\)\( T^{23} + \)\(37\!\cdots\!67\)\( T^{24} + \)\(32\!\cdots\!56\)\( T^{25} - \)\(22\!\cdots\!53\)\( T^{26} + \)\(47\!\cdots\!95\)\( T^{27} - \)\(27\!\cdots\!47\)\( T^{28} - \)\(40\!\cdots\!73\)\( T^{29} + \)\(70\!\cdots\!74\)\( T^{30} + \)\(10\!\cdots\!69\)\( T^{31} - \)\(85\!\cdots\!54\)\( T^{32} + \)\(36\!\cdots\!25\)\( T^{33} - \)\(35\!\cdots\!91\)\( T^{34} - \)\(35\!\cdots\!98\)\( T^{35} + \)\(24\!\cdots\!04\)\( T^{36} + \)\(11\!\cdots\!35\)\( T^{37} - \)\(78\!\cdots\!14\)\( T^{38} - \)\(14\!\cdots\!31\)\( T^{39} + \)\(10\!\cdots\!01\)\( T^{40} \)
$73$ \( 1 - 19 T - 37 T^{2} + 1911 T^{3} + 14984 T^{4} - 343095 T^{5} + 44886 T^{6} + 20424421 T^{7} + 38131320 T^{8} - 2107575143 T^{9} + 6644560253 T^{10} + 31127895089 T^{11} + 163541353759 T^{12} - 996108478075 T^{13} + 8406248222548 T^{14} - 631161430088471 T^{15} + 5329956394145463 T^{16} + 17871030714682627 T^{17} - 359037128943857193 T^{18} - 2008361662490082115 T^{19} + 51740292756835167037 T^{20} - \)\(14\!\cdots\!95\)\( T^{21} - \)\(19\!\cdots\!97\)\( T^{22} + \)\(69\!\cdots\!59\)\( T^{23} + \)\(15\!\cdots\!83\)\( T^{24} - \)\(13\!\cdots\!03\)\( T^{25} + \)\(12\!\cdots\!72\)\( T^{26} - \)\(11\!\cdots\!75\)\( T^{27} + \)\(13\!\cdots\!79\)\( T^{28} + \)\(18\!\cdots\!57\)\( T^{29} + \)\(28\!\cdots\!97\)\( T^{30} - \)\(66\!\cdots\!11\)\( T^{31} + \)\(87\!\cdots\!20\)\( T^{32} + \)\(34\!\cdots\!93\)\( T^{33} + \)\(54\!\cdots\!74\)\( T^{34} - \)\(30\!\cdots\!15\)\( T^{35} + \)\(97\!\cdots\!24\)\( T^{36} + \)\(90\!\cdots\!83\)\( T^{37} - \)\(12\!\cdots\!53\)\( T^{38} - \)\(48\!\cdots\!03\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$79$ \( 1 + 64 T + 1813 T^{2} + 28300 T^{3} + 220923 T^{4} - 272960 T^{5} - 26038614 T^{6} - 272781482 T^{7} - 1191792493 T^{8} - 1015821708 T^{9} + 10837883272 T^{10} + 850456415848 T^{11} + 23123314759077 T^{12} + 258076348414252 T^{13} + 716331205356126 T^{14} - 13571832494284164 T^{15} - 144937325341815863 T^{16} - 90759777606546922 T^{17} + 6424026371015525703 T^{18} + 21997663317142103344 T^{19} - \)\(14\!\cdots\!34\)\( T^{20} + \)\(17\!\cdots\!76\)\( T^{21} + \)\(40\!\cdots\!23\)\( T^{22} - \)\(44\!\cdots\!58\)\( T^{23} - \)\(56\!\cdots\!03\)\( T^{24} - \)\(41\!\cdots\!36\)\( T^{25} + \)\(17\!\cdots\!46\)\( T^{26} + \)\(49\!\cdots\!68\)\( T^{27} + \)\(35\!\cdots\!97\)\( T^{28} + \)\(10\!\cdots\!12\)\( T^{29} + \)\(10\!\cdots\!72\)\( T^{30} - \)\(75\!\cdots\!32\)\( T^{31} - \)\(70\!\cdots\!13\)\( T^{32} - \)\(12\!\cdots\!98\)\( T^{33} - \)\(96\!\cdots\!34\)\( T^{34} - \)\(79\!\cdots\!40\)\( T^{35} + \)\(50\!\cdots\!83\)\( T^{36} + \)\(51\!\cdots\!00\)\( T^{37} + \)\(26\!\cdots\!93\)\( T^{38} + \)\(72\!\cdots\!16\)\( T^{39} + \)\(89\!\cdots\!01\)\( T^{40} \)
$83$ \( 1 - 57 T + 1531 T^{2} - 25602 T^{3} + 283378 T^{4} - 1661127 T^{5} - 8691182 T^{6} + 349426040 T^{7} - 4444505934 T^{8} + 28616053427 T^{9} + 47050780495 T^{10} - 3583740129277 T^{11} + 45464734870983 T^{12} - 281124959027648 T^{13} - 395656472000900 T^{14} + 30726600590053373 T^{15} - 378959258485740125 T^{16} + 2393776145680167358 T^{17} + 572766742440286311 T^{18} - \)\(20\!\cdots\!17\)\( T^{19} + \)\(26\!\cdots\!27\)\( T^{20} - \)\(17\!\cdots\!11\)\( T^{21} + \)\(39\!\cdots\!79\)\( T^{22} + \)\(13\!\cdots\!46\)\( T^{23} - \)\(17\!\cdots\!25\)\( T^{24} + \)\(12\!\cdots\!39\)\( T^{25} - \)\(12\!\cdots\!00\)\( T^{26} - \)\(76\!\cdots\!96\)\( T^{27} + \)\(10\!\cdots\!03\)\( T^{28} - \)\(66\!\cdots\!31\)\( T^{29} + \)\(73\!\cdots\!55\)\( T^{30} + \)\(36\!\cdots\!09\)\( T^{31} - \)\(47\!\cdots\!74\)\( T^{32} + \)\(31\!\cdots\!20\)\( T^{33} - \)\(63\!\cdots\!78\)\( T^{34} - \)\(10\!\cdots\!89\)\( T^{35} + \)\(14\!\cdots\!18\)\( T^{36} - \)\(10\!\cdots\!46\)\( T^{37} + \)\(53\!\cdots\!79\)\( T^{38} - \)\(16\!\cdots\!79\)\( T^{39} + \)\(24\!\cdots\!01\)\( T^{40} \)
$89$ \( 1 + 6 T - 225 T^{2} - 2414 T^{3} + 10475 T^{4} + 275574 T^{5} + 2441593 T^{6} + 4668416 T^{7} - 252701912 T^{8} - 3560779949 T^{9} - 8899915733 T^{10} + 174253060095 T^{11} + 1975857342718 T^{12} + 12822124485500 T^{13} + 98844654551923 T^{14} - 556869091622998 T^{15} - 19095987187955715 T^{16} - 193768608185247050 T^{17} - 611163344428279505 T^{18} + 12505565957375472620 T^{19} + \)\(22\!\cdots\!40\)\( T^{20} + \)\(11\!\cdots\!80\)\( T^{21} - \)\(48\!\cdots\!05\)\( T^{22} - \)\(13\!\cdots\!50\)\( T^{23} - \)\(11\!\cdots\!15\)\( T^{24} - \)\(31\!\cdots\!02\)\( T^{25} + \)\(49\!\cdots\!03\)\( T^{26} + \)\(56\!\cdots\!00\)\( T^{27} + \)\(77\!\cdots\!58\)\( T^{28} + \)\(61\!\cdots\!55\)\( T^{29} - \)\(27\!\cdots\!33\)\( T^{30} - \)\(98\!\cdots\!61\)\( T^{31} - \)\(62\!\cdots\!52\)\( T^{32} + \)\(10\!\cdots\!04\)\( T^{33} + \)\(47\!\cdots\!13\)\( T^{34} + \)\(47\!\cdots\!26\)\( T^{35} + \)\(16\!\cdots\!75\)\( T^{36} - \)\(33\!\cdots\!06\)\( T^{37} - \)\(27\!\cdots\!25\)\( T^{38} + \)\(65\!\cdots\!54\)\( T^{39} + \)\(97\!\cdots\!01\)\( T^{40} \)
$97$ \( 1 + 18 T - 181 T^{2} - 5224 T^{3} - 12425 T^{4} + 240780 T^{5} + 3449805 T^{6} + 82687874 T^{7} + 326931282 T^{8} - 11449359402 T^{9} - 99823613357 T^{10} - 119817410092 T^{11} + 1031495320729 T^{12} + 126529834617016 T^{13} + 1496866997685305 T^{14} - 6508878292176770 T^{15} - 128528710934989580 T^{16} - 589248127879190526 T^{17} - 8372017197617105093 T^{18} + 50791694907309315928 T^{19} + \)\(20\!\cdots\!29\)\( T^{20} + \)\(49\!\cdots\!16\)\( T^{21} - \)\(78\!\cdots\!37\)\( T^{22} - \)\(53\!\cdots\!98\)\( T^{23} - \)\(11\!\cdots\!80\)\( T^{24} - \)\(55\!\cdots\!90\)\( T^{25} + \)\(12\!\cdots\!45\)\( T^{26} + \)\(10\!\cdots\!08\)\( T^{27} + \)\(80\!\cdots\!69\)\( T^{28} - \)\(91\!\cdots\!64\)\( T^{29} - \)\(73\!\cdots\!93\)\( T^{30} - \)\(81\!\cdots\!06\)\( T^{31} + \)\(22\!\cdots\!62\)\( T^{32} + \)\(55\!\cdots\!98\)\( T^{33} + \)\(22\!\cdots\!45\)\( T^{34} + \)\(15\!\cdots\!40\)\( T^{35} - \)\(76\!\cdots\!25\)\( T^{36} - \)\(31\!\cdots\!88\)\( T^{37} - \)\(10\!\cdots\!09\)\( T^{38} + \)\(10\!\cdots\!94\)\( T^{39} + \)\(54\!\cdots\!01\)\( T^{40} \)
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