Properties

Label 1003.2.a.i
Level 1003
Weight 2
Character orbit 1003.a
Self dual yes
Analytic conductor 8.009
Analytic rank 0
Dimension 18
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1003 = 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1003.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.00899532273\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 5 x^{17} - 16 x^{16} + 112 x^{15} + 52 x^{14} - 984 x^{13} + 431 x^{12} + 4304 x^{11} - 3825 x^{10} - 9826 x^{9} + 11533 x^{8} + 11182 x^{7} - 15697 x^{6} - 5604 x^{5} + 9201 x^{4} + 1189 x^{3} - 1952 x^{2} - 100 x + 28\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(1603205 \nu^{17} + 27603530 \nu^{16} - 195692906 \nu^{15} - 437835062 \nu^{14} + 3840968938 \nu^{13} + 1553634958 \nu^{12} - 31655264755 \nu^{11} + 8453508555 \nu^{10} + 130824516472 \nu^{9} - 75963419650 \nu^{8} - 276891915821 \nu^{7} + 202318455723 \nu^{6} + 273476087608 \nu^{5} - 212478348524 \nu^{4} - 93338201887 \nu^{3} + 64993586656 \nu^{2} + 5242626976 \nu - 1040316564\)\()/ 568793824 \)
\(\beta_{4}\)\(=\)\((\)\(4468293 \nu^{17} - 9046678 \nu^{16} - 104332442 \nu^{15} + 172115546 \nu^{14} + 1061307674 \nu^{13} - 1245763682 \nu^{12} - 6337233987 \nu^{11} + 4626655131 \nu^{10} + 23993616376 \nu^{9} - 11457941330 \nu^{8} - 54362020477 \nu^{7} + 23150614939 \nu^{6} + 63467758040 \nu^{5} - 30360885660 \nu^{4} - 29262232015 \nu^{3} + 16437157520 \nu^{2} + 2591601760 \nu - 1680834164\)\()/ 568793824 \)
\(\beta_{5}\)\(=\)\((\)\(-6526647 \nu^{17} + 84688962 \nu^{16} - 83459586 \nu^{15} - 1781428510 \nu^{14} + 3847479106 \nu^{13} + 14100829750 \nu^{12} - 39180669471 \nu^{11} - 50706085177 \nu^{10} + 180240532008 \nu^{9} + 72173081990 \nu^{8} - 411676152609 \nu^{7} + 12535086055 \nu^{6} + 440305730504 \nu^{5} - 99577814860 \nu^{4} - 176724978555 \nu^{3} + 33075248352 \nu^{2} + 19243183584 \nu + 3061678300\)\()/ 568793824 \)
\(\beta_{6}\)\(=\)\((\)\(7745729 \nu^{17} - 11556334 \nu^{16} - 262206418 \nu^{15} + 431026258 \nu^{14} + 3455427122 \nu^{13} - 6013468890 \nu^{12} - 22957637767 \nu^{11} + 41448412623 \nu^{10} + 82189416168 \nu^{9} - 152435041482 \nu^{8} - 154717191481 \nu^{7} + 295042593455 \nu^{6} + 132442588328 \nu^{5} - 267468538284 \nu^{4} - 28420424579 \nu^{3} + 78337756256 \nu^{2} - 4274427680 \nu - 599729124\)\()/ 568793824 \)
\(\beta_{7}\)\(=\)\((\)\(6002425 \nu^{17} - 5938506 \nu^{16} - 196373670 \nu^{15} + 220312190 \nu^{14} + 2543960726 \nu^{13} - 3055625518 \nu^{12} - 16838568259 \nu^{11} + 20921021015 \nu^{10} + 60917184580 \nu^{9} - 76407838686 \nu^{8} - 118650382749 \nu^{7} + 147120641455 \nu^{6} + 112629948452 \nu^{5} - 133664775808 \nu^{4} - 40703624959 \nu^{3} + 40558943600 \nu^{2} + 4552577048 \nu - 736631868\)\()/ 142198456 \)
\(\beta_{8}\)\(=\)\((\)\(-30052531 \nu^{17} + 84099226 \nu^{16} + 722209126 \nu^{15} - 2030406822 \nu^{14} - 6987875910 \nu^{13} + 19790303422 \nu^{12} + 34840412021 \nu^{11} - 100266293789 \nu^{10} - 94156691400 \nu^{9} + 282941145998 \nu^{8} + 127822374123 \nu^{7} - 438843100125 \nu^{6} - 55429121448 \nu^{5} + 336609343060 \nu^{4} - 33137195863 \nu^{3} - 91595317216 \nu^{2} + 19664554208 \nu + 1835590924\)\()/ 568793824 \)
\(\beta_{9}\)\(=\)\((\)\(33160165 \nu^{17} - 186790710 \nu^{16} - 379778058 \nu^{15} + 3887312362 \nu^{14} - 1510725526 \nu^{13} - 30205793170 \nu^{12} + 40801027885 \nu^{11} + 104573869707 \nu^{10} - 229270906824 \nu^{9} - 131568240834 \nu^{8} + 562653989427 \nu^{7} - 77018484181 \nu^{6} - 608004675912 \nu^{5} + 259080908628 \nu^{4} + 225691904993 \nu^{3} - 93309825920 \nu^{2} - 16222142464 \nu - 1088955028\)\()/ 568793824 \)
\(\beta_{10}\)\(=\)\((\)\(-37154163 \nu^{17} + 184167610 \nu^{16} + 566863078 \nu^{15} - 3965573350 \nu^{14} - 1494181414 \nu^{13} + 32718727454 \nu^{12} - 17567079211 \nu^{11} - 128256252797 \nu^{10} + 133661164920 \nu^{9} + 234252289166 \nu^{8} - 352535542229 \nu^{7} - 138565934845 \nu^{6} + 380890440888 \nu^{5} - 65264158156 \nu^{4} - 129377105239 \nu^{3} + 49161902080 \nu^{2} + 7531339520 \nu - 1527210676\)\()/ 568793824 \)
\(\beta_{11}\)\(=\)\((\)\(6040615 \nu^{17} - 22234052 \nu^{16} - 122475798 \nu^{15} + 499734910 \nu^{14} + 913780746 \nu^{13} - 4407747754 \nu^{12} - 2941782421 \nu^{11} + 19364549555 \nu^{10} + 3047670594 \nu^{9} - 44421256370 \nu^{8} + 2692357581 \nu^{7} + 50814226103 \nu^{6} - 3464216802 \nu^{5} - 25574527844 \nu^{4} - 4369624325 \nu^{3} + 5272197858 \nu^{2} + 2393923808 \nu - 231904632\)\()/71099228\)
\(\beta_{12}\)\(=\)\((\)\(6053175 \nu^{17} - 28282200 \nu^{16} - 97976644 \nu^{15} + 613684104 \nu^{14} + 360506880 \nu^{13} - 5128036560 \nu^{12} + 1941081393 \nu^{11} + 20581292633 \nu^{10} - 18409013782 \nu^{9} - 39689575104 \nu^{8} + 51907281475 \nu^{7} + 29244763413 \nu^{6} - 59382333630 \nu^{5} + 1840425726 \nu^{4} + 22352348333 \nu^{3} - 4229868864 \nu^{2} - 1770614060 \nu - 64456832\)\()/71099228\)
\(\beta_{13}\)\(=\)\((\)\(-25731019 \nu^{17} + 132425074 \nu^{16} + 392850758 \nu^{15} - 2917574854 \nu^{14} - 986596822 \nu^{13} + 24952854334 \nu^{12} - 13174676275 \nu^{11} - 104117203501 \nu^{10} + 100062459136 \nu^{9} + 216787275502 \nu^{8} - 269465073261 \nu^{7} - 198840292381 \nu^{6} + 303110008560 \nu^{5} + 47579502788 \nu^{4} - 113857594079 \nu^{3} + 4349495704 \nu^{2} + 8570974544 \nu - 301716580\)\()/ 284396912 \)
\(\beta_{14}\)\(=\)\((\)\(-27616971 \nu^{17} + 102983154 \nu^{16} + 576714958 \nu^{15} - 2389076942 \nu^{14} - 4512375294 \nu^{13} + 22083367798 \nu^{12} + 15817530869 \nu^{11} - 104218462469 \nu^{10} - 20090353488 \nu^{9} + 267515584966 \nu^{8} - 15371453797 \nu^{7} - 366506974869 \nu^{6} + 56580628080 \nu^{5} + 242495850636 \nu^{4} - 35961353735 \nu^{3} - 59079830624 \nu^{2} + 5349825328 \nu + 656205004\)\()/ 284396912 \)
\(\beta_{15}\)\(=\)\((\)\(-55864183 \nu^{17} + 316018786 \nu^{16} + 711941006 \nu^{15} - 6767846030 \nu^{14} + 963704754 \nu^{13} + 55256076454 \nu^{12} - 55249056879 \nu^{11} - 211764044169 \nu^{10} + 329590031528 \nu^{9} + 364158983542 \nu^{8} - 825445861681 \nu^{7} - 153858197321 \nu^{6} + 895533906920 \nu^{5} - 189752917180 \nu^{4} - 324651015307 \nu^{3} + 99470541936 \nu^{2} + 21482869568 \nu - 1512586372\)\()/ 568793824 \)
\(\beta_{16}\)\(=\)\((\)\(-55930331 \nu^{17} + 273896826 \nu^{16} + 890033958 \nu^{15} - 6007265254 \nu^{14} - 3034501318 \nu^{13} + 51151472350 \nu^{12} - 20012118563 \nu^{11} - 212861062133 \nu^{10} + 176131301896 \nu^{9} + 445032492334 \nu^{8} - 484568126045 \nu^{7} - 420831199701 \nu^{6} + 542752259080 \nu^{5} + 125519891236 \nu^{4} - 197884162319 \nu^{3} - 7738166112 \nu^{2} + 11706378208 \nu + 1077401004\)\()/ 568793824 \)
\(\beta_{17}\)\(=\)\((\)\(49926317 \nu^{17} - 203107614 \nu^{16} - 934344946 \nu^{15} + 4510183266 \nu^{14} + 5763599234 \nu^{13} - 39016219162 \nu^{12} - 8103174819 \nu^{11} + 165748033427 \nu^{10} - 48552214560 \nu^{9} - 356503271578 \nu^{8} + 199306839139 \nu^{7} + 352769386931 \nu^{6} - 244328770576 \nu^{5} - 115388668836 \nu^{4} + 81114719985 \nu^{3} + 3188992512 \nu^{2} - 1751857152 \nu + 555600236\)\()/ 284396912 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{16} - \beta_{13} + \beta_{4} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{16} - \beta_{14} - \beta_{13} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 6 \beta_{2} + \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(\beta_{17} + 8 \beta_{16} - \beta_{15} - \beta_{14} - 7 \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{8} + 9 \beta_{4} + \beta_{2} + 29 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{17} + 10 \beta_{16} - \beta_{15} - 11 \beta_{14} - 8 \beta_{13} + 12 \beta_{10} + 10 \beta_{9} + 11 \beta_{8} - 11 \beta_{7} + 11 \beta_{6} + 9 \beta_{5} + 11 \beta_{4} + 11 \beta_{3} + 36 \beta_{2} + 11 \beta_{1} + 76\)
\(\nu^{7}\)\(=\)\(13 \beta_{17} + 55 \beta_{16} - 13 \beta_{15} - 12 \beta_{14} - 42 \beta_{13} - 12 \beta_{12} - 11 \beta_{11} + 15 \beta_{10} + 12 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} + 68 \beta_{4} + 2 \beta_{3} + 12 \beta_{2} + 179 \beta_{1} + 2\)
\(\nu^{8}\)\(=\)\(15 \beta_{17} + 78 \beta_{16} - 17 \beta_{15} - 93 \beta_{14} - 48 \beta_{13} - 3 \beta_{12} + 106 \beta_{10} + 76 \beta_{9} + 91 \beta_{8} - 93 \beta_{7} + 92 \beta_{6} + 65 \beta_{5} + 91 \beta_{4} + 90 \beta_{3} + 224 \beta_{2} + 94 \beta_{1} + 446\)
\(\nu^{9}\)\(=\)\(122 \beta_{17} + 364 \beta_{16} - 127 \beta_{15} - 109 \beta_{14} - 241 \beta_{13} - 109 \beta_{12} - 88 \beta_{11} + 160 \beta_{10} - \beta_{9} + 106 \beta_{8} - 50 \beta_{7} + 44 \beta_{6} + 2 \beta_{5} + 486 \beta_{4} + 34 \beta_{3} + 109 \beta_{2} + 1143 \beta_{1} + 32\)
\(\nu^{10}\)\(=\)\(155 \beta_{17} + 564 \beta_{16} - 193 \beta_{15} - 717 \beta_{14} - 254 \beta_{13} - 56 \beta_{12} + 3 \beta_{11} + 844 \beta_{10} + 530 \beta_{9} + 681 \beta_{8} - 719 \beta_{7} + 700 \beta_{6} + 441 \beta_{5} + 688 \beta_{4} + 669 \beta_{3} + 1432 \beta_{2} + 739 \beta_{1} + 2735\)
\(\nu^{11}\)\(=\)\(1011 \beta_{17} + 2384 \beta_{16} - 1107 \beta_{15} - 899 \beta_{14} - 1352 \beta_{13} - 899 \beta_{12} - 625 \beta_{11} + 1474 \beta_{10} - 15 \beta_{9} + 836 \beta_{8} - 564 \beta_{7} + 443 \beta_{6} + 38 \beta_{5} + 3394 \beta_{4} + 393 \beta_{3} + 894 \beta_{2} + 7452 \beta_{1} + 354\)
\(\nu^{12}\)\(=\)\(1380 \beta_{17} + 3965 \beta_{16} - 1845 \beta_{15} - 5304 \beta_{14} - 1209 \beta_{13} - 688 \beta_{12} + 63 \beta_{11} + 6414 \beta_{10} + 3576 \beta_{9} + 4870 \beta_{8} - 5346 \beta_{7} + 5093 \beta_{6} + 2927 \beta_{5} + 5013 \beta_{4} + 4793 \beta_{3} + 9313 \beta_{2} + 5616 \beta_{1} + 17216\)
\(\nu^{13}\)\(=\)\(7879 \beta_{17} + 15598 \beta_{16} - 9065 \beta_{15} - 7081 \beta_{14} - 7445 \beta_{13} - 7094 \beta_{12} - 4182 \beta_{11} + 12518 \beta_{10} - 143 \beta_{9} + 6244 \beta_{8} - 5391 \beta_{7} + 3825 \beta_{6} + 464 \beta_{5} + 23452 \beta_{4} + 3845 \beta_{3} + 6972 \beta_{2} + 49257 \beta_{1} + 3370\)
\(\nu^{14}\)\(=\)\(11387 \beta_{17} + 27571 \beta_{16} - 16076 \beta_{15} - 38425 \beta_{14} - 4911 \beta_{13} - 7041 \beta_{12} + 850 \beta_{11} + 47581 \beta_{10} + 23809 \beta_{9} + 34021 \beta_{8} - 38996 \beta_{7} + 36135 \beta_{6} + 19272 \beta_{5} + 35895 \beta_{4} + 33854 \beta_{3} + 61221 \beta_{2} + 42021 \beta_{1} + 110171\)
\(\nu^{15}\)\(=\)\(59325 \beta_{17} + 102323 \beta_{16} - 71375 \beta_{15} - 54323 \beta_{14} - 40109 \beta_{13} - 54653 \beta_{12} - 26980 \beta_{11} + 101089 \beta_{10} - 1090 \beta_{9} + 45257 \beta_{8} - 47120 \beta_{7} + 30483 \beta_{6} + 4651 \beta_{5} + 161238 \beta_{4} + 34282 \beta_{3} + 52807 \beta_{2} + 328761 \beta_{1} + 29609\)
\(\nu^{16}\)\(=\)\(89935 \beta_{17} + 190938 \beta_{16} - 132420 \beta_{15} - 275216 \beta_{14} - 13036 \beta_{13} - 65140 \beta_{12} + 9396 \beta_{11} + 348233 \beta_{10} + 157735 \beta_{9} + 234565 \beta_{8} - 281620 \beta_{7} + 252377 \beta_{6} + 126578 \beta_{5} + 254710 \beta_{4} + 237978 \beta_{3} + 405279 \beta_{2} + 311854 \beta_{1} + 712952\)
\(\nu^{17}\)\(=\)\(437508 \beta_{17} + 673994 \beta_{16} - 547300 \beta_{15} - 409785 \beta_{14} - 209403 \beta_{13} - 414859 \beta_{12} - 169425 \beta_{11} + 789771 \beta_{10} - 7063 \beta_{9} + 322144 \beta_{8} - 390021 \beta_{7} + 231689 \beta_{6} + 41738 \beta_{5} + 1105998 \beta_{4} + 288301 \beta_{3} + 392490 \beta_{2} + 2210366 \beta_{1} + 247515\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.55956
−2.46111
−1.89397
−1.84714
−1.38222
−0.714378
−0.690432
−0.149899
0.101226
0.715815
1.14480
1.32090
1.64867
1.79692
2.26170
2.45136
2.58784
2.66948
−2.55956 −0.639116 4.55133 2.79521 1.63585 −2.94129 −6.53027 −2.59153 −7.15450
1.2 −2.46111 2.84715 4.05705 −0.414665 −7.00714 −0.439244 −5.06262 5.10626 1.02054
1.3 −1.89397 −0.0387693 1.58712 3.00387 0.0734279 0.643981 0.781977 −2.99850 −5.68925
1.4 −1.84714 −2.22957 1.41194 −0.984600 4.11833 2.90768 1.08623 1.97096 1.81870
1.5 −1.38222 1.10475 −0.0894560 −3.03767 −1.52702 −3.07306 2.88810 −1.77952 4.19874
1.6 −0.714378 −2.62504 −1.48966 0.882657 1.87527 −0.838797 2.49294 3.89086 −0.630551
1.7 −0.690432 1.58357 −1.52330 3.69903 −1.09335 2.15550 2.43260 −0.492301 −2.55393
1.8 −0.149899 −2.07180 −1.97753 4.08629 0.310561 5.08021 0.596228 1.29235 −0.612531
1.9 0.101226 −0.700214 −1.98975 0.497951 −0.0708800 −5.07659 −0.403867 −2.50970 0.0504057
1.10 0.715815 3.10981 −1.48761 0.799569 2.22605 −0.891306 −2.49648 6.67092 0.572344
1.11 1.14480 −1.66658 −0.689428 −2.63259 −1.90791 −1.31875 −3.07886 −0.222506 −3.01379
1.12 1.32090 −3.03201 −0.255233 3.39315 −4.00497 −0.854531 −2.97893 6.19308 4.48200
1.13 1.64867 2.75148 0.718114 4.35847 4.53629 −3.72637 −2.11341 4.57066 7.18568
1.14 1.79692 1.63689 1.22893 2.71894 2.94136 4.07184 −1.38555 −0.320597 4.88573
1.15 2.26170 −1.91993 3.11529 −0.621807 −4.34231 3.74378 2.52246 0.686140 −1.40634
1.16 2.45136 2.33620 4.00917 −2.57056 5.72687 2.10143 4.92520 2.45783 −6.30136
1.17 2.58784 −0.402595 4.69689 3.31536 −1.04185 −2.14234 6.97912 −2.83792 8.57961
1.18 2.66948 0.955769 5.12613 1.71138 2.55141 −0.402139 8.34515 −2.08651 4.56850
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1003.2.a.i 18
3.b odd 2 1 9027.2.a.q 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.2.a.i 18 1.a even 1 1 trivial
9027.2.a.q 18 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(17\) \(-1\)
\(59\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1003))\):

\(T_{2}^{18} - \cdots\)
\(T_{3}^{18} - \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5 T + 20 T^{2} - 58 T^{3} + 152 T^{4} - 344 T^{5} + 735 T^{6} - 1440 T^{7} + 2727 T^{8} - 4866 T^{9} + 8497 T^{10} - 14198 T^{11} + 23403 T^{12} - 37248 T^{13} + 58661 T^{14} - 89451 T^{15} + 134796 T^{16} - 196502 T^{17} + 282612 T^{18} - 393004 T^{19} + 539184 T^{20} - 715608 T^{21} + 938576 T^{22} - 1191936 T^{23} + 1497792 T^{24} - 1817344 T^{25} + 2175232 T^{26} - 2491392 T^{27} + 2792448 T^{28} - 2949120 T^{29} + 3010560 T^{30} - 2818048 T^{31} + 2490368 T^{32} - 1900544 T^{33} + 1310720 T^{34} - 655360 T^{35} + 262144 T^{36} \)
$3$ \( 1 - T + 19 T^{2} - 22 T^{3} + 201 T^{4} - 250 T^{5} + 1538 T^{6} - 1963 T^{7} + 9399 T^{8} - 11928 T^{9} + 48218 T^{10} - 59862 T^{11} + 213988 T^{12} - 257547 T^{13} + 838124 T^{14} - 971758 T^{15} + 2935787 T^{16} - 3258875 T^{17} + 9267386 T^{18} - 9776625 T^{19} + 26422083 T^{20} - 26237466 T^{21} + 67888044 T^{22} - 62583921 T^{23} + 155997252 T^{24} - 130918194 T^{25} + 316358298 T^{26} - 234778824 T^{27} + 555001551 T^{28} - 347739561 T^{29} + 817356258 T^{30} - 398580750 T^{31} + 961376769 T^{32} - 315675954 T^{33} + 817887699 T^{34} - 129140163 T^{35} + 387420489 T^{36} \)
$5$ \( 1 - 21 T + 248 T^{2} - 2112 T^{3} + 14390 T^{4} - 82850 T^{5} + 417181 T^{6} - 1880214 T^{7} + 7712027 T^{8} - 29143104 T^{9} + 102409048 T^{10} - 337007088 T^{11} + 1044205383 T^{12} - 3058958706 T^{13} + 8499119894 T^{14} - 22450225214 T^{15} + 56477307355 T^{16} - 135473844787 T^{17} + 310079023954 T^{18} - 677369223935 T^{19} + 1411932683875 T^{20} - 2806278151750 T^{21} + 5311949933750 T^{22} - 9559245956250 T^{23} + 16315709109375 T^{24} - 26328678750000 T^{25} + 40003534375000 T^{26} - 56920125000000 T^{27} + 75312763671875 T^{28} - 91807324218750 T^{29} + 101850830078125 T^{30} - 101135253906250 T^{31} + 87829589843750 T^{32} - 64453125000000 T^{33} + 37841796875000 T^{34} - 16021728515625 T^{35} + 3814697265625 T^{36} \)
$7$ \( 1 + T + 56 T^{2} + 34 T^{3} + 1518 T^{4} + 485 T^{5} + 26882 T^{6} + 3211 T^{7} + 350122 T^{8} - 1782 T^{9} + 3562855 T^{10} - 221519 T^{11} + 29467805 T^{12} - 1805704 T^{13} + 206767815 T^{14} - 3783304 T^{15} + 1329201842 T^{16} + 43203918 T^{17} + 8830674752 T^{18} + 302427426 T^{19} + 65130890258 T^{20} - 1297673272 T^{21} + 496449523815 T^{22} - 30348467128 T^{23} + 3466857790445 T^{24} - 182430421817 T^{25} + 20539150066855 T^{26} - 71910127674 T^{27} + 98900799130378 T^{28} + 6349196171773 T^{29} + 372081482537282 T^{30} + 46991170047395 T^{31} + 1029542624584782 T^{32} + 161417091338062 T^{33} + 1861044111897656 T^{34} + 232630513987207 T^{35} + 1628413597910449 T^{36} \)
$11$ \( 1 - 6 T + 127 T^{2} - 689 T^{3} + 7708 T^{4} - 38227 T^{5} + 298469 T^{6} - 1369387 T^{7} + 8327943 T^{8} - 35790520 T^{9} + 179869059 T^{10} - 732797929 T^{11} + 3164067020 T^{12} - 12337375884 T^{13} + 47169541671 T^{14} - 176813586395 T^{15} + 614445550848 T^{16} - 2207242164895 T^{17} + 7134211635492 T^{18} - 24279663813845 T^{19} + 74347911652608 T^{20} - 235338883491745 T^{21} + 690609259605111 T^{22} - 1986946723494084 T^{23} + 5605337734018220 T^{24} - 14280158550868859 T^{25} + 38556530213762979 T^{26} - 84392173993689320 T^{27} + 216005393643925743 T^{28} - 390702092682985457 T^{29} + 936723579171540149 T^{30} - 1319699717126050337 T^{31} + 2927111717259621628 T^{32} - 2878123988727383539 T^{33} + 5835615692673664447 T^{34} - 3032682170995762626 T^{35} + 5559917313492231481 T^{36} \)
$13$ \( 1 - 12 T + 195 T^{2} - 1722 T^{3} + 16570 T^{4} - 117747 T^{5} + 854099 T^{6} - 5136447 T^{7} + 30754314 T^{8} - 161864038 T^{9} + 839694908 T^{10} - 3965488172 T^{11} + 18391231959 T^{12} - 79463553929 T^{13} + 336854135767 T^{14} - 1350987506629 T^{15} + 5315299094844 T^{16} - 19964398048608 T^{17} + 73562236154478 T^{18} - 259537174631904 T^{19} + 898285547028636 T^{20} - 2968119552063913 T^{21} + 9620890971641287 T^{22} - 29504261328960197 T^{23} + 88770963940788831 T^{24} - 248828501974040924 T^{25} + 684964932722868668 T^{26} - 1716487089482248174 T^{27} + 4239743345890586586 T^{28} - 9205336879470166539 T^{29} + 19898871205025899619 T^{30} - 35662635175918013991 T^{31} + 65242326711037218730 T^{32} - 88142107770264283554 T^{33} + \)\(12\!\cdots\!95\)\( T^{34} - \)\(10\!\cdots\!96\)\( T^{35} + \)\(11\!\cdots\!29\)\( T^{36} \)
$17$ \( ( 1 - T )^{18} \)
$19$ \( 1 + 14 T + 290 T^{2} + 3029 T^{3} + 36817 T^{4} + 314426 T^{5} + 2870012 T^{6} + 21074543 T^{7} + 158491227 T^{8} + 1032221026 T^{9} + 6708312133 T^{10} + 39543233724 T^{11} + 228501358352 T^{12} + 1235145267619 T^{13} + 6463358843799 T^{14} + 32283670501184 T^{15} + 154818187749602 T^{16} + 716919841360198 T^{17} + 3174109718888434 T^{18} + 13621476985843762 T^{19} + 55889365777606322 T^{20} + 221433695967621056 T^{21} + 842311387882729479 T^{22} + 3058341962006138281 T^{23} + 10750047713366548112 T^{24} + 35346579094555326036 T^{25} + \)\(11\!\cdots\!53\)\( T^{26} + \)\(33\!\cdots\!54\)\( T^{27} + \)\(97\!\cdots\!27\)\( T^{28} + \)\(24\!\cdots\!17\)\( T^{29} + \)\(63\!\cdots\!32\)\( T^{30} + \)\(13\!\cdots\!34\)\( T^{31} + \)\(29\!\cdots\!57\)\( T^{32} + \)\(45\!\cdots\!71\)\( T^{33} + \)\(83\!\cdots\!90\)\( T^{34} + \)\(76\!\cdots\!46\)\( T^{35} + \)\(10\!\cdots\!41\)\( T^{36} \)
$23$ \( 1 - 7 T + 239 T^{2} - 1576 T^{3} + 28588 T^{4} - 179050 T^{5} + 2269277 T^{6} - 13561935 T^{7} + 133919327 T^{8} - 766089347 T^{9} + 6252142135 T^{10} - 34292124369 T^{11} + 240295781412 T^{12} - 1263196000211 T^{13} + 7818397648251 T^{14} - 39255760559581 T^{15} + 219632672936210 T^{16} - 1045456603448794 T^{17} + 5392265405260800 T^{18} - 24045501879322262 T^{19} + 116185683983255090 T^{20} - 477624838728422027 T^{21} + 2187908216284208091 T^{22} - 8130362733586068373 T^{23} + 35572399624275095268 T^{24} - \)\(11\!\cdots\!43\)\( T^{25} + \)\(48\!\cdots\!35\)\( T^{26} - \)\(13\!\cdots\!61\)\( T^{27} + \)\(55\!\cdots\!23\)\( T^{28} - \)\(12\!\cdots\!45\)\( T^{29} + \)\(49\!\cdots\!17\)\( T^{30} - \)\(90\!\cdots\!50\)\( T^{31} + \)\(33\!\cdots\!92\)\( T^{32} - \)\(42\!\cdots\!32\)\( T^{33} + \)\(14\!\cdots\!79\)\( T^{34} - \)\(98\!\cdots\!21\)\( T^{35} + \)\(32\!\cdots\!69\)\( T^{36} \)
$29$ \( 1 - 23 T + 563 T^{2} - 8615 T^{3} + 128432 T^{4} - 1522980 T^{5} + 17354219 T^{6} - 171102877 T^{7} + 1620106476 T^{8} - 13815187483 T^{9} + 113412360674 T^{10} - 857384208124 T^{11} + 6258622685406 T^{12} - 42644913205081 T^{13} + 281339577465084 T^{14} - 1747023215055564 T^{15} + 10525426577055857 T^{16} - 59964845786916469 T^{17} + 331827567818956736 T^{18} - 1738980527820577601 T^{19} + 8851883751303975737 T^{20} - 42608149191990150396 T^{21} + \)\(19\!\cdots\!04\)\( T^{22} - \)\(87\!\cdots\!69\)\( T^{23} + \)\(37\!\cdots\!26\)\( T^{24} - \)\(14\!\cdots\!16\)\( T^{25} + \)\(56\!\cdots\!14\)\( T^{26} - \)\(20\!\cdots\!27\)\( T^{27} + \)\(68\!\cdots\!76\)\( T^{28} - \)\(20\!\cdots\!33\)\( T^{29} + \)\(61\!\cdots\!79\)\( T^{30} - \)\(15\!\cdots\!20\)\( T^{31} + \)\(38\!\cdots\!92\)\( T^{32} - \)\(74\!\cdots\!35\)\( T^{33} + \)\(14\!\cdots\!23\)\( T^{34} - \)\(16\!\cdots\!07\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \)
$31$ \( 1 + T + 371 T^{2} + 375 T^{3} + 66747 T^{4} + 74684 T^{5} + 7774251 T^{6} + 10103299 T^{7} + 660259304 T^{8} + 1004583593 T^{9} + 43677801352 T^{10} + 76298900803 T^{11} + 2348382550731 T^{12} + 4538784174677 T^{13} + 105697196910147 T^{14} + 215218702414202 T^{15} + 4064525040452553 T^{16} + 8228165505228620 T^{17} + 135195912490404158 T^{18} + 255073130662087220 T^{19} + 3906008563874903433 T^{20} + 6411580363621491782 T^{21} + 97613580987655867587 T^{22} + \)\(12\!\cdots\!27\)\( T^{23} + \)\(20\!\cdots\!11\)\( T^{24} + \)\(20\!\cdots\!33\)\( T^{25} + \)\(37\!\cdots\!32\)\( T^{26} + \)\(26\!\cdots\!03\)\( T^{27} + \)\(54\!\cdots\!04\)\( T^{28} + \)\(25\!\cdots\!69\)\( T^{29} + \)\(61\!\cdots\!11\)\( T^{30} + \)\(18\!\cdots\!44\)\( T^{31} + \)\(50\!\cdots\!87\)\( T^{32} + \)\(87\!\cdots\!25\)\( T^{33} + \)\(26\!\cdots\!51\)\( T^{34} + \)\(22\!\cdots\!11\)\( T^{35} + \)\(69\!\cdots\!41\)\( T^{36} \)
$37$ \( 1 - 36 T + 1035 T^{2} - 21370 T^{3} + 381842 T^{4} - 5803940 T^{5} + 79432215 T^{6} - 976875470 T^{7} + 11061829355 T^{8} - 115471197904 T^{9} + 1125156678811 T^{10} - 10250379469620 T^{11} + 87935059995072 T^{12} - 711273924784364 T^{13} + 5448065421722451 T^{14} - 39546158667594450 T^{15} + 272747091667535146 T^{16} - 1787574464717241302 T^{17} + 11149407736420495504 T^{18} - 66140255194537928174 T^{19} + \)\(37\!\cdots\!74\)\( T^{20} - \)\(20\!\cdots\!50\)\( T^{21} + \)\(10\!\cdots\!11\)\( T^{22} - \)\(49\!\cdots\!48\)\( T^{23} + \)\(22\!\cdots\!48\)\( T^{24} - \)\(97\!\cdots\!60\)\( T^{25} + \)\(39\!\cdots\!31\)\( T^{26} - \)\(15\!\cdots\!08\)\( T^{27} + \)\(53\!\cdots\!95\)\( T^{28} - \)\(17\!\cdots\!10\)\( T^{29} + \)\(52\!\cdots\!15\)\( T^{30} - \)\(14\!\cdots\!80\)\( T^{31} + \)\(34\!\cdots\!38\)\( T^{32} - \)\(71\!\cdots\!10\)\( T^{33} + \)\(12\!\cdots\!35\)\( T^{34} - \)\(16\!\cdots\!12\)\( T^{35} + \)\(16\!\cdots\!29\)\( T^{36} \)
$41$ \( 1 - 37 T + 1162 T^{2} - 25639 T^{3} + 497024 T^{4} - 8108783 T^{5} + 119638831 T^{6} - 1574467807 T^{7} + 19095573312 T^{8} - 212257240486 T^{9} + 2201959688464 T^{10} - 21259177815952 T^{11} + 193234881336991 T^{12} - 1650317886320321 T^{13} + 13349672392428094 T^{14} - 102082179475824511 T^{15} + 742232061628877504 T^{16} - 5119197873181398524 T^{17} + 33636669577737855514 T^{18} - \)\(20\!\cdots\!84\)\( T^{19} + \)\(12\!\cdots\!24\)\( T^{20} - \)\(70\!\cdots\!31\)\( T^{21} + \)\(37\!\cdots\!34\)\( T^{22} - \)\(19\!\cdots\!21\)\( T^{23} + \)\(91\!\cdots\!31\)\( T^{24} - \)\(41\!\cdots\!12\)\( T^{25} + \)\(17\!\cdots\!44\)\( T^{26} - \)\(69\!\cdots\!46\)\( T^{27} + \)\(25\!\cdots\!12\)\( T^{28} - \)\(86\!\cdots\!87\)\( T^{29} + \)\(26\!\cdots\!11\)\( T^{30} - \)\(75\!\cdots\!43\)\( T^{31} + \)\(18\!\cdots\!64\)\( T^{32} - \)\(39\!\cdots\!39\)\( T^{33} + \)\(74\!\cdots\!42\)\( T^{34} - \)\(96\!\cdots\!97\)\( T^{35} + \)\(10\!\cdots\!21\)\( T^{36} \)
$43$ \( 1 + 15 T + 649 T^{2} + 7805 T^{3} + 190599 T^{4} + 1912780 T^{5} + 34448375 T^{6} + 295984148 T^{7} + 4370757901 T^{8} + 32739693264 T^{9} + 419942836228 T^{10} + 2782160636183 T^{11} + 32141059408382 T^{12} + 190837001390527 T^{13} + 2032298526457431 T^{14} + 10960026906223203 T^{15} + 108958866784162301 T^{16} + 540878278891610325 T^{17} + 5033504906601400058 T^{18} + 23257765992339243975 T^{19} + \)\(20\!\cdots\!49\)\( T^{20} + \)\(87\!\cdots\!21\)\( T^{21} + \)\(69\!\cdots\!31\)\( T^{22} + \)\(28\!\cdots\!61\)\( T^{23} + \)\(20\!\cdots\!18\)\( T^{24} + \)\(75\!\cdots\!81\)\( T^{25} + \)\(49\!\cdots\!28\)\( T^{26} + \)\(16\!\cdots\!52\)\( T^{27} + \)\(94\!\cdots\!49\)\( T^{28} + \)\(27\!\cdots\!36\)\( T^{29} + \)\(13\!\cdots\!75\)\( T^{30} + \)\(32\!\cdots\!40\)\( T^{31} + \)\(14\!\cdots\!51\)\( T^{32} + \)\(24\!\cdots\!35\)\( T^{33} + \)\(88\!\cdots\!49\)\( T^{34} + \)\(88\!\cdots\!45\)\( T^{35} + \)\(25\!\cdots\!49\)\( T^{36} \)
$47$ \( 1 - 23 T + 769 T^{2} - 13551 T^{3} + 269949 T^{4} - 3915438 T^{5} + 59208414 T^{6} - 735695639 T^{7} + 9217604329 T^{8} - 100563658366 T^{9} + 1089466533153 T^{10} - 10604131770061 T^{11} + 101737531943472 T^{12} - 892746822726844 T^{13} + 7694162352201876 T^{14} - 61273031451667859 T^{15} + 478443586728967441 T^{16} - 3470319267236089527 T^{17} + 24664505296042832200 T^{18} - \)\(16\!\cdots\!69\)\( T^{19} + \)\(10\!\cdots\!69\)\( T^{20} - \)\(63\!\cdots\!57\)\( T^{21} + \)\(37\!\cdots\!56\)\( T^{22} - \)\(20\!\cdots\!08\)\( T^{23} + \)\(10\!\cdots\!88\)\( T^{24} - \)\(53\!\cdots\!43\)\( T^{25} + \)\(25\!\cdots\!33\)\( T^{26} - \)\(11\!\cdots\!22\)\( T^{27} + \)\(48\!\cdots\!21\)\( T^{28} - \)\(18\!\cdots\!17\)\( T^{29} + \)\(68\!\cdots\!74\)\( T^{30} - \)\(21\!\cdots\!26\)\( T^{31} + \)\(69\!\cdots\!81\)\( T^{32} - \)\(16\!\cdots\!93\)\( T^{33} + \)\(43\!\cdots\!49\)\( T^{34} - \)\(61\!\cdots\!01\)\( T^{35} + \)\(12\!\cdots\!89\)\( T^{36} \)
$53$ \( 1 - 35 T + 1083 T^{2} - 22489 T^{3} + 425053 T^{4} - 6567248 T^{5} + 94678506 T^{6} - 1195535428 T^{7} + 14340016873 T^{8} - 155936505747 T^{9} + 1632034394994 T^{10} - 15801546478878 T^{11} + 148677841270600 T^{12} - 1310514992350566 T^{13} + 11302408728124255 T^{14} - 92025015377152967 T^{15} + 736342587091688232 T^{16} - 5586084118835796339 T^{17} + 41736989702520768260 T^{18} - \)\(29\!\cdots\!67\)\( T^{19} + \)\(20\!\cdots\!88\)\( T^{20} - \)\(13\!\cdots\!59\)\( T^{21} + \)\(89\!\cdots\!55\)\( T^{22} - \)\(54\!\cdots\!38\)\( T^{23} + \)\(32\!\cdots\!00\)\( T^{24} - \)\(18\!\cdots\!86\)\( T^{25} + \)\(10\!\cdots\!34\)\( T^{26} - \)\(51\!\cdots\!51\)\( T^{27} + \)\(25\!\cdots\!77\)\( T^{28} - \)\(11\!\cdots\!16\)\( T^{29} + \)\(46\!\cdots\!46\)\( T^{30} - \)\(17\!\cdots\!04\)\( T^{31} + \)\(58\!\cdots\!57\)\( T^{32} - \)\(16\!\cdots\!73\)\( T^{33} + \)\(41\!\cdots\!43\)\( T^{34} - \)\(71\!\cdots\!55\)\( T^{35} + \)\(10\!\cdots\!89\)\( T^{36} \)
$59$ \( ( 1 + T )^{18} \)
$61$ \( 1 - 39 T + 1471 T^{2} - 35818 T^{3} + 817118 T^{4} - 14960561 T^{5} + 258108425 T^{6} - 3854122514 T^{7} + 54695318545 T^{8} - 696771547696 T^{9} + 8499221240849 T^{10} - 95019195234761 T^{11} + 1023357880808331 T^{12} - 10232780871009974 T^{13} + 99036704888231764 T^{14} - 897188364949677630 T^{15} + 7893182830860804533 T^{16} - 65299249552356104475 T^{17} + \)\(52\!\cdots\!42\)\( T^{18} - \)\(39\!\cdots\!75\)\( T^{19} + \)\(29\!\cdots\!93\)\( T^{20} - \)\(20\!\cdots\!30\)\( T^{21} + \)\(13\!\cdots\!24\)\( T^{22} - \)\(86\!\cdots\!74\)\( T^{23} + \)\(52\!\cdots\!91\)\( T^{24} - \)\(29\!\cdots\!81\)\( T^{25} + \)\(16\!\cdots\!69\)\( T^{26} - \)\(81\!\cdots\!36\)\( T^{27} + \)\(39\!\cdots\!45\)\( T^{28} - \)\(16\!\cdots\!54\)\( T^{29} + \)\(68\!\cdots\!25\)\( T^{30} - \)\(24\!\cdots\!41\)\( T^{31} + \)\(80\!\cdots\!38\)\( T^{32} - \)\(21\!\cdots\!18\)\( T^{33} + \)\(54\!\cdots\!31\)\( T^{34} - \)\(87\!\cdots\!19\)\( T^{35} + \)\(13\!\cdots\!81\)\( T^{36} \)
$67$ \( 1 - 14 T + 584 T^{2} - 7402 T^{3} + 175998 T^{4} - 2019958 T^{5} + 35736509 T^{6} - 377347957 T^{7} + 5499794378 T^{8} - 54163644568 T^{9} + 685649961037 T^{10} - 6367484198644 T^{11} + 72217104791589 T^{12} - 636332394978799 T^{13} + 6595772574396989 T^{14} - 55209996682386672 T^{15} + 529915485627492618 T^{16} - 4201344704663207028 T^{17} + 37698537055203245858 T^{18} - \)\(28\!\cdots\!76\)\( T^{19} + \)\(23\!\cdots\!02\)\( T^{20} - \)\(16\!\cdots\!36\)\( T^{21} + \)\(13\!\cdots\!69\)\( T^{22} - \)\(85\!\cdots\!93\)\( T^{23} + \)\(65\!\cdots\!41\)\( T^{24} - \)\(38\!\cdots\!12\)\( T^{25} + \)\(27\!\cdots\!17\)\( T^{26} - \)\(14\!\cdots\!96\)\( T^{27} + \)\(10\!\cdots\!22\)\( T^{28} - \)\(46\!\cdots\!31\)\( T^{29} + \)\(29\!\cdots\!49\)\( T^{30} - \)\(11\!\cdots\!46\)\( T^{31} + \)\(64\!\cdots\!42\)\( T^{32} - \)\(18\!\cdots\!86\)\( T^{33} + \)\(96\!\cdots\!04\)\( T^{34} - \)\(15\!\cdots\!78\)\( T^{35} + \)\(74\!\cdots\!09\)\( T^{36} \)
$71$ \( 1 - 3 T + 537 T^{2} - 2435 T^{3} + 153205 T^{4} - 910500 T^{5} + 30959641 T^{6} - 215783078 T^{7} + 4973264314 T^{8} - 37289791104 T^{9} + 668654362394 T^{10} - 5085523989212 T^{11} + 76904950958463 T^{12} - 573811536150510 T^{13} + 7644694616954221 T^{14} - 54976086436920544 T^{15} + 661642485133273521 T^{16} - 4525499790997505672 T^{17} + 50144246014068624782 T^{18} - \)\(32\!\cdots\!12\)\( T^{19} + \)\(33\!\cdots\!61\)\( T^{20} - \)\(19\!\cdots\!84\)\( T^{21} + \)\(19\!\cdots\!01\)\( T^{22} - \)\(10\!\cdots\!10\)\( T^{23} + \)\(98\!\cdots\!23\)\( T^{24} - \)\(46\!\cdots\!92\)\( T^{25} + \)\(43\!\cdots\!34\)\( T^{26} - \)\(17\!\cdots\!24\)\( T^{27} + \)\(16\!\cdots\!14\)\( T^{28} - \)\(49\!\cdots\!38\)\( T^{29} + \)\(50\!\cdots\!81\)\( T^{30} - \)\(10\!\cdots\!00\)\( T^{31} + \)\(12\!\cdots\!05\)\( T^{32} - \)\(14\!\cdots\!85\)\( T^{33} + \)\(22\!\cdots\!77\)\( T^{34} - \)\(88\!\cdots\!73\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \)
$73$ \( 1 - 56 T + 2109 T^{2} - 58148 T^{3} + 1323233 T^{4} - 25545917 T^{5} + 434491514 T^{6} - 6603368704 T^{7} + 91350560633 T^{8} - 1160627321124 T^{9} + 13701049378596 T^{10} - 151273983137480 T^{11} + 1576247493655595 T^{12} - 15595164176234720 T^{13} + 147797237362272105 T^{14} - 1351002409324893588 T^{15} + 12022419854601633312 T^{16} - \)\(10\!\cdots\!47\)\( T^{17} + \)\(90\!\cdots\!48\)\( T^{18} - \)\(76\!\cdots\!31\)\( T^{19} + \)\(64\!\cdots\!48\)\( T^{20} - \)\(52\!\cdots\!96\)\( T^{21} + \)\(41\!\cdots\!05\)\( T^{22} - \)\(32\!\cdots\!60\)\( T^{23} + \)\(23\!\cdots\!55\)\( T^{24} - \)\(16\!\cdots\!60\)\( T^{25} + \)\(11\!\cdots\!76\)\( T^{26} - \)\(68\!\cdots\!12\)\( T^{27} + \)\(39\!\cdots\!17\)\( T^{28} - \)\(20\!\cdots\!08\)\( T^{29} + \)\(99\!\cdots\!94\)\( T^{30} - \)\(42\!\cdots\!61\)\( T^{31} + \)\(16\!\cdots\!97\)\( T^{32} - \)\(51\!\cdots\!36\)\( T^{33} + \)\(13\!\cdots\!49\)\( T^{34} - \)\(26\!\cdots\!68\)\( T^{35} + \)\(34\!\cdots\!69\)\( T^{36} \)
$79$ \( 1 + 22 T + 851 T^{2} + 14013 T^{3} + 321543 T^{4} + 4352824 T^{5} + 76480555 T^{6} + 899721359 T^{7} + 13400472097 T^{8} + 142374191076 T^{9} + 1896300817314 T^{10} + 18668917836386 T^{11} + 228746023792580 T^{12} + 2115817543963722 T^{13} + 24174321193637272 T^{14} + 211220187895134527 T^{15} + 2265598352861749407 T^{16} + 18723917741583607029 T^{17} + \)\(18\!\cdots\!20\)\( T^{18} + \)\(14\!\cdots\!91\)\( T^{19} + \)\(14\!\cdots\!87\)\( T^{20} + \)\(10\!\cdots\!53\)\( T^{21} + \)\(94\!\cdots\!32\)\( T^{22} + \)\(65\!\cdots\!78\)\( T^{23} + \)\(55\!\cdots\!80\)\( T^{24} + \)\(35\!\cdots\!74\)\( T^{25} + \)\(28\!\cdots\!54\)\( T^{26} + \)\(17\!\cdots\!44\)\( T^{27} + \)\(12\!\cdots\!97\)\( T^{28} + \)\(67\!\cdots\!61\)\( T^{29} + \)\(45\!\cdots\!55\)\( T^{30} + \)\(20\!\cdots\!36\)\( T^{31} + \)\(11\!\cdots\!83\)\( T^{32} + \)\(40\!\cdots\!87\)\( T^{33} + \)\(19\!\cdots\!71\)\( T^{34} + \)\(40\!\cdots\!98\)\( T^{35} + \)\(14\!\cdots\!61\)\( T^{36} \)
$83$ \( 1 - 12 T + 1042 T^{2} - 10723 T^{3} + 506124 T^{4} - 4414628 T^{5} + 152289937 T^{6} - 1096576246 T^{7} + 31815480079 T^{8} - 179334069678 T^{9} + 4910741731628 T^{10} - 19368015579723 T^{11} + 585423077938040 T^{12} - 1209725488576827 T^{13} + 56585014808837265 T^{14} - 6058433211885189 T^{15} + 4782661356230931324 T^{16} + 6581721606983553308 T^{17} + \)\(39\!\cdots\!52\)\( T^{18} + \)\(54\!\cdots\!64\)\( T^{19} + \)\(32\!\cdots\!36\)\( T^{20} - \)\(34\!\cdots\!43\)\( T^{21} + \)\(26\!\cdots\!65\)\( T^{22} - \)\(47\!\cdots\!61\)\( T^{23} + \)\(19\!\cdots\!60\)\( T^{24} - \)\(52\!\cdots\!21\)\( T^{25} + \)\(11\!\cdots\!48\)\( T^{26} - \)\(33\!\cdots\!34\)\( T^{27} + \)\(49\!\cdots\!71\)\( T^{28} - \)\(14\!\cdots\!82\)\( T^{29} + \)\(16\!\cdots\!57\)\( T^{30} - \)\(39\!\cdots\!64\)\( T^{31} + \)\(37\!\cdots\!96\)\( T^{32} - \)\(65\!\cdots\!61\)\( T^{33} + \)\(52\!\cdots\!02\)\( T^{34} - \)\(50\!\cdots\!76\)\( T^{35} + \)\(34\!\cdots\!09\)\( T^{36} \)
$89$ \( 1 - 34 T + 1437 T^{2} - 34838 T^{3} + 879809 T^{4} - 16884543 T^{5} + 322190914 T^{6} - 5167587423 T^{7} + 81202210588 T^{8} - 1125560659095 T^{9} + 15233128252595 T^{10} - 186980501670944 T^{11} + 2245952507904067 T^{12} - 24938842702007338 T^{13} + 272651064040594252 T^{14} - 2797664833595998600 T^{15} + 28496596163061382239 T^{16} - \)\(27\!\cdots\!65\)\( T^{17} + \)\(26\!\cdots\!56\)\( T^{18} - \)\(24\!\cdots\!85\)\( T^{19} + \)\(22\!\cdots\!19\)\( T^{20} - \)\(19\!\cdots\!00\)\( T^{21} + \)\(17\!\cdots\!32\)\( T^{22} - \)\(13\!\cdots\!62\)\( T^{23} + \)\(11\!\cdots\!87\)\( T^{24} - \)\(82\!\cdots\!76\)\( T^{25} + \)\(59\!\cdots\!95\)\( T^{26} - \)\(39\!\cdots\!55\)\( T^{27} + \)\(25\!\cdots\!88\)\( T^{28} - \)\(14\!\cdots\!47\)\( T^{29} + \)\(79\!\cdots\!94\)\( T^{30} - \)\(37\!\cdots\!67\)\( T^{31} + \)\(17\!\cdots\!69\)\( T^{32} - \)\(60\!\cdots\!62\)\( T^{33} + \)\(22\!\cdots\!57\)\( T^{34} - \)\(46\!\cdots\!86\)\( T^{35} + \)\(12\!\cdots\!81\)\( T^{36} \)
$97$ \( 1 - 31 T + 1150 T^{2} - 26640 T^{3} + 621033 T^{4} - 11795524 T^{5} + 217725486 T^{6} - 3569006410 T^{7} + 56489564240 T^{8} - 823771297356 T^{9} + 11600188070637 T^{10} - 153302569826193 T^{11} + 1959177410228710 T^{12} - 23739268881468945 T^{13} + 278562721657343088 T^{14} - 3117371594141937398 T^{15} + 33818200304155728284 T^{16} - \)\(35\!\cdots\!87\)\( T^{17} + \)\(35\!\cdots\!66\)\( T^{18} - \)\(34\!\cdots\!39\)\( T^{19} + \)\(31\!\cdots\!56\)\( T^{20} - \)\(28\!\cdots\!54\)\( T^{21} + \)\(24\!\cdots\!28\)\( T^{22} - \)\(20\!\cdots\!65\)\( T^{23} + \)\(16\!\cdots\!90\)\( T^{24} - \)\(12\!\cdots\!09\)\( T^{25} + \)\(90\!\cdots\!57\)\( T^{26} - \)\(62\!\cdots\!52\)\( T^{27} + \)\(41\!\cdots\!60\)\( T^{28} - \)\(25\!\cdots\!30\)\( T^{29} + \)\(15\!\cdots\!26\)\( T^{30} - \)\(79\!\cdots\!48\)\( T^{31} + \)\(40\!\cdots\!77\)\( T^{32} - \)\(16\!\cdots\!20\)\( T^{33} + \)\(70\!\cdots\!50\)\( T^{34} - \)\(18\!\cdots\!47\)\( T^{35} + \)\(57\!\cdots\!89\)\( T^{36} \)
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