Properties

Label 4029.2.a.e
Level 4029
Weight 2
Character orbit 4029.a
Self dual yes
Analytic conductor 32.172
Analytic rank 1
Dimension 18
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4029.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1717269744\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 6 x^{17} - 10 x^{16} + 120 x^{15} - 56 x^{14} - 921 x^{13} + 1181 x^{12} + 3316 x^{11} - 6280 x^{10} - 5249 x^{9} + 15005 x^{8} + 1809 x^{7} - 16711 x^{6} + 2434 x^{5} + 8758 x^{4} - 1858 x^{3} - 1942 x^{2} + 318 x + 138\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( 14 \nu^{17} - 113 \nu^{16} + 69 \nu^{15} + 1603 \nu^{14} - 3539 \nu^{13} - 6970 \nu^{12} + 25822 \nu^{11} + 4682 \nu^{10} - 73431 \nu^{9} + 30548 \nu^{8} + 84496 \nu^{7} - 51518 \nu^{6} - 39620 \nu^{5} + 26524 \nu^{4} + 3936 \nu^{3} - 5801 \nu^{2} + 550 \nu + 579 \)\()/65\)
\(\beta_{4}\)\(=\)\((\)\(-43 \nu^{17} + 309 \nu^{16} + 49 \nu^{15} - 4878 \nu^{14} + 7019 \nu^{13} + 27126 \nu^{12} - 60752 \nu^{11} - 60627 \nu^{10} + 200020 \nu^{9} + 40802 \nu^{8} - 292315 \nu^{7} + 2453 \nu^{6} + 202368 \nu^{5} - 263 \nu^{4} - 58891 \nu^{3} - 3355 \nu^{2} + 5717 \nu + 752\)\()/65\)
\(\beta_{5}\)\(=\)\((\)\(2 \nu^{17} - 44 \nu^{16} + 268 \nu^{15} - 57 \nu^{14} - 4166 \nu^{13} + 8732 \nu^{12} + 19508 \nu^{11} - 69325 \nu^{10} - 19774 \nu^{9} + 212910 \nu^{8} - 65151 \nu^{7} - 282150 \nu^{6} + 127434 \nu^{5} + 175573 \nu^{4} - 67372 \nu^{3} - 46221 \nu^{2} + 10356 \nu + 3901\)\()/65\)
\(\beta_{6}\)\(=\)\((\)\(29 \nu^{17} - 14 \nu^{16} - 1249 \nu^{15} + 2391 \nu^{14} + 14200 \nu^{13} - 37290 \nu^{12} - 61335 \nu^{11} + 222852 \nu^{10} + 75782 \nu^{9} - 603297 \nu^{8} + 117768 \nu^{7} + 733177 \nu^{6} - 264772 \nu^{5} - 417145 \nu^{4} + 142289 \nu^{3} + 100910 \nu^{2} - 22153 \nu - 8052\)\()/65\)
\(\beta_{7}\)\(=\)\((\)\(-135 \nu^{17} + 747 \nu^{16} + 1306 \nu^{15} - 13046 \nu^{14} + 2862 \nu^{13} + 86395 \nu^{12} - 73626 \nu^{11} - 273530 \nu^{10} + 308629 \nu^{9} + 444545 \nu^{8} - 532954 \nu^{7} - 409050 \nu^{6} + 427849 \nu^{5} + 219953 \nu^{4} - 148926 \nu^{3} - 58277 \nu^{2} + 18011 \nu + 5672\)\()/65\)
\(\beta_{8}\)\(=\)\((\)\(112 \nu^{17} - 553 \nu^{16} - 1554 \nu^{15} + 10770 \nu^{14} + 4760 \nu^{13} - 82553 \nu^{12} + 24680 \nu^{11} + 317060 \nu^{10} - 193847 \nu^{9} - 647065 \nu^{8} + 463002 \nu^{7} + 701215 \nu^{6} - 461754 \nu^{5} - 396078 \nu^{4} + 185785 \nu^{3} + 99153 \nu^{2} - 24551 \nu - 8173\)\()/65\)
\(\beta_{9}\)\(=\)\((\)\(-122 \nu^{17} + 604 \nu^{16} + 1683 \nu^{15} - 11759 \nu^{14} - 4938 \nu^{13} + 89970 \nu^{12} - 29231 \nu^{11} - 343886 \nu^{10} + 221633 \nu^{9} + 694431 \nu^{8} - 526493 \nu^{7} - 738496 \nu^{6} + 522775 \nu^{5} + 410208 \nu^{4} - 210078 \nu^{3} - 103387 \nu^{2} + 27605 \nu + 9013\)\()/65\)
\(\beta_{10}\)\(=\)\((\)\(-155 \nu^{17} + 862 \nu^{16} + 1603 \nu^{15} - 15648 \nu^{14} + 2259 \nu^{13} + 109679 \nu^{12} - 85432 \nu^{11} - 377687 \nu^{10} + 393867 \nu^{9} + 687867 \nu^{8} - 755317 \nu^{7} - 698762 \nu^{6} + 664122 \nu^{5} + 395880 \nu^{4} - 244741 \nu^{3} - 102963 \nu^{2} + 30528 \nu + 9380\)\()/65\)
\(\beta_{11}\)\(=\)\((\)\(144 \nu^{17} - 620 \nu^{16} - 2609 \nu^{15} + 13641 \nu^{14} + 15402 \nu^{13} - 118549 \nu^{12} - 15921 \nu^{11} + 513814 \nu^{10} - 165393 \nu^{9} - 1156469 \nu^{8} + 611373 \nu^{7} + 1307034 \nu^{6} - 715802 \nu^{5} - 734691 \nu^{4} + 312192 \nu^{3} + 178552 \nu^{2} - 43208 \nu - 14371\)\()/65\)
\(\beta_{12}\)\(=\)\((\)\(-216 \nu^{17} + 982 \nu^{16} + 3465 \nu^{15} - 20078 \nu^{14} - 16733 \nu^{13} + 162347 \nu^{12} - 3776 \nu^{11} - 658414 \nu^{10} + 265503 \nu^{9} + 1406109 \nu^{8} - 788223 \nu^{7} - 1549049 \nu^{6} + 853444 \nu^{5} + 864650 \nu^{4} - 357294 \nu^{3} - 210979 \nu^{2} + 48341 \nu + 16948\)\()/65\)
\(\beta_{13}\)\(=\)\((\)\(-196 \nu^{17} + 880 \nu^{16} + 3311 \nu^{15} - 18659 \nu^{14} - 17313 \nu^{13} + 156626 \nu^{12} + 2869 \nu^{11} - 658621 \nu^{10} + 258772 \nu^{9} + 1450321 \nu^{8} - 829032 \nu^{7} - 1627926 \nu^{6} + 933448 \nu^{5} + 921384 \nu^{4} - 403218 \nu^{3} - 228693 \nu^{2} + 56247 \nu + 18934\)\()/65\)
\(\beta_{14}\)\(=\)\((\)\(22 \nu^{17} - 100 \nu^{16} - 359 \nu^{15} + 2075 \nu^{14} + 1769 \nu^{13} - 17036 \nu^{12} + 363 \nu^{11} + 70104 \nu^{10} - 28892 \nu^{9} - 151444 \nu^{8} + 87357 \nu^{7} + 167684 \nu^{6} - 95454 \nu^{5} - 93920 \nu^{4} + 40201 \nu^{3} + 23037 \nu^{2} - 5501 \nu - 1881\)\()/5\)
\(\beta_{15}\)\(=\)\((\)\(-277 \nu^{17} + 1206 \nu^{16} + 4872 \nu^{15} - 26003 \nu^{14} - 27548 \nu^{13} + 221606 \nu^{12} + 21694 \nu^{11} - 942989 \nu^{10} + 322714 \nu^{9} + 2087964 \nu^{8} - 1129034 \nu^{7} - 2330124 \nu^{6} + 1296721 \nu^{5} + 1301219 \nu^{4} - 562199 \nu^{3} - 319398 \nu^{2} + 77984 \nu + 26492\)\()/65\)
\(\beta_{16}\)\(=\)\((\)\(-432 \nu^{17} + 2146 \nu^{16} + 5890 \nu^{15} - 41521 \nu^{14} - 16670 \nu^{13} + 315542 \nu^{12} - 105000 \nu^{11} - 1198437 \nu^{10} + 771181 \nu^{9} + 2411792 \nu^{8} - 1803881 \nu^{7} - 2573717 \nu^{6} + 1778193 \nu^{5} + 1435357 \nu^{4} - 712924 \nu^{3} - 358232 \nu^{2} + 95122 \nu + 30464\)\()/65\)
\(\beta_{17}\)\(=\)\((\)\(428 \nu^{17} - 1759 \nu^{16} - 8233 \nu^{15} + 39932 \nu^{14} + 53407 \nu^{13} - 356679 \nu^{12} - 90601 \nu^{11} + 1580811 \nu^{10} - 389681 \nu^{9} - 3612191 \nu^{8} + 1731721 \nu^{7} + 4100966 \nu^{6} - 2109774 \nu^{5} - 2302226 \nu^{4} + 935821 \nu^{3} + 560667 \nu^{2} - 131226 \nu - 45468\)\()/65\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{17} - \beta_{15} + \beta_{14} + \beta_{7} + \beta_{2} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{17} - \beta_{15} + \beta_{14} + \beta_{10} + \beta_{8} + \beta_{7} - \beta_{4} + 8 \beta_{2} + 14\)
\(\nu^{5}\)\(=\)\(-9 \beta_{17} + \beta_{16} - 8 \beta_{15} + 9 \beta_{14} + 2 \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 7 \beta_{7} + \beta_{6} - \beta_{4} + 11 \beta_{2} + 18 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(-11 \beta_{17} - \beta_{16} - 11 \beta_{15} + 13 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} + 11 \beta_{10} - 2 \beta_{9} + 10 \beta_{8} + 12 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 8 \beta_{4} + \beta_{3} + 58 \beta_{2} + 5 \beta_{1} + 74\)
\(\nu^{7}\)\(=\)\(-68 \beta_{17} + 9 \beta_{16} - 55 \beta_{15} + 72 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} + 24 \beta_{11} + 14 \beta_{10} - 14 \beta_{9} + 12 \beta_{8} + 46 \beta_{7} + 10 \beta_{6} - 3 \beta_{5} - 9 \beta_{4} + \beta_{3} + 95 \beta_{2} + 91 \beta_{1} + 31\)
\(\nu^{8}\)\(=\)\(-97 \beta_{17} - 14 \beta_{16} - 91 \beta_{15} + 128 \beta_{14} + 33 \beta_{13} + 31 \beta_{12} + 8 \beta_{11} + 95 \beta_{10} - 31 \beta_{9} + 83 \beta_{8} + 106 \beta_{7} - 23 \beta_{6} - 29 \beta_{5} - 51 \beta_{4} + 11 \beta_{3} + 413 \beta_{2} + 68 \beta_{1} + 423\)
\(\nu^{9}\)\(=\)\(-496 \beta_{17} + 59 \beta_{16} - 369 \beta_{15} + 564 \beta_{14} + 74 \beta_{13} + 68 \beta_{12} + 220 \beta_{11} + 142 \beta_{10} - 145 \beta_{9} + 121 \beta_{8} + 313 \beta_{7} + 77 \beta_{6} - 51 \beta_{5} - 59 \beta_{4} + 9 \beta_{3} + 762 \beta_{2} + 503 \beta_{1} + 320\)
\(\nu^{10}\)\(=\)\(-808 \beta_{17} - 135 \beta_{16} - 691 \beta_{15} + 1146 \beta_{14} + 380 \beta_{13} + 339 \beta_{12} + 158 \beta_{11} + 755 \beta_{10} - 341 \beta_{9} + 667 \beta_{8} + 845 \beta_{7} - 179 \beta_{6} - 300 \beta_{5} - 298 \beta_{4} + 76 \beta_{3} + 2941 \beta_{2} + 635 \beta_{1} + 2563\)
\(\nu^{11}\)\(=\)\(-3610 \beta_{17} + 329 \beta_{16} - 2486 \beta_{15} + 4402 \beta_{14} + 917 \beta_{13} + 797 \beta_{12} + 1853 \beta_{11} + 1278 \beta_{10} - 1331 \beta_{9} + 1156 \beta_{8} + 2212 \beta_{7} + 556 \beta_{6} - 592 \beta_{5} - 330 \beta_{4} + 28 \beta_{3} + 5927 \beta_{2} + 2965 \beta_{1} + 2811\)
\(\nu^{12}\)\(=\)\(-6584 \beta_{17} - 1131 \beta_{16} - 5089 \beta_{15} + 9796 \beta_{14} + 3786 \beta_{13} + 3242 \beta_{12} + 2049 \beta_{11} + 5792 \beta_{10} - 3264 \beta_{9} + 5352 \beta_{8} + 6450 \beta_{7} - 1147 \beta_{6} - 2732 \beta_{5} - 1624 \beta_{4} + 364 \beta_{3} + 21058 \beta_{2} + 5097 \beta_{1} + 16233\)
\(\nu^{13}\)\(=\)\(-26459 \beta_{17} + 1550 \beta_{16} - 16953 \beta_{15} + 34359 \beta_{14} + 9597 \beta_{13} + 8023 \beta_{12} + 15112 \beta_{11} + 10842 \beta_{10} - 11482 \beta_{9} + 10629 \beta_{8} + 16051 \beta_{7} + 3997 \beta_{6} - 5882 \beta_{5} - 1533 \beta_{4} - 334 \beta_{3} + 45407 \beta_{2} + 18285 \beta_{1} + 22809\)
\(\nu^{14}\)\(=\)\(-52969 \beta_{17} - 8906 \beta_{16} - 37087 \beta_{15} + 81465 \beta_{14} + 34992 \beta_{13} + 29017 \beta_{12} + 22094 \beta_{11} + 43760 \beta_{10} - 29041 \beta_{9} + 43080 \beta_{8} + 48256 \beta_{7} - 6221 \beta_{6} - 23424 \beta_{5} - 8004 \beta_{4} + 593 \beta_{3} + 151811 \beta_{2} + 37924 \beta_{1} + 106330\)
\(\nu^{15}\)\(=\)\(-195665 \beta_{17} + 5142 \beta_{16} - 117226 \beta_{15} + 268286 \beta_{14} + 91728 \beta_{13} + 74535 \beta_{12} + 121651 \beta_{11} + 88902 \beta_{10} - 95548 \beta_{9} + 94623 \beta_{8} + 118320 \beta_{7} + 29284 \beta_{6} - 53895 \beta_{5} - 4358 \beta_{4} - 7528 \beta_{3} + 345008 \beta_{2} + 116336 \beta_{1} + 177289\)
\(\nu^{16}\)\(=\)\(-422173 \beta_{17} - 68279 \beta_{16} - 269753 \beta_{15} + 665226 \beta_{14} + 309301 \beta_{13} + 250081 \beta_{12} + 215125 \beta_{11} + 328704 \beta_{10} - 247336 \beta_{9} + 347441 \beta_{8} + 357734 \beta_{7} - 26075 \beta_{6} - 194520 \beta_{5} - 31734 \beta_{4} - 13195 \beta_{3} + 1101869 \beta_{2} + 270608 \beta_{1} + 714456\)
\(\nu^{17}\)\(=\)\(-1459248 \beta_{17} - 4804 \beta_{16} - 821669 \beta_{15} + 2094864 \beta_{14} + 829624 \beta_{13} + 659726 \beta_{12} + 973948 \beta_{11} + 713477 \beta_{10} - 777349 \beta_{9} + 820353 \beta_{8} + 879716 \beta_{7} + 220389 \beta_{6} - 470656 \beta_{5} + 19846 \beta_{4} - 96717 \beta_{3} + 2609286 \beta_{2} + 756043 \beta_{1} + 1344775\)

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
2.77821
2.62992
2.53461
1.99378
1.64567
1.59625
1.47280
1.44853
0.615823
0.498176
−0.239152
−0.540222
−0.823313
−0.926931
−2.08696
−2.08785
−2.10151
−2.40783
−2.77821 1.00000 5.71847 1.71746 −2.77821 −2.08361 −10.3307 1.00000 −4.77148
1.2 −2.62992 1.00000 4.91649 0.484241 −2.62992 0.944316 −7.67014 1.00000 −1.27352
1.3 −2.53461 1.00000 4.42426 −2.65587 −2.53461 1.73204 −6.14455 1.00000 6.73160
1.4 −1.99378 1.00000 1.97517 1.24008 −1.99378 −4.06333 0.0495043 1.00000 −2.47246
1.5 −1.64567 1.00000 0.708234 −0.419467 −1.64567 1.49188 2.12582 1.00000 0.690305
1.6 −1.59625 1.00000 0.548008 −4.15667 −1.59625 −4.26341 2.31774 1.00000 6.63507
1.7 −1.47280 1.00000 0.169132 4.03939 −1.47280 0.912411 2.69650 1.00000 −5.94920
1.8 −1.44853 1.00000 0.0982249 1.34868 −1.44853 −1.91511 2.75477 1.00000 −1.95360
1.9 −0.615823 1.00000 −1.62076 −3.91500 −0.615823 4.24139 2.22975 1.00000 2.41095
1.10 −0.498176 1.00000 −1.75182 2.85615 −0.498176 −0.756391 1.86907 1.00000 −1.42286
1.11 0.239152 1.00000 −1.94281 −0.863011 0.239152 −3.18045 −0.942930 1.00000 −0.206391
1.12 0.540222 1.00000 −1.70816 3.09244 0.540222 −3.94821 −2.00323 1.00000 1.67061
1.13 0.823313 1.00000 −1.32216 0.963284 0.823313 4.26360 −2.73517 1.00000 0.793084
1.14 0.926931 1.00000 −1.14080 −2.07435 0.926931 1.24239 −2.91130 1.00000 −1.92278
1.15 2.08696 1.00000 2.35542 −2.36854 2.08696 −0.791085 0.741746 1.00000 −4.94306
1.16 2.08785 1.00000 2.35911 −3.16874 2.08785 0.308283 0.749773 1.00000 −6.61585
1.17 2.10151 1.00000 2.41637 −0.968130 2.10151 −2.79477 0.874998 1.00000 −2.03454
1.18 2.40783 1.00000 3.79762 −0.151954 2.40783 −4.33995 4.32836 1.00000 −0.365880
\(n\): 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 4029.2.a.e 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4029.2.a.e 18 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(-1\)
\(79\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4029))\):

\(T_{2}^{18} + \cdots\)
\(T_{5}^{18} + \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T + 26 T^{2} + 84 T^{3} + 236 T^{4} + 585 T^{5} + 1341 T^{6} + 2870 T^{7} + 5840 T^{8} + 11329 T^{9} + 21093 T^{10} + 37697 T^{11} + 64969 T^{12} + 108080 T^{13} + 174274 T^{14} + 272538 T^{15} + 414262 T^{16} + 611654 T^{17} + 877762 T^{18} + 1223308 T^{19} + 1657048 T^{20} + 2180304 T^{21} + 2788384 T^{22} + 3458560 T^{23} + 4158016 T^{24} + 4825216 T^{25} + 5399808 T^{26} + 5800448 T^{27} + 5980160 T^{28} + 5877760 T^{29} + 5492736 T^{30} + 4792320 T^{31} + 3866624 T^{32} + 2752512 T^{33} + 1703936 T^{34} + 786432 T^{35} + 262144 T^{36} \)
$3$ \( ( 1 - T )^{18} \)
$5$ \( 1 + 5 T + 51 T^{2} + 214 T^{3} + 1248 T^{4} + 4520 T^{5} + 19672 T^{6} + 62976 T^{7} + 226625 T^{8} + 653957 T^{9} + 2054349 T^{10} + 5430570 T^{11} + 15414869 T^{12} + 37859104 T^{13} + 99545331 T^{14} + 230161484 T^{15} + 571195125 T^{16} + 1257540163 T^{17} + 2981186163 T^{18} + 6287700815 T^{19} + 14279878125 T^{20} + 28770185500 T^{21} + 62215831875 T^{22} + 118309700000 T^{23} + 240857328125 T^{24} + 424263281250 T^{25} + 802480078125 T^{26} + 1277259765625 T^{27} + 2213134765625 T^{28} + 3075000000000 T^{29} + 4802734375000 T^{30} + 5517578125000 T^{31} + 7617187500000 T^{32} + 6530761718750 T^{33} + 7781982421875 T^{34} + 3814697265625 T^{35} + 3814697265625 T^{36} \)
$7$ \( 1 + 13 T + 140 T^{2} + 1061 T^{3} + 7102 T^{4} + 39918 T^{5} + 205104 T^{6} + 940029 T^{7} + 4027083 T^{8} + 15879338 T^{9} + 59503200 T^{10} + 209378398 T^{11} + 708772045 T^{12} + 2281855639 T^{13} + 7118854825 T^{14} + 21255808435 T^{15} + 61655778091 T^{16} + 171468343445 T^{17} + 463286448717 T^{18} + 1200278404115 T^{19} + 3021133126459 T^{20} + 7290742293205 T^{21} + 17092370434825 T^{22} + 38351147724673 T^{23} + 83386322322205 T^{24} + 172432114024114 T^{25} + 343024106863200 T^{26} + 640788565072166 T^{27} + 1137551273168667 T^{28} + 1858744480895547 T^{29} + 2838903370073904 T^{30} + 3867615517426626 T^{31} + 4816740263373598 T^{32} + 5037162762049523 T^{33} + 4652610279744140 T^{34} + 3024196681833691 T^{35} + 1628413597910449 T^{36} \)
$11$ \( 1 + 27 T + 474 T^{2} + 6124 T^{3} + 65105 T^{4} + 589676 T^{5} + 4709226 T^{6} + 33695597 T^{7} + 219440929 T^{8} + 1312203498 T^{9} + 7267204278 T^{10} + 37470574364 T^{11} + 180803719532 T^{12} + 819033857735 T^{13} + 3494184377912 T^{14} + 14064960404714 T^{15} + 53515932990553 T^{16} + 192638908889659 T^{17} + 656595064222556 T^{18} + 2119027997786249 T^{19} + 6475427891856913 T^{20} + 18720462298674334 T^{21} + 51158353477009592 T^{22} + 131906221822079485 T^{23} + 320304818177829452 T^{24} + 730195490099484244 T^{25} + 1557789777030492918 T^{26} + 3094107208231223118 T^{27} + 5691732550310894329 T^{28} + 9613747072304999767 T^{29} + 14779568510792327946 T^{30} + 20357214806184656356 T^{31} + 24723612915436905305 T^{32} + 25581467789501446724 T^{33} + 21780171955333204314 T^{34} + 13647069769480931817 T^{35} + 5559917313492231481 T^{36} \)
$13$ \( 1 + 4 T + 98 T^{2} + 250 T^{3} + 4210 T^{4} + 6297 T^{5} + 112437 T^{6} + 75818 T^{7} + 2173969 T^{8} + 18282 T^{9} + 31649589 T^{10} - 21621181 T^{11} + 320274648 T^{12} - 620969333 T^{13} + 1490415485 T^{14} - 11602397658 T^{15} - 14891727576 T^{16} - 171029557263 T^{17} - 373079673670 T^{18} - 2223384244419 T^{19} - 2516701960344 T^{20} - 25490467654626 T^{21} + 42567756667085 T^{22} - 230561566557569 T^{23} + 1545904553438232 T^{24} - 1356697043538577 T^{25} + 25817542054323669 T^{26} + 193871457537186 T^{27} + 299700087666478681 T^{28} + 135878016755097266 T^{29} + 2619566796916396197 T^{30} + 1907204546211417141 T^{31} + 16576354583794006690 T^{32} + 12796473253522689250 T^{33} + 65210827699951624418 T^{34} + 34601663677525351732 T^{35} + \)\(11\!\cdots\!29\)\( T^{36} \)
$17$ \( ( 1 - T )^{18} \)
$19$ \( 1 + 30 T + 613 T^{2} + 9328 T^{3} + 118519 T^{4} + 1295162 T^{5} + 12593080 T^{6} + 110549341 T^{7} + 889612918 T^{8} + 6617301111 T^{9} + 45875324298 T^{10} + 297919680698 T^{11} + 1821262153890 T^{12} + 10513859976151 T^{13} + 57486781497956 T^{14} + 298257016155968 T^{15} + 1470895681109265 T^{16} + 6901127118158326 T^{17} + 30827957285013563 T^{18} + 131121415245008194 T^{19} + 530993340880444665 T^{20} + 2045744873813784512 T^{21} + 7491734851595123876 T^{22} + 26033358173087514949 T^{23} + 85682882561712627090 T^{24} + \)\(26\!\cdots\!22\)\( T^{25} + \)\(77\!\cdots\!18\)\( T^{26} + \)\(21\!\cdots\!69\)\( T^{27} + \)\(54\!\cdots\!18\)\( T^{28} + \)\(12\!\cdots\!79\)\( T^{29} + \)\(27\!\cdots\!80\)\( T^{30} + \)\(54\!\cdots\!58\)\( T^{31} + \)\(94\!\cdots\!99\)\( T^{32} + \)\(14\!\cdots\!72\)\( T^{33} + \)\(17\!\cdots\!53\)\( T^{34} + \)\(16\!\cdots\!70\)\( T^{35} + \)\(10\!\cdots\!41\)\( T^{36} \)
$23$ \( 1 + 21 T + 399 T^{2} + 5294 T^{3} + 63915 T^{4} + 653182 T^{5} + 6204986 T^{6} + 53085837 T^{7} + 428229778 T^{8} + 3199234231 T^{9} + 22736118492 T^{10} + 151872330331 T^{11} + 970521249804 T^{12} + 5879138020655 T^{13} + 34195289160146 T^{14} + 189473534017856 T^{15} + 1010221396939261 T^{16} + 5144533999925194 T^{17} + 25234931256368241 T^{18} + 118324281998279462 T^{19} + 534407118980869069 T^{20} + 2305324488395253952 T^{21} + 9569243913864416786 T^{22} + 37840148845276664665 T^{23} + \)\(14\!\cdots\!56\)\( T^{24} + \)\(51\!\cdots\!57\)\( T^{25} + \)\(17\!\cdots\!52\)\( T^{26} + \)\(57\!\cdots\!53\)\( T^{27} + \)\(17\!\cdots\!22\)\( T^{28} + \)\(50\!\cdots\!99\)\( T^{29} + \)\(13\!\cdots\!06\)\( T^{30} + \)\(32\!\cdots\!06\)\( T^{31} + \)\(74\!\cdots\!35\)\( T^{32} + \)\(14\!\cdots\!58\)\( T^{33} + \)\(24\!\cdots\!39\)\( T^{34} + \)\(29\!\cdots\!63\)\( T^{35} + \)\(32\!\cdots\!69\)\( T^{36} \)
$29$ \( 1 + 47 T + 1313 T^{2} + 26430 T^{3} + 425112 T^{4} + 5721994 T^{5} + 66669440 T^{6} + 686460856 T^{7} + 6354540166 T^{8} + 53545945404 T^{9} + 415469607526 T^{10} + 2996718045254 T^{11} + 20286957428121 T^{12} + 130012385198156 T^{13} + 795771334712276 T^{14} + 4687735112736730 T^{15} + 26769931718718836 T^{16} + 148964538131643953 T^{17} + 810764484495779482 T^{18} + 4319971605817674637 T^{19} + 22513512575442541076 T^{20} + \)\(11\!\cdots\!70\)\( T^{21} + \)\(56\!\cdots\!56\)\( T^{22} + \)\(26\!\cdots\!44\)\( T^{23} + \)\(12\!\cdots\!41\)\( T^{24} + \)\(51\!\cdots\!86\)\( T^{25} + \)\(20\!\cdots\!86\)\( T^{26} + \)\(77\!\cdots\!76\)\( T^{27} + \)\(26\!\cdots\!66\)\( T^{28} + \)\(83\!\cdots\!24\)\( T^{29} + \)\(23\!\cdots\!40\)\( T^{30} + \)\(58\!\cdots\!66\)\( T^{31} + \)\(12\!\cdots\!72\)\( T^{32} + \)\(22\!\cdots\!70\)\( T^{33} + \)\(32\!\cdots\!73\)\( T^{34} + \)\(34\!\cdots\!23\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \)
$31$ \( 1 + 18 T + 395 T^{2} + 5537 T^{3} + 75131 T^{4} + 857931 T^{5} + 9199665 T^{6} + 89315185 T^{7} + 819667467 T^{8} + 6992698011 T^{9} + 56853618133 T^{10} + 435716674755 T^{11} + 3197795405136 T^{12} + 22314094581645 T^{13} + 149550626955265 T^{14} + 957665897398615 T^{15} + 5902456848639721 T^{16} + 34838474367831399 T^{17} + 198152534832019708 T^{18} + 1079992705402773369 T^{19} + 5672261031542771881 T^{20} + 28529824749402139465 T^{21} + \)\(13\!\cdots\!65\)\( T^{22} + \)\(63\!\cdots\!95\)\( T^{23} + \)\(28\!\cdots\!16\)\( T^{24} + \)\(11\!\cdots\!05\)\( T^{25} + \)\(48\!\cdots\!53\)\( T^{26} + \)\(18\!\cdots\!81\)\( T^{27} + \)\(67\!\cdots\!67\)\( T^{28} + \)\(22\!\cdots\!35\)\( T^{29} + \)\(72\!\cdots\!65\)\( T^{30} + \)\(20\!\cdots\!21\)\( T^{31} + \)\(56\!\cdots\!51\)\( T^{32} + \)\(12\!\cdots\!87\)\( T^{33} + \)\(28\!\cdots\!95\)\( T^{34} + \)\(40\!\cdots\!98\)\( T^{35} + \)\(69\!\cdots\!41\)\( T^{36} \)
$37$ \( 1 - T + 412 T^{2} - 354 T^{3} + 83444 T^{4} - 67883 T^{5} + 11105147 T^{6} - 9208591 T^{7} + 1094209425 T^{8} - 956006262 T^{9} + 85182936825 T^{10} - 78285856202 T^{11} + 5451516217666 T^{12} - 5153362534950 T^{13} + 294042385236328 T^{14} - 277041579062029 T^{15} + 13569080378057165 T^{16} - 12313692550669995 T^{17} + 539978839130965547 T^{18} - 455606624374789815 T^{19} + 18576071037560258885 T^{20} - 14032987104228954937 T^{21} + \)\(55\!\cdots\!08\)\( T^{22} - \)\(35\!\cdots\!50\)\( T^{23} + \)\(13\!\cdots\!94\)\( T^{24} - \)\(74\!\cdots\!66\)\( T^{25} + \)\(29\!\cdots\!25\)\( T^{26} - \)\(12\!\cdots\!74\)\( T^{27} + \)\(52\!\cdots\!25\)\( T^{28} - \)\(16\!\cdots\!83\)\( T^{29} + \)\(73\!\cdots\!07\)\( T^{30} - \)\(16\!\cdots\!51\)\( T^{31} + \)\(75\!\cdots\!16\)\( T^{32} - \)\(11\!\cdots\!22\)\( T^{33} + \)\(50\!\cdots\!92\)\( T^{34} - \)\(45\!\cdots\!17\)\( T^{35} + \)\(16\!\cdots\!29\)\( T^{36} \)
$41$ \( 1 + 18 T + 448 T^{2} + 5983 T^{3} + 89359 T^{4} + 969897 T^{5} + 11107708 T^{6} + 102855327 T^{7} + 989203059 T^{8} + 8062923195 T^{9} + 68334607194 T^{10} + 502708222870 T^{11} + 3881844521818 T^{12} + 26379771465264 T^{13} + 190864206279602 T^{14} + 1225857147599468 T^{15} + 8513820028353411 T^{16} + 52756216822433594 T^{17} + 357325936283022160 T^{18} + 2163004889719777354 T^{19} + 14311731467662083891 T^{20} + 84487300469702934028 T^{21} + \)\(53\!\cdots\!22\)\( T^{22} + \)\(30\!\cdots\!64\)\( T^{23} + \)\(18\!\cdots\!38\)\( T^{24} + \)\(97\!\cdots\!70\)\( T^{25} + \)\(54\!\cdots\!74\)\( T^{26} + \)\(26\!\cdots\!95\)\( T^{27} + \)\(13\!\cdots\!59\)\( T^{28} + \)\(56\!\cdots\!07\)\( T^{29} + \)\(25\!\cdots\!48\)\( T^{30} + \)\(89\!\cdots\!37\)\( T^{31} + \)\(33\!\cdots\!99\)\( T^{32} + \)\(93\!\cdots\!83\)\( T^{33} + \)\(28\!\cdots\!68\)\( T^{34} + \)\(47\!\cdots\!58\)\( T^{35} + \)\(10\!\cdots\!21\)\( T^{36} \)
$43$ \( 1 + 39 T + 1183 T^{2} + 26421 T^{3} + 505467 T^{4} + 8316424 T^{5} + 122655302 T^{6} + 1633709544 T^{7} + 19991129432 T^{8} + 226023061513 T^{9} + 2382134596419 T^{10} + 23495313142778 T^{11} + 217992338769957 T^{12} + 1907475481819399 T^{13} + 15790821650006012 T^{14} + 123862438254265247 T^{15} + 922306024038269159 T^{16} + 6523797732987249839 T^{17} + 43874778897622692696 T^{18} + \)\(28\!\cdots\!77\)\( T^{19} + \)\(17\!\cdots\!91\)\( T^{20} + \)\(98\!\cdots\!29\)\( T^{21} + \)\(53\!\cdots\!12\)\( T^{22} + \)\(28\!\cdots\!57\)\( T^{23} + \)\(13\!\cdots\!93\)\( T^{24} + \)\(63\!\cdots\!46\)\( T^{25} + \)\(27\!\cdots\!19\)\( T^{26} + \)\(11\!\cdots\!59\)\( T^{27} + \)\(43\!\cdots\!68\)\( T^{28} + \)\(15\!\cdots\!08\)\( T^{29} + \)\(49\!\cdots\!02\)\( T^{30} + \)\(14\!\cdots\!32\)\( T^{31} + \)\(37\!\cdots\!83\)\( T^{32} + \)\(83\!\cdots\!47\)\( T^{33} + \)\(16\!\cdots\!83\)\( T^{34} + \)\(22\!\cdots\!77\)\( T^{35} + \)\(25\!\cdots\!49\)\( T^{36} \)
$47$ \( 1 + 462 T^{2} - 280 T^{3} + 106315 T^{4} - 119557 T^{5} + 16392446 T^{6} - 25076987 T^{7} + 1913187274 T^{8} - 3465273660 T^{9} + 180069700066 T^{10} - 355945625134 T^{11} + 14161759331307 T^{12} - 28952630487794 T^{13} + 950120856387563 T^{14} - 1931670701944198 T^{15} + 55045695249395495 T^{16} - 107724675191143418 T^{17} + 2772053073615354805 T^{18} - 5063059733983740646 T^{19} + \)\(12\!\cdots\!55\)\( T^{20} - \)\(20\!\cdots\!54\)\( T^{21} + \)\(46\!\cdots\!03\)\( T^{22} - \)\(66\!\cdots\!58\)\( T^{23} + \)\(15\!\cdots\!03\)\( T^{24} - \)\(18\!\cdots\!42\)\( T^{25} + \)\(42\!\cdots\!26\)\( T^{26} - \)\(38\!\cdots\!20\)\( T^{27} + \)\(10\!\cdots\!26\)\( T^{28} - \)\(61\!\cdots\!61\)\( T^{29} + \)\(19\!\cdots\!86\)\( T^{30} - \)\(65\!\cdots\!39\)\( T^{31} + \)\(27\!\cdots\!35\)\( T^{32} - \)\(33\!\cdots\!40\)\( T^{33} + \)\(26\!\cdots\!02\)\( T^{34} + \)\(12\!\cdots\!89\)\( T^{36} \)
$53$ \( 1 + 9 T + 539 T^{2} + 3791 T^{3} + 137142 T^{4} + 736084 T^{5} + 22291906 T^{6} + 86968448 T^{7} + 2659984730 T^{8} + 6982771218 T^{9} + 254874594104 T^{10} + 407890373453 T^{11} + 20842966776803 T^{12} + 18727247042940 T^{13} + 1502309104097726 T^{14} + 771732602740687 T^{15} + 96105659181356924 T^{16} + 34132637516256824 T^{17} + 5429441445485800205 T^{18} + 1809029788361611672 T^{19} + \)\(26\!\cdots\!16\)\( T^{20} + \)\(11\!\cdots\!99\)\( T^{21} + \)\(11\!\cdots\!06\)\( T^{22} + \)\(78\!\cdots\!20\)\( T^{23} + \)\(46\!\cdots\!87\)\( T^{24} + \)\(47\!\cdots\!61\)\( T^{25} + \)\(15\!\cdots\!44\)\( T^{26} + \)\(23\!\cdots\!94\)\( T^{27} + \)\(46\!\cdots\!70\)\( T^{28} + \)\(80\!\cdots\!56\)\( T^{29} + \)\(10\!\cdots\!46\)\( T^{30} + \)\(19\!\cdots\!32\)\( T^{31} + \)\(18\!\cdots\!98\)\( T^{32} + \)\(27\!\cdots\!87\)\( T^{33} + \)\(20\!\cdots\!19\)\( T^{34} + \)\(18\!\cdots\!17\)\( T^{35} + \)\(10\!\cdots\!89\)\( T^{36} \)
$59$ \( 1 + 42 T + 1360 T^{2} + 31025 T^{3} + 612208 T^{4} + 10145493 T^{5} + 152736852 T^{6} + 2055827762 T^{7} + 25836509031 T^{8} + 299604939052 T^{9} + 3298991321185 T^{10} + 34132677066868 T^{11} + 338901264367870 T^{12} + 3196245623845163 T^{13} + 29112314717343492 T^{14} + 253410850248199794 T^{15} + 2137331268684060507 T^{16} + 17277688776576138780 T^{17} + \)\(13\!\cdots\!09\)\( T^{18} + \)\(10\!\cdots\!20\)\( T^{19} + \)\(74\!\cdots\!67\)\( T^{20} + \)\(52\!\cdots\!26\)\( T^{21} + \)\(35\!\cdots\!12\)\( T^{22} + \)\(22\!\cdots\!37\)\( T^{23} + \)\(14\!\cdots\!70\)\( T^{24} + \)\(84\!\cdots\!92\)\( T^{25} + \)\(48\!\cdots\!85\)\( T^{26} + \)\(25\!\cdots\!28\)\( T^{27} + \)\(13\!\cdots\!31\)\( T^{28} + \)\(61\!\cdots\!58\)\( T^{29} + \)\(27\!\cdots\!12\)\( T^{30} + \)\(10\!\cdots\!47\)\( T^{31} + \)\(37\!\cdots\!88\)\( T^{32} + \)\(11\!\cdots\!75\)\( T^{33} + \)\(29\!\cdots\!60\)\( T^{34} + \)\(53\!\cdots\!98\)\( T^{35} + \)\(75\!\cdots\!21\)\( T^{36} \)
$61$ \( 1 + 43 T + 1236 T^{2} + 25823 T^{3} + 454265 T^{4} + 6917604 T^{5} + 96199752 T^{6} + 1232649011 T^{7} + 14874379074 T^{8} + 169027953844 T^{9} + 1829604952738 T^{10} + 18856237695016 T^{11} + 186566959862853 T^{12} + 1770499415004349 T^{13} + 16203255957729501 T^{14} + 142747671198317857 T^{15} + 1215480166445993409 T^{16} + 9983323912115608909 T^{17} + 79357672547095432243 T^{18} + \)\(60\!\cdots\!49\)\( T^{19} + \)\(45\!\cdots\!89\)\( T^{20} + \)\(32\!\cdots\!17\)\( T^{21} + \)\(22\!\cdots\!41\)\( T^{22} + \)\(14\!\cdots\!49\)\( T^{23} + \)\(96\!\cdots\!33\)\( T^{24} + \)\(59\!\cdots\!36\)\( T^{25} + \)\(35\!\cdots\!78\)\( T^{26} + \)\(19\!\cdots\!04\)\( T^{27} + \)\(10\!\cdots\!74\)\( T^{28} + \)\(53\!\cdots\!71\)\( T^{29} + \)\(25\!\cdots\!92\)\( T^{30} + \)\(11\!\cdots\!24\)\( T^{31} + \)\(44\!\cdots\!65\)\( T^{32} + \)\(15\!\cdots\!23\)\( T^{33} + \)\(45\!\cdots\!96\)\( T^{34} + \)\(96\!\cdots\!03\)\( T^{35} + \)\(13\!\cdots\!81\)\( T^{36} \)
$67$ \( 1 + 620 T^{2} - 694 T^{3} + 195835 T^{4} - 414839 T^{5} + 41966104 T^{6} - 124017117 T^{7} + 6854173238 T^{8} - 24649096438 T^{9} + 906665321488 T^{10} - 3652672065348 T^{11} + 100553749300529 T^{12} - 428309205455551 T^{13} + 9542743300208266 T^{14} - 41079869245922669 T^{15} + 784417051405701689 T^{16} - 3280455697875365096 T^{17} + 56222487252751681224 T^{18} - \)\(21\!\cdots\!32\)\( T^{19} + \)\(35\!\cdots\!21\)\( T^{20} - \)\(12\!\cdots\!47\)\( T^{21} + \)\(19\!\cdots\!86\)\( T^{22} - \)\(57\!\cdots\!57\)\( T^{23} + \)\(90\!\cdots\!01\)\( T^{24} - \)\(22\!\cdots\!04\)\( T^{25} + \)\(36\!\cdots\!08\)\( T^{26} - \)\(67\!\cdots\!86\)\( T^{27} + \)\(12\!\cdots\!62\)\( T^{28} - \)\(15\!\cdots\!11\)\( T^{29} + \)\(34\!\cdots\!44\)\( T^{30} - \)\(22\!\cdots\!93\)\( T^{31} + \)\(71\!\cdots\!15\)\( T^{32} - \)\(17\!\cdots\!42\)\( T^{33} + \)\(10\!\cdots\!20\)\( T^{34} + \)\(74\!\cdots\!09\)\( T^{36} \)
$71$ \( 1 - 9 T + 291 T^{2} - 1590 T^{3} + 50941 T^{4} - 266473 T^{5} + 7865537 T^{6} - 35758854 T^{7} + 969153546 T^{8} - 4041636128 T^{9} + 108417058876 T^{10} - 429588450768 T^{11} + 10657558124541 T^{12} - 38510988047844 T^{13} + 946786285111541 T^{14} - 3333395129584239 T^{15} + 77466044278610343 T^{16} - 254903646300667229 T^{17} + 5700836094854813860 T^{18} - 18098158887347373259 T^{19} + \)\(39\!\cdots\!63\)\( T^{20} - \)\(11\!\cdots\!29\)\( T^{21} + \)\(24\!\cdots\!21\)\( T^{22} - \)\(69\!\cdots\!44\)\( T^{23} + \)\(13\!\cdots\!61\)\( T^{24} - \)\(39\!\cdots\!88\)\( T^{25} + \)\(70\!\cdots\!36\)\( T^{26} - \)\(18\!\cdots\!68\)\( T^{27} + \)\(31\!\cdots\!46\)\( T^{28} - \)\(82\!\cdots\!34\)\( T^{29} + \)\(12\!\cdots\!17\)\( T^{30} - \)\(31\!\cdots\!03\)\( T^{31} + \)\(42\!\cdots\!21\)\( T^{32} - \)\(93\!\cdots\!90\)\( T^{33} + \)\(12\!\cdots\!11\)\( T^{34} - \)\(26\!\cdots\!19\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \)
$73$ \( 1 - 19 T + 749 T^{2} - 10601 T^{3} + 244073 T^{4} - 2830323 T^{5} + 49093824 T^{6} - 494314534 T^{7} + 7050835601 T^{8} - 63967945732 T^{9} + 774857518306 T^{10} - 6520491473601 T^{11} + 67787832545490 T^{12} - 548291694762135 T^{13} + 4932260733670244 T^{14} - 40370827287664084 T^{15} + 324419015683971685 T^{16} - 2838905010118790909 T^{17} + 22206332169056364108 T^{18} - \)\(20\!\cdots\!57\)\( T^{19} + \)\(17\!\cdots\!65\)\( T^{20} - \)\(15\!\cdots\!28\)\( T^{21} + \)\(14\!\cdots\!04\)\( T^{22} - \)\(11\!\cdots\!55\)\( T^{23} + \)\(10\!\cdots\!10\)\( T^{24} - \)\(72\!\cdots\!97\)\( T^{25} + \)\(62\!\cdots\!86\)\( T^{26} - \)\(37\!\cdots\!16\)\( T^{27} + \)\(30\!\cdots\!49\)\( T^{28} - \)\(15\!\cdots\!18\)\( T^{29} + \)\(11\!\cdots\!04\)\( T^{30} - \)\(47\!\cdots\!59\)\( T^{31} + \)\(29\!\cdots\!57\)\( T^{32} - \)\(94\!\cdots\!57\)\( T^{33} + \)\(48\!\cdots\!89\)\( T^{34} - \)\(90\!\cdots\!07\)\( T^{35} + \)\(34\!\cdots\!69\)\( T^{36} \)
$79$ \( ( 1 + T )^{18} \)
$83$ \( 1 + 61 T + 2659 T^{2} + 85042 T^{3} + 2290011 T^{4} + 52717719 T^{5} + 1082078720 T^{6} + 19989335932 T^{7} + 338697934016 T^{8} + 5294435238489 T^{9} + 77142424786147 T^{10} + 1051473457010882 T^{11} + 13490168591220219 T^{12} + 163260992813337951 T^{13} + 1871016380051029068 T^{14} + 20327143595012317476 T^{15} + \)\(20\!\cdots\!15\)\( T^{16} + \)\(20\!\cdots\!46\)\( T^{17} + \)\(19\!\cdots\!94\)\( T^{18} + \)\(17\!\cdots\!18\)\( T^{19} + \)\(14\!\cdots\!35\)\( T^{20} + \)\(11\!\cdots\!12\)\( T^{21} + \)\(88\!\cdots\!28\)\( T^{22} + \)\(64\!\cdots\!93\)\( T^{23} + \)\(44\!\cdots\!11\)\( T^{24} + \)\(28\!\cdots\!14\)\( T^{25} + \)\(17\!\cdots\!27\)\( T^{26} + \)\(98\!\cdots\!67\)\( T^{27} + \)\(52\!\cdots\!84\)\( T^{28} + \)\(25\!\cdots\!44\)\( T^{29} + \)\(11\!\cdots\!20\)\( T^{30} + \)\(46\!\cdots\!97\)\( T^{31} + \)\(16\!\cdots\!19\)\( T^{32} + \)\(51\!\cdots\!94\)\( T^{33} + \)\(13\!\cdots\!79\)\( T^{34} + \)\(25\!\cdots\!03\)\( T^{35} + \)\(34\!\cdots\!09\)\( T^{36} \)
$89$ \( 1 - 10 T + 836 T^{2} - 5722 T^{3} + 323153 T^{4} - 1317081 T^{5} + 79815303 T^{6} - 119637155 T^{7} + 14743271160 T^{8} + 12653680757 T^{9} + 2244215899085 T^{10} + 6303048874800 T^{11} + 297230779694685 T^{12} + 1238330185305012 T^{13} + 35028745646944059 T^{14} + 171155973199386036 T^{15} + 3692138215615098409 T^{16} + 18646992610590805947 T^{17} + \)\(34\!\cdots\!22\)\( T^{18} + \)\(16\!\cdots\!83\)\( T^{19} + \)\(29\!\cdots\!89\)\( T^{20} + \)\(12\!\cdots\!84\)\( T^{21} + \)\(21\!\cdots\!19\)\( T^{22} + \)\(69\!\cdots\!88\)\( T^{23} + \)\(14\!\cdots\!85\)\( T^{24} + \)\(27\!\cdots\!00\)\( T^{25} + \)\(88\!\cdots\!85\)\( T^{26} + \)\(44\!\cdots\!13\)\( T^{27} + \)\(45\!\cdots\!60\)\( T^{28} - \)\(33\!\cdots\!95\)\( T^{29} + \)\(19\!\cdots\!63\)\( T^{30} - \)\(28\!\cdots\!89\)\( T^{31} + \)\(63\!\cdots\!73\)\( T^{32} - \)\(99\!\cdots\!78\)\( T^{33} + \)\(12\!\cdots\!96\)\( T^{34} - \)\(13\!\cdots\!90\)\( T^{35} + \)\(12\!\cdots\!81\)\( T^{36} \)
$97$ \( 1 + 9 T + 1028 T^{2} + 8118 T^{3} + 518160 T^{4} + 3592116 T^{5} + 172002077 T^{6} + 1049394365 T^{7} + 42525205864 T^{8} + 229573122687 T^{9} + 8373979285693 T^{10} + 40339892166513 T^{11} + 1367347895433451 T^{12} + 5941223599913869 T^{13} + 189747033224810021 T^{14} + 752519037609646140 T^{15} + 22704059575290630160 T^{16} + 83151135478609032095 T^{17} + \)\(23\!\cdots\!88\)\( T^{18} + \)\(80\!\cdots\!15\)\( T^{19} + \)\(21\!\cdots\!40\)\( T^{20} + \)\(68\!\cdots\!20\)\( T^{21} + \)\(16\!\cdots\!01\)\( T^{22} + \)\(51\!\cdots\!33\)\( T^{23} + \)\(11\!\cdots\!79\)\( T^{24} + \)\(32\!\cdots\!69\)\( T^{25} + \)\(65\!\cdots\!73\)\( T^{26} + \)\(17\!\cdots\!79\)\( T^{27} + \)\(31\!\cdots\!36\)\( T^{28} + \)\(75\!\cdots\!45\)\( T^{29} + \)\(11\!\cdots\!57\)\( T^{30} + \)\(24\!\cdots\!32\)\( T^{31} + \)\(33\!\cdots\!40\)\( T^{32} + \)\(51\!\cdots\!74\)\( T^{33} + \)\(63\!\cdots\!88\)\( T^{34} + \)\(53\!\cdots\!33\)\( T^{35} + \)\(57\!\cdots\!89\)\( T^{36} \)
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