Properties

Label 4923.2.a.l
Level $4923$
Weight $2$
Character orbit 4923.a
Self dual yes
Analytic conductor $39.310$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4923 = 3^{2} \cdot 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4923.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.3103529151\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} - 137 x^{10} - 7703 x^{9} + 2068 x^{8} + 11068 x^{7} - 4274 x^{6} - 9021 x^{5} + 4048 x^{4} + 3834 x^{3} - 1851 x^{2} - 654 x + 328\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 547)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} - 137 x^{10} - 7703 x^{9} + 2068 x^{8} + 11068 x^{7} - 4274 x^{6} - 9021 x^{5} + 4048 x^{4} + 3834 x^{3} - 1851 x^{2} - 654 x + 328\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(73030826 \nu^{17} - 256120405 \nu^{16} - 1356797574 \nu^{15} + 5113465236 \nu^{14} + 9822519058 \nu^{13} - 40388724797 \nu^{12} - 37177126598 \nu^{11} + 163882987462 \nu^{10} + 92262525211 \nu^{9} - 377723445077 \nu^{8} - 191420819452 \nu^{7} + 529318619097 \nu^{6} + 308337281661 \nu^{5} - 473115271532 \nu^{4} - 270072920673 \nu^{3} + 252013415040 \nu^{2} + 84141702217 \nu - 59655036146\)\()/ 2338032863 \)
\(\beta_{4}\)\(=\)\((\)\(-197952229 \nu^{17} - 3918595304 \nu^{16} + 22600044356 \nu^{15} + 63480744292 \nu^{14} - 406133708186 \nu^{13} - 324605514650 \nu^{12} + 3091850363118 \nu^{11} + 212070080346 \nu^{10} - 12110742676833 \nu^{9} + 3292313272555 \nu^{8} + 25761837332094 \nu^{7} - 11293715285784 \nu^{6} - 29356638606926 \nu^{5} + 15052674500601 \nu^{4} + 16464410672688 \nu^{3} - 8838109885626 \nu^{2} - 3494627834537 \nu + 1867879847532\)\()/ 4676065726 \)
\(\beta_{5}\)\(=\)\((\)\(377442591 \nu^{17} - 3691417402 \nu^{16} + 2195758068 \nu^{15} + 67864507870 \nu^{14} - 136147624382 \nu^{13} - 463002782106 \nu^{12} + 1307571395468 \nu^{11} + 1393851573734 \nu^{10} - 5634532502087 \nu^{9} - 1407900557235 \nu^{8} + 12578032685850 \nu^{7} - 1636390467968 \nu^{6} - 14752318588402 \nu^{5} + 4796920139089 \nu^{4} + 8479774846946 \nu^{3} - 3614898714428 \nu^{2} - 1849586438863 \nu + 875081238676\)\()/ 4676065726 \)
\(\beta_{6}\)\(=\)\((\)\(280914143 \nu^{17} - 701368576 \nu^{16} - 6767277373 \nu^{15} + 16268509010 \nu^{14} + 67383774797 \nu^{13} - 153950006775 \nu^{12} - 359494769204 \nu^{11} + 767990167544 \nu^{10} + 1112789084625 \nu^{9} - 2178751678709 \nu^{8} - 2022050999427 \nu^{7} + 3538625257235 \nu^{6} + 2083530847860 \nu^{5} - 3158025449217 \nu^{4} - 1112397543996 \nu^{3} + 1407786874034 \nu^{2} + 239108215524 \nu - 245184253071\)\()/ 2338032863 \)
\(\beta_{7}\)\(=\)\((\)\(918896525 \nu^{17} - 5543735306 \nu^{16} - 8092976468 \nu^{15} + 105501268198 \nu^{14} - 60933186040 \nu^{13} - 765818676414 \nu^{12} + 1039858492108 \nu^{11} + 2637470974952 \nu^{10} - 5149492335763 \nu^{9} - 4229517789413 \nu^{8} + 12152589498648 \nu^{7} + 1974660952940 \nu^{6} - 14594279986318 \nu^{5} + 2221552590097 \nu^{4} + 8435391103120 \nu^{3} - 2660699161670 \nu^{2} - 1826393345453 \nu + 742560783748\)\()/ 4676065726 \)
\(\beta_{8}\)\(=\)\((\)\(1002969289 \nu^{17} - 5357577374 \nu^{16} - 11353710084 \nu^{15} + 102147035436 \nu^{14} - 15077828450 \nu^{13} - 742460530448 \nu^{12} + 719668444006 \nu^{11} + 2554781465752 \nu^{10} - 3915968147965 \nu^{9} - 4059181425139 \nu^{8} + 9456059751276 \nu^{7} + 1761924457730 \nu^{6} - 11330964484020 \nu^{5} + 2327689707151 \nu^{4} + 6443984989758 \nu^{3} - 2595584128412 \nu^{2} - 1361950041733 \nu + 677849390104\)\()/ 4676065726 \)
\(\beta_{9}\)\(=\)\((\)\(-685736770 \nu^{17} + 2668474422 \nu^{16} + 11910611780 \nu^{15} - 53542188699 \nu^{14} - 72646310628 \nu^{13} + 421926099202 \nu^{12} + 159397792583 \nu^{11} - 1669988944760 \nu^{10} + 105700448546 \nu^{9} + 3530406219995 \nu^{8} - 1020369463139 \nu^{7} - 3904262274773 \nu^{6} + 1630638051902 \nu^{5} + 2002598616766 \nu^{4} - 1055239106380 \nu^{3} - 288673423392 \nu^{2} + 241733576401 \nu - 42018281445\)\()/ 2338032863 \)
\(\beta_{10}\)\(=\)\((\)\(-1386154783 \nu^{17} + 6430527688 \nu^{16} + 19874972614 \nu^{15} - 125741499426 \nu^{14} - 62775758566 \nu^{13} + 953649814168 \nu^{12} - 338251157070 \nu^{11} - 3555019516198 \nu^{10} + 2820111137503 \nu^{9} + 6789169061169 \nu^{8} - 7571956262232 \nu^{7} - 6130162137078 \nu^{6} + 9466541681870 \nu^{5} + 1619918836209 \nu^{4} - 5505720453480 \nu^{3} + 800314810142 \nu^{2} + 1176511343261 \nu - 374413808574\)\()/ 4676065726 \)
\(\beta_{11}\)\(=\)\((\)\(1665088329 \nu^{17} - 6609375872 \nu^{16} - 28840683926 \nu^{15} + 133345072732 \nu^{14} + 175522841140 \nu^{13} - 1060816401736 \nu^{12} - 388928496010 \nu^{11} + 4271913813578 \nu^{10} - 195622849629 \nu^{9} - 9337524855313 \nu^{8} + 2171494179692 \nu^{7} + 11072418009430 \nu^{6} - 3313170350702 \nu^{5} - 6711664631477 \nu^{4} + 1978357909406 \nu^{3} + 1751178628584 \nu^{2} - 412599628507 \nu - 107954465010\)\()/ 4676065726 \)
\(\beta_{12}\)\(=\)\((\)\(1832524509 \nu^{17} - 6639698786 \nu^{16} - 33972458906 \nu^{15} + 135523924258 \nu^{14} + 236411293988 \nu^{13} - 1095061846114 \nu^{12} - 751627852732 \nu^{11} + 4503367193936 \nu^{10} + 969054521373 \nu^{9} - 10127247887873 \nu^{8} + 172575097992 \nu^{7} + 12485993487236 \nu^{6} - 1620756524402 \nu^{5} - 8006299422797 \nu^{4} + 1401444986280 \nu^{3} + 2295956452378 \nu^{2} - 373320812211 \nu - 180269480248\)\()/ 4676065726 \)
\(\beta_{13}\)\(=\)\((\)\(1216895826 \nu^{17} - 5844979198 \nu^{16} - 17002403395 \nu^{15} + 114758539026 \nu^{14} + 46183633539 \nu^{13} - 876187343173 \nu^{12} + 367014413536 \nu^{11} + 3304628929445 \nu^{10} - 2751485250448 \nu^{9} - 6454959292553 \nu^{8} + 7227541456661 \nu^{7} + 6146488078100 \nu^{6} - 8956043701859 \nu^{5} - 2059176049323 \nu^{4} + 5183979508206 \nu^{3} - 465598650622 \nu^{2} - 1108167717886 \nu + 306757426577\)\()/ 2338032863 \)
\(\beta_{14}\)\(=\)\((\)\(-2694174873 \nu^{17} + 8898905406 \nu^{16} + 54157234818 \nu^{15} - 186122321980 \nu^{14} - 433999335340 \nu^{13} + 1556547553040 \nu^{12} + 1809525369456 \nu^{11} - 6725442689912 \nu^{10} - 4359336517881 \nu^{9} + 16254679484487 \nu^{8} + 6453241055578 \nu^{7} - 22305169708038 \nu^{6} - 5981629649466 \nu^{5} + 16869186956217 \nu^{4} + 3171431762500 \nu^{3} - 6422460662454 \nu^{2} - 696794541829 \nu + 952441749148\)\()/ 4676065726 \)
\(\beta_{15}\)\(=\)\((\)\(1410676829 \nu^{17} - 5054336188 \nu^{16} - 26608639913 \nu^{15} + 103627914575 \nu^{14} + 192515702928 \nu^{13} - 843825093768 \nu^{12} - 677013696028 \nu^{11} + 3518625852588 \nu^{10} + 1225025755998 \nu^{9} - 8120029442208 \nu^{8} - 1160492675804 \nu^{7} + 10522113418083 \nu^{6} + 652018872805 \nu^{5} - 7446708263691 \nu^{4} - 293835473034 \nu^{3} + 2636488005199 \nu^{2} + 81529793478 \nu - 361693721127\)\()/ 2338032863 \)
\(\beta_{16}\)\(=\)\((\)\(1422023073 \nu^{17} - 4606595223 \nu^{16} - 28914727965 \nu^{15} + 96834645667 \nu^{14} + 234949310539 \nu^{13} - 814866392181 \nu^{12} - 993274871485 \nu^{11} + 3546564720314 \nu^{10} + 2406800470543 \nu^{9} - 8638012215790 \nu^{8} - 3497637260481 \nu^{7} + 11927671515175 \nu^{6} + 3064559023173 \nu^{5} - 9027042005382 \nu^{4} - 1501425031058 \nu^{3} + 3394560268895 \nu^{2} + 308210347484 \nu - 485286282032\)\()/ 2338032863 \)
\(\beta_{17}\)\(=\)\((\)\(3038182645 \nu^{17} - 11663900722 \nu^{16} - 54084465652 \nu^{15} + 236680213784 \nu^{14} + 347894361946 \nu^{13} - 1897444758324 \nu^{12} - 908861840806 \nu^{11} + 7720593308970 \nu^{10} + 328593632691 \nu^{9} - 17114585269829 \nu^{8} + 2814844117860 \nu^{7} + 20702464661638 \nu^{6} - 5221618643208 \nu^{5} - 12970037049775 \nu^{4} + 3467300081708 \nu^{3} + 3665906661752 \nu^{2} - 785496755045 \nu - 317577900352\)\()/ 4676065726 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{14} - \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} + \beta_{2} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{15} + 2 \beta_{11} - \beta_{10} + \beta_{9} - \beta_{5} - \beta_{3} + 8 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(-\beta_{17} - \beta_{16} - \beta_{15} - 9 \beta_{14} - \beta_{13} + \beta_{12} - 5 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} + 7 \beta_{7} - 6 \beta_{6} - 2 \beta_{5} + \beta_{4} - 3 \beta_{3} + 10 \beta_{2} + 21 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(2 \beta_{17} + \beta_{16} - 11 \beta_{15} - 2 \beta_{13} - 3 \beta_{12} + 24 \beta_{11} - 13 \beta_{10} + 12 \beta_{9} - \beta_{8} - 2 \beta_{6} - 12 \beta_{5} + \beta_{4} - 15 \beta_{3} + 57 \beta_{2} + 2 \beta_{1} + 87\)
\(\nu^{7}\)\(=\)\(-13 \beta_{17} - 14 \beta_{16} - 11 \beta_{15} - 69 \beta_{14} - 16 \beta_{13} + 18 \beta_{12} - 15 \beta_{11} + 6 \beta_{10} + 31 \beta_{9} + 2 \beta_{8} + 41 \beta_{7} - 31 \beta_{6} - 21 \beta_{5} + 13 \beta_{4} - 38 \beta_{3} + 84 \beta_{2} + 122 \beta_{1} + 16\)
\(\nu^{8}\)\(=\)\(19 \beta_{17} + 10 \beta_{16} - 88 \beta_{15} - 5 \beta_{14} - 35 \beta_{13} - 27 \beta_{12} + 218 \beta_{11} - 125 \beta_{10} + 117 \beta_{9} - 10 \beta_{8} - 5 \beta_{7} - 22 \beta_{6} - 104 \beta_{5} + 19 \beta_{4} - 154 \beta_{3} + 401 \beta_{2} + 31 \beta_{1} + 532\)
\(\nu^{9}\)\(=\)\(-129 \beta_{17} - 142 \beta_{16} - 90 \beta_{15} - 504 \beta_{14} - 177 \beta_{13} + 215 \beta_{12} + 34 \beta_{11} - 82 \beta_{10} + 337 \beta_{9} + 38 \beta_{8} + 222 \beta_{7} - 142 \beta_{6} - 170 \beta_{5} + 128 \beta_{4} - 358 \beta_{3} + 665 \beta_{2} + 746 \beta_{1} + 177\)
\(\nu^{10}\)\(=\)\(110 \beta_{17} + 52 \beta_{16} - 631 \beta_{15} - 100 \beta_{14} - 405 \beta_{13} - 123 \beta_{12} + 1793 \beta_{11} - 1069 \beta_{10} + 1050 \beta_{9} - 61 \beta_{8} - 96 \beta_{7} - 157 \beta_{6} - 807 \beta_{5} + 233 \beta_{4} - 1363 \beta_{3} + 2824 \beta_{2} + 329 \beta_{1} + 3349\)
\(\nu^{11}\)\(=\)\(-1143 \beta_{17} - 1271 \beta_{16} - 668 \beta_{15} - 3612 \beta_{14} - 1672 \beta_{13} + 2131 \beta_{12} + 1154 \beta_{11} - 1273 \beta_{10} + 3144 \beta_{9} + 452 \beta_{8} + 1111 \beta_{7} - 513 \beta_{6} - 1286 \beta_{5} + 1138 \beta_{4} - 3044 \beta_{3} + 5110 \beta_{2} + 4710 \beta_{1} + 1668\)
\(\nu^{12}\)\(=\)\(363 \beta_{17} + 9 \beta_{16} - 4324 \beta_{15} - 1303 \beta_{14} - 3942 \beta_{13} + 112 \beta_{12} + 14074 \beta_{11} - 8613 \beta_{10} + 8939 \beta_{9} - 226 \beta_{8} - 1201 \beta_{7} - 839 \beta_{6} - 5980 \beta_{5} + 2372 \beta_{4} - 11202 \beta_{3} + 19938 \beta_{2} + 2999 \beta_{1} + 21527\)
\(\nu^{13}\)\(=\)\(-9516 \beta_{17} - 10684 \beta_{16} - 4773 \beta_{15} - 25720 \beta_{14} - 14522 \beta_{13} + 19102 \beta_{12} + 14051 \beta_{11} - 12989 \beta_{10} + 27005 \beta_{9} + 4393 \beta_{8} + 4867 \beta_{7} - 555 \beta_{6} - 9572 \beta_{5} + 9620 \beta_{4} - 24686 \beta_{3} + 38571 \beta_{2} + 30427 \beta_{1} + 14428\)
\(\nu^{14}\)\(=\)\(-1271 \beta_{17} - 3636 \beta_{16} - 29054 \beta_{15} - 14050 \beta_{14} - 35039 \beta_{13} + 9835 \beta_{12} + 107749 \beta_{11} - 67069 \beta_{10} + 73347 \beta_{9} + 373 \beta_{8} - 12477 \beta_{7} - 2680 \beta_{6} - 43413 \beta_{5} + 21875 \beta_{4} - 88303 \beta_{3} + 141110 \beta_{2} + 25288 \beta_{1} + 140724\)
\(\nu^{15}\)\(=\)\(-76316 \beta_{17} - 86672 \beta_{16} - 33603 \beta_{15} - 183102 \beta_{14} - 120050 \beta_{13} + 161457 \beta_{12} + 137444 \beta_{11} - 115631 \beta_{10} + 220799 \beta_{9} + 38344 \beta_{8} + 14983 \beta_{7} + 15344 \beta_{6} - 71221 \beta_{5} + 78982 \beta_{4} - 195039 \beta_{3} + 287784 \beta_{2} + 200115 \beta_{1} + 118635\)
\(\nu^{16}\)\(=\)\(-39322 \beta_{17} - 57635 \beta_{16} - 193666 \beta_{15} - 136266 \beta_{14} - 295219 \beta_{13} + 142040 \beta_{12} + 813230 \beta_{11} - 511554 \beta_{10} + 586247 \beta_{9} + 18257 \beta_{8} - 117170 \beta_{7} + 10258 \beta_{6} - 312410 \beta_{5} + 190191 \beta_{4} - 678773 \beta_{3} + 1001004 \beta_{2} + 203810 \beta_{1} + 933162\)
\(\nu^{17}\)\(=\)\(-597910 \beta_{17} - 688126 \beta_{16} - 235431 \beta_{15} - 1307213 \beta_{14} - 962160 \beta_{13} + 1316142 \beta_{12} + 1220338 \beta_{11} - 964358 \beta_{10} + 1749909 \beta_{9} + 314188 \beta_{8} - 21463 \beta_{7} + 230986 \beta_{6} - 531815 \beta_{5} + 636273 \beta_{4} - 1515957 \beta_{3} + 2130566 \beta_{2} + 1335686 \beta_{1} + 944762\)

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
−2.50138
−2.24960
−1.87675
−1.52216
−1.15793
−0.957552
−0.924759
−0.735255
0.523506
0.763493
0.826129
1.04467
1.35726
1.74487
1.98431
2.35947
2.59964
2.72204
−2.50138 0 4.25691 3.57921 0 1.44216 −5.64540 0 −8.95298
1.2 −2.24960 0 3.06069 3.96974 0 −4.97706 −2.38611 0 −8.93031
1.3 −1.87675 0 1.52220 1.30620 0 −1.71403 0.896704 0 −2.45142
1.4 −1.52216 0 0.316965 −1.24712 0 −0.899316 2.56184 0 1.89831
1.5 −1.15793 0 −0.659200 0.421419 0 −0.645304 3.07917 0 −0.487974
1.6 −0.957552 0 −1.08309 4.10274 0 4.97202 2.95222 0 −3.92859
1.7 −0.924759 0 −1.14482 3.95421 0 −3.08028 2.90820 0 −3.65669
1.8 −0.735255 0 −1.45940 −0.962787 0 −3.25298 2.54354 0 0.707894
1.9 0.523506 0 −1.72594 1.35183 0 3.60927 −1.95055 0 0.707691
1.10 0.763493 0 −1.41708 1.51218 0 1.20167 −2.60892 0 1.15454
1.11 0.826129 0 −1.31751 0.786316 0 −5.06179 −2.74069 0 0.649598
1.12 1.04467 0 −0.908666 −0.714085 0 −2.03236 −3.03859 0 −0.745983
1.13 1.35726 0 −0.157846 1.46929 0 0.194696 −2.92876 0 1.99421
1.14 1.74487 0 1.04455 3.61409 0 4.28084 −1.66712 0 6.30610
1.15 1.98431 0 1.93749 −2.87852 0 −2.68467 −0.124041 0 −5.71188
1.16 2.35947 0 3.56710 4.18174 0 −2.82979 3.69754 0 9.86669
1.17 2.59964 0 4.75812 3.02323 0 −0.561390 7.17011 0 7.85930
1.18 2.72204 0 5.40952 −0.469688 0 1.03831 9.28087 0 −1.27851
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(547\) \(1\)

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 4923.2.a.l 18
3.b odd 2 1 547.2.a.b 18
12.b even 2 1 8752.2.a.s 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.2.a.b 18 3.b odd 2 1
4923.2.a.l 18 1.a even 1 1 trivial
8752.2.a.s 18 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4923))\):

\(T_{2}^{18} - \cdots\)
\(T_{11}^{18} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T + 18 T^{2} - 52 T^{3} + 152 T^{4} - 364 T^{5} + 854 T^{6} - 1784 T^{7} + 3639 T^{8} - 6871 T^{9} + 12688 T^{10} - 22114 T^{11} + 37818 T^{12} - 61733 T^{13} + 99096 T^{14} - 152728 T^{15} + 231789 T^{16} - 338874 T^{17} + 488092 T^{18} - 677748 T^{19} + 927156 T^{20} - 1221824 T^{21} + 1585536 T^{22} - 1975456 T^{23} + 2420352 T^{24} - 2830592 T^{25} + 3248128 T^{26} - 3517952 T^{27} + 3726336 T^{28} - 3653632 T^{29} + 3497984 T^{30} - 2981888 T^{31} + 2490368 T^{32} - 1703936 T^{33} + 1179648 T^{34} - 524288 T^{35} + 262144 T^{36} \)
$3$ 1
$5$ \( 1 - 27 T + 394 T^{2} - 4066 T^{3} + 33017 T^{4} - 223247 T^{5} + 1301478 T^{6} - 6696595 T^{7} + 30923803 T^{8} - 129765410 T^{9} + 499584606 T^{10} - 1777920166 T^{11} + 5884067150 T^{12} - 18197020803 T^{13} + 52790609192 T^{14} - 144100415815 T^{15} + 370957947126 T^{16} - 902062261929 T^{17} + 2074042607038 T^{18} - 4510311309645 T^{19} + 9273948678150 T^{20} - 18012551976875 T^{21} + 32994130745000 T^{22} - 56865690009375 T^{23} + 91938549218750 T^{24} - 138900012968750 T^{25} + 195150236718750 T^{26} - 253448066406250 T^{27} + 301990263671875 T^{28} - 326982177734375 T^{29} + 317743652343750 T^{30} - 272518310546875 T^{31} + 201519775390625 T^{32} - 124084472656250 T^{33} + 60119628906250 T^{34} - 20599365234375 T^{35} + 3814697265625 T^{36} \)
$7$ \( 1 + 11 T + 109 T^{2} + 718 T^{3} + 4269 T^{4} + 20685 T^{5} + 92163 T^{6} + 360502 T^{7} + 1322737 T^{8} + 4452817 T^{9} + 14363197 T^{10} + 43877097 T^{11} + 130743026 T^{12} + 375593700 T^{13} + 1062872977 T^{14} + 2919179505 T^{15} + 7963303304 T^{16} + 21214026883 T^{17} + 56674382450 T^{18} + 148498188181 T^{19} + 390201861896 T^{20} + 1001278570215 T^{21} + 2551958017777 T^{22} + 6312603315900 T^{23} + 15381786265874 T^{24} + 36134676094671 T^{25} + 82800972428797 T^{26} + 179687227260919 T^{27} + 373640463436513 T^{28} + 712830245504986 T^{29} + 1275654552305763 T^{30} + 2004149180268795 T^{31} + 2895334297992381 T^{32} + 3408749164139074 T^{33} + 3622389432086509 T^{34} + 2558935653859277 T^{35} + 1628413597910449 T^{36} \)
$11$ \( 1 + 2 T + 85 T^{2} + 150 T^{3} + 3546 T^{4} + 5526 T^{5} + 98861 T^{6} + 141478 T^{7} + 2126155 T^{8} + 2969051 T^{9} + 38313335 T^{10} + 54369337 T^{11} + 603595390 T^{12} + 870619543 T^{13} + 8472071478 T^{14} + 12151342611 T^{15} + 107165383120 T^{16} + 149528311671 T^{17} + 1233316997354 T^{18} + 1644811428381 T^{19} + 12967011357520 T^{20} + 16173437015241 T^{21} + 124039598509398 T^{22} + 140214148019693 T^{23} + 1069306052703790 T^{24} + 1059504567275627 T^{25} + 8212803617978135 T^{26} + 7000866949911241 T^{27} + 55146985002539155 T^{28} + 40365324534703058 T^{29} + 310268167751014781 T^{30} + 190772507307362706 T^{31} + 1346592909886172586 T^{32} + 626587225412347650 T^{33} + 3905727038403633685 T^{34} + 1010894056998587542 T^{35} + 5559917313492231481 T^{36} \)
$13$ \( 1 + 25 T + 419 T^{2} + 5077 T^{3} + 50838 T^{4} + 430037 T^{5} + 3215885 T^{6} + 21488134 T^{7} + 131328937 T^{8} + 738626108 T^{9} + 3880155769 T^{10} + 19109203885 T^{11} + 89183382015 T^{12} + 395302024323 T^{13} + 1678118625851 T^{14} + 6828960297885 T^{15} + 26814195655203 T^{16} + 101525943915242 T^{17} + 372390103391781 T^{18} + 1319837270898146 T^{19} + 4531599065729307 T^{20} + 15003225774453345 T^{21} + 47928746072930411 T^{22} + 146772874516959639 T^{23} + 430471150960440135 T^{24} + 1199074204834388545 T^{25} + 3165162263038679449 T^{26} + 7832760099167430284 T^{27} + 18104809190952334513 T^{28} + 38510182696559856958 T^{29} + 74923962474109810685 T^{30} + \)\(13\!\cdots\!61\)\( T^{31} + \)\(20\!\cdots\!82\)\( T^{32} + \)\(25\!\cdots\!89\)\( T^{33} + \)\(27\!\cdots\!79\)\( T^{34} + \)\(21\!\cdots\!25\)\( T^{35} + \)\(11\!\cdots\!29\)\( T^{36} \)
$17$ \( 1 - 30 T + 584 T^{2} - 8365 T^{3} + 98434 T^{4} - 985682 T^{5} + 8696277 T^{6} - 68768626 T^{7} + 495900647 T^{8} - 3296047078 T^{9} + 20410580082 T^{10} - 118632779583 T^{11} + 651877922855 T^{12} - 3403006356186 T^{13} + 16950930016609 T^{14} - 80772988719250 T^{15} + 368969665979574 T^{16} - 1617067599179214 T^{17} + 6804533522448758 T^{18} - 27490149186046638 T^{19} + 106632233468096886 T^{20} - 396837693577675250 T^{21} + 1415758625917200289 T^{22} - 4831782395875185402 T^{23} + 15734748342489239495 T^{24} - 48679617348389713359 T^{25} + \)\(14\!\cdots\!62\)\( T^{26} - \)\(39\!\cdots\!66\)\( T^{27} + \)\(99\!\cdots\!03\)\( T^{28} - \)\(23\!\cdots\!58\)\( T^{29} + \)\(50\!\cdots\!97\)\( T^{30} - \)\(97\!\cdots\!34\)\( T^{31} + \)\(16\!\cdots\!86\)\( T^{32} - \)\(23\!\cdots\!45\)\( T^{33} + \)\(28\!\cdots\!04\)\( T^{34} - \)\(24\!\cdots\!10\)\( T^{35} + \)\(14\!\cdots\!09\)\( T^{36} \)
$19$ \( 1 - 4 T + 130 T^{2} - 516 T^{3} + 8958 T^{4} - 36501 T^{5} + 441125 T^{6} - 1838457 T^{7} + 17317003 T^{8} - 72845222 T^{9} + 571934352 T^{10} - 2393239283 T^{11} + 16359891658 T^{12} - 67164262668 T^{13} + 412288983795 T^{14} - 1640265737352 T^{15} + 9248689375227 T^{16} - 35240518894600 T^{17} + 185627099738938 T^{18} - 669569858997400 T^{19} + 3338776864456947 T^{20} - 11250582692497368 T^{21} + 53729912657148195 T^{22} - 166305363627972132 T^{23} + 769665516115160698 T^{24} - 2139248959738323137 T^{25} + 9713483122505484432 T^{26} - 23506256981380161938 T^{27} + \)\(10\!\cdots\!03\)\( T^{28} - \)\(21\!\cdots\!83\)\( T^{29} + \)\(97\!\cdots\!25\)\( T^{30} - \)\(15\!\cdots\!59\)\( T^{31} + \)\(71\!\cdots\!18\)\( T^{32} - \)\(78\!\cdots\!84\)\( T^{33} + \)\(37\!\cdots\!30\)\( T^{34} - \)\(21\!\cdots\!56\)\( T^{35} + \)\(10\!\cdots\!41\)\( T^{36} \)
$23$ \( 1 - 26 T + 516 T^{2} - 7395 T^{3} + 90928 T^{4} - 951516 T^{5} + 8960211 T^{6} - 75889920 T^{7} + 593605359 T^{8} - 4293455106 T^{9} + 29167458828 T^{10} - 186453853585 T^{11} + 1133632710313 T^{12} - 6568187578396 T^{13} + 36553007673681 T^{14} - 195688621929534 T^{15} + 1013582264018988 T^{16} - 5081312191544094 T^{17} + 24743429994041086 T^{18} - 116870180405514162 T^{19} + 536185017666044652 T^{20} - 2380943463016640178 T^{21} + 10229030220410564721 T^{22} - 42275108142896045828 T^{23} + \)\(16\!\cdots\!57\)\( T^{24} - \)\(63\!\cdots\!95\)\( T^{25} + \)\(22\!\cdots\!68\)\( T^{26} - \)\(77\!\cdots\!78\)\( T^{27} + \)\(24\!\cdots\!91\)\( T^{28} - \)\(72\!\cdots\!40\)\( T^{29} + \)\(19\!\cdots\!31\)\( T^{30} - \)\(47\!\cdots\!28\)\( T^{31} + \)\(10\!\cdots\!52\)\( T^{32} - \)\(19\!\cdots\!65\)\( T^{33} + \)\(31\!\cdots\!76\)\( T^{34} - \)\(36\!\cdots\!78\)\( T^{35} + \)\(32\!\cdots\!69\)\( T^{36} \)
$29$ \( 1 - 18 T + 531 T^{2} - 7242 T^{3} + 124429 T^{4} - 1388519 T^{5} + 17800604 T^{6} - 169607253 T^{7} + 1778477154 T^{8} - 14850454505 T^{9} + 133353152337 T^{10} - 992168503525 T^{11} + 7832929693192 T^{12} - 52481764441320 T^{13} + 370156555204319 T^{14} - 2247849800307197 T^{15} + 14304279024344391 T^{16} - 78983107272313558 T^{17} + 456054513586518578 T^{18} - 2290510110897093182 T^{19} + 12029898659473632831 T^{20} - 54822808779692227633 T^{21} + \)\(26\!\cdots\!39\)\( T^{22} - \)\(10\!\cdots\!80\)\( T^{23} + \)\(46\!\cdots\!32\)\( T^{24} - \)\(17\!\cdots\!25\)\( T^{25} + \)\(66\!\cdots\!57\)\( T^{26} - \)\(21\!\cdots\!45\)\( T^{27} + \)\(74\!\cdots\!54\)\( T^{28} - \)\(20\!\cdots\!37\)\( T^{29} + \)\(62\!\cdots\!64\)\( T^{30} - \)\(14\!\cdots\!91\)\( T^{31} + \)\(37\!\cdots\!49\)\( T^{32} - \)\(62\!\cdots\!58\)\( T^{33} + \)\(13\!\cdots\!51\)\( T^{34} - \)\(13\!\cdots\!62\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \)
$31$ \( 1 + 5 T + 226 T^{2} + 736 T^{3} + 25466 T^{4} + 48427 T^{5} + 1947515 T^{6} + 1181005 T^{7} + 114803460 T^{8} - 68748799 T^{9} + 5711179330 T^{10} - 8784337491 T^{11} + 254611322915 T^{12} - 519109476169 T^{13} + 10413960551250 T^{14} - 22094105896997 T^{15} + 386758640569635 T^{16} - 776800932680999 T^{17} + 12771866096424756 T^{18} - 24080828913110969 T^{19} + 371675053587419235 T^{20} - 658205508777437627 T^{21} + 9617511262250951250 T^{22} - 14861663578773202519 T^{23} + \)\(22\!\cdots\!15\)\( T^{24} - \)\(24\!\cdots\!01\)\( T^{25} + \)\(48\!\cdots\!30\)\( T^{26} - \)\(18\!\cdots\!29\)\( T^{27} + \)\(94\!\cdots\!60\)\( T^{28} + \)\(30\!\cdots\!55\)\( T^{29} + \)\(15\!\cdots\!15\)\( T^{30} + \)\(11\!\cdots\!57\)\( T^{31} + \)\(19\!\cdots\!86\)\( T^{32} + \)\(17\!\cdots\!36\)\( T^{33} + \)\(16\!\cdots\!06\)\( T^{34} + \)\(11\!\cdots\!55\)\( T^{35} + \)\(69\!\cdots\!41\)\( T^{36} \)
$37$ \( 1 + 18 T + 477 T^{2} + 6182 T^{3} + 98542 T^{4} + 1025448 T^{5} + 12472914 T^{6} + 109789868 T^{7} + 1115943998 T^{8} + 8578978218 T^{9} + 76645978734 T^{10} + 526915335411 T^{11} + 4285047049335 T^{12} + 26892539266031 T^{13} + 204473204412967 T^{14} + 1194720790323650 T^{15} + 8656318678483359 T^{16} + 47928410626044380 T^{17} + 333914579074632830 T^{18} + 1773351193163642060 T^{19} + 11850500270843718471 T^{20} + 60516192192263843450 T^{21} + \)\(38\!\cdots\!87\)\( T^{22} + \)\(18\!\cdots\!67\)\( T^{23} + \)\(10\!\cdots\!15\)\( T^{24} + \)\(50\!\cdots\!63\)\( T^{25} + \)\(26\!\cdots\!14\)\( T^{26} + \)\(11\!\cdots\!86\)\( T^{27} + \)\(53\!\cdots\!02\)\( T^{28} + \)\(19\!\cdots\!84\)\( T^{29} + \)\(82\!\cdots\!34\)\( T^{30} + \)\(24\!\cdots\!56\)\( T^{31} + \)\(88\!\cdots\!38\)\( T^{32} + \)\(20\!\cdots\!26\)\( T^{33} + \)\(58\!\cdots\!57\)\( T^{34} + \)\(82\!\cdots\!06\)\( T^{35} + \)\(16\!\cdots\!29\)\( T^{36} \)
$41$ \( 1 - 17 T + 500 T^{2} - 6935 T^{3} + 118746 T^{4} - 1411049 T^{5} + 18245075 T^{6} - 191387044 T^{7} + 2060476099 T^{8} - 19451816951 T^{9} + 182957350222 T^{10} - 1574307116444 T^{11} + 13284361461403 T^{12} - 105065275805631 T^{13} + 807813171251088 T^{14} - 5903001862494918 T^{15} + 41738761877644030 T^{16} - 282544368422135749 T^{17} + 1846894667477774252 T^{18} - 11584319105307565709 T^{19} + 70162858716319614430 T^{20} - \)\(40\!\cdots\!78\)\( T^{21} + \)\(22\!\cdots\!68\)\( T^{22} - \)\(12\!\cdots\!31\)\( T^{23} + \)\(63\!\cdots\!23\)\( T^{24} - \)\(30\!\cdots\!64\)\( T^{25} + \)\(14\!\cdots\!62\)\( T^{26} - \)\(63\!\cdots\!11\)\( T^{27} + \)\(27\!\cdots\!99\)\( T^{28} - \)\(10\!\cdots\!04\)\( T^{29} + \)\(41\!\cdots\!75\)\( T^{30} - \)\(13\!\cdots\!29\)\( T^{31} + \)\(45\!\cdots\!06\)\( T^{32} - \)\(10\!\cdots\!35\)\( T^{33} + \)\(31\!\cdots\!00\)\( T^{34} - \)\(44\!\cdots\!77\)\( T^{35} + \)\(10\!\cdots\!21\)\( T^{36} \)
$43$ \( 1 - 8 T + 492 T^{2} - 3407 T^{3} + 117668 T^{4} - 715275 T^{5} + 18347951 T^{6} - 99049565 T^{7} + 2107507666 T^{8} - 10209475962 T^{9} + 190684760633 T^{10} - 837060417712 T^{11} + 14161353767533 T^{12} - 56841478497834 T^{13} + 886000874851046 T^{14} - 3276977381712634 T^{15} + 47444588926667291 T^{16} - 162652673998347287 T^{17} + 2192977866661315750 T^{18} - 6994064981928933341 T^{19} + 87725044925407821059 T^{20} - \)\(26\!\cdots\!38\)\( T^{21} + \)\(30\!\cdots\!46\)\( T^{22} - \)\(83\!\cdots\!62\)\( T^{23} + \)\(89\!\cdots\!17\)\( T^{24} - \)\(22\!\cdots\!84\)\( T^{25} + \)\(22\!\cdots\!33\)\( T^{26} - \)\(51\!\cdots\!66\)\( T^{27} + \)\(45\!\cdots\!34\)\( T^{28} - \)\(92\!\cdots\!55\)\( T^{29} + \)\(73\!\cdots\!51\)\( T^{30} - \)\(12\!\cdots\!25\)\( T^{31} + \)\(86\!\cdots\!32\)\( T^{32} - \)\(10\!\cdots\!49\)\( T^{33} + \)\(67\!\cdots\!92\)\( T^{34} - \)\(46\!\cdots\!44\)\( T^{35} + \)\(25\!\cdots\!49\)\( T^{36} \)
$47$ \( 1 - 52 T + 1865 T^{2} - 49131 T^{3} + 1077176 T^{4} - 20125023 T^{5} + 332676442 T^{6} - 4927129277 T^{7} + 66491991801 T^{8} - 823722681768 T^{9} + 9453432637832 T^{10} - 100973005595782 T^{11} + 1009215441829873 T^{12} - 9466179137600663 T^{13} + 83603559634490998 T^{14} - 696394998474434729 T^{15} + 5481817752006743685 T^{16} - 40807789859381398323 T^{17} + \)\(28\!\cdots\!97\)\( T^{18} - \)\(19\!\cdots\!81\)\( T^{19} + \)\(12\!\cdots\!65\)\( T^{20} - \)\(72\!\cdots\!67\)\( T^{21} + \)\(40\!\cdots\!38\)\( T^{22} - \)\(21\!\cdots\!41\)\( T^{23} + \)\(10\!\cdots\!17\)\( T^{24} - \)\(51\!\cdots\!66\)\( T^{25} + \)\(22\!\cdots\!52\)\( T^{26} - \)\(92\!\cdots\!56\)\( T^{27} + \)\(34\!\cdots\!49\)\( T^{28} - \)\(12\!\cdots\!31\)\( T^{29} + \)\(38\!\cdots\!22\)\( T^{30} - \)\(10\!\cdots\!21\)\( T^{31} + \)\(27\!\cdots\!44\)\( T^{32} - \)\(59\!\cdots\!33\)\( T^{33} + \)\(10\!\cdots\!65\)\( T^{34} - \)\(13\!\cdots\!24\)\( T^{35} + \)\(12\!\cdots\!89\)\( T^{36} \)
$53$ \( 1 - 60 T + 2292 T^{2} - 64439 T^{3} + 1482802 T^{4} - 29045053 T^{5} + 500611675 T^{6} - 7729740473 T^{7} + 108589762152 T^{8} - 1401743191155 T^{9} + 16770166121672 T^{10} - 187053090421054 T^{11} + 1955182548973064 T^{12} - 19221087500660183 T^{13} + 178277342948315803 T^{14} - 1563393906027613236 T^{15} + 12985863111074957420 T^{16} - \)\(10\!\cdots\!72\)\( T^{17} + \)\(76\!\cdots\!76\)\( T^{18} - \)\(54\!\cdots\!16\)\( T^{19} + \)\(36\!\cdots\!80\)\( T^{20} - \)\(23\!\cdots\!72\)\( T^{21} + \)\(14\!\cdots\!43\)\( T^{22} - \)\(80\!\cdots\!19\)\( T^{23} + \)\(43\!\cdots\!56\)\( T^{24} - \)\(21\!\cdots\!98\)\( T^{25} + \)\(10\!\cdots\!92\)\( T^{26} - \)\(46\!\cdots\!15\)\( T^{27} + \)\(18\!\cdots\!48\)\( T^{28} - \)\(71\!\cdots\!81\)\( T^{29} + \)\(24\!\cdots\!75\)\( T^{30} - \)\(75\!\cdots\!69\)\( T^{31} + \)\(20\!\cdots\!38\)\( T^{32} - \)\(47\!\cdots\!23\)\( T^{33} + \)\(88\!\cdots\!32\)\( T^{34} - \)\(12\!\cdots\!80\)\( T^{35} + \)\(10\!\cdots\!89\)\( T^{36} \)
$59$ \( 1 - 8 T + 547 T^{2} - 4886 T^{3} + 151709 T^{4} - 1426288 T^{5} + 28528730 T^{6} - 268965857 T^{7} + 4083691371 T^{8} - 37250203024 T^{9} + 471936791901 T^{10} - 4078707707888 T^{11} + 45548114306989 T^{12} - 370272133366719 T^{13} + 3754286226421140 T^{14} - 28722234855750938 T^{15} + 268673863454487823 T^{16} - 1936169927933708248 T^{17} + 16882310523288642930 T^{18} - \)\(11\!\cdots\!32\)\( T^{19} + \)\(93\!\cdots\!63\)\( T^{20} - \)\(58\!\cdots\!02\)\( T^{21} + \)\(45\!\cdots\!40\)\( T^{22} - \)\(26\!\cdots\!81\)\( T^{23} + \)\(19\!\cdots\!49\)\( T^{24} - \)\(10\!\cdots\!72\)\( T^{25} + \)\(69\!\cdots\!21\)\( T^{26} - \)\(32\!\cdots\!36\)\( T^{27} + \)\(20\!\cdots\!71\)\( T^{28} - \)\(81\!\cdots\!63\)\( T^{29} + \)\(50\!\cdots\!30\)\( T^{30} - \)\(14\!\cdots\!52\)\( T^{31} + \)\(93\!\cdots\!49\)\( T^{32} - \)\(17\!\cdots\!14\)\( T^{33} + \)\(11\!\cdots\!27\)\( T^{34} - \)\(10\!\cdots\!52\)\( T^{35} + \)\(75\!\cdots\!21\)\( T^{36} \)
$61$ \( 1 + 26 T + 1092 T^{2} + 21198 T^{3} + 517593 T^{4} + 8124611 T^{5} + 147632317 T^{6} + 1959197923 T^{7} + 29069634275 T^{8} + 335550327593 T^{9} + 4266607342662 T^{10} + 43706374725751 T^{11} + 490749289239130 T^{12} + 4530429185717696 T^{13} + 45852509636769219 T^{14} + 386091138768948475 T^{15} + 3572270707620359676 T^{16} + 27670900627019165169 T^{17} + \)\(23\!\cdots\!26\)\( T^{18} + \)\(16\!\cdots\!09\)\( T^{19} + \)\(13\!\cdots\!96\)\( T^{20} + \)\(87\!\cdots\!75\)\( T^{21} + \)\(63\!\cdots\!79\)\( T^{22} + \)\(38\!\cdots\!96\)\( T^{23} + \)\(25\!\cdots\!30\)\( T^{24} + \)\(13\!\cdots\!71\)\( T^{25} + \)\(81\!\cdots\!22\)\( T^{26} + \)\(39\!\cdots\!13\)\( T^{27} + \)\(20\!\cdots\!75\)\( T^{28} + \)\(85\!\cdots\!03\)\( T^{29} + \)\(39\!\cdots\!57\)\( T^{30} + \)\(13\!\cdots\!91\)\( T^{31} + \)\(51\!\cdots\!13\)\( T^{32} + \)\(12\!\cdots\!98\)\( T^{33} + \)\(40\!\cdots\!12\)\( T^{34} + \)\(58\!\cdots\!46\)\( T^{35} + \)\(13\!\cdots\!81\)\( T^{36} \)
$67$ \( 1 - 12 T + 743 T^{2} - 8061 T^{3} + 273924 T^{4} - 2736258 T^{5} + 67007649 T^{6} - 622982958 T^{7} + 12221255375 T^{8} - 106355514112 T^{9} + 1765865096326 T^{10} - 14415408887134 T^{11} + 209411947253484 T^{12} - 1603069573249209 T^{13} + 20828295096643948 T^{14} - 149173768364533151 T^{15} + 1760455572360289685 T^{16} - 11748171421911845627 T^{17} + \)\(12\!\cdots\!21\)\( T^{18} - \)\(78\!\cdots\!09\)\( T^{19} + \)\(79\!\cdots\!65\)\( T^{20} - \)\(44\!\cdots\!13\)\( T^{21} + \)\(41\!\cdots\!08\)\( T^{22} - \)\(21\!\cdots\!63\)\( T^{23} + \)\(18\!\cdots\!96\)\( T^{24} - \)\(87\!\cdots\!82\)\( T^{25} + \)\(71\!\cdots\!66\)\( T^{26} - \)\(28\!\cdots\!64\)\( T^{27} + \)\(22\!\cdots\!75\)\( T^{28} - \)\(76\!\cdots\!14\)\( T^{29} + \)\(54\!\cdots\!89\)\( T^{30} - \)\(15\!\cdots\!46\)\( T^{31} + \)\(10\!\cdots\!96\)\( T^{32} - \)\(19\!\cdots\!23\)\( T^{33} + \)\(12\!\cdots\!83\)\( T^{34} - \)\(13\!\cdots\!24\)\( T^{35} + \)\(74\!\cdots\!09\)\( T^{36} \)
$71$ \( 1 - T + 570 T^{2} + 395 T^{3} + 165367 T^{4} + 402481 T^{5} + 32659200 T^{6} + 135249850 T^{7} + 4976102531 T^{8} + 28212173634 T^{9} + 627081345260 T^{10} + 4301680526387 T^{11} + 68002974751364 T^{12} + 517699284019560 T^{13} + 6477452704212298 T^{14} + 51294377851106202 T^{15} + 547109894053610156 T^{16} + 4280850522490499670 T^{17} + 41143559904963586258 T^{18} + \)\(30\!\cdots\!70\)\( T^{19} + \)\(27\!\cdots\!96\)\( T^{20} + \)\(18\!\cdots\!22\)\( T^{21} + \)\(16\!\cdots\!38\)\( T^{22} + \)\(93\!\cdots\!60\)\( T^{23} + \)\(87\!\cdots\!44\)\( T^{24} + \)\(39\!\cdots\!17\)\( T^{25} + \)\(40\!\cdots\!60\)\( T^{26} + \)\(12\!\cdots\!54\)\( T^{27} + \)\(16\!\cdots\!31\)\( T^{28} + \)\(31\!\cdots\!50\)\( T^{29} + \)\(53\!\cdots\!00\)\( T^{30} + \)\(46\!\cdots\!91\)\( T^{31} + \)\(13\!\cdots\!27\)\( T^{32} + \)\(23\!\cdots\!45\)\( T^{33} + \)\(23\!\cdots\!70\)\( T^{34} - \)\(29\!\cdots\!91\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \)
$73$ \( 1 + 2 T + 402 T^{2} + 2100 T^{3} + 91054 T^{4} + 664143 T^{5} + 16192289 T^{6} + 128526512 T^{7} + 2389630448 T^{8} + 19163681612 T^{9} + 298745214234 T^{10} + 2361654820993 T^{11} + 32513178733431 T^{12} + 249000864582652 T^{13} + 3122389883437745 T^{14} + 23022354077637360 T^{15} + 267483657127421832 T^{16} + 1885230323459498199 T^{17} + 20590875702168070970 T^{18} + \)\(13\!\cdots\!27\)\( T^{19} + \)\(14\!\cdots\!28\)\( T^{20} + \)\(89\!\cdots\!20\)\( T^{21} + \)\(88\!\cdots\!45\)\( T^{22} + \)\(51\!\cdots\!36\)\( T^{23} + \)\(49\!\cdots\!59\)\( T^{24} + \)\(26\!\cdots\!21\)\( T^{25} + \)\(24\!\cdots\!54\)\( T^{26} + \)\(11\!\cdots\!56\)\( T^{27} + \)\(10\!\cdots\!52\)\( T^{28} + \)\(40\!\cdots\!24\)\( T^{29} + \)\(37\!\cdots\!69\)\( T^{30} + \)\(11\!\cdots\!19\)\( T^{31} + \)\(11\!\cdots\!86\)\( T^{32} + \)\(18\!\cdots\!00\)\( T^{33} + \)\(26\!\cdots\!22\)\( T^{34} + \)\(94\!\cdots\!06\)\( T^{35} + \)\(34\!\cdots\!69\)\( T^{36} \)
$79$ \( 1 - 18 T + 1297 T^{2} - 20031 T^{3} + 786188 T^{4} - 10631617 T^{5} + 298506186 T^{6} - 3585561802 T^{7} + 80086048911 T^{8} - 863338284200 T^{9} + 16209659796490 T^{10} - 157996153097097 T^{11} + 2576626887472372 T^{12} - 22823417890305456 T^{13} + 330077405985329133 T^{14} - 2664890406520132266 T^{15} + 34639279636509263037 T^{16} - \)\(25\!\cdots\!59\)\( T^{17} + \)\(30\!\cdots\!66\)\( T^{18} - \)\(20\!\cdots\!61\)\( T^{19} + \)\(21\!\cdots\!17\)\( T^{20} - \)\(13\!\cdots\!74\)\( T^{21} + \)\(12\!\cdots\!73\)\( T^{22} - \)\(70\!\cdots\!44\)\( T^{23} + \)\(62\!\cdots\!12\)\( T^{24} - \)\(30\!\cdots\!23\)\( T^{25} + \)\(24\!\cdots\!90\)\( T^{26} - \)\(10\!\cdots\!00\)\( T^{27} + \)\(75\!\cdots\!11\)\( T^{28} - \)\(26\!\cdots\!58\)\( T^{29} + \)\(17\!\cdots\!26\)\( T^{30} - \)\(49\!\cdots\!63\)\( T^{31} + \)\(28\!\cdots\!28\)\( T^{32} - \)\(58\!\cdots\!69\)\( T^{33} + \)\(29\!\cdots\!37\)\( T^{34} - \)\(32\!\cdots\!62\)\( T^{35} + \)\(14\!\cdots\!61\)\( T^{36} \)
$83$ \( 1 - 43 T + 1214 T^{2} - 27029 T^{3} + 517233 T^{4} - 8734852 T^{5} + 134157792 T^{6} - 1904363732 T^{7} + 25324885272 T^{8} - 318326987644 T^{9} + 3816043239532 T^{10} - 43881446080752 T^{11} + 486363584266611 T^{12} - 5212562566603327 T^{13} + 54157832160852686 T^{14} - 546073362822746053 T^{15} + 5347456077270647030 T^{16} - 50872297755235833738 T^{17} + \)\(47\!\cdots\!82\)\( T^{18} - \)\(42\!\cdots\!54\)\( T^{19} + \)\(36\!\cdots\!70\)\( T^{20} - \)\(31\!\cdots\!11\)\( T^{21} + \)\(25\!\cdots\!06\)\( T^{22} - \)\(20\!\cdots\!61\)\( T^{23} + \)\(15\!\cdots\!59\)\( T^{24} - \)\(11\!\cdots\!04\)\( T^{25} + \)\(85\!\cdots\!12\)\( T^{26} - \)\(59\!\cdots\!32\)\( T^{27} + \)\(39\!\cdots\!28\)\( T^{28} - \)\(24\!\cdots\!44\)\( T^{29} + \)\(14\!\cdots\!12\)\( T^{30} - \)\(77\!\cdots\!76\)\( T^{31} + \)\(38\!\cdots\!57\)\( T^{32} - \)\(16\!\cdots\!03\)\( T^{33} + \)\(61\!\cdots\!34\)\( T^{34} - \)\(18\!\cdots\!89\)\( T^{35} + \)\(34\!\cdots\!09\)\( T^{36} \)
$89$ \( 1 - 28 T + 1278 T^{2} - 27910 T^{3} + 750883 T^{4} - 13674851 T^{5} + 278327874 T^{6} - 4395094780 T^{7} + 74207320481 T^{8} - 1041079823079 T^{9} + 15262444839605 T^{10} - 193236416956553 T^{11} + 2524149327752443 T^{12} - 29133832029172985 T^{13} + 344418211589959806 T^{14} - 3646576704043225682 T^{15} + 39387762497694988269 T^{16} - \)\(38\!\cdots\!80\)\( T^{17} + \)\(38\!\cdots\!72\)\( T^{18} - \)\(34\!\cdots\!20\)\( T^{19} + \)\(31\!\cdots\!49\)\( T^{20} - \)\(25\!\cdots\!58\)\( T^{21} + \)\(21\!\cdots\!46\)\( T^{22} - \)\(16\!\cdots\!65\)\( T^{23} + \)\(12\!\cdots\!23\)\( T^{24} - \)\(85\!\cdots\!37\)\( T^{25} + \)\(60\!\cdots\!05\)\( T^{26} - \)\(36\!\cdots\!11\)\( T^{27} + \)\(23\!\cdots\!81\)\( T^{28} - \)\(12\!\cdots\!20\)\( T^{29} + \)\(68\!\cdots\!54\)\( T^{30} - \)\(30\!\cdots\!19\)\( T^{31} + \)\(14\!\cdots\!03\)\( T^{32} - \)\(48\!\cdots\!90\)\( T^{33} + \)\(19\!\cdots\!58\)\( T^{34} - \)\(38\!\cdots\!12\)\( T^{35} + \)\(12\!\cdots\!81\)\( T^{36} \)
$97$ \( 1 + 34 T + 1734 T^{2} + 44982 T^{3} + 1350738 T^{4} + 28687000 T^{5} + 647807944 T^{6} + 11728222048 T^{7} + 217762436551 T^{8} + 3446769515305 T^{9} + 54984423618660 T^{10} + 773527117230350 T^{11} + 10868994800485399 T^{12} + 137384482180690541 T^{13} + 1726110187080262341 T^{14} + 19737225141617642237 T^{15} + \)\(22\!\cdots\!37\)\( T^{16} + \)\(23\!\cdots\!18\)\( T^{17} + \)\(23\!\cdots\!52\)\( T^{18} + \)\(22\!\cdots\!46\)\( T^{19} + \)\(21\!\cdots\!33\)\( T^{20} + \)\(18\!\cdots\!01\)\( T^{21} + \)\(15\!\cdots\!21\)\( T^{22} + \)\(11\!\cdots\!37\)\( T^{23} + \)\(90\!\cdots\!71\)\( T^{24} + \)\(62\!\cdots\!50\)\( T^{25} + \)\(43\!\cdots\!60\)\( T^{26} + \)\(26\!\cdots\!85\)\( T^{27} + \)\(16\!\cdots\!99\)\( T^{28} + \)\(83\!\cdots\!44\)\( T^{29} + \)\(44\!\cdots\!04\)\( T^{30} + \)\(19\!\cdots\!00\)\( T^{31} + \)\(88\!\cdots\!22\)\( T^{32} + \)\(28\!\cdots\!26\)\( T^{33} + \)\(10\!\cdots\!14\)\( T^{34} + \)\(20\!\cdots\!58\)\( T^{35} + \)\(57\!\cdots\!89\)\( T^{36} \)
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