Properties

Label 1339.2.a.d
Level $1339$
Weight $2$
Character orbit 1339.a
Self dual yes
Analytic conductor $10.692$
Analytic rank $1$
Dimension $19$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1339 = 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1339.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.6919688306\)
Analytic rank: \(1\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
Defining polynomial: \(x^{19} - 9 x^{18} + 13 x^{17} + 106 x^{16} - 351 x^{15} - 279 x^{14} + 2337 x^{13} - 1079 x^{12} - 6826 x^{11} + 7199 x^{10} + 9364 x^{9} - 14841 x^{8} - 5183 x^{7} + 14037 x^{6} - 108 x^{5} - 6099 x^{4} + 855 x^{3} + 1062 x^{2} - 135 x - 63\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19} - 9 x^{18} + 13 x^{17} + 106 x^{16} - 351 x^{15} - 279 x^{14} + 2337 x^{13} - 1079 x^{12} - 6826 x^{11} + 7199 x^{10} + 9364 x^{9} - 14841 x^{8} - 5183 x^{7} + 14037 x^{6} - 108 x^{5} - 6099 x^{4} + 855 x^{3} + 1062 x^{2} - 135 x - 63\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\((\)\(518861 \nu^{18} + 866534 \nu^{17} - 35333866 \nu^{16} + 64448440 \nu^{15} + 439605017 \nu^{14} - 1193838810 \nu^{13} - 2146010844 \nu^{12} + 7874399570 \nu^{11} + 4425829614 \nu^{10} - 25338218398 \nu^{9} - 2547741249 \nu^{8} + 43197066608 \nu^{7} - 2587790527 \nu^{6} - 38810104606 \nu^{5} + 2390504031 \nu^{4} + 16419032754 \nu^{3} + 522952632 \nu^{2} - 2239694676 \nu - 367316328\)\()/24433851\)
\(\beta_{4}\)\(=\)\((\)\(266872 \nu^{18} - 1733205 \nu^{17} - 1918661 \nu^{16} + 30046194 \nu^{15} - 9244957 \nu^{14} - 224139921 \nu^{13} + 126785428 \nu^{12} + 988608896 \nu^{11} - 475558874 \nu^{10} - 2876506439 \nu^{9} + 912123174 \nu^{8} + 5323234806 \nu^{7} - 1014594649 \nu^{6} - 5515646682 \nu^{5} + 484643920 \nu^{4} + 2631800879 \nu^{3} + 73057679 \nu^{2} - 352301263 \nu - 52348346\)\()/8144617\)
\(\beta_{5}\)\(=\)\((\)\(1351897 \nu^{18} - 5921985 \nu^{17} - 25217420 \nu^{16} + 131218405 \nu^{15} + 170797761 \nu^{14} - 1175948955 \nu^{13} - 461151195 \nu^{12} + 5484083056 \nu^{11} + 90805985 \nu^{10} - 14303955112 \nu^{9} + 1743397843 \nu^{8} + 20944458162 \nu^{7} - 2506879364 \nu^{6} - 16348498764 \nu^{5} + 290078265 \nu^{4} + 5929583586 \nu^{3} + 991928916 \nu^{2} - 729624204 \nu - 256932042\)\()/24433851\)
\(\beta_{6}\)\(=\)\((\)\(-1733783 \nu^{18} + 12425812 \nu^{17} + 5062687 \nu^{16} - 219039295 \nu^{15} + 287327512 \nu^{14} + 1441567245 \nu^{13} - 3119479608 \nu^{12} - 4136363897 \nu^{11} + 13490967474 \nu^{10} + 3594678784 \nu^{9} - 28454019576 \nu^{8} + 5268963067 \nu^{7} + 29212585114 \nu^{6} - 10754227640 \nu^{5} - 13454741109 \nu^{4} + 4816071849 \nu^{3} + 2472441105 \nu^{2} - 369112455 \nu - 176479527\)\()/24433851\)
\(\beta_{7}\)\(=\)\((\)\(-1774029 \nu^{18} + 16696447 \nu^{17} - 26659212 \nu^{16} - 188001263 \nu^{15} + 628906693 \nu^{14} + 494081487 \nu^{13} - 3796144359 \nu^{12} + 1163695290 \nu^{11} + 9809723083 \nu^{10} - 6745738768 \nu^{9} - 11949471883 \nu^{8} + 8856919174 \nu^{7} + 7551460911 \nu^{6} - 3234689276 \nu^{5} - 2943638814 \nu^{4} - 337108071 \nu^{3} + 547247193 \nu^{2} + 80522568 \nu + 11552727\)\()/24433851\)
\(\beta_{8}\)\(=\)\((\)\(-2967169 \nu^{18} + 22214635 \nu^{17} - 9319981 \nu^{16} - 292838892 \nu^{15} + 557989561 \nu^{14} + 1273304997 \nu^{13} - 3878467977 \nu^{12} - 1667724457 \nu^{11} + 11125918067 \nu^{10} - 1694028624 \nu^{9} - 15359073416 \nu^{8} + 5325699157 \nu^{7} + 11072810483 \nu^{6} - 3874829504 \nu^{5} - 4541442723 \nu^{4} + 1341841695 \nu^{3} + 959132472 \nu^{2} - 354761775 \nu - 72057150\)\()/24433851\)
\(\beta_{9}\)\(=\)\((\)\(2975640 \nu^{18} - 27847177 \nu^{17} + 41511942 \nu^{16} + 340341434 \nu^{15} - 1104351670 \nu^{14} - 1087566852 \nu^{13} + 7497899679 \nu^{12} - 1724394570 \nu^{11} - 23045098663 \nu^{10} + 16410444493 \nu^{9} + 35953796293 \nu^{8} - 34577299945 \nu^{7} - 29224136445 \nu^{6} + 32040083114 \nu^{5} + 12278066430 \nu^{4} - 13208855523 \nu^{3} - 2877998025 \nu^{2} + 1767136542 \nu + 396383271\)\()/24433851\)
\(\beta_{10}\)\(=\)\((\)\(-2977490 \nu^{18} + 28229263 \nu^{17} - 39139232 \nu^{16} - 377869468 \nu^{15} + 1157175646 \nu^{14} + 1527145143 \nu^{13} - 8621118225 \nu^{12} - 13650818 \nu^{11} + 29906142930 \nu^{10} - 14778103997 \nu^{9} - 54462269205 \nu^{8} + 38723604598 \nu^{7} + 52932730804 \nu^{6} - 40591999526 \nu^{5} - 26415907989 \nu^{4} + 17502771159 \nu^{3} + 6285394443 \nu^{2} - 2240429361 \nu - 594935301\)\()/24433851\)
\(\beta_{11}\)\(=\)\((\)\(3222355 \nu^{18} - 23242464 \nu^{17} - 4157897 \nu^{16} + 370547824 \nu^{15} - 503026266 \nu^{14} - 2232289599 \nu^{13} + 4641677091 \nu^{12} + 6248040703 \nu^{11} - 17717082622 \nu^{10} - 7761853987 \nu^{9} + 34008428146 \nu^{8} + 2244994815 \nu^{7} - 33239325248 \nu^{6} + 2468241711 \nu^{5} + 15320518098 \nu^{4} - 1311577932 \nu^{3} - 2847026964 \nu^{2} + 165420858 \nu + 179541513\)\()/24433851\)
\(\beta_{12}\)\(=\)\((\)\(3242789 \nu^{18} - 28246403 \nu^{17} + 32898422 \nu^{16} + 365202966 \nu^{15} - 1060064717 \nu^{14} - 1329885306 \nu^{13} + 7728033165 \nu^{12} - 1139127976 \nu^{11} - 25462459450 \nu^{10} + 17991728124 \nu^{9} + 42616920457 \nu^{8} - 43261149383 \nu^{7} - 36360422821 \nu^{6} + 43775529148 \nu^{5} + 15066984189 \nu^{4} - 18421513458 \nu^{3} - 3103113504 \nu^{2} + 2255746206 \nu + 380675985\)\()/24433851\)
\(\beta_{13}\)\(=\)\((\)\(3916792 \nu^{18} - 35489841 \nu^{17} + 45831970 \nu^{16} + 460096249 \nu^{15} - 1382033019 \nu^{14} - 1718293269 \nu^{13} + 10037043489 \nu^{12} - 917532422 \nu^{11} - 33486398635 \nu^{10} + 20223944111 \nu^{9} + 58206182989 \nu^{8} - 49254184047 \nu^{7} - 54269913053 \nu^{6} + 50255632128 \nu^{5} + 26954603601 \nu^{4} - 21633770148 \nu^{3} - 7161240999 \nu^{2} + 2826082806 \nu + 854546172\)\()/24433851\)
\(\beta_{14}\)\(=\)\((\)\(-4506342 \nu^{18} + 35302225 \nu^{17} - 14137671 \nu^{16} - 521843402 \nu^{15} + 1009915663 \nu^{14} + 2731646766 \nu^{13} - 8288751996 \nu^{12} - 5372183439 \nu^{11} + 29962810723 \nu^{10} - 1069577854 \nu^{9} - 55839515491 \nu^{8} + 18383790961 \nu^{7} + 54803623152 \nu^{6} - 23941531913 \nu^{5} - 27167273565 \nu^{4} + 11076515934 \nu^{3} + 6282432237 \nu^{2} - 1542363603 \nu - 538160700\)\()/24433851\)
\(\beta_{15}\)\(=\)\((\)\(-4982707 \nu^{18} + 40706596 \nu^{17} - 33835177 \nu^{16} - 537568329 \nu^{15} + 1312656961 \nu^{14} + 2193453195 \nu^{13} - 9524314713 \nu^{12} - 895515472 \nu^{11} + 30469391777 \nu^{10} - 14808048921 \nu^{9} - 49097203763 \nu^{8} + 37822024414 \nu^{7} + 40741389017 \nu^{6} - 36892221704 \nu^{5} - 17035177710 \nu^{4} + 14632675959 \nu^{3} + 3588750066 \nu^{2} - 1686760458 \nu - 312204861\)\()/24433851\)
\(\beta_{16}\)\(=\)\((\)\(-5349330 \nu^{18} + 44295845 \nu^{17} - 42791964 \nu^{16} - 563967892 \nu^{15} + 1478773274 \nu^{14} + 2093135799 \nu^{13} - 10388315715 \nu^{12} + 488579049 \nu^{11} + 32287372634 \nu^{10} - 19202427860 \nu^{9} - 50947890482 \nu^{8} + 43796725379 \nu^{7} + 42992247621 \nu^{6} - 41201640961 \nu^{5} - 20129603628 \nu^{4} + 16743331674 \nu^{3} + 5448393864 \nu^{2} - 2222818413 \nu - 661897668\)\()/24433851\)
\(\beta_{17}\)\(=\)\((\)\(5831874 \nu^{18} - 45112259 \nu^{17} + 15470457 \nu^{16} + 666919645 \nu^{15} - 1253274389 \nu^{14} - 3488842920 \nu^{13} + 10301983554 \nu^{12} + 6828072783 \nu^{11} - 36950019419 \nu^{10} + 1580203424 \nu^{9} + 67636343651 \nu^{8} - 23988369473 \nu^{7} - 64012151559 \nu^{6} + 30962880442 \nu^{5} + 29484011664 \nu^{4} - 14013135555 \nu^{3} - 5996011284 \nu^{2} + 1811028675 \nu + 507806415\)\()/24433851\)
\(\beta_{18}\)\(=\)\((\)\(7258260 \nu^{18} - 53818736 \nu^{17} + 10811283 \nu^{16} + 785351959 \nu^{15} - 1378655723 \nu^{14} - 4042041579 \nu^{13} + 11324010768 \nu^{12} + 7773012714 \nu^{11} - 39967012229 \nu^{10} + 1569302456 \nu^{9} + 71956966178 \nu^{8} - 25530330101 \nu^{7} - 67703514507 \nu^{6} + 32655029953 \nu^{5} + 31499657322 \nu^{4} - 14803792149 \nu^{3} - 6412415181 \nu^{2} + 1875025842 \nu + 532661505\)\()/24433851\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{13} + \beta_{12} + \beta_{10} + \beta_{8} + \beta_{6} + \beta_{3} + \beta_{2} + 6 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{15} + 2 \beta_{13} + 2 \beta_{12} + 2 \beta_{10} + \beta_{8} + \beta_{6} + \beta_{4} + \beta_{3} + 8 \beta_{2} + 9 \beta_{1} + 8\)
\(\nu^{5}\)\(=\)\(\beta_{17} + \beta_{15} - \beta_{14} + 8 \beta_{13} + 9 \beta_{12} + 9 \beta_{10} - \beta_{9} + 9 \beta_{8} - \beta_{7} + 9 \beta_{6} + \beta_{4} + 7 \beta_{3} + 10 \beta_{2} + 38 \beta_{1} + 9\)
\(\nu^{6}\)\(=\)\(2 \beta_{17} + 9 \beta_{15} + 21 \beta_{13} + 21 \beta_{12} + 21 \beta_{10} - 2 \beta_{9} + 14 \beta_{8} - \beta_{7} + 14 \beta_{6} + \beta_{5} + 11 \beta_{4} + 10 \beta_{3} + 56 \beta_{2} + 72 \beta_{1} + 42\)
\(\nu^{7}\)\(=\)\(14 \beta_{17} - \beta_{16} + 12 \beta_{15} - 7 \beta_{14} + 63 \beta_{13} + 69 \beta_{12} + 71 \beta_{10} - 13 \beta_{9} + 73 \beta_{8} - 11 \beta_{7} + 71 \beta_{6} + 3 \beta_{5} + 17 \beta_{4} + 45 \beta_{3} + 87 \beta_{2} + 257 \beta_{1} + 67\)
\(\nu^{8}\)\(=\)\(\beta_{18} + 34 \beta_{17} - 2 \beta_{16} + 65 \beta_{15} + 5 \beta_{14} + 177 \beta_{13} + 170 \beta_{12} + 2 \beta_{11} + 176 \beta_{10} - 30 \beta_{9} + 144 \beta_{8} - 15 \beta_{7} + 141 \beta_{6} + 15 \beta_{5} + 96 \beta_{4} + 78 \beta_{3} + 387 \beta_{2} + 559 \beta_{1} + 248\)
\(\nu^{9}\)\(=\)\(2 \beta_{18} + 152 \beta_{17} - 18 \beta_{16} + 107 \beta_{15} - 23 \beta_{14} + 490 \beta_{13} + 503 \beta_{12} + 3 \beta_{11} + 537 \beta_{10} - 127 \beta_{9} + 579 \beta_{8} - 92 \beta_{7} + 551 \beta_{6} + 44 \beta_{5} + 187 \beta_{4} + 289 \beta_{3} + 711 \beta_{2} + 1808 \beta_{1} + 472\)
\(\nu^{10}\)\(=\)\(19 \beta_{18} + 404 \beta_{17} - 40 \beta_{16} + 447 \beta_{15} + 100 \beta_{14} + 1393 \beta_{13} + 1272 \beta_{12} + 31 \beta_{11} + 1377 \beta_{10} - 318 \beta_{9} + 1310 \beta_{8} - 156 \beta_{7} + 1259 \beta_{6} + 154 \beta_{5} + 782 \beta_{4} + 567 \beta_{3} + 2702 \beta_{2} + 4284 \beta_{1} + 1547\)
\(\nu^{11}\)\(=\)\(41 \beta_{18} + 1491 \beta_{17} - 218 \beta_{16} + 857 \beta_{15} + 106 \beta_{14} + 3768 \beta_{13} + 3605 \beta_{12} + 49 \beta_{11} + 4008 \beta_{10} - 1123 \beta_{9} + 4565 \beta_{8} - 708 \beta_{7} + 4286 \beta_{6} + 443 \beta_{5} + 1747 \beta_{4} + 1889 \beta_{3} + 5611 \beta_{2} + 13046 \beta_{1} + 3260\)
\(\nu^{12}\)\(=\)\(234 \beta_{18} + 4133 \beta_{17} - 522 \beta_{16} + 3060 \beta_{15} + 1298 \beta_{14} + 10683 \beta_{13} + 9272 \beta_{12} + 304 \beta_{11} + 10514 \beta_{10} - 2945 \beta_{9} + 11209 \beta_{8} - 1393 \beta_{7} + 10634 \beta_{6} + 1351 \beta_{5} + 6216 \beta_{4} + 4053 \beta_{3} + 19159 \beta_{2} + 32651 \beta_{1} + 9941\)
\(\nu^{13}\)\(=\)\(545 \beta_{18} + 13798 \beta_{17} - 2231 \beta_{16} + 6543 \beta_{15} + 2887 \beta_{14} + 28779 \beta_{13} + 25772 \beta_{12} + 481 \beta_{11} + 29886 \beta_{10} - 9462 \beta_{9} + 35920 \beta_{8} - 5278 \beta_{7} + 33483 \beta_{6} + 3814 \beta_{5} + 15107 \beta_{4} + 12659 \beta_{3} + 43427 \beta_{2} + 95817 \beta_{1} + 22354\)
\(\nu^{14}\)\(=\)\(2419 \beta_{18} + 38988 \beta_{17} - 5631 \beta_{16} + 21149 \beta_{15} + 13943 \beta_{14} + 81135 \beta_{13} + 67172 \beta_{12} + 2333 \beta_{11} + 79694 \beta_{10} - 25462 \beta_{9} + 92685 \beta_{8} - 11474 \beta_{7} + 87234 \beta_{6} + 10865 \beta_{5} + 48959 \beta_{4} + 29096 \beta_{3} + 137960 \beta_{2} + 248496 \beta_{1} + 65089\)
\(\nu^{15}\)\(=\)\(6000 \beta_{18} + 122834 \beta_{17} - 20876 \beta_{16} + 48855 \beta_{15} + 36192 \beta_{14} + 219080 \beta_{13} + 185027 \beta_{12} + 3432 \beta_{11} + 223775 \beta_{10} - 77481 \beta_{9} + 282297 \beta_{8} - 38770 \beta_{7} + 262367 \beta_{6} + 30125 \beta_{5} + 125383 \beta_{4} + 87231 \beta_{3} + 332640 \beta_{2} + 712910 \beta_{1} + 152983\)
\(\nu^{16}\)\(=\)\(22926 \beta_{18} + 349895 \beta_{17} - 54661 \beta_{16} + 148178 \beta_{15} + 135002 \beta_{14} + 614336 \beta_{13} + 487699 \beta_{12} + 14377 \beta_{11} + 603938 \beta_{10} - 211469 \beta_{9} + 750839 \beta_{8} - 90067 \beta_{7} + 703698 \beta_{6} + 82498 \beta_{5} + 384188 \beta_{4} + 211600 \beta_{3} + 1007006 \beta_{2} + 1892611 \beta_{1} + 431911\)
\(\nu^{17}\)\(=\)\(59620 \beta_{18} + 1063691 \beta_{17} - 185221 \beta_{16} + 361372 \beta_{15} + 369551 \beta_{14} + 1665656 \beta_{13} + 1338048 \beta_{12} + 16154 \beta_{11} + 1685333 \beta_{10} - 622570 \beta_{9} + 2215902 \beta_{8} - 282326 \beta_{7} + 2058923 \beta_{6} + 224796 \beta_{5} + 1016495 \beta_{4} + 618079 \beta_{3} + 2535172 \beta_{2} + 5356086 \beta_{1} + 1047559\)
\(\nu^{18}\)\(=\)\(206988 \beta_{18} + 3037776 \beta_{17} - 497337 \beta_{16} + 1052762 \beta_{15} + 1226490 \beta_{14} + 4650745 \beta_{13} + 3560905 \beta_{12} + 61479 \beta_{11} + 4588833 \beta_{10} - 1712090 \beta_{9} + 6004268 \beta_{8} - 685364 \beta_{7} + 5618254 \beta_{6} + 599547 \beta_{5} + 3009237 \beta_{4} + 1563378 \beta_{3} + 7434039 \beta_{2} + 14440520 \beta_{1} + 2896095\)

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
2.78640
2.64132
2.49720
2.47938
1.93727
1.68626
1.49342
1.26598
0.979850
0.550251
0.493077
−0.234290
−0.427526
−0.794128
−1.14213
−1.16528
−1.63365
−2.17587
−2.23754
−2.78640 −0.0654083 5.76405 −3.53830 0.182254 1.33123 −10.4882 −2.99572 9.85914
1.2 −2.64132 −0.762312 4.97659 1.72196 2.01351 1.20730 −7.86215 −2.41888 −4.54826
1.3 −2.49720 1.20166 4.23600 1.95194 −3.00079 −3.56552 −5.58375 −1.55601 −4.87439
1.4 −2.47938 2.80838 4.14734 −2.79508 −6.96305 0.439709 −5.32409 4.88699 6.93007
1.5 −1.93727 −2.42478 1.75303 2.12296 4.69746 2.19023 0.478451 2.87955 −4.11276
1.6 −1.68626 0.556240 0.843466 −0.525039 −0.937964 4.12488 1.95021 −2.69060 0.885350
1.7 −1.49342 2.98467 0.230315 −3.68204 −4.45737 −3.54305 2.64289 5.90824 5.49884
1.8 −1.26598 −1.64384 −0.397300 −2.39473 2.08107 −1.52311 3.03493 −0.297791 3.03168
1.9 −0.979850 0.723336 −1.03989 −1.69086 −0.708761 1.13749 2.97864 −2.47678 1.65679
1.10 −0.550251 −2.20461 −1.69722 −3.11514 1.21309 −3.99436 2.03440 1.86029 1.71411
1.11 −0.493077 0.177153 −1.75687 2.02635 −0.0873502 −2.21543 1.85243 −2.96862 −0.999149
1.12 0.234290 1.28880 −1.94511 2.43495 0.301953 −2.52468 −0.924301 −1.33900 0.570486
1.13 0.427526 −2.84080 −1.81722 −0.938760 −1.21451 1.20823 −1.63196 5.07012 −0.401345
1.14 0.794128 −2.28908 −1.36936 −3.96939 −1.81782 4.40368 −2.67570 2.23989 −3.15220
1.15 1.14213 −2.60408 −0.695533 2.07586 −2.97420 −0.495396 −3.07866 3.78123 2.37091
1.16 1.16528 1.72020 −0.642117 −0.583086 2.00452 −1.01939 −3.07881 −0.0409067 −0.679460
1.17 1.63365 2.37722 0.668797 −3.50821 3.88353 −4.23045 −2.17471 2.65117 −5.73118
1.18 2.17587 −0.456273 2.73443 −0.352848 −0.992793 −3.48073 1.59803 −2.79181 −0.767754
1.19 2.23754 −0.546477 3.00660 −3.24055 −1.22277 2.54937 2.25232 −2.70136 −7.25087
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \(-1\)
\(103\) \(-1\)

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 1339.2.a.d 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1339.2.a.d 19 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1339))\):

\(T_{2}^{19} + \cdots\)
\(T_{3}^{19} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 9 T + 51 T^{2} + 218 T^{3} + 775 T^{4} + 2395 T^{5} + 6631 T^{6} + 16763 T^{7} + 39252 T^{8} + 86013 T^{9} + 177832 T^{10} + 349029 T^{11} + 653521 T^{12} + 1171711 T^{13} + 2017506 T^{14} + 3343319 T^{15} + 5340883 T^{16} + 8233894 T^{17} + 12259899 T^{18} + 17637423 T^{19} + 24519798 T^{20} + 32935576 T^{21} + 42727064 T^{22} + 53493104 T^{23} + 64560192 T^{24} + 74989504 T^{25} + 83650688 T^{26} + 89351424 T^{27} + 91049984 T^{28} + 88077312 T^{29} + 80388096 T^{30} + 68661248 T^{31} + 54321152 T^{32} + 39239680 T^{33} + 25395200 T^{34} + 14286848 T^{35} + 6684672 T^{36} + 2359296 T^{37} + 524288 T^{38} \)
$3$ \( 1 + 2 T + 27 T^{2} + 49 T^{3} + 367 T^{4} + 599 T^{5} + 3359 T^{6} + 4938 T^{7} + 23407 T^{8} + 31253 T^{9} + 133113 T^{10} + 163160 T^{11} + 644570 T^{12} + 732606 T^{13} + 2725815 T^{14} + 2896510 T^{15} + 10213924 T^{16} + 10211683 T^{17} + 34183497 T^{18} + 32306849 T^{19} + 102550491 T^{20} + 91905147 T^{21} + 275775948 T^{22} + 234617310 T^{23} + 662373045 T^{24} + 534069774 T^{25} + 1409674590 T^{26} + 1070492760 T^{27} + 2620063179 T^{28} + 1845458397 T^{29} + 4146479829 T^{30} + 2624255658 T^{31} + 5355330957 T^{32} + 2864998431 T^{33} + 5266048869 T^{34} + 2109289329 T^{35} + 3486784401 T^{36} + 774840978 T^{37} + 1162261467 T^{38} \)
$5$ \( 1 + 18 T + 198 T^{2} + 1614 T^{3} + 10808 T^{4} + 62204 T^{5} + 317677 T^{6} + 1466774 T^{7} + 6211678 T^{8} + 24361049 T^{9} + 89168408 T^{10} + 306318752 T^{11} + 992201397 T^{12} + 3040669028 T^{13} + 8841466196 T^{14} + 24443775380 T^{15} + 64364984588 T^{16} + 161611789234 T^{17} + 387282779146 T^{18} + 886146712711 T^{19} + 1936413895730 T^{20} + 4040294730850 T^{21} + 8045623073500 T^{22} + 15277359612500 T^{23} + 27629581862500 T^{24} + 47510453562500 T^{25} + 77515734140625 T^{26} + 119655762500000 T^{27} + 174157046875000 T^{28} + 237900869140625 T^{29} + 303304589843750 T^{30} + 358099121093750 T^{31} + 387789306640625 T^{32} + 379663085937500 T^{33} + 329833984375000 T^{34} + 246276855468750 T^{35} + 151062011718750 T^{36} + 68664550781250 T^{37} + 19073486328125 T^{38} \)
$7$ \( 1 + 8 T + 95 T^{2} + 571 T^{3} + 4076 T^{4} + 20029 T^{5} + 109297 T^{6} + 460860 T^{7} + 2104889 T^{8} + 7864691 T^{9} + 31542666 T^{10} + 106877131 T^{11} + 387697905 T^{12} + 1211158179 T^{13} + 4048265203 T^{14} + 11784473502 T^{15} + 36684532288 T^{16} + 100051268292 T^{17} + 291598402452 T^{18} + 746445593183 T^{19} + 2041188817164 T^{20} + 4902512146308 T^{21} + 12582794574784 T^{22} + 28294520878302 T^{23} + 68039193266821 T^{24} + 142491548601171 T^{25} + 319285895777415 T^{26} + 616125391665931 T^{27} + 1272860347496262 T^{28} + 2221580548533059 T^{29} + 4162053310746527 T^{30} + 6378895619452860 T^{31} + 10589678170453879 T^{32} + 13584129926092621 T^{33} + 19351060714527668 T^{34} + 18976003355242171 T^{35} + 22099898828784665 T^{36} + 13027308783283592 T^{37} + 11398895185373143 T^{38} \)
$11$ \( 1 + T + 102 T^{2} + 20 T^{3} + 5325 T^{4} - 2238 T^{5} + 191228 T^{6} - 172743 T^{7} + 5296369 T^{8} - 6689313 T^{9} + 119990771 T^{10} - 180484782 T^{11} + 2297389068 T^{12} - 3764423433 T^{13} + 37873614375 T^{14} - 63756252789 T^{15} + 543203736624 T^{16} - 900459102983 T^{17} + 6815121096055 T^{18} - 10750324909520 T^{19} + 74966332056605 T^{20} - 108955551460943 T^{21} + 723004173446544 T^{22} - 933455297083749 T^{23} + 6099583468708125 T^{24} - 6668905741388913 T^{25} + 44769613621646628 T^{26} - 38688515907048942 T^{27} + 282931961420759761 T^{28} - 173503551569989113 T^{29} + 1511115887562311459 T^{30} - 542141533079915703 T^{31} + 6601709197859637268 T^{32} - 849880127559293358 T^{33} + 22243846502138341575 T^{34} + 918994597271443220 T^{35} + 51555596906927964642 T^{36} + 5559917313492231481 T^{37} + 61159090448414546291 T^{38} \)
$13$ \( ( 1 - T )^{19} \)
$17$ \( 1 + 16 T + 237 T^{2} + 2387 T^{3} + 22324 T^{4} + 171768 T^{5} + 1251784 T^{6} + 8021223 T^{7} + 49508712 T^{8} + 278086011 T^{9} + 1528413633 T^{10} + 7817335897 T^{11} + 39671992438 T^{12} + 190172110843 T^{13} + 912524107173 T^{14} + 4164222470908 T^{15} + 19049662981015 T^{16} + 82877635457990 T^{17} + 360409944876147 T^{18} + 1487644492080962 T^{19} + 6126969062894499 T^{20} + 23951636647359110 T^{21} + 93590994225726695 T^{22} + 347800024992707068 T^{23} + 1295653741238334261 T^{24} + 4590292447348560667 T^{25} + 16278952732274954774 T^{26} + 54531839052294159577 T^{27} + \)\(18\!\cdots\!01\)\( T^{28} + \)\(56\!\cdots\!39\)\( T^{29} + \)\(16\!\cdots\!96\)\( T^{30} + \)\(46\!\cdots\!03\)\( T^{31} + \)\(12\!\cdots\!08\)\( T^{32} + \)\(28\!\cdots\!72\)\( T^{33} + \)\(63\!\cdots\!32\)\( T^{34} + \)\(11\!\cdots\!47\)\( T^{35} + \)\(19\!\cdots\!49\)\( T^{36} + \)\(22\!\cdots\!44\)\( T^{37} + \)\(23\!\cdots\!53\)\( T^{38} \)
$19$ \( 1 + 10 T + 227 T^{2} + 1845 T^{3} + 23672 T^{4} + 163985 T^{5} + 1554982 T^{6} + 9528630 T^{7} + 74258347 T^{8} + 414683275 T^{9} + 2808265383 T^{10} + 14604064023 T^{11} + 88732459543 T^{12} + 435306881316 T^{13} + 2418896170307 T^{14} + 11258826849766 T^{15} + 57955252169470 T^{16} + 256162313195053 T^{17} + 1233417003633455 T^{18} + 5164377869779015 T^{19} + 23434923069035645 T^{20} + 92474595063414133 T^{21} + 397515074630394730 T^{22} + 1467261573888354886 T^{23} + 5989426388400992393 T^{24} + 20479395736873659396 T^{25} + 79315437917448555277 T^{26} + \)\(24\!\cdots\!43\)\( T^{27} + \)\(90\!\cdots\!57\)\( T^{28} + \)\(25\!\cdots\!75\)\( T^{29} + \)\(86\!\cdots\!93\)\( T^{30} + \)\(21\!\cdots\!30\)\( T^{31} + \)\(65\!\cdots\!38\)\( T^{32} + \)\(13\!\cdots\!85\)\( T^{33} + \)\(35\!\cdots\!28\)\( T^{34} + \)\(53\!\cdots\!45\)\( T^{35} + \)\(12\!\cdots\!53\)\( T^{36} + \)\(10\!\cdots\!10\)\( T^{37} + \)\(19\!\cdots\!79\)\( T^{38} \)
$23$ \( 1 + 14 T + 331 T^{2} + 3530 T^{3} + 48881 T^{4} + 429270 T^{5} + 4479056 T^{6} + 33901160 T^{7} + 293378798 T^{8} + 1971113777 T^{9} + 14868705538 T^{10} + 90488082120 T^{11} + 613003027963 T^{12} + 3428113140801 T^{13} + 21264972897606 T^{14} + 110411454229169 T^{15} + 635275663902722 T^{16} + 3083914591895232 T^{17} + 16590554705250677 T^{18} + 75576994646226703 T^{19} + 381582758220765571 T^{20} + 1631390819112577728 T^{21} + 7729399002704418574 T^{22} + 30897651762944882129 T^{23} + \)\(13\!\cdots\!58\)\( T^{24} + \)\(50\!\cdots\!89\)\( T^{25} + \)\(20\!\cdots\!61\)\( T^{26} + \)\(70\!\cdots\!20\)\( T^{27} + \)\(26\!\cdots\!94\)\( T^{28} + \)\(81\!\cdots\!73\)\( T^{29} + \)\(27\!\cdots\!46\)\( T^{30} + \)\(74\!\cdots\!60\)\( T^{31} + \)\(22\!\cdots\!48\)\( T^{32} + \)\(49\!\cdots\!30\)\( T^{33} + \)\(13\!\cdots\!67\)\( T^{34} + \)\(21\!\cdots\!30\)\( T^{35} + \)\(46\!\cdots\!93\)\( T^{36} + \)\(45\!\cdots\!66\)\( T^{37} + \)\(74\!\cdots\!87\)\( T^{38} \)
$29$ \( 1 + 22 T + 540 T^{2} + 8186 T^{3} + 121345 T^{4} + 1431795 T^{5} + 16088985 T^{6} + 156972174 T^{7} + 1451579214 T^{8} + 12137034530 T^{9} + 96331162859 T^{10} + 707549615806 T^{11} + 4954535363860 T^{12} + 32641088584875 T^{13} + 206426609056547 T^{14} + 1246567426174602 T^{15} + 7295654087388475 T^{16} + 41359631525384506 T^{17} + 229531297445482986 T^{18} + 1246418917742693551 T^{19} + 6656407625919006594 T^{20} + 34783450112848369546 T^{21} + \)\(17\!\cdots\!75\)\( T^{22} + \)\(88\!\cdots\!62\)\( T^{23} + \)\(42\!\cdots\!03\)\( T^{24} + \)\(19\!\cdots\!75\)\( T^{25} + \)\(85\!\cdots\!40\)\( T^{26} + \)\(35\!\cdots\!66\)\( T^{27} + \)\(13\!\cdots\!71\)\( T^{28} + \)\(51\!\cdots\!30\)\( T^{29} + \)\(17\!\cdots\!06\)\( T^{30} + \)\(55\!\cdots\!34\)\( T^{31} + \)\(16\!\cdots\!65\)\( T^{32} + \)\(42\!\cdots\!95\)\( T^{33} + \)\(10\!\cdots\!05\)\( T^{34} + \)\(20\!\cdots\!06\)\( T^{35} + \)\(39\!\cdots\!60\)\( T^{36} + \)\(46\!\cdots\!42\)\( T^{37} + \)\(61\!\cdots\!69\)\( T^{38} \)
$31$ \( 1 + 3 T + 253 T^{2} + 640 T^{3} + 33363 T^{4} + 81675 T^{5} + 3055527 T^{6} + 7831686 T^{7} + 216386675 T^{8} + 600299111 T^{9} + 12554953422 T^{10} + 37926394245 T^{11} + 619593171382 T^{12} + 2011968340151 T^{13} + 26676397096080 T^{14} + 90725449910352 T^{15} + 1019771506063363 T^{16} + 3509117681597213 T^{17} + 35009319078125048 T^{18} + 117045637610599947 T^{19} + 1085288891421876488 T^{20} + 3372262092014921693 T^{21} + 30380012937133647133 T^{22} + 83786858226658189392 T^{23} + \)\(76\!\cdots\!80\)\( T^{24} + \)\(17\!\cdots\!31\)\( T^{25} + \)\(17\!\cdots\!02\)\( T^{26} + \)\(32\!\cdots\!45\)\( T^{27} + \)\(33\!\cdots\!62\)\( T^{28} + \)\(49\!\cdots\!11\)\( T^{29} + \)\(54\!\cdots\!25\)\( T^{30} + \)\(61\!\cdots\!46\)\( T^{31} + \)\(74\!\cdots\!57\)\( T^{32} + \)\(61\!\cdots\!75\)\( T^{33} + \)\(78\!\cdots\!13\)\( T^{34} + \)\(46\!\cdots\!40\)\( T^{35} + \)\(57\!\cdots\!83\)\( T^{36} + \)\(20\!\cdots\!23\)\( T^{37} + \)\(21\!\cdots\!71\)\( T^{38} \)
$37$ \( 1 + 19 T + 491 T^{2} + 6616 T^{3} + 101441 T^{4} + 1094503 T^{5} + 12741626 T^{6} + 117650153 T^{7} + 1144023030 T^{8} + 9413283687 T^{9} + 80288050644 T^{10} + 603447940356 T^{11} + 4647014085427 T^{12} + 32444198183946 T^{13} + 230442803839096 T^{14} + 1516351289851025 T^{15} + 10121875623025439 T^{16} + 63575162012785243 T^{17} + 404839650047853747 T^{18} + 2445727359407079655 T^{19} + 14979067051770588639 T^{20} + 87034396795502997667 T^{21} + \)\(51\!\cdots\!67\)\( T^{22} + \)\(28\!\cdots\!25\)\( T^{23} + \)\(15\!\cdots\!72\)\( T^{24} + \)\(83\!\cdots\!14\)\( T^{25} + \)\(44\!\cdots\!91\)\( T^{26} + \)\(21\!\cdots\!76\)\( T^{27} + \)\(10\!\cdots\!88\)\( T^{28} + \)\(45\!\cdots\!63\)\( T^{29} + \)\(20\!\cdots\!90\)\( T^{30} + \)\(77\!\cdots\!93\)\( T^{31} + \)\(31\!\cdots\!22\)\( T^{32} + \)\(98\!\cdots\!67\)\( T^{33} + \)\(33\!\cdots\!13\)\( T^{34} + \)\(81\!\cdots\!56\)\( T^{35} + \)\(22\!\cdots\!47\)\( T^{36} + \)\(32\!\cdots\!51\)\( T^{37} + \)\(62\!\cdots\!73\)\( T^{38} \)
$41$ \( 1 + 52 T + 1810 T^{2} + 46437 T^{3} + 985714 T^{4} + 17849959 T^{5} + 285501303 T^{6} + 4094702052 T^{7} + 53491353711 T^{8} + 642062053101 T^{9} + 7141858587350 T^{10} + 74019764499369 T^{11} + 718475303462794 T^{12} + 6554265725851788 T^{13} + 56372661730637615 T^{14} + 458151324923142410 T^{15} + 3525196730700775759 T^{16} + 25712523033730701312 T^{17} + \)\(17\!\cdots\!63\)\( T^{18} + \)\(11\!\cdots\!04\)\( T^{19} + \)\(72\!\cdots\!83\)\( T^{20} + \)\(43\!\cdots\!72\)\( T^{21} + \)\(24\!\cdots\!39\)\( T^{22} + \)\(12\!\cdots\!10\)\( T^{23} + \)\(65\!\cdots\!15\)\( T^{24} + \)\(31\!\cdots\!08\)\( T^{25} + \)\(13\!\cdots\!14\)\( T^{26} + \)\(59\!\cdots\!49\)\( T^{27} + \)\(23\!\cdots\!50\)\( T^{28} + \)\(86\!\cdots\!01\)\( T^{29} + \)\(29\!\cdots\!51\)\( T^{30} + \)\(92\!\cdots\!12\)\( T^{31} + \)\(26\!\cdots\!63\)\( T^{32} + \)\(67\!\cdots\!99\)\( T^{33} + \)\(15\!\cdots\!14\)\( T^{34} + \)\(29\!\cdots\!17\)\( T^{35} + \)\(47\!\cdots\!10\)\( T^{36} + \)\(55\!\cdots\!92\)\( T^{37} + \)\(43\!\cdots\!61\)\( T^{38} \)
$43$ \( 1 + 2 T + 383 T^{2} + 499 T^{3} + 72479 T^{4} + 54797 T^{5} + 9254511 T^{6} + 3648207 T^{7} + 912332337 T^{8} + 177826268 T^{9} + 74327957909 T^{10} + 6676426712 T^{11} + 5180277499429 T^{12} + 122770366735 T^{13} + 315196434995199 T^{14} - 5854795133954 T^{15} + 16977218156735673 T^{16} - 633445494713960 T^{17} + 815911630068533768 T^{18} - 32657981578676819 T^{19} + 35084200092946952024 T^{20} - 1171240719726112040 T^{21} + \)\(13\!\cdots\!11\)\( T^{22} - 20016379458757069154 T^{23} + \)\(46\!\cdots\!57\)\( T^{24} + \)\(77\!\cdots\!15\)\( T^{25} + \)\(14\!\cdots\!03\)\( T^{26} + \)\(78\!\cdots\!12\)\( T^{27} + \)\(37\!\cdots\!87\)\( T^{28} + \)\(38\!\cdots\!32\)\( T^{29} + \)\(84\!\cdots\!59\)\( T^{30} + \)\(14\!\cdots\!07\)\( T^{31} + \)\(15\!\cdots\!73\)\( T^{32} + \)\(40\!\cdots\!53\)\( T^{33} + \)\(23\!\cdots\!53\)\( T^{34} + \)\(68\!\cdots\!99\)\( T^{35} + \)\(22\!\cdots\!69\)\( T^{36} + \)\(50\!\cdots\!98\)\( T^{37} + \)\(10\!\cdots\!07\)\( T^{38} \)
$47$ \( 1 + 24 T + 736 T^{2} + 12508 T^{3} + 229100 T^{4} + 3086296 T^{5} + 42868352 T^{6} + 486306407 T^{7} + 5611672880 T^{8} + 55741857359 T^{9} + 561561912097 T^{10} + 5023694239757 T^{11} + 45619469431483 T^{12} + 375249712914668 T^{13} + 3138006134470401 T^{14} + 24070514052589038 T^{15} + 187849034602241882 T^{16} + 1354348522899912987 T^{17} + 9934013028503139578 T^{18} + 67525744118247356353 T^{19} + \)\(46\!\cdots\!66\)\( T^{20} + \)\(29\!\cdots\!83\)\( T^{21} + \)\(19\!\cdots\!86\)\( T^{22} + \)\(11\!\cdots\!78\)\( T^{23} + \)\(71\!\cdots\!07\)\( T^{24} + \)\(40\!\cdots\!72\)\( T^{25} + \)\(23\!\cdots\!29\)\( T^{26} + \)\(11\!\cdots\!77\)\( T^{27} + \)\(62\!\cdots\!99\)\( T^{28} + \)\(29\!\cdots\!91\)\( T^{29} + \)\(13\!\cdots\!40\)\( T^{30} + \)\(56\!\cdots\!87\)\( T^{31} + \)\(23\!\cdots\!04\)\( T^{32} + \)\(79\!\cdots\!24\)\( T^{33} + \)\(27\!\cdots\!00\)\( T^{34} + \)\(70\!\cdots\!68\)\( T^{35} + \)\(19\!\cdots\!32\)\( T^{36} + \)\(30\!\cdots\!36\)\( T^{37} + \)\(58\!\cdots\!83\)\( T^{38} \)
$53$ \( 1 + 11 T + 346 T^{2} + 3101 T^{3} + 58649 T^{4} + 423672 T^{5} + 6448166 T^{6} + 37221598 T^{7} + 524392233 T^{8} + 2258001028 T^{9} + 33490123240 T^{10} + 80013840075 T^{11} + 1667255041334 T^{12} - 1345414395294 T^{13} + 60022530683787 T^{14} - 475207785558431 T^{15} + 1241988887046962 T^{16} - 43340472265016245 T^{17} - 2993396699206258 T^{18} - 2666790097581265998 T^{19} - 158650025057931674 T^{20} - \)\(12\!\cdots\!05\)\( T^{21} + \)\(18\!\cdots\!74\)\( T^{22} - \)\(37\!\cdots\!11\)\( T^{23} + \)\(25\!\cdots\!91\)\( T^{24} - \)\(29\!\cdots\!26\)\( T^{25} + \)\(19\!\cdots\!58\)\( T^{26} + \)\(49\!\cdots\!75\)\( T^{27} + \)\(11\!\cdots\!20\)\( T^{28} + \)\(39\!\cdots\!72\)\( T^{29} + \)\(48\!\cdots\!01\)\( T^{30} + \)\(18\!\cdots\!18\)\( T^{31} + \)\(16\!\cdots\!18\)\( T^{32} + \)\(58\!\cdots\!68\)\( T^{33} + \)\(42\!\cdots\!93\)\( T^{34} + \)\(12\!\cdots\!21\)\( T^{35} + \)\(71\!\cdots\!98\)\( T^{36} + \)\(11\!\cdots\!79\)\( T^{37} + \)\(57\!\cdots\!17\)\( T^{38} \)
$59$ \( 1 + 52 T + 1833 T^{2} + 47729 T^{3} + 1032141 T^{4} + 19098634 T^{5} + 313379464 T^{6} + 4632546659 T^{7} + 62746478699 T^{8} + 786525069382 T^{9} + 9215801486064 T^{10} + 101629912303778 T^{11} + 1061738143585558 T^{12} + 10557201164192470 T^{13} + 100334808606830921 T^{14} + 914064493119892775 T^{15} + 8001838078369489353 T^{16} + 67408678550362988011 T^{17} + \)\(54\!\cdots\!10\)\( T^{18} + \)\(42\!\cdots\!67\)\( T^{19} + \)\(32\!\cdots\!90\)\( T^{20} + \)\(23\!\cdots\!91\)\( T^{21} + \)\(16\!\cdots\!87\)\( T^{22} + \)\(11\!\cdots\!75\)\( T^{23} + \)\(71\!\cdots\!79\)\( T^{24} + \)\(44\!\cdots\!70\)\( T^{25} + \)\(26\!\cdots\!02\)\( T^{26} + \)\(14\!\cdots\!38\)\( T^{27} + \)\(79\!\cdots\!96\)\( T^{28} + \)\(40\!\cdots\!82\)\( T^{29} + \)\(18\!\cdots\!41\)\( T^{30} + \)\(82\!\cdots\!79\)\( T^{31} + \)\(32\!\cdots\!56\)\( T^{32} + \)\(11\!\cdots\!74\)\( T^{33} + \)\(37\!\cdots\!59\)\( T^{34} + \)\(10\!\cdots\!89\)\( T^{35} + \)\(23\!\cdots\!27\)\( T^{36} + \)\(39\!\cdots\!92\)\( T^{37} + \)\(44\!\cdots\!39\)\( T^{38} \)
$61$ \( 1 + 25 T + 788 T^{2} + 14186 T^{3} + 274407 T^{4} + 3969590 T^{5} + 59172970 T^{6} + 725803271 T^{7} + 9050446396 T^{8} + 97062790561 T^{9} + 1055748693085 T^{10} + 10107098503889 T^{11} + 98581868115422 T^{12} + 857664462279724 T^{13} + 7686856190526509 T^{14} + 62047091242085191 T^{15} + 524987853136928919 T^{16} + 4037647559460115732 T^{17} + 33174419612092286184 T^{18} + \)\(24\!\cdots\!53\)\( T^{19} + \)\(20\!\cdots\!24\)\( T^{20} + \)\(15\!\cdots\!72\)\( T^{21} + \)\(11\!\cdots\!39\)\( T^{22} + \)\(85\!\cdots\!31\)\( T^{23} + \)\(64\!\cdots\!09\)\( T^{24} + \)\(44\!\cdots\!64\)\( T^{25} + \)\(30\!\cdots\!62\)\( T^{26} + \)\(19\!\cdots\!09\)\( T^{27} + \)\(12\!\cdots\!85\)\( T^{28} + \)\(69\!\cdots\!61\)\( T^{29} + \)\(39\!\cdots\!56\)\( T^{30} + \)\(19\!\cdots\!91\)\( T^{31} + \)\(95\!\cdots\!70\)\( T^{32} + \)\(39\!\cdots\!90\)\( T^{33} + \)\(16\!\cdots\!07\)\( T^{34} + \)\(52\!\cdots\!46\)\( T^{35} + \)\(17\!\cdots\!48\)\( T^{36} + \)\(34\!\cdots\!25\)\( T^{37} + \)\(83\!\cdots\!41\)\( T^{38} \)
$67$ \( 1 + 2 T + 868 T^{2} + 2853 T^{3} + 364854 T^{4} + 1616840 T^{5} + 99909914 T^{6} + 534048354 T^{7} + 20180882955 T^{8} + 120314182078 T^{9} + 3212581366838 T^{10} + 20143838084142 T^{11} + 418325138096849 T^{12} + 2642468537361910 T^{13} + 45504654788990413 T^{14} + 280772572166108335 T^{15} + 4186007272566093824 T^{16} + 24644805687647069749 T^{17} + \)\(32\!\cdots\!08\)\( T^{18} + \)\(18\!\cdots\!45\)\( T^{19} + \)\(21\!\cdots\!36\)\( T^{20} + \)\(11\!\cdots\!61\)\( T^{21} + \)\(12\!\cdots\!12\)\( T^{22} + \)\(56\!\cdots\!35\)\( T^{23} + \)\(61\!\cdots\!91\)\( T^{24} + \)\(23\!\cdots\!90\)\( T^{25} + \)\(25\!\cdots\!27\)\( T^{26} + \)\(81\!\cdots\!22\)\( T^{27} + \)\(87\!\cdots\!86\)\( T^{28} + \)\(21\!\cdots\!22\)\( T^{29} + \)\(24\!\cdots\!65\)\( T^{30} + \)\(43\!\cdots\!94\)\( T^{31} + \)\(54\!\cdots\!18\)\( T^{32} + \)\(59\!\cdots\!60\)\( T^{33} + \)\(89\!\cdots\!22\)\( T^{34} + \)\(47\!\cdots\!93\)\( T^{35} + \)\(95\!\cdots\!36\)\( T^{36} + \)\(14\!\cdots\!18\)\( T^{37} + \)\(49\!\cdots\!03\)\( T^{38} \)
$71$ \( 1 + 44 T + 1569 T^{2} + 41292 T^{3} + 942998 T^{4} + 18695128 T^{5} + 334976798 T^{6} + 5472812256 T^{7} + 82751616460 T^{8} + 1166224261931 T^{9} + 15432560275106 T^{10} + 192646148591907 T^{11} + 2278709585632134 T^{12} + 25614036867383486 T^{13} + 274402521434013453 T^{14} + 2806579876908761827 T^{15} + 27456070636853745989 T^{16} + \)\(25\!\cdots\!96\)\( T^{17} + \)\(23\!\cdots\!04\)\( T^{18} + \)\(19\!\cdots\!73\)\( T^{19} + \)\(16\!\cdots\!84\)\( T^{20} + \)\(12\!\cdots\!36\)\( T^{21} + \)\(98\!\cdots\!79\)\( T^{22} + \)\(71\!\cdots\!87\)\( T^{23} + \)\(49\!\cdots\!03\)\( T^{24} + \)\(32\!\cdots\!06\)\( T^{25} + \)\(20\!\cdots\!94\)\( T^{26} + \)\(12\!\cdots\!27\)\( T^{27} + \)\(70\!\cdots\!86\)\( T^{28} + \)\(37\!\cdots\!31\)\( T^{29} + \)\(19\!\cdots\!60\)\( T^{30} + \)\(89\!\cdots\!96\)\( T^{31} + \)\(39\!\cdots\!78\)\( T^{32} + \)\(15\!\cdots\!68\)\( T^{33} + \)\(55\!\cdots\!98\)\( T^{34} + \)\(17\!\cdots\!32\)\( T^{35} + \)\(46\!\cdots\!79\)\( T^{36} + \)\(92\!\cdots\!84\)\( T^{37} + \)\(14\!\cdots\!31\)\( T^{38} \)
$73$ \( 1 + 39 T + 1112 T^{2} + 24931 T^{3} + 493693 T^{4} + 8689833 T^{5} + 140554412 T^{6} + 2104339625 T^{7} + 29567117227 T^{8} + 391478589428 T^{9} + 4919293990386 T^{10} + 58843806089045 T^{11} + 672740598448990 T^{12} + 7364523462401056 T^{13} + 77390887280479850 T^{14} + 781593317122606142 T^{15} + 7597892314073826869 T^{16} + 71132010810500523366 T^{17} + \)\(64\!\cdots\!23\)\( T^{18} + \)\(55\!\cdots\!13\)\( T^{19} + \)\(46\!\cdots\!79\)\( T^{20} + \)\(37\!\cdots\!14\)\( T^{21} + \)\(29\!\cdots\!73\)\( T^{22} + \)\(22\!\cdots\!22\)\( T^{23} + \)\(16\!\cdots\!50\)\( T^{24} + \)\(11\!\cdots\!84\)\( T^{25} + \)\(74\!\cdots\!30\)\( T^{26} + \)\(47\!\cdots\!45\)\( T^{27} + \)\(28\!\cdots\!18\)\( T^{28} + \)\(16\!\cdots\!72\)\( T^{29} + \)\(92\!\cdots\!79\)\( T^{30} + \)\(48\!\cdots\!25\)\( T^{31} + \)\(23\!\cdots\!96\)\( T^{32} + \)\(10\!\cdots\!97\)\( T^{33} + \)\(43\!\cdots\!01\)\( T^{34} + \)\(16\!\cdots\!91\)\( T^{35} + \)\(52\!\cdots\!36\)\( T^{36} + \)\(13\!\cdots\!91\)\( T^{37} + \)\(25\!\cdots\!37\)\( T^{38} \)
$79$ \( 1 - T + 577 T^{2} + 596 T^{3} + 175861 T^{4} + 388455 T^{5} + 38203016 T^{6} + 110694848 T^{7} + 6538158329 T^{8} + 21888460413 T^{9} + 930494535617 T^{10} + 3386357535816 T^{11} + 114035864303267 T^{12} + 433973964247942 T^{13} + 12305046386628366 T^{14} + 47612801071271311 T^{15} + 1186719959617893527 T^{16} + 4551239196498462969 T^{17} + \)\(10\!\cdots\!89\)\( T^{18} + \)\(38\!\cdots\!90\)\( T^{19} + \)\(81\!\cdots\!31\)\( T^{20} + \)\(28\!\cdots\!29\)\( T^{21} + \)\(58\!\cdots\!53\)\( T^{22} + \)\(18\!\cdots\!91\)\( T^{23} + \)\(37\!\cdots\!34\)\( T^{24} + \)\(10\!\cdots\!82\)\( T^{25} + \)\(21\!\cdots\!53\)\( T^{26} + \)\(51\!\cdots\!76\)\( T^{27} + \)\(11\!\cdots\!23\)\( T^{28} + \)\(20\!\cdots\!13\)\( T^{29} + \)\(48\!\cdots\!91\)\( T^{30} + \)\(65\!\cdots\!68\)\( T^{31} + \)\(17\!\cdots\!24\)\( T^{32} + \)\(14\!\cdots\!55\)\( T^{33} + \)\(51\!\cdots\!39\)\( T^{34} + \)\(13\!\cdots\!16\)\( T^{35} + \)\(10\!\cdots\!43\)\( T^{36} - \)\(14\!\cdots\!61\)\( T^{37} + \)\(11\!\cdots\!19\)\( T^{38} \)
$83$ \( 1 + 27 T + 1061 T^{2} + 21652 T^{3} + 529707 T^{4} + 9076993 T^{5} + 173187629 T^{6} + 2600117442 T^{7} + 41947522047 T^{8} + 565262861053 T^{9} + 8025293161899 T^{10} + 98500674340986 T^{11} + 1258813864291879 T^{12} + 14204784751823140 T^{13} + 165694154500131550 T^{14} + 1729082023699801251 T^{15} + 18570173171829956670 T^{16} + \)\(17\!\cdots\!63\)\( T^{17} + \)\(17\!\cdots\!87\)\( T^{18} + \)\(16\!\cdots\!47\)\( T^{19} + \)\(14\!\cdots\!21\)\( T^{20} + \)\(12\!\cdots\!07\)\( T^{21} + \)\(10\!\cdots\!90\)\( T^{22} + \)\(82\!\cdots\!71\)\( T^{23} + \)\(65\!\cdots\!50\)\( T^{24} + \)\(46\!\cdots\!60\)\( T^{25} + \)\(34\!\cdots\!33\)\( T^{26} + \)\(22\!\cdots\!26\)\( T^{27} + \)\(15\!\cdots\!97\)\( T^{28} + \)\(87\!\cdots\!97\)\( T^{29} + \)\(54\!\cdots\!49\)\( T^{30} + \)\(27\!\cdots\!62\)\( T^{31} + \)\(15\!\cdots\!27\)\( T^{32} + \)\(66\!\cdots\!97\)\( T^{33} + \)\(32\!\cdots\!49\)\( T^{34} + \)\(10\!\cdots\!12\)\( T^{35} + \)\(44\!\cdots\!03\)\( T^{36} + \)\(94\!\cdots\!43\)\( T^{37} + \)\(29\!\cdots\!47\)\( T^{38} \)
$89$ \( 1 + 86 T + 4808 T^{2} + 198690 T^{3} + 6733357 T^{4} + 194194126 T^{5} + 4916860345 T^{6} + 111181073018 T^{7} + 2277381535342 T^{8} + 42660026197638 T^{9} + 736545846785157 T^{10} + 11788777804402798 T^{11} + 175751139094968731 T^{12} + 2449424687074653956 T^{13} + 32008239306383930487 T^{14} + \)\(39\!\cdots\!23\)\( T^{15} + \)\(45\!\cdots\!14\)\( T^{16} + \)\(49\!\cdots\!14\)\( T^{17} + \)\(50\!\cdots\!63\)\( T^{18} + \)\(49\!\cdots\!41\)\( T^{19} + \)\(45\!\cdots\!07\)\( T^{20} + \)\(39\!\cdots\!94\)\( T^{21} + \)\(32\!\cdots\!66\)\( T^{22} + \)\(24\!\cdots\!43\)\( T^{23} + \)\(17\!\cdots\!63\)\( T^{24} + \)\(12\!\cdots\!16\)\( T^{25} + \)\(77\!\cdots\!99\)\( T^{26} + \)\(46\!\cdots\!38\)\( T^{27} + \)\(25\!\cdots\!13\)\( T^{28} + \)\(13\!\cdots\!38\)\( T^{29} + \)\(63\!\cdots\!38\)\( T^{30} + \)\(27\!\cdots\!78\)\( T^{31} + \)\(10\!\cdots\!05\)\( T^{32} + \)\(37\!\cdots\!66\)\( T^{33} + \)\(11\!\cdots\!93\)\( T^{34} + \)\(30\!\cdots\!90\)\( T^{35} + \)\(66\!\cdots\!32\)\( T^{36} + \)\(10\!\cdots\!66\)\( T^{37} + \)\(10\!\cdots\!09\)\( T^{38} \)
$97$ \( 1 + 20 T + 1242 T^{2} + 20932 T^{3} + 726793 T^{4} + 10658516 T^{5} + 270594271 T^{6} + 3529219650 T^{7} + 72782811720 T^{8} + 858250232818 T^{9} + 15213391467171 T^{10} + 164375428993321 T^{11} + 2594760312225812 T^{12} + 25971791784188106 T^{13} + 373876286436159441 T^{14} + 3496162540583458560 T^{15} + 46627609607879495423 T^{16} + \)\(40\!\cdots\!24\)\( T^{17} + \)\(51\!\cdots\!56\)\( T^{18} + \)\(42\!\cdots\!27\)\( T^{19} + \)\(49\!\cdots\!32\)\( T^{20} + \)\(38\!\cdots\!16\)\( T^{21} + \)\(42\!\cdots\!79\)\( T^{22} + \)\(30\!\cdots\!60\)\( T^{23} + \)\(32\!\cdots\!37\)\( T^{24} + \)\(21\!\cdots\!74\)\( T^{25} + \)\(20\!\cdots\!56\)\( T^{26} + \)\(12\!\cdots\!81\)\( T^{27} + \)\(11\!\cdots\!07\)\( T^{28} + \)\(63\!\cdots\!82\)\( T^{29} + \)\(52\!\cdots\!60\)\( T^{30} + \)\(24\!\cdots\!50\)\( T^{31} + \)\(18\!\cdots\!67\)\( T^{32} + \)\(69\!\cdots\!04\)\( T^{33} + \)\(46\!\cdots\!49\)\( T^{34} + \)\(12\!\cdots\!72\)\( T^{35} + \)\(74\!\cdots\!54\)\( T^{36} + \)\(11\!\cdots\!80\)\( T^{37} + \)\(56\!\cdots\!33\)\( T^{38} \)
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