Properties

Label 1849.2.a.r
Level $1849$
Weight $2$
Character orbit 1849.a
Self dual yes
Analytic conductor $14.764$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7643393337\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 5 x^{19} - 19 x^{18} + 127 x^{17} + 95 x^{16} - 1293 x^{15} + 329 x^{14} + 6765 x^{13} - 5068 x^{12} - 19360 x^{11} + 20357 x^{10} + 29618 x^{9} - 37889 x^{8} - 21700 x^{7} + 32885 x^{6} + 6093 x^{5} - 11330 x^{4} - 1004 x^{3} + 1386 x^{2} + 87 x - 43\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 5 x^{19} - 19 x^{18} + 127 x^{17} + 95 x^{16} - 1293 x^{15} + 329 x^{14} + 6765 x^{13} - 5068 x^{12} - 19360 x^{11} + 20357 x^{10} + 29618 x^{9} - 37889 x^{8} - 21700 x^{7} + 32885 x^{6} + 6093 x^{5} - 11330 x^{4} - 1004 x^{3} + 1386 x^{2} + 87 x - 43\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-14230071983 \nu^{19} + 46615997016 \nu^{18} + 330972065901 \nu^{17} - 1171616127168 \nu^{16} - 2910628173617 \nu^{15} + 11732620801942 \nu^{14} + 11484657262735 \nu^{13} - 59715411593224 \nu^{12} - 14901877981764 \nu^{11} + 162485765268892 \nu^{10} - 28900607748863 \nu^{9} - 222851714467217 \nu^{8} + 106811883992698 \nu^{7} + 115971133895630 \nu^{6} - 99110234446389 \nu^{5} + 14545772761564 \nu^{4} + 23111814188338 \nu^{3} - 15588667852490 \nu^{2} - 1145388621680 \nu + 943797497903\)\()/ 210503587696 \)
\(\beta_{4}\)\(=\)\((\)\(-40192831361 \nu^{19} + 111749991106 \nu^{18} + 989932636217 \nu^{17} - 2828666427292 \nu^{16} - 9605341367555 \nu^{15} + 28657156909476 \nu^{14} + 46317478630483 \nu^{13} - 148861241315388 \nu^{12} - 113171607664568 \nu^{11} + 421393038907800 \nu^{10} + 115848818962819 \nu^{9} - 632498558445865 \nu^{8} + 12082802759062 \nu^{7} + 440852989621230 \nu^{6} - 82784184830195 \nu^{5} - 94138817905982 \nu^{4} + 22368332065168 \nu^{3} - 1917001809428 \nu^{2} - 2841030729550 \nu + 1077447576383\)\()/ 210503587696 \)
\(\beta_{5}\)\(=\)\((\)\(14228637027 \nu^{19} - 34498144612 \nu^{18} - 366459861464 \nu^{17} + 883733858774 \nu^{16} + 3804260095709 \nu^{15} - 9117046895340 \nu^{14} - 20470715366044 \nu^{13} + 48751376649190 \nu^{12} + 61123803690812 \nu^{11} - 144977562345826 \nu^{10} - 100364878110005 \nu^{9} + 238528299976955 \nu^{8} + 84716752424831 \nu^{7} - 202549945339379 \nu^{6} - 33672691110928 \nu^{5} + 76217290556585 \nu^{4} + 7671932147992 \nu^{3} - 10227894614659 \nu^{2} - 599774238618 \nu + 327793978196\)\()/ 52625896924 \)
\(\beta_{6}\)\(=\)\((\)\(37344752748 \nu^{19} - 94788187709 \nu^{18} - 952275906137 \nu^{17} + 2426492911022 \nu^{16} + 9747060758626 \nu^{15} - 25003477843045 \nu^{14} - 51369446049423 \nu^{13} + 133418948270550 \nu^{12} + 148499808956902 \nu^{11} - 395191025815830 \nu^{10} - 230797776420126 \nu^{9} + 645124894290285 \nu^{8} + 175530105599254 \nu^{7} - 539094925429208 \nu^{6} - 57782923544782 \nu^{5} + 196382266801699 \nu^{4} + 14208138715755 \nu^{3} - 25874132256391 \nu^{2} - 1162405628057 \nu + 927358146403\)\()/ 105251793848 \)
\(\beta_{7}\)\(=\)\((\)\(-40076363964 \nu^{19} + 95400656607 \nu^{18} + 1041101282307 \nu^{17} - 2448128554906 \nu^{16} - 10953640803422 \nu^{15} + 25317574032743 \nu^{14} + 60244560176733 \nu^{13} - 135847791045306 \nu^{12} - 186812864123554 \nu^{11} + 406000279232914 \nu^{10} + 329108224951442 \nu^{9} - 672856286458559 \nu^{8} - 320073159163538 \nu^{7} + 577328926635576 \nu^{6} + 167900179824962 \nu^{5} - 219404535918529 \nu^{4} - 49064096880753 \nu^{3} + 28207382303301 \nu^{2} + 4247919237747 \nu - 775838268705\)\()/ 105251793848 \)
\(\beta_{8}\)\(=\)\((\)\(-22565527133 \nu^{19} + 52329290331 \nu^{18} + 589133963225 \nu^{17} - 1346077054352 \nu^{16} - 6235731993181 \nu^{15} + 13972083923691 \nu^{14} + 34530559209167 \nu^{13} - 75422166461600 \nu^{12} - 107767360467754 \nu^{11} + 227770708090896 \nu^{10} + 190202416504421 \nu^{9} - 384942866151136 \nu^{8} - 181685495337785 \nu^{7} + 343948035834559 \nu^{6} + 87259234746636 \nu^{5} - 143607640371362 \nu^{4} - 20242958750477 \nu^{3} + 23096313662912 \nu^{2} + 1534054231361 \nu - 1038060570543\)\()/ 52625896924 \)
\(\beta_{9}\)\(=\)\((\)\(12222313595 \nu^{19} - 35149906384 \nu^{18} - 301069644714 \nu^{17} + 897048356248 \nu^{16} + 2922147432589 \nu^{15} - 9199313526368 \nu^{14} - 14098932791770 \nu^{13} + 48702101793264 \nu^{12} + 34481376336088 \nu^{11} - 142288509684814 \nu^{10} - 35291390286993 \nu^{9} + 226414828641029 \nu^{8} - 4135372453827 \nu^{7} - 179754151581493 \nu^{6} + 26408030509110 \nu^{5} + 58657760598743 \nu^{4} - 7361814401786 \nu^{3} - 6725786608347 \nu^{2} + 365413200906 \nu + 148383897244\)\()/ 26312948462 \)
\(\beta_{10}\)\(=\)\((\)\(-111811015775 \nu^{19} + 285669062450 \nu^{18} + 2847790709171 \nu^{17} - 7314751041244 \nu^{16} - 29101168611029 \nu^{15} + 75391201461040 \nu^{14} + 153026166000465 \nu^{13} - 402330962018060 \nu^{12} - 441081865090352 \nu^{11} + 1191445061787792 \nu^{10} + 683609133637797 \nu^{9} - 1943082102381195 \nu^{8} - 522147572120342 \nu^{7} + 1619784588572106 \nu^{6} + 183783042442835 \nu^{5} - 587136620124678 \nu^{4} - 56463860416428 \nu^{3} + 76935016739816 \nu^{2} + 6874916674146 \nu - 2574044282979\)\()/ 210503587696 \)
\(\beta_{11}\)\(=\)\((\)\(71868444078 \nu^{19} - 192655709901 \nu^{18} - 1799642357535 \nu^{17} + 4914739593678 \nu^{16} + 17917648163456 \nu^{15} - 50381737694825 \nu^{14} - 90276060091025 \nu^{13} + 266678782011486 \nu^{12} + 240735491623326 \nu^{11} - 779577845684462 \nu^{10} - 314953377096812 \nu^{9} + 1243989143864211 \nu^{8} + 140421681280074 \nu^{7} - 997254553913076 \nu^{6} + 30186985595208 \nu^{5} + 337784614049419 \nu^{4} - 5290998649529 \nu^{3} - 45026666286195 \nu^{2} - 1356886460249 \nu + 1778222415337\)\()/ 105251793848 \)
\(\beta_{12}\)\(=\)\((\)\(76368492269 \nu^{19} - 185555230662 \nu^{18} - 1971310193269 \nu^{17} + 4754809536988 \nu^{16} + 20544990365311 \nu^{15} - 49054829077416 \nu^{14} - 111354631835791 \nu^{13} + 262139873422644 \nu^{12} + 337239578662104 \nu^{11} - 777835241874840 \nu^{10} - 570751820635207 \nu^{9} + 1272593937059985 \nu^{8} + 517697984612838 \nu^{7} - 1066622998787002 \nu^{6} - 246006347238045 \nu^{5} + 389532750589718 \nu^{4} + 71370077753908 \nu^{3} - 49961865912316 \nu^{2} - 6505454572630 \nu + 1789038068597\)\()/ 105251793848 \)
\(\beta_{13}\)\(=\)\((\)\(155548732541 \nu^{19} - 402386853510 \nu^{18} - 3945463645905 \nu^{17} + 10289851728316 \nu^{16} + 40072962939383 \nu^{15} - 105854505673304 \nu^{14} - 208715018002883 \nu^{13} + 563309181684564 \nu^{12} + 591855997923712 \nu^{11} - 1660824137675056 \nu^{10} - 888494600749711 \nu^{9} + 2688865535032593 \nu^{8} + 629069854853162 \nu^{7} - 2212656513798606 \nu^{6} - 180984174704881 \nu^{5} + 783716582169282 \nu^{4} + 55146394966836 \nu^{3} - 101653872424720 \nu^{2} - 6157506806398 \nu + 3615445347609\)\()/ 210503587696 \)
\(\beta_{14}\)\(=\)\((\)\(-91935576031 \nu^{19} + 242725603925 \nu^{18} + 2316290687974 \nu^{17} - 6199309247566 \nu^{16} - 23283287460119 \nu^{15} + 63655869703515 \nu^{14} + 119218304069670 \nu^{13} - 337763015883766 \nu^{12} - 327803387385450 \nu^{11} + 991024908111162 \nu^{10} + 460951992599979 \nu^{9} - 1590485442468226 \nu^{8} - 271286209202392 \nu^{7} + 1285865117662762 \nu^{6} + 31159311691865 \nu^{5} - 437324187350595 \nu^{4} - 11619999502601 \nu^{3} + 52922232936785 \nu^{2} + 2321635342673 \nu - 1605824368164\)\()/ 105251793848 \)
\(\beta_{15}\)\(=\)\((\)\(-93833087507 \nu^{19} + 255780936564 \nu^{18} + 2344007607053 \nu^{17} - 6529262514856 \nu^{16} - 23257513775093 \nu^{15} + 66984208582074 \nu^{14} + 116572703951159 \nu^{13} - 354861426319336 \nu^{12} - 308141908987604 \nu^{11} + 1038068318046268 \nu^{10} + 395637183429693 \nu^{9} - 1655777594146369 \nu^{8} - 163225402993358 \nu^{7} + 1320398144934142 \nu^{6} - 51206133702961 \nu^{5} - 433878137753528 \nu^{4} + 10161021045118 \nu^{3} + 49326510654306 \nu^{2} + 528468761140 \nu - 1234373979961\)\()/ 105251793848 \)
\(\beta_{16}\)\(=\)\((\)\(273386016425 \nu^{19} - 723381409446 \nu^{18} - 6885247962181 \nu^{17} + 18479122112404 \nu^{16} + 69180441936035 \nu^{15} - 189811990190440 \nu^{14} - 354070559699815 \nu^{13} + 1007740093038052 \nu^{12} + 973216203616208 \nu^{11} - 2959745981769472 \nu^{10} - 1368536562842755 \nu^{9} + 4758555148819317 \nu^{8} + 806514453745394 \nu^{7} - 3860688296203710 \nu^{6} - 94127898992397 \nu^{5} + 1323282669147178 \nu^{4} + 35323243098284 \nu^{3} - 161844984538296 \nu^{2} - 7153514089446 \nu + 4807873678325\)\()/ 210503587696 \)
\(\beta_{17}\)\(=\)\((\)\(375356809195 \nu^{19} - 990037727626 \nu^{18} - 9464837304391 \nu^{17} + 25295833347988 \nu^{16} + 95270005502209 \nu^{15} - 259890551008872 \nu^{14} - 488977993955301 \nu^{13} + 1380176974441500 \nu^{12} + 1350599324318704 \nu^{11} - 4055000149086848 \nu^{10} - 1918176825366745 \nu^{9} + 6522655782099111 \nu^{8} + 1162750982341094 \nu^{7} - 5296204244315666 \nu^{6} - 164041845203031 \nu^{5} + 1817513534656366 \nu^{4} + 54517055046444 \nu^{3} - 221748050108224 \nu^{2} - 9917294383458 \nu + 6688595499263\)\()/ 210503587696 \)
\(\beta_{18}\)\(=\)\((\)\(305725917825 \nu^{19} - 806215480513 \nu^{18} - 7705358032622 \nu^{17} + 20593247638918 \nu^{16} + 77503097820145 \nu^{15} - 211499274791995 \nu^{14} - 397316678434818 \nu^{13} + 1122664590143306 \nu^{12} + 1095082077575794 \nu^{11} - 3296382641454286 \nu^{10} - 1548195644482233 \nu^{9} + 5297771184472656 \nu^{8} + 925769465133894 \nu^{7} - 4295743992084572 \nu^{6} - 118881150103701 \nu^{5} + 1470838693878981 \nu^{4} + 41415463490661 \nu^{3} - 179515614702555 \nu^{2} - 8139215573933 \nu + 5366073004952\)\()/ 105251793848 \)
\(\beta_{19}\)\(=\)\((\)\(-633296538777 \nu^{19} + 1639968208882 \nu^{18} + 16054290614973 \nu^{17} - 41932542212588 \nu^{16} - 162900388536371 \nu^{15} + 431290860199524 \nu^{14} + 846930284261719 \nu^{13} - 2294382320058436 \nu^{12} - 2392916580413464 \nu^{11} + 6760409747236640 \nu^{10} + 3561178192248795 \nu^{9} - 10930634469065993 \nu^{8} - 2455097015073230 \nu^{7} + 8965145688721578 \nu^{6} + 633271755666169 \nu^{5} - 3142535764262346 \nu^{4} - 183939248550712 \nu^{3} + 392460878843472 \nu^{2} + 23976035504562 \nu - 11881100268157\)\()/ 210503587696 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{13} + \beta_{10} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{19} + \beta_{18} + \beta_{17} - \beta_{16} - \beta_{13} + \beta_{12} + \beta_{10} + \beta_{6} + \beta_{4} + 7 \beta_{2} + \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(\beta_{18} - 2 \beta_{17} - \beta_{16} - \beta_{15} + 11 \beta_{13} - \beta_{11} + 8 \beta_{10} + \beta_{7} - 9 \beta_{6} + 9 \beta_{5} + 8 \beta_{4} - 8 \beta_{3} + 8 \beta_{2} + 29 \beta_{1} - 1\)
\(\nu^{6}\)\(=\)\(11 \beta_{19} + 11 \beta_{18} + 10 \beta_{17} - 11 \beta_{16} - \beta_{15} - 11 \beta_{13} + 11 \beta_{12} + 10 \beta_{10} + \beta_{9} - \beta_{7} + 7 \beta_{6} + 3 \beta_{5} + 10 \beta_{4} + \beta_{3} + 46 \beta_{2} + 10 \beta_{1} + 95\)
\(\nu^{7}\)\(=\)\(15 \beta_{18} - 26 \beta_{17} - 16 \beta_{16} - 13 \beta_{15} - \beta_{14} + 87 \beta_{13} + \beta_{12} - 11 \beta_{11} + 56 \beta_{10} + \beta_{9} - \beta_{8} + 10 \beta_{7} - 69 \beta_{6} + 66 \beta_{5} + 57 \beta_{4} - 56 \beta_{3} + 55 \beta_{2} + 181 \beta_{1} - 12\)
\(\nu^{8}\)\(=\)\(94 \beta_{19} + 95 \beta_{18} + 77 \beta_{17} - 96 \beta_{16} - 17 \beta_{15} - 93 \beta_{13} + 95 \beta_{12} + 2 \beta_{11} + 83 \beta_{10} + 14 \beta_{9} - 15 \beta_{7} + 41 \beta_{6} + 42 \beta_{5} + 82 \beta_{4} + 13 \beta_{3} + 304 \beta_{2} + 77 \beta_{1} + 594\)
\(\nu^{9}\)\(=\)\(3 \beta_{19} + 155 \beta_{18} - 242 \beta_{17} - 170 \beta_{16} - 127 \beta_{15} - 16 \beta_{14} + 625 \beta_{13} + 17 \beta_{12} - 94 \beta_{11} + 387 \beta_{10} + 13 \beta_{9} - 19 \beta_{8} + 76 \beta_{7} - 504 \beta_{6} + 456 \beta_{5} + 398 \beta_{4} - 381 \beta_{3} + 367 \beta_{2} + 1174 \beta_{1} - 113\)
\(\nu^{10}\)\(=\)\(734 \beta_{19} + 752 \beta_{18} + 547 \beta_{17} - 774 \beta_{16} - 190 \beta_{15} + 4 \beta_{14} - 708 \beta_{13} + 747 \beta_{12} + 32 \beta_{11} + 643 \beta_{10} + 137 \beta_{9} + 2 \beta_{8} - 155 \beta_{7} + 228 \beta_{6} + 423 \beta_{5} + 631 \beta_{4} + 117 \beta_{3} + 2030 \beta_{2} + 545 \beta_{1} + 3820\)
\(\nu^{11}\)\(=\)\(52 \beta_{19} + 1378 \beta_{18} - 1981 \beta_{17} - 1536 \beta_{16} - 1101 \beta_{15} - 173 \beta_{14} + 4334 \beta_{13} + 188 \beta_{12} - 740 \beta_{11} + 2692 \beta_{10} + 114 \beta_{9} - 228 \beta_{8} + 526 \beta_{7} - 3618 \beta_{6} + 3081 \beta_{5} + 2772 \beta_{4} - 2575 \beta_{3} + 2439 \beta_{2} + 7784 \beta_{1} - 960\)
\(\nu^{12}\)\(=\)\(5493 \beta_{19} + 5696 \beta_{18} + 3785 \beta_{17} - 5997 \beta_{16} - 1780 \beta_{15} + 88 \beta_{14} - 5099 \beta_{13} + 5598 \beta_{12} + 349 \beta_{11} + 4808 \beta_{10} + 1163 \beta_{9} + 40 \beta_{8} - 1380 \beta_{7} + 1214 \beta_{6} + 3748 \beta_{5} + 4715 \beta_{4} + 905 \beta_{3} + 13674 \beta_{2} + 3740 \beta_{1} + 24971\)
\(\nu^{13}\)\(=\)\(581 \beta_{19} + 11329 \beta_{18} - 15223 \beta_{17} - 12776 \beta_{16} - 8944 \beta_{15} - 1575 \beta_{14} + 29649 \beta_{13} + 1722 \beta_{12} - 5621 \beta_{11} + 18871 \beta_{10} + 831 \beta_{9} - 2257 \beta_{8} + 3500 \beta_{7} - 25783 \beta_{6} + 20607 \beta_{5} + 19322 \beta_{4} - 17404 \beta_{3} + 16265 \beta_{2} + 52307 \beta_{1} - 7659\)
\(\nu^{14}\)\(=\)\(40166 \beta_{19} + 42021 \beta_{18} + 26061 \beta_{17} - 45334 \beta_{16} - 15198 \beta_{15} + 1230 \beta_{14} - 35541 \beta_{13} + 40801 \beta_{12} + 3251 \beta_{11} + 35239 \beta_{10} + 9168 \beta_{9} + 509 \beta_{8} - 11385 \beta_{7} + 6042 \beta_{6} + 31075 \beta_{5} + 34662 \beta_{4} + 6445 \beta_{3} + 92738 \beta_{2} + 25427 \beta_{1} + 164944\)
\(\nu^{15}\)\(=\)\(5350 \beta_{19} + 88929 \beta_{18} - 112921 \beta_{17} - 101259 \beta_{16} - 69834 \beta_{15} - 13023 \beta_{14} + 201925 \beta_{13} + 14212 \beta_{12} - 41905 \beta_{11} + 133007 \beta_{10} + 5322 \beta_{9} - 20188 \beta_{8} + 22885 \beta_{7} - 183067 \beta_{6} + 137072 \beta_{5} + 134821 \beta_{4} - 117931 \beta_{3} + 109104 \beta_{2} + 354560 \beta_{1} - 58534\)
\(\nu^{16}\)\(=\)\(289647 \beta_{19} + 304697 \beta_{18} + 180112 \beta_{17} - 336911 \beta_{16} - 122855 \beta_{15} + 13980 \beta_{14} - 242685 \beta_{13} + 292291 \beta_{12} + 27861 \beta_{11} + 255230 \beta_{10} + 69112 \beta_{9} + 5282 \beta_{8} - 89837 \beta_{7} + 26358 \beta_{6} + 247507 \beta_{5} + 252268 \beta_{4} + 43491 \beta_{3} + 632362 \beta_{2} + 172946 \beta_{1} + 1097545\)
\(\nu^{17}\)\(=\)\(44295 \beta_{19} + 677703 \beta_{18} - 819862 \beta_{17} - 778323 \beta_{16} - 531369 \beta_{15} - 101326 \beta_{14} + 1374649 \beta_{13} + 109939 \beta_{12} - 308909 \beta_{11} + 940396 \beta_{10} + 29915 \beta_{9} - 169978 \beta_{8} + 148558 \beta_{7} - 1297065 \beta_{6} + 908677 \beta_{5} + 941412 \beta_{4} - 801903 \beta_{3} + 736600 \beta_{2} + 2417758 \beta_{1} - 433825\)
\(\nu^{18}\)\(=\)\(2070183 \beta_{19} + 2183393 \beta_{18} + 1253150 \beta_{17} - 2472918 \beta_{16} - 959188 \beta_{15} + 141091 \beta_{14} - 1634648 \beta_{13} + 2070528 \beta_{12} + 227009 \beta_{11} + 1835151 \beta_{10} + 505794 \beta_{9} + 48844 \beta_{8} - 689020 \beta_{7} + 80203 \beta_{6} + 1919528 \beta_{5} + 1823832 \beta_{4} + 281672 \beta_{3} + 4330814 \beta_{2} + 1182030 \beta_{1} + 7344074\)
\(\nu^{19}\)\(=\)\(343673 \beta_{19} + 5062454 \beta_{18} - 5871055 \beta_{17} - 5860285 \beta_{16} - 3972692 \beta_{15} - 756262 \beta_{14} + 9371430 \beta_{13} + 814491 \beta_{12} - 2260118 \beta_{11} + 6658786 \beta_{10} + 138039 \beta_{9} - 1376132 \beta_{8} + 962561 \beta_{7} - 9177327 \beta_{6} + 6010491 \beta_{5} + 6576723 \beta_{4} - 5473052 \beta_{3} + 5004508 \beta_{2} + 16558451 \beta_{1} - 3143881\)

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
−2.64020
−2.55076
−1.97560
−1.90109
−1.59121
−1.22059
−0.528833
−0.369089
−0.277120
0.173392
0.415785
0.881156
1.14959
1.54977
1.62391
2.31191
2.37008
2.37157
2.54512
2.66221
−2.64020 −0.354022 4.97064 −1.23228 0.934688 −3.87263 −7.84307 −2.87467 3.25345
1.2 −2.55076 −2.76473 4.50637 0.117709 7.05217 −0.649562 −6.39315 4.64375 −0.300248
1.3 −1.97560 0.955865 1.90300 −3.57694 −1.88841 −2.56717 0.191632 −2.08632 7.06661
1.4 −1.90109 2.82058 1.61414 3.37266 −5.36217 1.68862 0.733548 4.95566 −6.41173
1.5 −1.59121 −2.69692 0.531955 −1.39893 4.29137 −3.40103 2.33597 4.27339 2.22600
1.6 −1.22059 1.36553 −0.510154 2.60720 −1.66675 5.05996 3.06388 −1.13534 −3.18232
1.7 −0.528833 0.931794 −1.72034 0.425701 −0.492764 2.15726 1.96744 −2.13176 −0.225125
1.8 −0.369089 0.00229289 −1.86377 −2.37597 −0.000846282 0 4.42543 1.42608 −2.99999 0.876946
1.9 −0.277120 −3.15485 −1.92320 0.330136 0.874272 −1.68241 1.08720 6.95305 −0.0914873
1.10 0.173392 0.896458 −1.96994 −3.74986 0.155439 −1.28742 −0.688355 −2.19636 −0.650196
1.11 0.415785 −2.94504 −1.82712 2.53117 −1.22450 3.71316 −1.59126 5.67325 1.05242
1.12 0.881156 2.54519 −1.22356 2.37402 2.24271 −3.19714 −2.84046 3.47799 2.09188
1.13 1.14959 1.67501 −0.678443 −3.60838 1.92557 −0.890099 −3.07911 −0.194345 −4.14816
1.14 1.54977 3.21447 0.401778 2.07911 4.98169 −0.871568 −2.47687 7.33285 3.22213
1.15 1.62391 −2.30521 0.637097 3.05171 −3.74346 −2.09950 −2.21324 2.31400 4.95571
1.16 2.31191 0.910509 3.34494 2.04187 2.10502 −2.45270 3.10939 −2.17097 4.72063
1.17 2.37008 −1.76176 3.61728 −2.72492 −4.17550 −3.47356 3.83309 0.103783 −6.45828
1.18 2.37157 −0.630839 3.62433 −1.61545 −1.49608 3.60874 3.85219 −2.60204 −3.83115
1.19 2.54512 2.55827 4.47761 −1.89319 6.51109 1.19543 6.30581 3.54474 −4.81839
1.20 2.66221 −2.26260 5.08738 0.244650 −6.02353 2.59618 8.21928 2.11936 0.651312
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(43\) \(-1\)

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 1849.2.a.r yes 20
43.b odd 2 1 1849.2.a.p 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1849.2.a.p 20 43.b odd 2 1
1849.2.a.r yes 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{20} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1849))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5 T + 21 T^{2} - 63 T^{3} + 171 T^{4} - 395 T^{5} + 861 T^{6} - 1697 T^{7} + 3232 T^{8} - 5730 T^{9} + 9969 T^{10} - 16462 T^{11} + 27043 T^{12} - 42784 T^{13} + 67961 T^{14} - 104635 T^{15} + 161930 T^{16} - 242650 T^{17} + 363878 T^{18} - 526409 T^{19} + 758669 T^{20} - 1052818 T^{21} + 1455512 T^{22} - 1941200 T^{23} + 2590880 T^{24} - 3348320 T^{25} + 4349504 T^{26} - 5476352 T^{27} + 6923008 T^{28} - 8428544 T^{29} + 10208256 T^{30} - 11735040 T^{31} + 13238272 T^{32} - 13901824 T^{33} + 14106624 T^{34} - 12943360 T^{35} + 11206656 T^{36} - 8257536 T^{37} + 5505024 T^{38} - 2621440 T^{39} + 1048576 T^{40} \)
$3$ \( 1 + T + 17 T^{2} + 23 T^{3} + 167 T^{4} + 232 T^{5} + 1222 T^{6} + 1630 T^{7} + 7167 T^{8} + 9204 T^{9} + 35773 T^{10} + 43758 T^{11} + 157158 T^{12} + 182474 T^{13} + 618648 T^{14} + 684729 T^{15} + 2226544 T^{16} + 2352876 T^{17} + 7416388 T^{18} + 7539203 T^{19} + 23008870 T^{20} + 22617609 T^{21} + 66747492 T^{22} + 63527652 T^{23} + 180350064 T^{24} + 166389147 T^{25} + 450994392 T^{26} + 399070638 T^{27} + 1031113638 T^{28} + 861288714 T^{29} + 2112359877 T^{30} + 1630460988 T^{31} + 3808837647 T^{32} + 2598746490 T^{33} + 5844788118 T^{34} + 3328946424 T^{35} + 7188802407 T^{36} + 2970223749 T^{37} + 6586148313 T^{38} + 1162261467 T^{39} + 3486784401 T^{40} \)
$5$ \( 1 + 3 T + 49 T^{2} + 144 T^{3} + 1263 T^{4} + 3607 T^{5} + 22534 T^{6} + 61984 T^{7} + 309298 T^{8} + 813710 T^{9} + 3448115 T^{10} + 8635650 T^{11} + 32236817 T^{12} + 76617151 T^{13} + 257923006 T^{14} + 580352696 T^{15} + 1789501390 T^{16} + 3803096299 T^{17} + 10857165918 T^{18} + 21729721248 T^{19} + 57872565353 T^{20} + 108648606240 T^{21} + 271429147950 T^{22} + 475387037375 T^{23} + 1118438368750 T^{24} + 1813602175000 T^{25} + 4030046968750 T^{26} + 5985714921875 T^{27} + 12592506640625 T^{28} + 16866503906250 T^{29} + 33672998046875 T^{30} + 39731933593750 T^{31} + 75512207031250 T^{32} + 75664062500000 T^{33} + 137536621093750 T^{34} + 110076904296875 T^{35} + 192718505859375 T^{36} + 109863281250000 T^{37} + 186920166015625 T^{38} + 57220458984375 T^{39} + 95367431640625 T^{40} \)
$7$ \( 1 + 2 T + 62 T^{2} + 65 T^{3} + 1889 T^{4} + 519 T^{5} + 39331 T^{6} - 15569 T^{7} + 639716 T^{8} - 616487 T^{9} + 8694547 T^{10} - 12252311 T^{11} + 102295606 T^{12} - 176127090 T^{13} + 1061310639 T^{14} - 2021411595 T^{15} + 9802789013 T^{16} - 19335318101 T^{17} + 80959704741 T^{18} - 157473424267 T^{19} + 598780926318 T^{20} - 1102313969869 T^{21} + 3967025532309 T^{22} - 6632014108643 T^{23} + 23536496420213 T^{24} - 33973864677165 T^{25} + 124862135367711 T^{26} - 145048232079870 T^{27} + 589713811764406 T^{28} - 494424942935777 T^{29} + 2455994328767203 T^{30} - 1218996231811841 T^{31} + 8854492883074916 T^{32} - 1508465003026583 T^{33} + 26675191678224019 T^{34} + 2463984423660417 T^{35} + 62777005845976289 T^{36} + 15120983409168455 T^{37} + 100961643070447838 T^{38} + 22797790370746286 T^{39} + 79792266297612001 T^{40} \)
$11$ \( 1 - 21 T + 332 T^{2} - 3830 T^{3} + 37841 T^{4} - 320137 T^{5} + 2436666 T^{6} - 16742723 T^{7} + 106159161 T^{8} - 623189861 T^{9} + 3426949878 T^{10} - 17699462467 T^{11} + 86459926113 T^{12} - 400240623516 T^{13} + 1763631303304 T^{14} - 7407316029817 T^{15} + 29739414733198 T^{16} - 114225849306908 T^{17} + 420481686442108 T^{18} - 1483819660190874 T^{19} + 5024676407893132 T^{20} - 16322016262099614 T^{21} + 50878284059495068 T^{22} - 152034605427494548 T^{23} + 435414771108751918 T^{24} - 1192955653918057667 T^{25} + 3124380435312537544 T^{26} - 7799557471602913236 T^{27} + 18533453012925359553 T^{28} - 41734406656003813697 T^{29} + 88886254072031148678 T^{30} - \)\(17\!\cdots\!71\)\( T^{31} + \)\(33\!\cdots\!81\)\( T^{32} - \)\(57\!\cdots\!13\)\( T^{33} + \)\(92\!\cdots\!06\)\( T^{34} - \)\(13\!\cdots\!87\)\( T^{35} + \)\(17\!\cdots\!01\)\( T^{36} - \)\(19\!\cdots\!30\)\( T^{37} + \)\(18\!\cdots\!92\)\( T^{38} - \)\(12\!\cdots\!11\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \)
$13$ \( 1 - 11 T + 196 T^{2} - 1713 T^{3} + 18160 T^{4} - 133662 T^{5} + 1079881 T^{6} - 6932011 T^{7} + 46713697 T^{8} - 267489078 T^{9} + 1571773670 T^{10} - 8150101646 T^{11} + 42825515208 T^{12} - 203146993039 T^{13} + 969462848845 T^{14} - 4235770049611 T^{15} + 18539666968204 T^{16} - 74930665858929 T^{17} + 302660024363703 T^{18} - 1134071638747758 T^{19} + 4242290418159309 T^{20} - 14742931303720854 T^{21} + 51149544117465807 T^{22} - 164622672892067013 T^{23} + 529511428278874444 T^{24} - 1572711769030217023 T^{25} + 4679412003970685605 T^{26} - 12747172546206573163 T^{27} + 34934088397818304968 T^{28} - 86427747794893267958 T^{29} + \)\(21\!\cdots\!30\)\( T^{30} - \)\(47\!\cdots\!86\)\( T^{31} + \)\(10\!\cdots\!57\)\( T^{32} - \)\(20\!\cdots\!83\)\( T^{33} + \)\(42\!\cdots\!09\)\( T^{34} - \)\(68\!\cdots\!34\)\( T^{35} + \)\(12\!\cdots\!60\)\( T^{36} - \)\(14\!\cdots\!29\)\( T^{37} + \)\(22\!\cdots\!84\)\( T^{38} - \)\(16\!\cdots\!47\)\( T^{39} + \)\(19\!\cdots\!01\)\( T^{40} \)
$17$ \( 1 - 15 T + 313 T^{2} - 3573 T^{3} + 44424 T^{4} - 414231 T^{5} + 3917027 T^{6} - 31128563 T^{7} + 244617241 T^{8} - 1703384965 T^{9} + 11619991396 T^{10} - 72257120664 T^{11} + 438711655391 T^{12} - 2468350927685 T^{13} + 13546454355779 T^{14} - 69584613261749 T^{15} + 348603683348992 T^{16} - 1644463538759774 T^{17} + 7566533178075647 T^{18} - 32888168110340244 T^{19} + 139446945683457267 T^{20} - 559098857875784148 T^{21} + 2186728088463861983 T^{22} - 8079249365926769662 T^{23} + 29115728236991160832 T^{24} - 98800200231987149893 T^{25} + \)\(32\!\cdots\!51\)\( T^{26} - \)\(10\!\cdots\!05\)\( T^{27} + \)\(30\!\cdots\!31\)\( T^{28} - \)\(85\!\cdots\!08\)\( T^{29} + \)\(23\!\cdots\!04\)\( T^{30} - \)\(58\!\cdots\!45\)\( T^{31} + \)\(14\!\cdots\!01\)\( T^{32} - \)\(30\!\cdots\!31\)\( T^{33} + \)\(65\!\cdots\!83\)\( T^{34} - \)\(11\!\cdots\!83\)\( T^{35} + \)\(21\!\cdots\!44\)\( T^{36} - \)\(29\!\cdots\!21\)\( T^{37} + \)\(44\!\cdots\!17\)\( T^{38} - \)\(35\!\cdots\!95\)\( T^{39} + \)\(40\!\cdots\!01\)\( T^{40} \)
$19$ \( 1 + 7 T + 231 T^{2} + 1374 T^{3} + 25024 T^{4} + 130485 T^{5} + 1715592 T^{6} + 8012436 T^{7} + 84195440 T^{8} + 357387230 T^{9} + 3159874579 T^{10} + 12319711684 T^{11} + 94598235142 T^{12} + 342087763955 T^{13} + 2339303113735 T^{14} + 7951572377870 T^{15} + 49783554013471 T^{16} + 162524198764971 T^{17} + 966128590282455 T^{18} + 3115875295016196 T^{19} + 18283539240533924 T^{20} + 59201630605307724 T^{21} + 348772421091966255 T^{22} + 1114753479328936089 T^{23} + 6487842542589554191 T^{24} + 19688880413271529130 T^{25} + \)\(11\!\cdots\!35\)\( T^{26} + \)\(30\!\cdots\!45\)\( T^{27} + \)\(16\!\cdots\!22\)\( T^{28} + \)\(39\!\cdots\!36\)\( T^{29} + \)\(19\!\cdots\!79\)\( T^{30} + \)\(41\!\cdots\!70\)\( T^{31} + \)\(18\!\cdots\!40\)\( T^{32} + \)\(33\!\cdots\!24\)\( T^{33} + \)\(13\!\cdots\!32\)\( T^{34} + \)\(19\!\cdots\!15\)\( T^{35} + \)\(72\!\cdots\!44\)\( T^{36} + \)\(75\!\cdots\!86\)\( T^{37} + \)\(24\!\cdots\!71\)\( T^{38} + \)\(13\!\cdots\!53\)\( T^{39} + \)\(37\!\cdots\!01\)\( T^{40} \)
$23$ \( 1 - 26 T + 593 T^{2} - 9245 T^{3} + 129577 T^{4} - 1511139 T^{5} + 16221238 T^{6} - 154957328 T^{7} + 1384484378 T^{8} - 11370338208 T^{9} + 88282943166 T^{10} - 641718697525 T^{11} + 4442215536019 T^{12} - 29114088050089 T^{13} + 182617431976691 T^{14} - 1092252819275905 T^{15} + 6272762653585341 T^{16} - 34502594610117177 T^{17} + 182584167899486144 T^{18} - 927641876546830932 T^{19} + 4538624829212785064 T^{20} - 21335763160577111436 T^{21} + 96587024818828170176 T^{22} - \)\(41\!\cdots\!59\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} - \)\(70\!\cdots\!15\)\( T^{25} + \)\(27\!\cdots\!99\)\( T^{26} - \)\(99\!\cdots\!83\)\( T^{27} + \)\(34\!\cdots\!39\)\( T^{28} - \)\(11\!\cdots\!75\)\( T^{29} + \)\(36\!\cdots\!34\)\( T^{30} - \)\(10\!\cdots\!16\)\( T^{31} + \)\(30\!\cdots\!38\)\( T^{32} - \)\(78\!\cdots\!24\)\( T^{33} + \)\(18\!\cdots\!42\)\( T^{34} - \)\(40\!\cdots\!73\)\( T^{35} + \)\(79\!\cdots\!97\)\( T^{36} - \)\(13\!\cdots\!35\)\( T^{37} + \)\(19\!\cdots\!17\)\( T^{38} - \)\(19\!\cdots\!62\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$29$ \( 1 + 12 T + 389 T^{2} + 3720 T^{3} + 68933 T^{4} + 541228 T^{5} + 7485382 T^{6} + 48633620 T^{7} + 560017432 T^{8} + 2961118542 T^{9} + 30502328697 T^{10} + 123368346582 T^{11} + 1233796038955 T^{12} + 3114560706046 T^{13} + 36613152683904 T^{14} + 7331957410626 T^{15} + 751283965485840 T^{16} - 3447445803430502 T^{17} + 9273961941957196 T^{18} - 180596486019813978 T^{19} + 99640287316737551 T^{20} - 5237298094574605362 T^{21} + 7799401993186001836 T^{22} - 84079755699866513278 T^{23} + \)\(53\!\cdots\!40\)\( T^{24} + \)\(15\!\cdots\!74\)\( T^{25} + \)\(21\!\cdots\!84\)\( T^{26} + \)\(53\!\cdots\!14\)\( T^{27} + \)\(61\!\cdots\!55\)\( T^{28} + \)\(17\!\cdots\!58\)\( T^{29} + \)\(12\!\cdots\!97\)\( T^{30} + \)\(36\!\cdots\!18\)\( T^{31} + \)\(19\!\cdots\!12\)\( T^{32} + \)\(49\!\cdots\!80\)\( T^{33} + \)\(22\!\cdots\!42\)\( T^{34} + \)\(46\!\cdots\!72\)\( T^{35} + \)\(17\!\cdots\!93\)\( T^{36} + \)\(26\!\cdots\!80\)\( T^{37} + \)\(81\!\cdots\!29\)\( T^{38} + \)\(73\!\cdots\!28\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$31$ \( 1 - 26 T + 699 T^{2} - 11988 T^{3} + 197159 T^{4} - 2606034 T^{5} + 32787948 T^{6} - 358975262 T^{7} + 3752975389 T^{8} - 35462352562 T^{9} + 321545092610 T^{10} - 2692085875236 T^{11} + 21727037022202 T^{12} - 164069990196176 T^{13} + 1198764838137908 T^{14} - 8264662661616974 T^{15} + 55280859892476778 T^{16} - 350744272756180416 T^{17} + 2162834271239151139 T^{18} - 12687257023224161930 T^{19} + 72392713215803297198 T^{20} - \)\(39\!\cdots\!30\)\( T^{21} + \)\(20\!\cdots\!79\)\( T^{22} - \)\(10\!\cdots\!56\)\( T^{23} + \)\(51\!\cdots\!38\)\( T^{24} - \)\(23\!\cdots\!74\)\( T^{25} + \)\(10\!\cdots\!48\)\( T^{26} - \)\(45\!\cdots\!36\)\( T^{27} + \)\(18\!\cdots\!82\)\( T^{28} - \)\(71\!\cdots\!56\)\( T^{29} + \)\(26\!\cdots\!10\)\( T^{30} - \)\(90\!\cdots\!22\)\( T^{31} + \)\(29\!\cdots\!29\)\( T^{32} - \)\(87\!\cdots\!42\)\( T^{33} + \)\(24\!\cdots\!08\)\( T^{34} - \)\(61\!\cdots\!34\)\( T^{35} + \)\(14\!\cdots\!79\)\( T^{36} - \)\(27\!\cdots\!68\)\( T^{37} + \)\(48\!\cdots\!59\)\( T^{38} - \)\(56\!\cdots\!46\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \)
$37$ \( 1 + 19 T + 674 T^{2} + 10602 T^{3} + 214944 T^{4} + 2880765 T^{5} + 43380980 T^{6} + 506355490 T^{7} + 6241509967 T^{8} + 64506815954 T^{9} + 682665477543 T^{10} + 6324666556184 T^{11} + 59031781806262 T^{12} + 494695786144745 T^{13} + 4139373674392274 T^{14} + 31576171482499992 T^{15} + 239318805423602520 T^{16} + 1668642371011655963 T^{17} + 11528995954659392273 T^{18} + 73635723896403709878 T^{19} + \)\(46\!\cdots\!96\)\( T^{20} + \)\(27\!\cdots\!86\)\( T^{21} + \)\(15\!\cdots\!37\)\( T^{22} + \)\(84\!\cdots\!39\)\( T^{23} + \)\(44\!\cdots\!20\)\( T^{24} + \)\(21\!\cdots\!44\)\( T^{25} + \)\(10\!\cdots\!66\)\( T^{26} + \)\(46\!\cdots\!85\)\( T^{27} + \)\(20\!\cdots\!02\)\( T^{28} + \)\(82\!\cdots\!68\)\( T^{29} + \)\(32\!\cdots\!07\)\( T^{30} + \)\(11\!\cdots\!02\)\( T^{31} + \)\(41\!\cdots\!27\)\( T^{32} + \)\(12\!\cdots\!30\)\( T^{33} + \)\(39\!\cdots\!20\)\( T^{34} + \)\(96\!\cdots\!45\)\( T^{35} + \)\(26\!\cdots\!04\)\( T^{36} + \)\(48\!\cdots\!34\)\( T^{37} + \)\(11\!\cdots\!46\)\( T^{38} + \)\(11\!\cdots\!87\)\( T^{39} + \)\(23\!\cdots\!01\)\( T^{40} \)
$41$ \( 1 - 27 T + 869 T^{2} - 16530 T^{3} + 319061 T^{4} - 4759485 T^{5} + 69410492 T^{6} - 858796296 T^{7} + 10290120942 T^{8} - 109274591678 T^{9} + 1122948542213 T^{10} - 10479929878806 T^{11} + 94920152010059 T^{12} - 793170034277493 T^{13} + 6465307666310432 T^{14} - 49195805087779524 T^{15} + 367588572508585940 T^{16} - 2588560044156107325 T^{17} + 18023329911574085292 T^{18} - \)\(11\!\cdots\!60\)\( T^{19} + \)\(78\!\cdots\!59\)\( T^{20} - \)\(48\!\cdots\!60\)\( T^{21} + \)\(30\!\cdots\!52\)\( T^{22} - \)\(17\!\cdots\!25\)\( T^{23} + \)\(10\!\cdots\!40\)\( T^{24} - \)\(56\!\cdots\!24\)\( T^{25} + \)\(30\!\cdots\!12\)\( T^{26} - \)\(15\!\cdots\!33\)\( T^{27} + \)\(75\!\cdots\!39\)\( T^{28} - \)\(34\!\cdots\!66\)\( T^{29} + \)\(15\!\cdots\!13\)\( T^{30} - \)\(60\!\cdots\!98\)\( T^{31} + \)\(23\!\cdots\!02\)\( T^{32} - \)\(79\!\cdots\!16\)\( T^{33} + \)\(26\!\cdots\!12\)\( T^{34} - \)\(74\!\cdots\!85\)\( T^{35} + \)\(20\!\cdots\!01\)\( T^{36} - \)\(43\!\cdots\!30\)\( T^{37} + \)\(93\!\cdots\!49\)\( T^{38} - \)\(11\!\cdots\!47\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$43$ 1
$47$ \( 1 - 45 T + 1511 T^{2} - 36937 T^{3} + 770808 T^{4} - 13736882 T^{5} + 219798824 T^{6} - 3167477674 T^{7} + 42058010785 T^{8} - 516011995467 T^{9} + 5923697054548 T^{10} - 63764307510743 T^{11} + 648591155045327 T^{12} - 6243318059242073 T^{13} + 57164463799000625 T^{14} - 498296134307400984 T^{15} + 4149644163892787994 T^{16} - 33024657601896825176 T^{17} + \)\(25\!\cdots\!32\)\( T^{18} - \)\(18\!\cdots\!69\)\( T^{19} + \)\(12\!\cdots\!66\)\( T^{20} - \)\(86\!\cdots\!43\)\( T^{21} + \)\(55\!\cdots\!88\)\( T^{22} - \)\(34\!\cdots\!48\)\( T^{23} + \)\(20\!\cdots\!14\)\( T^{24} - \)\(11\!\cdots\!88\)\( T^{25} + \)\(61\!\cdots\!25\)\( T^{26} - \)\(31\!\cdots\!99\)\( T^{27} + \)\(15\!\cdots\!47\)\( T^{28} - \)\(71\!\cdots\!81\)\( T^{29} + \)\(31\!\cdots\!52\)\( T^{30} - \)\(12\!\cdots\!01\)\( T^{31} + \)\(48\!\cdots\!85\)\( T^{32} - \)\(17\!\cdots\!98\)\( T^{33} + \)\(56\!\cdots\!56\)\( T^{34} - \)\(16\!\cdots\!26\)\( T^{35} + \)\(43\!\cdots\!68\)\( T^{36} - \)\(98\!\cdots\!19\)\( T^{37} + \)\(18\!\cdots\!79\)\( T^{38} - \)\(26\!\cdots\!35\)\( T^{39} + \)\(27\!\cdots\!01\)\( T^{40} \)
$53$ \( 1 - 3 T + 613 T^{2} - 1912 T^{3} + 186977 T^{4} - 607031 T^{5} + 37799602 T^{6} - 127535552 T^{7} + 5694013788 T^{8} - 19875117064 T^{9} + 681406942949 T^{10} - 2441370740336 T^{11} + 67442970127123 T^{12} - 245242274964199 T^{13} + 5672683623486178 T^{14} - 20630924101460484 T^{15} + 413096049704419240 T^{16} - 1476192887327507123 T^{17} + 26367839384243376638 T^{18} - 90699112484413664918 T^{19} + \)\(14\!\cdots\!27\)\( T^{20} - \)\(48\!\cdots\!54\)\( T^{21} + \)\(74\!\cdots\!42\)\( T^{22} - \)\(21\!\cdots\!71\)\( T^{23} + \)\(32\!\cdots\!40\)\( T^{24} - \)\(86\!\cdots\!12\)\( T^{25} + \)\(12\!\cdots\!62\)\( T^{26} - \)\(28\!\cdots\!63\)\( T^{27} + \)\(41\!\cdots\!03\)\( T^{28} - \)\(80\!\cdots\!88\)\( T^{29} + \)\(11\!\cdots\!01\)\( T^{30} - \)\(18\!\cdots\!08\)\( T^{31} + \)\(27\!\cdots\!08\)\( T^{32} - \)\(33\!\cdots\!96\)\( T^{33} + \)\(52\!\cdots\!38\)\( T^{34} - \)\(44\!\cdots\!67\)\( T^{35} + \)\(72\!\cdots\!17\)\( T^{36} - \)\(39\!\cdots\!56\)\( T^{37} + \)\(66\!\cdots\!57\)\( T^{38} - \)\(17\!\cdots\!51\)\( T^{39} + \)\(30\!\cdots\!01\)\( T^{40} \)
$59$ \( 1 - 66 T + 2815 T^{2} - 88361 T^{3} + 2275828 T^{4} - 49883249 T^{5} + 962488724 T^{6} - 16630945847 T^{7} + 261502441352 T^{8} - 3778845268886 T^{9} + 50649777859055 T^{10} - 633662617235908 T^{11} + 7443048277471240 T^{12} - 82425509095651651 T^{13} + 863910544933685903 T^{14} - 8593161165219341990 T^{15} + 81320010966712563121 T^{16} - \)\(73\!\cdots\!79\)\( T^{17} + \)\(63\!\cdots\!87\)\( T^{18} - \)\(51\!\cdots\!89\)\( T^{19} + \)\(40\!\cdots\!16\)\( T^{20} - \)\(30\!\cdots\!51\)\( T^{21} + \)\(21\!\cdots\!47\)\( T^{22} - \)\(15\!\cdots\!41\)\( T^{23} + \)\(98\!\cdots\!81\)\( T^{24} - \)\(61\!\cdots\!10\)\( T^{25} + \)\(36\!\cdots\!23\)\( T^{26} - \)\(20\!\cdots\!69\)\( T^{27} + \)\(10\!\cdots\!40\)\( T^{28} - \)\(54\!\cdots\!12\)\( T^{29} + \)\(25\!\cdots\!55\)\( T^{30} - \)\(11\!\cdots\!74\)\( T^{31} + \)\(46\!\cdots\!12\)\( T^{32} - \)\(17\!\cdots\!13\)\( T^{33} + \)\(59\!\cdots\!64\)\( T^{34} - \)\(18\!\cdots\!51\)\( T^{35} + \)\(49\!\cdots\!48\)\( T^{36} - \)\(11\!\cdots\!59\)\( T^{37} + \)\(21\!\cdots\!15\)\( T^{38} - \)\(29\!\cdots\!74\)\( T^{39} + \)\(26\!\cdots\!01\)\( T^{40} \)
$61$ \( 1 - 30 T + 922 T^{2} - 17978 T^{3} + 345594 T^{4} - 5305743 T^{5} + 80142022 T^{6} - 1050374877 T^{7} + 13559425579 T^{8} - 158226315366 T^{9} + 1819771079659 T^{10} - 19368804087758 T^{11} + 203166699504050 T^{12} - 2001054980174280 T^{13} + 19412282312992375 T^{14} - 178471358955453852 T^{15} + 1615021614573728110 T^{16} - 13931247955336092241 T^{17} + \)\(11\!\cdots\!07\)\( T^{18} - \)\(95\!\cdots\!05\)\( T^{19} + \)\(76\!\cdots\!03\)\( T^{20} - \)\(58\!\cdots\!05\)\( T^{21} + \)\(43\!\cdots\!47\)\( T^{22} - \)\(31\!\cdots\!21\)\( T^{23} + \)\(22\!\cdots\!10\)\( T^{24} - \)\(15\!\cdots\!52\)\( T^{25} + \)\(10\!\cdots\!75\)\( T^{26} - \)\(62\!\cdots\!80\)\( T^{27} + \)\(38\!\cdots\!50\)\( T^{28} - \)\(22\!\cdots\!78\)\( T^{29} + \)\(12\!\cdots\!59\)\( T^{30} - \)\(68\!\cdots\!26\)\( T^{31} + \)\(35\!\cdots\!59\)\( T^{32} - \)\(17\!\cdots\!37\)\( T^{33} + \)\(79\!\cdots\!02\)\( T^{34} - \)\(31\!\cdots\!43\)\( T^{35} + \)\(12\!\cdots\!34\)\( T^{36} - \)\(40\!\cdots\!38\)\( T^{37} + \)\(12\!\cdots\!82\)\( T^{38} - \)\(25\!\cdots\!30\)\( T^{39} + \)\(50\!\cdots\!01\)\( T^{40} \)
$67$ \( 1 + 6 T + 713 T^{2} + 4865 T^{3} + 256067 T^{4} + 1917283 T^{5} + 61955432 T^{6} + 492179942 T^{7} + 11362573071 T^{8} + 93015796636 T^{9} + 1680301496756 T^{10} + 13848234931211 T^{11} + 207778921318378 T^{12} + 1693635064497520 T^{13} + 21986354136885960 T^{14} + 174788972094032572 T^{15} + 2021682240803889472 T^{16} + 15486466214090967294 T^{17} + \)\(16\!\cdots\!83\)\( T^{18} + \)\(11\!\cdots\!21\)\( T^{19} + \)\(11\!\cdots\!62\)\( T^{20} + \)\(79\!\cdots\!07\)\( T^{21} + \)\(73\!\cdots\!87\)\( T^{22} + \)\(46\!\cdots\!22\)\( T^{23} + \)\(40\!\cdots\!12\)\( T^{24} + \)\(23\!\cdots\!04\)\( T^{25} + \)\(19\!\cdots\!40\)\( T^{26} + \)\(10\!\cdots\!60\)\( T^{27} + \)\(84\!\cdots\!98\)\( T^{28} + \)\(37\!\cdots\!17\)\( T^{29} + \)\(30\!\cdots\!44\)\( T^{30} + \)\(11\!\cdots\!88\)\( T^{31} + \)\(92\!\cdots\!31\)\( T^{32} + \)\(26\!\cdots\!54\)\( T^{33} + \)\(22\!\cdots\!28\)\( T^{34} + \)\(47\!\cdots\!69\)\( T^{35} + \)\(42\!\cdots\!27\)\( T^{36} + \)\(53\!\cdots\!55\)\( T^{37} + \)\(52\!\cdots\!17\)\( T^{38} + \)\(29\!\cdots\!18\)\( T^{39} + \)\(33\!\cdots\!01\)\( T^{40} \)
$71$ \( 1 + 31 T + 1322 T^{2} + 31320 T^{3} + 798364 T^{4} + 15452678 T^{5} + 299944301 T^{6} + 4942514823 T^{7} + 79706547295 T^{8} + 1148868791647 T^{9} + 16066350850621 T^{10} + 206368868523456 T^{11} + 2564945996247701 T^{12} + 29747128248307392 T^{13} + 333708285975141834 T^{14} + 3526768418326305726 T^{15} + 36069118289707342370 T^{16} + \)\(34\!\cdots\!13\)\( T^{17} + \)\(32\!\cdots\!14\)\( T^{18} + \)\(29\!\cdots\!04\)\( T^{19} + \)\(25\!\cdots\!70\)\( T^{20} + \)\(20\!\cdots\!84\)\( T^{21} + \)\(16\!\cdots\!74\)\( T^{22} + \)\(12\!\cdots\!43\)\( T^{23} + \)\(91\!\cdots\!70\)\( T^{24} + \)\(63\!\cdots\!26\)\( T^{25} + \)\(42\!\cdots\!14\)\( T^{26} + \)\(27\!\cdots\!72\)\( T^{27} + \)\(16\!\cdots\!61\)\( T^{28} + \)\(94\!\cdots\!36\)\( T^{29} + \)\(52\!\cdots\!21\)\( T^{30} + \)\(26\!\cdots\!37\)\( T^{31} + \)\(13\!\cdots\!95\)\( T^{32} + \)\(57\!\cdots\!53\)\( T^{33} + \)\(24\!\cdots\!81\)\( T^{34} + \)\(90\!\cdots\!78\)\( T^{35} + \)\(33\!\cdots\!44\)\( T^{36} + \)\(92\!\cdots\!20\)\( T^{37} + \)\(27\!\cdots\!42\)\( T^{38} + \)\(46\!\cdots\!61\)\( T^{39} + \)\(10\!\cdots\!01\)\( T^{40} \)
$73$ \( 1 + 26 T + 1291 T^{2} + 27722 T^{3} + 783640 T^{4} + 14380525 T^{5} + 300791939 T^{6} + 4827381656 T^{7} + 82453306014 T^{8} + 1176453930867 T^{9} + 17238057078813 T^{10} + 221303711720328 T^{11} + 2861428005828764 T^{12} + 33345380649687651 T^{13} + 387137229185140814 T^{14} + 4120923032974543647 T^{15} + 43437310660293055700 T^{16} + \)\(42\!\cdots\!70\)\( T^{17} + \)\(40\!\cdots\!62\)\( T^{18} + \)\(36\!\cdots\!52\)\( T^{19} + \)\(32\!\cdots\!57\)\( T^{20} + \)\(26\!\cdots\!96\)\( T^{21} + \)\(21\!\cdots\!98\)\( T^{22} + \)\(16\!\cdots\!90\)\( T^{23} + \)\(12\!\cdots\!00\)\( T^{24} + \)\(85\!\cdots\!71\)\( T^{25} + \)\(58\!\cdots\!46\)\( T^{26} + \)\(36\!\cdots\!47\)\( T^{27} + \)\(23\!\cdots\!84\)\( T^{28} + \)\(13\!\cdots\!64\)\( T^{29} + \)\(74\!\cdots\!37\)\( T^{30} + \)\(36\!\cdots\!59\)\( T^{31} + \)\(18\!\cdots\!94\)\( T^{32} + \)\(80\!\cdots\!48\)\( T^{33} + \)\(36\!\cdots\!51\)\( T^{34} + \)\(12\!\cdots\!25\)\( T^{35} + \)\(50\!\cdots\!40\)\( T^{36} + \)\(13\!\cdots\!66\)\( T^{37} + \)\(44\!\cdots\!79\)\( T^{38} + \)\(65\!\cdots\!62\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$79$ \( 1 - 39 T + 1449 T^{2} - 35558 T^{3} + 812562 T^{4} - 15192159 T^{5} + 267563164 T^{6} - 4155506201 T^{7} + 61579924350 T^{8} - 834905764293 T^{9} + 10910568449802 T^{10} - 133193600438580 T^{11} + 1578428678113741 T^{12} - 17695120582970783 T^{13} + 193525885871052352 T^{14} - 2018288570921152211 T^{15} + 20606670235364380219 T^{16} - \)\(20\!\cdots\!02\)\( T^{17} + \)\(19\!\cdots\!13\)\( T^{18} - \)\(17\!\cdots\!42\)\( T^{19} + \)\(16\!\cdots\!26\)\( T^{20} - \)\(14\!\cdots\!18\)\( T^{21} + \)\(12\!\cdots\!33\)\( T^{22} - \)\(99\!\cdots\!78\)\( T^{23} + \)\(80\!\cdots\!39\)\( T^{24} - \)\(62\!\cdots\!89\)\( T^{25} + \)\(47\!\cdots\!92\)\( T^{26} - \)\(33\!\cdots\!97\)\( T^{27} + \)\(23\!\cdots\!01\)\( T^{28} - \)\(15\!\cdots\!20\)\( T^{29} + \)\(10\!\cdots\!02\)\( T^{30} - \)\(62\!\cdots\!47\)\( T^{31} + \)\(36\!\cdots\!50\)\( T^{32} - \)\(19\!\cdots\!39\)\( T^{33} + \)\(98\!\cdots\!84\)\( T^{34} - \)\(44\!\cdots\!41\)\( T^{35} + \)\(18\!\cdots\!02\)\( T^{36} - \)\(64\!\cdots\!22\)\( T^{37} + \)\(20\!\cdots\!89\)\( T^{38} - \)\(44\!\cdots\!41\)\( T^{39} + \)\(89\!\cdots\!01\)\( T^{40} \)
$83$ \( 1 - 13 T + 940 T^{2} - 11084 T^{3} + 440421 T^{4} - 4812065 T^{5} + 137986658 T^{6} - 1414251517 T^{7} + 32565902809 T^{8} - 315087302047 T^{9} + 6164011043080 T^{10} - 56448298149495 T^{11} + 970896449158067 T^{12} - 8418999649219894 T^{13} + 130224284451608500 T^{14} - 1068214850843374843 T^{15} + 15093540496126075108 T^{16} - \)\(11\!\cdots\!18\)\( T^{17} + \)\(15\!\cdots\!46\)\( T^{18} - \)\(11\!\cdots\!02\)\( T^{19} + \)\(13\!\cdots\!36\)\( T^{20} - \)\(92\!\cdots\!66\)\( T^{21} + \)\(10\!\cdots\!94\)\( T^{22} - \)\(66\!\cdots\!66\)\( T^{23} + \)\(71\!\cdots\!68\)\( T^{24} - \)\(42\!\cdots\!49\)\( T^{25} + \)\(42\!\cdots\!00\)\( T^{26} - \)\(22\!\cdots\!38\)\( T^{27} + \)\(21\!\cdots\!47\)\( T^{28} - \)\(10\!\cdots\!85\)\( T^{29} + \)\(95\!\cdots\!20\)\( T^{30} - \)\(40\!\cdots\!49\)\( T^{31} + \)\(34\!\cdots\!49\)\( T^{32} - \)\(12\!\cdots\!71\)\( T^{33} + \)\(10\!\cdots\!82\)\( T^{34} - \)\(29\!\cdots\!55\)\( T^{35} + \)\(22\!\cdots\!01\)\( T^{36} - \)\(46\!\cdots\!32\)\( T^{37} + \)\(32\!\cdots\!60\)\( T^{38} - \)\(37\!\cdots\!11\)\( T^{39} + \)\(24\!\cdots\!01\)\( T^{40} \)
$89$ \( 1 + 4 T + 1028 T^{2} + 4088 T^{3} + 523306 T^{4} + 2066896 T^{5} + 176115864 T^{6} + 687204558 T^{7} + 44130361600 T^{8} + 168664694084 T^{9} + 8788135497517 T^{10} + 32564477960810 T^{11} + 1448948480228385 T^{12} + 5153430855025836 T^{13} + 203304254976165983 T^{14} + 688359262903258760 T^{15} + 24738386446626412881 T^{16} + 79304421418613845292 T^{17} + \)\(26\!\cdots\!32\)\( T^{18} + \)\(79\!\cdots\!92\)\( T^{19} + \)\(24\!\cdots\!25\)\( T^{20} + \)\(71\!\cdots\!88\)\( T^{21} + \)\(20\!\cdots\!72\)\( T^{22} + \)\(55\!\cdots\!48\)\( T^{23} + \)\(15\!\cdots\!21\)\( T^{24} + \)\(38\!\cdots\!40\)\( T^{25} + \)\(10\!\cdots\!63\)\( T^{26} + \)\(22\!\cdots\!44\)\( T^{27} + \)\(57\!\cdots\!85\)\( T^{28} + \)\(11\!\cdots\!90\)\( T^{29} + \)\(27\!\cdots\!17\)\( T^{30} + \)\(46\!\cdots\!76\)\( T^{31} + \)\(10\!\cdots\!00\)\( T^{32} + \)\(15\!\cdots\!02\)\( T^{33} + \)\(34\!\cdots\!24\)\( T^{34} + \)\(35\!\cdots\!04\)\( T^{35} + \)\(81\!\cdots\!66\)\( T^{36} + \)\(56\!\cdots\!52\)\( T^{37} + \)\(12\!\cdots\!68\)\( T^{38} + \)\(43\!\cdots\!36\)\( T^{39} + \)\(97\!\cdots\!01\)\( T^{40} \)
$97$ \( 1 + 2 T + 1189 T^{2} + 3583 T^{3} + 696112 T^{4} + 2780155 T^{5} + 268164947 T^{6} + 1317002885 T^{7} + 76580507893 T^{8} + 438837705949 T^{9} + 17300229878303 T^{10} + 110945920301791 T^{11} + 3218716322383712 T^{12} + 22271189365451886 T^{13} + 506456339637067704 T^{14} + 3652951957138677156 T^{15} + 68610447555198868641 T^{16} + \)\(49\!\cdots\!58\)\( T^{17} + \)\(80\!\cdots\!66\)\( T^{18} + \)\(57\!\cdots\!15\)\( T^{19} + \)\(83\!\cdots\!29\)\( T^{20} + \)\(55\!\cdots\!55\)\( T^{21} + \)\(76\!\cdots\!94\)\( T^{22} + \)\(45\!\cdots\!34\)\( T^{23} + \)\(60\!\cdots\!21\)\( T^{24} + \)\(31\!\cdots\!92\)\( T^{25} + \)\(42\!\cdots\!16\)\( T^{26} + \)\(17\!\cdots\!18\)\( T^{27} + \)\(25\!\cdots\!32\)\( T^{28} + \)\(84\!\cdots\!47\)\( T^{29} + \)\(12\!\cdots\!47\)\( T^{30} + \)\(31\!\cdots\!97\)\( T^{31} + \)\(53\!\cdots\!13\)\( T^{32} + \)\(88\!\cdots\!45\)\( T^{33} + \)\(17\!\cdots\!43\)\( T^{34} + \)\(17\!\cdots\!15\)\( T^{35} + \)\(42\!\cdots\!52\)\( T^{36} + \)\(21\!\cdots\!71\)\( T^{37} + \)\(68\!\cdots\!21\)\( T^{38} + \)\(11\!\cdots\!66\)\( T^{39} + \)\(54\!\cdots\!01\)\( T^{40} \)
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