Properties

Label 1045.4.a.c
Level $1045$
Weight $4$
Character orbit 1045.a
Self dual yes
Analytic conductor $61.657$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.6569959560\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + 1317797 x^{12} - 1160501 x^{11} - 9914845 x^{10} + 7570653 x^{9} + 42786958 x^{8} - 23777633 x^{7} - 102801526 x^{6} + 28436356 x^{5} + 122325928 x^{4} + 411232 x^{3} - 47350496 x^{2} - 4782848 x + 150528\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + \beta_{5} q^{3} + ( 3 + \beta_{2} ) q^{4} -5 q^{5} + ( -3 + \beta_{1} + \beta_{6} ) q^{6} + ( 2 + \beta_{7} ) q^{7} + ( 1 - 3 \beta_{1} - \beta_{3} ) q^{8} + ( 7 - \beta_{5} - \beta_{10} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + \beta_{5} q^{3} + ( 3 + \beta_{2} ) q^{4} -5 q^{5} + ( -3 + \beta_{1} + \beta_{6} ) q^{6} + ( 2 + \beta_{7} ) q^{7} + ( 1 - 3 \beta_{1} - \beta_{3} ) q^{8} + ( 7 - \beta_{5} - \beta_{10} ) q^{9} + 5 \beta_{1} q^{10} -11 q^{11} + ( -2 + 3 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{17} ) q^{12} + ( 3 - \beta_{1} - \beta_{5} + \beta_{18} ) q^{13} + ( -5 - 2 \beta_{1} - 2 \beta_{2} - \beta_{7} - \beta_{12} ) q^{14} -5 \beta_{5} q^{15} + ( 15 + 3 \beta_{2} + \beta_{3} - 2 \beta_{6} + \beta_{9} + \beta_{10} + \beta_{12} - \beta_{17} - \beta_{18} ) q^{16} + ( -9 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{17} + ( -9 \beta_{1} - \beta_{2} + \beta_{3} - 5 \beta_{5} - 3 \beta_{6} + \beta_{11} + \beta_{13} - \beta_{17} - \beta_{18} ) q^{18} + 19 q^{19} + ( -15 - 5 \beta_{2} ) q^{20} + ( 5 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{17} ) q^{21} + 11 \beta_{1} q^{22} + ( -9 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{13} - \beta_{14} - \beta_{17} - \beta_{18} ) q^{23} + ( -25 + 7 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} + 6 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} - 3 \beta_{13} + \beta_{14} - \beta_{15} + \beta_{17} + 2 \beta_{18} ) q^{24} + 25 q^{25} + ( 7 - 7 \beta_{1} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 2 \beta_{8} + 3 \beta_{10} + \beta_{12} + \beta_{15} - \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{26} + ( -12 + 11 \beta_{1} - 5 \beta_{2} + 3 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{13} + \beta_{14} - \beta_{18} ) q^{27} + ( 7 + 14 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} + \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} + 3 \beta_{10} + \beta_{12} - \beta_{14} + 2 \beta_{15} - 2 \beta_{17} + 2 \beta_{19} ) q^{28} + ( -12 + 7 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + 4 \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{16} - 3 \beta_{18} ) q^{29} + ( 15 - 5 \beta_{1} - 5 \beta_{6} ) q^{30} + ( -41 - 8 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{6} - 2 \beta_{8} + \beta_{11} - 3 \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{16} + \beta_{17} + 3 \beta_{18} + \beta_{19} ) q^{31} + ( -20 - 6 \beta_{1} - 7 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} - 12 \beta_{5} + 5 \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 6 \beta_{10} - \beta_{12} - 2 \beta_{13} + \beta_{14} - 3 \beta_{15} + 3 \beta_{17} + 3 \beta_{18} - 3 \beta_{19} ) q^{32} -11 \beta_{5} q^{33} + ( -32 + 24 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{34} + ( -10 - 5 \beta_{7} ) q^{35} + ( 33 + 3 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 22 \beta_{5} - 4 \beta_{6} - \beta_{9} - 6 \beta_{10} - \beta_{11} - 2 \beta_{12} + 5 \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{16} + 3 \beta_{18} + \beta_{19} ) q^{36} + ( 28 + 15 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} - 10 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} - 4 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{18} ) q^{37} -19 \beta_{1} q^{38} + ( -41 + 17 \beta_{1} - 3 \beta_{2} - 4 \beta_{5} - 4 \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} - 3 \beta_{12} + 3 \beta_{13} - 2 \beta_{14} - \beta_{15} - \beta_{16} + 5 \beta_{17} - 3 \beta_{19} ) q^{39} + ( -5 + 15 \beta_{1} + 5 \beta_{3} ) q^{40} + ( 30 + 8 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} - 5 \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{11} + \beta_{12} + 4 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} + \beta_{16} - 3 \beta_{18} + 2 \beta_{19} ) q^{41} + ( 8 + 12 \beta_{1} - 2 \beta_{2} - \beta_{3} - 5 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} - 6 \beta_{10} - 3 \beta_{11} + 3 \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{42} + ( -47 + 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} + 5 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{15} - \beta_{16} - \beta_{18} - 2 \beta_{19} ) q^{43} + ( -33 - 11 \beta_{2} ) q^{44} + ( -35 + 5 \beta_{5} + 5 \beta_{10} ) q^{45} + ( -21 + 23 \beta_{1} - 10 \beta_{2} - 4 \beta_{3} - \beta_{4} - 14 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - 11 \beta_{10} + 2 \beta_{11} - 3 \beta_{12} + 4 \beta_{13} - \beta_{14} - 2 \beta_{15} + 3 \beta_{17} + 4 \beta_{18} - \beta_{19} ) q^{46} + ( -38 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} + 7 \beta_{6} - 4 \beta_{7} - \beta_{8} - 2 \beta_{9} - 3 \beta_{10} - \beta_{11} - 4 \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} - 3 \beta_{16} + \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{47} + ( -21 + 39 \beta_{1} + 13 \beta_{2} + 6 \beta_{3} + \beta_{4} + 18 \beta_{5} - \beta_{6} + 2 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} + 9 \beta_{10} + 4 \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} - 3 \beta_{16} - 5 \beta_{17} - 6 \beta_{18} - \beta_{19} ) q^{48} + ( -4 + 34 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 15 \beta_{5} - 3 \beta_{6} + 7 \beta_{7} - \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - 5 \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} + 4 \beta_{17} + \beta_{18} - 3 \beta_{19} ) q^{49} -25 \beta_{1} q^{50} + ( -49 + 7 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + \beta_{4} - 14 \beta_{5} - \beta_{6} + 4 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 6 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - 2 \beta_{13} - 4 \beta_{14} + 4 \beta_{15} + \beta_{16} - \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{51} + ( 37 + 16 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 16 \beta_{5} + 2 \beta_{6} + 8 \beta_{7} + \beta_{8} + \beta_{9} - 6 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{17} + \beta_{18} + 4 \beta_{19} ) q^{52} + ( -23 + 26 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - \beta_{6} - 2 \beta_{7} - 3 \beta_{9} + \beta_{10} + 5 \beta_{12} - 4 \beta_{13} + 2 \beta_{14} - 4 \beta_{15} + 4 \beta_{16} - \beta_{19} ) q^{53} + ( -123 + 54 \beta_{1} - 7 \beta_{2} + 5 \beta_{3} + 3 \beta_{4} + 10 \beta_{5} + 5 \beta_{6} + 10 \beta_{7} + 8 \beta_{10} - \beta_{11} + 4 \beta_{12} - 7 \beta_{13} + 2 \beta_{14} + \beta_{15} - 2 \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{54} + 55 q^{55} + ( -139 - 12 \beta_{1} - 26 \beta_{2} - 7 \beta_{3} - 15 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} + 5 \beta_{8} - 5 \beta_{9} - 6 \beta_{10} - \beta_{12} - \beta_{13} - 2 \beta_{14} - \beta_{16} + 5 \beta_{17} + 6 \beta_{18} - \beta_{19} ) q^{56} + 19 \beta_{5} q^{57} + ( -84 + 19 \beta_{1} - 11 \beta_{2} - 17 \beta_{3} - 3 \beta_{4} - 15 \beta_{5} + 17 \beta_{6} + 5 \beta_{7} - 8 \beta_{8} - 5 \beta_{9} - 12 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - 7 \beta_{13} + 4 \beta_{14} - 5 \beta_{15} + 3 \beta_{16} + 2 \beta_{17} + 8 \beta_{18} - \beta_{19} ) q^{58} + ( -144 + \beta_{1} - 13 \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} + \beta_{10} - 2 \beta_{11} + 3 \beta_{12} + 2 \beta_{13} - \beta_{14} - 3 \beta_{15} + 3 \beta_{16} + 3 \beta_{17} + \beta_{19} ) q^{59} + ( 10 - 15 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} - 15 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} - 5 \beta_{10} - 5 \beta_{17} ) q^{60} + ( -57 + 10 \beta_{1} - 18 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} - 3 \beta_{8} - 6 \beta_{10} + 3 \beta_{11} - 3 \beta_{12} + \beta_{13} + 4 \beta_{14} - 5 \beta_{15} + \beta_{16} - \beta_{17} + \beta_{18} + 3 \beta_{19} ) q^{61} + ( 72 + 94 \beta_{1} + 24 \beta_{2} + 12 \beta_{3} + 4 \beta_{4} - 7 \beta_{5} + 9 \beta_{6} - \beta_{7} + 5 \beta_{8} + \beta_{9} + 8 \beta_{10} - \beta_{11} - 4 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} - 3 \beta_{16} - \beta_{17} - 5 \beta_{18} - 2 \beta_{19} ) q^{62} + ( -26 + 10 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} + \beta_{4} - 4 \beta_{5} + 13 \beta_{6} + 2 \beta_{7} + 5 \beta_{9} - \beta_{10} + 2 \beta_{11} - 3 \beta_{12} + \beta_{13} - 4 \beta_{16} - 5 \beta_{17} - 6 \beta_{18} ) q^{63} + ( 14 + 43 \beta_{1} + 9 \beta_{2} + 10 \beta_{3} + 2 \beta_{4} + 23 \beta_{5} - 14 \beta_{6} + 2 \beta_{7} - \beta_{8} + 11 \beta_{10} + \beta_{11} + 6 \beta_{12} + 4 \beta_{13} + 2 \beta_{14} + 4 \beta_{15} - \beta_{16} - 6 \beta_{18} ) q^{64} + ( -15 + 5 \beta_{1} + 5 \beta_{5} - 5 \beta_{18} ) q^{65} + ( 33 - 11 \beta_{1} - 11 \beta_{6} ) q^{66} + ( -58 + 45 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 6 \beta_{4} - 8 \beta_{5} + 16 \beta_{6} - 3 \beta_{7} + \beta_{8} - 3 \beta_{10} - \beta_{11} + 3 \beta_{12} + 2 \beta_{13} - \beta_{14} + 4 \beta_{15} - 2 \beta_{18} ) q^{67} + ( -238 + 33 \beta_{1} - 36 \beta_{2} - 6 \beta_{3} - 7 \beta_{4} - 19 \beta_{5} + 7 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} - 8 \beta_{10} + 6 \beta_{11} - 7 \beta_{12} - \beta_{13} - 3 \beta_{14} - \beta_{16} + \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{68} + ( -29 + 24 \beta_{1} + 9 \beta_{2} + 8 \beta_{3} + 8 \beta_{4} + 3 \beta_{5} + 13 \beta_{6} - \beta_{7} + 4 \beta_{8} - 2 \beta_{9} + 8 \beta_{10} - 4 \beta_{11} + 5 \beta_{12} + 2 \beta_{13} + \beta_{14} + 5 \beta_{15} - 3 \beta_{17} - 2 \beta_{18} ) q^{69} + ( 25 + 10 \beta_{1} + 10 \beta_{2} + 5 \beta_{7} + 5 \beta_{12} ) q^{70} + ( -1 + 24 \beta_{1} + 7 \beta_{2} - 15 \beta_{3} - 2 \beta_{4} - 20 \beta_{5} + 16 \beta_{6} - 2 \beta_{7} - 5 \beta_{8} - 2 \beta_{9} - 9 \beta_{10} + 4 \beta_{11} + \beta_{14} + 2 \beta_{15} - 5 \beta_{16} - 2 \beta_{17} - 6 \beta_{19} ) q^{71} + ( 12 - 44 \beta_{1} + 10 \beta_{2} + 9 \beta_{3} - 20 \beta_{5} - 30 \beta_{6} + 6 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} + 10 \beta_{13} - 3 \beta_{14} + 5 \beta_{15} + 5 \beta_{16} - 6 \beta_{17} - 9 \beta_{18} - 2 \beta_{19} ) q^{72} + ( 31 + 70 \beta_{1} + 27 \beta_{2} + 10 \beta_{3} - 3 \beta_{4} - 10 \beta_{5} + 6 \beta_{6} - 5 \beta_{7} + 9 \beta_{8} + \beta_{9} + 11 \beta_{10} + 3 \beta_{12} - 4 \beta_{14} + 5 \beta_{15} - 3 \beta_{16} - 5 \beta_{18} ) q^{73} + ( -187 - 77 \beta_{1} - 32 \beta_{2} - 11 \beta_{3} - 4 \beta_{4} - 26 \beta_{5} - 9 \beta_{6} + 12 \beta_{7} - 3 \beta_{8} - 9 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} + 6 \beta_{13} - 2 \beta_{14} + 4 \beta_{16} + \beta_{17} + 2 \beta_{19} ) q^{74} + 25 \beta_{5} q^{75} + ( 57 + 19 \beta_{2} ) q^{76} + ( -22 - 11 \beta_{7} ) q^{77} + ( -210 + 79 \beta_{1} - 10 \beta_{2} - 6 \beta_{3} - 32 \beta_{5} + 26 \beta_{6} + 8 \beta_{7} - 13 \beta_{8} - \beta_{9} - 12 \beta_{10} - \beta_{11} + \beta_{12} - 4 \beta_{13} + \beta_{14} - 3 \beta_{15} + \beta_{16} - 7 \beta_{17} + 4 \beta_{18} + 2 \beta_{19} ) q^{78} + ( 40 + 48 \beta_{1} + 24 \beta_{2} + 12 \beta_{3} + 3 \beta_{5} + 7 \beta_{6} - 6 \beta_{7} + 12 \beta_{8} + 8 \beta_{9} + 8 \beta_{10} - 4 \beta_{11} - 6 \beta_{12} + 5 \beta_{13} - 12 \beta_{14} + 3 \beta_{15} - 9 \beta_{16} - \beta_{17} + \beta_{18} + 4 \beta_{19} ) q^{79} + ( -75 - 15 \beta_{2} - 5 \beta_{3} + 10 \beta_{6} - 5 \beta_{9} - 5 \beta_{10} - 5 \beta_{12} + 5 \beta_{17} + 5 \beta_{18} ) q^{80} + ( -87 - 46 \beta_{1} + 11 \beta_{2} - 11 \beta_{3} - 6 \beta_{4} + 13 \beta_{5} - 18 \beta_{6} + 3 \beta_{7} + 5 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + 6 \beta_{13} + 3 \beta_{14} - 5 \beta_{15} + \beta_{16} - 9 \beta_{17} - 3 \beta_{18} + \beta_{19} ) q^{81} + ( -100 - 40 \beta_{1} - 21 \beta_{2} - 6 \beta_{3} - \beta_{4} - 57 \beta_{5} + 10 \beta_{6} + 7 \beta_{7} - 2 \beta_{8} - 10 \beta_{9} - 20 \beta_{10} + 4 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} + 4 \beta_{14} - 9 \beta_{15} + 5 \beta_{16} + 3 \beta_{17} + 9 \beta_{18} - 2 \beta_{19} ) q^{82} + ( -217 - 5 \beta_{1} - 16 \beta_{2} - \beta_{4} - 14 \beta_{5} - 8 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 3 \beta_{12} + 13 \beta_{13} + 5 \beta_{14} - 6 \beta_{15} + 6 \beta_{16} + 5 \beta_{17} - 4 \beta_{18} + 3 \beta_{19} ) q^{83} + ( -108 + 29 \beta_{1} + 15 \beta_{2} + 12 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - 17 \beta_{6} + 7 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 9 \beta_{10} + 2 \beta_{11} - 3 \beta_{12} + 9 \beta_{13} + \beta_{15} - \beta_{16} - \beta_{17} - 13 \beta_{18} - 3 \beta_{19} ) q^{84} + ( 45 - 20 \beta_{1} + 10 \beta_{2} + 10 \beta_{5} + 5 \beta_{7} + 5 \beta_{9} + 5 \beta_{11} ) q^{85} + ( -43 + 95 \beta_{1} + 15 \beta_{2} - 13 \beta_{3} + 42 \beta_{5} + 6 \beta_{6} - \beta_{7} - 6 \beta_{8} - 2 \beta_{9} + 9 \beta_{10} - \beta_{11} + 8 \beta_{12} - 15 \beta_{13} + 5 \beta_{14} - 7 \beta_{15} + \beta_{17} - 7 \beta_{19} ) q^{86} + ( -47 + 37 \beta_{1} + 28 \beta_{2} - \beta_{4} - 67 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} - 7 \beta_{8} + 2 \beta_{9} - 6 \beta_{10} - 4 \beta_{12} - 6 \beta_{13} + \beta_{14} - 5 \beta_{15} - 4 \beta_{17} + 6 \beta_{18} + 5 \beta_{19} ) q^{87} + ( -11 + 33 \beta_{1} + 11 \beta_{3} ) q^{88} + ( -250 + 32 \beta_{1} - 26 \beta_{2} - \beta_{3} - 4 \beta_{4} - 58 \beta_{5} + 22 \beta_{6} + 6 \beta_{7} - 12 \beta_{8} - 10 \beta_{9} - 14 \beta_{10} - 4 \beta_{11} - 8 \beta_{12} - 3 \beta_{13} + 2 \beta_{14} - 7 \beta_{15} - 2 \beta_{16} + 5 \beta_{17} + 10 \beta_{18} + 3 \beta_{19} ) q^{89} + ( 45 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} + 25 \beta_{5} + 15 \beta_{6} - 5 \beta_{11} - 5 \beta_{13} + 5 \beta_{17} + 5 \beta_{18} ) q^{90} + ( -5 - 19 \beta_{1} + 21 \beta_{2} - 10 \beta_{3} - \beta_{4} - 2 \beta_{5} + 11 \beta_{6} + 6 \beta_{7} - 5 \beta_{8} - 9 \beta_{9} + 5 \beta_{10} - \beta_{11} + 3 \beta_{12} - 11 \beta_{13} + 5 \beta_{14} - 5 \beta_{15} + 8 \beta_{16} - 4 \beta_{17} + 5 \beta_{18} - 5 \beta_{19} ) q^{91} + ( -156 + 44 \beta_{1} + 4 \beta_{2} + 20 \beta_{3} - 4 \beta_{4} - 18 \beta_{5} - 32 \beta_{6} + 9 \beta_{7} - \beta_{8} + 2 \beta_{9} - 3 \beta_{10} + 8 \beta_{11} + 7 \beta_{12} + 11 \beta_{13} + 7 \beta_{14} + 3 \beta_{15} + 5 \beta_{16} - 8 \beta_{17} - 14 \beta_{18} - \beta_{19} ) q^{92} + ( 43 + 34 \beta_{1} + 34 \beta_{2} - 11 \beta_{3} - \beta_{4} - 63 \beta_{5} + \beta_{6} - 6 \beta_{7} + 6 \beta_{8} - 8 \beta_{9} - 6 \beta_{10} - 5 \beta_{11} + 7 \beta_{12} + 11 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} + 7 \beta_{16} + 3 \beta_{17} + 4 \beta_{18} - 6 \beta_{19} ) q^{93} + ( 65 + 20 \beta_{1} + 20 \beta_{2} + 21 \beta_{3} + 4 \beta_{4} + 18 \beta_{5} - 12 \beta_{6} + 12 \beta_{7} + 7 \beta_{8} + 2 \beta_{9} + 20 \beta_{10} + 3 \beta_{12} - 7 \beta_{13} - 3 \beta_{14} + 5 \beta_{15} - 5 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} + 6 \beta_{19} ) q^{94} -95 q^{95} + ( -277 - 53 \beta_{1} - 12 \beta_{2} - 21 \beta_{3} - 3 \beta_{4} + 41 \beta_{5} - 9 \beta_{6} + \beta_{7} + 12 \beta_{8} + 9 \beta_{9} + 5 \beta_{10} - 5 \beta_{12} + 3 \beta_{13} - 5 \beta_{14} - 3 \beta_{16} + 7 \beta_{17} + 9 \beta_{18} - 3 \beta_{19} ) q^{96} + ( 101 - 11 \beta_{1} + 45 \beta_{2} + 9 \beta_{3} - 4 \beta_{4} - 10 \beta_{5} + 3 \beta_{6} - 20 \beta_{7} + 6 \beta_{9} + 5 \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - 9 \beta_{14} + 3 \beta_{15} - 7 \beta_{16} + \beta_{17} + 7 \beta_{18} + 6 \beta_{19} ) q^{97} + ( -396 + 30 \beta_{1} - 23 \beta_{2} + \beta_{3} + 5 \beta_{4} - 7 \beta_{5} + 8 \beta_{6} - 15 \beta_{7} - \beta_{8} + 6 \beta_{9} - 3 \beta_{10} - 11 \beta_{11} - \beta_{13} + 3 \beta_{14} + 2 \beta_{15} - 5 \beta_{17} + 2 \beta_{18} + 8 \beta_{19} ) q^{98} + ( -77 + 11 \beta_{5} + 11 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + O(q^{10}) \) \( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + 1317797 x^{12} - 1160501 x^{11} - 9914845 x^{10} + 7570653 x^{9} + 42786958 x^{8} - 23777633 x^{7} - 102801526 x^{6} + 28436356 x^{5} + 122325928 x^{4} + 411232 x^{3} - 47350496 x^{2} - 4782848 x + 150528\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 11 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 19 \nu + 1 \)
\(\beta_{4}\)\(=\)\((\)\(\)\(17\!\cdots\!19\)\( \nu^{19} - \)\(27\!\cdots\!52\)\( \nu^{18} - \)\(16\!\cdots\!19\)\( \nu^{17} + \)\(28\!\cdots\!34\)\( \nu^{16} + \)\(60\!\cdots\!10\)\( \nu^{15} - \)\(11\!\cdots\!89\)\( \nu^{14} - \)\(96\!\cdots\!97\)\( \nu^{13} + \)\(25\!\cdots\!15\)\( \nu^{12} + \)\(42\!\cdots\!38\)\( \nu^{11} - \)\(30\!\cdots\!13\)\( \nu^{10} + \)\(58\!\cdots\!88\)\( \nu^{9} + \)\(20\!\cdots\!35\)\( \nu^{8} - \)\(72\!\cdots\!27\)\( \nu^{7} - \)\(67\!\cdots\!42\)\( \nu^{6} + \)\(24\!\cdots\!48\)\( \nu^{5} + \)\(10\!\cdots\!68\)\( \nu^{4} - \)\(29\!\cdots\!88\)\( \nu^{3} - \)\(48\!\cdots\!24\)\( \nu^{2} + \)\(16\!\cdots\!08\)\( \nu - \)\(27\!\cdots\!72\)\(\)\()/ \)\(54\!\cdots\!76\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(33\!\cdots\!79\)\( \nu^{19} + \)\(67\!\cdots\!15\)\( \nu^{18} + \)\(33\!\cdots\!76\)\( \nu^{17} - \)\(68\!\cdots\!06\)\( \nu^{16} - \)\(13\!\cdots\!92\)\( \nu^{15} + \)\(28\!\cdots\!57\)\( \nu^{14} + \)\(29\!\cdots\!12\)\( \nu^{13} - \)\(62\!\cdots\!67\)\( \nu^{12} - \)\(34\!\cdots\!87\)\( \nu^{11} + \)\(74\!\cdots\!60\)\( \nu^{10} + \)\(21\!\cdots\!20\)\( \nu^{9} - \)\(48\!\cdots\!15\)\( \nu^{8} - \)\(69\!\cdots\!90\)\( \nu^{7} + \)\(15\!\cdots\!24\)\( \nu^{6} + \)\(97\!\cdots\!04\)\( \nu^{5} - \)\(23\!\cdots\!32\)\( \nu^{4} - \)\(32\!\cdots\!20\)\( \nu^{3} + \)\(12\!\cdots\!96\)\( \nu^{2} - \)\(23\!\cdots\!28\)\( \nu - \)\(18\!\cdots\!84\)\(\)\()/ \)\(37\!\cdots\!56\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(48\!\cdots\!48\)\( \nu^{19} + \)\(13\!\cdots\!17\)\( \nu^{18} + \)\(49\!\cdots\!67\)\( \nu^{17} - \)\(14\!\cdots\!44\)\( \nu^{16} - \)\(20\!\cdots\!86\)\( \nu^{15} + \)\(59\!\cdots\!52\)\( \nu^{14} + \)\(43\!\cdots\!57\)\( \nu^{13} - \)\(12\!\cdots\!68\)\( \nu^{12} - \)\(51\!\cdots\!83\)\( \nu^{11} + \)\(15\!\cdots\!05\)\( \nu^{10} + \)\(33\!\cdots\!04\)\( \nu^{9} - \)\(10\!\cdots\!56\)\( \nu^{8} - \)\(11\!\cdots\!31\)\( \nu^{7} + \)\(34\!\cdots\!50\)\( \nu^{6} + \)\(20\!\cdots\!44\)\( \nu^{5} - \)\(53\!\cdots\!56\)\( \nu^{4} - \)\(18\!\cdots\!32\)\( \nu^{3} + \)\(25\!\cdots\!16\)\( \nu^{2} + \)\(24\!\cdots\!60\)\( \nu + \)\(89\!\cdots\!08\)\(\)\()/ \)\(53\!\cdots\!08\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(18\!\cdots\!01\)\( \nu^{19} - \)\(41\!\cdots\!56\)\( \nu^{18} - \)\(19\!\cdots\!05\)\( \nu^{17} + \)\(42\!\cdots\!06\)\( \nu^{16} + \)\(81\!\cdots\!14\)\( \nu^{15} - \)\(17\!\cdots\!03\)\( \nu^{14} - \)\(17\!\cdots\!43\)\( \nu^{13} + \)\(38\!\cdots\!57\)\( \nu^{12} + \)\(22\!\cdots\!30\)\( \nu^{11} - \)\(47\!\cdots\!75\)\( \nu^{10} - \)\(15\!\cdots\!92\)\( \nu^{9} + \)\(31\!\cdots\!53\)\( \nu^{8} + \)\(60\!\cdots\!47\)\( \nu^{7} - \)\(10\!\cdots\!42\)\( \nu^{6} - \)\(12\!\cdots\!64\)\( \nu^{5} + \)\(15\!\cdots\!68\)\( \nu^{4} + \)\(13\!\cdots\!68\)\( \nu^{3} - \)\(68\!\cdots\!12\)\( \nu^{2} - \)\(38\!\cdots\!44\)\( \nu - \)\(22\!\cdots\!32\)\(\)\()/ \)\(10\!\cdots\!52\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(19\!\cdots\!41\)\( \nu^{19} + \)\(58\!\cdots\!16\)\( \nu^{18} + \)\(19\!\cdots\!37\)\( \nu^{17} - \)\(60\!\cdots\!02\)\( \nu^{16} - \)\(77\!\cdots\!70\)\( \nu^{15} + \)\(25\!\cdots\!79\)\( \nu^{14} + \)\(16\!\cdots\!39\)\( \nu^{13} - \)\(54\!\cdots\!69\)\( \nu^{12} - \)\(17\!\cdots\!26\)\( \nu^{11} + \)\(65\!\cdots\!75\)\( \nu^{10} + \)\(10\!\cdots\!68\)\( \nu^{9} - \)\(43\!\cdots\!61\)\( \nu^{8} - \)\(24\!\cdots\!07\)\( \nu^{7} + \)\(14\!\cdots\!22\)\( \nu^{6} + \)\(11\!\cdots\!84\)\( \nu^{5} - \)\(22\!\cdots\!36\)\( \nu^{4} + \)\(35\!\cdots\!24\)\( \nu^{3} + \)\(11\!\cdots\!64\)\( \nu^{2} - \)\(37\!\cdots\!68\)\( \nu + \)\(10\!\cdots\!48\)\(\)\()/ \)\(10\!\cdots\!52\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(19\!\cdots\!57\)\( \nu^{19} - \)\(13\!\cdots\!36\)\( \nu^{18} - \)\(20\!\cdots\!69\)\( \nu^{17} + \)\(13\!\cdots\!46\)\( \nu^{16} + \)\(84\!\cdots\!54\)\( \nu^{15} - \)\(58\!\cdots\!91\)\( \nu^{14} - \)\(18\!\cdots\!31\)\( \nu^{13} + \)\(13\!\cdots\!53\)\( \nu^{12} + \)\(21\!\cdots\!78\)\( \nu^{11} - \)\(16\!\cdots\!23\)\( \nu^{10} - \)\(14\!\cdots\!60\)\( \nu^{9} + \)\(10\!\cdots\!25\)\( \nu^{8} + \)\(46\!\cdots\!27\)\( \nu^{7} - \)\(34\!\cdots\!50\)\( \nu^{6} - \)\(62\!\cdots\!72\)\( \nu^{5} + \)\(53\!\cdots\!88\)\( \nu^{4} + \)\(67\!\cdots\!68\)\( \nu^{3} - \)\(44\!\cdots\!52\)\( \nu^{2} + \)\(14\!\cdots\!52\)\( \nu + \)\(80\!\cdots\!48\)\(\)\()/ \)\(10\!\cdots\!52\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(22\!\cdots\!35\)\( \nu^{19} + \)\(77\!\cdots\!72\)\( \nu^{18} + \)\(23\!\cdots\!43\)\( \nu^{17} - \)\(79\!\cdots\!38\)\( \nu^{16} - \)\(97\!\cdots\!26\)\( \nu^{15} + \)\(32\!\cdots\!53\)\( \nu^{14} + \)\(21\!\cdots\!17\)\( \nu^{13} - \)\(71\!\cdots\!07\)\( \nu^{12} - \)\(25\!\cdots\!38\)\( \nu^{11} + \)\(86\!\cdots\!13\)\( \nu^{10} + \)\(17\!\cdots\!32\)\( \nu^{9} - \)\(56\!\cdots\!35\)\( \nu^{8} - \)\(65\!\cdots\!29\)\( \nu^{7} + \)\(19\!\cdots\!10\)\( \nu^{6} + \)\(14\!\cdots\!56\)\( \nu^{5} - \)\(29\!\cdots\!68\)\( \nu^{4} - \)\(16\!\cdots\!88\)\( \nu^{3} + \)\(13\!\cdots\!20\)\( \nu^{2} + \)\(51\!\cdots\!12\)\( \nu - \)\(14\!\cdots\!28\)\(\)\()/ \)\(97\!\cdots\!56\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(27\!\cdots\!89\)\( \nu^{19} - \)\(36\!\cdots\!12\)\( \nu^{18} - \)\(28\!\cdots\!25\)\( \nu^{17} + \)\(38\!\cdots\!78\)\( \nu^{16} + \)\(11\!\cdots\!58\)\( \nu^{15} - \)\(17\!\cdots\!99\)\( \nu^{14} - \)\(24\!\cdots\!43\)\( \nu^{13} + \)\(41\!\cdots\!13\)\( \nu^{12} + \)\(28\!\cdots\!94\)\( \nu^{11} - \)\(48\!\cdots\!27\)\( \nu^{10} - \)\(17\!\cdots\!60\)\( \nu^{9} + \)\(20\!\cdots\!53\)\( \nu^{8} + \)\(50\!\cdots\!19\)\( \nu^{7} + \)\(20\!\cdots\!62\)\( \nu^{6} - \)\(45\!\cdots\!60\)\( \nu^{5} - \)\(13\!\cdots\!40\)\( \nu^{4} - \)\(42\!\cdots\!08\)\( \nu^{3} - \)\(17\!\cdots\!08\)\( \nu^{2} + \)\(38\!\cdots\!80\)\( \nu + \)\(62\!\cdots\!84\)\(\)\()/ \)\(10\!\cdots\!52\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(51\!\cdots\!07\)\( \nu^{19} + \)\(81\!\cdots\!07\)\( \nu^{18} + \)\(53\!\cdots\!26\)\( \nu^{17} - \)\(84\!\cdots\!24\)\( \nu^{16} - \)\(22\!\cdots\!84\)\( \nu^{15} + \)\(35\!\cdots\!13\)\( \nu^{14} + \)\(48\!\cdots\!76\)\( \nu^{13} - \)\(77\!\cdots\!53\)\( \nu^{12} - \)\(59\!\cdots\!63\)\( \nu^{11} + \)\(94\!\cdots\!66\)\( \nu^{10} + \)\(41\!\cdots\!74\)\( \nu^{9} - \)\(62\!\cdots\!83\)\( \nu^{8} - \)\(15\!\cdots\!82\)\( \nu^{7} + \)\(21\!\cdots\!38\)\( \nu^{6} + \)\(29\!\cdots\!72\)\( \nu^{5} - \)\(31\!\cdots\!16\)\( \nu^{4} - \)\(25\!\cdots\!64\)\( \nu^{3} + \)\(14\!\cdots\!20\)\( \nu^{2} + \)\(56\!\cdots\!20\)\( \nu + \)\(48\!\cdots\!36\)\(\)\()/ \)\(13\!\cdots\!44\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(15\!\cdots\!86\)\( \nu^{19} + \)\(48\!\cdots\!03\)\( \nu^{18} + \)\(15\!\cdots\!11\)\( \nu^{17} - \)\(49\!\cdots\!68\)\( \nu^{16} - \)\(65\!\cdots\!78\)\( \nu^{15} + \)\(20\!\cdots\!98\)\( \nu^{14} + \)\(14\!\cdots\!25\)\( \nu^{13} - \)\(44\!\cdots\!26\)\( \nu^{12} - \)\(16\!\cdots\!65\)\( \nu^{11} + \)\(53\!\cdots\!93\)\( \nu^{10} + \)\(10\!\cdots\!44\)\( \nu^{9} - \)\(35\!\cdots\!30\)\( \nu^{8} - \)\(36\!\cdots\!39\)\( \nu^{7} + \)\(11\!\cdots\!46\)\( \nu^{6} + \)\(65\!\cdots\!80\)\( \nu^{5} - \)\(18\!\cdots\!16\)\( \nu^{4} - \)\(56\!\cdots\!88\)\( \nu^{3} + \)\(93\!\cdots\!64\)\( \nu^{2} + \)\(73\!\cdots\!68\)\( \nu - \)\(20\!\cdots\!96\)\(\)\()/ \)\(27\!\cdots\!88\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(69\!\cdots\!63\)\( \nu^{19} - \)\(32\!\cdots\!16\)\( \nu^{18} - \)\(72\!\cdots\!63\)\( \nu^{17} + \)\(33\!\cdots\!98\)\( \nu^{16} + \)\(30\!\cdots\!34\)\( \nu^{15} - \)\(14\!\cdots\!45\)\( \nu^{14} - \)\(67\!\cdots\!93\)\( \nu^{13} + \)\(33\!\cdots\!83\)\( \nu^{12} + \)\(83\!\cdots\!18\)\( \nu^{11} - \)\(42\!\cdots\!21\)\( \nu^{10} - \)\(58\!\cdots\!04\)\( \nu^{9} + \)\(28\!\cdots\!71\)\( \nu^{8} + \)\(22\!\cdots\!57\)\( \nu^{7} - \)\(88\!\cdots\!26\)\( \nu^{6} - \)\(44\!\cdots\!12\)\( \nu^{5} + \)\(11\!\cdots\!96\)\( \nu^{4} + \)\(37\!\cdots\!76\)\( \nu^{3} - \)\(54\!\cdots\!40\)\( \nu^{2} - \)\(10\!\cdots\!24\)\( \nu - \)\(87\!\cdots\!92\)\(\)\()/ \)\(10\!\cdots\!52\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(89\!\cdots\!99\)\( \nu^{19} + \)\(12\!\cdots\!64\)\( \nu^{18} + \)\(91\!\cdots\!19\)\( \nu^{17} - \)\(13\!\cdots\!58\)\( \nu^{16} - \)\(38\!\cdots\!54\)\( \nu^{15} + \)\(54\!\cdots\!33\)\( \nu^{14} + \)\(82\!\cdots\!29\)\( \nu^{13} - \)\(11\!\cdots\!23\)\( \nu^{12} - \)\(99\!\cdots\!62\)\( \nu^{11} + \)\(14\!\cdots\!81\)\( \nu^{10} + \)\(66\!\cdots\!12\)\( \nu^{9} - \)\(92\!\cdots\!43\)\( \nu^{8} - \)\(23\!\cdots\!73\)\( \nu^{7} + \)\(29\!\cdots\!38\)\( \nu^{6} + \)\(41\!\cdots\!08\)\( \nu^{5} - \)\(41\!\cdots\!20\)\( \nu^{4} - \)\(31\!\cdots\!92\)\( \nu^{3} + \)\(18\!\cdots\!56\)\( \nu^{2} + \)\(41\!\cdots\!88\)\( \nu - \)\(25\!\cdots\!64\)\(\)\()/ \)\(10\!\cdots\!52\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(11\!\cdots\!93\)\( \nu^{19} + \)\(16\!\cdots\!88\)\( \nu^{18} + \)\(11\!\cdots\!99\)\( \nu^{17} - \)\(17\!\cdots\!68\)\( \nu^{16} - \)\(48\!\cdots\!90\)\( \nu^{15} + \)\(71\!\cdots\!19\)\( \nu^{14} + \)\(10\!\cdots\!47\)\( \nu^{13} - \)\(15\!\cdots\!07\)\( \nu^{12} - \)\(12\!\cdots\!62\)\( \nu^{11} + \)\(18\!\cdots\!89\)\( \nu^{10} + \)\(86\!\cdots\!50\)\( \nu^{9} - \)\(11\!\cdots\!65\)\( \nu^{8} - \)\(31\!\cdots\!99\)\( \nu^{7} + \)\(37\!\cdots\!64\)\( \nu^{6} + \)\(60\!\cdots\!72\)\( \nu^{5} - \)\(51\!\cdots\!68\)\( \nu^{4} - \)\(55\!\cdots\!40\)\( \nu^{3} + \)\(20\!\cdots\!08\)\( \nu^{2} + \)\(15\!\cdots\!36\)\( \nu + \)\(10\!\cdots\!72\)\(\)\()/ \)\(13\!\cdots\!44\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(12\!\cdots\!71\)\( \nu^{19} - \)\(28\!\cdots\!00\)\( \nu^{18} - \)\(12\!\cdots\!75\)\( \nu^{17} + \)\(29\!\cdots\!22\)\( \nu^{16} + \)\(51\!\cdots\!78\)\( \nu^{15} - \)\(12\!\cdots\!05\)\( \nu^{14} - \)\(11\!\cdots\!05\)\( \nu^{13} + \)\(26\!\cdots\!51\)\( \nu^{12} + \)\(13\!\cdots\!10\)\( \nu^{11} - \)\(32\!\cdots\!57\)\( \nu^{10} - \)\(85\!\cdots\!72\)\( \nu^{9} + \)\(21\!\cdots\!07\)\( \nu^{8} + \)\(28\!\cdots\!77\)\( \nu^{7} - \)\(70\!\cdots\!74\)\( \nu^{6} - \)\(47\!\cdots\!36\)\( \nu^{5} + \)\(10\!\cdots\!16\)\( \nu^{4} + \)\(33\!\cdots\!56\)\( \nu^{3} - \)\(53\!\cdots\!88\)\( \nu^{2} - \)\(78\!\cdots\!60\)\( \nu + \)\(43\!\cdots\!48\)\(\)\()/ \)\(10\!\cdots\!52\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(37\!\cdots\!79\)\( \nu^{19} + \)\(92\!\cdots\!12\)\( \nu^{18} + \)\(38\!\cdots\!31\)\( \nu^{17} - \)\(95\!\cdots\!98\)\( \nu^{16} - \)\(15\!\cdots\!30\)\( \nu^{15} + \)\(39\!\cdots\!17\)\( \nu^{14} + \)\(34\!\cdots\!45\)\( \nu^{13} - \)\(86\!\cdots\!83\)\( \nu^{12} - \)\(41\!\cdots\!82\)\( \nu^{11} + \)\(10\!\cdots\!17\)\( \nu^{10} + \)\(27\!\cdots\!24\)\( \nu^{9} - \)\(68\!\cdots\!47\)\( \nu^{8} - \)\(97\!\cdots\!49\)\( \nu^{7} + \)\(22\!\cdots\!62\)\( \nu^{6} + \)\(18\!\cdots\!80\)\( \nu^{5} - \)\(34\!\cdots\!84\)\( \nu^{4} - \)\(15\!\cdots\!32\)\( \nu^{3} + \)\(16\!\cdots\!08\)\( \nu^{2} + \)\(28\!\cdots\!52\)\( \nu - \)\(81\!\cdots\!76\)\(\)\()/ \)\(27\!\cdots\!88\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(51\!\cdots\!03\)\( \nu^{19} - \)\(97\!\cdots\!92\)\( \nu^{18} - \)\(52\!\cdots\!27\)\( \nu^{17} + \)\(10\!\cdots\!74\)\( \nu^{16} + \)\(21\!\cdots\!62\)\( \nu^{15} - \)\(41\!\cdots\!17\)\( \nu^{14} - \)\(46\!\cdots\!45\)\( \nu^{13} + \)\(91\!\cdots\!19\)\( \nu^{12} + \)\(55\!\cdots\!46\)\( \nu^{11} - \)\(10\!\cdots\!33\)\( \nu^{10} - \)\(36\!\cdots\!96\)\( \nu^{9} + \)\(71\!\cdots\!75\)\( \nu^{8} + \)\(12\!\cdots\!13\)\( \nu^{7} - \)\(23\!\cdots\!70\)\( \nu^{6} - \)\(22\!\cdots\!04\)\( \nu^{5} + \)\(35\!\cdots\!96\)\( \nu^{4} + \)\(16\!\cdots\!16\)\( \nu^{3} - \)\(17\!\cdots\!36\)\( \nu^{2} - \)\(16\!\cdots\!68\)\( \nu + \)\(13\!\cdots\!12\)\(\)\()/ \)\(27\!\cdots\!88\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 11\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 19 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-\beta_{18} - \beta_{17} + \beta_{12} + \beta_{10} + \beta_{9} - 2 \beta_{6} + \beta_{3} + 27 \beta_{2} + 215\)
\(\nu^{5}\)\(=\)\(3 \beta_{19} - 3 \beta_{18} - 3 \beta_{17} + 3 \beta_{15} - \beta_{14} + 2 \beta_{13} + \beta_{12} + 6 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + \beta_{7} - 5 \beta_{6} + 12 \beta_{5} + 2 \beta_{4} + 38 \beta_{3} + 7 \beta_{2} + 422 \beta_{1} - 12\)
\(\nu^{6}\)\(=\)\(-46 \beta_{18} - 40 \beta_{17} - \beta_{16} + 4 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 46 \beta_{12} + \beta_{11} + 51 \beta_{10} + 40 \beta_{9} - \beta_{8} + 2 \beta_{7} - 94 \beta_{6} + 23 \beta_{5} + 2 \beta_{4} + 50 \beta_{3} + 705 \beta_{2} + 43 \beta_{1} + 4902\)
\(\nu^{7}\)\(=\)\(130 \beta_{19} - 159 \beta_{18} - 149 \beta_{17} - 6 \beta_{16} + 147 \beta_{15} - 46 \beta_{14} + 93 \beta_{13} + 59 \beta_{12} + 13 \beta_{11} + 313 \beta_{10} + 104 \beta_{9} + 95 \beta_{8} + 72 \beta_{7} - 301 \beta_{6} + 562 \beta_{5} + 95 \beta_{4} + 1206 \beta_{3} + 383 \beta_{2} + 10091 \beta_{1} + 579\)
\(\nu^{8}\)\(=\)\(11 \beta_{19} - 1661 \beta_{18} - 1292 \beta_{17} - 54 \beta_{16} + 236 \beta_{15} + 84 \beta_{14} + 223 \beta_{13} + 1607 \beta_{12} + 87 \beta_{11} + 1982 \beta_{10} + 1308 \beta_{9} - 22 \beta_{8} + 95 \beta_{7} - 3408 \beta_{6} + 1660 \beta_{5} + 104 \beta_{4} + 2000 \beta_{3} + 18733 \beta_{2} + 2815 \beta_{1} + 120340\)
\(\nu^{9}\)\(=\)\(4271 \beta_{19} - 6297 \beta_{18} - 5567 \beta_{17} - 340 \beta_{16} + 5316 \beta_{15} - 1575 \beta_{14} + 3305 \beta_{13} + 2549 \beta_{12} + 795 \beta_{11} + 12086 \beta_{10} + 4017 \beta_{9} + 3441 \beta_{8} + 2969 \beta_{7} - 12944 \beta_{6} + 20022 \beta_{5} + 3359 \beta_{4} + 36429 \beta_{3} + 16044 \beta_{2} + 253158 \beta_{1} + 45091\)
\(\nu^{10}\)\(=\)\(846 \beta_{19} - 55036 \beta_{18} - 39243 \beta_{17} - 2185 \beta_{16} + 9967 \beta_{15} + 2533 \beta_{14} + 8997 \beta_{13} + 50974 \beta_{12} + 4270 \beta_{11} + 69417 \beta_{10} + 40328 \beta_{9} + 335 \beta_{8} + 3010 \beta_{7} - 113033 \beta_{6} + 77689 \beta_{5} + 4192 \beta_{4} + 73125 \beta_{3} + 507722 \beta_{2} + 130915 \beta_{1} + 3094240\)
\(\nu^{11}\)\(=\)\(128035 \beta_{19} - 223257 \beta_{18} - 187549 \beta_{17} - 13615 \beta_{16} + 172773 \beta_{15} - 48817 \beta_{14} + 108006 \beta_{13} + 96255 \beta_{12} + 33885 \beta_{11} + 416453 \beta_{10} + 139410 \beta_{9} + 112773 \beta_{8} + 100193 \beta_{7} - 483313 \beta_{6} + 654251 \beta_{5} + 107032 \beta_{4} + 1081059 \beta_{3} + 605069 \beta_{2} + 6576971 \beta_{1} + 2118774\)
\(\nu^{12}\)\(=\)\(44522 \beta_{19} - 1750196 \beta_{18} - 1168530 \beta_{17} - 79459 \beta_{16} + 370735 \beta_{15} + 66442 \beta_{14} + 320437 \beta_{13} + 1550768 \beta_{12} + 168694 \beta_{11} + 2306043 \beta_{10} + 1217241 \beta_{9} + 51056 \beta_{8} + 82758 \beta_{7} - 3601153 \beta_{6} + 3033677 \beta_{5} + 155343 \beta_{4} + 2536300 \beta_{3} + 13995466 \beta_{2} + 5289799 \beta_{1} + 82177470\)
\(\nu^{13}\)\(=\)\(3699520 \beta_{19} - 7480110 \beta_{18} - 6013437 \beta_{17} - 476114 \beta_{16} + 5353570 \beta_{15} - 1450529 \beta_{14} + 3410026 \beta_{13} + 3383330 \beta_{12} + 1248054 \beta_{11} + 13571362 \beta_{10} + 4595583 \beta_{9} + 3518967 \beta_{8} + 3085952 \beta_{7} - 16732248 \beta_{6} + 20648724 \beta_{5} + 3254967 \beta_{4} + 31868126 \beta_{3} + 21518924 \beta_{2} + 175471987 \beta_{1} + 84526989\)
\(\nu^{14}\)\(=\)\(1979265 \beta_{19} - 54415322 \beta_{18} - 34629199 \beta_{17} - 2731437 \beta_{16} + 12932089 \beta_{15} + 1583656 \beta_{14} + 10731240 \beta_{13} + 46273786 \beta_{12} + 5994451 \beta_{11} + 74268574 \beta_{10} + 36449766 \beta_{9} + 2873370 \beta_{8} + 2216123 \beta_{7} - 112339735 \beta_{6} + 107698235 \beta_{5} + 5509703 \beta_{4} + 84955450 \beta_{3} + 391175908 \beta_{2} + 197618710 \beta_{1} + 2234952776\)
\(\nu^{15}\)\(=\)\(105330882 \beta_{19} - 242415793 \beta_{18} - 187736926 \beta_{17} - 15577910 \beta_{16} + 162009101 \beta_{15} - 42255623 \beta_{14} + 105779221 \beta_{13} + 114007801 \beta_{12} + 42618649 \beta_{11} + 428907363 \beta_{10} + 147202031 \beta_{9} + 106865206 \beta_{8} + 90830278 \beta_{7} - 553637757 \beta_{6} + 641288574 \beta_{5} + 96745170 \beta_{4} + 937523228 \beta_{3} + 736670315 \beta_{2} + 4779653805 \beta_{1} + 3105704128\)
\(\nu^{16}\)\(=\)\(79748010 \beta_{19} - 1669451069 \beta_{18} - 1027307523 \beta_{17} - 90564337 \beta_{16} + 434268389 \beta_{15} + 34046648 \beta_{14} + 347412829 \beta_{13} + 1368553291 \beta_{12} + 200901098 \beta_{11} + 2345031162 \beta_{10} + 1088563318 \beta_{9} + 125942194 \beta_{8} + 61859116 \beta_{7} - 3462360151 \beta_{6} + 3614575653 \beta_{5} + 189587399 \beta_{4} + 2777539760 \beta_{3} + 11058415360 \beta_{2} + 7017447453 \beta_{1} + 61887295479\)
\(\nu^{17}\)\(=\)\(2985292839 \beta_{19} - 7692052264 \beta_{18} - 5772398627 \beta_{17} - 491796238 \beta_{16} + 4843493623 \beta_{15} - 1219174744 \beta_{14} + 3245700414 \beta_{13} + 3740267536 \beta_{12} + 1392392528 \beta_{11} + 13314754333 \beta_{10} + 4633178476 \beta_{9} + 3194635139 \beta_{8} + 2613283213 \beta_{7} - 17796810975 \beta_{6} + 19750725668 \beta_{5} + 2843341885 \beta_{4} + 27584139284 \beta_{3} + 24552647349 \beta_{2} + 132357450898 \beta_{1} + 108549113772\)
\(\nu^{18}\)\(=\)\(3010521868 \beta_{19} - 50805181552 \beta_{18} - 30570060429 \beta_{17} - 2927121960 \beta_{16} + 14224935444 \beta_{15} + 618279097 \beta_{14} + 11027591910 \beta_{13} + 40328797096 \beta_{12} + 6503247994 \beta_{11} + 73064881348 \beta_{10} + 32494166277 \beta_{9} + 4895561507 \beta_{8} + 1853595974 \beta_{7} - 105928178786 \beta_{6} + 117087940064 \beta_{5} + 6371745715 \beta_{4} + 89244718500 \beta_{3} + 315567136391 \beta_{2} + 240494948454 \beta_{1} + 1737872315738\)
\(\nu^{19}\)\(=\)\(84647898327 \beta_{19} - 240671770704 \beta_{18} - 175897812579 \beta_{17} - 15221997041 \beta_{16} + 143922218163 \beta_{15} - 35017992312 \beta_{14} + 98835719252 \beta_{13} + 120512667360 \beta_{12} + 44247618407 \beta_{11} + 408904689734 \beta_{10} + 144185574772 \beta_{9} + 94634149732 \beta_{8} + 74391093409 \beta_{7} - 561170924129 \beta_{6} + 605386684927 \beta_{5} + 83154238433 \beta_{4} + 812532498323 \beta_{3} + 802373727677 \beta_{2} + 3714437856979 \beta_{1} + 3668728755334\)

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
5.49185
4.63319
4.18815
3.91893
2.81607
2.35645
2.10393
2.01989
0.783853
0.0251982
−0.130697
−1.09490
−1.13531
−1.56126
−1.78835
−3.55600
−3.59432
−4.38362
−4.96274
−5.13031
−5.49185 3.98529 22.1605 −5.00000 −21.8866 23.4740 −77.7672 −11.1175 27.4593
1.2 −4.63319 −7.91113 13.4664 −5.00000 36.6537 11.8333 −25.3269 35.5860 23.1659
1.3 −4.18815 8.40510 9.54062 −5.00000 −35.2018 −15.6625 −6.45234 43.6456 20.9408
1.4 −3.91893 −3.32066 7.35803 −5.00000 13.0134 −28.1182 2.51584 −15.9732 19.5947
1.5 −2.81607 3.33662 −0.0697562 −5.00000 −9.39614 35.3821 22.7250 −15.8670 14.0803
1.6 −2.35645 −6.45217 −2.44713 −5.00000 15.2042 27.1536 24.6182 14.6305 11.7823
1.7 −2.10393 −1.35130 −3.57349 −5.00000 2.84303 −13.9401 24.3498 −25.1740 10.5196
1.8 −2.01989 7.09682 −3.92003 −5.00000 −14.3348 −8.46588 24.0772 23.3649 10.0995
1.9 −0.783853 0.0603595 −7.38557 −5.00000 −0.0473130 −1.09978 12.0600 −26.9964 3.91926
1.10 −0.0251982 −6.38249 −7.99937 −5.00000 0.160827 −27.7864 0.403155 13.7361 0.125991
1.11 0.130697 8.99140 −7.98292 −5.00000 1.17515 14.2539 −2.08892 53.8454 −0.653486
1.12 1.09490 3.72657 −6.80118 −5.00000 4.08024 3.04995 −16.2059 −13.1127 −5.47452
1.13 1.13531 −7.07177 −6.71106 −5.00000 −8.02868 20.7485 −16.7017 23.0099 −5.67657
1.14 1.56126 −1.47971 −5.56246 −5.00000 −2.31022 22.4493 −21.1746 −24.8104 −7.80632
1.15 1.78835 −7.45188 −4.80180 −5.00000 −13.3266 −20.9424 −22.8941 28.5305 −8.94175
1.16 3.55600 6.62353 4.64511 −5.00000 23.5532 −2.03165 −11.9300 16.8711 −17.7800
1.17 3.59432 4.02265 4.91912 −5.00000 14.4587 10.5448 −11.0737 −10.8183 −17.9716
1.18 4.38362 −4.80730 11.2161 −5.00000 −21.0733 18.1824 14.0981 −3.88988 −21.9181
1.19 4.96274 −9.59476 16.6288 −5.00000 −47.6163 −3.47238 42.8227 65.0593 −24.8137
1.20 5.13031 1.57482 18.3201 −5.00000 8.07930 −16.5525 52.9453 −24.5200 −25.6516
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(1\)
\(19\) \(-1\)

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 1045.4.a.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.4.a.c 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1045))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 150528 + 4782848 T - 47350496 T^{2} - 411232 T^{3} + 122325928 T^{4} - 28436356 T^{5} - 102801526 T^{6} + 23777633 T^{7} + 42786958 T^{8} - 7570653 T^{9} - 9914845 T^{10} + 1160501 T^{11} + 1317797 T^{12} - 95592 T^{13} - 101844 T^{14} + 4345 T^{15} + 4500 T^{16} - 103 T^{17} - 105 T^{18} + T^{19} + T^{20} \)
$3$ \( 353918232320 - 5680081523264 T - 3347090800224 T^{2} + 4981234806264 T^{3} + 2186993166324 T^{4} - 1556721423766 T^{5} - 458826991589 T^{6} + 226203325202 T^{7} + 48148768465 T^{8} - 17546651366 T^{9} - 2985194551 T^{10} + 780417038 T^{11} + 115594664 T^{12} - 20368658 T^{13} - 2784185 T^{14} + 306356 T^{15} + 40006 T^{16} - 2446 T^{17} - 311 T^{18} + 8 T^{19} + T^{20} \)
$5$ \( ( 5 + T )^{20} \)
$7$ \( \)\(40\!\cdots\!00\)\( + \)\(51\!\cdots\!24\)\( T + \)\(68\!\cdots\!00\)\( T^{2} - \)\(84\!\cdots\!32\)\( T^{3} - \)\(14\!\cdots\!80\)\( T^{4} + 34380181686418242504 T^{5} + 3799704075495518456 T^{6} - 630848237624266849 T^{7} - 37091233561933415 T^{8} + 5932103950132976 T^{9} + 163776804213452 T^{10} - 31019508355629 T^{11} - 265383501723 T^{12} + 93106689223 T^{13} - 333682289 T^{14} - 157745180 T^{15} + 1820196 T^{16} + 138798 T^{17} - 2336 T^{18} - 49 T^{19} + T^{20} \)
$11$ \( ( 11 + T )^{20} \)
$13$ \( \)\(30\!\cdots\!92\)\( - \)\(35\!\cdots\!60\)\( T - \)\(74\!\cdots\!92\)\( T^{2} + \)\(38\!\cdots\!20\)\( T^{3} + \)\(59\!\cdots\!96\)\( T^{4} + \)\(29\!\cdots\!04\)\( T^{5} - \)\(21\!\cdots\!97\)\( T^{6} - \)\(24\!\cdots\!08\)\( T^{7} + \)\(37\!\cdots\!12\)\( T^{8} + 50370927825104373604 T^{9} - 3617239721761390826 T^{10} - 47841248510126736 T^{11} + 2120772112189691 T^{12} + 24375754318106 T^{13} - 763260295368 T^{14} - 6868131904 T^{15} + 165001888 T^{16} + 1007666 T^{17} - 19689 T^{18} - 60 T^{19} + T^{20} \)
$17$ \( -\)\(11\!\cdots\!56\)\( - \)\(70\!\cdots\!52\)\( T - \)\(43\!\cdots\!76\)\( T^{2} + \)\(19\!\cdots\!20\)\( T^{3} + \)\(35\!\cdots\!28\)\( T^{4} + \)\(23\!\cdots\!84\)\( T^{5} + \)\(59\!\cdots\!52\)\( T^{6} - \)\(76\!\cdots\!95\)\( T^{7} - \)\(66\!\cdots\!55\)\( T^{8} - \)\(65\!\cdots\!17\)\( T^{9} + \)\(21\!\cdots\!49\)\( T^{10} + 4292748483516459781 T^{11} - 25090591437229807 T^{12} - 1018170693817862 T^{13} - 878851574870 T^{14} + 119101933922 T^{15} + 458715080 T^{16} - 6866284 T^{17} - 37646 T^{18} + 155 T^{19} + T^{20} \)
$19$ \( ( -19 + T )^{20} \)
$23$ \( -\)\(79\!\cdots\!36\)\( - \)\(21\!\cdots\!92\)\( T + \)\(18\!\cdots\!32\)\( T^{2} + \)\(51\!\cdots\!16\)\( T^{3} - \)\(10\!\cdots\!24\)\( T^{4} - \)\(35\!\cdots\!16\)\( T^{5} + \)\(22\!\cdots\!91\)\( T^{6} + \)\(11\!\cdots\!92\)\( T^{7} - \)\(11\!\cdots\!66\)\( T^{8} - \)\(21\!\cdots\!50\)\( T^{9} - \)\(41\!\cdots\!42\)\( T^{10} + \)\(24\!\cdots\!26\)\( T^{11} + 901994866106385512 T^{12} - 17524813687293420 T^{13} - 82984003188875 T^{14} + 731292883768 T^{15} + 4032188455 T^{16} - 16504574 T^{17} - 100272 T^{18} + 154 T^{19} + T^{20} \)
$29$ \( -\)\(23\!\cdots\!04\)\( + \)\(17\!\cdots\!40\)\( T - \)\(59\!\cdots\!92\)\( T^{2} - \)\(52\!\cdots\!40\)\( T^{3} + \)\(63\!\cdots\!72\)\( T^{4} + \)\(67\!\cdots\!88\)\( T^{5} + \)\(71\!\cdots\!02\)\( T^{6} - \)\(15\!\cdots\!45\)\( T^{7} - \)\(32\!\cdots\!69\)\( T^{8} - \)\(16\!\cdots\!23\)\( T^{9} + \)\(38\!\cdots\!03\)\( T^{10} + \)\(23\!\cdots\!21\)\( T^{11} - 14236007951392198087 T^{12} - 163547875866376954 T^{13} + 47647259150696 T^{14} + 4676032227598 T^{15} + 7823143432 T^{16} - 61243026 T^{17} - 163378 T^{18} + 305 T^{19} + T^{20} \)
$31$ \( -\)\(22\!\cdots\!16\)\( - \)\(21\!\cdots\!88\)\( T - \)\(30\!\cdots\!56\)\( T^{2} - \)\(43\!\cdots\!00\)\( T^{3} + \)\(67\!\cdots\!52\)\( T^{4} + \)\(22\!\cdots\!12\)\( T^{5} - \)\(17\!\cdots\!88\)\( T^{6} - \)\(15\!\cdots\!12\)\( T^{7} - \)\(13\!\cdots\!26\)\( T^{8} + \)\(15\!\cdots\!41\)\( T^{9} + \)\(22\!\cdots\!43\)\( T^{10} - \)\(28\!\cdots\!50\)\( T^{11} - \)\(12\!\cdots\!32\)\( T^{12} - 239407522451168767 T^{13} + 3041737153480207 T^{14} + 11282144957771 T^{15} - 24953735237 T^{16} - 162266427 T^{17} - 44159 T^{18} + 759 T^{19} + T^{20} \)
$37$ \( \)\(15\!\cdots\!52\)\( - \)\(23\!\cdots\!76\)\( T + \)\(34\!\cdots\!44\)\( T^{2} + \)\(77\!\cdots\!60\)\( T^{3} - \)\(16\!\cdots\!52\)\( T^{4} - \)\(50\!\cdots\!72\)\( T^{5} + \)\(21\!\cdots\!68\)\( T^{6} - \)\(24\!\cdots\!48\)\( T^{7} - \)\(13\!\cdots\!60\)\( T^{8} + \)\(42\!\cdots\!02\)\( T^{9} + \)\(45\!\cdots\!91\)\( T^{10} - \)\(19\!\cdots\!26\)\( T^{11} - \)\(76\!\cdots\!19\)\( T^{12} + 4553149741474022836 T^{13} + 5174225246063838 T^{14} - 54483998468012 T^{15} + 12642672774 T^{16} + 316637386 T^{17} - 312449 T^{18} - 698 T^{19} + T^{20} \)
$41$ \( \)\(26\!\cdots\!80\)\( - \)\(22\!\cdots\!16\)\( T + \)\(63\!\cdots\!92\)\( T^{2} - \)\(42\!\cdots\!80\)\( T^{3} - \)\(86\!\cdots\!96\)\( T^{4} + \)\(11\!\cdots\!12\)\( T^{5} + \)\(37\!\cdots\!52\)\( T^{6} - \)\(91\!\cdots\!28\)\( T^{7} - \)\(30\!\cdots\!58\)\( T^{8} + \)\(35\!\cdots\!99\)\( T^{9} - \)\(25\!\cdots\!13\)\( T^{10} - \)\(77\!\cdots\!31\)\( T^{11} + \)\(93\!\cdots\!97\)\( T^{12} + 10099638478816682665 T^{13} - 14861270048132789 T^{14} - 76997085769720 T^{15} + 126230571864 T^{16} + 317760051 T^{17} - 556361 T^{18} - 547 T^{19} + T^{20} \)
$43$ \( \)\(57\!\cdots\!88\)\( + \)\(11\!\cdots\!16\)\( T + \)\(81\!\cdots\!60\)\( T^{2} + \)\(26\!\cdots\!60\)\( T^{3} + \)\(42\!\cdots\!36\)\( T^{4} + \)\(27\!\cdots\!96\)\( T^{5} - \)\(11\!\cdots\!36\)\( T^{6} - \)\(26\!\cdots\!82\)\( T^{7} - \)\(96\!\cdots\!58\)\( T^{8} + \)\(51\!\cdots\!07\)\( T^{9} + \)\(42\!\cdots\!63\)\( T^{10} + \)\(18\!\cdots\!75\)\( T^{11} - \)\(61\!\cdots\!19\)\( T^{12} - 13627551057269983988 T^{13} + 35083514954320342 T^{14} + 141754863232579 T^{15} - 33251386025 T^{16} - 600377497 T^{17} - 380677 T^{18} + 925 T^{19} + T^{20} \)
$47$ \( -\)\(30\!\cdots\!20\)\( - \)\(26\!\cdots\!68\)\( T - \)\(96\!\cdots\!92\)\( T^{2} + \)\(13\!\cdots\!92\)\( T^{3} + \)\(56\!\cdots\!36\)\( T^{4} - \)\(23\!\cdots\!72\)\( T^{5} - \)\(66\!\cdots\!28\)\( T^{6} + \)\(18\!\cdots\!84\)\( T^{7} + \)\(31\!\cdots\!80\)\( T^{8} - \)\(70\!\cdots\!35\)\( T^{9} - \)\(91\!\cdots\!99\)\( T^{10} + \)\(14\!\cdots\!96\)\( T^{11} + \)\(17\!\cdots\!06\)\( T^{12} - 18117831175987435065 T^{13} - 21240588169374837 T^{14} + 125100222288906 T^{15} + 156842300844 T^{16} - 456486238 T^{17} - 620460 T^{18} + 681 T^{19} + T^{20} \)
$53$ \( -\)\(58\!\cdots\!96\)\( + \)\(17\!\cdots\!12\)\( T + \)\(10\!\cdots\!16\)\( T^{2} - \)\(18\!\cdots\!84\)\( T^{3} - \)\(13\!\cdots\!60\)\( T^{4} - \)\(14\!\cdots\!32\)\( T^{5} + \)\(43\!\cdots\!94\)\( T^{6} + \)\(34\!\cdots\!47\)\( T^{7} - \)\(19\!\cdots\!25\)\( T^{8} - \)\(21\!\cdots\!61\)\( T^{9} + \)\(91\!\cdots\!91\)\( T^{10} + \)\(45\!\cdots\!07\)\( T^{11} + \)\(44\!\cdots\!05\)\( T^{12} - 43148647921780194638 T^{13} - 67904985654517610 T^{14} + 203751209554650 T^{15} + 391562661130 T^{16} - 469681692 T^{17} - 1019638 T^{18} + 419 T^{19} + T^{20} \)
$59$ \( \)\(39\!\cdots\!56\)\( + \)\(75\!\cdots\!28\)\( T - \)\(38\!\cdots\!28\)\( T^{2} - \)\(11\!\cdots\!96\)\( T^{3} - \)\(13\!\cdots\!28\)\( T^{4} + \)\(27\!\cdots\!40\)\( T^{5} + \)\(40\!\cdots\!14\)\( T^{6} + \)\(25\!\cdots\!25\)\( T^{7} + \)\(44\!\cdots\!77\)\( T^{8} - \)\(31\!\cdots\!17\)\( T^{9} - \)\(19\!\cdots\!15\)\( T^{10} - \)\(25\!\cdots\!81\)\( T^{11} + \)\(10\!\cdots\!07\)\( T^{12} + \)\(42\!\cdots\!00\)\( T^{13} + 285596159963315880 T^{14} - 1151658814763994 T^{15} - 2556527912286 T^{16} - 1058840870 T^{17} + 2199310 T^{18} + 2829 T^{19} + T^{20} \)
$61$ \( \)\(88\!\cdots\!88\)\( - \)\(97\!\cdots\!28\)\( T + \)\(12\!\cdots\!64\)\( T^{2} + \)\(57\!\cdots\!76\)\( T^{3} - \)\(10\!\cdots\!80\)\( T^{4} - \)\(46\!\cdots\!80\)\( T^{5} + \)\(11\!\cdots\!00\)\( T^{6} + \)\(43\!\cdots\!52\)\( T^{7} - \)\(30\!\cdots\!60\)\( T^{8} - \)\(14\!\cdots\!61\)\( T^{9} + \)\(35\!\cdots\!15\)\( T^{10} + \)\(20\!\cdots\!88\)\( T^{11} - \)\(15\!\cdots\!00\)\( T^{12} - \)\(15\!\cdots\!95\)\( T^{13} - 18873823433332637 T^{14} + 616837259325677 T^{15} + 363733900475 T^{16} - 1227804825 T^{17} - 1065903 T^{18} + 959 T^{19} + T^{20} \)
$67$ \( -\)\(57\!\cdots\!64\)\( - \)\(44\!\cdots\!20\)\( T - \)\(85\!\cdots\!28\)\( T^{2} + \)\(44\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!92\)\( T^{4} + \)\(19\!\cdots\!70\)\( T^{5} - \)\(18\!\cdots\!67\)\( T^{6} - \)\(81\!\cdots\!68\)\( T^{7} - \)\(26\!\cdots\!72\)\( T^{8} + \)\(64\!\cdots\!44\)\( T^{9} + \)\(52\!\cdots\!91\)\( T^{10} + \)\(57\!\cdots\!80\)\( T^{11} - \)\(23\!\cdots\!05\)\( T^{12} - \)\(58\!\cdots\!64\)\( T^{13} + 212052125319292648 T^{14} + 1745310721963490 T^{15} + 731487420062 T^{16} - 2212693660 T^{17} - 1705210 T^{18} + 1020 T^{19} + T^{20} \)
$71$ \( -\)\(52\!\cdots\!64\)\( + \)\(59\!\cdots\!44\)\( T + \)\(26\!\cdots\!36\)\( T^{2} - \)\(20\!\cdots\!60\)\( T^{3} + \)\(29\!\cdots\!00\)\( T^{4} + \)\(26\!\cdots\!04\)\( T^{5} - \)\(55\!\cdots\!84\)\( T^{6} - \)\(18\!\cdots\!48\)\( T^{7} + \)\(44\!\cdots\!10\)\( T^{8} + \)\(71\!\cdots\!76\)\( T^{9} - \)\(19\!\cdots\!43\)\( T^{10} - \)\(16\!\cdots\!08\)\( T^{11} + \)\(52\!\cdots\!20\)\( T^{12} + \)\(23\!\cdots\!90\)\( T^{13} - 8508362807136585621 T^{14} - 1930765755003804 T^{15} + 8307193812790 T^{16} + 775578218 T^{17} - 4443156 T^{18} - 106 T^{19} + T^{20} \)
$73$ \( -\)\(20\!\cdots\!12\)\( + \)\(13\!\cdots\!12\)\( T + \)\(32\!\cdots\!32\)\( T^{2} - \)\(62\!\cdots\!32\)\( T^{3} - \)\(13\!\cdots\!84\)\( T^{4} - \)\(41\!\cdots\!18\)\( T^{5} + \)\(29\!\cdots\!81\)\( T^{6} + \)\(17\!\cdots\!50\)\( T^{7} - \)\(46\!\cdots\!60\)\( T^{8} - \)\(18\!\cdots\!10\)\( T^{9} - \)\(23\!\cdots\!23\)\( T^{10} + \)\(71\!\cdots\!78\)\( T^{11} + \)\(15\!\cdots\!49\)\( T^{12} - \)\(10\!\cdots\!54\)\( T^{13} - 4158998607266899802 T^{14} + 84908779477088 T^{15} + 5381699243322 T^{16} + 931958912 T^{17} - 3558072 T^{18} - 558 T^{19} + T^{20} \)
$79$ \( \)\(37\!\cdots\!72\)\( + \)\(84\!\cdots\!12\)\( T - \)\(84\!\cdots\!20\)\( T^{2} + \)\(26\!\cdots\!84\)\( T^{3} - \)\(32\!\cdots\!72\)\( T^{4} + \)\(91\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!72\)\( T^{6} - \)\(77\!\cdots\!16\)\( T^{7} - \)\(10\!\cdots\!46\)\( T^{8} + \)\(80\!\cdots\!64\)\( T^{9} - \)\(65\!\cdots\!07\)\( T^{10} - \)\(33\!\cdots\!36\)\( T^{11} + \)\(37\!\cdots\!13\)\( T^{12} + \)\(66\!\cdots\!40\)\( T^{13} - 8578551456381772678 T^{14} - 6689572670696524 T^{15} + 9558910943148 T^{16} + 3137027616 T^{17} - 5030707 T^{18} - 536 T^{19} + T^{20} \)
$83$ \( -\)\(79\!\cdots\!76\)\( - \)\(88\!\cdots\!72\)\( T + \)\(30\!\cdots\!28\)\( T^{2} + \)\(76\!\cdots\!84\)\( T^{3} + \)\(26\!\cdots\!24\)\( T^{4} - \)\(82\!\cdots\!16\)\( T^{5} - \)\(65\!\cdots\!44\)\( T^{6} - \)\(57\!\cdots\!42\)\( T^{7} + \)\(46\!\cdots\!42\)\( T^{8} + \)\(11\!\cdots\!79\)\( T^{9} - \)\(71\!\cdots\!79\)\( T^{10} - \)\(53\!\cdots\!60\)\( T^{11} - \)\(35\!\cdots\!68\)\( T^{12} + \)\(91\!\cdots\!89\)\( T^{13} + 13972629396592921329 T^{14} - 2069922812524859 T^{15} - 15472611931687 T^{16} - 7798408089 T^{17} + 3582069 T^{18} + 4179 T^{19} + T^{20} \)
$89$ \( -\)\(28\!\cdots\!80\)\( + \)\(42\!\cdots\!04\)\( T + \)\(52\!\cdots\!72\)\( T^{2} + \)\(10\!\cdots\!84\)\( T^{3} - \)\(13\!\cdots\!08\)\( T^{4} - \)\(86\!\cdots\!68\)\( T^{5} - \)\(16\!\cdots\!92\)\( T^{6} + \)\(19\!\cdots\!88\)\( T^{7} + \)\(13\!\cdots\!50\)\( T^{8} + \)\(18\!\cdots\!08\)\( T^{9} - \)\(88\!\cdots\!05\)\( T^{10} - \)\(48\!\cdots\!96\)\( T^{11} - \)\(35\!\cdots\!92\)\( T^{12} + \)\(29\!\cdots\!68\)\( T^{13} + 55410021092706367038 T^{14} + 12454410640585344 T^{15} - 25359832529023 T^{16} - 16880642912 T^{17} + 1430432 T^{18} + 4120 T^{19} + T^{20} \)
$97$ \( -\)\(18\!\cdots\!40\)\( + \)\(58\!\cdots\!84\)\( T - \)\(21\!\cdots\!64\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(59\!\cdots\!48\)\( T^{4} + \)\(17\!\cdots\!32\)\( T^{5} - \)\(52\!\cdots\!24\)\( T^{6} - \)\(99\!\cdots\!68\)\( T^{7} + \)\(22\!\cdots\!78\)\( T^{8} + \)\(31\!\cdots\!34\)\( T^{9} - \)\(51\!\cdots\!37\)\( T^{10} - \)\(56\!\cdots\!36\)\( T^{11} + \)\(70\!\cdots\!88\)\( T^{12} + \)\(59\!\cdots\!88\)\( T^{13} - 60041751111401558854 T^{14} - 35480732782145018 T^{15} + 30594086099817 T^{16} + 11145649200 T^{17} - 8562788 T^{18} - 1414 T^{19} + T^{20} \)
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