Properties

Label 1045.4.a.b
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: \(x^{20} - 8 x^{19} - 82 x^{18} + 700 x^{17} + 2826 x^{16} - 25467 x^{15} - 53768 x^{14} + 498499 x^{13} + 623613 x^{12} - 5673747 x^{11} - 4539454 x^{10} + 37893109 x^{9} + 19879768 x^{8} - 143049638 x^{7} - 45064360 x^{6} + 280461480 x^{5} + 43097920 x^{4} - 240447168 x^{3} - 36068096 x^{2} + 74703872 x + 17756160\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 7 \)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + ( -1 + \beta_{5} ) q^{3} + ( 4 - \beta_{1} + \beta_{2} ) q^{4} + 5 q^{5} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{6} + ( -7 - \beta_{7} ) q^{7} + ( -7 + 4 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{8} + ( 8 + \beta_{1} - \beta_{5} + \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + ( -1 + \beta_{5} ) q^{3} + ( 4 - \beta_{1} + \beta_{2} ) q^{4} + 5 q^{5} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{6} + ( -7 - \beta_{7} ) q^{7} + ( -7 + 4 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{8} + ( 8 + \beta_{1} - \beta_{5} + \beta_{9} ) q^{9} + ( -5 + 5 \beta_{1} ) q^{10} + 11 q^{11} + ( -10 - 2 \beta_{1} - 2 \beta_{2} + 5 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{10} - \beta_{13} - \beta_{14} - \beta_{17} - \beta_{19} ) q^{12} + ( -11 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{17} ) q^{13} + ( 4 - 9 \beta_{1} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} ) q^{14} + ( -5 + 5 \beta_{5} ) q^{15} + ( 23 - 6 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{12} + \beta_{16} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{16} + ( -23 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{19} ) q^{17} + ( 6 + 7 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} + 3 \beta_{13} + 2 \beta_{14} + 2 \beta_{17} + \beta_{18} + \beta_{19} ) q^{18} + 19 q^{19} + ( 20 - 5 \beta_{1} + 5 \beta_{2} ) q^{20} + ( 1 - 8 \beta_{1} - 2 \beta_{3} + \beta_{4} - 12 \beta_{5} + 4 \beta_{7} + \beta_{8} - 3 \beta_{9} + \beta_{11} + 2 \beta_{14} - 2 \beta_{16} - \beta_{17} - \beta_{18} ) q^{21} + ( -11 + 11 \beta_{1} ) q^{22} + ( -30 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{15} - \beta_{16} + \beta_{18} - \beta_{19} ) q^{23} + ( -18 - 17 \beta_{1} - 5 \beta_{2} + \beta_{3} - 3 \beta_{4} - 7 \beta_{5} + 2 \beta_{7} - \beta_{9} + 2 \beta_{10} - \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} - \beta_{16} + 3 \beta_{17} + 2 \beta_{18} ) q^{24} + 25 q^{25} + ( 1 - 21 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + 7 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{26} + ( -5 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} + 7 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} + \beta_{18} ) q^{27} + ( -45 + 8 \beta_{1} - 11 \beta_{2} + \beta_{3} - \beta_{4} + 7 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + 2 \beta_{9} + 2 \beta_{11} + \beta_{12} - \beta_{13} - 3 \beta_{14} + \beta_{15} + \beta_{16} - 2 \beta_{19} ) q^{28} + ( 8 - 4 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} + \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} - \beta_{16} + 2 \beta_{17} + \beta_{19} ) q^{29} + ( -10 - 10 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} - 5 \beta_{5} ) q^{30} + ( -5 - 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{6} - 5 \beta_{7} - 3 \beta_{8} + \beta_{9} - \beta_{10} + 3 \beta_{12} - 3 \beta_{13} - 4 \beta_{14} + \beta_{15} + 5 \beta_{16} - 4 \beta_{17} - \beta_{19} ) q^{31} + ( -23 + 21 \beta_{1} - 7 \beta_{2} - \beta_{3} + 3 \beta_{4} - 11 \beta_{5} + 2 \beta_{6} + 7 \beta_{7} - \beta_{9} + \beta_{10} + 3 \beta_{11} - 3 \beta_{12} + 4 \beta_{13} + 2 \beta_{14} - \beta_{15} - 2 \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{32} + ( -11 + 11 \beta_{5} ) q^{33} + ( 41 - 22 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - 2 \beta_{9} - 3 \beta_{10} - \beta_{12} - 2 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} - \beta_{16} - 3 \beta_{18} + \beta_{19} ) q^{34} + ( -35 - 5 \beta_{7} ) q^{35} + ( 48 + 11 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{12} - 2 \beta_{13} + \beta_{14} - 3 \beta_{15} - \beta_{16} - 3 \beta_{17} - 3 \beta_{18} + 2 \beta_{19} ) q^{36} + ( -8 - 6 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - \beta_{6} - 6 \beta_{7} + 5 \beta_{9} + 2 \beta_{10} - \beta_{11} + 5 \beta_{12} - 4 \beta_{13} - 5 \beta_{14} - \beta_{15} + 3 \beta_{16} - \beta_{17} - \beta_{19} ) q^{37} + ( -19 + 19 \beta_{1} ) q^{38} + ( -2 + 13 \beta_{1} + 4 \beta_{4} - 26 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + \beta_{9} + 2 \beta_{10} - 3 \beta_{11} + 4 \beta_{12} + 2 \beta_{13} - 3 \beta_{14} + \beta_{15} + 3 \beta_{16} + \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{39} + ( -35 + 20 \beta_{1} - 5 \beta_{2} + 5 \beta_{4} - 5 \beta_{5} ) q^{40} + ( -54 - 18 \beta_{1} - 6 \beta_{2} - 5 \beta_{3} - \beta_{4} - 5 \beta_{5} + \beta_{6} + 3 \beta_{7} - 2 \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} - 2 \beta_{16} + \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{41} + ( -43 + 26 \beta_{1} + 10 \beta_{2} - 11 \beta_{3} + 5 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} - 5 \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + 3 \beta_{12} - 4 \beta_{13} - 6 \beta_{14} + \beta_{15} + 4 \beta_{16} - 2 \beta_{18} + 2 \beta_{19} ) q^{42} + ( -85 + 7 \beta_{1} - 7 \beta_{2} + \beta_{3} + 2 \beta_{4} - 6 \beta_{5} - 3 \beta_{6} - 7 \beta_{7} - 2 \beta_{10} - 4 \beta_{11} + \beta_{13} - 4 \beta_{14} + \beta_{15} + 2 \beta_{16} - \beta_{17} - 2 \beta_{18} ) q^{43} + ( 44 - 11 \beta_{1} + 11 \beta_{2} ) q^{44} + ( 40 + 5 \beta_{1} - 5 \beta_{5} + 5 \beta_{9} ) q^{45} + ( -3 - 31 \beta_{1} - 5 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} + 10 \beta_{5} + 3 \beta_{6} - 18 \beta_{7} + 2 \beta_{8} + 6 \beta_{9} + \beta_{10} - 2 \beta_{11} + 3 \beta_{12} - 4 \beta_{13} - 5 \beta_{14} + 3 \beta_{15} + 3 \beta_{16} + \beta_{18} - 2 \beta_{19} ) q^{46} + ( -102 + 4 \beta_{1} - 10 \beta_{2} + \beta_{3} + \beta_{4} + 8 \beta_{5} - 4 \beta_{6} + 6 \beta_{7} + 3 \beta_{8} - 7 \beta_{9} - 4 \beta_{10} - 4 \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{16} - 4 \beta_{17} + \beta_{18} - 4 \beta_{19} ) q^{47} + ( -70 - 26 \beta_{1} - 13 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 24 \beta_{5} - 10 \beta_{6} - 26 \beta_{7} - 16 \beta_{8} + 9 \beta_{9} - 8 \beta_{10} - 3 \beta_{11} - \beta_{12} - 13 \beta_{13} - 6 \beta_{14} + 2 \beta_{16} - 2 \beta_{17} - \beta_{18} - 4 \beta_{19} ) q^{48} + ( 49 - 12 \beta_{1} - 11 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 12 \beta_{5} + 4 \beta_{6} + 12 \beta_{7} + 4 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} + 2 \beta_{12} + 7 \beta_{13} + 6 \beta_{14} - 2 \beta_{15} + 2 \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{49} + ( -25 + 25 \beta_{1} ) q^{50} + ( -15 - 8 \beta_{1} - 11 \beta_{2} + 10 \beta_{3} + \beta_{4} - 46 \beta_{5} - 3 \beta_{6} - 7 \beta_{7} - 3 \beta_{8} + 2 \beta_{10} + 2 \beta_{12} + 5 \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{51} + ( -168 - 4 \beta_{1} - 43 \beta_{2} + 9 \beta_{3} - 4 \beta_{4} - 15 \beta_{5} + \beta_{8} - 4 \beta_{9} + \beta_{10} + 5 \beta_{11} - 5 \beta_{12} + 5 \beta_{13} + 6 \beta_{14} - 6 \beta_{16} + 6 \beta_{17} + 2 \beta_{18} - 3 \beta_{19} ) q^{52} + ( -35 - 11 \beta_{1} - 9 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 17 \beta_{5} - \beta_{6} + 10 \beta_{7} - 2 \beta_{8} + 6 \beta_{9} + 7 \beta_{10} + 2 \beta_{11} - 3 \beta_{12} + 7 \beta_{13} + 8 \beta_{14} - 2 \beta_{15} - 8 \beta_{16} + 2 \beta_{17} + 7 \beta_{18} + \beta_{19} ) q^{53} + ( -40 - 2 \beta_{1} - 36 \beta_{2} + 16 \beta_{3} + 7 \beta_{4} - 10 \beta_{5} - 6 \beta_{6} - 4 \beta_{7} - 9 \beta_{8} + 9 \beta_{9} - 3 \beta_{11} - 7 \beta_{12} - 4 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} - 3 \beta_{16} + \beta_{17} + 3 \beta_{18} - 2 \beta_{19} ) q^{54} + 55 q^{55} + ( 75 - 72 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} - 6 \beta_{6} + 8 \beta_{7} - 5 \beta_{9} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} + 9 \beta_{14} - 3 \beta_{15} - 5 \beta_{16} - 2 \beta_{17} + 3 \beta_{18} ) q^{56} + ( -19 + 19 \beta_{5} ) q^{57} + ( -23 + 10 \beta_{1} - 7 \beta_{2} - 6 \beta_{3} - 5 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - 4 \beta_{10} + 8 \beta_{11} + 3 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} + \beta_{16} - \beta_{17} + 2 \beta_{18} ) q^{58} + ( -16 - 41 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} + 8 \beta_{4} - 23 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 8 \beta_{8} - 4 \beta_{10} + \beta_{11} + \beta_{12} - 4 \beta_{13} - 5 \beta_{14} + 2 \beta_{16} + \beta_{17} + 5 \beta_{18} - \beta_{19} ) q^{59} + ( -50 - 10 \beta_{1} - 10 \beta_{2} + 25 \beta_{5} - 5 \beta_{6} - 10 \beta_{7} - 5 \beta_{8} - 5 \beta_{10} - 5 \beta_{13} - 5 \beta_{14} - 5 \beta_{17} - 5 \beta_{19} ) q^{60} + ( 15 - 15 \beta_{2} + 6 \beta_{3} + 5 \beta_{4} - 17 \beta_{5} + 7 \beta_{6} + 4 \beta_{7} + 4 \beta_{8} - 5 \beta_{9} - 4 \beta_{10} + 9 \beta_{11} + 3 \beta_{12} + \beta_{13} + 6 \beta_{14} - \beta_{15} - 5 \beta_{16} - 7 \beta_{17} - 4 \beta_{18} - 4 \beta_{19} ) q^{61} + ( -40 - 5 \beta_{1} - 25 \beta_{2} + 6 \beta_{3} + 7 \beta_{4} - 37 \beta_{5} + 4 \beta_{6} - \beta_{7} - 4 \beta_{8} - 4 \beta_{9} - \beta_{10} + 5 \beta_{11} - 5 \beta_{12} + 4 \beta_{13} + 9 \beta_{14} + \beta_{15} - 3 \beta_{16} + 2 \beta_{17} - \beta_{18} - 3 \beta_{19} ) q^{62} + ( -204 - 12 \beta_{1} + \beta_{3} + 6 \beta_{4} - 7 \beta_{5} - 13 \beta_{6} - 18 \beta_{7} - 15 \beta_{8} - 6 \beta_{9} - \beta_{10} - 8 \beta_{11} - 2 \beta_{12} - 10 \beta_{13} - 4 \beta_{14} - 5 \beta_{17} - \beta_{18} - 5 \beta_{19} ) q^{63} + ( 134 - 5 \beta_{1} + 46 \beta_{2} - 7 \beta_{3} - 10 \beta_{4} + 4 \beta_{5} - \beta_{6} - 6 \beta_{7} + 9 \beta_{8} + 10 \beta_{9} + 9 \beta_{10} - 8 \beta_{11} + 2 \beta_{12} - \beta_{14} + \beta_{15} + 5 \beta_{16} + 3 \beta_{17} + \beta_{18} + 6 \beta_{19} ) q^{64} + ( -55 - 5 \beta_{1} - 5 \beta_{2} - 5 \beta_{5} + 5 \beta_{17} ) q^{65} + ( -22 - 22 \beta_{1} - 11 \beta_{2} + 11 \beta_{3} - 11 \beta_{5} ) q^{66} + ( -80 - 13 \beta_{1} + 11 \beta_{2} + \beta_{3} - 2 \beta_{4} - 15 \beta_{5} + 8 \beta_{6} + 4 \beta_{7} + 13 \beta_{8} - 5 \beta_{9} + \beta_{10} - 10 \beta_{11} + 3 \beta_{12} + 5 \beta_{13} + 3 \beta_{14} + \beta_{15} + \beta_{16} + 5 \beta_{17} + 2 \beta_{18} + 10 \beta_{19} ) q^{67} + ( -98 + 5 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 13 \beta_{4} - 27 \beta_{5} + 11 \beta_{6} - 2 \beta_{7} + 21 \beta_{8} + 9 \beta_{9} + 14 \beta_{10} - 2 \beta_{11} + 5 \beta_{12} + 13 \beta_{13} - 2 \beta_{14} + 6 \beta_{15} + 2 \beta_{16} + 13 \beta_{17} + 8 \beta_{18} + 6 \beta_{19} ) q^{68} + ( 14 \beta_{1} + 22 \beta_{2} + 4 \beta_{3} + \beta_{4} - 42 \beta_{5} - 3 \beta_{7} + 7 \beta_{8} + 3 \beta_{9} - \beta_{10} + 5 \beta_{11} - \beta_{13} - 9 \beta_{14} + 3 \beta_{15} + 2 \beta_{16} - 4 \beta_{17} - 2 \beta_{19} ) q^{69} + ( 20 - 45 \beta_{1} - 10 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} - 5 \beta_{8} - 5 \beta_{11} ) q^{70} + ( -9 - 17 \beta_{1} - 4 \beta_{2} + 11 \beta_{3} - \beta_{4} - 16 \beta_{5} - 16 \beta_{7} - 6 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - 11 \beta_{11} - 5 \beta_{12} - 9 \beta_{13} - 11 \beta_{14} + 2 \beta_{15} + 9 \beta_{16} - 2 \beta_{17} - 4 \beta_{18} - \beta_{19} ) q^{71} + ( 36 + 25 \beta_{1} + 18 \beta_{2} - 2 \beta_{3} + 7 \beta_{4} - 97 \beta_{5} + 19 \beta_{6} + 12 \beta_{7} + 20 \beta_{8} + 15 \beta_{10} - 2 \beta_{11} + 9 \beta_{12} + 7 \beta_{13} - 3 \beta_{14} + 4 \beta_{15} + 4 \beta_{16} + 8 \beta_{17} - 6 \beta_{18} + 13 \beta_{19} ) q^{72} + ( -183 - 23 \beta_{1} + 12 \beta_{2} - 10 \beta_{3} - 3 \beta_{4} - 21 \beta_{5} - 4 \beta_{6} + 9 \beta_{8} - 7 \beta_{9} - 3 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} + 6 \beta_{14} + 5 \beta_{15} - 8 \beta_{16} + 5 \beta_{17} - \beta_{18} - 5 \beta_{19} ) q^{73} + ( -39 - 17 \beta_{1} - 5 \beta_{2} - 11 \beta_{3} + 7 \beta_{4} - 31 \beta_{5} + 2 \beta_{6} + 37 \beta_{7} + 9 \beta_{8} - 6 \beta_{9} + 7 \beta_{10} + 9 \beta_{11} - 3 \beta_{12} + 20 \beta_{13} + 16 \beta_{14} - 8 \beta_{15} - 19 \beta_{16} + 8 \beta_{17} - 3 \beta_{18} ) q^{74} + ( -25 + 25 \beta_{5} ) q^{75} + ( 76 - 19 \beta_{1} + 19 \beta_{2} ) q^{76} + ( -77 - 11 \beta_{7} ) q^{77} + ( 211 + 16 \beta_{1} + 50 \beta_{2} - 22 \beta_{3} + 18 \beta_{4} + 24 \beta_{5} + 3 \beta_{6} - 13 \beta_{7} + \beta_{9} - 3 \beta_{10} + 4 \beta_{11} + 7 \beta_{12} - 7 \beta_{14} - 4 \beta_{15} - 2 \beta_{16} - 16 \beta_{17} - 10 \beta_{18} - 4 \beta_{19} ) q^{78} + ( 6 + 20 \beta_{2} - 5 \beta_{3} + 4 \beta_{4} + 10 \beta_{5} + 7 \beta_{6} + \beta_{7} - \beta_{8} - 8 \beta_{9} - 6 \beta_{10} + 10 \beta_{11} - 6 \beta_{12} - 6 \beta_{13} + 7 \beta_{14} - 5 \beta_{15} + 6 \beta_{16} - 5 \beta_{17} - 4 \beta_{18} + \beta_{19} ) q^{79} + ( 115 - 30 \beta_{1} + 25 \beta_{2} - 15 \beta_{3} - 5 \beta_{4} - 10 \beta_{5} + 10 \beta_{6} + 10 \beta_{7} + 10 \beta_{8} - 5 \beta_{9} + 5 \beta_{12} + 5 \beta_{16} - 5 \beta_{17} - 5 \beta_{18} + 5 \beta_{19} ) q^{80} + ( -3 - 17 \beta_{1} - 7 \beta_{2} + 32 \beta_{3} + 15 \beta_{4} - 15 \beta_{5} - 5 \beta_{6} - 23 \beta_{7} - 12 \beta_{8} + 9 \beta_{9} - 7 \beta_{10} - 6 \beta_{11} - 6 \beta_{12} - 15 \beta_{13} - 15 \beta_{14} + 2 \beta_{15} + 3 \beta_{16} - 5 \beta_{17} - 6 \beta_{18} - 4 \beta_{19} ) q^{81} + ( -147 - 117 \beta_{1} - 22 \beta_{2} - 4 \beta_{3} - 13 \beta_{4} - 15 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} + 6 \beta_{8} - 10 \beta_{9} + 6 \beta_{10} + 3 \beta_{11} + \beta_{12} + 7 \beta_{13} + 5 \beta_{14} + \beta_{15} - \beta_{16} + 9 \beta_{17} + 7 \beta_{18} + 4 \beta_{19} ) q^{82} + ( -140 - 61 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} - 7 \beta_{4} + 16 \beta_{5} - 11 \beta_{6} + 3 \beta_{7} - 11 \beta_{8} - 8 \beta_{9} - 6 \beta_{10} + 9 \beta_{11} - 2 \beta_{12} - 10 \beta_{13} + 4 \beta_{14} - 3 \beta_{15} + 8 \beta_{16} - 10 \beta_{17} + 3 \beta_{18} + 8 \beta_{19} ) q^{83} + ( 323 - 5 \beta_{1} + 57 \beta_{2} - 6 \beta_{3} + 4 \beta_{4} - 64 \beta_{5} + 9 \beta_{6} + 42 \beta_{7} + 8 \beta_{8} + 2 \beta_{9} + 8 \beta_{10} - 6 \beta_{11} + 3 \beta_{12} + 17 \beta_{13} + 14 \beta_{14} - 7 \beta_{15} + 3 \beta_{16} + 14 \beta_{17} + 2 \beta_{18} + 19 \beta_{19} ) q^{84} + ( -115 - 5 \beta_{3} - 5 \beta_{4} - 5 \beta_{5} + 5 \beta_{6} + 10 \beta_{7} - 5 \beta_{9} + 5 \beta_{10} + 5 \beta_{11} + 5 \beta_{19} ) q^{85} + ( 151 - 161 \beta_{1} + 13 \beta_{2} + 3 \beta_{3} - 12 \beta_{4} - 12 \beta_{5} + 14 \beta_{6} - 3 \beta_{7} + 9 \beta_{8} + 3 \beta_{9} + 10 \beta_{10} + 3 \beta_{11} + 11 \beta_{12} + 13 \beta_{13} + \beta_{14} + \beta_{15} - 2 \beta_{16} - \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{86} + ( -126 - 24 \beta_{1} - 20 \beta_{2} + 4 \beta_{3} + 16 \beta_{4} + 8 \beta_{5} - \beta_{6} + \beta_{7} - 4 \beta_{8} - 12 \beta_{9} + 3 \beta_{11} - 2 \beta_{12} - 6 \beta_{13} + 10 \beta_{14} - 7 \beta_{15} + \beta_{16} - 3 \beta_{17} + 2 \beta_{19} ) q^{87} + ( -77 + 44 \beta_{1} - 11 \beta_{2} + 11 \beta_{4} - 11 \beta_{5} ) q^{88} + ( 120 - 77 \beta_{1} - 7 \beta_{2} + \beta_{3} + 17 \beta_{4} - 20 \beta_{5} - 11 \beta_{6} + 7 \beta_{7} - 16 \beta_{8} - 4 \beta_{9} + 5 \beta_{10} - 12 \beta_{11} + 5 \beta_{12} - 13 \beta_{13} + \beta_{14} + 6 \beta_{16} - 7 \beta_{17} - 7 \beta_{18} - 2 \beta_{19} ) q^{89} + ( 30 + 35 \beta_{1} + 20 \beta_{2} - 15 \beta_{3} - 15 \beta_{4} - 10 \beta_{5} + 10 \beta_{6} + 20 \beta_{7} + 15 \beta_{8} - 5 \beta_{9} + 10 \beta_{10} + 5 \beta_{11} - 5 \beta_{12} + 15 \beta_{13} + 10 \beta_{14} + 10 \beta_{17} + 5 \beta_{18} + 5 \beta_{19} ) q^{90} + ( 98 - 8 \beta_{1} + 16 \beta_{2} + 11 \beta_{3} - 8 \beta_{4} - 15 \beta_{5} + 13 \beta_{6} + 15 \beta_{7} + 11 \beta_{8} - 8 \beta_{9} + 20 \beta_{11} - 4 \beta_{12} + 11 \beta_{13} + 6 \beta_{14} - 3 \beta_{15} - 6 \beta_{16} - 9 \beta_{17} - 2 \beta_{19} ) q^{91} + ( -178 - 39 \beta_{1} - 97 \beta_{2} + 20 \beta_{3} + 16 \beta_{4} + 43 \beta_{5} - 25 \beta_{6} + 7 \beta_{7} - 20 \beta_{8} - 3 \beta_{9} - 10 \beta_{10} + 6 \beta_{11} + 5 \beta_{13} + 15 \beta_{14} - 2 \beta_{15} - 12 \beta_{16} - 2 \beta_{17} + 2 \beta_{18} - 10 \beta_{19} ) q^{92} + ( -60 - 89 \beta_{1} + 38 \beta_{2} - 18 \beta_{3} + 15 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} + 10 \beta_{7} - 16 \beta_{8} - 3 \beta_{9} - 10 \beta_{10} - 4 \beta_{11} - \beta_{12} - 13 \beta_{13} - 10 \beta_{14} - 4 \beta_{15} + 6 \beta_{16} + 3 \beta_{17} - 8 \beta_{18} + 16 \beta_{19} ) q^{93} + ( 181 - 158 \beta_{1} - 4 \beta_{2} + 26 \beta_{3} - 2 \beta_{4} + 13 \beta_{5} + 23 \beta_{6} + 5 \beta_{8} + 11 \beta_{9} - 2 \beta_{10} + 14 \beta_{11} + 7 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} + 2 \beta_{15} + \beta_{16} - 4 \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{94} + 95 q^{95} + ( -287 - 96 \beta_{1} - 27 \beta_{2} + 21 \beta_{3} - 33 \beta_{4} + 13 \beta_{5} - 11 \beta_{6} - 5 \beta_{8} - 16 \beta_{9} + 5 \beta_{10} + \beta_{11} - 6 \beta_{12} + 2 \beta_{13} + 3 \beta_{14} + 4 \beta_{15} + 5 \beta_{16} + 6 \beta_{17} + 16 \beta_{18} + 2 \beta_{19} ) q^{96} + ( -51 - 78 \beta_{1} + 23 \beta_{2} - 2 \beta_{3} - 17 \beta_{4} - 17 \beta_{5} + 5 \beta_{6} + 30 \beta_{7} + 17 \beta_{8} - 22 \beta_{9} + 3 \beta_{10} - 14 \beta_{12} + 8 \beta_{13} + 21 \beta_{14} - 2 \beta_{15} - 3 \beta_{16} + 7 \beta_{17} - 10 \beta_{18} + 10 \beta_{19} ) q^{97} + ( -130 + 10 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} - 11 \beta_{4} + 79 \beta_{5} - 5 \beta_{6} - 41 \beta_{7} - 4 \beta_{8} + 25 \beta_{9} + 3 \beta_{10} - 9 \beta_{11} - 8 \beta_{12} - 16 \beta_{13} - 24 \beta_{14} + 7 \beta_{15} + 12 \beta_{16} - 11 \beta_{17} + 2 \beta_{18} - 5 \beta_{19} ) q^{98} + ( 88 + 11 \beta_{1} - 11 \beta_{5} + 11 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} + O(q^{10}) \) \( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 8 x^{19} - 82 x^{18} + 700 x^{17} + 2826 x^{16} - 25467 x^{15} - 53768 x^{14} + 498499 x^{13} + 623613 x^{12} - 5673747 x^{11} - 4539454 x^{10} + 37893109 x^{9} + 19879768 x^{8} - 143049638 x^{7} - 45064360 x^{6} + 280461480 x^{5} + 43097920 x^{4} - 240447168 x^{3} - 36068096 x^{2} + 74703872 x + 17756160\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 11 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(39\!\cdots\!97\)\( \nu^{19} - \)\(20\!\cdots\!68\)\( \nu^{18} + \)\(62\!\cdots\!50\)\( \nu^{17} - \)\(92\!\cdots\!04\)\( \nu^{16} - \)\(31\!\cdots\!58\)\( \nu^{15} + \)\(77\!\cdots\!43\)\( \nu^{14} + \)\(74\!\cdots\!68\)\( \nu^{13} - \)\(24\!\cdots\!19\)\( \nu^{12} - \)\(81\!\cdots\!57\)\( \nu^{11} + \)\(39\!\cdots\!95\)\( \nu^{10} + \)\(27\!\cdots\!98\)\( \nu^{9} - \)\(34\!\cdots\!41\)\( \nu^{8} + \)\(19\!\cdots\!88\)\( \nu^{7} + \)\(15\!\cdots\!62\)\( \nu^{6} - \)\(17\!\cdots\!12\)\( \nu^{5} - \)\(32\!\cdots\!64\)\( \nu^{4} + \)\(42\!\cdots\!80\)\( \nu^{3} + \)\(21\!\cdots\!08\)\( \nu^{2} - \)\(23\!\cdots\!32\)\( \nu - \)\(10\!\cdots\!64\)\(\)\()/ \)\(34\!\cdots\!88\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(16\!\cdots\!09\)\( \nu^{19} - \)\(13\!\cdots\!60\)\( \nu^{18} - \)\(13\!\cdots\!10\)\( \nu^{17} + \)\(11\!\cdots\!00\)\( \nu^{16} + \)\(46\!\cdots\!18\)\( \nu^{15} - \)\(43\!\cdots\!35\)\( \nu^{14} - \)\(86\!\cdots\!40\)\( \nu^{13} + \)\(86\!\cdots\!63\)\( \nu^{12} + \)\(94\!\cdots\!41\)\( \nu^{11} - \)\(97\!\cdots\!51\)\( \nu^{10} - \)\(59\!\cdots\!06\)\( \nu^{9} + \)\(64\!\cdots\!73\)\( \nu^{8} + \)\(19\!\cdots\!48\)\( \nu^{7} - \)\(23\!\cdots\!90\)\( \nu^{6} - \)\(12\!\cdots\!92\)\( \nu^{5} + \)\(39\!\cdots\!72\)\( \nu^{4} - \)\(57\!\cdots\!24\)\( \nu^{3} - \)\(23\!\cdots\!96\)\( \nu^{2} + \)\(14\!\cdots\!80\)\( \nu + \)\(46\!\cdots\!92\)\(\)\()/ \)\(13\!\cdots\!52\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(16\!\cdots\!09\)\( \nu^{19} - \)\(13\!\cdots\!60\)\( \nu^{18} - \)\(13\!\cdots\!10\)\( \nu^{17} + \)\(11\!\cdots\!00\)\( \nu^{16} + \)\(46\!\cdots\!18\)\( \nu^{15} - \)\(43\!\cdots\!35\)\( \nu^{14} - \)\(86\!\cdots\!40\)\( \nu^{13} + \)\(86\!\cdots\!63\)\( \nu^{12} + \)\(94\!\cdots\!41\)\( \nu^{11} - \)\(97\!\cdots\!51\)\( \nu^{10} - \)\(59\!\cdots\!06\)\( \nu^{9} + \)\(64\!\cdots\!73\)\( \nu^{8} + \)\(19\!\cdots\!48\)\( \nu^{7} - \)\(23\!\cdots\!90\)\( \nu^{6} - \)\(12\!\cdots\!92\)\( \nu^{5} + \)\(39\!\cdots\!72\)\( \nu^{4} - \)\(58\!\cdots\!76\)\( \nu^{3} - \)\(23\!\cdots\!92\)\( \nu^{2} + \)\(26\!\cdots\!16\)\( \nu + \)\(31\!\cdots\!20\)\(\)\()/ \)\(13\!\cdots\!52\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(26\!\cdots\!93\)\( \nu^{19} + \)\(74\!\cdots\!20\)\( \nu^{18} - \)\(32\!\cdots\!10\)\( \nu^{17} - \)\(50\!\cdots\!32\)\( \nu^{16} + \)\(35\!\cdots\!18\)\( \nu^{15} + \)\(12\!\cdots\!43\)\( \nu^{14} - \)\(12\!\cdots\!16\)\( \nu^{13} - \)\(11\!\cdots\!91\)\( \nu^{12} + \)\(22\!\cdots\!07\)\( \nu^{11} - \)\(18\!\cdots\!45\)\( \nu^{10} - \)\(21\!\cdots\!74\)\( \nu^{9} + \)\(11\!\cdots\!35\)\( \nu^{8} + \)\(11\!\cdots\!76\)\( \nu^{7} - \)\(80\!\cdots\!58\)\( \nu^{6} - \)\(27\!\cdots\!96\)\( \nu^{5} + \)\(19\!\cdots\!76\)\( \nu^{4} + \)\(26\!\cdots\!32\)\( \nu^{3} - \)\(10\!\cdots\!76\)\( \nu^{2} - \)\(10\!\cdots\!00\)\( \nu - \)\(13\!\cdots\!08\)\(\)\()/ \)\(98\!\cdots\!68\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(12\!\cdots\!65\)\( \nu^{19} - \)\(11\!\cdots\!36\)\( \nu^{18} - \)\(79\!\cdots\!46\)\( \nu^{17} + \)\(97\!\cdots\!88\)\( \nu^{16} + \)\(17\!\cdots\!54\)\( \nu^{15} - \)\(32\!\cdots\!07\)\( \nu^{14} - \)\(10\!\cdots\!76\)\( \nu^{13} + \)\(57\!\cdots\!19\)\( \nu^{12} - \)\(16\!\cdots\!23\)\( \nu^{11} - \)\(56\!\cdots\!39\)\( \nu^{10} + \)\(30\!\cdots\!38\)\( \nu^{9} + \)\(31\!\cdots\!09\)\( \nu^{8} - \)\(19\!\cdots\!68\)\( \nu^{7} - \)\(90\!\cdots\!94\)\( \nu^{6} + \)\(53\!\cdots\!32\)\( \nu^{5} + \)\(11\!\cdots\!04\)\( \nu^{4} - \)\(59\!\cdots\!84\)\( \nu^{3} - \)\(71\!\cdots\!08\)\( \nu^{2} + \)\(26\!\cdots\!96\)\( \nu + \)\(15\!\cdots\!32\)\(\)\()/ \)\(19\!\cdots\!36\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(26\!\cdots\!21\)\( \nu^{19} - \)\(39\!\cdots\!00\)\( \nu^{18} - \)\(29\!\cdots\!62\)\( \nu^{17} + \)\(29\!\cdots\!20\)\( \nu^{16} - \)\(73\!\cdots\!50\)\( \nu^{15} - \)\(85\!\cdots\!35\)\( \nu^{14} + \)\(33\!\cdots\!72\)\( \nu^{13} + \)\(12\!\cdots\!39\)\( \nu^{12} - \)\(64\!\cdots\!03\)\( \nu^{11} - \)\(84\!\cdots\!71\)\( \nu^{10} + \)\(64\!\cdots\!66\)\( \nu^{9} + \)\(17\!\cdots\!13\)\( \nu^{8} - \)\(33\!\cdots\!80\)\( \nu^{7} + \)\(76\!\cdots\!62\)\( \nu^{6} + \)\(83\!\cdots\!76\)\( \nu^{5} - \)\(36\!\cdots\!00\)\( \nu^{4} - \)\(78\!\cdots\!92\)\( \nu^{3} + \)\(20\!\cdots\!12\)\( \nu^{2} + \)\(27\!\cdots\!84\)\( \nu + \)\(37\!\cdots\!12\)\(\)\()/ \)\(34\!\cdots\!88\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(17\!\cdots\!43\)\( \nu^{19} + \)\(17\!\cdots\!56\)\( \nu^{18} + \)\(10\!\cdots\!66\)\( \nu^{17} - \)\(14\!\cdots\!52\)\( \nu^{16} - \)\(21\!\cdots\!26\)\( \nu^{15} + \)\(46\!\cdots\!17\)\( \nu^{14} + \)\(31\!\cdots\!28\)\( \nu^{13} - \)\(78\!\cdots\!49\)\( \nu^{12} + \)\(42\!\cdots\!65\)\( \nu^{11} + \)\(74\!\cdots\!13\)\( \nu^{10} - \)\(58\!\cdots\!54\)\( \nu^{9} - \)\(37\!\cdots\!31\)\( \nu^{8} + \)\(31\!\cdots\!32\)\( \nu^{7} + \)\(90\!\cdots\!94\)\( \nu^{6} - \)\(64\!\cdots\!60\)\( \nu^{5} - \)\(70\!\cdots\!68\)\( \nu^{4} + \)\(10\!\cdots\!36\)\( \nu^{3} + \)\(12\!\cdots\!88\)\( \nu^{2} + \)\(20\!\cdots\!20\)\( \nu + \)\(91\!\cdots\!52\)\(\)\()/ \)\(13\!\cdots\!52\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(27\!\cdots\!79\)\( \nu^{19} - \)\(15\!\cdots\!88\)\( \nu^{18} - \)\(27\!\cdots\!06\)\( \nu^{17} + \)\(14\!\cdots\!32\)\( \nu^{16} + \)\(12\!\cdots\!74\)\( \nu^{15} - \)\(56\!\cdots\!85\)\( \nu^{14} - \)\(29\!\cdots\!24\)\( \nu^{13} + \)\(11\!\cdots\!81\)\( \nu^{12} + \)\(43\!\cdots\!39\)\( \nu^{11} - \)\(13\!\cdots\!69\)\( \nu^{10} - \)\(38\!\cdots\!02\)\( \nu^{9} + \)\(93\!\cdots\!83\)\( \nu^{8} + \)\(19\!\cdots\!32\)\( \nu^{7} - \)\(33\!\cdots\!14\)\( \nu^{6} - \)\(49\!\cdots\!00\)\( \nu^{5} + \)\(54\!\cdots\!44\)\( \nu^{4} + \)\(52\!\cdots\!48\)\( \nu^{3} - \)\(22\!\cdots\!60\)\( \nu^{2} - \)\(22\!\cdots\!00\)\( \nu - \)\(40\!\cdots\!20\)\(\)\()/ \)\(19\!\cdots\!36\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(60\!\cdots\!03\)\( \nu^{19} + \)\(57\!\cdots\!32\)\( \nu^{18} + \)\(40\!\cdots\!06\)\( \nu^{17} - \)\(47\!\cdots\!96\)\( \nu^{16} - \)\(94\!\cdots\!38\)\( \nu^{15} + \)\(16\!\cdots\!29\)\( \nu^{14} + \)\(74\!\cdots\!76\)\( \nu^{13} - \)\(28\!\cdots\!21\)\( \nu^{12} + \)\(49\!\cdots\!61\)\( \nu^{11} + \)\(29\!\cdots\!13\)\( \nu^{10} - \)\(12\!\cdots\!62\)\( \nu^{9} - \)\(16\!\cdots\!51\)\( \nu^{8} + \)\(86\!\cdots\!48\)\( \nu^{7} + \)\(51\!\cdots\!42\)\( \nu^{6} - \)\(24\!\cdots\!12\)\( \nu^{5} - \)\(72\!\cdots\!24\)\( \nu^{4} + \)\(27\!\cdots\!64\)\( \nu^{3} + \)\(38\!\cdots\!88\)\( \nu^{2} - \)\(64\!\cdots\!52\)\( \nu - \)\(45\!\cdots\!08\)\(\)\()/ \)\(34\!\cdots\!88\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(25\!\cdots\!31\)\( \nu^{19} + \)\(26\!\cdots\!40\)\( \nu^{18} + \)\(13\!\cdots\!38\)\( \nu^{17} - \)\(21\!\cdots\!80\)\( \nu^{16} - \)\(16\!\cdots\!14\)\( \nu^{15} + \)\(68\!\cdots\!61\)\( \nu^{14} - \)\(39\!\cdots\!72\)\( \nu^{13} - \)\(11\!\cdots\!45\)\( \nu^{12} + \)\(13\!\cdots\!09\)\( \nu^{11} + \)\(10\!\cdots\!93\)\( \nu^{10} - \)\(15\!\cdots\!62\)\( \nu^{9} - \)\(51\!\cdots\!43\)\( \nu^{8} + \)\(84\!\cdots\!12\)\( \nu^{7} + \)\(11\!\cdots\!38\)\( \nu^{6} - \)\(20\!\cdots\!68\)\( \nu^{5} - \)\(86\!\cdots\!56\)\( \nu^{4} + \)\(16\!\cdots\!96\)\( \nu^{3} + \)\(19\!\cdots\!00\)\( \nu^{2} - \)\(37\!\cdots\!84\)\( \nu + \)\(30\!\cdots\!16\)\(\)\()/ \)\(13\!\cdots\!52\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(32\!\cdots\!39\)\( \nu^{19} + \)\(25\!\cdots\!85\)\( \nu^{18} + \)\(26\!\cdots\!42\)\( \nu^{17} - \)\(21\!\cdots\!38\)\( \nu^{16} - \)\(90\!\cdots\!54\)\( \nu^{15} + \)\(75\!\cdots\!87\)\( \nu^{14} + \)\(17\!\cdots\!89\)\( \nu^{13} - \)\(14\!\cdots\!21\)\( \nu^{12} - \)\(21\!\cdots\!00\)\( \nu^{11} + \)\(15\!\cdots\!86\)\( \nu^{10} + \)\(17\!\cdots\!75\)\( \nu^{9} - \)\(91\!\cdots\!93\)\( \nu^{8} - \)\(85\!\cdots\!11\)\( \nu^{7} + \)\(29\!\cdots\!90\)\( \nu^{6} + \)\(22\!\cdots\!02\)\( \nu^{5} - \)\(43\!\cdots\!56\)\( \nu^{4} - \)\(26\!\cdots\!32\)\( \nu^{3} + \)\(18\!\cdots\!28\)\( \nu^{2} + \)\(12\!\cdots\!32\)\( \nu + \)\(16\!\cdots\!72\)\(\)\()/ \)\(17\!\cdots\!44\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(27\!\cdots\!13\)\( \nu^{19} - \)\(25\!\cdots\!84\)\( \nu^{18} - \)\(18\!\cdots\!06\)\( \nu^{17} + \)\(21\!\cdots\!52\)\( \nu^{16} + \)\(42\!\cdots\!42\)\( \nu^{15} - \)\(72\!\cdots\!27\)\( \nu^{14} - \)\(31\!\cdots\!64\)\( \nu^{13} + \)\(12\!\cdots\!59\)\( \nu^{12} - \)\(27\!\cdots\!15\)\( \nu^{11} - \)\(13\!\cdots\!03\)\( \nu^{10} + \)\(66\!\cdots\!34\)\( \nu^{9} + \)\(75\!\cdots\!25\)\( \nu^{8} - \)\(47\!\cdots\!84\)\( \nu^{7} - \)\(23\!\cdots\!70\)\( \nu^{6} + \)\(15\!\cdots\!40\)\( \nu^{5} + \)\(33\!\cdots\!40\)\( \nu^{4} - \)\(22\!\cdots\!20\)\( \nu^{3} - \)\(17\!\cdots\!36\)\( \nu^{2} + \)\(83\!\cdots\!88\)\( \nu + \)\(32\!\cdots\!48\)\(\)\()/ \)\(13\!\cdots\!52\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(19\!\cdots\!71\)\( \nu^{19} + \)\(16\!\cdots\!72\)\( \nu^{18} + \)\(14\!\cdots\!14\)\( \nu^{17} - \)\(13\!\cdots\!68\)\( \nu^{16} - \)\(42\!\cdots\!58\)\( \nu^{15} + \)\(44\!\cdots\!97\)\( \nu^{14} + \)\(70\!\cdots\!08\)\( \nu^{13} - \)\(76\!\cdots\!77\)\( \nu^{12} - \)\(77\!\cdots\!51\)\( \nu^{11} + \)\(74\!\cdots\!33\)\( \nu^{10} + \)\(64\!\cdots\!18\)\( \nu^{9} - \)\(38\!\cdots\!15\)\( \nu^{8} - \)\(41\!\cdots\!96\)\( \nu^{7} + \)\(91\!\cdots\!30\)\( \nu^{6} + \)\(15\!\cdots\!88\)\( \nu^{5} - \)\(65\!\cdots\!60\)\( \nu^{4} - \)\(30\!\cdots\!84\)\( \nu^{3} - \)\(15\!\cdots\!28\)\( \nu^{2} + \)\(16\!\cdots\!16\)\( \nu + \)\(41\!\cdots\!76\)\(\)\()/ \)\(69\!\cdots\!76\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(58\!\cdots\!27\)\( \nu^{19} - \)\(61\!\cdots\!44\)\( \nu^{18} - \)\(32\!\cdots\!62\)\( \nu^{17} + \)\(48\!\cdots\!08\)\( \nu^{16} + \)\(40\!\cdots\!58\)\( \nu^{15} - \)\(15\!\cdots\!93\)\( \nu^{14} + \)\(86\!\cdots\!20\)\( \nu^{13} + \)\(26\!\cdots\!77\)\( \nu^{12} - \)\(30\!\cdots\!41\)\( \nu^{11} - \)\(23\!\cdots\!61\)\( \nu^{10} + \)\(35\!\cdots\!66\)\( \nu^{9} + \)\(11\!\cdots\!15\)\( \nu^{8} - \)\(18\!\cdots\!76\)\( \nu^{7} - \)\(25\!\cdots\!14\)\( \nu^{6} + \)\(44\!\cdots\!76\)\( \nu^{5} + \)\(15\!\cdots\!04\)\( \nu^{4} - \)\(32\!\cdots\!88\)\( \nu^{3} - \)\(63\!\cdots\!96\)\( \nu^{2} + \)\(74\!\cdots\!04\)\( \nu + \)\(25\!\cdots\!88\)\(\)\()/ \)\(13\!\cdots\!52\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(62\!\cdots\!67\)\( \nu^{19} - \)\(64\!\cdots\!72\)\( \nu^{18} - \)\(35\!\cdots\!02\)\( \nu^{17} + \)\(51\!\cdots\!60\)\( \nu^{16} + \)\(54\!\cdots\!74\)\( \nu^{15} - \)\(16\!\cdots\!89\)\( \nu^{14} + \)\(59\!\cdots\!08\)\( \nu^{13} + \)\(28\!\cdots\!25\)\( \nu^{12} - \)\(27\!\cdots\!61\)\( \nu^{11} - \)\(26\!\cdots\!41\)\( \nu^{10} + \)\(33\!\cdots\!30\)\( \nu^{9} + \)\(13\!\cdots\!91\)\( \nu^{8} - \)\(18\!\cdots\!88\)\( \nu^{7} - \)\(35\!\cdots\!18\)\( \nu^{6} + \)\(46\!\cdots\!04\)\( \nu^{5} + \)\(36\!\cdots\!40\)\( \nu^{4} - \)\(41\!\cdots\!24\)\( \nu^{3} - \)\(19\!\cdots\!52\)\( \nu^{2} + \)\(11\!\cdots\!80\)\( \nu + \)\(46\!\cdots\!20\)\(\)\()/ \)\(13\!\cdots\!52\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(35\!\cdots\!61\)\( \nu^{19} + \)\(37\!\cdots\!16\)\( \nu^{18} + \)\(19\!\cdots\!82\)\( \nu^{17} - \)\(29\!\cdots\!60\)\( \nu^{16} - \)\(26\!\cdots\!18\)\( \nu^{15} + \)\(96\!\cdots\!71\)\( \nu^{14} - \)\(48\!\cdots\!40\)\( \nu^{13} - \)\(16\!\cdots\!59\)\( \nu^{12} + \)\(18\!\cdots\!03\)\( \nu^{11} + \)\(15\!\cdots\!03\)\( \nu^{10} - \)\(22\!\cdots\!34\)\( \nu^{9} - \)\(81\!\cdots\!57\)\( \nu^{8} + \)\(12\!\cdots\!24\)\( \nu^{7} + \)\(21\!\cdots\!10\)\( \nu^{6} - \)\(33\!\cdots\!00\)\( \nu^{5} - \)\(23\!\cdots\!64\)\( \nu^{4} + \)\(35\!\cdots\!48\)\( \nu^{3} + \)\(12\!\cdots\!48\)\( \nu^{2} - \)\(12\!\cdots\!60\)\( \nu - \)\(35\!\cdots\!28\)\(\)\()/ \)\(69\!\cdots\!76\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(85\!\cdots\!05\)\( \nu^{19} + \)\(85\!\cdots\!68\)\( \nu^{18} + \)\(52\!\cdots\!42\)\( \nu^{17} - \)\(68\!\cdots\!84\)\( \nu^{16} - \)\(99\!\cdots\!34\)\( \nu^{15} + \)\(22\!\cdots\!87\)\( \nu^{14} + \)\(15\!\cdots\!40\)\( \nu^{13} - \)\(39\!\cdots\!43\)\( \nu^{12} + \)\(24\!\cdots\!31\)\( \nu^{11} + \)\(37\!\cdots\!07\)\( \nu^{10} - \)\(33\!\cdots\!26\)\( \nu^{9} - \)\(20\!\cdots\!09\)\( \nu^{8} + \)\(19\!\cdots\!20\)\( \nu^{7} + \)\(53\!\cdots\!90\)\( \nu^{6} - \)\(49\!\cdots\!52\)\( \nu^{5} - \)\(59\!\cdots\!88\)\( \nu^{4} + \)\(45\!\cdots\!92\)\( \nu^{3} + \)\(28\!\cdots\!92\)\( \nu^{2} - \)\(13\!\cdots\!68\)\( \nu - \)\(48\!\cdots\!48\)\(\)\()/ \)\(13\!\cdots\!52\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 11\)
\(\nu^{3}\)\(=\)\(-\beta_{5} + \beta_{4} + 2 \beta_{2} + 20 \beta_{1} + 11\)
\(\nu^{4}\)\(=\)\(\beta_{19} - \beta_{18} - \beta_{17} + \beta_{16} + \beta_{12} - \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 6 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 31 \beta_{2} + 48 \beta_{1} + 224\)
\(\nu^{5}\)\(=\)\(6 \beta_{19} - 6 \beta_{18} - 4 \beta_{17} + 3 \beta_{16} - \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 2 \beta_{12} + 3 \beta_{11} + \beta_{10} - 6 \beta_{9} + 10 \beta_{8} + 17 \beta_{7} + 12 \beta_{6} - 63 \beta_{5} + 40 \beta_{4} - 16 \beta_{3} + 106 \beta_{2} + 514 \beta_{1} + 554\)
\(\nu^{6}\)\(=\)\(67 \beta_{19} - 60 \beta_{18} - 46 \beta_{17} + 48 \beta_{16} - 5 \beta_{15} + 11 \beta_{14} + 24 \beta_{13} + 39 \beta_{12} + 10 \beta_{11} + 15 \beta_{10} - 51 \beta_{9} + 119 \beta_{8} + 146 \beta_{7} + 121 \beta_{6} - 384 \beta_{5} + 165 \beta_{4} - 178 \beta_{3} + 1018 \beta_{2} + 1934 \beta_{1} + 5936\)
\(\nu^{7}\)\(=\)\(420 \beta_{19} - 372 \beta_{18} - 204 \beta_{17} + 187 \beta_{16} - 80 \beta_{15} + 161 \beta_{14} + 310 \beta_{13} + 130 \beta_{12} + 175 \beta_{11} + 137 \beta_{10} - 298 \beta_{9} + 709 \beta_{8} + 1098 \beta_{7} + 769 \beta_{6} - 2995 \beta_{5} + 1432 \beta_{4} - 1082 \beta_{3} + 4692 \beta_{2} + 15686 \beta_{1} + 23126\)
\(\nu^{8}\)\(=\)\(3377 \beta_{19} - 2777 \beta_{18} - 1582 \beta_{17} + 1819 \beta_{16} - 462 \beta_{15} + 962 \beta_{14} + 2014 \beta_{13} + 1367 \beta_{12} + 798 \beta_{11} + 1287 \beta_{10} - 1936 \beta_{9} + 5850 \beta_{8} + 7763 \beta_{7} + 5806 \beta_{6} - 18926 \beta_{5} + 7029 \beta_{4} - 8613 \beta_{3} + 36287 \beta_{2} + 75513 \beta_{1} + 187075\)
\(\nu^{9}\)\(=\)\(21643 \beta_{19} - 17550 \beta_{18} - 7475 \beta_{17} + 8277 \beta_{16} - 4543 \beta_{15} + 9024 \beta_{14} + 17486 \beta_{13} + 6093 \beta_{12} + 8123 \beta_{11} + 9641 \beta_{10} - 11268 \beta_{9} + 37020 \beta_{8} + 54723 \beta_{7} + 37382 \beta_{6} - 133060 \beta_{5} + 51216 \beta_{4} - 55004 \beta_{3} + 196028 \beta_{2} + 535659 \beta_{1} + 927526\)
\(\nu^{10}\)\(=\)\(155005 \beta_{19} - 119346 \beta_{18} - 48702 \beta_{17} + 64603 \beta_{16} - 28242 \beta_{15} + 56925 \beta_{14} + 116132 \beta_{13} + 49187 \beta_{12} + 44159 \beta_{11} + 76038 \beta_{10} - 67755 \beta_{9} + 268431 \beta_{8} + 367689 \beta_{7} + 257613 \beta_{6} - 855328 \beta_{5} + 278022 \beta_{4} - 390486 \beta_{3} + 1367484 \beta_{2} + 2942312 \beta_{1} + 6575409\)
\(\nu^{11}\)\(=\)\(1002188 \beta_{19} - 760767 \beta_{18} - 237489 \beta_{17} + 325670 \beta_{16} - 226505 \beta_{15} + 444272 \beta_{14} + 867742 \beta_{13} + 258394 \beta_{12} + 356225 \beta_{11} + 533366 \beta_{10} - 395006 \beta_{9} + 1731410 \beta_{8} + 2497120 \beta_{7} + 1660952 \beta_{6} - 5749689 \beta_{5} + 1868228 \beta_{4} - 2532261 \beta_{3} + 8016625 \beta_{2} + 19587974 \beta_{1} + 36909652\)
\(\nu^{12}\)\(=\)\(6818979 \beta_{19} - 4993113 \beta_{18} - 1399582 \beta_{17} + 2267171 \beta_{16} - 1454239 \beta_{15} + 2871821 \beta_{14} + 5767424 \beta_{13} + 1840975 \beta_{12} + 2121360 \beta_{11} + 3861377 \beta_{10} - 2324714 \beta_{9} + 11860541 \beta_{8} + 16462214 \beta_{7} + 11034047 \beta_{6} - 37158178 \beta_{5} + 10769635 \beta_{4} - 17127309 \beta_{3} + 53260663 \beta_{2} + 115281671 \beta_{1} + 246131231\)
\(\nu^{13}\)\(=\)\(44172983 \beta_{19} - 31931267 \beta_{18} - 6866475 \beta_{17} + 12257988 \beta_{16} - 10557314 \beta_{15} + 20512025 \beta_{14} + 40228592 \beta_{13} + 10584947 \beta_{12} + 15307535 \beta_{11} + 26242611 \beta_{10} - 13623343 \beta_{9} + 76850649 \beta_{8} + 109363028 \beta_{7} + 71063145 \beta_{6} - 244471932 \beta_{5} + 69688104 \beta_{4} - 111336599 \beta_{3} + 325435245 \beta_{2} + 745752558 \beta_{1} + 1470459851\)
\(\nu^{14}\)\(=\)\(292956562 \beta_{19} - 206552694 \beta_{18} - 37604907 \beta_{17} + 80298409 \beta_{16} - 68567621 \beta_{15} + 133630846 \beta_{14} + 265796356 \beta_{13} + 71030556 \beta_{12} + 95255305 \beta_{11} + 181276484 \beta_{10} - 80113203 \beta_{9} + 511718184 \beta_{8} + 714137498 \beta_{7} + 464126576 \beta_{6} - 1580171067 \beta_{5} + 416689596 \beta_{4} - 735735190 \beta_{3} + 2113622146 \beta_{2} + 4549853093 \beta_{1} + 9540322513\)
\(\nu^{15}\)\(=\)\(1895755282 \beta_{19} - 1322082746 \beta_{18} - 180712613 \beta_{17} + 455725798 \beta_{16} - 472381015 \beta_{15} + 911624935 \beta_{14} + 1790194732 \beta_{13} + 428662136 \beta_{12} + 650152984 \beta_{11} + 1209292641 \beta_{10} - 473193653 \beta_{9} + 3314046503 \beta_{8} + 4681232506 \beta_{7} + 2984876407 \beta_{6} - 10280855513 \beta_{5} + 2652174471 \beta_{4} - 4773459408 \beta_{3} + 13188346944 \beta_{2} + 29080446235 \beta_{1} + 58802396908\)
\(\nu^{16}\)\(=\)\(12398941408 \beta_{19} - 8498991098 \beta_{18} - 916947006 \beta_{17} + 2894980946 \beta_{16} - 3074530997 \beta_{15} + 5940401999 \beta_{14} + 11744810552 \beta_{13} + 2796051752 \beta_{12} + 4126432764 \beta_{11} + 8127198734 \beta_{10} - 2803197894 \beta_{9} + 21728800737 \beta_{8} + 30387721045 \beta_{7} + 19325332347 \beta_{6} - 66340371287 \beta_{5} + 16221419329 \beta_{4} - 31157082602 \beta_{3} + 84779646240 \beta_{2} + 180840900372 \beta_{1} + 377068449485\)
\(\nu^{17}\)\(=\)\(80070537363 \beta_{19} - 54399311658 \beta_{18} - 4121592075 \beta_{17} + 16995242528 \beta_{16} - 20576464899 \beta_{15} + 39520278569 \beta_{14} + 77595876524 \beta_{13} + 17312985945 \beta_{12} + 27369324172 \beta_{11} + 53558709704 \beta_{10} - 16717187852 \beta_{9} + 140435089999 \beta_{8} + 197542432955 \beta_{7} + 124122672371 \beta_{6} - 428932027776 \beta_{5} + 102623940677 \beta_{4} - 201551550238 \beta_{3} + 534704936912 \beta_{2} + 1150821558901 \beta_{1} + 2361234897287\)
\(\nu^{18}\)\(=\)\(519479969859 \beta_{19} - 348726275330 \beta_{18} - 18319042043 \beta_{17} + 106513938337 \beta_{16} - 133699833268 \beta_{15} + 256854127331 \beta_{14} + 505880846194 \beta_{13} + 111459257695 \beta_{12} + 175114958137 \beta_{11} + 353942650543 \beta_{10} - 100059208652 \beta_{9} + 912550545267 \beta_{8} + 1277034190244 \beta_{7} + 799746522597 \beta_{6} - 2762223527248 \beta_{5} + 636710515242 \beta_{4} - 1306214168320 \beta_{3} + 3422011678922 \beta_{2} + 7231840541983 \beta_{1} + 15072375790850\)
\(\nu^{19}\)\(=\)\(3347512819449 \beta_{19} - 2231480755745 \beta_{18} - 65788200504 \beta_{17} + 640316557656 \beta_{16} - 879941448401 \beta_{15} + 1684327872066 \beta_{14} + 3304721860176 \beta_{13} + 699646109461 \beta_{12} + 1143917844363 \beta_{11} + 2312813829456 \beta_{10} - 602918825526 \beta_{9} + 5884464268952 \beta_{8} + 8257407329996 \beta_{7} + 5130938800314 \beta_{6} - 17792617984594 \beta_{5} + 4023551267326 \beta_{4} - 8426069532731 \beta_{3} + 21704012621537 \beta_{2} + 45975789289745 \beta_{1} + 95172808147204\)

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
−4.47888
−4.19424
−3.95689
−3.42835
−2.55697
−2.53684
−2.41218
−0.822685
−0.731640
−0.259637
0.999641
1.12533
1.38435
2.13989
3.57546
3.74913
4.23917
4.30367
5.47116
6.39053
−5.47888 −2.61641 22.0181 5.00000 14.3350 −5.28814 −76.8037 −20.1544 −27.3944
1.2 −5.19424 8.38824 18.9801 5.00000 −43.5706 −4.76095 −57.0335 43.3626 −25.9712
1.3 −4.95689 −6.17893 16.5708 5.00000 30.6283 −36.0727 −42.4843 11.1792 −24.7845
1.4 −4.42835 −1.78158 11.6103 5.00000 7.88948 24.1111 −15.9878 −23.8260 −22.1418
1.5 −3.55697 3.83974 4.65205 5.00000 −13.6579 −6.66228 11.9086 −12.2564 −17.7849
1.6 −3.53684 1.54358 4.50922 5.00000 −5.45941 −15.5595 12.3463 −24.6173 −17.6842
1.7 −3.41218 −9.99772 3.64299 5.00000 34.1140 −3.40924 14.8669 72.9544 −17.0609
1.8 −1.82269 5.45780 −4.67782 5.00000 −9.94786 19.7082 23.1077 2.78762 −9.11343
1.9 −1.73164 −4.39818 −5.00142 5.00000 7.61606 22.5500 22.5138 −7.65602 −8.65820
1.10 −1.25964 7.52739 −6.41331 5.00000 −9.48178 −31.9473 18.1555 29.6616 −6.29819
1.11 −0.000359199 0 −7.63742 −8.00000 5.00000 0.00274335 −26.4307 0.00574718 31.3302 −0.00179599
1.12 0.125334 −3.53244 −7.98429 5.00000 −0.442735 −25.7104 −2.00338 −14.5219 0.626671
1.13 0.384350 −2.95996 −7.85228 5.00000 −1.13766 3.33433 −6.09282 −18.2387 1.92175
1.14 1.13989 3.45749 −6.70066 5.00000 3.94115 1.89071 −16.7571 −15.0457 5.69943
1.15 2.57546 8.83286 −1.36700 5.00000 22.7487 −30.7821 −24.1243 51.0194 12.8773
1.16 2.74913 −7.66455 −0.442295 5.00000 −21.0708 25.2191 −23.2090 31.7453 13.7456
1.17 3.23917 2.30947 2.49220 5.00000 7.48077 −5.43134 −17.8407 −21.6663 16.1958
1.18 3.30367 −7.25904 2.91425 5.00000 −23.9815 −9.92986 −16.8016 25.6936 16.5184
1.19 4.47116 0.109820 11.9912 5.00000 0.491023 −7.59378 17.8455 −26.9879 22.3558
1.20 5.39053 −8.44019 21.0578 5.00000 −45.4970 −18.2352 70.3882 44.2368 26.9526
\(n\): 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.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.4.a.b 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

$p$ $F_p(T)$
$2$ \( -3840 - 10653408 T + 103133888 T^{2} - 105488216 T^{3} - 313740532 T^{4} + 95862386 T^{5} + 290994701 T^{6} + 6991187 T^{7} - 105293959 T^{8} - 16253206 T^{9} + 18581116 T^{10} + 4319428 T^{11} - 1700735 T^{12} - 521840 T^{13} + 74163 T^{14} + 32533 T^{15} - 727 T^{16} - 1004 T^{17} - 44 T^{18} + 12 T^{19} + T^{20} \)
$3$ \( -751355754688 + 6546935756032 T + 3245121732600 T^{2} - 4852029212044 T^{3} - 2470333974350 T^{4} + 1110994396207 T^{5} + 680348157975 T^{6} - 85952322749 T^{7} - 85146120393 T^{8} - 879630279 T^{9} + 5338344637 T^{10} + 448234176 T^{11} - 169301996 T^{12} - 23269115 T^{13} + 2490347 T^{14} + 518698 T^{15} - 9296 T^{16} - 5359 T^{17} - 129 T^{18} + 21 T^{19} + T^{20} \)
$5$ \( ( -5 + T )^{20} \)
$7$ \( \)\(27\!\cdots\!72\)\( + \)\(14\!\cdots\!24\)\( T - \)\(47\!\cdots\!96\)\( T^{2} - \)\(51\!\cdots\!76\)\( T^{3} - \)\(10\!\cdots\!12\)\( T^{4} + 13885623239193427348 T^{5} + 9888923992309032868 T^{6} + 1822500696615854631 T^{7} + 139511419770550337 T^{8} - 1021118753288800 T^{9} - 955810807759832 T^{10} - 54737452834889 T^{11} + 706803126729 T^{12} + 181482944023 T^{13} + 4810481527 T^{14} - 153640844 T^{15} - 9978704 T^{16} - 89106 T^{17} + 4768 T^{18} + 131 T^{19} + T^{20} \)
$11$ \( ( -11 + T )^{20} \)
$13$ \( -\)\(33\!\cdots\!20\)\( - \)\(24\!\cdots\!12\)\( T - \)\(77\!\cdots\!24\)\( T^{2} - \)\(13\!\cdots\!04\)\( T^{3} - \)\(13\!\cdots\!88\)\( T^{4} - \)\(79\!\cdots\!99\)\( T^{5} - \)\(19\!\cdots\!17\)\( T^{6} + \)\(59\!\cdots\!04\)\( T^{7} + \)\(57\!\cdots\!12\)\( T^{8} + \)\(13\!\cdots\!30\)\( T^{9} - 1511328372061636154 T^{10} - 136023233227595811 T^{11} - 1814523620894927 T^{12} + 32146623555154 T^{13} + 1097984433560 T^{14} + 3977472520 T^{15} - 195688238 T^{16} - 2334313 T^{17} + 5503 T^{18} + 223 T^{19} + T^{20} \)
$17$ \( \)\(92\!\cdots\!60\)\( - \)\(32\!\cdots\!84\)\( T + \)\(90\!\cdots\!92\)\( T^{2} + \)\(43\!\cdots\!08\)\( T^{3} + \)\(59\!\cdots\!76\)\( T^{4} - \)\(11\!\cdots\!56\)\( T^{5} - \)\(41\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!29\)\( T^{7} + \)\(29\!\cdots\!49\)\( T^{8} + \)\(37\!\cdots\!39\)\( T^{9} - \)\(57\!\cdots\!55\)\( T^{10} - 16206653083062733783 T^{11} - 35925217272644271 T^{12} + 2338056098727270 T^{13} + 22184319320722 T^{14} - 82353887526 T^{15} - 2198707632 T^{16} - 7385584 T^{17} + 55614 T^{18} + 471 T^{19} + T^{20} \)
$19$ \( ( -19 + T )^{20} \)
$23$ \( \)\(18\!\cdots\!64\)\( + \)\(23\!\cdots\!60\)\( T - \)\(24\!\cdots\!12\)\( T^{2} - \)\(22\!\cdots\!72\)\( T^{3} + \)\(60\!\cdots\!74\)\( T^{4} + \)\(52\!\cdots\!49\)\( T^{5} + \)\(20\!\cdots\!15\)\( T^{6} - \)\(24\!\cdots\!82\)\( T^{7} - \)\(23\!\cdots\!44\)\( T^{8} + \)\(43\!\cdots\!08\)\( T^{9} + \)\(58\!\cdots\!28\)\( T^{10} - \)\(32\!\cdots\!78\)\( T^{11} - 6598534286015700338 T^{12} + 2646637577097979 T^{13} + 370954968467263 T^{14} + 883915205333 T^{15} - 9590385871 T^{16} - 44317394 T^{17} + 60536 T^{18} + 653 T^{19} + T^{20} \)
$29$ \( \)\(19\!\cdots\!60\)\( + \)\(69\!\cdots\!08\)\( T + \)\(97\!\cdots\!56\)\( T^{2} + \)\(64\!\cdots\!48\)\( T^{3} + \)\(16\!\cdots\!12\)\( T^{4} - \)\(25\!\cdots\!84\)\( T^{5} - \)\(21\!\cdots\!54\)\( T^{6} - \)\(18\!\cdots\!29\)\( T^{7} + \)\(74\!\cdots\!59\)\( T^{8} + \)\(85\!\cdots\!49\)\( T^{9} - \)\(13\!\cdots\!93\)\( T^{10} - \)\(11\!\cdots\!39\)\( T^{11} + 13834442443116827493 T^{12} + 58882672099520210 T^{13} - 698276491226140 T^{14} - 1397610679762 T^{15} + 17681345044 T^{16} + 14485758 T^{17} - 214826 T^{18} - 51 T^{19} + T^{20} \)
$31$ \( \)\(31\!\cdots\!00\)\( + \)\(27\!\cdots\!00\)\( T - \)\(25\!\cdots\!20\)\( T^{2} - \)\(15\!\cdots\!04\)\( T^{3} + \)\(80\!\cdots\!76\)\( T^{4} + \)\(31\!\cdots\!16\)\( T^{5} - \)\(13\!\cdots\!24\)\( T^{6} - \)\(28\!\cdots\!14\)\( T^{7} + \)\(12\!\cdots\!63\)\( T^{8} + \)\(10\!\cdots\!66\)\( T^{9} - \)\(68\!\cdots\!71\)\( T^{10} - \)\(18\!\cdots\!98\)\( T^{11} + \)\(22\!\cdots\!85\)\( T^{12} - 89644971838297306 T^{13} - 4628738217957814 T^{14} + 2378221543664 T^{15} + 55572084332 T^{16} - 24445120 T^{17} - 364260 T^{18} + 90 T^{19} + T^{20} \)
$37$ \( \)\(10\!\cdots\!96\)\( - \)\(37\!\cdots\!80\)\( T - \)\(44\!\cdots\!52\)\( T^{2} - \)\(11\!\cdots\!92\)\( T^{3} - \)\(12\!\cdots\!00\)\( T^{4} - \)\(27\!\cdots\!12\)\( T^{5} + \)\(10\!\cdots\!48\)\( T^{6} + \)\(69\!\cdots\!24\)\( T^{7} - \)\(14\!\cdots\!88\)\( T^{8} - \)\(30\!\cdots\!32\)\( T^{9} - \)\(45\!\cdots\!85\)\( T^{10} + \)\(57\!\cdots\!28\)\( T^{11} + \)\(17\!\cdots\!85\)\( T^{12} - 5677557686004184968 T^{13} - 24512068251269658 T^{14} + 30984797082576 T^{15} + 177154278886 T^{16} - 86643788 T^{17} - 659085 T^{18} + 96 T^{19} + T^{20} \)
$41$ \( \)\(62\!\cdots\!60\)\( + \)\(90\!\cdots\!04\)\( T + \)\(41\!\cdots\!36\)\( T^{2} + \)\(64\!\cdots\!80\)\( T^{3} - \)\(55\!\cdots\!92\)\( T^{4} - \)\(30\!\cdots\!84\)\( T^{5} - \)\(29\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!32\)\( T^{7} + \)\(44\!\cdots\!97\)\( T^{8} + \)\(23\!\cdots\!62\)\( T^{9} - \)\(11\!\cdots\!52\)\( T^{10} - \)\(15\!\cdots\!30\)\( T^{11} - \)\(29\!\cdots\!78\)\( T^{12} + 2653202852127870042 T^{13} + 13181314789472619 T^{14} - 3108408820566 T^{15} - 140580631601 T^{16} - 242291350 T^{17} + 322136 T^{18} + 1284 T^{19} + T^{20} \)
$43$ \( \)\(77\!\cdots\!20\)\( - \)\(28\!\cdots\!76\)\( T - \)\(79\!\cdots\!52\)\( T^{2} + \)\(85\!\cdots\!72\)\( T^{3} + \)\(30\!\cdots\!80\)\( T^{4} + \)\(12\!\cdots\!72\)\( T^{5} - \)\(19\!\cdots\!38\)\( T^{6} - \)\(16\!\cdots\!38\)\( T^{7} + \)\(27\!\cdots\!59\)\( T^{8} + \)\(64\!\cdots\!18\)\( T^{9} + \)\(92\!\cdots\!56\)\( T^{10} - \)\(11\!\cdots\!70\)\( T^{11} - \)\(37\!\cdots\!95\)\( T^{12} + 8107994607618419872 T^{13} + 51333602172421359 T^{14} + 10279389990818 T^{15} - 303465279240 T^{16} - 433442378 T^{17} + 512138 T^{18} + 1592 T^{19} + T^{20} \)
$47$ \( -\)\(48\!\cdots\!00\)\( - \)\(91\!\cdots\!68\)\( T + \)\(36\!\cdots\!64\)\( T^{2} + \)\(10\!\cdots\!32\)\( T^{3} + \)\(26\!\cdots\!72\)\( T^{4} - \)\(10\!\cdots\!92\)\( T^{5} - \)\(90\!\cdots\!72\)\( T^{6} + \)\(37\!\cdots\!78\)\( T^{7} + \)\(33\!\cdots\!43\)\( T^{8} + \)\(12\!\cdots\!14\)\( T^{9} - \)\(20\!\cdots\!75\)\( T^{10} - \)\(21\!\cdots\!00\)\( T^{11} - \)\(29\!\cdots\!05\)\( T^{12} + \)\(10\!\cdots\!22\)\( T^{13} + 345779164601745805 T^{14} + 33432675372610 T^{15} - 1087408721750 T^{16} - 1251096744 T^{17} + 693759 T^{18} + 2030 T^{19} + T^{20} \)
$53$ \( -\)\(13\!\cdots\!16\)\( + \)\(11\!\cdots\!72\)\( T + \)\(28\!\cdots\!76\)\( T^{2} - \)\(84\!\cdots\!32\)\( T^{3} - \)\(28\!\cdots\!44\)\( T^{4} - \)\(14\!\cdots\!88\)\( T^{5} + \)\(10\!\cdots\!86\)\( T^{6} + \)\(19\!\cdots\!55\)\( T^{7} - \)\(16\!\cdots\!17\)\( T^{8} - \)\(48\!\cdots\!25\)\( T^{9} + \)\(13\!\cdots\!71\)\( T^{10} + \)\(54\!\cdots\!91\)\( T^{11} - \)\(44\!\cdots\!31\)\( T^{12} - \)\(31\!\cdots\!42\)\( T^{13} - 24988078024601638 T^{14} + 998545668422478 T^{15} + 572833379166 T^{16} - 1559546076 T^{17} - 1355578 T^{18} + 943 T^{19} + T^{20} \)
$59$ \( -\)\(18\!\cdots\!20\)\( + \)\(17\!\cdots\!48\)\( T - \)\(41\!\cdots\!60\)\( T^{2} + \)\(72\!\cdots\!44\)\( T^{3} + \)\(39\!\cdots\!16\)\( T^{4} - \)\(63\!\cdots\!32\)\( T^{5} - \)\(13\!\cdots\!76\)\( T^{6} + \)\(23\!\cdots\!27\)\( T^{7} + \)\(23\!\cdots\!23\)\( T^{8} - \)\(38\!\cdots\!07\)\( T^{9} - \)\(22\!\cdots\!01\)\( T^{10} + \)\(34\!\cdots\!89\)\( T^{11} + \)\(14\!\cdots\!19\)\( T^{12} - \)\(17\!\cdots\!84\)\( T^{13} - 546582146854537096 T^{14} + 511774703807058 T^{15} + 1301718016814 T^{16} - 800881814 T^{17} - 1742128 T^{18} + 515 T^{19} + T^{20} \)
$61$ \( \)\(99\!\cdots\!80\)\( + \)\(32\!\cdots\!64\)\( T - \)\(13\!\cdots\!68\)\( T^{2} - \)\(24\!\cdots\!12\)\( T^{3} + \)\(54\!\cdots\!48\)\( T^{4} + \)\(23\!\cdots\!80\)\( T^{5} - \)\(49\!\cdots\!48\)\( T^{6} - \)\(99\!\cdots\!74\)\( T^{7} + \)\(16\!\cdots\!89\)\( T^{8} + \)\(24\!\cdots\!72\)\( T^{9} - \)\(23\!\cdots\!41\)\( T^{10} - \)\(29\!\cdots\!66\)\( T^{11} + \)\(16\!\cdots\!51\)\( T^{12} + \)\(17\!\cdots\!04\)\( T^{13} - 691217104214199638 T^{14} - 529006094510728 T^{15} + 1611423905624 T^{16} + 786974016 T^{17} - 1991906 T^{18} - 446 T^{19} + T^{20} \)
$67$ \( \)\(55\!\cdots\!28\)\( + \)\(58\!\cdots\!96\)\( T - \)\(12\!\cdots\!00\)\( T^{2} - \)\(16\!\cdots\!80\)\( T^{3} - \)\(12\!\cdots\!78\)\( T^{4} + \)\(43\!\cdots\!23\)\( T^{5} + \)\(13\!\cdots\!23\)\( T^{6} - \)\(28\!\cdots\!74\)\( T^{7} - \)\(17\!\cdots\!22\)\( T^{8} - \)\(25\!\cdots\!03\)\( T^{9} + \)\(90\!\cdots\!43\)\( T^{10} + \)\(94\!\cdots\!47\)\( T^{11} - \)\(21\!\cdots\!31\)\( T^{12} - \)\(38\!\cdots\!06\)\( T^{13} + 2020237517858163196 T^{14} + 6907089919837582 T^{15} + 747215388652 T^{16} - 5644516812 T^{17} - 2319164 T^{18} + 1719 T^{19} + T^{20} \)
$71$ \( \)\(25\!\cdots\!40\)\( - \)\(64\!\cdots\!44\)\( T - \)\(38\!\cdots\!16\)\( T^{2} + \)\(90\!\cdots\!60\)\( T^{3} - \)\(98\!\cdots\!12\)\( T^{4} - \)\(52\!\cdots\!20\)\( T^{5} + \)\(40\!\cdots\!64\)\( T^{6} + \)\(13\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!22\)\( T^{8} - \)\(12\!\cdots\!36\)\( T^{9} - \)\(25\!\cdots\!15\)\( T^{10} + \)\(54\!\cdots\!16\)\( T^{11} + \)\(13\!\cdots\!96\)\( T^{12} - \)\(11\!\cdots\!42\)\( T^{13} - 3426416043808919277 T^{14} + 1306781761507668 T^{15} + 4828483494874 T^{16} - 648459866 T^{17} - 3473104 T^{18} + 90 T^{19} + T^{20} \)
$73$ \( -\)\(68\!\cdots\!16\)\( + \)\(61\!\cdots\!12\)\( T + \)\(21\!\cdots\!92\)\( T^{2} + \)\(16\!\cdots\!88\)\( T^{3} - \)\(12\!\cdots\!40\)\( T^{4} - \)\(65\!\cdots\!55\)\( T^{5} - \)\(22\!\cdots\!71\)\( T^{6} + \)\(23\!\cdots\!76\)\( T^{7} + \)\(28\!\cdots\!86\)\( T^{8} + \)\(40\!\cdots\!43\)\( T^{9} - \)\(93\!\cdots\!87\)\( T^{10} - \)\(31\!\cdots\!51\)\( T^{11} - \)\(93\!\cdots\!51\)\( T^{12} + \)\(67\!\cdots\!50\)\( T^{13} + 8627402593964996684 T^{14} - 2269189029561176 T^{15} - 11099372283712 T^{16} - 5614207094 T^{17} + 3062570 T^{18} + 3763 T^{19} + T^{20} \)
$79$ \( -\)\(39\!\cdots\!00\)\( - \)\(11\!\cdots\!00\)\( T - \)\(42\!\cdots\!00\)\( T^{2} + \)\(11\!\cdots\!04\)\( T^{3} + \)\(79\!\cdots\!76\)\( T^{4} - \)\(40\!\cdots\!76\)\( T^{5} - \)\(33\!\cdots\!76\)\( T^{6} + \)\(53\!\cdots\!08\)\( T^{7} + \)\(54\!\cdots\!18\)\( T^{8} - \)\(39\!\cdots\!38\)\( T^{9} - \)\(43\!\cdots\!11\)\( T^{10} + \)\(16\!\cdots\!34\)\( T^{11} + \)\(18\!\cdots\!61\)\( T^{12} - \)\(40\!\cdots\!56\)\( T^{13} - 4531158546718322310 T^{14} + 489293693874728 T^{15} + 6010831960020 T^{16} - 211671870 T^{17} - 3963811 T^{18} - 2 T^{19} + T^{20} \)
$83$ \( \)\(82\!\cdots\!00\)\( + \)\(59\!\cdots\!60\)\( T + \)\(39\!\cdots\!48\)\( T^{2} - \)\(10\!\cdots\!76\)\( T^{3} - \)\(59\!\cdots\!92\)\( T^{4} + \)\(13\!\cdots\!44\)\( T^{5} + \)\(97\!\cdots\!94\)\( T^{6} - \)\(37\!\cdots\!38\)\( T^{7} - \)\(61\!\cdots\!99\)\( T^{8} - \)\(15\!\cdots\!48\)\( T^{9} + \)\(18\!\cdots\!61\)\( T^{10} + \)\(11\!\cdots\!58\)\( T^{11} - \)\(30\!\cdots\!47\)\( T^{12} - \)\(25\!\cdots\!24\)\( T^{13} + 25097860601041254198 T^{14} + 29008672328017916 T^{15} - 7944625110896 T^{16} - 15976937310 T^{17} - 1332452 T^{18} + 3436 T^{19} + T^{20} \)
$89$ \( \)\(43\!\cdots\!08\)\( + \)\(54\!\cdots\!32\)\( T + \)\(10\!\cdots\!44\)\( T^{2} - \)\(11\!\cdots\!96\)\( T^{3} - \)\(39\!\cdots\!40\)\( T^{4} + \)\(35\!\cdots\!28\)\( T^{5} + \)\(34\!\cdots\!20\)\( T^{6} - \)\(34\!\cdots\!88\)\( T^{7} - \)\(77\!\cdots\!90\)\( T^{8} + \)\(15\!\cdots\!12\)\( T^{9} - \)\(18\!\cdots\!93\)\( T^{10} - \)\(36\!\cdots\!80\)\( T^{11} + \)\(11\!\cdots\!32\)\( T^{12} + \)\(46\!\cdots\!24\)\( T^{13} - 20317822759934755454 T^{14} - 33378795965946896 T^{15} + 16944085747525 T^{16} + 12029147916 T^{17} - 6691300 T^{18} - 1700 T^{19} + T^{20} \)
$97$ \( -\)\(10\!\cdots\!24\)\( + \)\(15\!\cdots\!00\)\( T - \)\(60\!\cdots\!92\)\( T^{2} + \)\(18\!\cdots\!84\)\( T^{3} + \)\(17\!\cdots\!28\)\( T^{4} + \)\(10\!\cdots\!96\)\( T^{5} + \)\(13\!\cdots\!28\)\( T^{6} - \)\(57\!\cdots\!64\)\( T^{7} - \)\(18\!\cdots\!34\)\( T^{8} - \)\(55\!\cdots\!80\)\( T^{9} + \)\(44\!\cdots\!31\)\( T^{10} + \)\(48\!\cdots\!52\)\( T^{11} - \)\(33\!\cdots\!60\)\( T^{12} - \)\(74\!\cdots\!36\)\( T^{13} - 1670064890355759030 T^{14} + 49984758708410448 T^{15} + 14557498102265 T^{16} - 15897946152 T^{17} - 6830900 T^{18} + 1956 T^{19} + T^{20} \)
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