Properties

Label 63.4.f.c
Level $63$
Weight $4$
Character orbit 63.f
Analytic conductor $3.717$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 63.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.71712033036\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 3 x^{17} + 6 x^{16} - 23 x^{15} - 6 x^{14} + 255 x^{13} - 56 x^{12} - 81 x^{11} + 5832 x^{10} - 32373 x^{9} + 52488 x^{8} - 6561 x^{7} - 40824 x^{6} + 1673055 x^{5} - 354294 x^{4} - 12223143 x^{3} + 28697814 x^{2} - 129140163 x + 387420489\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{2} - \beta_{3} - \beta_{5} ) q^{2} + ( 1 + \beta_{8} ) q^{3} + ( -\beta_{3} - 4 \beta_{5} + \beta_{7} - \beta_{15} ) q^{4} + ( 3 \beta_{5} + \beta_{12} ) q^{5} + ( 3 - \beta_{2} - 4 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} ) q^{6} + ( -7 + 7 \beta_{5} ) q^{7} + ( -9 + \beta_{1} + 4 \beta_{2} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} + \beta_{16} + \beta_{17} ) q^{8} + ( 2 - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{2} - \beta_{3} - \beta_{5} ) q^{2} + ( 1 + \beta_{8} ) q^{3} + ( -\beta_{3} - 4 \beta_{5} + \beta_{7} - \beta_{15} ) q^{4} + ( 3 \beta_{5} + \beta_{12} ) q^{5} + ( 3 - \beta_{2} - 4 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} ) q^{6} + ( -7 + 7 \beta_{5} ) q^{7} + ( -9 + \beta_{1} + 4 \beta_{2} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} + \beta_{16} + \beta_{17} ) q^{8} + ( 2 - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{9} + ( 5 - \beta_{1} - 7 \beta_{2} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{15} - 2 \beta_{16} ) q^{10} + ( 10 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 10 \beta_{5} + \beta_{7} - 2 \beta_{8} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{11} + ( 6 + 4 \beta_{2} + \beta_{3} + \beta_{4} - 10 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + 3 \beta_{10} - \beta_{12} + 2 \beta_{13} - 4 \beta_{15} + \beta_{17} ) q^{12} + ( 3 + \beta_{1} + \beta_{3} - 3 \beta_{4} - 4 \beta_{5} + \beta_{6} + \beta_{7} + 4 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - \beta_{16} ) q^{13} + ( 7 \beta_{3} + 7 \beta_{5} ) q^{14} + ( -7 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 8 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} + \beta_{12} + 4 \beta_{15} - \beta_{17} ) q^{15} + ( -27 - \beta_{1} + 12 \beta_{2} + 12 \beta_{3} - 3 \beta_{4} + 24 \beta_{5} + \beta_{6} + 2 \beta_{7} - 7 \beta_{8} - 6 \beta_{10} + 3 \beta_{11} + \beta_{12} - 2 \beta_{13} - 3 \beta_{14} + 5 \beta_{15} - \beta_{16} ) q^{16} + ( -24 - 2 \beta_{1} - 5 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{7} + \beta_{8} - 4 \beta_{9} - \beta_{10} - 4 \beta_{11} - 2 \beta_{16} - 3 \beta_{17} ) q^{17} + ( 17 - 3 \beta_{1} - 15 \beta_{2} - 3 \beta_{3} + \beta_{4} - 27 \beta_{5} - 3 \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} - 3 \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} + 3 \beta_{16} - \beta_{17} ) q^{18} + ( 6 + 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{4} + \beta_{5} + 2 \beta_{7} - \beta_{8} + 3 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} - 3 \beta_{16} ) q^{19} + ( 50 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 6 \beta_{4} - 47 \beta_{5} - \beta_{7} + 2 \beta_{8} - 3 \beta_{9} + 9 \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} - 8 \beta_{15} ) q^{20} + ( -14 + 7 \beta_{5} - 7 \beta_{8} - 7 \beta_{10} ) q^{21} + ( 8 - 2 \beta_{1} - 20 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} - 3 \beta_{6} - 7 \beta_{7} + 10 \beta_{8} - 6 \beta_{9} - \beta_{10} - \beta_{11} + 3 \beta_{14} + 4 \beta_{15} + 3 \beta_{16} + 3 \beta_{17} ) q^{22} + ( -19 + 5 \beta_{1} - 9 \beta_{3} + 6 \beta_{4} + 41 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} - 10 \beta_{8} + \beta_{9} - 8 \beta_{10} + \beta_{11} - 2 \beta_{12} + 6 \beta_{13} + 3 \beta_{16} ) q^{23} + ( -29 + 6 \beta_{1} + 27 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} + 25 \beta_{5} - 8 \beta_{6} + 6 \beta_{7} - 5 \beta_{8} + 12 \beta_{9} + 3 \beta_{11} + 6 \beta_{12} + 5 \beta_{13} - 3 \beta_{15} + 9 \beta_{16} + 3 \beta_{17} ) q^{24} + ( -45 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + 37 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - 7 \beta_{8} - 3 \beta_{9} - 13 \beta_{10} - 4 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 6 \beta_{14} - 4 \beta_{15} + 4 \beta_{16} ) q^{25} + ( 14 + 2 \beta_{1} + 12 \beta_{2} + 6 \beta_{4} - 7 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 17 \beta_{8} - 4 \beta_{9} + 8 \beta_{10} + 5 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{26} + ( 33 + 3 \beta_{1} - 26 \beta_{2} - 13 \beta_{3} - 7 \beta_{4} - 18 \beta_{5} + \beta_{6} - 6 \beta_{7} + 8 \beta_{9} - 3 \beta_{10} + 6 \beta_{11} + \beta_{12} - 3 \beta_{13} + 2 \beta_{14} + 7 \beta_{15} + \beta_{17} ) q^{27} + ( 28 - 7 \beta_{2} - 7 \beta_{7} ) q^{28} + ( 47 - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 44 \beta_{5} - 6 \beta_{6} + 3 \beta_{8} + 6 \beta_{10} + 5 \beta_{12} + 6 \beta_{14} - 6 \beta_{15} + 5 \beta_{16} ) q^{29} + ( -49 - 3 \beta_{1} - 15 \beta_{2} + 29 \beta_{3} - 4 \beta_{4} + 30 \beta_{5} + 7 \beta_{6} + 2 \beta_{7} + \beta_{8} + 6 \beta_{9} - \beta_{10} - 6 \beta_{11} - 2 \beta_{12} - 7 \beta_{13} + \beta_{14} - 2 \beta_{15} - 9 \beta_{16} - 6 \beta_{17} ) q^{30} + ( 1 - \beta_{1} - \beta_{4} - 7 \beta_{5} + 12 \beta_{6} - \beta_{8} - 2 \beta_{10} + \beta_{11} - 9 \beta_{12} - 6 \beta_{13} - 9 \beta_{14} + 12 \beta_{15} - 12 \beta_{16} - 9 \beta_{17} ) q^{31} + ( -5 - 7 \beta_{1} + 20 \beta_{3} + 6 \beta_{4} + 99 \beta_{5} + 8 \beta_{6} - 16 \beta_{7} - \beta_{8} - 18 \beta_{9} - 24 \beta_{10} - \beta_{12} - 14 \beta_{13} - \beta_{14} + 24 \beta_{15} - 8 \beta_{16} - \beta_{17} ) q^{32} + ( 8 + 27 \beta_{2} - 3 \beta_{4} + 40 \beta_{5} + 2 \beta_{6} - \beta_{7} + 12 \beta_{8} + 10 \beta_{9} + 7 \beta_{10} + 6 \beta_{11} - 6 \beta_{12} - 4 \beta_{13} + 2 \beta_{14} + 6 \beta_{15} - 9 \beta_{16} - \beta_{17} ) q^{33} + ( 12 + 3 \beta_{1} + 38 \beta_{2} + 38 \beta_{3} - 7 \beta_{4} - 16 \beta_{5} - \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - 11 \beta_{10} - 5 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 3 \beta_{14} - 15 \beta_{15} + 4 \beta_{16} ) q^{34} + ( -21 + 7 \beta_{16} ) q^{35} + ( 89 + 9 \beta_{1} + 2 \beta_{2} - 22 \beta_{3} - 2 \beta_{4} - 148 \beta_{5} - 17 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} + 18 \beta_{9} - 3 \beta_{10} + 8 \beta_{12} + 9 \beta_{13} + 5 \beta_{14} - 25 \beta_{15} + 9 \beta_{16} + 6 \beta_{17} ) q^{36} + ( 19 - 2 \beta_{1} - 30 \beta_{2} - 5 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} - 5 \beta_{8} - 9 \beta_{9} + 17 \beta_{10} - 7 \beta_{11} + 3 \beta_{12} + 3 \beta_{13} - 3 \beta_{15} + 3 \beta_{16} - 3 \beta_{17} ) q^{37} + ( 4 - \beta_{1} - 10 \beta_{2} - 10 \beta_{3} + 6 \beta_{4} - 13 \beta_{5} + 6 \beta_{6} + 2 \beta_{7} - 28 \beta_{8} - 15 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} - 6 \beta_{12} - 2 \beta_{13} - 8 \beta_{14} + 13 \beta_{15} - 8 \beta_{16} ) q^{38} + ( -78 + 3 \beta_{1} + 17 \beta_{2} + 38 \beta_{3} + 2 \beta_{4} + 58 \beta_{5} + 4 \beta_{6} - 8 \beta_{7} + 5 \beta_{8} + 4 \beta_{9} + 6 \beta_{10} + 6 \beta_{11} - 11 \beta_{12} + \beta_{13} - 3 \beta_{14} + \beta_{15} - \beta_{17} ) q^{39} + ( 20 - 59 \beta_{3} - 5 \beta_{4} - 62 \beta_{5} - 16 \beta_{6} + 6 \beta_{7} + 12 \beta_{8} + 8 \beta_{9} + 28 \beta_{10} - 5 \beta_{11} + 17 \beta_{12} + 20 \beta_{13} + 6 \beta_{14} - 22 \beta_{15} + 16 \beta_{16} + 6 \beta_{17} ) q^{40} + ( -19 - 11 \beta_{1} - 40 \beta_{3} + 24 \beta_{4} + 52 \beta_{5} - 8 \beta_{6} - 2 \beta_{7} - 26 \beta_{8} - 15 \beta_{9} - 12 \beta_{10} - 9 \beta_{11} + 14 \beta_{13} + \beta_{14} - 6 \beta_{15} + 8 \beta_{16} + \beta_{17} ) q^{41} + ( 7 + 7 \beta_{2} + 7 \beta_{3} + 14 \beta_{5} - 7 \beta_{10} - 7 \beta_{13} ) q^{42} + ( 9 - 8 \beta_{1} + 35 \beta_{2} + 35 \beta_{3} + 9 \beta_{4} - 9 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} - 5 \beta_{8} - 9 \beta_{9} + 9 \beta_{10} + 6 \beta_{11} - 4 \beta_{12} + 2 \beta_{13} - 3 \beta_{14} - 2 \beta_{16} ) q^{43} + ( -68 - 6 \beta_{1} - 37 \beta_{2} + 9 \beta_{4} - 18 \beta_{5} + 16 \beta_{6} + 8 \beta_{7} + 15 \beta_{8} - 16 \beta_{9} + 5 \beta_{10} - 4 \beta_{11} - 8 \beta_{12} - 8 \beta_{13} + 8 \beta_{15} - 9 \beta_{16} - 2 \beta_{17} ) q^{44} + ( 23 - 9 \beta_{1} - 49 \beta_{2} - 17 \beta_{3} - \beta_{4} - 81 \beta_{5} + 11 \beta_{6} + 9 \beta_{7} - 14 \beta_{8} - 10 \beta_{9} - 4 \beta_{10} - 6 \beta_{11} - 5 \beta_{12} - 5 \beta_{13} - 10 \beta_{14} + 25 \beta_{15} - 24 \beta_{16} - 8 \beta_{17} ) q^{45} + ( -64 + 2 \beta_{1} - 34 \beta_{2} - 13 \beta_{4} + 22 \beta_{5} - 16 \beta_{6} + 15 \beta_{7} - 31 \beta_{8} + 16 \beta_{9} - 13 \beta_{10} - 4 \beta_{11} + 8 \beta_{12} + 8 \beta_{13} - 8 \beta_{15} + 37 \beta_{16} + 6 \beta_{17} ) q^{46} + ( 80 - 2 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} - 84 \beta_{5} + 9 \beta_{6} - 6 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} - 4 \beta_{10} - 4 \beta_{11} - 12 \beta_{12} + 6 \beta_{13} - 3 \beta_{14} + 30 \beta_{15} - 6 \beta_{16} ) q^{47} + ( -287 + 3 \beta_{1} + 57 \beta_{2} + 60 \beta_{3} - 15 \beta_{4} + 277 \beta_{5} - 18 \beta_{6} + 15 \beta_{7} - \beta_{8} + 21 \beta_{9} + 2 \beta_{10} + 15 \beta_{11} + 24 \beta_{12} - 6 \beta_{13} + 15 \beta_{15} + 9 \beta_{16} + 12 \beta_{17} ) q^{48} -49 \beta_{5} q^{49} + ( -1 + 15 \beta_{1} + 27 \beta_{3} - 18 \beta_{4} + 62 \beta_{5} - \beta_{6} + 26 \beta_{7} + 15 \beta_{8} + 14 \beta_{9} - \beta_{10} + 11 \beta_{11} - \beta_{12} - 14 \beta_{13} + 8 \beta_{14} - 27 \beta_{15} + \beta_{16} + 8 \beta_{17} ) q^{50} + ( 83 - 9 \beta_{1} - 13 \beta_{3} - 11 \beta_{4} - 23 \beta_{5} + 27 \beta_{6} + 10 \beta_{7} - 30 \beta_{8} - 9 \beta_{9} + \beta_{10} - 6 \beta_{11} - 25 \beta_{12} - 15 \beta_{13} - 16 \beta_{14} + 11 \beta_{15} - 36 \beta_{16} - 9 \beta_{17} ) q^{51} + ( -50 - 3 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} + 9 \beta_{4} + 56 \beta_{5} + 15 \beta_{6} - 9 \beta_{7} + 21 \beta_{8} - 3 \beta_{9} + 15 \beta_{10} - 6 \beta_{11} - 21 \beta_{12} + 9 \beta_{13} - 6 \beta_{14} + 22 \beta_{15} - 12 \beta_{16} ) q^{52} + ( -137 + 6 \beta_{1} + 47 \beta_{2} + 3 \beta_{4} + 12 \beta_{5} - 10 \beta_{6} - 14 \beta_{7} + 3 \beta_{8} + 13 \beta_{9} - 11 \beta_{10} + 7 \beta_{11} + 5 \beta_{12} + 5 \beta_{13} - 5 \beta_{15} + 8 \beta_{17} ) q^{53} + ( 189 - 3 \beta_{1} - 63 \beta_{2} - 27 \beta_{3} + 12 \beta_{4} - 330 \beta_{5} + 24 \beta_{6} + 15 \beta_{7} + 3 \beta_{8} - 18 \beta_{9} + 18 \beta_{10} + 3 \beta_{11} - 3 \beta_{12} - 18 \beta_{13} - 9 \beta_{14} - 12 \beta_{15} - 18 \beta_{16} - 6 \beta_{17} ) q^{54} + ( -8 + 8 \beta_{1} + 16 \beta_{2} + 14 \beta_{4} + \beta_{5} - 18 \beta_{6} + 7 \beta_{7} + 29 \beta_{8} + 10 \beta_{10} + 7 \beta_{11} + 9 \beta_{12} + 9 \beta_{13} - 9 \beta_{15} - 6 \beta_{16} + 3 \beta_{17} ) q^{55} + ( 56 - 28 \beta_{2} - 28 \beta_{3} - 63 \beta_{5} - 7 \beta_{7} - 7 \beta_{9} - 7 \beta_{10} - 7 \beta_{11} + 7 \beta_{13} + 7 \beta_{14} - 14 \beta_{15} + 7 \beta_{16} ) q^{56} + ( -122 + 57 \beta_{2} + 38 \beta_{3} - 10 \beta_{4} + 99 \beta_{5} - 14 \beta_{6} - 10 \beta_{7} + 18 \beta_{8} + 5 \beta_{10} - 9 \beta_{11} + 22 \beta_{12} + 11 \beta_{13} + 10 \beta_{14} - 23 \beta_{15} + 36 \beta_{16} + 6 \beta_{17} ) q^{57} + ( -17 + 13 \beta_{1} - 66 \beta_{3} - \beta_{4} - 33 \beta_{5} - 7 \beta_{6} + 10 \beta_{7} + 10 \beta_{8} - \beta_{9} - 9 \beta_{10} + 3 \beta_{11} + 20 \beta_{12} + 14 \beta_{13} - 17 \beta_{15} + 7 \beta_{16} ) q^{58} + ( 57 + 5 \beta_{1} + 27 \beta_{3} - 36 \beta_{4} + 96 \beta_{5} + 19 \beta_{6} + 37 \beta_{7} + 26 \beta_{8} + 41 \beta_{9} + 62 \beta_{10} + 5 \beta_{11} - 2 \beta_{12} - 10 \beta_{13} - 14 \beta_{14} - 18 \beta_{15} - 19 \beta_{16} - 14 \beta_{17} ) q^{59} + ( 370 - 27 \beta_{3} + 21 \beta_{4} - 121 \beta_{5} - 11 \beta_{6} - 8 \beta_{7} + 24 \beta_{8} - 10 \beta_{9} + 20 \beta_{10} - 6 \beta_{11} + 15 \beta_{12} + 13 \beta_{13} + 7 \beta_{14} - 51 \beta_{15} + 9 \beta_{16} + \beta_{17} ) q^{60} + ( 19 + 12 \beta_{1} + 51 \beta_{2} + 51 \beta_{3} - 8 \beta_{4} - 3 \beta_{5} - 11 \beta_{6} + 20 \beta_{7} + 24 \beta_{9} + 8 \beta_{10} + 8 \beta_{11} + 13 \beta_{12} - 20 \beta_{13} - 9 \beta_{14} + 8 \beta_{15} - 7 \beta_{16} ) q^{61} + ( -21 - 13 \beta_{1} - 25 \beta_{2} + 12 \beta_{4} - 16 \beta_{5} + 12 \beta_{6} - 33 \beta_{7} + 2 \beta_{8} - 23 \beta_{9} - 5 \beta_{10} - 20 \beta_{11} - 6 \beta_{12} - 6 \beta_{13} + 6 \beta_{15} + 12 \beta_{16} - 3 \beta_{17} ) q^{62} + ( -28 + 7 \beta_{3} + 14 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} - 7 \beta_{9} + 7 \beta_{14} + 7 \beta_{17} ) q^{63} + ( 338 - 18 \beta_{1} - 141 \beta_{2} + 24 \beta_{4} - 57 \beta_{5} + 74 \beta_{6} - 65 \beta_{7} + 45 \beta_{8} - 38 \beta_{9} - 2 \beta_{10} + \beta_{11} - 37 \beta_{12} - 37 \beta_{13} + 37 \beta_{15} - 29 \beta_{16} - 6 \beta_{17} ) q^{64} + ( 322 + \beta_{1} + 74 \beta_{2} + 74 \beta_{3} + 6 \beta_{4} - 283 \beta_{5} + 6 \beta_{6} + 7 \beta_{7} + 58 \beta_{8} + 33 \beta_{9} + 45 \beta_{10} + 6 \beta_{11} + 10 \beta_{12} - 7 \beta_{13} - 13 \beta_{14} + 17 \beta_{15} + 3 \beta_{16} ) q^{65} + ( -281 + 9 \beta_{1} - 58 \beta_{2} + 5 \beta_{3} + 32 \beta_{4} + 294 \beta_{5} + 13 \beta_{6} - 5 \beta_{7} - 13 \beta_{8} - 20 \beta_{9} + 7 \beta_{10} + 4 \beta_{12} + 13 \beta_{13} - 15 \beta_{14} + 34 \beta_{15} - 9 \beta_{16} - 10 \beta_{17} ) q^{66} + ( -34 - 21 \beta_{3} + 13 \beta_{4} + 53 \beta_{5} - \beta_{6} + 7 \beta_{7} - 39 \beta_{8} + 5 \beta_{9} - 20 \beta_{10} + 4 \beta_{11} + 23 \beta_{12} - 16 \beta_{13} + 9 \beta_{14} - 8 \beta_{15} + \beta_{16} + 9 \beta_{17} ) q^{67} + ( 35 - 5 \beta_{1} - 64 \beta_{3} - 9 \beta_{4} + 133 \beta_{5} - 8 \beta_{6} + 16 \beta_{7} + 34 \beta_{8} - 9 \beta_{9} + 18 \beta_{10} - 6 \beta_{11} + 8 \beta_{12} + 14 \beta_{13} + \beta_{14} - 24 \beta_{15} + 8 \beta_{16} + \beta_{17} ) q^{68} + ( 161 + 15 \beta_{1} - 60 \beta_{2} - 126 \beta_{3} - 31 \beta_{4} - 22 \beta_{5} - 26 \beta_{6} - 5 \beta_{7} + 5 \beta_{8} - 5 \beta_{9} - 39 \beta_{10} + 6 \beta_{11} - 2 \beta_{12} + 15 \beta_{13} + 27 \beta_{14} - 17 \beta_{15} + 36 \beta_{16} + 20 \beta_{17} ) q^{69} + ( 49 \beta_{2} + 49 \beta_{3} - 7 \beta_{4} + 14 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} + 21 \beta_{8} + 21 \beta_{9} + 7 \beta_{10} + 7 \beta_{11} + 14 \beta_{12} - 7 \beta_{13} - 7 \beta_{15} + 7 \beta_{16} ) q^{70} + ( -360 + 10 \beta_{1} - 80 \beta_{2} - 36 \beta_{4} + 43 \beta_{5} - 16 \beta_{6} - 14 \beta_{7} - 59 \beta_{8} + 45 \beta_{9} - 24 \beta_{10} + 12 \beta_{11} + 8 \beta_{12} + 8 \beta_{13} - 8 \beta_{15} - 10 \beta_{16} - 4 \beta_{17} ) q^{71} + ( -143 + 15 \beta_{1} + 60 \beta_{2} - 99 \beta_{3} - 19 \beta_{4} - 219 \beta_{5} - 39 \beta_{6} + 39 \beta_{7} + 5 \beta_{8} + 32 \beta_{9} - 20 \beta_{10} + 18 \beta_{11} + 10 \beta_{12} + 29 \beta_{13} + 8 \beta_{14} - 8 \beta_{15} + 33 \beta_{16} + 10 \beta_{17} ) q^{72} + ( 4 + 2 \beta_{1} + 86 \beta_{2} + 17 \beta_{4} - 5 \beta_{5} + 14 \beta_{6} + 21 \beta_{7} + 26 \beta_{8} + 4 \beta_{9} - 19 \beta_{10} + 11 \beta_{11} - 7 \beta_{12} - 7 \beta_{13} + 7 \beta_{15} - 35 \beta_{16} + 3 \beta_{17} ) q^{73} + ( 171 + \beta_{1} - 23 \beta_{2} - 23 \beta_{3} - 39 \beta_{4} - 202 \beta_{5} - 15 \beta_{6} + 6 \beta_{7} - 35 \beta_{8} + 8 \beta_{9} - 70 \beta_{10} + 5 \beta_{11} - 3 \beta_{12} - 6 \beta_{13} + 9 \beta_{14} - 24 \beta_{15} - 9 \beta_{16} ) q^{74} + ( -327 - 12 \beta_{1} - 34 \beta_{2} - 39 \beta_{3} + 30 \beta_{4} + 181 \beta_{5} + 9 \beta_{6} + 11 \beta_{7} - 30 \beta_{8} - 71 \beta_{9} - 26 \beta_{10} - 24 \beta_{11} - 42 \beta_{12} + 2 \beta_{13} + 5 \beta_{14} + 6 \beta_{15} - 18 \beta_{16} - 7 \beta_{17} ) q^{75} + ( 7 - 9 \beta_{1} + 48 \beta_{3} - \beta_{4} + 6 \beta_{5} + 25 \beta_{6} - 22 \beta_{7} - 9 \beta_{8} - 2 \beta_{9} - 7 \beta_{10} + 2 \beta_{11} - 26 \beta_{12} - 26 \beta_{13} - 12 \beta_{14} + 47 \beta_{15} - 25 \beta_{16} - 12 \beta_{17} ) q^{76} + ( 21 - 7 \beta_{1} - 14 \beta_{3} + 70 \beta_{5} - 14 \beta_{6} - 14 \beta_{7} + 14 \beta_{8} - 7 \beta_{9} + 14 \beta_{10} - 7 \beta_{11} + 14 \beta_{13} + 7 \beta_{14} + 14 \beta_{16} + 7 \beta_{17} ) q^{77} + ( 205 + 3 \beta_{1} - 3 \beta_{2} + 16 \beta_{3} + 18 \beta_{4} + 244 \beta_{5} - 20 \beta_{6} - 48 \beta_{7} - 11 \beta_{8} + 9 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} + 15 \beta_{12} + 29 \beta_{13} + 6 \beta_{15} + 45 \beta_{16} + 3 \beta_{17} ) q^{78} + ( -93 + 4 \beta_{1} - 57 \beta_{2} - 57 \beta_{3} + 36 \beta_{4} + 96 \beta_{5} - \beta_{6} + 4 \beta_{7} - 38 \beta_{8} - 33 \beta_{9} + 39 \beta_{10} - 19 \beta_{12} - 4 \beta_{13} - 3 \beta_{14} + 22 \beta_{15} - 23 \beta_{16} ) q^{79} + ( -498 + 24 \beta_{1} + 66 \beta_{2} - 69 \beta_{4} + 39 \beta_{5} - 36 \beta_{6} + 81 \beta_{7} - 60 \beta_{8} + 45 \beta_{9} + 51 \beta_{10} + 30 \beta_{11} + 18 \beta_{12} + 18 \beta_{13} - 18 \beta_{15} - 17 \beta_{16} + 3 \beta_{17} ) q^{80} + ( -18 - 9 \beta_{1} + 9 \beta_{2} + 90 \beta_{3} + 18 \beta_{4} - 96 \beta_{5} + 18 \beta_{6} - 27 \beta_{7} + 48 \beta_{8} - 45 \beta_{9} + 42 \beta_{10} - 9 \beta_{11} + 18 \beta_{12} + 9 \beta_{14} + 18 \beta_{15} + 45 \beta_{16} ) q^{81} + ( -445 + \beta_{1} - 6 \beta_{2} - 17 \beta_{4} + 74 \beta_{5} - 36 \beta_{6} + 45 \beta_{7} - 89 \beta_{8} + 57 \beta_{9} - 94 \beta_{10} - 16 \beta_{11} + 18 \beta_{12} + 18 \beta_{13} - 18 \beta_{15} + 27 \beta_{16} - 15 \beta_{17} ) q^{82} + ( 261 - 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 239 \beta_{5} - 13 \beta_{7} + 73 \beta_{8} + 25 \beta_{9} + 19 \beta_{10} - 8 \beta_{11} + 23 \beta_{12} + 13 \beta_{13} + 13 \beta_{14} + 7 \beta_{15} + 36 \beta_{16} ) q^{83} + ( 42 - 21 \beta_{2} - 28 \beta_{3} + 14 \beta_{4} + 21 \beta_{5} + 7 \beta_{6} - 28 \beta_{7} + 14 \beta_{8} - 21 \beta_{9} - 7 \beta_{10} + 7 \beta_{12} - 7 \beta_{13} + 7 \beta_{14} + 7 \beta_{15} ) q^{84} + ( -93 - 11 \beta_{1} - 35 \beta_{3} + 57 \beta_{4} + 82 \beta_{5} - 17 \beta_{6} - 17 \beta_{7} - 62 \beta_{8} - 53 \beta_{9} - 95 \beta_{10} - 5 \beta_{11} - 56 \beta_{12} - 2 \beta_{13} + 18 \beta_{14} + 17 \beta_{16} + 18 \beta_{17} ) q^{85} + ( -49 - 17 \beta_{1} - 42 \beta_{3} + 45 \beta_{4} + 419 \beta_{5} + 10 \beta_{6} - 56 \beta_{7} - 68 \beta_{8} - 15 \beta_{9} - 24 \beta_{10} - 12 \beta_{11} - 8 \beta_{12} + 14 \beta_{13} - 17 \beta_{14} + 66 \beta_{15} - 10 \beta_{16} - 17 \beta_{17} ) q^{86} + ( 346 - 15 \beta_{1} + 70 \beta_{2} + 21 \beta_{3} + 48 \beta_{4} - 174 \beta_{5} + 15 \beta_{6} + 22 \beta_{7} + 29 \beta_{8} - 70 \beta_{9} + 40 \beta_{10} - 18 \beta_{11} + 3 \beta_{12} + 7 \beta_{13} - 14 \beta_{14} - 27 \beta_{15} - 17 \beta_{17} ) q^{87} + ( 309 - 18 \beta_{1} + 31 \beta_{2} + 31 \beta_{3} + 8 \beta_{4} - 286 \beta_{5} - 4 \beta_{6} - 14 \beta_{7} + 66 \beta_{8} + 15 \beta_{9} + 31 \beta_{10} + 4 \beta_{11} - 37 \beta_{12} + 14 \beta_{13} + 18 \beta_{14} - 19 \beta_{15} - 23 \beta_{16} ) q^{88} + ( -209 - 16 \beta_{1} - 53 \beta_{2} - 21 \beta_{4} + 17 \beta_{5} + 4 \beta_{6} - 91 \beta_{7} - 70 \beta_{8} + 3 \beta_{9} - 33 \beta_{10} - 30 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} + 30 \beta_{16} - 11 \beta_{17} ) q^{89} + ( 207 - 36 \beta_{1} + 37 \beta_{2} + 104 \beta_{3} + 14 \beta_{4} - 489 \beta_{5} + 55 \beta_{6} - 24 \beta_{7} - 66 \beta_{8} - 49 \beta_{9} - 51 \beta_{10} - 27 \beta_{11} - 62 \beta_{12} - 48 \beta_{13} - 16 \beta_{14} + 16 \beta_{15} - 72 \beta_{16} - 26 \beta_{17} ) q^{90} + ( -14 + 7 \beta_{2} + 21 \beta_{5} - 14 \beta_{6} - 21 \beta_{8} + 14 \beta_{9} - 28 \beta_{10} - 7 \beta_{11} + 7 \beta_{12} + 7 \beta_{13} - 7 \beta_{15} - 7 \beta_{16} ) q^{91} + ( 1 + 22 \beta_{1} + 181 \beta_{2} + 181 \beta_{3} - 60 \beta_{4} + 3 \beta_{5} - 48 \beta_{6} + 53 \beta_{7} - 38 \beta_{8} + 64 \beta_{9} - 56 \beta_{10} + 31 \beta_{11} + 41 \beta_{12} - 53 \beta_{13} - 5 \beta_{14} - 17 \beta_{15} - 12 \beta_{16} ) q^{92} + ( -25 - 9 \beta_{1} - 27 \beta_{2} + 15 \beta_{3} - 75 \beta_{4} + 118 \beta_{5} + 21 \beta_{6} - 48 \beta_{7} + 18 \beta_{8} + 27 \beta_{9} + 23 \beta_{10} + 27 \beta_{11} - 24 \beta_{12} - 12 \beta_{13} - 6 \beta_{14} + 30 \beta_{15} - 18 \beta_{16} + 9 \beta_{17} ) q^{93} + ( 24 - 24 \beta_{1} + 131 \beta_{3} + 3 \beta_{4} - 27 \beta_{5} + 39 \beta_{6} + 10 \beta_{7} - 63 \beta_{8} + 39 \beta_{9} + 45 \beta_{10} - 3 \beta_{11} - 51 \beta_{12} - 36 \beta_{13} - 21 \beta_{14} + 29 \beta_{15} - 39 \beta_{16} - 21 \beta_{17} ) q^{94} + ( 104 + 42 \beta_{1} + 29 \beta_{3} - 93 \beta_{4} + 319 \beta_{5} - 25 \beta_{6} - \beta_{7} + 102 \beta_{8} + 86 \beta_{9} + 119 \beta_{10} + 20 \beta_{11} + 34 \beta_{12} + 4 \beta_{13} + 23 \beta_{14} - 24 \beta_{15} + 25 \beta_{16} + 23 \beta_{17} ) q^{95} + ( -96 - 21 \beta_{1} + 258 \beta_{3} + 39 \beta_{4} + 903 \beta_{5} + 51 \beta_{6} - 45 \beta_{7} - 21 \beta_{8} - 66 \beta_{9} - 39 \beta_{10} + 3 \beta_{11} - 9 \beta_{12} - 51 \beta_{13} - 27 \beta_{14} + 153 \beta_{15} - 63 \beta_{16} - 18 \beta_{17} ) q^{96} + ( 176 + 11 \beta_{1} - 139 \beta_{2} - 139 \beta_{3} + 20 \beta_{4} - 186 \beta_{5} - 27 \beta_{6} - 3 \beta_{7} - 34 \beta_{8} - 30 \beta_{9} + 10 \beta_{10} - 14 \beta_{11} + 75 \beta_{12} + 3 \beta_{13} + 30 \beta_{14} - 91 \beta_{15} + 78 \beta_{16} ) q^{97} + ( -49 + 49 \beta_{2} ) q^{98} + ( 106 + 3 \beta_{1} + 73 \beta_{2} + 125 \beta_{3} + \beta_{4} - 198 \beta_{5} - 35 \beta_{6} - 36 \beta_{7} + 2 \beta_{8} - 29 \beta_{9} - 32 \beta_{10} - 15 \beta_{11} + 23 \beta_{12} + 50 \beta_{13} - 8 \beta_{14} - 16 \beta_{15} + 42 \beta_{16} - \beta_{17} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 6q^{2} + 9q^{3} - 36q^{4} + 24q^{5} - 63q^{7} - 150q^{8} + 63q^{9} + O(q^{10}) \) \( 18q + 6q^{2} + 9q^{3} - 36q^{4} + 24q^{5} - 63q^{7} - 150q^{8} + 63q^{9} + 111q^{11} - 18q^{13} + 42q^{14} - 36q^{15} - 144q^{16} - 546q^{17} - 45q^{18} + 90q^{19} + 402q^{20} - 63q^{21} + 162q^{22} + 312q^{23} - 36q^{24} - 279q^{25} + 102q^{26} + 432q^{27} + 504q^{28} + 378q^{29} - 864q^{30} - 18q^{31} + 891q^{32} + 513q^{33} + 324q^{34} - 336q^{35} + 414q^{36} - 72q^{37} + 147q^{38} - 810q^{39} - 405q^{40} + 477q^{41} + 315q^{42} + 171q^{43} - 1896q^{44} - 720q^{45} - 756q^{46} + 654q^{47} - 2709q^{48} - 441q^{49} + 429q^{50} + 1341q^{51} - 747q^{52} - 1896q^{53} - 108q^{54} - 432q^{55} + 525q^{56} - 1143q^{57} - 297q^{58} + 957q^{59} + 5400q^{60} + 198q^{61} - 600q^{62} - 504q^{63} + 4770q^{64} + 2478q^{65} - 2646q^{66} + 333q^{67} + 1443q^{68} + 3366q^{69} - 5652q^{71} - 3681q^{72} + 306q^{73} + 2100q^{74} - 4113q^{75} + 144q^{76} + 777q^{77} + 6336q^{78} - 1152q^{79} - 8418q^{80} - 1917q^{81} - 6048q^{82} + 1890q^{83} + 1008q^{84} + 648q^{85} + 3837q^{86} + 4212q^{87} + 2268q^{88} - 2604q^{89} - 135q^{90} + 252q^{91} + 987q^{92} + 378q^{93} - 324q^{94} + 3144q^{95} + 5643q^{96} + 1737q^{97} - 588q^{98} + 720q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 3 x^{17} + 6 x^{16} - 23 x^{15} - 6 x^{14} + 255 x^{13} - 56 x^{12} - 81 x^{11} + 5832 x^{10} - 32373 x^{9} + 52488 x^{8} - 6561 x^{7} - 40824 x^{6} + 1673055 x^{5} - 354294 x^{4} - 12223143 x^{3} + 28697814 x^{2} - 129140163 x + 387420489\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(35137211695 \nu^{17} - 1622378379288 \nu^{16} - 1349491982943 \nu^{15} + 12548313313075 \nu^{14} + 1945915263777 \nu^{13} + 103812466017549 \nu^{12} - 114437325703658 \nu^{11} - 845973725242347 \nu^{10} + 672217844702184 \nu^{9} - 1991355894202257 \nu^{8} - 21389655609235926 \nu^{7} + 80555477706327447 \nu^{6} - 144556973598898449 \nu^{5} - 85601791788605451 \nu^{4} - 1850113016931392397 \nu^{3} - 5210825550067431396 \nu^{2} + 5904511107602186655 \nu + 51846424989580967652\)\()/ 4707313149254413698 \)
\(\beta_{2}\)\(=\)\((\)\(-33291362 \nu^{17} + 49543827 \nu^{16} - 1047096165 \nu^{15} + 321711301 \nu^{14} + 2245610217 \nu^{13} - 14517513141 \nu^{12} + 70878228223 \nu^{11} - 117898249752 \nu^{10} - 884932522146 \nu^{9} + 1145459391516 \nu^{8} - 6854232029553 \nu^{7} + 5013864571491 \nu^{6} + 59917096469565 \nu^{5} - 83734083298719 \nu^{4} - 144988358417469 \nu^{3} - 519422332644837 \nu^{2} - 5923936985556078 \nu + 11267435141907696\)\()/ 3217575631752846 \)
\(\beta_{3}\)\(=\)\((\)\(-100884080039 \nu^{17} + 189431315142 \nu^{16} + 536203636539 \nu^{15} + 1055016162484 \nu^{14} + 4015788303360 \nu^{13} - 2994552155139 \nu^{12} - 81562840503560 \nu^{11} + 182148515794692 \nu^{10} - 461602825792911 \nu^{9} + 1769313657466872 \nu^{8} + 9615103941684375 \nu^{7} - 172487360299626 \nu^{6} - 43193748914859267 \nu^{5} - 44639606072364003 \nu^{4} - 545745309113024532 \nu^{3} + 1976707349836874793 \nu^{2} + 3676744989816290505 \nu + 3413597358957560028\)\()/ 4707313149254413698 \)
\(\beta_{4}\)\(=\)\((\)\(-934928795 \nu^{17} + 1529236356 \nu^{16} - 4688064201 \nu^{15} + 16147747558 \nu^{14} + 65485822377 \nu^{13} - 138335706870 \nu^{12} - 80550057695 \nu^{11} + 258976968156 \nu^{10} - 5718092158647 \nu^{9} + 23057641759086 \nu^{8} + 5135240793609 \nu^{7} + 306432172764 \nu^{6} + 275726496456693 \nu^{5} - 1049359101957288 \nu^{4} - 2273118270370209 \nu^{3} + 10006703489641230 \nu^{2} + 98984780345132307 \nu + 102514065773416128\)\()/ 36490799606623362 \)
\(\beta_{5}\)\(=\)\((\)\(-934928795 \nu^{17} + 1529236356 \nu^{16} - 4688064201 \nu^{15} + 16147747558 \nu^{14} + 65485822377 \nu^{13} - 138335706870 \nu^{12} - 80550057695 \nu^{11} + 258976968156 \nu^{10} - 5718092158647 \nu^{9} + 23057641759086 \nu^{8} + 5135240793609 \nu^{7} + 306432172764 \nu^{6} + 275726496456693 \nu^{5} - 1049359101957288 \nu^{4} - 2273118270370209 \nu^{3} + 10006703489641230 \nu^{2} - 10487618474737779 \nu + 139004865380039490\)\()/ 36490799606623362 \)
\(\beta_{6}\)\(=\)\((\)\(43671897251 \nu^{17} + 431314232919 \nu^{16} - 1386353129901 \nu^{15} - 533363823130 \nu^{14} + 717615664749 \nu^{13} - 12844962658524 \nu^{12} + 95749953197663 \nu^{11} + 139583945033697 \nu^{10} - 691871012613297 \nu^{9} + 4342883091564006 \nu^{8} - 8848681826373504 \nu^{7} - 26307119375892954 \nu^{6} + 36448106791784427 \nu^{5} + 121992553145083806 \nu^{4} + 207453679969747347 \nu^{3} + 2741951792703581982 \nu^{2} - 8641986127456840182 \nu + 2485512948659080068\)\()/ 1569104383084804566 \)
\(\beta_{7}\)\(=\)\((\)\(-125740550 \nu^{17} - 239756565 \nu^{16} - 3019695132 \nu^{15} + 6332440258 \nu^{14} + 11007706557 \nu^{13} - 9903504579 \nu^{12} + 11411973058 \nu^{11} - 712049569587 \nu^{10} - 2496324550491 \nu^{9} + 4186029859569 \nu^{8} - 8137056375558 \nu^{7} + 28169477094393 \nu^{6} + 66801823059249 \nu^{5} - 415051571753154 \nu^{4} - 232601207419728 \nu^{3} - 615245817374109 \nu^{2} - 8779936681283850 \nu + 39991178219510790\)\()/ 3217575631752846 \)
\(\beta_{8}\)\(=\)\((\)\(4601274938 \nu^{17} + 11439252651 \nu^{16} - 13681731984 \nu^{15} + 20748409853 \nu^{14} - 463596833694 \nu^{13} - 594792094989 \nu^{12} + 3477392688962 \nu^{11} + 1802148287787 \nu^{10} + 19842257298204 \nu^{9} + 5431414715595 \nu^{8} - 381044608549578 \nu^{7} - 168840466295661 \nu^{6} - 196116116733540 \nu^{5} + 253570637064879 \nu^{4} + 26702491649963004 \nu^{3} + 5132151750505509 \nu^{2} - 138134461886727678 \nu - 420516095503084947\)\()/ 109472398819870086 \)
\(\beta_{9}\)\(=\)\((\)\(2891721292 \nu^{17} - 1235289351 \nu^{16} + 11378779980 \nu^{15} - 163481001263 \nu^{14} - 234727585188 \nu^{13} + 260382503775 \nu^{12} - 630293264978 \nu^{11} + 1055739246777 \nu^{10} + 15908332205700 \nu^{9} - 139565708397621 \nu^{8} + 22618147747302 \nu^{7} - 712370457816075 \nu^{6} - 1268758083027618 \nu^{5} + 6763714696640529 \nu^{4} + 1990076329014156 \nu^{3} - 39021679072607823 \nu^{2} + 44180755144094592 \nu - 947626847437202775\)\()/ 36490799606623362 \)
\(\beta_{10}\)\(=\)\((\)\(9425808251 \nu^{17} - 6860997375 \nu^{16} + 18029993601 \nu^{15} - 106282700527 \nu^{14} - 312915284751 \nu^{13} + 935677307391 \nu^{12} + 2808500612789 \nu^{11} + 1961104296717 \nu^{10} + 47182173300999 \nu^{9} - 172379626590501 \nu^{8} + 34808846139873 \nu^{7} - 217977136315947 \nu^{6} + 319604025285387 \nu^{5} + 9869764798530405 \nu^{4} + 17180114645769165 \nu^{3} - 58102003441941615 \nu^{2} + 49347480328799157 \nu - 988753088741797251\)\()/ 109472398819870086 \)
\(\beta_{11}\)\(=\)\((\)\(64909219150 \nu^{17} + 559430453634 \nu^{16} + 1212366559218 \nu^{15} + 3983935173979 \nu^{14} + 2582756456304 \nu^{13} + 5535038007021 \nu^{12} + 67701024923002 \nu^{11} + 274492934463807 \nu^{10} + 739083059269038 \nu^{9} + 4163084466819300 \nu^{8} + 4367746858671216 \nu^{7} + 9641924605489086 \nu^{6} + 48414173149570887 \nu^{5} + 76918468530814974 \nu^{4} + 829058515550406477 \nu^{3} + 2343923519230650384 \nu^{2} + 3373252182885495105 \nu + 10876885373246469840\)\()/ 672473307036344814 \)
\(\beta_{12}\)\(=\)\((\)\(471729716684 \nu^{17} - 1671338975199 \nu^{16} + 124452258888 \nu^{15} - 14759801768083 \nu^{14} - 72466589790912 \nu^{13} + 72039367299345 \nu^{12} - 169488840757678 \nu^{11} - 440347159243647 \nu^{10} + 3519525718901412 \nu^{9} - 18853037614659897 \nu^{8} - 14334366612763584 \nu^{7} - 10610125980958341 \nu^{6} - 227700912741884154 \nu^{5} + 884632689717653907 \nu^{4} - 44836319963101518 \nu^{3} - 9832279729966328421 \nu^{2} - 3671517536651688012 \nu - 93408456005016574797\)\()/ 4707313149254413698 \)
\(\beta_{13}\)\(=\)\((\)\(-573834086020 \nu^{17} + 1839711567324 \nu^{16} - 4474830173874 \nu^{15} - 8020513339717 \nu^{14} + 12311447948295 \nu^{13} - 126081051918753 \nu^{12} - 135456296173342 \nu^{11} + 1734985354078341 \nu^{10} - 7456418123673024 \nu^{9} + 8751778489906551 \nu^{8} - 16529150660190939 \nu^{7} - 167591107338217443 \nu^{6} + 159999377948694036 \nu^{5} + 367157627300670606 \nu^{4} - 1644288303851142606 \nu^{3} + 9221725341409284657 \nu^{2} - 39686251108195466496 \nu - 24635093532100855029\)\()/ 4707313149254413698 \)
\(\beta_{14}\)\(=\)\((\)\(-690830509814 \nu^{17} - 685272601038 \nu^{16} - 6766420033119 \nu^{15} + 23031052498687 \nu^{14} + 17954398393824 \nu^{13} + 34729375256076 \nu^{12} + 282348177152974 \nu^{11} - 1828057379869431 \nu^{10} - 3885674808558906 \nu^{9} + 16748752246119036 \nu^{8} - 4924713931955226 \nu^{7} + 76743495374118219 \nu^{6} + 87572289239205363 \nu^{5} - 1080413715196985040 \nu^{4} - 433274254296064164 \nu^{3} - 2290783118194996398 \nu^{2} - 3474228106562651595 \nu + 151773268929646401234\)\()/ 4707313149254413698 \)
\(\beta_{15}\)\(=\)\((\)\(705247659434 \nu^{17} - 1068128624931 \nu^{16} + 2710590961395 \nu^{15} - 9283779652789 \nu^{14} - 48907930159686 \nu^{13} + 39166828660761 \nu^{12} - 66330624557083 \nu^{11} - 572242087209636 \nu^{10} + 3945431785070952 \nu^{9} - 15071541223029687 \nu^{8} - 3808088466554214 \nu^{7} - 32734770644266632 \nu^{6} - 204939247335550506 \nu^{5} + 660305229890229171 \nu^{4} + 1448002762893790344 \nu^{3} - 9136310861396200140 \nu^{2} + 1449788136648507387 \nu - 96897859370502363963\)\()/ 4707313149254413698 \)
\(\beta_{16}\)\(=\)\((\)\(193556779 \nu^{17} + 276829698 \nu^{16} + 2133154329 \nu^{15} + 3107835478 \nu^{14} + 2588401863 \nu^{13} + 16790388852 \nu^{12} + 67356098923 \nu^{11} + 88562160132 \nu^{10} + 1910169643113 \nu^{9} + 1190126292660 \nu^{8} + 6447970888551 \nu^{7} + 16531719218676 \nu^{6} - 22075011824337 \nu^{5} + 214183935388710 \nu^{4} + 1123070492704041 \nu^{3} + 1359093202135182 \nu^{2} + 8119839759351435 \nu - 2931149838578904\)\()/ 1072525210584282 \)
\(\beta_{17}\)\(=\)\((\)\(-806254553 \nu^{17} - 655552110 \nu^{16} - 2598929994 \nu^{15} - 8913639005 \nu^{14} + 31465705485 \nu^{13} - 104208855882 \nu^{12} - 439901079998 \nu^{11} - 202231748547 \nu^{10} - 6205309088742 \nu^{9} + 584782971849 \nu^{8} + 10170952652862 \nu^{7} - 93781849873482 \nu^{6} + 58512572203329 \nu^{5} - 780862350400713 \nu^{4} - 4348227918448890 \nu^{3} + 5719803142312638 \nu^{2} - 14966129314477557 \nu + 13646186182687446\)\()/ 3217575631752846 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{5} + \beta_{4} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{17} + \beta_{16} - 2 \beta_{15} + \beta_{12} + \beta_{10} + 2 \beta_{8} + \beta_{7} - 5 \beta_{5} - \beta_{2} + 4\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{17} + \beta_{16} - 2 \beta_{15} + 2 \beta_{13} - \beta_{11} - 3 \beta_{10} - 2 \beta_{9} - \beta_{8} + 4 \beta_{7} - \beta_{6} - 19 \beta_{5} + \beta_{4} - 13 \beta_{3} - 13 \beta_{2} - 3 \beta_{1} + 19\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(15 \beta_{16} - 3 \beta_{15} + 3 \beta_{14} + 9 \beta_{13} + 15 \beta_{12} - 3 \beta_{11} + \beta_{9} - 12 \beta_{8} - 3 \beta_{6} + 33 \beta_{5} + 8 \beta_{4} + 3 \beta_{3} + 30 \beta_{2} + 5\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(17 \beta_{17} - 11 \beta_{16} - 4 \beta_{15} - 4 \beta_{14} - 6 \beta_{13} + 23 \beta_{12} + 27 \beta_{11} + 28 \beta_{10} + 9 \beta_{9} - 16 \beta_{8} + 10 \beta_{7} - 11 \beta_{6} + 95 \beta_{5} + 18 \beta_{4} - 91 \beta_{3} + 21 \beta_{2} + 3 \beta_{1} - 202\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-38 \beta_{17} + 37 \beta_{16} - 120 \beta_{15} - 56 \beta_{14} - 22 \beta_{13} + 92 \beta_{12} - 29 \beta_{11} - 61 \beta_{10} - 37 \beta_{9} - 36 \beta_{8} + 79 \beta_{7} - 11 \beta_{6} - 128 \beta_{5} - 15 \beta_{4} + 183 \beta_{3} + 178 \beta_{2} - 47 \beta_{1} - 546\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-60 \beta_{17} + 102 \beta_{15} + 99 \beta_{14} - 6 \beta_{13} + 60 \beta_{12} - 57 \beta_{11} + 534 \beta_{10} - 118 \beta_{9} - 174 \beta_{8} + 39 \beta_{7} - 24 \beta_{6} + 1204 \beta_{5} - 201 \beta_{4} + 840 \beta_{3} - 123 \beta_{2} - 231 \beta_{1} + 231\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(-420 \beta_{17} - 683 \beta_{16} + 1233 \beta_{15} - 143 \beta_{14} - 471 \beta_{13} - 210 \beta_{12} + 78 \beta_{11} - 1481 \beta_{10} - 804 \beta_{9} - 2143 \beta_{8} - 767 \beta_{7} + 1148 \beta_{6} - 489 \beta_{5} + 648 \beta_{4} + 196 \beta_{3} + 154 \beta_{2} - 315 \beta_{1} - 9930\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(-81 \beta_{17} - 389 \beta_{16} - 475 \beta_{15} + 32 \beta_{14} - 1753 \beta_{13} + 1165 \beta_{12} - 3 \beta_{11} + 508 \beta_{10} + 993 \beta_{9} + 718 \beta_{8} - 2459 \beta_{7} + 846 \beta_{6} + 30000 \beta_{5} - 3634 \beta_{4} - 6656 \beta_{3} - 5226 \beta_{2} + 290 \beta_{1} - 19318\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(-5367 \beta_{17} - 4704 \beta_{16} + 216 \beta_{15} - 2235 \beta_{14} + 3786 \beta_{13} - 4575 \beta_{12} - 3507 \beta_{11} + 8382 \beta_{10} - 12681 \beta_{9} - 2808 \beta_{8} - 5412 \beta_{7} - 306 \beta_{6} - 19699 \beta_{5} + 3625 \beta_{4} - 600 \beta_{3} - 18474 \beta_{2} - 2061 \beta_{1} + 28210\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(3529 \beta_{17} + 12574 \beta_{16} - 5303 \beta_{15} + 18801 \beta_{14} + 5103 \beta_{13} - 11825 \beta_{12} - 10836 \beta_{11} + 28099 \beta_{10} - 9198 \beta_{9} + 18956 \beta_{8} - 35990 \beta_{7} - 1602 \beta_{6} - 121109 \beta_{5} + 1314 \beta_{4} + 52056 \beta_{3} + 73709 \beta_{2} - 2457 \beta_{1} - 2174\)\()/3\)
\(\nu^{12}\)\(=\)\((\)\(1628 \beta_{17} - 86318 \beta_{16} - 46361 \beta_{15} + 4536 \beta_{14} - 34837 \beta_{13} - 52011 \beta_{12} + 63512 \beta_{11} - 45273 \beta_{10} + 98305 \beta_{9} - 42247 \beta_{8} + 36967 \beta_{7} + 31337 \beta_{6} - 411121 \beta_{5} - 101465 \beta_{4} - 78295 \beta_{3} - 161473 \beta_{2} + 54465 \beta_{1} - 238958\)\()/3\)
\(\nu^{13}\)\(=\)\((\)\(-86211 \beta_{17} + 58911 \beta_{16} - 11847 \beta_{15} - 77784 \beta_{14} + 5229 \beta_{13} - 193962 \beta_{12} - 111702 \beta_{11} - 59526 \beta_{10} + 55990 \beta_{9} - 232698 \beta_{8} + 25092 \beta_{7} + 49281 \beta_{6} + 669291 \beta_{5} - 148294 \beta_{4} - 446757 \beta_{3} - 244464 \beta_{2} + 40779 \beta_{1} - 993532\)\()/3\)
\(\nu^{14}\)\(=\)\((\)\(224765 \beta_{17} + 413332 \beta_{16} + 244841 \beta_{15} + 367079 \beta_{14} + 614037 \beta_{13} + 5495 \beta_{12} + 31023 \beta_{11} + 1313254 \beta_{10} - 44325 \beta_{9} + 1176194 \beta_{8} + 41995 \beta_{7} - 727589 \beta_{6} + 2621453 \beta_{5} - 76500 \beta_{4} + 323585 \beta_{3} - 147849 \beta_{2} + 499350 \beta_{1} - 2927218\)\()/3\)
\(\nu^{15}\)\(=\)\((\)\(1219543 \beta_{17} + 340426 \beta_{16} + 1357989 \beta_{15} + 184588 \beta_{14} + 248207 \beta_{13} - 1151323 \beta_{12} + 975400 \beta_{11} - 3438835 \beta_{10} + 183860 \beta_{9} - 252792 \beta_{8} - 1634744 \beta_{7} - 716753 \beta_{6} - 16873238 \beta_{5} - 425715 \beta_{4} + 341013 \beta_{3} + 7304632 \beta_{2} + 959011 \beta_{1} + 3716139\)\()/3\)
\(\nu^{16}\)\(=\)\((\)\(-1012533 \beta_{17} - 2860344 \beta_{16} - 1287222 \beta_{15} - 1252890 \beta_{14} - 3570288 \beta_{13} - 1167861 \beta_{12} + 4068258 \beta_{11} - 6157068 \beta_{10} + 15450839 \beta_{9} + 6536382 \beta_{8} + 4417707 \beta_{7} - 1419648 \beta_{6} + 42316540 \beta_{5} - 8551101 \beta_{4} + 12209412 \beta_{3} + 9273855 \beta_{2} + 3544950 \beta_{1} + 3248052\)\()/3\)
\(\nu^{17}\)\(=\)\((\)\(-4016220 \beta_{17} - 10996082 \beta_{16} + 21339252 \beta_{15} - 12737096 \beta_{14} - 8017464 \beta_{13} - 13475010 \beta_{12} + 420504 \beta_{11} + 21820765 \beta_{10} - 9926670 \beta_{9} - 5808007 \beta_{8} + 14328268 \beta_{7} + 13130960 \beta_{6} + 601845 \beta_{5} + 12122244 \beta_{4} + 51109504 \beta_{3} - 142568 \beta_{2} + 1404576 \beta_{1} + 150484434\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(1\) \(-\beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
0.960810 + 2.84198i
−2.90795 0.737429i
−0.0718325 2.99914i
−0.831471 + 2.88247i
2.85089 0.934028i
2.81021 + 1.05012i
−2.61694 + 1.46684i
−1.07135 2.80218i
2.37763 + 1.82944i
0.960810 2.84198i
−2.90795 + 0.737429i
−0.0718325 + 2.99914i
−0.831471 2.88247i
2.85089 + 0.934028i
2.81021 1.05012i
−2.61694 1.46684i
−1.07135 + 2.80218i
2.37763 1.82944i
−2.32254 4.02275i 3.90244 3.43088i −6.78835 + 11.7578i 8.24268 14.2767i −22.8651 7.73020i −3.50000 6.06218i 25.9042 3.45809 26.7776i −76.5757
22.2 −1.61753 2.80164i −5.00056 1.41222i −1.23279 + 2.13526i −4.95111 + 8.57557i 4.13202 + 16.2941i −3.50000 6.06218i −17.9041 23.0113 + 14.1238i 32.0342
22.3 −1.21836 2.11026i −2.70508 + 4.43650i 1.03120 1.78609i 2.61762 4.53384i 12.6579 + 0.303171i −3.50000 6.06218i −24.5192 −12.3651 24.0022i −12.7568
22.4 −0.231183 0.400421i 1.24909 5.04379i 3.89311 6.74306i −2.26496 + 3.92303i −2.30841 + 0.665877i −3.50000 6.06218i −7.29902 −23.8796 12.6003i 2.09449
22.5 0.377604 + 0.654029i 3.46745 + 3.86999i 3.71483 6.43428i 6.62462 11.4742i −1.22176 + 3.72913i −3.50000 6.06218i 11.6526 −2.95361 + 26.8380i 10.0059
22.6 1.32666 + 2.29785i 5.12474 + 0.858536i 0.479932 0.831266i −10.0300 + 17.3725i 4.82601 + 12.9148i −3.50000 6.06218i 23.7734 25.5258 + 8.79954i −53.2258
22.7 1.66614 + 2.88585i −2.65509 4.46660i −1.55207 + 2.68826i 8.37356 14.5034i 8.46616 15.1042i −3.50000 6.06218i 16.3144 −12.9010 + 23.7184i 55.8062
22.8 2.21232 + 3.83185i −4.03378 + 3.27546i −5.78871 + 10.0263i −1.92707 + 3.33779i −21.4751 8.21047i −3.50000 6.06218i −15.8288 5.54273 26.4250i −17.0532
22.9 2.80688 + 4.86166i 5.15079 0.685073i −11.7571 + 20.3640i 5.31469 9.20530i 17.7882 + 23.1185i −3.50000 6.06218i −87.0935 26.0614 7.05734i 59.6707
43.1 −2.32254 + 4.02275i 3.90244 + 3.43088i −6.78835 11.7578i 8.24268 + 14.2767i −22.8651 + 7.73020i −3.50000 + 6.06218i 25.9042 3.45809 + 26.7776i −76.5757
43.2 −1.61753 + 2.80164i −5.00056 + 1.41222i −1.23279 2.13526i −4.95111 8.57557i 4.13202 16.2941i −3.50000 + 6.06218i −17.9041 23.0113 14.1238i 32.0342
43.3 −1.21836 + 2.11026i −2.70508 4.43650i 1.03120 + 1.78609i 2.61762 + 4.53384i 12.6579 0.303171i −3.50000 + 6.06218i −24.5192 −12.3651 + 24.0022i −12.7568
43.4 −0.231183 + 0.400421i 1.24909 + 5.04379i 3.89311 + 6.74306i −2.26496 3.92303i −2.30841 0.665877i −3.50000 + 6.06218i −7.29902 −23.8796 + 12.6003i 2.09449
43.5 0.377604 0.654029i 3.46745 3.86999i 3.71483 + 6.43428i 6.62462 + 11.4742i −1.22176 3.72913i −3.50000 + 6.06218i 11.6526 −2.95361 26.8380i 10.0059
43.6 1.32666 2.29785i 5.12474 0.858536i 0.479932 + 0.831266i −10.0300 17.3725i 4.82601 12.9148i −3.50000 + 6.06218i 23.7734 25.5258 8.79954i −53.2258
43.7 1.66614 2.88585i −2.65509 + 4.46660i −1.55207 2.68826i 8.37356 + 14.5034i 8.46616 + 15.1042i −3.50000 + 6.06218i 16.3144 −12.9010 23.7184i 55.8062
43.8 2.21232 3.83185i −4.03378 3.27546i −5.78871 10.0263i −1.92707 3.33779i −21.4751 + 8.21047i −3.50000 + 6.06218i −15.8288 5.54273 + 26.4250i −17.0532
43.9 2.80688 4.86166i 5.15079 + 0.685073i −11.7571 20.3640i 5.31469 + 9.20530i 17.7882 23.1185i −3.50000 + 6.06218i −87.0935 26.0614 + 7.05734i 59.6707
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.4.f.c 18
3.b odd 2 1 189.4.f.c 18
9.c even 3 1 inner 63.4.f.c 18
9.c even 3 1 567.4.a.j 9
9.d odd 6 1 189.4.f.c 18
9.d odd 6 1 567.4.a.k 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.f.c 18 1.a even 1 1 trivial
63.4.f.c 18 9.c even 3 1 inner
189.4.f.c 18 3.b odd 2 1
189.4.f.c 18 9.d odd 6 1
567.4.a.j 9 9.c even 3 1
567.4.a.k 9 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{18} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(63, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 7884864 + 5458752 T + 29708208 T^{2} - 21337344 T^{3} + 76740372 T^{4} - 16711110 T^{5} + 23915007 T^{6} - 5084775 T^{7} + 5055453 T^{8} - 932256 T^{9} + 591381 T^{10} - 96714 T^{11} + 48753 T^{12} - 6831 T^{13} + 2340 T^{14} - 246 T^{15} + 72 T^{16} - 6 T^{17} + T^{18} \)
$3$ \( 7625597484987 - 2541865828329 T + 94143178827 T^{2} + 6973568802 T^{3} + 18208762983 T^{4} - 4347718821 T^{5} - 87156324 T^{6} + 79184709 T^{7} + 11160261 T^{8} - 5025726 T^{9} + 413343 T^{10} + 108621 T^{11} - 4428 T^{12} - 8181 T^{13} + 1269 T^{14} + 18 T^{15} + 9 T^{16} - 9 T^{17} + T^{18} \)
$5$ \( 498326127039641124 + 74804938004844306 T + 52211668245675633 T^{2} + 395382749613912 T^{3} + 2738408940718407 T^{4} - 64538852282586 T^{5} + 94555533842622 T^{6} - 7922197927080 T^{7} + 1852897087809 T^{8} - 135756759078 T^{9} + 20471440878 T^{10} - 1437274530 T^{11} + 146602296 T^{12} - 7347294 T^{13} + 496197 T^{14} - 16716 T^{15} + 990 T^{16} - 24 T^{17} + T^{18} \)
$7$ \( ( 49 + 7 T + T^{2} )^{9} \)
$11$ \( \)\(10\!\cdots\!84\)\( + \)\(68\!\cdots\!08\)\( T + \)\(73\!\cdots\!56\)\( T^{2} - \)\(22\!\cdots\!40\)\( T^{3} + \)\(57\!\cdots\!01\)\( T^{4} - 70654914403390479627 T^{5} + 7764244374557466234 T^{6} - 555515645646282003 T^{7} + 45366544248364566 T^{8} - 2683050192185589 T^{9} + 162997085266569 T^{10} - 5631504570504 T^{11} + 166666254471 T^{12} - 3080305179 T^{13} + 58009761 T^{14} - 759156 T^{15} + 12681 T^{16} - 111 T^{17} + T^{18} \)
$13$ \( \)\(10\!\cdots\!16\)\( - \)\(74\!\cdots\!56\)\( T + \)\(43\!\cdots\!76\)\( T^{2} - \)\(67\!\cdots\!20\)\( T^{3} + \)\(76\!\cdots\!88\)\( T^{4} - \)\(51\!\cdots\!30\)\( T^{5} + \)\(28\!\cdots\!37\)\( T^{6} - 9587064743391385962 T^{7} + 329026695818582619 T^{8} - 7021204423796068 T^{9} + 247147113650346 T^{10} - 3468450829986 T^{11} + 120028362207 T^{12} - 539975556 T^{13} + 40879386 T^{14} - 57018 T^{15} + 7983 T^{16} + 18 T^{17} + T^{18} \)
$17$ \( ( 5020282494076716 - 549339878122704 T - 950735023749 T^{2} + 718718485203 T^{3} + 4683127833 T^{4} - 242323227 T^{5} - 2957754 T^{6} + 9954 T^{7} + 273 T^{8} + T^{9} )^{2} \)
$19$ \( ( -278256660857 - 4975695962487 T - 1347895653579 T^{2} - 77545772169 T^{3} + 2455510023 T^{4} + 110526219 T^{5} - 132108 T^{6} - 22806 T^{7} - 45 T^{8} + T^{9} )^{2} \)
$23$ \( \)\(55\!\cdots\!36\)\( + \)\(52\!\cdots\!76\)\( T + \)\(39\!\cdots\!89\)\( T^{2} + \)\(10\!\cdots\!14\)\( T^{3} + \)\(25\!\cdots\!14\)\( T^{4} + \)\(25\!\cdots\!00\)\( T^{5} + \)\(37\!\cdots\!47\)\( T^{6} + \)\(15\!\cdots\!98\)\( T^{7} + \)\(40\!\cdots\!48\)\( T^{8} + 28822488502827554100 T^{9} + 2826388092605987070 T^{10} - 6055524105245280 T^{11} + 153610722142335 T^{12} - 458719800804 T^{13} + 5459390244 T^{14} - 18979140 T^{15} + 133155 T^{16} - 312 T^{17} + T^{18} \)
$29$ \( \)\(31\!\cdots\!36\)\( - \)\(11\!\cdots\!12\)\( T + \)\(41\!\cdots\!32\)\( T^{2} - \)\(77\!\cdots\!48\)\( T^{3} + \)\(16\!\cdots\!72\)\( T^{4} - \)\(24\!\cdots\!16\)\( T^{5} + \)\(43\!\cdots\!85\)\( T^{6} - \)\(49\!\cdots\!60\)\( T^{7} + \)\(62\!\cdots\!34\)\( T^{8} - \)\(54\!\cdots\!36\)\( T^{9} + 5683746558529893447 T^{10} - 40111408285896582 T^{11} + 320055471757908 T^{12} - 1529565257214 T^{13} + 8883855387 T^{14} - 32190030 T^{15} + 159111 T^{16} - 378 T^{17} + T^{18} \)
$31$ \( \)\(10\!\cdots\!24\)\( - \)\(46\!\cdots\!96\)\( T + \)\(24\!\cdots\!96\)\( T^{2} - \)\(18\!\cdots\!44\)\( T^{3} + \)\(10\!\cdots\!92\)\( T^{4} - \)\(94\!\cdots\!50\)\( T^{5} + \)\(31\!\cdots\!37\)\( T^{6} - \)\(10\!\cdots\!02\)\( T^{7} + \)\(23\!\cdots\!22\)\( T^{8} - \)\(43\!\cdots\!26\)\( T^{9} + 12643518097389632481 T^{10} - 12294153764732700 T^{11} + 396466584893160 T^{12} - 125834870970 T^{13} + 9040299903 T^{14} - 772302 T^{15} + 116415 T^{16} + 18 T^{17} + T^{18} \)
$37$ \( ( 1222496532105697432 + 41731167377448108 T - 5910471677746638 T^{2} - 88948378105503 T^{3} + 456605766864 T^{4} + 8139492702 T^{5} - 6232920 T^{6} - 174303 T^{7} + 36 T^{8} + T^{9} )^{2} \)
$41$ \( \)\(57\!\cdots\!44\)\( - \)\(21\!\cdots\!32\)\( T + \)\(16\!\cdots\!84\)\( T^{2} + \)\(33\!\cdots\!56\)\( T^{3} + \)\(13\!\cdots\!49\)\( T^{4} + \)\(52\!\cdots\!63\)\( T^{5} + \)\(10\!\cdots\!34\)\( T^{6} - \)\(36\!\cdots\!45\)\( T^{7} + \)\(65\!\cdots\!20\)\( T^{8} - \)\(18\!\cdots\!35\)\( T^{9} + \)\(13\!\cdots\!43\)\( T^{10} - 2663814102137341668 T^{11} + 16448552545154823 T^{12} - 27849172034295 T^{13} + 115730147493 T^{14} - 129947670 T^{15} + 493263 T^{16} - 477 T^{17} + T^{18} \)
$43$ \( \)\(93\!\cdots\!96\)\( - \)\(49\!\cdots\!68\)\( T + \)\(29\!\cdots\!04\)\( T^{2} - \)\(11\!\cdots\!28\)\( T^{3} + \)\(71\!\cdots\!05\)\( T^{4} - \)\(29\!\cdots\!91\)\( T^{5} + \)\(47\!\cdots\!33\)\( T^{6} - \)\(16\!\cdots\!32\)\( T^{7} + \)\(20\!\cdots\!51\)\( T^{8} - \)\(65\!\cdots\!31\)\( T^{9} + \)\(47\!\cdots\!19\)\( T^{10} - 995689031888893536 T^{11} + 6129124309824621 T^{12} - 10019355163659 T^{13} + 55758271410 T^{14} - 53990085 T^{15} + 289080 T^{16} - 171 T^{17} + T^{18} \)
$47$ \( \)\(23\!\cdots\!04\)\( - \)\(31\!\cdots\!80\)\( T + \)\(68\!\cdots\!32\)\( T^{2} - \)\(80\!\cdots\!96\)\( T^{3} + \)\(12\!\cdots\!84\)\( T^{4} - \)\(13\!\cdots\!24\)\( T^{5} + \)\(11\!\cdots\!89\)\( T^{6} - \)\(71\!\cdots\!26\)\( T^{7} + \)\(38\!\cdots\!92\)\( T^{8} - \)\(14\!\cdots\!88\)\( T^{9} + \)\(55\!\cdots\!15\)\( T^{10} - 16155227899322109702 T^{11} + 51296915012972526 T^{12} - 107781794226684 T^{13} + 247921291557 T^{14} - 321645846 T^{15} + 683361 T^{16} - 654 T^{17} + T^{18} \)
$53$ \( ( -84265412689816836624 - 6036406494071921604 T - 173470908732652440 T^{2} - 2602374112241811 T^{3} - 21981354006150 T^{4} - 102488663079 T^{5} - 217318746 T^{6} + 49698 T^{7} + 948 T^{8} + T^{9} )^{2} \)
$59$ \( \)\(53\!\cdots\!96\)\( + \)\(27\!\cdots\!92\)\( T + \)\(14\!\cdots\!16\)\( T^{2} + \)\(39\!\cdots\!16\)\( T^{3} + \)\(34\!\cdots\!53\)\( T^{4} + \)\(63\!\cdots\!13\)\( T^{5} + \)\(46\!\cdots\!57\)\( T^{6} + \)\(70\!\cdots\!36\)\( T^{7} + \)\(36\!\cdots\!73\)\( T^{8} + \)\(25\!\cdots\!63\)\( T^{9} + \)\(15\!\cdots\!73\)\( T^{10} + 19385609442553381824 T^{11} + 495094731831726003 T^{12} - 151756061674653 T^{13} + 1060773872958 T^{14} - 640356837 T^{15} + 1750050 T^{16} - 957 T^{17} + T^{18} \)
$61$ \( \)\(33\!\cdots\!84\)\( + \)\(93\!\cdots\!52\)\( T + \)\(32\!\cdots\!61\)\( T^{2} + \)\(17\!\cdots\!34\)\( T^{3} + \)\(31\!\cdots\!10\)\( T^{4} - \)\(29\!\cdots\!84\)\( T^{5} + \)\(50\!\cdots\!69\)\( T^{6} - \)\(32\!\cdots\!16\)\( T^{7} + \)\(18\!\cdots\!32\)\( T^{8} - \)\(54\!\cdots\!62\)\( T^{9} + \)\(14\!\cdots\!70\)\( T^{10} - \)\(23\!\cdots\!34\)\( T^{11} + 462825798030657105 T^{12} - 507355529245218 T^{13} + 1067698317546 T^{14} - 602032926 T^{15} + 1155051 T^{16} - 198 T^{17} + T^{18} \)
$67$ \( \)\(11\!\cdots\!16\)\( - \)\(79\!\cdots\!32\)\( T + \)\(94\!\cdots\!88\)\( T^{2} - \)\(22\!\cdots\!32\)\( T^{3} + \)\(25\!\cdots\!73\)\( T^{4} - \)\(39\!\cdots\!69\)\( T^{5} + \)\(48\!\cdots\!28\)\( T^{6} - \)\(32\!\cdots\!19\)\( T^{7} + \)\(53\!\cdots\!70\)\( T^{8} - \)\(27\!\cdots\!77\)\( T^{9} + \)\(41\!\cdots\!25\)\( T^{10} - 12809386756277396100 T^{11} + 192137494913260287 T^{12} - 88711387519365 T^{13} + 589825960227 T^{14} - 172655544 T^{15} + 974637 T^{16} - 333 T^{17} + T^{18} \)
$71$ \( ( \)\(32\!\cdots\!96\)\( + \)\(82\!\cdots\!61\)\( T + 52422234683323826538 T^{2} - 49418156279877543 T^{3} - 981095862050982 T^{4} - 2270401638417 T^{5} - 1167567156 T^{6} + 2041848 T^{7} + 2826 T^{8} + T^{9} )^{2} \)
$73$ \( ( \)\(34\!\cdots\!64\)\( + \)\(52\!\cdots\!20\)\( T - 282861283492595907 T^{2} - 31652703309044445 T^{3} - 66063856098978 T^{4} + 375381079398 T^{5} + 419085309 T^{6} - 1419381 T^{7} - 153 T^{8} + T^{9} )^{2} \)
$79$ \( \)\(10\!\cdots\!96\)\( + \)\(94\!\cdots\!38\)\( T + \)\(31\!\cdots\!89\)\( T^{2} + \)\(11\!\cdots\!62\)\( T^{3} + \)\(97\!\cdots\!95\)\( T^{4} + \)\(20\!\cdots\!08\)\( T^{5} + \)\(38\!\cdots\!04\)\( T^{6} + \)\(15\!\cdots\!22\)\( T^{7} + \)\(60\!\cdots\!99\)\( T^{8} + \)\(13\!\cdots\!28\)\( T^{9} + \)\(34\!\cdots\!90\)\( T^{10} + \)\(59\!\cdots\!60\)\( T^{11} + 1256196344203275900 T^{12} + 1502028871524432 T^{13} + 2176894598499 T^{14} + 1512558636 T^{15} + 2127078 T^{16} + 1152 T^{17} + T^{18} \)
$83$ \( \)\(45\!\cdots\!44\)\( + \)\(72\!\cdots\!72\)\( T + \)\(11\!\cdots\!92\)\( T^{2} + \)\(45\!\cdots\!36\)\( T^{3} + \)\(29\!\cdots\!64\)\( T^{4} - \)\(11\!\cdots\!56\)\( T^{5} + \)\(78\!\cdots\!77\)\( T^{6} - \)\(18\!\cdots\!44\)\( T^{7} + \)\(53\!\cdots\!07\)\( T^{8} - \)\(93\!\cdots\!36\)\( T^{9} + \)\(20\!\cdots\!47\)\( T^{10} - \)\(29\!\cdots\!64\)\( T^{11} + 4665253846773825684 T^{12} - 4604659984790496 T^{13} + 5336395273773 T^{14} - 3999407544 T^{15} + 3887082 T^{16} - 1890 T^{17} + T^{18} \)
$89$ \( ( -\)\(15\!\cdots\!08\)\( - \)\(13\!\cdots\!60\)\( T - \)\(30\!\cdots\!90\)\( T^{2} + 603436160810412567 T^{3} + 7425601308907074 T^{4} + 3455975380041 T^{5} - 5558213076 T^{6} - 3706974 T^{7} + 1302 T^{8} + T^{9} )^{2} \)
$97$ \( \)\(11\!\cdots\!64\)\( - \)\(20\!\cdots\!04\)\( T + \)\(28\!\cdots\!28\)\( T^{2} - \)\(12\!\cdots\!32\)\( T^{3} + \)\(49\!\cdots\!73\)\( T^{4} - \)\(90\!\cdots\!77\)\( T^{5} + \)\(19\!\cdots\!83\)\( T^{6} - \)\(18\!\cdots\!42\)\( T^{7} + \)\(47\!\cdots\!81\)\( T^{8} - \)\(32\!\cdots\!31\)\( T^{9} + \)\(60\!\cdots\!03\)\( T^{10} - \)\(34\!\cdots\!08\)\( T^{11} + 53764810514738992275 T^{12} - 25500210625431603 T^{13} + 25227096748668 T^{14} - 7833175221 T^{15} + 6980166 T^{16} - 1737 T^{17} + T^{18} \)
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