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Properties

Label 169.4.c.l
Level $169$
Weight $4$
Character orbit 169.c
Analytic conductor $9.971$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 4 x^{17} + 62 x^{16} - 106 x^{15} + 2016 x^{14} - 2731 x^{13} + 39895 x^{12} - 21896 x^{11} + 490851 x^{10} - 318018 x^{9} + 3391249 x^{8} - 990441 x^{7} + 13815033 x^{6} - 7349264 x^{5} + 28218112 x^{4} - 9093120 x^{3} + 32440320 x^{2} - 14680064 x + 16777216\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 13^{4} \)
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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{2} + ( \beta_{11} - \beta_{14} ) q^{3} + ( -5 + 5 \beta_{3} - \beta_{6} - \beta_{10} + \beta_{12} ) q^{4} + ( -2 - 3 \beta_{2} + \beta_{5} + \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{16} - \beta_{17} ) q^{5} + ( 6 + \beta_{1} - 6 \beta_{3} + 2 \beta_{6} - \beta_{7} + 2 \beta_{10} - \beta_{14} ) q^{6} + ( 4 + \beta_{1} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{12} - \beta_{14} ) q^{7} + ( -9 + 4 \beta_{2} - 4 \beta_{7} + 4 \beta_{9} + \beta_{11} + \beta_{12} - \beta_{17} ) q^{8} + ( -7 + \beta_{1} + 7 \beta_{3} - \beta_{4} + \beta_{6} + 3 \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{14} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{2} + ( \beta_{11} - \beta_{14} ) q^{3} + ( -5 + 5 \beta_{3} - \beta_{6} - \beta_{10} + \beta_{12} ) q^{4} + ( -2 - 3 \beta_{2} + \beta_{5} + \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{16} - \beta_{17} ) q^{5} + ( 6 + \beta_{1} - 6 \beta_{3} + 2 \beta_{6} - \beta_{7} + 2 \beta_{10} - \beta_{14} ) q^{6} + ( 4 + \beta_{1} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{12} - \beta_{14} ) q^{7} + ( -9 + 4 \beta_{2} - 4 \beta_{7} + 4 \beta_{9} + \beta_{11} + \beta_{12} - \beta_{17} ) q^{8} + ( -7 + \beta_{1} + 7 \beta_{3} - \beta_{4} + \beta_{6} + 3 \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{14} ) q^{9} + ( -6 \beta_{1} + 6 \beta_{2} + 16 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + 2 \beta_{13} - \beta_{14} + 2 \beta_{15} + \beta_{16} + \beta_{17} ) q^{10} + ( 2 \beta_{1} - 2 \beta_{2} + 21 \beta_{3} - \beta_{6} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} - \beta_{16} - 3 \beta_{17} ) q^{11} + ( 10 - 14 \beta_{2} + 3 \beta_{5} + 3 \beta_{7} - 3 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - 3 \beta_{13} - 2 \beta_{17} ) q^{12} + ( -10 - 9 \beta_{2} + 4 \beta_{5} + 2 \beta_{7} - 2 \beta_{9} + \beta_{11} + 3 \beta_{12} + 4 \beta_{13} + 4 \beta_{15} + 4 \beta_{16} - 3 \beta_{17} ) q^{14} + ( 3 \beta_{1} - 3 \beta_{2} + 20 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} + \beta_{6} - \beta_{8} + 3 \beta_{9} + 9 \beta_{10} - \beta_{11} - 9 \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{15} + ( -7 \beta_{1} + 7 \beta_{2} - 37 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} + 2 \beta_{6} + 8 \beta_{9} + 7 \beta_{10} - 4 \beta_{11} - 7 \beta_{13} + 4 \beta_{14} + 2 \beta_{16} - \beta_{17} ) q^{16} + ( 7 + 8 \beta_{1} - 7 \beta_{3} + 5 \beta_{4} + 7 \beta_{6} - 2 \beta_{7} + 5 \beta_{8} + 7 \beta_{10} + \beta_{12} + 2 \beta_{14} ) q^{17} + ( -5 + 3 \beta_{2} - 5 \beta_{5} + 6 \beta_{7} - 6 \beta_{9} - 2 \beta_{11} - \beta_{12} + 9 \beta_{13} + 2 \beta_{15} - 5 \beta_{16} + \beta_{17} ) q^{18} + ( 14 + 4 \beta_{1} - 14 \beta_{3} - 10 \beta_{6} + \beta_{8} + \beta_{10} + 3 \beta_{12} + \beta_{14} ) q^{19} + ( 36 + 15 \beta_{1} - 36 \beta_{3} + 10 \beta_{4} + 9 \beta_{6} - 6 \beta_{7} - 6 \beta_{8} + 5 \beta_{10} - 3 \beta_{12} + 8 \beta_{14} ) q^{20} + ( -24 + 4 \beta_{2} - 5 \beta_{5} - 14 \beta_{7} + 14 \beta_{9} - 2 \beta_{11} - 10 \beta_{12} - \beta_{13} - 3 \beta_{15} - 2 \beta_{16} + 10 \beta_{17} ) q^{21} + ( -43 + 15 \beta_{1} + 43 \beta_{3} + 8 \beta_{4} - 3 \beta_{6} + 3 \beta_{7} + 4 \beta_{10} - 3 \beta_{12} + 10 \beta_{14} ) q^{22} + ( 8 \beta_{1} - 8 \beta_{2} + 22 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{10} - 6 \beta_{11} - 2 \beta_{13} + 6 \beta_{14} - 3 \beta_{16} - 14 \beta_{17} ) q^{23} + ( -3 \beta_{1} + 3 \beta_{2} + 94 \beta_{3} - \beta_{4} - \beta_{5} + 6 \beta_{6} - 6 \beta_{8} - 5 \beta_{9} - 7 \beta_{10} + 11 \beta_{11} + 7 \beta_{13} - 11 \beta_{14} + 6 \beta_{15} + 6 \beta_{16} + 9 \beta_{17} ) q^{24} + ( 25 + 23 \beta_{2} - 4 \beta_{5} - 3 \beta_{7} + 3 \beta_{9} - 9 \beta_{11} - 3 \beta_{12} + \beta_{13} + 7 \beta_{15} - 3 \beta_{16} + 3 \beta_{17} ) q^{25} + ( -83 + 29 \beta_{2} - 4 \beta_{5} + \beta_{7} - \beta_{9} - 6 \beta_{11} + 6 \beta_{12} + 6 \beta_{13} - 19 \beta_{16} - 6 \beta_{17} ) q^{27} + ( \beta_{1} - \beta_{2} + 46 \beta_{3} + 25 \beta_{4} + 25 \beta_{5} - 5 \beta_{6} - 8 \beta_{8} - 10 \beta_{9} + 4 \beta_{11} - 4 \beta_{14} + 8 \beta_{15} - 5 \beta_{16} - 12 \beta_{17} ) q^{28} + ( 29 \beta_{1} - 29 \beta_{2} - 16 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} - 8 \beta_{6} - 3 \beta_{9} + 6 \beta_{10} + 3 \beta_{11} - 6 \beta_{13} - 3 \beta_{14} - 8 \beta_{16} + 5 \beta_{17} ) q^{29} + ( -50 - 22 \beta_{1} + 50 \beta_{3} - 16 \beta_{4} - 17 \beta_{6} + 34 \beta_{7} + 8 \beta_{8} - 11 \beta_{10} + 6 \beta_{12} - 9 \beta_{14} ) q^{30} + ( -82 + 21 \beta_{2} + 2 \beta_{5} + \beta_{7} - \beta_{9} - 7 \beta_{11} + \beta_{12} + 12 \beta_{13} + 8 \beta_{15} + 4 \beta_{16} - \beta_{17} ) q^{31} + ( 89 - 18 \beta_{1} - 89 \beta_{3} + 7 \beta_{4} + 12 \beta_{6} + 10 \beta_{7} - 10 \beta_{8} - \beta_{10} - 18 \beta_{12} + 17 \beta_{14} ) q^{32} + ( 59 - 24 \beta_{1} - 59 \beta_{3} - 4 \beta_{4} - 5 \beta_{6} - 22 \beta_{7} - 5 \beta_{8} - 3 \beta_{10} - 2 \beta_{12} - 23 \beta_{14} ) q^{33} + ( -19 - 37 \beta_{2} + 25 \beta_{5} + 23 \beta_{7} - 23 \beta_{9} - 10 \beta_{11} + 6 \beta_{12} - 29 \beta_{13} - 10 \beta_{15} + 9 \beta_{16} - 6 \beta_{17} ) q^{34} + ( 20 - 40 \beta_{1} - 20 \beta_{3} - \beta_{4} - 8 \beta_{6} - 12 \beta_{7} + \beta_{8} + 17 \beta_{10} - 4 \beta_{12} - 12 \beta_{14} ) q^{35} + ( 31 \beta_{1} - 31 \beta_{2} + 115 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{8} - 16 \beta_{9} + 12 \beta_{10} + 12 \beta_{11} - 12 \beta_{13} - 12 \beta_{14} - 2 \beta_{15} - 3 \beta_{16} + 12 \beta_{17} ) q^{36} + ( -28 \beta_{1} + 28 \beta_{2} + 12 \beta_{3} - 9 \beta_{4} - 9 \beta_{5} - 12 \beta_{6} + 11 \beta_{8} - 14 \beta_{9} + 13 \beta_{10} - 16 \beta_{11} - 13 \beta_{13} + 16 \beta_{14} - 11 \beta_{15} - 12 \beta_{16} - 4 \beta_{17} ) q^{37} + ( -90 - 4 \beta_{2} + 20 \beta_{5} + 2 \beta_{7} - 2 \beta_{9} + 7 \beta_{11} + 8 \beta_{12} - 21 \beta_{13} + 18 \beta_{16} - 8 \beta_{17} ) q^{38} + ( -8 - 18 \beta_{2} - 22 \beta_{5} + 24 \beta_{7} - 24 \beta_{9} + 9 \beta_{11} - 16 \beta_{12} - 35 \beta_{13} - 4 \beta_{15} + 15 \beta_{16} + 16 \beta_{17} ) q^{40} + ( 14 \beta_{1} - 14 \beta_{2} + 150 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 7 \beta_{6} - 2 \beta_{8} - 6 \beta_{9} - 12 \beta_{10} + 3 \beta_{11} + 12 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} + 7 \beta_{16} + 21 \beta_{17} ) q^{41} + ( -6 \beta_{1} + 6 \beta_{2} - 20 \beta_{3} - 25 \beta_{4} - 25 \beta_{5} - 2 \beta_{6} + 10 \beta_{8} + 26 \beta_{9} + 19 \beta_{10} - 2 \beta_{11} - 19 \beta_{13} + 2 \beta_{14} - 10 \beta_{15} - 2 \beta_{16} + 5 \beta_{17} ) q^{42} + ( 70 - 14 \beta_{1} - 70 \beta_{3} - 25 \beta_{4} + 11 \beta_{6} - 2 \beta_{7} - 5 \beta_{8} + 5 \beta_{10} + 21 \beta_{12} + 3 \beta_{14} ) q^{43} + ( -63 - 11 \beta_{5} + 15 \beta_{7} - 15 \beta_{9} + 7 \beta_{11} + 17 \beta_{12} - 29 \beta_{13} - 16 \beta_{15} + 15 \beta_{16} - 17 \beta_{17} ) q^{44} + ( -26 + 42 \beta_{1} + 26 \beta_{3} + 13 \beta_{4} + 11 \beta_{6} - 14 \beta_{7} + 10 \beta_{8} + 20 \beta_{10} + 8 \beta_{12} - 42 \beta_{14} ) q^{45} + ( -82 - 39 \beta_{1} + 82 \beta_{3} + 5 \beta_{4} - 27 \beta_{6} + 26 \beta_{7} + 2 \beta_{8} - 12 \beta_{10} - 4 \beta_{12} + 20 \beta_{14} ) q^{46} + ( -92 - 25 \beta_{2} - 25 \beta_{5} - \beta_{7} + \beta_{9} + 23 \beta_{11} - 7 \beta_{12} + 10 \beta_{13} - 4 \beta_{15} - 5 \beta_{16} + 7 \beta_{17} ) q^{47} + ( -6 + 42 \beta_{1} + 6 \beta_{3} + 10 \beta_{4} + 37 \beta_{6} - 33 \beta_{7} + 2 \beta_{8} + 3 \beta_{10} + 23 \beta_{12} - 9 \beta_{14} ) q^{48} + ( -25 \beta_{1} + 25 \beta_{2} - 55 \beta_{3} - 39 \beta_{4} - 39 \beta_{5} + 6 \beta_{6} - 7 \beta_{8} + 41 \beta_{9} - 19 \beta_{10} - 5 \beta_{11} + 19 \beta_{13} + 5 \beta_{14} + 7 \beta_{15} + 6 \beta_{16} + 27 \beta_{17} ) q^{49} + ( 53 \beta_{1} - 53 \beta_{2} - 113 \beta_{3} + 26 \beta_{4} + 26 \beta_{5} + 5 \beta_{6} + 8 \beta_{8} + 2 \beta_{9} + 11 \beta_{10} - 9 \beta_{11} - 11 \beta_{13} + 9 \beta_{14} - 8 \beta_{15} + 5 \beta_{16} - 22 \beta_{17} ) q^{50} + ( 49 + 44 \beta_{2} - 8 \beta_{5} - 4 \beta_{7} + 4 \beta_{9} + 11 \beta_{11} + 24 \beta_{12} + 40 \beta_{13} - 8 \beta_{15} - 30 \beta_{16} - 24 \beta_{17} ) q^{51} + ( -18 - 18 \beta_{2} + 36 \beta_{5} + 12 \beta_{7} - 12 \beta_{9} + 12 \beta_{11} + 6 \beta_{12} + 13 \beta_{13} - 3 \beta_{15} + 43 \beta_{16} - 6 \beta_{17} ) q^{53} + ( -49 \beta_{1} + 49 \beta_{2} - 325 \beta_{3} - 10 \beta_{4} - 10 \beta_{5} - 5 \beta_{6} + 8 \beta_{8} - 8 \beta_{9} + 18 \beta_{10} - 32 \beta_{11} - 18 \beta_{13} + 32 \beta_{14} - 8 \beta_{15} - 5 \beta_{16} - 26 \beta_{17} ) q^{54} + ( 37 \beta_{1} - 37 \beta_{2} + 50 \beta_{3} + 38 \beta_{4} + 38 \beta_{5} + 37 \beta_{6} - 7 \beta_{8} - 83 \beta_{9} + 7 \beta_{10} + 19 \beta_{11} - 7 \beta_{13} - 19 \beta_{14} + 7 \beta_{15} + 37 \beta_{16} - 19 \beta_{17} ) q^{55} + ( -30 + 4 \beta_{1} + 30 \beta_{3} + 55 \beta_{4} + 18 \beta_{6} - 4 \beta_{7} - 18 \beta_{8} + \beta_{10} - 35 \beta_{12} + 36 \beta_{14} ) q^{56} + ( 113 - 34 \beta_{2} - 4 \beta_{5} - 40 \beta_{7} + 40 \beta_{9} + 43 \beta_{11} - 21 \beta_{12} - \beta_{13} + \beta_{15} - 18 \beta_{16} + 21 \beta_{17} ) q^{57} + ( -346 - 25 \beta_{1} + 346 \beta_{3} - 16 \beta_{4} - 28 \beta_{6} + 4 \beta_{7} - 12 \beta_{8} - 60 \beta_{10} + 25 \beta_{12} - 3 \beta_{14} ) q^{58} + ( 158 + 49 \beta_{1} - 158 \beta_{3} + 8 \beta_{4} + 34 \beta_{6} + \beta_{7} + 22 \beta_{8} + 22 \beta_{10} - 13 \beta_{12} + 30 \beta_{14} ) q^{59} + ( 354 + 85 \beta_{2} + 7 \beta_{5} + 4 \beta_{7} - 4 \beta_{9} + 23 \beta_{11} - 2 \beta_{12} + 51 \beta_{13} + 40 \beta_{15} - 20 \beta_{16} + 2 \beta_{17} ) q^{60} + ( -4 - 6 \beta_{1} + 4 \beta_{3} + 17 \beta_{4} - 18 \beta_{6} + 52 \beta_{7} - 13 \beta_{8} - 51 \beta_{10} - 18 \beta_{12} + 16 \beta_{14} ) q^{61} + ( -27 \beta_{1} + 27 \beta_{2} - 248 \beta_{3} + 72 \beta_{4} + 72 \beta_{5} + 34 \beta_{6} - 4 \beta_{8} - 38 \beta_{9} + 32 \beta_{10} - 17 \beta_{11} - 32 \beta_{13} + 17 \beta_{14} + 4 \beta_{15} + 34 \beta_{16} - 33 \beta_{17} ) q^{62} + ( 27 \beta_{1} - 27 \beta_{2} - 86 \beta_{3} - 34 \beta_{4} - 34 \beta_{5} + 17 \beta_{6} - 13 \beta_{8} + 15 \beta_{9} - 15 \beta_{10} + 27 \beta_{11} + 15 \beta_{13} - 27 \beta_{14} + 13 \beta_{15} + 17 \beta_{16} + 33 \beta_{17} ) q^{63} + ( 49 - 71 \beta_{2} - 66 \beta_{5} - 22 \beta_{7} + 22 \beta_{9} + 9 \beta_{11} - 62 \beta_{12} - 27 \beta_{13} - 14 \beta_{15} - 15 \beta_{16} + 62 \beta_{17} ) q^{64} + ( 381 - 7 \beta_{2} + 2 \beta_{5} - 79 \beta_{7} + 79 \beta_{9} - 39 \beta_{11} - \beta_{12} + 33 \beta_{13} + 8 \beta_{15} - 43 \beta_{16} + \beta_{17} ) q^{66} + ( 45 \beta_{1} - 45 \beta_{2} - 102 \beta_{3} - 37 \beta_{4} - 37 \beta_{5} + 22 \beta_{6} + 31 \beta_{8} + 57 \beta_{9} - 9 \beta_{10} + 28 \beta_{11} + 9 \beta_{13} - 28 \beta_{14} - 31 \beta_{15} + 22 \beta_{16} + 3 \beta_{17} ) q^{67} + ( -76 \beta_{1} + 76 \beta_{2} + 185 \beta_{3} - 26 \beta_{4} - 26 \beta_{5} - 4 \beta_{6} - 10 \beta_{8} + 51 \beta_{9} - 115 \beta_{10} + 4 \beta_{11} + 115 \beta_{13} - 4 \beta_{14} + 10 \beta_{15} - 4 \beta_{16} + 22 \beta_{17} ) q^{68} + ( 212 + 15 \beta_{1} - 212 \beta_{3} + 15 \beta_{4} - 27 \beta_{6} - 101 \beta_{7} - 6 \beta_{8} + 10 \beta_{10} - 57 \beta_{12} - 39 \beta_{14} ) q^{69} + ( 576 - 26 \beta_{2} + 57 \beta_{5} + 20 \beta_{7} - 20 \beta_{9} - 30 \beta_{11} - 29 \beta_{12} + 3 \beta_{13} + 2 \beta_{15} - 24 \beta_{16} + 29 \beta_{17} ) q^{70} + ( 264 - 11 \beta_{1} - 264 \beta_{3} - 27 \beta_{4} + 26 \beta_{6} - 11 \beta_{7} + 23 \beta_{8} + 27 \beta_{10} + 17 \beta_{12} + 71 \beta_{14} ) q^{71} + ( -543 + 117 \beta_{1} + 543 \beta_{3} - 29 \beta_{4} - 4 \beta_{6} + 30 \beta_{7} + 20 \beta_{8} - 4 \beta_{10} + 57 \beta_{12} - 67 \beta_{14} ) q^{72} + ( -52 + 85 \beta_{2} - 45 \beta_{5} - 5 \beta_{7} + 5 \beta_{9} - 50 \beta_{11} + 39 \beta_{12} + 19 \beta_{13} - 15 \beta_{15} - 2 \beta_{16} - 39 \beta_{17} ) q^{73} + ( 232 - 44 \beta_{1} - 232 \beta_{3} - 121 \beta_{4} - 72 \beta_{6} + 42 \beta_{7} + 18 \beta_{8} - 57 \beta_{10} + 11 \beta_{12} - 4 \beta_{14} ) q^{74} + ( 15 \beta_{1} - 15 \beta_{2} - 226 \beta_{3} + 18 \beta_{4} + 18 \beta_{5} - 34 \beta_{6} + 36 \beta_{8} + 3 \beta_{9} - 64 \beta_{10} + 36 \beta_{11} + 64 \beta_{13} - 36 \beta_{14} - 36 \beta_{15} - 34 \beta_{16} - 11 \beta_{17} ) q^{75} + ( -120 \beta_{1} + 120 \beta_{2} - 434 \beta_{3} + 22 \beta_{4} + 22 \beta_{5} - 35 \beta_{6} - 32 \beta_{8} + 60 \beta_{9} - 22 \beta_{10} + 12 \beta_{11} + 22 \beta_{13} - 12 \beta_{14} + 32 \beta_{15} - 35 \beta_{16} - 18 \beta_{17} ) q^{76} + ( -32 - 40 \beta_{2} + 31 \beta_{5} + 50 \beta_{7} - 50 \beta_{9} - 24 \beta_{11} - 22 \beta_{12} + 65 \beta_{13} + 15 \beta_{15} + 76 \beta_{16} + 22 \beta_{17} ) q^{77} + ( 44 - 5 \beta_{2} + 3 \beta_{5} - 77 \beta_{7} + 77 \beta_{9} + 3 \beta_{11} - 9 \beta_{12} - 5 \beta_{13} - 21 \beta_{15} - 38 \beta_{16} + 9 \beta_{17} ) q^{79} + ( -17 \beta_{1} + 17 \beta_{2} - 56 \beta_{3} - 79 \beta_{4} - 79 \beta_{5} - 20 \beta_{6} - 4 \beta_{8} - 12 \beta_{9} - 57 \beta_{10} + 57 \beta_{11} + 57 \beta_{13} - 57 \beta_{14} + 4 \beta_{15} - 20 \beta_{16} + 82 \beta_{17} ) q^{80} + ( -124 \beta_{1} + 124 \beta_{2} + 62 \beta_{3} - 27 \beta_{4} - 27 \beta_{5} - 81 \beta_{6} - 32 \beta_{8} + 8 \beta_{9} - 72 \beta_{10} - 85 \beta_{11} + 72 \beta_{13} + 85 \beta_{14} + 32 \beta_{15} - 81 \beta_{16} - 74 \beta_{17} ) q^{81} + ( -438 + 237 \beta_{1} + 438 \beta_{3} + 18 \beta_{4} + 11 \beta_{6} - 74 \beta_{7} - 4 \beta_{8} - 12 \beta_{10} + 35 \beta_{12} - 23 \beta_{14} ) q^{82} + ( -444 - 21 \beta_{2} - 10 \beta_{5} - 85 \beta_{7} + 85 \beta_{9} - 88 \beta_{11} + 16 \beta_{12} - 25 \beta_{13} - 13 \beta_{15} + 10 \beta_{16} - 16 \beta_{17} ) q^{83} + ( -100 - 84 \beta_{1} + 100 \beta_{3} - 79 \beta_{4} - 37 \beta_{6} + 4 \beta_{7} + 26 \beta_{8} + 26 \beta_{10} - 47 \beta_{12} - 35 \beta_{14} ) q^{84} + ( -182 - 167 \beta_{1} + 182 \beta_{3} - 90 \beta_{4} + 24 \beta_{6} + 65 \beta_{7} - 38 \beta_{8} - 88 \beta_{10} + 43 \beta_{12} + 21 \beta_{14} ) q^{85} + ( 14 - 67 \beta_{2} + 47 \beta_{5} - 12 \beta_{7} + 12 \beta_{9} - 61 \beta_{11} + 54 \beta_{12} + 9 \beta_{13} + 50 \beta_{15} + 33 \beta_{16} - 54 \beta_{17} ) q^{86} + ( 132 + 30 \beta_{1} - 132 \beta_{3} - 3 \beta_{4} + 88 \beta_{6} - 38 \beta_{7} + 11 \beta_{8} + 71 \beta_{10} - 18 \beta_{12} + 36 \beta_{14} ) q^{87} + ( -127 \beta_{1} + 127 \beta_{2} - 3 \beta_{3} - 79 \beta_{4} - 79 \beta_{5} - 77 \beta_{6} + 22 \beta_{8} + 107 \beta_{9} + \beta_{10} - 20 \beta_{11} - \beta_{13} + 20 \beta_{14} - 22 \beta_{15} - 77 \beta_{16} + 12 \beta_{17} ) q^{88} + ( 42 \beta_{1} - 42 \beta_{2} + 210 \beta_{3} + 18 \beta_{4} + 18 \beta_{5} + 13 \beta_{6} + 29 \beta_{8} + 44 \beta_{9} - 17 \beta_{10} - 9 \beta_{11} + 17 \beta_{13} + 9 \beta_{14} - 29 \beta_{15} + 13 \beta_{16} + 46 \beta_{17} ) q^{89} + ( -158 - 5 \beta_{2} + 75 \beta_{5} - 8 \beta_{7} + 8 \beta_{9} - 82 \beta_{11} + 48 \beta_{12} - 36 \beta_{13} - 26 \beta_{15} - 35 \beta_{16} - 48 \beta_{17} ) q^{90} + ( 418 + 96 \beta_{2} - 35 \beta_{5} + 42 \beta_{7} - 42 \beta_{9} + 46 \beta_{11} + 35 \beta_{12} + 31 \beta_{13} - 10 \beta_{15} - 42 \beta_{16} - 35 \beta_{17} ) q^{92} + ( -86 \beta_{1} + 86 \beta_{2} - 150 \beta_{3} - 35 \beta_{4} - 35 \beta_{5} - 82 \beta_{6} + 27 \beta_{8} + 68 \beta_{9} - 67 \beta_{10} - 96 \beta_{11} + 67 \beta_{13} + 96 \beta_{14} - 27 \beta_{15} - 82 \beta_{16} + 18 \beta_{17} ) q^{93} + ( -92 \beta_{1} + 92 \beta_{2} + 226 \beta_{3} - 61 \beta_{4} - 61 \beta_{5} - 27 \beta_{6} + 50 \beta_{8} - 54 \beta_{9} + 56 \beta_{10} + 39 \beta_{11} - 56 \beta_{13} - 39 \beta_{14} - 50 \beta_{15} - 27 \beta_{16} + 85 \beta_{17} ) q^{94} + ( 206 - 140 \beta_{1} - 206 \beta_{3} - 16 \beta_{4} - 37 \beta_{6} - 4 \beta_{7} - 55 \beta_{8} - 37 \beta_{10} + 14 \beta_{12} + 12 \beta_{14} ) q^{95} + ( 298 + 38 \beta_{2} + 7 \beta_{5} - 47 \beta_{7} + 47 \beta_{9} - 7 \beta_{11} + 8 \beta_{12} - 36 \beta_{13} + 28 \beta_{15} + 53 \beta_{16} - 8 \beta_{17} ) q^{96} + ( -157 + 96 \beta_{1} + 157 \beta_{3} + 87 \beta_{4} - 8 \beta_{6} + 68 \beta_{7} + 6 \beta_{8} + 20 \beta_{10} - 17 \beta_{12} + 54 \beta_{14} ) q^{97} + ( 33 + 26 \beta_{1} - 33 \beta_{3} + 15 \beta_{4} + 10 \beta_{6} - 4 \beta_{7} + 78 \beta_{8} + 153 \beta_{10} + 32 \beta_{12} - 7 \beta_{14} ) q^{98} + ( -121 - 50 \beta_{2} + 20 \beta_{5} + 70 \beta_{7} - 70 \beta_{9} + 38 \beta_{11} - 18 \beta_{12} + 14 \beta_{13} + 14 \beta_{15} + 100 \beta_{16} + 18 \beta_{17} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 5 q^{2} - q^{3} - 37 q^{4} - 60 q^{5} + 48 q^{6} + 38 q^{7} - 120 q^{8} - 66 q^{9} + O(q^{10}) \) \( 18 q + 5 q^{2} - q^{3} - 37 q^{4} - 60 q^{5} + 48 q^{6} + 38 q^{7} - 120 q^{8} - 66 q^{9} + 147 q^{10} + 181 q^{11} + 78 q^{12} - 294 q^{14} + 218 q^{15} - 269 q^{16} + 55 q^{17} - 158 q^{18} + 161 q^{19} + 370 q^{20} - 376 q^{21} - 340 q^{22} + 204 q^{23} + 798 q^{24} + 614 q^{25} - 1336 q^{27} + 344 q^{28} - 280 q^{29} - 521 q^{30} - 1412 q^{31} + 680 q^{32} + 500 q^{33} - 432 q^{34} - 20 q^{35} + 909 q^{36} + 298 q^{37} - 1478 q^{38} + 26 q^{40} + 1201 q^{41} + 4 q^{42} + 533 q^{43} - 710 q^{44} - 90 q^{45} - 840 q^{46} - 1912 q^{47} + 132 q^{48} - 403 q^{49} - 1156 q^{50} + 940 q^{51} - 556 q^{53} - 2555 q^{54} + 250 q^{55} - 250 q^{56} + 1620 q^{57} - 2877 q^{58} + 1377 q^{59} + 6314 q^{60} + 136 q^{61} - 2035 q^{62} - 944 q^{63} + 568 q^{64} + 6558 q^{66} - 931 q^{67} + 1536 q^{68} + 2050 q^{69} + 9708 q^{70} + 2046 q^{71} - 4342 q^{72} + 90 q^{73} + 1990 q^{74} - 2393 q^{75} - 3608 q^{76} - 1436 q^{77} + 824 q^{79} - 787 q^{80} + 835 q^{81} - 2757 q^{82} - 7418 q^{83} - 1539 q^{84} - 2106 q^{85} - 250 q^{86} + 786 q^{87} + 636 q^{88} + 1663 q^{89} - 2560 q^{90} + 8020 q^{92} - 1186 q^{93} + 2531 q^{94} + 1614 q^{95} + 6168 q^{96} - 1087 q^{97} - 282 q^{98} - 2714 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 4 x^{17} + 62 x^{16} - 106 x^{15} + 2016 x^{14} - 2731 x^{13} + 39895 x^{12} - 21896 x^{11} + 490851 x^{10} - 318018 x^{9} + 3391249 x^{8} - 990441 x^{7} + 13815033 x^{6} - 7349264 x^{5} + 28218112 x^{4} - 9093120 x^{3} + 32440320 x^{2} - 14680064 x + 16777216\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(71\!\cdots\!85\)\( \nu^{17} + \)\(42\!\cdots\!76\)\( \nu^{16} - \)\(50\!\cdots\!42\)\( \nu^{15} + \)\(16\!\cdots\!58\)\( \nu^{14} - \)\(15\!\cdots\!08\)\( \nu^{13} + \)\(46\!\cdots\!55\)\( \nu^{12} - \)\(32\!\cdots\!27\)\( \nu^{11} + \)\(68\!\cdots\!76\)\( \nu^{10} - \)\(38\!\cdots\!79\)\( \nu^{9} + \)\(83\!\cdots\!98\)\( \nu^{8} - \)\(29\!\cdots\!61\)\( \nu^{7} + \)\(44\!\cdots\!05\)\( \nu^{6} - \)\(11\!\cdots\!29\)\( \nu^{5} + \)\(18\!\cdots\!76\)\( \nu^{4} - \)\(37\!\cdots\!60\)\( \nu^{3} + \)\(25\!\cdots\!64\)\( \nu^{2} - \)\(15\!\cdots\!52\)\( \nu + \)\(26\!\cdots\!56\)\(\)\()/ \)\(32\!\cdots\!52\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(12\!\cdots\!53\)\( \nu^{17} + \)\(44\!\cdots\!32\)\( \nu^{16} - \)\(75\!\cdots\!78\)\( \nu^{15} + \)\(93\!\cdots\!82\)\( \nu^{14} - \)\(24\!\cdots\!84\)\( \nu^{13} + \)\(21\!\cdots\!79\)\( \nu^{12} - \)\(46\!\cdots\!95\)\( \nu^{11} + \)\(15\!\cdots\!72\)\( \nu^{10} - \)\(56\!\cdots\!95\)\( \nu^{9} + \)\(94\!\cdots\!22\)\( \nu^{8} - \)\(36\!\cdots\!13\)\( \nu^{7} - \)\(11\!\cdots\!15\)\( \nu^{6} - \)\(13\!\cdots\!09\)\( \nu^{5} + \)\(12\!\cdots\!60\)\( \nu^{4} - \)\(21\!\cdots\!28\)\( \nu^{3} - \)\(18\!\cdots\!20\)\( \nu^{2} - \)\(20\!\cdots\!48\)\( \nu + \)\(61\!\cdots\!76\)\(\)\()/ \)\(25\!\cdots\!16\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(11\!\cdots\!41\)\( \nu^{17} - \)\(32\!\cdots\!00\)\( \nu^{16} + \)\(66\!\cdots\!34\)\( \nu^{15} - \)\(28\!\cdots\!26\)\( \nu^{14} + \)\(21\!\cdots\!44\)\( \nu^{13} - \)\(15\!\cdots\!95\)\( \nu^{12} + \)\(41\!\cdots\!03\)\( \nu^{11} + \)\(34\!\cdots\!08\)\( \nu^{10} + \)\(52\!\cdots\!55\)\( \nu^{9} + \)\(32\!\cdots\!58\)\( \nu^{8} + \)\(33\!\cdots\!69\)\( \nu^{7} + \)\(35\!\cdots\!19\)\( \nu^{6} + \)\(14\!\cdots\!89\)\( \nu^{5} + \)\(69\!\cdots\!36\)\( \nu^{4} + \)\(22\!\cdots\!28\)\( \nu^{3} + \)\(27\!\cdots\!04\)\( \nu^{2} + \)\(65\!\cdots\!84\)\( \nu + \)\(25\!\cdots\!24\)\(\)\()/ \)\(20\!\cdots\!16\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(10\!\cdots\!51\)\( \nu^{17} - \)\(58\!\cdots\!04\)\( \nu^{16} + \)\(70\!\cdots\!62\)\( \nu^{15} - \)\(21\!\cdots\!54\)\( \nu^{14} + \)\(22\!\cdots\!76\)\( \nu^{13} - \)\(63\!\cdots\!29\)\( \nu^{12} + \)\(46\!\cdots\!65\)\( \nu^{11} - \)\(91\!\cdots\!96\)\( \nu^{10} + \)\(55\!\cdots\!01\)\( \nu^{9} - \)\(11\!\cdots\!22\)\( \nu^{8} + \)\(41\!\cdots\!67\)\( \nu^{7} - \)\(63\!\cdots\!11\)\( \nu^{6} + \)\(16\!\cdots\!19\)\( \nu^{5} - \)\(25\!\cdots\!72\)\( \nu^{4} + \)\(44\!\cdots\!04\)\( \nu^{3} - \)\(35\!\cdots\!12\)\( \nu^{2} + \)\(21\!\cdots\!80\)\( \nu - \)\(56\!\cdots\!96\)\(\)\()/ \)\(16\!\cdots\!28\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(17\!\cdots\!07\)\( \nu^{17} + \)\(24\!\cdots\!88\)\( \nu^{16} - \)\(15\!\cdots\!18\)\( \nu^{15} + \)\(11\!\cdots\!94\)\( \nu^{14} - \)\(41\!\cdots\!92\)\( \nu^{13} + \)\(36\!\cdots\!57\)\( \nu^{12} - \)\(77\!\cdots\!89\)\( \nu^{11} + \)\(64\!\cdots\!88\)\( \nu^{10} - \)\(46\!\cdots\!01\)\( \nu^{9} + \)\(80\!\cdots\!54\)\( \nu^{8} - \)\(21\!\cdots\!79\)\( \nu^{7} + \)\(47\!\cdots\!47\)\( \nu^{6} + \)\(19\!\cdots\!89\)\( \nu^{5} + \)\(19\!\cdots\!40\)\( \nu^{4} + \)\(43\!\cdots\!04\)\( \nu^{3} + \)\(18\!\cdots\!08\)\( \nu^{2} + \)\(94\!\cdots\!28\)\( \nu + \)\(13\!\cdots\!44\)\(\)\()/ \)\(20\!\cdots\!16\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(28\!\cdots\!09\)\( \nu^{17} + \)\(15\!\cdots\!56\)\( \nu^{16} - \)\(18\!\cdots\!54\)\( \nu^{15} + \)\(52\!\cdots\!38\)\( \nu^{14} - \)\(56\!\cdots\!76\)\( \nu^{13} + \)\(15\!\cdots\!79\)\( \nu^{12} - \)\(10\!\cdots\!43\)\( \nu^{11} + \)\(20\!\cdots\!84\)\( \nu^{10} - \)\(11\!\cdots\!11\)\( \nu^{9} + \)\(26\!\cdots\!06\)\( \nu^{8} - \)\(72\!\cdots\!65\)\( \nu^{7} + \)\(12\!\cdots\!05\)\( \nu^{6} - \)\(18\!\cdots\!05\)\( \nu^{5} + \)\(60\!\cdots\!12\)\( \nu^{4} - \)\(24\!\cdots\!04\)\( \nu^{3} + \)\(25\!\cdots\!56\)\( \nu^{2} + \)\(12\!\cdots\!64\)\( \nu + \)\(74\!\cdots\!32\)\(\)\()/ \)\(31\!\cdots\!64\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(12\!\cdots\!01\)\( \nu^{17} + \)\(20\!\cdots\!81\)\( \nu^{16} + \)\(59\!\cdots\!42\)\( \nu^{15} + \)\(27\!\cdots\!56\)\( \nu^{14} + \)\(23\!\cdots\!06\)\( \nu^{13} + \)\(10\!\cdots\!37\)\( \nu^{12} + \)\(49\!\cdots\!60\)\( \nu^{11} + \)\(24\!\cdots\!35\)\( \nu^{10} + \)\(80\!\cdots\!23\)\( \nu^{9} + \)\(30\!\cdots\!45\)\( \nu^{8} + \)\(60\!\cdots\!63\)\( \nu^{7} + \)\(21\!\cdots\!68\)\( \nu^{6} + \)\(32\!\cdots\!84\)\( \nu^{5} + \)\(84\!\cdots\!33\)\( \nu^{4} + \)\(55\!\cdots\!40\)\( \nu^{3} + \)\(10\!\cdots\!80\)\( \nu^{2} + \)\(78\!\cdots\!52\)\( \nu + \)\(90\!\cdots\!72\)\(\)\()/ \)\(10\!\cdots\!08\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(11\!\cdots\!13\)\( \nu^{17} - \)\(57\!\cdots\!84\)\( \nu^{16} + \)\(78\!\cdots\!98\)\( \nu^{15} - \)\(19\!\cdots\!42\)\( \nu^{14} + \)\(25\!\cdots\!24\)\( \nu^{13} - \)\(52\!\cdots\!59\)\( \nu^{12} + \)\(51\!\cdots\!63\)\( \nu^{11} - \)\(66\!\cdots\!68\)\( \nu^{10} + \)\(63\!\cdots\!67\)\( \nu^{9} - \)\(84\!\cdots\!98\)\( \nu^{8} + \)\(47\!\cdots\!97\)\( \nu^{7} - \)\(45\!\cdots\!77\)\( \nu^{6} + \)\(19\!\cdots\!33\)\( \nu^{5} - \)\(20\!\cdots\!28\)\( \nu^{4} + \)\(51\!\cdots\!28\)\( \nu^{3} - \)\(45\!\cdots\!20\)\( \nu^{2} + \)\(63\!\cdots\!64\)\( \nu - \)\(33\!\cdots\!56\)\(\)\()/ \)\(81\!\cdots\!64\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(38\!\cdots\!47\)\( \nu^{17} + \)\(54\!\cdots\!82\)\( \nu^{16} - \)\(20\!\cdots\!86\)\( \nu^{15} - \)\(17\!\cdots\!42\)\( \nu^{14} - \)\(71\!\cdots\!92\)\( \nu^{13} - \)\(87\!\cdots\!59\)\( \nu^{12} - \)\(14\!\cdots\!95\)\( \nu^{11} - \)\(29\!\cdots\!26\)\( \nu^{10} - \)\(19\!\cdots\!57\)\( \nu^{9} - \)\(35\!\cdots\!48\)\( \nu^{8} - \)\(13\!\cdots\!23\)\( \nu^{7} - \)\(27\!\cdots\!87\)\( \nu^{6} - \)\(63\!\cdots\!25\)\( \nu^{5} - \)\(99\!\cdots\!78\)\( \nu^{4} - \)\(10\!\cdots\!28\)\( \nu^{3} - \)\(16\!\cdots\!20\)\( \nu^{2} - \)\(57\!\cdots\!48\)\( \nu - \)\(14\!\cdots\!88\)\(\)\()/ \)\(20\!\cdots\!16\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(28\!\cdots\!69\)\( \nu^{17} + \)\(92\!\cdots\!72\)\( \nu^{16} - \)\(15\!\cdots\!34\)\( \nu^{15} + \)\(14\!\cdots\!90\)\( \nu^{14} - \)\(49\!\cdots\!64\)\( \nu^{13} + \)\(30\!\cdots\!19\)\( \nu^{12} - \)\(91\!\cdots\!19\)\( \nu^{11} - \)\(28\!\cdots\!72\)\( \nu^{10} - \)\(10\!\cdots\!83\)\( \nu^{9} + \)\(25\!\cdots\!50\)\( \nu^{8} - \)\(53\!\cdots\!25\)\( \nu^{7} - \)\(29\!\cdots\!27\)\( \nu^{6} - \)\(16\!\cdots\!37\)\( \nu^{5} + \)\(14\!\cdots\!48\)\( \nu^{4} + \)\(89\!\cdots\!48\)\( \nu^{3} + \)\(23\!\cdots\!16\)\( \nu^{2} - \)\(17\!\cdots\!68\)\( \nu + \)\(11\!\cdots\!84\)\(\)\()/ \)\(12\!\cdots\!56\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(54\!\cdots\!69\)\( \nu^{17} - \)\(90\!\cdots\!38\)\( \nu^{16} + \)\(28\!\cdots\!98\)\( \nu^{15} + \)\(19\!\cdots\!02\)\( \nu^{14} + \)\(98\!\cdots\!64\)\( \nu^{13} + \)\(97\!\cdots\!41\)\( \nu^{12} + \)\(18\!\cdots\!53\)\( \nu^{11} + \)\(35\!\cdots\!50\)\( \nu^{10} + \)\(25\!\cdots\!63\)\( \nu^{9} + \)\(40\!\cdots\!60\)\( \nu^{8} + \)\(16\!\cdots\!69\)\( \nu^{7} + \)\(31\!\cdots\!57\)\( \nu^{6} + \)\(79\!\cdots\!59\)\( \nu^{5} + \)\(10\!\cdots\!62\)\( \nu^{4} + \)\(12\!\cdots\!00\)\( \nu^{3} + \)\(19\!\cdots\!44\)\( \nu^{2} + \)\(26\!\cdots\!60\)\( \nu + \)\(17\!\cdots\!56\)\(\)\()/ \)\(20\!\cdots\!16\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(25\!\cdots\!97\)\( \nu^{17} - \)\(97\!\cdots\!00\)\( \nu^{16} + \)\(15\!\cdots\!66\)\( \nu^{15} - \)\(23\!\cdots\!78\)\( \nu^{14} + \)\(48\!\cdots\!96\)\( \nu^{13} - \)\(62\!\cdots\!39\)\( \nu^{12} + \)\(91\!\cdots\!27\)\( \nu^{11} - \)\(43\!\cdots\!56\)\( \nu^{10} + \)\(10\!\cdots\!47\)\( \nu^{9} - \)\(80\!\cdots\!86\)\( \nu^{8} + \)\(64\!\cdots\!45\)\( \nu^{7} - \)\(29\!\cdots\!45\)\( \nu^{6} + \)\(21\!\cdots\!09\)\( \nu^{5} - \)\(27\!\cdots\!72\)\( \nu^{4} + \)\(16\!\cdots\!24\)\( \nu^{3} - \)\(39\!\cdots\!08\)\( \nu^{2} + \)\(26\!\cdots\!68\)\( \nu - \)\(63\!\cdots\!36\)\(\)\()/ \)\(81\!\cdots\!64\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(82\!\cdots\!31\)\( \nu^{17} - \)\(25\!\cdots\!18\)\( \nu^{16} + \)\(48\!\cdots\!62\)\( \nu^{15} - \)\(39\!\cdots\!78\)\( \nu^{14} + \)\(15\!\cdots\!08\)\( \nu^{13} - \)\(66\!\cdots\!17\)\( \nu^{12} + \)\(31\!\cdots\!71\)\( \nu^{11} + \)\(13\!\cdots\!18\)\( \nu^{10} + \)\(39\!\cdots\!65\)\( \nu^{9} + \)\(14\!\cdots\!20\)\( \nu^{8} + \)\(26\!\cdots\!39\)\( \nu^{7} + \)\(21\!\cdots\!43\)\( \nu^{6} + \)\(10\!\cdots\!25\)\( \nu^{5} + \)\(72\!\cdots\!26\)\( \nu^{4} + \)\(16\!\cdots\!60\)\( \nu^{3} + \)\(18\!\cdots\!08\)\( \nu^{2} + \)\(88\!\cdots\!12\)\( \nu + \)\(18\!\cdots\!88\)\(\)\()/ \)\(20\!\cdots\!16\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(65\!\cdots\!29\)\( \nu^{17} + \)\(24\!\cdots\!60\)\( \nu^{16} - \)\(38\!\cdots\!42\)\( \nu^{15} + \)\(55\!\cdots\!10\)\( \nu^{14} - \)\(12\!\cdots\!60\)\( \nu^{13} + \)\(14\!\cdots\!55\)\( \nu^{12} - \)\(23\!\cdots\!47\)\( \nu^{11} + \)\(75\!\cdots\!24\)\( \nu^{10} - \)\(27\!\cdots\!35\)\( \nu^{9} + \)\(16\!\cdots\!58\)\( \nu^{8} - \)\(15\!\cdots\!49\)\( \nu^{7} + \)\(52\!\cdots\!69\)\( \nu^{6} - \)\(52\!\cdots\!81\)\( \nu^{5} + \)\(63\!\cdots\!00\)\( \nu^{4} - \)\(29\!\cdots\!52\)\( \nu^{3} + \)\(93\!\cdots\!92\)\( \nu^{2} - \)\(62\!\cdots\!20\)\( \nu + \)\(10\!\cdots\!20\)\(\)\()/ \)\(12\!\cdots\!56\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(11\!\cdots\!21\)\( \nu^{17} - \)\(50\!\cdots\!84\)\( \nu^{16} + \)\(70\!\cdots\!78\)\( \nu^{15} - \)\(15\!\cdots\!42\)\( \nu^{14} + \)\(22\!\cdots\!92\)\( \nu^{13} - \)\(41\!\cdots\!11\)\( \nu^{12} + \)\(44\!\cdots\!71\)\( \nu^{11} - \)\(45\!\cdots\!80\)\( \nu^{10} + \)\(52\!\cdots\!79\)\( \nu^{9} - \)\(64\!\cdots\!74\)\( \nu^{8} + \)\(34\!\cdots\!45\)\( \nu^{7} - \)\(30\!\cdots\!89\)\( \nu^{6} + \)\(12\!\cdots\!85\)\( \nu^{5} - \)\(17\!\cdots\!44\)\( \nu^{4} + \)\(18\!\cdots\!88\)\( \nu^{3} - \)\(24\!\cdots\!04\)\( \nu^{2} + \)\(15\!\cdots\!68\)\( \nu - \)\(19\!\cdots\!44\)\(\)\()/ \)\(16\!\cdots\!28\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(12\!\cdots\!63\)\( \nu^{17} - \)\(45\!\cdots\!24\)\( \nu^{16} + \)\(73\!\cdots\!78\)\( \nu^{15} - \)\(11\!\cdots\!66\)\( \nu^{14} + \)\(23\!\cdots\!16\)\( \nu^{13} - \)\(28\!\cdots\!73\)\( \nu^{12} + \)\(46\!\cdots\!21\)\( \nu^{11} - \)\(17\!\cdots\!12\)\( \nu^{10} + \)\(57\!\cdots\!53\)\( \nu^{9} - \)\(27\!\cdots\!10\)\( \nu^{8} + \)\(39\!\cdots\!95\)\( \nu^{7} - \)\(47\!\cdots\!19\)\( \nu^{6} + \)\(16\!\cdots\!55\)\( \nu^{5} - \)\(62\!\cdots\!92\)\( \nu^{4} + \)\(30\!\cdots\!16\)\( \nu^{3} - \)\(49\!\cdots\!84\)\( \nu^{2} + \)\(33\!\cdots\!92\)\( \nu - \)\(14\!\cdots\!88\)\(\)\()/ \)\(16\!\cdots\!28\)\( \)

Display \(\nu^j\) in terms of \(\beta_i\)

\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{17} + \beta_{16} - \beta_{13} + \beta_{10} + \beta_{6} - 12 \beta_{3} - 2 \beta_{2} + 2 \beta_{1}\)
\(\nu^{3}\)\(=\)\(-2 \beta_{17} + 3 \beta_{16} - 3 \beta_{13} + 2 \beta_{12} - \beta_{11} - 4 \beta_{9} + 4 \beta_{7} - 23 \beta_{2} - 12\)
\(\nu^{4}\)\(=\)\(-8 \beta_{14} + 27 \beta_{12} - 37 \beta_{10} + 8 \beta_{7} - 32 \beta_{6} - 5 \beta_{4} + 260 \beta_{3} - 77 \beta_{1} - 260\)
\(\nu^{5}\)\(=\)\(75 \beta_{17} - 128 \beta_{16} + 10 \beta_{15} - 45 \beta_{14} + 166 \beta_{13} + 45 \beta_{11} - 166 \beta_{10} + 138 \beta_{9} - 10 \beta_{8} - 128 \beta_{6} - 18 \beta_{5} - 18 \beta_{4} + 604 \beta_{3} + 636 \beta_{2} - 636 \beta_{1}\)
\(\nu^{6}\)\(=\)\(748 \beta_{17} - 1004 \beta_{16} + 46 \beta_{15} + 1315 \beta_{13} - 748 \beta_{12} + 339 \beta_{11} + 490 \beta_{9} - 490 \beta_{7} - 299 \beta_{5} + 2746 \beta_{2} + 6752\)
\(\nu^{7}\)\(=\)\(1762 \beta_{14} - 2684 \beta_{12} + 6896 \beta_{10} + 644 \beta_{8} - 4588 \beta_{7} + 4677 \beta_{6} + 1299 \beta_{4} - 23360 \beta_{3} + 19224 \beta_{1} + 23360\)
\(\nu^{8}\)\(=\)\(-21858 \beta_{17} + 32278 \beta_{16} - 3242 \beta_{15} + 12178 \beta_{14} - 46550 \beta_{13} - 12178 \beta_{11} + 46550 \beta_{10} - 21190 \beta_{9} + 3242 \beta_{8} + 32278 \beta_{6} + 12940 \beta_{5} + 12940 \beta_{4} - 192772 \beta_{3} - 95033 \beta_{2} + 95033 \beta_{1}\)
\(\nu^{9}\)\(=\)\(-93585 \beta_{17} + 164491 \beta_{16} - 29122 \beta_{15} - 260223 \beta_{13} + 93585 \beta_{12} - 65220 \beta_{11} - 155738 \beta_{9} + 155738 \beta_{7} + 63424 \beta_{5} - 609880 \beta_{2} - 831272\)
\(\nu^{10}\)\(=\)\(-423443 \beta_{14} + 667892 \beta_{12} - 1637813 \beta_{10} - 155970 \beta_{8} + 812858 \beta_{7} - 1058555 \beta_{6} - 500294 \beta_{4} + 5853008 \beta_{3} - 3244441 \beta_{1} - 5853008\)
\(\nu^{11}\)\(=\)\(3212681 \beta_{17} - 5697380 \beta_{16} + 1156558 \beta_{15} - 2341040 \beta_{14} + 9404329 \beta_{13} + 2341040 \beta_{11} - 9404329 \beta_{10} + 5347886 \beta_{9} - 1156558 \beta_{8} - 5697380 \beta_{6} - 2641593 \beta_{5} - 2641593 \beta_{4} + 28645472 \beta_{3} + 19896385 \beta_{2} - 19896385 \beta_{1}\)
\(\nu^{12}\)\(=\)\(21117643 \beta_{17} - 35204052 \beta_{16} + 6439744 \beta_{15} + 57218756 \beta_{13} - 21117643 \beta_{12} + 14595321 \beta_{11} + 29575380 \beta_{9} - 29575380 \beta_{7} - 18387680 \beta_{5} + 110201064 \beta_{2} + 185118648\)
\(\nu^{13}\)\(=\)\(82492865 \beta_{14} - 109440828 \beta_{12} + 332264719 \beta_{10} + 43215104 \beta_{8} - 184233596 \beta_{7} + 195850226 \beta_{6} + 101502237 \beta_{4} - 974986692 \beta_{3} + 659993730 \beta_{1} + 974986692\)
\(\nu^{14}\)\(=\)\(-684437956 \beta_{17} + 1181688405 \beta_{16} - 246219578 \beta_{15} + 501405476 \beta_{14} - 1986451992 \beta_{13} - 501405476 \beta_{11} + 1986451992 \beta_{10} - 1047709358 \beta_{9} + 246219578 \beta_{8} + 1181688405 \beta_{6} + 658045263 \beta_{5} + 658045263 \beta_{4} - 6013474260 \beta_{3} - 3738644514 \beta_{2} + 3738644514 \beta_{1}\)
\(\nu^{15}\)\(=\)\(-3716482924 \beta_{17} + 6705745868 \beta_{16} - 1562310104 \beta_{15} - 11588149060 \beta_{13} + 3716482924 \beta_{12} - 2873366536 \beta_{11} - 6343709628 \beta_{9} + 6343709628 \beta_{7} + 3727539608 \beta_{5} - 22116656065 \beta_{2} - 33051801880\)
\(\nu^{16}\)\(=\)\(-17193582024 \beta_{14} + 22567596581 \beta_{12} - 68620566389 \beta_{10} - 9017389320 \beta_{8} + 36572897116 \beta_{7} - 39902478713 \beta_{6} - 23181789868 \beta_{4} + 198745293060 \beta_{3} - 126885353254 \beta_{1} - 198745293060\)
\(\nu^{17}\)\(=\)\(126112059954 \beta_{17} - 229119790391 \beta_{16} + 55380969056 \beta_{15} - 99346555769 \beta_{14} + 400991725275 \beta_{13} + 99346555769 \beta_{11} - 400991725275 \beta_{10} + 218067369976 \beta_{9} - 55380969056 \beta_{8} - 229119790391 \beta_{6} - 133249111332 \beta_{5} - 133249111332 \beta_{4} + 1119843402916 \beta_{3} + 745830358799 \beta_{2} - 745830358799 \beta_{1}\)

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{3}\)

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
2.92009 + 5.05774i
2.41719 + 4.18670i
1.36382 + 2.36220i
0.695058 + 1.20388i
0.425471 + 0.736937i
−0.613994 − 1.06347i
−1.08068 − 1.87179i
−1.91278 − 3.31303i
−2.21418 − 3.83507i
2.92009 − 5.05774i
2.41719 − 4.18670i
1.36382 − 2.36220i
0.695058 − 1.20388i
0.425471 − 0.736937i
−0.613994 + 1.06347i
−1.08068 + 1.87179i
−1.91278 + 3.31303i
−2.21418 + 3.83507i
−2.42009 + 4.19172i 3.09831 − 5.36643i −7.71365 − 13.3604i −15.2399 14.9964 + 25.9745i 2.15810 + 3.73794i 35.9495 −5.69903 − 9.87102i 36.8818 − 63.8811i
22.2 −1.91719 + 3.32067i 0.139581 − 0.241762i −3.35124 − 5.80452i −11.3710 0.535209 + 0.927008i 15.5311 + 26.9007i −4.97517 13.4610 + 23.3152i 21.8004 − 37.7594i
22.3 −0.863817 + 1.49617i −3.44796 + 5.97204i 2.50764 + 4.34336i −20.8281 −5.95681 − 10.3175i 3.78283 + 6.55206i −22.4856 −10.2768 − 17.8000i 17.9916 − 31.1624i
22.4 −0.195058 + 0.337850i −1.80483 + 3.12606i 3.92391 + 6.79640i 7.52136 −0.704093 − 1.21953i 9.77228 + 16.9261i −6.18247 6.98515 + 12.0986i −1.46710 + 2.54109i
22.5 0.0745292 − 0.129088i 3.24429 − 5.61927i 3.98889 + 6.90896i 10.2526 −0.483588 − 0.837600i −14.8372 − 25.6987i 2.38162 −7.55083 − 13.0784i 0.764114 − 1.32348i
22.6 1.11399 − 1.92949i 4.87434 − 8.44260i 1.51803 + 2.62931i 8.20685 −10.8600 − 18.8100i 4.17747 + 7.23560i 24.5882 −34.0183 − 58.9214i 9.14239 − 15.8351i
22.7 1.58068 − 2.73781i −3.54442 + 6.13911i −0.997073 − 1.72698i −13.6039 11.2051 + 19.4079i −7.16574 − 12.4114i 18.9866 −11.6258 − 20.1364i −21.5034 + 37.2450i
22.8 2.41278 − 4.17905i −2.22176 + 3.84820i −7.64299 − 13.2380i 12.7712 10.7212 + 18.5697i 13.0936 + 22.6787i −35.1589 3.62756 + 6.28312i 30.8140 − 53.3715i
22.9 2.71418 − 4.70109i −0.837548 + 1.45068i −10.7335 − 18.5910i −7.70909 4.54651 + 7.87478i −7.51249 − 13.0120i −73.1038 12.0970 + 20.9527i −20.9238 + 36.2412i
146.1 −2.42009 − 4.19172i 3.09831 + 5.36643i −7.71365 + 13.3604i −15.2399 14.9964 − 25.9745i 2.15810 − 3.73794i 35.9495 −5.69903 + 9.87102i 36.8818 + 63.8811i
146.2 −1.91719 − 3.32067i 0.139581 + 0.241762i −3.35124 + 5.80452i −11.3710 0.535209 − 0.927008i 15.5311 − 26.9007i −4.97517 13.4610 − 23.3152i 21.8004 + 37.7594i
146.3 −0.863817 − 1.49617i −3.44796 − 5.97204i 2.50764 − 4.34336i −20.8281 −5.95681 + 10.3175i 3.78283 − 6.55206i −22.4856 −10.2768 + 17.8000i 17.9916 + 31.1624i
146.4 −0.195058 − 0.337850i −1.80483 − 3.12606i 3.92391 − 6.79640i 7.52136 −0.704093 + 1.21953i 9.77228 − 16.9261i −6.18247 6.98515 − 12.0986i −1.46710 − 2.54109i
146.5 0.0745292 + 0.129088i 3.24429 + 5.61927i 3.98889 − 6.90896i 10.2526 −0.483588 + 0.837600i −14.8372 + 25.6987i 2.38162 −7.55083 + 13.0784i 0.764114 + 1.32348i
146.6 1.11399 + 1.92949i 4.87434 + 8.44260i 1.51803 − 2.62931i 8.20685 −10.8600 + 18.8100i 4.17747 − 7.23560i 24.5882 −34.0183 + 58.9214i 9.14239 + 15.8351i
146.7 1.58068 + 2.73781i −3.54442 − 6.13911i −0.997073 + 1.72698i −13.6039 11.2051 − 19.4079i −7.16574 + 12.4114i 18.9866 −11.6258 + 20.1364i −21.5034 − 37.2450i
146.8 2.41278 + 4.17905i −2.22176 − 3.84820i −7.64299 + 13.2380i 12.7712 10.7212 − 18.5697i 13.0936 − 22.6787i −35.1589 3.62756 − 6.28312i 30.8140 + 53.3715i
146.9 2.71418 + 4.70109i −0.837548 − 1.45068i −10.7335 + 18.5910i −7.70909 4.54651 − 7.87478i −7.51249 + 13.0120i −73.1038 12.0970 − 20.9527i −20.9238 − 36.2412i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 146.9
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.c.l 18
13.b even 2 1 169.4.c.k 18
13.c even 3 1 169.4.a.k ✓ 9
13.c even 3 1 inner 169.4.c.l 18
13.d odd 4 2 169.4.e.h 36
13.e even 6 1 169.4.a.l yes 9
13.e even 6 1 169.4.c.k 18
13.f odd 12 2 169.4.b.g 18
13.f odd 12 2 169.4.e.h 36
39.h odd 6 1 1521.4.a.bg 9
39.i odd 6 1 1521.4.a.bh 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.4.a.k ✓ 9 13.c even 3 1
169.4.a.l yes 9 13.e even 6 1
169.4.b.g 18 13.f odd 12 2
169.4.c.k 18 13.b even 2 1
169.4.c.k 18 13.e even 6 1
169.4.c.l 18 1.a even 1 1 trivial
169.4.c.l 18 13.c even 3 1 inner
169.4.e.h 36 13.d odd 4 2
169.4.e.h 36 13.f odd 12 2
1521.4.a.bg 9 39.h odd 6 1
1521.4.a.bh 9 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{18} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 118336 - 503616 T + 4172208 T^{2} + 7769856 T^{3} + 35832012 T^{4} - 817290 T^{5} + 14351885 T^{6} - 2514618 T^{7} + 4613216 T^{8} - 861638 T^{9} + 596508 T^{10} - 82227 T^{11} + 48373 T^{12} - 6060 T^{13} + 2303 T^{14} - 200 T^{15} + 67 T^{16} - 5 T^{17} + T^{18} \)
$3$ \( 20654576089 - 54013160110 T + 209182639932 T^{2} + 205685338530 T^{3} + 183274375865 T^{4} + 65521249045 T^{5} + 23689660457 T^{6} + 4577376767 T^{7} + 1250431893 T^{8} + 179270955 T^{9} + 45680240 T^{10} + 4360348 T^{11} + 1025108 T^{12} + 65249 T^{13} + 16830 T^{14} + 564 T^{15} + 155 T^{16} + T^{17} + T^{18} \)
$5$ \( ( 3059376152 - 79067576 T - 128609278 T^{2} + 1389429 T^{3} + 1945112 T^{4} + 11193 T^{5} - 12686 T^{6} - 266 T^{7} + 30 T^{8} + T^{9} )^{2} \)
$7$ \( \)\(76\!\cdots\!76\)\( - \)\(33\!\cdots\!32\)\( T + \)\(11\!\cdots\!16\)\( T^{2} - 19224338739789056640 T^{3} + 2893409191989213108 T^{4} - 268619366451258348 T^{5} + 28521475416470489 T^{6} - 1972561707995316 T^{7} + 195493360188416 T^{8} - 9517797571076 T^{9} + 755535359241 T^{10} - 30777655758 T^{11} + 2045718934 T^{12} - 61943904 T^{13} + 2866673 T^{14} - 59438 T^{15} + 2467 T^{16} - 38 T^{17} + T^{18} \)
$11$ \( \)\(76\!\cdots\!61\)\( + \)\(49\!\cdots\!14\)\( T + \)\(24\!\cdots\!68\)\( T^{2} + \)\(52\!\cdots\!62\)\( T^{3} + \)\(90\!\cdots\!29\)\( T^{4} + 41659892052777689127 T^{5} + 4077823442728502777 T^{6} - 164280923855853411 T^{7} + 23826237278602657 T^{8} - 990214581519071 T^{9} + 63666167961744 T^{10} - 2813297344768 T^{11} + 122331582028 T^{12} - 3739044709 T^{13} + 93306698 T^{14} - 1630424 T^{15} + 21927 T^{16} - 181 T^{17} + T^{18} \)
$13$ \( T^{18} \)
$17$ \( \)\(31\!\cdots\!49\)\( + \)\(75\!\cdots\!80\)\( T + \)\(23\!\cdots\!96\)\( T^{2} - \)\(11\!\cdots\!74\)\( T^{3} + \)\(67\!\cdots\!05\)\( T^{4} - \)\(20\!\cdots\!71\)\( T^{5} + \)\(10\!\cdots\!27\)\( T^{6} - \)\(98\!\cdots\!25\)\( T^{7} + 69974812476298858573 T^{8} - 682920665683323197 T^{9} + 23462918876298010 T^{10} - 126956287652076 T^{11} + 3958932521362 T^{12} - 17359119377 T^{13} + 464386442 T^{14} - 1003250 T^{15} + 26461 T^{16} - 55 T^{17} + T^{18} \)
$19$ \( \)\(74\!\cdots\!61\)\( - \)\(15\!\cdots\!74\)\( T + \)\(67\!\cdots\!18\)\( T^{2} + \)\(11\!\cdots\!90\)\( T^{3} + \)\(49\!\cdots\!07\)\( T^{4} + \)\(17\!\cdots\!09\)\( T^{5} + \)\(48\!\cdots\!99\)\( T^{6} - \)\(19\!\cdots\!83\)\( T^{7} + 33089443442948329229 T^{8} - 193806745823502525 T^{9} + 12762004883407470 T^{10} - 106029853059354 T^{11} + 3565893354702 T^{12} - 24282726391 T^{13} + 447681636 T^{14} - 2415378 T^{15} + 37213 T^{16} - 161 T^{17} + T^{18} \)
$23$ \( \)\(46\!\cdots\!64\)\( - \)\(11\!\cdots\!12\)\( T + \)\(93\!\cdots\!36\)\( T^{2} - \)\(74\!\cdots\!48\)\( T^{3} + \)\(94\!\cdots\!92\)\( T^{4} - \)\(53\!\cdots\!52\)\( T^{5} + \)\(57\!\cdots\!17\)\( T^{6} + \)\(14\!\cdots\!96\)\( T^{7} + \)\(17\!\cdots\!47\)\( T^{8} + 1014270614343138636 T^{9} + 259831752914550378 T^{10} - 33868408759588 T^{11} + 27695381664963 T^{12} - 53353907516 T^{13} + 1594777466 T^{14} - 4872604 T^{15} + 70747 T^{16} - 204 T^{17} + T^{18} \)
$29$ \( \)\(31\!\cdots\!16\)\( - \)\(32\!\cdots\!52\)\( T + \)\(30\!\cdots\!36\)\( T^{2} - \)\(11\!\cdots\!76\)\( T^{3} + \)\(50\!\cdots\!52\)\( T^{4} - \)\(11\!\cdots\!42\)\( T^{5} + \)\(51\!\cdots\!29\)\( T^{6} - \)\(88\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!74\)\( T^{8} - 41816555190287721978 T^{9} + 2239335757883924443 T^{10} - 1435571713780738 T^{11} + 154332745752460 T^{12} + 195704076088 T^{13} + 4898684195 T^{14} + 11546440 T^{15} + 126935 T^{16} + 280 T^{17} + T^{18} \)
$31$ \( ( -3096277304854445656 - 65749435529097852 T + 1985161101259952 T^{2} + 25565731686363 T^{3} - 228326104668 T^{4} - 3609315492 T^{5} - 1130104 T^{6} + 148229 T^{7} + 706 T^{8} + T^{9} )^{2} \)
$37$ \( \)\(96\!\cdots\!04\)\( - \)\(27\!\cdots\!04\)\( T + \)\(64\!\cdots\!48\)\( T^{2} - \)\(77\!\cdots\!60\)\( T^{3} + \)\(93\!\cdots\!60\)\( T^{4} - \)\(82\!\cdots\!90\)\( T^{5} + \)\(76\!\cdots\!65\)\( T^{6} - \)\(53\!\cdots\!80\)\( T^{7} + \)\(35\!\cdots\!55\)\( T^{8} - \)\(16\!\cdots\!52\)\( T^{9} + \)\(73\!\cdots\!07\)\( T^{10} - 2351148959751075498 T^{11} + 8897795109502448 T^{12} - 21949929757352 T^{13} + 72098604943 T^{14} - 106714272 T^{15} + 320902 T^{16} - 298 T^{17} + T^{18} \)
$41$ \( \)\(38\!\cdots\!09\)\( - \)\(10\!\cdots\!64\)\( T + \)\(27\!\cdots\!58\)\( T^{2} - \)\(49\!\cdots\!10\)\( T^{3} + \)\(97\!\cdots\!10\)\( T^{4} - \)\(14\!\cdots\!29\)\( T^{5} + \)\(20\!\cdots\!41\)\( T^{6} - \)\(20\!\cdots\!49\)\( T^{7} + \)\(17\!\cdots\!12\)\( T^{8} - \)\(11\!\cdots\!02\)\( T^{9} + \)\(66\!\cdots\!14\)\( T^{10} - 3286174082313580957 T^{11} + 14959878253000499 T^{12} - 57390036989386 T^{13} + 190067461824 T^{14} - 479813819 T^{15} + 948266 T^{16} - 1201 T^{17} + T^{18} \)
$43$ \( \)\(12\!\cdots\!29\)\( - \)\(16\!\cdots\!94\)\( T + \)\(47\!\cdots\!49\)\( T^{2} - \)\(67\!\cdots\!52\)\( T^{3} + \)\(13\!\cdots\!96\)\( T^{4} - \)\(15\!\cdots\!39\)\( T^{5} + \)\(16\!\cdots\!36\)\( T^{6} - \)\(10\!\cdots\!26\)\( T^{7} + \)\(64\!\cdots\!98\)\( T^{8} - \)\(28\!\cdots\!37\)\( T^{9} + \)\(12\!\cdots\!33\)\( T^{10} - 4555308418580028902 T^{11} + 16267885437294289 T^{12} - 42136908293233 T^{13} + 110886289945 T^{14} - 192054941 T^{15} + 437168 T^{16} - 533 T^{17} + T^{18} \)
$47$ \( ( 11464353104027851384 + 1329362167362062820 T + 32303753105700258 T^{2} - 49762716955757 T^{3} - 6545051933164 T^{4} - 53107853183 T^{5} - 127738074 T^{6} + 136612 T^{7} + 956 T^{8} + T^{9} )^{2} \)
$53$ \( ( -\)\(12\!\cdots\!76\)\( + \)\(24\!\cdots\!24\)\( T - 412869644476868336 T^{2} - 9184309039838179 T^{3} + 15501811028404 T^{4} + 112216880490 T^{5} - 123482118 T^{6} - 566375 T^{7} + 278 T^{8} + T^{9} )^{2} \)
$59$ \( \)\(75\!\cdots\!29\)\( - \)\(27\!\cdots\!52\)\( T + \)\(69\!\cdots\!90\)\( T^{2} - \)\(97\!\cdots\!74\)\( T^{3} + \)\(10\!\cdots\!20\)\( T^{4} - \)\(83\!\cdots\!79\)\( T^{5} + \)\(58\!\cdots\!25\)\( T^{6} - \)\(34\!\cdots\!83\)\( T^{7} + \)\(18\!\cdots\!36\)\( T^{8} - \)\(84\!\cdots\!12\)\( T^{9} + \)\(32\!\cdots\!98\)\( T^{10} - 95993064355437258163 T^{11} + 236806641363115491 T^{12} - 438914700029708 T^{13} + 780836396140 T^{14} - 1092489505 T^{15} + 1643196 T^{16} - 1377 T^{17} + T^{18} \)
$61$ \( \)\(75\!\cdots\!24\)\( - \)\(31\!\cdots\!56\)\( T + \)\(96\!\cdots\!80\)\( T^{2} - \)\(13\!\cdots\!32\)\( T^{3} + \)\(13\!\cdots\!44\)\( T^{4} - \)\(78\!\cdots\!16\)\( T^{5} + \)\(35\!\cdots\!13\)\( T^{6} - \)\(73\!\cdots\!46\)\( T^{7} + \)\(25\!\cdots\!87\)\( T^{8} - \)\(35\!\cdots\!86\)\( T^{9} + \)\(12\!\cdots\!27\)\( T^{10} - \)\(10\!\cdots\!30\)\( T^{11} + 393351043130390100 T^{12} - 192408323394130 T^{13} + 901450183727 T^{14} - 221280596 T^{15} + 1149502 T^{16} - 136 T^{17} + T^{18} \)
$67$ \( \)\(10\!\cdots\!01\)\( - \)\(17\!\cdots\!74\)\( T + \)\(23\!\cdots\!45\)\( T^{2} - \)\(11\!\cdots\!68\)\( T^{3} + \)\(67\!\cdots\!88\)\( T^{4} - \)\(16\!\cdots\!99\)\( T^{5} + \)\(90\!\cdots\!08\)\( T^{6} - \)\(15\!\cdots\!62\)\( T^{7} + \)\(79\!\cdots\!90\)\( T^{8} - \)\(30\!\cdots\!17\)\( T^{9} + \)\(32\!\cdots\!09\)\( T^{10} - 3856375795997781174 T^{11} + 956985045477793677 T^{12} + 328870743200719 T^{13} + 1637041859981 T^{14} + 719289659 T^{15} + 2008096 T^{16} + 931 T^{17} + T^{18} \)
$71$ \( \)\(19\!\cdots\!84\)\( - \)\(85\!\cdots\!64\)\( T + \)\(58\!\cdots\!04\)\( T^{2} + \)\(10\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!80\)\( T^{4} + \)\(54\!\cdots\!16\)\( T^{5} + \)\(26\!\cdots\!73\)\( T^{6} + \)\(14\!\cdots\!10\)\( T^{7} + \)\(12\!\cdots\!76\)\( T^{8} - \)\(38\!\cdots\!08\)\( T^{9} + \)\(54\!\cdots\!47\)\( T^{10} - \)\(49\!\cdots\!06\)\( T^{11} + 1559491110884811770 T^{12} - 1737784965963112 T^{13} + 3493641737169 T^{14} - 3527965926 T^{15} + 3877829 T^{16} - 2046 T^{17} + T^{18} \)
$73$ \( ( -\)\(31\!\cdots\!21\)\( - \)\(15\!\cdots\!50\)\( T + 17310298486778544639 T^{2} - 32424645086193109 T^{3} - 181643944920255 T^{4} + 609692221608 T^{5} + 173876306 T^{6} - 1609687 T^{7} - 45 T^{8} + T^{9} )^{2} \)
$79$ \( ( \)\(63\!\cdots\!88\)\( + \)\(79\!\cdots\!44\)\( T + 556477632229543614 T^{2} - 21981799979683177 T^{3} - 36065298040498 T^{4} + 234613432336 T^{5} + 282452944 T^{6} - 1032469 T^{7} - 412 T^{8} + T^{9} )^{2} \)
$83$ \( ( \)\(14\!\cdots\!61\)\( - \)\(26\!\cdots\!46\)\( T - \)\(47\!\cdots\!03\)\( T^{2} - 2443290848701223317 T^{3} - 5791016389917361 T^{4} - 6314534308340 T^{5} - 1123208654 T^{6} + 4107377 T^{7} + 3709 T^{8} + T^{9} )^{2} \)
$89$ \( \)\(19\!\cdots\!29\)\( - \)\(24\!\cdots\!62\)\( T + \)\(18\!\cdots\!19\)\( T^{2} - \)\(99\!\cdots\!48\)\( T^{3} + \)\(40\!\cdots\!72\)\( T^{4} - \)\(12\!\cdots\!57\)\( T^{5} + \)\(34\!\cdots\!98\)\( T^{6} - \)\(74\!\cdots\!50\)\( T^{7} + \)\(13\!\cdots\!42\)\( T^{8} - \)\(22\!\cdots\!59\)\( T^{9} + \)\(32\!\cdots\!31\)\( T^{10} - \)\(41\!\cdots\!86\)\( T^{11} + 4971428757385931487 T^{12} - 5127706544590427 T^{13} + 4912234593447 T^{14} - 3817038699 T^{15} + 3036832 T^{16} - 1663 T^{17} + T^{18} \)
$97$ \( \)\(43\!\cdots\!21\)\( - \)\(23\!\cdots\!14\)\( T + \)\(15\!\cdots\!19\)\( T^{2} - \)\(38\!\cdots\!48\)\( T^{3} + \)\(17\!\cdots\!32\)\( T^{4} - \)\(36\!\cdots\!63\)\( T^{5} + \)\(13\!\cdots\!74\)\( T^{6} - \)\(17\!\cdots\!58\)\( T^{7} + \)\(43\!\cdots\!26\)\( T^{8} - \)\(36\!\cdots\!65\)\( T^{9} + \)\(93\!\cdots\!59\)\( T^{10} - \)\(51\!\cdots\!22\)\( T^{11} + 11449991778395028411 T^{12} - 1083140963365609 T^{13} + 8116369698711 T^{14} + 563189279 T^{15} + 4260740 T^{16} + 1087 T^{17} + T^{18} \)
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