Properties

Label 38.4.e.b
Level 38
Weight 4
Character orbit 38.e
Analytic conductor 2.242
Analytic rank 0
Dimension 18
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.24207258022\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial: \(x^{18} + 165 x^{16} - 56 x^{15} + 18435 x^{14} - 11748 x^{13} + 1092662 x^{12} - 1833567 x^{11} + 46842546 x^{10} - 93643115 x^{9} + 1273086000 x^{8} - 3159018183 x^{7} + 24752942108 x^{6} - 51265274250 x^{5} + 247917053361 x^{4} - 421854393530 x^{3} + 1634375353605 x^{2} - 2066139456366 x + 3892796082289\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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.

\(f(q)\) \(=\) \( q -2 \beta_{8} q^{2} + ( \beta_{3} + \beta_{4} + \beta_{6} + \beta_{9} ) q^{3} + 4 \beta_{7} q^{4} + ( \beta_{5} - \beta_{16} ) q^{5} + ( -2 - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{6} + ( -4 + 5 \beta_{3} + 4 \beta_{4} - \beta_{5} + \beta_{6} + 4 \beta_{7} - \beta_{8} - 5 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{7} + 8 \beta_{4} q^{8} + ( 6 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{6} - 12 \beta_{7} - 6 \beta_{8} + 12 \beta_{9} - 3 \beta_{11} + \beta_{12} + \beta_{13} ) q^{9} +O(q^{10})\) \( q -2 \beta_{8} q^{2} + ( \beta_{3} + \beta_{4} + \beta_{6} + \beta_{9} ) q^{3} + 4 \beta_{7} q^{4} + ( \beta_{5} - \beta_{16} ) q^{5} + ( -2 - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{6} + ( -4 + 5 \beta_{3} + 4 \beta_{4} - \beta_{5} + \beta_{6} + 4 \beta_{7} - \beta_{8} - 5 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{7} + 8 \beta_{4} q^{8} + ( 6 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{6} - 12 \beta_{7} - 6 \beta_{8} + 12 \beta_{9} - 3 \beta_{11} + \beta_{12} + \beta_{13} ) q^{9} + 2 \beta_{12} q^{10} + ( -2 \beta_{1} + 2 \beta_{3} - 10 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - 7 \beta_{8} + 7 \beta_{9} + 3 \beta_{10} - \beta_{14} + 2 \beta_{17} ) q^{11} + ( 4 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{8} + 4 \beta_{9} ) q^{12} + ( 3 - 3 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{6} + 6 \beta_{8} - 13 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{13} + ( 10 + 2 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + 8 \beta_{8} - 10 \beta_{9} + 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} - 2 \beta_{17} ) q^{14} + ( -6 + 3 \beta_{1} + 6 \beta_{3} - 9 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} + 9 \beta_{8} + 2 \beta_{10} + 3 \beta_{13} + 3 \beta_{14} + 3 \beta_{15} ) q^{15} -16 \beta_{3} q^{16} + ( -5 + 6 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} - 3 \beta_{7} + 26 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} - 6 \beta_{11} - 2 \beta_{12} + 3 \beta_{13} + 2 \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{17} + ( -24 - 6 \beta_{2} + 8 \beta_{3} - 6 \beta_{6} + 12 \beta_{7} - 12 \beta_{8} - 4 \beta_{9} - 6 \beta_{10} + 4 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{18} + ( -10 + 6 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} - 26 \beta_{4} - 4 \beta_{6} - 14 \beta_{7} + 15 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + 6 \beta_{11} + \beta_{12} - 3 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{19} + ( 4 \beta_{14} + 4 \beta_{15} ) q^{20} + ( -8 + 5 \beta_{1} - 5 \beta_{4} - 13 \beta_{7} - 34 \beta_{8} + 5 \beta_{9} - 5 \beta_{11} - 4 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} + 6 \beta_{16} + 2 \beta_{17} ) q^{21} + ( -14 - 6 \beta_{1} + 20 \beta_{3} + 10 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 14 \beta_{7} - 4 \beta_{9} + 6 \beta_{10} - 6 \beta_{11} + 4 \beta_{12} - 4 \beta_{14} + 4 \beta_{17} ) q^{22} + ( 31 - 3 \beta_{1} - 5 \beta_{2} - 31 \beta_{3} - 43 \beta_{4} + \beta_{5} + 5 \beta_{6} + 35 \beta_{7} + 43 \beta_{8} + 7 \beta_{10} - 3 \beta_{13} + 2 \beta_{14} - \beta_{15} - 3 \beta_{16} - 6 \beta_{17} ) q^{23} + ( -8 + 8 \beta_{3} + 8 \beta_{4} + 8 \beta_{6} - 8 \beta_{7} - 8 \beta_{8} + 8 \beta_{9} - 8 \beta_{13} ) q^{24} + ( 11 - 10 \beta_{1} + 5 \beta_{2} - 44 \beta_{3} + 44 \beta_{4} - 3 \beta_{5} - 10 \beta_{6} + 55 \beta_{8} - 60 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} + 15 \beta_{13} - 3 \beta_{15} + 3 \beta_{17} ) q^{25} + ( 26 - 6 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} - 26 \beta_{4} - 6 \beta_{6} - 12 \beta_{7} - 6 \beta_{8} + 6 \beta_{9} + 10 \beta_{10} - 6 \beta_{11} + 10 \beta_{13} - 2 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} ) q^{26} + ( -6 \beta_{1} + 69 \beta_{3} + 83 \beta_{4} - 2 \beta_{5} + 12 \beta_{6} - 69 \beta_{7} - 100 \beta_{8} + 100 \beta_{9} - 8 \beta_{10} - 4 \beta_{11} - 4 \beta_{13} - 3 \beta_{14} - 2 \beta_{17} ) q^{27} + ( 20 + 4 \beta_{2} + 4 \beta_{3} - 16 \beta_{4} + 4 \beta_{6} - 16 \beta_{7} - 20 \beta_{8} + 16 \beta_{9} - 4 \beta_{12} - 4 \beta_{13} - 4 \beta_{15} + 4 \beta_{16} ) q^{28} + ( 24 + 40 \beta_{3} + 16 \beta_{4} + 2 \beta_{5} + 19 \beta_{7} - 24 \beta_{8} - 19 \beta_{9} + 15 \beta_{11} - 4 \beta_{12} + 2 \beta_{14} - \beta_{15} + \beta_{16} ) q^{29} + ( -4 \beta_{1} + 18 \beta_{3} + 4 \beta_{4} + 6 \beta_{6} - 18 \beta_{7} + 12 \beta_{8} - 12 \beta_{9} - 6 \beta_{10} - 6 \beta_{12} - 6 \beta_{16} - 6 \beta_{17} ) q^{30} + ( -13 - 7 \beta_{1} - 7 \beta_{2} - 36 \beta_{3} + 13 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} - 47 \beta_{7} - 11 \beta_{8} + 36 \beta_{9} - 21 \beta_{10} + 6 \beta_{11} + 6 \beta_{12} - 21 \beta_{13} - 2 \beta_{15} - 4 \beta_{16} + 4 \beta_{17} ) q^{31} + 32 \beta_{9} q^{32} + ( 52 + 12 \beta_{1} + 16 \beta_{2} - 2 \beta_{3} - 77 \beta_{4} + 4 \beta_{5} - 13 \beta_{6} + 77 \beta_{7} + 2 \beta_{8} - 52 \beta_{9} - 4 \beta_{10} + 16 \beta_{11} + 6 \beta_{12} + 13 \beta_{13} + 3 \beta_{14} + 6 \beta_{15} - 4 \beta_{16} + 3 \beta_{17} ) q^{33} + ( 4 - 6 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} - 10 \beta_{4} + 4 \beta_{5} + 12 \beta_{6} - 52 \beta_{7} + 10 \beta_{8} - 6 \beta_{10} - 6 \beta_{13} - 4 \beta_{15} - 4 \beta_{16} + 2 \beta_{17} ) q^{34} + ( -176 + 3 \beta_{1} - 62 \beta_{3} + 139 \beta_{4} - 5 \beta_{5} + 28 \beta_{6} + 176 \beta_{7} - 37 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} - 3 \beta_{14} + 2 \beta_{15} + 5 \beta_{17} ) q^{35} + ( 8 + 12 \beta_{1} + 8 \beta_{2} + 16 \beta_{4} + 24 \beta_{7} + 48 \beta_{8} - 16 \beta_{9} + 8 \beta_{10} - 12 \beta_{11} + 12 \beta_{13} - 4 \beta_{16} ) q^{36} + ( -50 - 16 \beta_{2} + 134 \beta_{3} + 4 \beta_{5} - 12 \beta_{6} + 63 \beta_{7} - 63 \beta_{8} + 71 \beta_{9} - 12 \beta_{10} - 7 \beta_{11} + \beta_{12} + 19 \beta_{13} - 5 \beta_{16} - 4 \beta_{17} ) q^{37} + ( 4 + 6 \beta_{1} + 12 \beta_{2} + 52 \beta_{3} - 32 \beta_{4} + 2 \beta_{5} + 12 \beta_{6} - 30 \beta_{7} + 20 \beta_{8} + 8 \beta_{9} + 14 \beta_{10} - 8 \beta_{11} - 12 \beta_{13} + 2 \beta_{15} + 4 \beta_{16} - 4 \beta_{17} ) q^{38} + ( 85 - 17 \beta_{2} - 99 \beta_{3} + 2 \beta_{5} + 7 \beta_{6} + 90 \beta_{7} - 90 \beta_{8} - 189 \beta_{9} + 7 \beta_{10} + 23 \beta_{11} - 5 \beta_{12} - 30 \beta_{13} - 6 \beta_{14} - 6 \beta_{15} + 3 \beta_{16} - 2 \beta_{17} ) q^{39} -8 \beta_{16} q^{40} + ( 14 + 21 \beta_{1} + 21 \beta_{2} - 25 \beta_{3} - 46 \beta_{4} - 6 \beta_{5} + 11 \beta_{6} - 14 \beta_{7} - 32 \beta_{9} - 21 \beta_{10} - 12 \beta_{12} + 12 \beta_{14} + 4 \beta_{15} - 8 \beta_{17} ) q^{41} + ( -10 - 10 \beta_{2} + 10 \beta_{3} - 16 \beta_{4} + 10 \beta_{6} + 68 \beta_{7} + 16 \beta_{8} + 8 \beta_{10} - 4 \beta_{14} + 4 \beta_{16} + 12 \beta_{17} ) q^{42} + ( -75 + \beta_{1} + 13 \beta_{2} - 21 \beta_{3} + 109 \beta_{4} - 6 \beta_{5} - 14 \beta_{6} - 109 \beta_{7} + 21 \beta_{8} + 75 \beta_{9} - 12 \beta_{10} + 13 \beta_{11} - 7 \beta_{12} + 14 \beta_{13} - 3 \beta_{14} - 7 \beta_{15} + 6 \beta_{16} - 3 \beta_{17} ) q^{43} + ( 8 - 12 \beta_{1} - 12 \beta_{2} - 20 \beta_{3} + 20 \beta_{4} + 8 \beta_{5} - 12 \beta_{6} + 28 \beta_{8} - 40 \beta_{9} + 8 \beta_{10} - 8 \beta_{11} + 4 \beta_{12} + 8 \beta_{15} + 4 \beta_{17} ) q^{44} + ( 176 + 12 \beta_{1} + 12 \beta_{2} - 84 \beta_{3} - 176 \beta_{4} + 3 \beta_{5} - 20 \beta_{6} - 31 \beta_{7} + 53 \beta_{8} + 84 \beta_{9} + 17 \beta_{10} - 20 \beta_{11} + 3 \beta_{12} + 17 \beta_{13} + 7 \beta_{15} + 6 \beta_{16} - 6 \beta_{17} ) q^{45} + ( -14 \beta_{1} + 86 \beta_{3} + 70 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} - 86 \beta_{7} - 62 \beta_{8} + 62 \beta_{9} - 4 \beta_{10} + 10 \beta_{11} + 2 \beta_{12} + 10 \beta_{13} + 12 \beta_{14} + 2 \beta_{16} - 4 \beta_{17} ) q^{46} + ( 63 - 9 \beta_{1} - 5 \beta_{2} + 62 \beta_{3} - \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 91 \beta_{7} - 63 \beta_{8} + 91 \beta_{9} + 11 \beta_{11} + 2 \beta_{12} + 5 \beta_{13} - 4 \beta_{14} + 4 \beta_{15} - 4 \beta_{16} ) q^{47} + ( -16 - 16 \beta_{3} + 16 \beta_{7} + 16 \beta_{8} - 16 \beta_{9} + 16 \beta_{11} ) q^{48} + ( -21 \beta_{1} + 96 \beta_{3} - 108 \beta_{4} - 7 \beta_{5} - 2 \beta_{6} - 96 \beta_{7} + 22 \beta_{8} - 22 \beta_{9} + 30 \beta_{10} - 28 \beta_{11} + 9 \beta_{12} - 28 \beta_{13} + 6 \beta_{14} + 9 \beta_{16} + 2 \beta_{17} ) q^{49} + ( 120 + 6 \beta_{1} + 6 \beta_{2} - 88 \beta_{3} - 120 \beta_{4} - 6 \beta_{5} - 20 \beta_{6} - 110 \beta_{7} - 22 \beta_{8} + 88 \beta_{9} - 10 \beta_{10} - 20 \beta_{11} - 6 \beta_{12} - 10 \beta_{13} + 6 \beta_{15} + 6 \beta_{16} - 6 \beta_{17} ) q^{50} + ( 111 - 6 \beta_{1} - 12 \beta_{2} - 85 \beta_{3} + 85 \beta_{4} - 6 \beta_{6} + 196 \beta_{8} + 73 \beta_{9} + 51 \beta_{10} - 51 \beta_{11} + 9 \beta_{12} - 6 \beta_{13} - 3 \beta_{14} - 3 \beta_{15} - 3 \beta_{16} + 9 \beta_{17} ) q^{51} + ( -12 - 20 \beta_{1} - 12 \beta_{2} + 52 \beta_{3} - 12 \beta_{4} - 4 \beta_{5} - 12 \beta_{6} + 12 \beta_{7} - 52 \beta_{8} + 12 \beta_{9} - 8 \beta_{10} - 12 \beta_{11} + 12 \beta_{13} - 4 \beta_{14} + 4 \beta_{16} - 4 \beta_{17} ) q^{52} + ( -23 + 18 \beta_{1} + 9 \beta_{2} + 23 \beta_{3} - 10 \beta_{4} + 5 \beta_{5} - 9 \beta_{6} - 110 \beta_{7} + 10 \beta_{8} + 8 \beta_{10} + 18 \beta_{13} - 19 \beta_{14} - 5 \beta_{15} + 14 \beta_{16} + \beta_{17} ) q^{53} + ( -200 + 16 \beta_{1} - 8 \beta_{2} - 166 \beta_{3} + 62 \beta_{4} + 6 \beta_{5} - 12 \beta_{6} + 200 \beta_{7} - 138 \beta_{9} - 16 \beta_{10} + 24 \beta_{11} - 4 \beta_{12} + 4 \beta_{14} - 4 \beta_{17} ) q^{54} + ( -241 - 14 \beta_{1} + 19 \beta_{2} + 72 \beta_{4} - 169 \beta_{7} + 51 \beta_{8} - 72 \beta_{9} + 19 \beta_{10} + 14 \beta_{11} + 10 \beta_{12} - 21 \beta_{13} - 5 \beta_{14} - 15 \beta_{15} + 6 \beta_{16} + 15 \beta_{17} ) q^{55} + ( -32 + 32 \beta_{3} - 8 \beta_{5} + 40 \beta_{7} - 40 \beta_{8} - 8 \beta_{9} + 8 \beta_{11} - 8 \beta_{13} - 8 \beta_{14} - 8 \beta_{15} + 8 \beta_{16} + 8 \beta_{17} ) q^{56} + ( -114 + 25 \beta_{1} - 6 \beta_{2} + 19 \beta_{3} - 71 \beta_{4} - 3 \beta_{5} - 34 \beta_{6} + 209 \beta_{7} + 236 \beta_{8} - 262 \beta_{9} + 10 \beta_{10} - 19 \beta_{11} - 4 \beta_{12} + 37 \beta_{13} + 15 \beta_{14} + 12 \beta_{15} - 3 \beta_{16} + 3 \beta_{17} ) q^{57} + ( 38 + 30 \beta_{2} - 32 \beta_{3} - 6 \beta_{5} + 48 \beta_{7} - 48 \beta_{8} - 80 \beta_{9} + 4 \beta_{12} - 8 \beta_{14} - 8 \beta_{15} + 2 \beta_{16} + 6 \beta_{17} ) q^{58} + ( 47 - 2 \beta_{1} - 40 \beta_{2} - 87 \beta_{4} - 40 \beta_{7} - 117 \beta_{8} + 87 \beta_{9} - 40 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + \beta_{13} + 18 \beta_{14} + 16 \beta_{15} - 20 \beta_{16} - 16 \beta_{17} ) q^{59} + ( 24 + 12 \beta_{1} - 8 \beta_{3} - 60 \beta_{4} - 8 \beta_{6} - 24 \beta_{7} - 36 \beta_{9} - 12 \beta_{10} + 12 \beta_{11} - 12 \beta_{15} - 12 \beta_{17} ) q^{60} + ( -120 - 29 \beta_{1} + 120 \beta_{3} - 227 \beta_{4} - 4 \beta_{5} + 102 \beta_{7} + 227 \beta_{8} - 30 \beta_{10} - 29 \beta_{13} + \beta_{14} + 4 \beta_{15} + 3 \beta_{16} + 7 \beta_{17} ) q^{61} + ( -72 + 42 \beta_{1} + 12 \beta_{2} - 26 \beta_{3} - 22 \beta_{4} - 4 \beta_{5} - 14 \beta_{6} + 22 \beta_{7} + 26 \beta_{8} + 72 \beta_{9} + 30 \beta_{10} + 12 \beta_{11} + 12 \beta_{12} + 14 \beta_{13} + 4 \beta_{14} + 12 \beta_{15} + 4 \beta_{16} + 4 \beta_{17} ) q^{62} + ( -47 - 3 \beta_{1} - 9 \beta_{2} - 215 \beta_{3} + 215 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + 168 \beta_{8} - 105 \beta_{9} - 17 \beta_{10} + 17 \beta_{11} - 25 \beta_{12} - 6 \beta_{13} + 4 \beta_{14} + 6 \beta_{15} + 4 \beta_{16} - 25 \beta_{17} ) q^{63} + ( -64 + 64 \beta_{4} ) q^{64} + ( -\beta_{1} + 146 \beta_{3} + 191 \beta_{4} + 25 \beta_{5} - 146 \beta_{7} - 253 \beta_{8} + 253 \beta_{9} + 17 \beta_{10} - 17 \beta_{11} - 4 \beta_{12} - 17 \beta_{13} - 5 \beta_{14} - 4 \beta_{16} + 21 \beta_{17} ) q^{65} + ( 104 + 8 \beta_{1} + 32 \beta_{2} + 154 \beta_{3} + 50 \beta_{4} + 6 \beta_{5} + 24 \beta_{6} - 4 \beta_{7} - 104 \beta_{8} + 4 \beta_{9} - 26 \beta_{11} + 8 \beta_{12} - 32 \beta_{13} + 6 \beta_{14} + 12 \beta_{15} - 12 \beta_{16} ) q^{66} + ( -38 - 28 \beta_{1} - 37 \beta_{2} + \beta_{3} + 39 \beta_{4} - 9 \beta_{5} - 9 \beta_{6} - 213 \beta_{7} + 38 \beta_{8} + 213 \beta_{9} - 25 \beta_{11} - \beta_{12} + 37 \beta_{13} - 9 \beta_{14} + 10 \beta_{15} - 10 \beta_{16} ) q^{67} + ( 12 \beta_{1} + 20 \beta_{3} - 104 \beta_{4} - 8 \beta_{5} - 12 \beta_{6} - 20 \beta_{7} - 8 \beta_{8} + 8 \beta_{9} - 12 \beta_{10} + 24 \beta_{11} + 8 \beta_{12} + 24 \beta_{13} - 4 \beta_{14} + 8 \beta_{16} ) q^{68} + ( 416 + 30 \beta_{1} + 30 \beta_{2} - 35 \beta_{3} - 416 \beta_{4} - 25 \beta_{5} + 29 \beta_{6} - 95 \beta_{7} - 60 \beta_{8} + 35 \beta_{9} + 36 \beta_{10} + 29 \beta_{11} - 25 \beta_{12} + 36 \beta_{13} - 10 \beta_{15} + 9 \beta_{16} - 9 \beta_{17} ) q^{69} + ( 74 + 6 \beta_{1} + 6 \beta_{2} - 278 \beta_{3} + 278 \beta_{4} + 10 \beta_{5} + 6 \beta_{6} + 352 \beta_{8} + 124 \beta_{9} - 56 \beta_{10} + 56 \beta_{11} - 10 \beta_{12} - 4 \beta_{14} + 6 \beta_{15} - 4 \beta_{16} - 10 \beta_{17} ) q^{70} + ( 198 + 19 \beta_{1} + 16 \beta_{2} + 217 \beta_{3} - 91 \beta_{4} + 10 \beta_{5} + 41 \beta_{6} + 91 \beta_{7} - 217 \beta_{8} - 198 \beta_{9} + 3 \beta_{10} + 16 \beta_{11} - 14 \beta_{12} - 41 \beta_{13} + 11 \beta_{14} - 14 \beta_{15} - 10 \beta_{16} + 11 \beta_{17} ) q^{71} + ( 32 - 16 \beta_{1} - 24 \beta_{2} - 32 \beta_{3} + 16 \beta_{4} + 24 \beta_{6} - 96 \beta_{7} - 16 \beta_{8} - 24 \beta_{10} - 16 \beta_{13} - 8 \beta_{17} ) q^{72} + ( -159 - 31 \beta_{1} - 16 \beta_{2} - 489 \beta_{3} + 101 \beta_{4} + 21 \beta_{5} + 5 \beta_{6} + 159 \beta_{7} - 58 \beta_{9} + 31 \beta_{10} - 15 \beta_{11} - \beta_{12} + \beta_{14} - 14 \beta_{15} - 15 \beta_{17} ) q^{73} + ( -142 + 24 \beta_{1} - 14 \beta_{2} + 268 \beta_{4} + 126 \beta_{7} + 100 \beta_{8} - 268 \beta_{9} - 14 \beta_{10} - 24 \beta_{11} + 8 \beta_{12} + 32 \beta_{13} + 10 \beta_{14} + 2 \beta_{15} - 2 \beta_{17} ) q^{74} + ( -248 + 28 \beta_{2} + 705 \beta_{3} - 6 \beta_{5} + 45 \beta_{6} + 454 \beta_{7} - 454 \beta_{8} + 251 \beta_{9} + 45 \beta_{10} - 18 \beta_{11} - 4 \beta_{12} - 27 \beta_{13} - 6 \beta_{14} - 6 \beta_{15} + 10 \beta_{16} + 6 \beta_{17} ) q^{75} + ( -16 - 28 \beta_{1} - 16 \beta_{2} + 64 \beta_{3} - 44 \beta_{4} + 4 \beta_{5} + 12 \beta_{6} - 40 \beta_{7} - 8 \beta_{8} - 104 \beta_{9} + 24 \beta_{11} + 4 \beta_{12} - 24 \beta_{13} + 8 \beta_{14} - 4 \beta_{16} + 12 \beta_{17} ) q^{76} + ( 357 - 29 \beta_{2} + 29 \beta_{3} + 4 \beta_{5} + \beta_{6} + 441 \beta_{7} - 441 \beta_{8} - 412 \beta_{9} + \beta_{10} - 52 \beta_{11} + 21 \beta_{12} + 51 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} - 25 \beta_{16} - 4 \beta_{17} ) q^{77} + ( 378 - 14 \beta_{1} + 46 \beta_{2} - 198 \beta_{4} + 180 \beta_{7} - 170 \beta_{8} + 198 \beta_{9} + 46 \beta_{10} + 14 \beta_{11} + 4 \beta_{12} + 34 \beta_{13} - 6 \beta_{14} - 10 \beta_{15} + 12 \beta_{16} + 10 \beta_{17} ) q^{78} + ( 151 - 37 \beta_{1} + 3 \beta_{2} - 227 \beta_{3} - 217 \beta_{4} + 15 \beta_{5} + 9 \beta_{6} - 151 \beta_{7} - 66 \beta_{9} + 37 \beta_{10} - 40 \beta_{11} + 24 \beta_{12} - 24 \beta_{14} + 2 \beta_{15} + 26 \beta_{17} ) q^{79} -16 \beta_{17} q^{80} + ( -198 - 86 \beta_{1} - 35 \beta_{2} + 87 \beta_{3} + 609 \beta_{4} + 15 \beta_{5} + 40 \beta_{6} - 609 \beta_{7} - 87 \beta_{8} + 198 \beta_{9} - 51 \beta_{10} - 35 \beta_{11} + 11 \beta_{12} - 40 \beta_{13} + 5 \beta_{14} + 11 \beta_{15} - 15 \beta_{16} + 5 \beta_{17} ) q^{81} + ( 64 + 42 \beta_{1} + 92 \beta_{3} - 92 \beta_{4} - 16 \beta_{5} + 42 \beta_{6} - 28 \beta_{8} + 50 \beta_{9} - 22 \beta_{10} + 22 \beta_{11} - 12 \beta_{12} - 42 \beta_{13} - 8 \beta_{14} - 24 \beta_{15} - 8 \beta_{16} - 12 \beta_{17} ) q^{82} + ( -2 + 20 \beta_{1} + 20 \beta_{2} - 176 \beta_{3} + 2 \beta_{4} + 11 \beta_{5} + 57 \beta_{6} + 94 \beta_{7} + 270 \beta_{8} + 176 \beta_{9} - 68 \beta_{10} + 57 \beta_{11} + 11 \beta_{12} - 68 \beta_{13} + 8 \beta_{15} - 15 \beta_{16} + 15 \beta_{17} ) q^{83} + ( -16 \beta_{1} + 32 \beta_{3} + 136 \beta_{4} + 8 \beta_{5} - 32 \beta_{7} + 20 \beta_{8} - 20 \beta_{9} - 20 \beta_{10} + 20 \beta_{11} + 20 \beta_{13} - 24 \beta_{14} + 8 \beta_{17} ) q^{84} + ( -114 + 10 \beta_{1} - 32 \beta_{2} + 232 \beta_{3} + 346 \beta_{4} + 7 \beta_{5} - 42 \beta_{6} - 101 \beta_{7} + 114 \beta_{8} + 101 \beta_{9} + 43 \beta_{11} - 5 \beta_{12} + 32 \beta_{13} + 7 \beta_{14} - 3 \beta_{15} + 3 \beta_{16} ) q^{85} + ( -150 + 24 \beta_{1} + 26 \beta_{2} - 218 \beta_{3} - 68 \beta_{4} - 8 \beta_{5} + 2 \beta_{6} - 42 \beta_{7} + 150 \beta_{8} + 42 \beta_{9} - 28 \beta_{11} - 12 \beta_{12} - 26 \beta_{13} - 8 \beta_{14} - 14 \beta_{15} + 14 \beta_{16} ) q^{86} + ( 60 \beta_{1} - 36 \beta_{3} - 488 \beta_{4} + 3 \beta_{5} + 36 \beta_{7} - 113 \beta_{8} + 113 \beta_{9} - 41 \beta_{10} + 41 \beta_{11} + 3 \beta_{12} + 41 \beta_{13} + 3 \beta_{14} + 3 \beta_{16} + 6 \beta_{17} ) q^{87} + ( 80 - 16 \beta_{1} - 16 \beta_{2} - 40 \beta_{3} - 80 \beta_{4} + 16 \beta_{5} - 24 \beta_{6} - 56 \beta_{7} - 16 \beta_{8} + 40 \beta_{9} + 24 \beta_{10} - 24 \beta_{11} + 16 \beta_{12} + 24 \beta_{13} + 8 \beta_{15} - 16 \beta_{16} + 16 \beta_{17} ) q^{88} + ( 260 + 14 \beta_{2} + 38 \beta_{3} - 38 \beta_{4} - 27 \beta_{5} + 222 \beta_{8} + 342 \beta_{9} - 67 \beta_{10} + 67 \beta_{11} - 6 \beta_{12} + 14 \beta_{13} + 2 \beta_{14} - 25 \beta_{15} + 2 \beta_{16} - 6 \beta_{17} ) q^{89} + ( -168 - 34 \beta_{1} - 40 \beta_{2} + 352 \beta_{3} + 106 \beta_{4} + 14 \beta_{5} + 24 \beta_{6} - 106 \beta_{7} - 352 \beta_{8} + 168 \beta_{9} + 6 \beta_{10} - 40 \beta_{11} + 6 \beta_{12} - 24 \beta_{13} + 18 \beta_{14} + 6 \beta_{15} - 14 \beta_{16} + 18 \beta_{17} ) q^{90} + ( -109 + 22 \beta_{1} + 51 \beta_{2} + 109 \beta_{3} + 46 \beta_{4} + 2 \beta_{5} - 51 \beta_{6} - 201 \beta_{7} - 46 \beta_{8} + 17 \beta_{10} + 22 \beta_{13} + 42 \beta_{14} - 2 \beta_{15} - 44 \beta_{16} - 17 \beta_{17} ) q^{91} + ( -124 + 8 \beta_{1} + 20 \beta_{2} - 140 \beta_{3} - 48 \beta_{4} - 24 \beta_{5} - 28 \beta_{6} + 124 \beta_{7} - 172 \beta_{9} - 8 \beta_{10} - 12 \beta_{11} - 12 \beta_{12} + 12 \beta_{14} + 4 \beta_{15} - 8 \beta_{17} ) q^{92} + ( -825 - 66 \beta_{1} - 85 \beta_{2} + 630 \beta_{4} - 195 \beta_{7} - 53 \beta_{8} - 630 \beta_{9} - 85 \beta_{10} + 66 \beta_{11} - 27 \beta_{12} - 47 \beta_{13} + 12 \beta_{14} + 39 \beta_{15} - 17 \beta_{16} - 39 \beta_{17} ) q^{93} + ( -182 + 22 \beta_{2} + 2 \beta_{3} + 16 \beta_{5} - 18 \beta_{6} + 126 \beta_{7} - 126 \beta_{8} - 124 \beta_{9} - 18 \beta_{10} + 8 \beta_{11} - 8 \beta_{12} + 10 \beta_{13} + 4 \beta_{14} + 4 \beta_{15} - 8 \beta_{16} - 16 \beta_{17} ) q^{94} + ( 88 - 58 \beta_{1} - 2 \beta_{2} - 274 \beta_{3} - 58 \beta_{4} - 19 \beta_{5} + 24 \beta_{6} + 322 \beta_{7} + 498 \beta_{8} - 569 \beta_{9} - 69 \beta_{10} + 47 \beta_{11} - 15 \beta_{12} - 16 \beta_{13} - 30 \beta_{14} - 50 \beta_{15} + 41 \beta_{16} - 21 \beta_{17} ) q^{95} + ( 32 + 32 \beta_{2} - 32 \beta_{7} + 32 \beta_{8} + 32 \beta_{9} ) q^{96} + ( 373 + 5 \beta_{1} + 43 \beta_{2} + 84 \beta_{4} + 457 \beta_{7} + 274 \beta_{8} - 84 \beta_{9} + 43 \beta_{10} - 5 \beta_{11} - 24 \beta_{12} - 38 \beta_{13} - 45 \beta_{14} - 21 \beta_{15} + 36 \beta_{16} + 21 \beta_{17} ) q^{97} + ( 44 - 60 \beta_{1} - 56 \beta_{2} + 216 \beta_{3} - 236 \beta_{4} - 12 \beta_{5} - 42 \beta_{6} - 44 \beta_{7} - 192 \beta_{9} + 60 \beta_{10} - 4 \beta_{11} - 14 \beta_{12} + 14 \beta_{14} + 18 \beta_{15} + 4 \beta_{17} ) q^{98} + ( 35 + 92 \beta_{1} + 77 \beta_{2} - 35 \beta_{3} - 603 \beta_{4} - 7 \beta_{5} - 77 \beta_{6} + 302 \beta_{7} + 603 \beta_{8} + 31 \beta_{10} + 92 \beta_{13} - 17 \beta_{14} + 7 \beta_{15} + 24 \beta_{16} + 13 \beta_{17} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 6q^{3} - 12q^{6} - 33q^{7} + 72q^{8} + 42q^{9} + O(q^{10}) \) \( 18q + 6q^{3} - 12q^{6} - 33q^{7} + 72q^{8} + 42q^{9} - 75q^{11} + 36q^{12} + 99q^{13} + 162q^{14} - 183q^{15} - 111q^{17} - 408q^{18} - 372q^{19} + 24q^{20} - 207q^{21} - 180q^{22} + 198q^{23} - 48q^{24} + 534q^{25} + 180q^{26} + 678q^{27} + 216q^{28} + 669q^{29} - 42q^{31} + 315q^{33} - 48q^{34} - 1995q^{35} + 168q^{36} - 1056q^{37} - 180q^{38} + 1812q^{39} - 210q^{41} - 342q^{42} - 399q^{43} + 360q^{44} + 1494q^{45} + 672q^{46} + 1149q^{47} - 192q^{48} - 858q^{49} + 1068q^{50} + 2646q^{51} - 468q^{52} - 633q^{53} - 2898q^{54} - 3483q^{55} - 528q^{56} - 2814q^{57} + 636q^{58} + 51q^{59} - 84q^{60} - 4104q^{61} - 1326q^{62} + 1215q^{63} - 576q^{64} + 1755q^{65} + 2340q^{66} - 675q^{67} - 948q^{68} + 3693q^{69} + 3990q^{70} + 2964q^{71} + 672q^{72} - 2004q^{73} - 486q^{74} - 4446q^{75} - 408q^{76} + 5820q^{77} + 4992q^{78} + 543q^{79} + 1722q^{81} + 420q^{82} + 381q^{83} + 1092q^{84} + 1266q^{85} - 3396q^{86} - 4506q^{87} + 600q^{88} + 4386q^{89} - 2148q^{90} - 1356q^{91} - 2628q^{92} - 8604q^{93} - 3264q^{94} + 921q^{95} + 576q^{96} + 7599q^{97} - 954q^{98} - 5055q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} + 165 x^{16} - 56 x^{15} + 18435 x^{14} - 11748 x^{13} + 1092662 x^{12} - 1833567 x^{11} + 46842546 x^{10} - 93643115 x^{9} + 1273086000 x^{8} - 3159018183 x^{7} + 24752942108 x^{6} - 51265274250 x^{5} + 247917053361 x^{4} - 421854393530 x^{3} + 1634375353605 x^{2} - 2066139456366 x + 3892796082289\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(64\!\cdots\!63\)\( \nu^{17} - \)\(51\!\cdots\!51\)\( \nu^{16} - \)\(78\!\cdots\!56\)\( \nu^{15} - \)\(77\!\cdots\!98\)\( \nu^{14} - \)\(73\!\cdots\!22\)\( \nu^{13} - \)\(83\!\cdots\!58\)\( \nu^{12} - \)\(17\!\cdots\!58\)\( \nu^{11} - \)\(41\!\cdots\!99\)\( \nu^{10} + \)\(40\!\cdots\!37\)\( \nu^{9} - \)\(19\!\cdots\!81\)\( \nu^{8} + \)\(65\!\cdots\!38\)\( \nu^{7} - \)\(50\!\cdots\!28\)\( \nu^{6} + \)\(22\!\cdots\!72\)\( \nu^{5} - \)\(12\!\cdots\!66\)\( \nu^{4} + \)\(53\!\cdots\!39\)\( \nu^{3} - \)\(10\!\cdots\!77\)\( \nu^{2} + \)\(14\!\cdots\!23\)\( \nu - \)\(68\!\cdots\!38\)\(\)\()/ \)\(31\!\cdots\!93\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(23\!\cdots\!97\)\( \nu^{17} - \)\(27\!\cdots\!99\)\( \nu^{16} - \)\(47\!\cdots\!23\)\( \nu^{15} - \)\(44\!\cdots\!54\)\( \nu^{14} - \)\(55\!\cdots\!14\)\( \nu^{13} - \)\(47\!\cdots\!70\)\( \nu^{12} - \)\(36\!\cdots\!58\)\( \nu^{11} - \)\(24\!\cdots\!09\)\( \nu^{10} - \)\(13\!\cdots\!67\)\( \nu^{9} - \)\(87\!\cdots\!11\)\( \nu^{8} - \)\(34\!\cdots\!66\)\( \nu^{7} - \)\(18\!\cdots\!28\)\( \nu^{6} - \)\(45\!\cdots\!24\)\( \nu^{5} - \)\(26\!\cdots\!58\)\( \nu^{4} - \)\(47\!\cdots\!67\)\( \nu^{3} - \)\(17\!\cdots\!41\)\( \nu^{2} - \)\(16\!\cdots\!59\)\( \nu - \)\(84\!\cdots\!58\)\(\)\()/ \)\(41\!\cdots\!44\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(18\!\cdots\!06\)\( \nu^{17} - \)\(67\!\cdots\!09\)\( \nu^{16} + \)\(29\!\cdots\!97\)\( \nu^{15} - \)\(18\!\cdots\!44\)\( \nu^{14} + \)\(32\!\cdots\!96\)\( \nu^{13} - \)\(29\!\cdots\!34\)\( \nu^{12} + \)\(19\!\cdots\!78\)\( \nu^{11} - \)\(35\!\cdots\!96\)\( \nu^{10} + \)\(81\!\cdots\!19\)\( \nu^{9} - \)\(16\!\cdots\!99\)\( \nu^{8} + \)\(21\!\cdots\!17\)\( \nu^{7} - \)\(50\!\cdots\!64\)\( \nu^{6} + \)\(39\!\cdots\!44\)\( \nu^{5} - \)\(70\!\cdots\!04\)\( \nu^{4} + \)\(32\!\cdots\!28\)\( \nu^{3} - \)\(21\!\cdots\!03\)\( \nu^{2} + \)\(19\!\cdots\!19\)\( \nu + \)\(97\!\cdots\!92\)\(\)\()/ \)\(32\!\cdots\!99\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(19\!\cdots\!39\)\( \nu^{17} - \)\(11\!\cdots\!99\)\( \nu^{16} - \)\(13\!\cdots\!23\)\( \nu^{15} - \)\(19\!\cdots\!66\)\( \nu^{14} - \)\(32\!\cdots\!14\)\( \nu^{13} - \)\(21\!\cdots\!90\)\( \nu^{12} - \)\(48\!\cdots\!78\)\( \nu^{11} - \)\(12\!\cdots\!65\)\( \nu^{10} - \)\(18\!\cdots\!27\)\( \nu^{9} - \)\(49\!\cdots\!99\)\( \nu^{8} - \)\(66\!\cdots\!10\)\( \nu^{7} - \)\(11\!\cdots\!24\)\( \nu^{6} - \)\(88\!\cdots\!84\)\( \nu^{5} - \)\(17\!\cdots\!46\)\( \nu^{4} - \)\(20\!\cdots\!99\)\( \nu^{3} - \)\(11\!\cdots\!57\)\( \nu^{2} - \)\(64\!\cdots\!63\)\( \nu - \)\(52\!\cdots\!50\)\(\)\()/ \)\(20\!\cdots\!72\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(50\!\cdots\!79\)\( \nu^{17} + \)\(87\!\cdots\!45\)\( \nu^{16} + \)\(10\!\cdots\!33\)\( \nu^{15} + \)\(14\!\cdots\!74\)\( \nu^{14} + \)\(12\!\cdots\!42\)\( \nu^{13} + \)\(15\!\cdots\!74\)\( \nu^{12} + \)\(86\!\cdots\!90\)\( \nu^{11} + \)\(81\!\cdots\!23\)\( \nu^{10} + \)\(30\!\cdots\!81\)\( \nu^{9} + \)\(30\!\cdots\!77\)\( \nu^{8} + \)\(69\!\cdots\!30\)\( \nu^{7} + \)\(67\!\cdots\!56\)\( \nu^{6} + \)\(57\!\cdots\!12\)\( \nu^{5} + \)\(10\!\cdots\!74\)\( \nu^{4} + \)\(28\!\cdots\!05\)\( \nu^{3} + \)\(81\!\cdots\!15\)\( \nu^{2} - \)\(23\!\cdots\!47\)\( \nu + \)\(31\!\cdots\!66\)\(\)\()/ \)\(40\!\cdots\!08\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(55\!\cdots\!10\)\( \nu^{17} + \)\(35\!\cdots\!61\)\( \nu^{16} + \)\(95\!\cdots\!73\)\( \nu^{15} + \)\(53\!\cdots\!47\)\( \nu^{14} + \)\(10\!\cdots\!16\)\( \nu^{13} + \)\(55\!\cdots\!14\)\( \nu^{12} + \)\(63\!\cdots\!42\)\( \nu^{11} + \)\(24\!\cdots\!72\)\( \nu^{10} + \)\(23\!\cdots\!17\)\( \nu^{9} + \)\(86\!\cdots\!25\)\( \nu^{8} + \)\(53\!\cdots\!79\)\( \nu^{7} + \)\(16\!\cdots\!08\)\( \nu^{6} + \)\(65\!\cdots\!08\)\( \nu^{5} + \)\(28\!\cdots\!00\)\( \nu^{4} + \)\(34\!\cdots\!96\)\( \nu^{3} + \)\(20\!\cdots\!35\)\( \nu^{2} + \)\(80\!\cdots\!31\)\( \nu + \)\(10\!\cdots\!79\)\(\)\()/ \)\(41\!\cdots\!44\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(83\!\cdots\!42\)\( \nu^{17} - \)\(52\!\cdots\!97\)\( \nu^{16} + \)\(12\!\cdots\!95\)\( \nu^{15} - \)\(15\!\cdots\!71\)\( \nu^{14} + \)\(13\!\cdots\!88\)\( \nu^{13} - \)\(22\!\cdots\!22\)\( \nu^{12} + \)\(74\!\cdots\!22\)\( \nu^{11} - \)\(24\!\cdots\!84\)\( \nu^{10} + \)\(30\!\cdots\!43\)\( \nu^{9} - \)\(10\!\cdots\!13\)\( \nu^{8} + \)\(74\!\cdots\!89\)\( \nu^{7} - \)\(33\!\cdots\!76\)\( \nu^{6} + \)\(13\!\cdots\!28\)\( \nu^{5} - \)\(48\!\cdots\!16\)\( \nu^{4} + \)\(99\!\cdots\!80\)\( \nu^{3} - \)\(38\!\cdots\!75\)\( \nu^{2} + \)\(50\!\cdots\!65\)\( \nu - \)\(14\!\cdots\!51\)\(\)\()/ \)\(41\!\cdots\!44\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(11\!\cdots\!57\)\( \nu^{17} + \)\(20\!\cdots\!15\)\( \nu^{16} + \)\(18\!\cdots\!59\)\( \nu^{15} + \)\(24\!\cdots\!38\)\( \nu^{14} + \)\(19\!\cdots\!22\)\( \nu^{13} + \)\(19\!\cdots\!26\)\( \nu^{12} + \)\(11\!\cdots\!18\)\( \nu^{11} - \)\(37\!\cdots\!95\)\( \nu^{10} + \)\(43\!\cdots\!15\)\( \nu^{9} - \)\(35\!\cdots\!69\)\( \nu^{8} + \)\(10\!\cdots\!74\)\( \nu^{7} - \)\(18\!\cdots\!28\)\( \nu^{6} + \)\(17\!\cdots\!24\)\( \nu^{5} - \)\(23\!\cdots\!70\)\( \nu^{4} + \)\(12\!\cdots\!19\)\( \nu^{3} - \)\(22\!\cdots\!19\)\( \nu^{2} + \)\(75\!\cdots\!99\)\( \nu - \)\(71\!\cdots\!30\)\(\)\()/ \)\(41\!\cdots\!44\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(14\!\cdots\!22\)\( \nu^{17} - \)\(86\!\cdots\!83\)\( \nu^{16} - \)\(25\!\cdots\!39\)\( \nu^{15} - \)\(13\!\cdots\!73\)\( \nu^{14} - \)\(28\!\cdots\!24\)\( \nu^{13} - \)\(13\!\cdots\!66\)\( \nu^{12} - \)\(17\!\cdots\!26\)\( \nu^{11} - \)\(60\!\cdots\!48\)\( \nu^{10} - \)\(65\!\cdots\!23\)\( \nu^{9} - \)\(19\!\cdots\!35\)\( \nu^{8} - \)\(15\!\cdots\!45\)\( \nu^{7} - \)\(33\!\cdots\!20\)\( \nu^{6} - \)\(21\!\cdots\!40\)\( \nu^{5} - \)\(49\!\cdots\!08\)\( \nu^{4} - \)\(17\!\cdots\!08\)\( \nu^{3} - \)\(25\!\cdots\!69\)\( \nu^{2} - \)\(58\!\cdots\!49\)\( \nu - \)\(21\!\cdots\!81\)\(\)\()/ \)\(40\!\cdots\!08\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(19\!\cdots\!05\)\( \nu^{17} - \)\(40\!\cdots\!22\)\( \nu^{16} + \)\(29\!\cdots\!10\)\( \nu^{15} - \)\(84\!\cdots\!11\)\( \nu^{14} + \)\(31\!\cdots\!34\)\( \nu^{13} - \)\(10\!\cdots\!12\)\( \nu^{12} + \)\(16\!\cdots\!68\)\( \nu^{11} - \)\(86\!\cdots\!49\)\( \nu^{10} + \)\(67\!\cdots\!02\)\( \nu^{9} - \)\(36\!\cdots\!82\)\( \nu^{8} + \)\(16\!\cdots\!21\)\( \nu^{7} - \)\(10\!\cdots\!24\)\( \nu^{6} + \)\(32\!\cdots\!60\)\( \nu^{5} - \)\(14\!\cdots\!06\)\( \nu^{4} + \)\(24\!\cdots\!37\)\( \nu^{3} - \)\(10\!\cdots\!02\)\( \nu^{2} + \)\(15\!\cdots\!24\)\( \nu - \)\(42\!\cdots\!11\)\(\)\()/ \)\(40\!\cdots\!08\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(10\!\cdots\!35\)\( \nu^{17} + \)\(15\!\cdots\!86\)\( \nu^{16} - \)\(13\!\cdots\!46\)\( \nu^{15} + \)\(25\!\cdots\!97\)\( \nu^{14} - \)\(13\!\cdots\!26\)\( \nu^{13} + \)\(29\!\cdots\!88\)\( \nu^{12} - \)\(59\!\cdots\!48\)\( \nu^{11} + \)\(17\!\cdots\!23\)\( \nu^{10} - \)\(38\!\cdots\!38\)\( \nu^{9} + \)\(72\!\cdots\!66\)\( \nu^{8} - \)\(11\!\cdots\!71\)\( \nu^{7} + \)\(18\!\cdots\!12\)\( \nu^{6} - \)\(36\!\cdots\!48\)\( \nu^{5} + \)\(29\!\cdots\!98\)\( \nu^{4} - \)\(35\!\cdots\!71\)\( \nu^{3} + \)\(25\!\cdots\!58\)\( \nu^{2} - \)\(36\!\cdots\!04\)\( \nu + \)\(11\!\cdots\!41\)\(\)\()/ \)\(20\!\cdots\!72\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(21\!\cdots\!14\)\( \nu^{17} + \)\(18\!\cdots\!19\)\( \nu^{16} - \)\(32\!\cdots\!69\)\( \nu^{15} + \)\(25\!\cdots\!41\)\( \nu^{14} - \)\(35\!\cdots\!40\)\( \nu^{13} + \)\(42\!\cdots\!66\)\( \nu^{12} - \)\(18\!\cdots\!66\)\( \nu^{11} + \)\(54\!\cdots\!88\)\( \nu^{10} - \)\(74\!\cdots\!81\)\( \nu^{9} + \)\(24\!\cdots\!15\)\( \nu^{8} - \)\(17\!\cdots\!23\)\( \nu^{7} + \)\(79\!\cdots\!20\)\( \nu^{6} - \)\(31\!\cdots\!44\)\( \nu^{5} + \)\(11\!\cdots\!24\)\( \nu^{4} - \)\(23\!\cdots\!52\)\( \nu^{3} + \)\(10\!\cdots\!97\)\( \nu^{2} - \)\(15\!\cdots\!67\)\( \nu + \)\(39\!\cdots\!97\)\(\)\()/ \)\(40\!\cdots\!08\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(10\!\cdots\!39\)\( \nu^{17} - \)\(11\!\cdots\!88\)\( \nu^{16} + \)\(13\!\cdots\!96\)\( \nu^{15} - \)\(18\!\cdots\!03\)\( \nu^{14} + \)\(14\!\cdots\!62\)\( \nu^{13} - \)\(21\!\cdots\!20\)\( \nu^{12} + \)\(69\!\cdots\!04\)\( \nu^{11} - \)\(12\!\cdots\!51\)\( \nu^{10} + \)\(39\!\cdots\!40\)\( \nu^{9} - \)\(50\!\cdots\!28\)\( \nu^{8} + \)\(11\!\cdots\!81\)\( \nu^{7} - \)\(12\!\cdots\!80\)\( \nu^{6} + \)\(31\!\cdots\!28\)\( \nu^{5} - \)\(16\!\cdots\!58\)\( \nu^{4} + \)\(26\!\cdots\!55\)\( \nu^{3} - \)\(91\!\cdots\!20\)\( \nu^{2} + \)\(17\!\cdots\!02\)\( \nu - \)\(20\!\cdots\!55\)\(\)\()/ \)\(17\!\cdots\!56\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(13\!\cdots\!44\)\( \nu^{17} + \)\(96\!\cdots\!07\)\( \nu^{16} + \)\(21\!\cdots\!83\)\( \nu^{15} + \)\(13\!\cdots\!03\)\( \nu^{14} + \)\(23\!\cdots\!08\)\( \nu^{13} + \)\(13\!\cdots\!62\)\( \nu^{12} + \)\(13\!\cdots\!10\)\( \nu^{11} + \)\(51\!\cdots\!50\)\( \nu^{10} + \)\(41\!\cdots\!51\)\( \nu^{9} + \)\(15\!\cdots\!99\)\( \nu^{8} + \)\(81\!\cdots\!59\)\( \nu^{7} + \)\(15\!\cdots\!12\)\( \nu^{6} + \)\(55\!\cdots\!40\)\( \nu^{5} + \)\(18\!\cdots\!52\)\( \nu^{4} + \)\(40\!\cdots\!18\)\( \nu^{3} - \)\(27\!\cdots\!63\)\( \nu^{2} - \)\(80\!\cdots\!67\)\( \nu + \)\(56\!\cdots\!27\)\(\)\()/ \)\(17\!\cdots\!56\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(55\!\cdots\!46\)\( \nu^{17} - \)\(39\!\cdots\!89\)\( \nu^{16} + \)\(88\!\cdots\!15\)\( \nu^{15} - \)\(45\!\cdots\!55\)\( \nu^{14} + \)\(96\!\cdots\!08\)\( \nu^{13} - \)\(83\!\cdots\!78\)\( \nu^{12} + \)\(54\!\cdots\!02\)\( \nu^{11} - \)\(11\!\cdots\!88\)\( \nu^{10} + \)\(22\!\cdots\!43\)\( \nu^{9} - \)\(56\!\cdots\!37\)\( \nu^{8} + \)\(55\!\cdots\!13\)\( \nu^{7} - \)\(18\!\cdots\!68\)\( \nu^{6} + \)\(97\!\cdots\!08\)\( \nu^{5} - \)\(27\!\cdots\!76\)\( \nu^{4} + \)\(68\!\cdots\!76\)\( \nu^{3} - \)\(21\!\cdots\!03\)\( \nu^{2} + \)\(31\!\cdots\!37\)\( \nu - \)\(77\!\cdots\!15\)\(\)\()/ \)\(20\!\cdots\!72\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(55\!\cdots\!90\)\( \nu^{17} - \)\(20\!\cdots\!29\)\( \nu^{16} - \)\(91\!\cdots\!01\)\( \nu^{15} - \)\(28\!\cdots\!11\)\( \nu^{14} - \)\(99\!\cdots\!16\)\( \nu^{13} - \)\(28\!\cdots\!86\)\( \nu^{12} - \)\(56\!\cdots\!50\)\( \nu^{11} - \)\(94\!\cdots\!68\)\( \nu^{10} - \)\(21\!\cdots\!13\)\( \nu^{9} - \)\(30\!\cdots\!33\)\( \nu^{8} - \)\(47\!\cdots\!67\)\( \nu^{7} - \)\(34\!\cdots\!76\)\( \nu^{6} - \)\(65\!\cdots\!60\)\( \nu^{5} - \)\(10\!\cdots\!56\)\( \nu^{4} - \)\(36\!\cdots\!16\)\( \nu^{3} - \)\(72\!\cdots\!31\)\( \nu^{2} - \)\(13\!\cdots\!23\)\( \nu - \)\(49\!\cdots\!95\)\(\)\()/ \)\(20\!\cdots\!72\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{15} - 3 \beta_{11} - 5 \beta_{9} + 5 \beta_{7} - 3 \beta_{6} + 36 \beta_{4} + 5 \beta_{3} - \beta_{2} - \beta_{1} - 36\)
\(\nu^{3}\)\(=\)\(\beta_{16} + 2 \beta_{13} - \beta_{12} + 6 \beta_{11} - 8 \beta_{10} + 24 \beta_{9} - 88 \beta_{8} + 88 \beta_{7} - 8 \beta_{6} + 112 \beta_{3} + 57 \beta_{2} + 8\)
\(\nu^{4}\)\(=\)\(5 \beta_{17} + 7 \beta_{16} + 56 \beta_{14} + 296 \beta_{13} + 7 \beta_{12} + 296 \beta_{11} - 274 \beta_{10} + 622 \beta_{9} - 622 \beta_{8} - 222 \beta_{7} - 22 \beta_{6} - 2 \beta_{5} - 2078 \beta_{4} + 222 \beta_{3} + 29 \beta_{1}\)
\(\nu^{5}\)\(=\)\(40 \beta_{17} - 40 \beta_{16} + 6 \beta_{15} + 879 \beta_{13} + 156 \beta_{12} + 268 \beta_{11} + 879 \beta_{10} + 9012 \beta_{9} - 2569 \beta_{8} - 11581 \beta_{7} + 268 \beta_{6} + 156 \beta_{5} - 1595 \beta_{4} - 9012 \beta_{3} - 3938 \beta_{2} - 3938 \beta_{1} + 1595\)
\(\nu^{6}\)\(=\)\(-893 \beta_{17} - 400 \beta_{16} - 3312 \beta_{15} - 3312 \beta_{14} - 23021 \beta_{13} - 493 \beta_{12} - 2780 \beta_{11} + 25801 \beta_{10} - 31030 \beta_{9} + 35155 \beta_{8} - 35155 \beta_{7} + 25801 \beta_{6} + 893 \beta_{5} - 66185 \beta_{3} - 2478 \beta_{2} + 145459\)
\(\nu^{7}\)\(=\)\(-14918 \beta_{17} - 14572 \beta_{16} - 1818 \beta_{14} - 128486 \beta_{13} - 14572 \beta_{12} - 128486 \beta_{11} + 37444 \beta_{10} - 1038778 \beta_{9} + 1038778 \beta_{8} + 233570 \beta_{7} + 91042 \beta_{6} - 346 \beta_{5} + 319773 \beta_{4} - 233570 \beta_{3} + 295792 \beta_{1}\)
\(\nu^{8}\)\(=\)\(58696 \beta_{17} - 58696 \beta_{16} + 224404 \beta_{15} - 292430 \beta_{13} - 34846 \beta_{12} - 1918802 \beta_{11} - 292430 \beta_{10} - 3361386 \beta_{9} + 3261130 \beta_{8} + 6622516 \beta_{7} - 1918802 \beta_{6} - 34846 \beta_{5} + 11160210 \beta_{4} + 3361386 \beta_{3} + 571199 \beta_{2} + 571199 \beta_{1} - 11160210\)
\(\nu^{9}\)\(=\)\(1409632 \beta_{17} + 1293906 \beta_{16} + 386589 \beta_{15} + 386589 \beta_{14} + 4671837 \beta_{13} + 115726 \beta_{12} + 8449912 \beta_{11} - 13121749 \beta_{10} + 20972780 \beta_{9} - 69920703 \beta_{8} + 69920703 \beta_{7} - 13121749 \beta_{6} - 1409632 \beta_{5} + 90893483 \beta_{3} + 23356371 \beta_{2} - 40208848\)
\(\nu^{10}\)\(=\)\(3551481 \beta_{17} + 9258444 \beta_{16} + 16810526 \beta_{14} + 190372536 \beta_{13} + 9258444 \beta_{12} + 190372536 \beta_{11} - 160533576 \beta_{10} + 640135440 \beta_{9} - 640135440 \beta_{8} - 310353742 \beta_{7} - 29838960 \beta_{6} - 5706963 \beta_{5} - 902199260 \beta_{4} + 310353742 \beta_{3} - 74540953 \beta_{1}\)
\(\nu^{11}\)\(=\)\(-17339398 \beta_{17} + 17339398 \beta_{16} - 50766222 \beta_{15} + 750873190 \beta_{13} + 109791635 \beta_{12} + 529267958 \beta_{11} + 750873190 \beta_{10} + 6039569000 \beta_{9} - 1912116366 \beta_{8} - 7951685366 \beta_{7} + 529267958 \beta_{6} + 109791635 \beta_{5} - 4346923604 \beta_{4} - 6039569000 \beta_{3} - 1907531249 \beta_{2} - 1907531249 \beta_{1} + 4346923604\)
\(\nu^{12}\)\(=\)\(-890919814 \beta_{17} - 375717320 \beta_{16} - 1339981704 \beta_{15} - 1339981704 \beta_{14} - 13528021390 \beta_{13} - 515202494 \beta_{12} - 3012820279 \beta_{11} + 16540841669 \beta_{10} - 28297062725 \beta_{9} + 32248680410 \beta_{8} - 32248680410 \beta_{7} + 16540841669 \beta_{6} + 890919814 \beta_{5} - 60545743135 \beta_{3} - 8246508828 \beta_{2} + 75363031909\)
\(\nu^{13}\)\(=\)\(-9302450500 \beta_{17} - 11254259423 \beta_{16} - 5647093956 \beta_{14} - 121694400067 \beta_{13} - 11254259423 \beta_{12} - 121694400067 \beta_{11} + 56142874133 \beta_{10} - 699602981387 \beta_{9} + 699602981387 \beta_{8} + 176858485096 \beta_{7} + 65551525934 \beta_{6} + 1951808923 \beta_{5} + 438194213457 \beta_{4} - 176858485096 \beta_{3} + 159584620144 \beta_{1}\)
\(\nu^{14}\)\(=\)\(45203148824 \beta_{17} - 45203148824 \beta_{16} + 110912225908 \beta_{15} - 300460695808 \beta_{13} - 38993259452 \beta_{12} - 1149648891760 \beta_{11} - 300460695808 \beta_{10} - 3115524089800 \beta_{9} + 2530386494240 \beta_{8} + 5645910584040 \beta_{7} - 1149648891760 \beta_{6} - 38993259452 \beta_{5} + 6433442474541 \beta_{4} + 3115524089800 \beta_{3} + 845467528556 \beta_{2} + 845467528556 \beta_{1} - 6433442474541\)
\(\nu^{15}\)\(=\)\(994183242116 \beta_{17} + 792214993472 \beta_{16} + 580600861908 \beta_{15} + 580600861908 \beta_{14} + 5697403289404 \beta_{13} + 201968248644 \beta_{12} + 5689131318464 \beta_{11} - 11386534607868 \beta_{10} + 16480906012216 \beta_{9} - 45455051759780 \beta_{8} + 45455051759780 \beta_{7} - 11386534607868 \beta_{6} - 994183242116 \beta_{5} + 61935957771996 \beta_{3} + 13590445167669 \beta_{2} - 42530549970820\)
\(\nu^{16}\)\(=\)\(3930296819656 \beta_{17} + 7858333786556 \beta_{16} + 9402098067641 \beta_{14} + 128074633720903 \beta_{13} + 7858333786556 \beta_{12} + 128074633720903 \beta_{11} - 98545965521027 \beta_{10} + 521545359414761 \beta_{9} - 521545359414761 \beta_{8} - 224416276031732 \beta_{7} - 29528668199876 \beta_{6} - 3928036966900 \beta_{5} - 557483903681648 \beta_{4} + 224416276031732 \beta_{3} - 83123860453267 \beta_{1}\)
\(\nu^{17}\)\(=\)\(-20122050426723 \beta_{17} + 20122050426723 \beta_{16} - 57162326555028 \beta_{15} + 493750285485482 \beta_{13} + 67957658485045 \beta_{12} + 560816701204698 \beta_{11} + 493750285485482 \beta_{10} + 3973829786944264 \beta_{9} - 1538217906478700 \beta_{8} - 5512047693422964 \beta_{7} + 560816701204698 \beta_{6} + 67957658485045 \beta_{5} - 4036005654079904 \beta_{4} - 3973829786944264 \beta_{3} - 1173032816647721 \beta_{2} - 1173032816647721 \beta_{1} + 4036005654079904\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\) \(21\)
\(\chi(n)\) \(\beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−4.21239 + 7.29608i
1.11938 1.93882i
2.15332 3.72966i
−3.03492 5.25663i
−1.54550 2.67688i
4.75406 + 8.23428i
−3.03492 + 5.25663i
−1.54550 + 2.67688i
4.75406 8.23428i
−4.21239 7.29608i
1.11938 + 1.93882i
2.15332 + 3.72966i
2.89395 + 5.01246i
1.58432 + 2.74412i
−3.71222 6.42975i
2.89395 5.01246i
1.58432 2.74412i
−3.71222 + 6.42975i
1.87939 0.684040i −1.31024 + 7.43077i 3.06418 2.57115i 3.45478 + 2.89891i 2.62049 + 14.8615i 4.21085 + 7.29340i 4.00000 6.92820i −28.1279 10.2377i 8.47584 + 3.08495i
5.2 1.87939 0.684040i 0.541460 3.07077i 3.06418 2.57115i 8.27240 + 6.94137i −1.08292 6.14154i −13.7717 23.8533i 4.00000 6.92820i 16.2353 + 5.90915i 20.2952 + 7.38685i
5.3 1.87939 0.684040i 0.900544 5.10724i 3.06418 2.57115i −10.9611 9.19749i −1.80109 10.2145i 16.0725 + 27.8383i 4.00000 6.92820i 0.0987659 + 0.0359478i −26.8917 9.78776i
9.1 −0.347296 1.96962i −5.68185 + 4.76764i −3.75877 + 1.36808i 18.4168 + 6.70318i 11.3637 + 9.53528i −15.6039 + 27.0267i 4.00000 + 6.92820i 4.86455 27.5882i 6.80659 38.6021i
9.2 −0.347296 1.96962i −3.39993 + 2.85288i −3.75877 + 1.36808i −20.2092 7.35554i 6.79986 + 5.70576i 2.98693 5.17352i 4.00000 + 6.92820i −1.26790 + 7.19064i −7.46901 + 42.3588i
9.3 −0.347296 1.96962i 6.25156 5.24568i −3.75877 + 1.36808i 0.852642 + 0.310336i −12.5031 10.4914i −4.68031 + 8.10654i 4.00000 + 6.92820i 6.87632 38.9975i 0.315124 1.78716i
17.1 −0.347296 + 1.96962i −5.68185 4.76764i −3.75877 1.36808i 18.4168 6.70318i 11.3637 9.53528i −15.6039 27.0267i 4.00000 6.92820i 4.86455 + 27.5882i 6.80659 + 38.6021i
17.2 −0.347296 + 1.96962i −3.39993 2.85288i −3.75877 1.36808i −20.2092 + 7.35554i 6.79986 5.70576i 2.98693 + 5.17352i 4.00000 6.92820i −1.26790 7.19064i −7.46901 42.3588i
17.3 −0.347296 + 1.96962i 6.25156 + 5.24568i −3.75877 1.36808i 0.852642 0.310336i −12.5031 + 10.4914i −4.68031 8.10654i 4.00000 6.92820i 6.87632 + 38.9975i 0.315124 + 1.78716i
23.1 1.87939 + 0.684040i −1.31024 7.43077i 3.06418 + 2.57115i 3.45478 2.89891i 2.62049 14.8615i 4.21085 7.29340i 4.00000 + 6.92820i −28.1279 + 10.2377i 8.47584 3.08495i
23.2 1.87939 + 0.684040i 0.541460 + 3.07077i 3.06418 + 2.57115i 8.27240 6.94137i −1.08292 + 6.14154i −13.7717 + 23.8533i 4.00000 + 6.92820i 16.2353 5.90915i 20.2952 7.38685i
23.3 1.87939 + 0.684040i 0.900544 + 5.10724i 3.06418 + 2.57115i −10.9611 + 9.19749i −1.80109 + 10.2145i 16.0725 27.8383i 4.00000 + 6.92820i 0.0987659 0.0359478i −26.8917 + 9.78776i
25.1 −1.53209 + 1.28558i −3.05945 1.11355i 0.694593 3.93923i −1.91957 10.8864i 6.11891 2.22710i 9.80910 16.9899i 4.00000 + 6.92820i −12.5629 10.5416i 16.9363 + 14.2112i
25.2 −1.53209 + 1.28558i −0.598158 0.217712i 0.694593 3.93923i 3.08282 + 17.4836i 1.19632 0.435424i −11.6036 + 20.0980i 4.00000 + 6.92820i −20.3728 17.0948i −27.1996 22.8232i
25.3 −1.53209 + 1.28558i 9.35607 + 3.40533i 0.694593 3.93923i −0.989599 5.61230i −18.7121 + 6.81067i −3.91986 + 6.78940i 4.00000 + 6.92820i 55.2566 + 46.3658i 8.73118 + 7.32633i
35.1 −1.53209 1.28558i −3.05945 + 1.11355i 0.694593 + 3.93923i −1.91957 + 10.8864i 6.11891 + 2.22710i 9.80910 + 16.9899i 4.00000 6.92820i −12.5629 + 10.5416i 16.9363 14.2112i
35.2 −1.53209 1.28558i −0.598158 + 0.217712i 0.694593 + 3.93923i 3.08282 17.4836i 1.19632 + 0.435424i −11.6036 20.0980i 4.00000 6.92820i −20.3728 + 17.0948i −27.1996 + 22.8232i
35.3 −1.53209 1.28558i 9.35607 3.40533i 0.694593 + 3.93923i −0.989599 + 5.61230i −18.7121 6.81067i −3.91986 6.78940i 4.00000 6.92820i 55.2566 46.3658i 8.73118 7.32633i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.4.e.b 18
19.e even 9 1 inner 38.4.e.b 18
19.e even 9 1 722.4.a.t 9
19.f odd 18 1 722.4.a.u 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.e.b 18 1.a even 1 1 trivial
38.4.e.b 18 19.e even 9 1 inner
722.4.a.t 9 19.e even 9 1
722.4.a.u 9 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{18} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 8 T^{3} + 64 T^{6} )^{3} \)
$3$ \( 1 - 6 T - 3 T^{2} - 55 T^{3} - 231 T^{4} - 3153 T^{5} + 58718 T^{6} + 7050 T^{7} + 45123 T^{8} - 349408 T^{9} - 12376779 T^{10} - 151094280 T^{11} + 96439322 T^{12} + 2499516561 T^{13} - 1081271199 T^{14} + 62400324401 T^{15} - 90523254507 T^{16} + 582443874150 T^{17} - 14120123123546 T^{18} + 15725984602050 T^{19} - 65991452535603 T^{20} + 1228225585184883 T^{21} - 574631847267759 T^{22} + 35865330678748827 T^{23} + 37362569288068458 T^{24} - 1580499535752978840 T^{25} - 3495567956097774699 T^{26} - 2664444766034337696 T^{27} + 9290425553506846827 T^{28} + 39191376994216437150 T^{29} + \)\(88\!\cdots\!78\)\( T^{30} - \)\(12\!\cdots\!51\)\( T^{31} - \)\(25\!\cdots\!79\)\( T^{32} - \)\(16\!\cdots\!65\)\( T^{33} - \)\(23\!\cdots\!83\)\( T^{34} - \)\(12\!\cdots\!82\)\( T^{35} + \)\(58\!\cdots\!69\)\( T^{36} \)
$5$ \( 1 - 267 T^{2} + 1258 T^{3} + 32868 T^{4} + 141147 T^{5} - 2046432 T^{6} - 86261805 T^{7} + 585293631 T^{8} + 12218034308 T^{9} - 44359445814 T^{10} - 602925144675 T^{11} - 8772752315690 T^{12} + 87388971513021 T^{13} + 2160066082643325 T^{14} - 15867986376441242 T^{15} - 141389265386101980 T^{16} + 1117288237691422860 T^{17} + 7410539066753057324 T^{18} + \)\(13\!\cdots\!00\)\( T^{19} - \)\(22\!\cdots\!00\)\( T^{20} - \)\(30\!\cdots\!50\)\( T^{21} + \)\(52\!\cdots\!25\)\( T^{22} + \)\(26\!\cdots\!25\)\( T^{23} - \)\(33\!\cdots\!50\)\( T^{24} - \)\(28\!\cdots\!75\)\( T^{25} - \)\(26\!\cdots\!50\)\( T^{26} + \)\(91\!\cdots\!00\)\( T^{27} + \)\(54\!\cdots\!75\)\( T^{28} - \)\(10\!\cdots\!25\)\( T^{29} - \)\(29\!\cdots\!00\)\( T^{30} + \)\(25\!\cdots\!75\)\( T^{31} + \)\(74\!\cdots\!00\)\( T^{32} + \)\(35\!\cdots\!50\)\( T^{33} - \)\(94\!\cdots\!75\)\( T^{34} + \)\(55\!\cdots\!25\)\( T^{36} \)
$7$ \( 1 + 33 T - 570 T^{2} - 23827 T^{3} + 231582 T^{4} + 6957321 T^{5} - 91968137 T^{6} + 896424150 T^{7} + 42932196516 T^{8} - 1678083932540 T^{9} - 16051674667956 T^{10} + 765830563983678 T^{11} + 2876241864331831 T^{12} - 184132608423416247 T^{13} + 1325739588289481910 T^{14} + 16457490302970427475 T^{15} - \)\(15\!\cdots\!82\)\( T^{16} - 79209425745890952231 T^{17} + \)\(66\!\cdots\!02\)\( T^{18} - \)\(27\!\cdots\!33\)\( T^{19} - \)\(17\!\cdots\!18\)\( T^{20} + \)\(66\!\cdots\!25\)\( T^{21} + \)\(18\!\cdots\!10\)\( T^{22} - \)\(87\!\cdots\!21\)\( T^{23} + \)\(46\!\cdots\!19\)\( T^{24} + \)\(42\!\cdots\!46\)\( T^{25} - \)\(30\!\cdots\!56\)\( T^{26} - \)\(11\!\cdots\!20\)\( T^{27} + \)\(96\!\cdots\!84\)\( T^{28} + \)\(69\!\cdots\!50\)\( T^{29} - \)\(24\!\cdots\!37\)\( T^{30} + \)\(63\!\cdots\!03\)\( T^{31} + \)\(72\!\cdots\!18\)\( T^{32} - \)\(25\!\cdots\!89\)\( T^{33} - \)\(20\!\cdots\!70\)\( T^{34} + \)\(41\!\cdots\!19\)\( T^{35} + \)\(43\!\cdots\!49\)\( T^{36} \)
$11$ \( 1 + 75 T - 2889 T^{2} - 357560 T^{3} + 1645863 T^{4} + 927823443 T^{5} + 6218307630 T^{6} - 2171309612715 T^{7} - 39599721354387 T^{8} + 4135376583802898 T^{9} + 123750708119658015 T^{10} - 6171479294557549923 T^{11} - \)\(27\!\cdots\!38\)\( T^{12} + \)\(81\!\cdots\!31\)\( T^{13} + \)\(52\!\cdots\!61\)\( T^{14} - \)\(78\!\cdots\!04\)\( T^{15} - \)\(85\!\cdots\!31\)\( T^{16} + \)\(34\!\cdots\!75\)\( T^{17} + \)\(11\!\cdots\!06\)\( T^{18} + \)\(45\!\cdots\!25\)\( T^{19} - \)\(15\!\cdots\!91\)\( T^{20} - \)\(18\!\cdots\!64\)\( T^{21} + \)\(16\!\cdots\!81\)\( T^{22} + \)\(33\!\cdots\!81\)\( T^{23} - \)\(15\!\cdots\!78\)\( T^{24} - \)\(45\!\cdots\!53\)\( T^{25} + \)\(12\!\cdots\!15\)\( T^{26} + \)\(54\!\cdots\!58\)\( T^{27} - \)\(69\!\cdots\!87\)\( T^{28} - \)\(50\!\cdots\!65\)\( T^{29} + \)\(19\!\cdots\!30\)\( T^{30} + \)\(38\!\cdots\!13\)\( T^{31} + \)\(90\!\cdots\!23\)\( T^{32} - \)\(26\!\cdots\!60\)\( T^{33} - \)\(28\!\cdots\!09\)\( T^{34} + \)\(96\!\cdots\!25\)\( T^{35} + \)\(17\!\cdots\!41\)\( T^{36} \)
$13$ \( 1 - 99 T + 6633 T^{2} - 463206 T^{3} + 34523361 T^{4} - 2511744471 T^{5} + 152308509276 T^{6} - 8698832445249 T^{7} + 528812388841689 T^{8} - 32148958673469604 T^{9} + 1764643684881532491 T^{10} - 92665293196837026267 T^{11} + \)\(48\!\cdots\!44\)\( T^{12} - \)\(26\!\cdots\!53\)\( T^{13} + \)\(13\!\cdots\!19\)\( T^{14} - \)\(66\!\cdots\!14\)\( T^{15} + \)\(32\!\cdots\!55\)\( T^{16} - \)\(16\!\cdots\!81\)\( T^{17} + \)\(77\!\cdots\!06\)\( T^{18} - \)\(35\!\cdots\!57\)\( T^{19} + \)\(15\!\cdots\!95\)\( T^{20} - \)\(70\!\cdots\!22\)\( T^{21} + \)\(32\!\cdots\!39\)\( T^{22} - \)\(13\!\cdots\!21\)\( T^{23} + \)\(55\!\cdots\!76\)\( T^{24} - \)\(22\!\cdots\!71\)\( T^{25} + \)\(95\!\cdots\!51\)\( T^{26} - \)\(38\!\cdots\!68\)\( T^{27} + \)\(13\!\cdots\!61\)\( T^{28} - \)\(50\!\cdots\!97\)\( T^{29} + \)\(19\!\cdots\!16\)\( T^{30} - \)\(69\!\cdots\!67\)\( T^{31} + \)\(21\!\cdots\!09\)\( T^{32} - \)\(62\!\cdots\!58\)\( T^{33} + \)\(19\!\cdots\!93\)\( T^{34} - \)\(64\!\cdots\!63\)\( T^{35} + \)\(14\!\cdots\!89\)\( T^{36} \)
$17$ \( 1 + 111 T + 7893 T^{2} - 249246 T^{3} - 103167585 T^{4} - 6990209259 T^{5} - 53066514354 T^{6} + 54651070131375 T^{7} + 5641811241072309 T^{8} + 160668296353458864 T^{9} - 16273510166757181149 T^{10} - \)\(27\!\cdots\!99\)\( T^{11} - \)\(15\!\cdots\!62\)\( T^{12} + \)\(20\!\cdots\!23\)\( T^{13} + \)\(94\!\cdots\!37\)\( T^{14} + \)\(77\!\cdots\!22\)\( T^{15} + \)\(15\!\cdots\!27\)\( T^{16} - \)\(28\!\cdots\!63\)\( T^{17} - \)\(30\!\cdots\!90\)\( T^{18} - \)\(13\!\cdots\!19\)\( T^{19} + \)\(36\!\cdots\!63\)\( T^{20} + \)\(92\!\cdots\!34\)\( T^{21} + \)\(54\!\cdots\!57\)\( T^{22} + \)\(59\!\cdots\!39\)\( T^{23} - \)\(21\!\cdots\!58\)\( T^{24} - \)\(19\!\cdots\!83\)\( T^{25} - \)\(55\!\cdots\!29\)\( T^{26} + \)\(26\!\cdots\!72\)\( T^{27} + \)\(46\!\cdots\!41\)\( T^{28} + \)\(21\!\cdots\!75\)\( T^{29} - \)\(10\!\cdots\!74\)\( T^{30} - \)\(67\!\cdots\!27\)\( T^{31} - \)\(49\!\cdots\!65\)\( T^{32} - \)\(58\!\cdots\!22\)\( T^{33} + \)\(90\!\cdots\!13\)\( T^{34} + \)\(62\!\cdots\!63\)\( T^{35} + \)\(27\!\cdots\!29\)\( T^{36} \)
$19$ \( 1 + 372 T + 46743 T^{2} - 48370 T^{3} - 656902341 T^{4} - 74163224628 T^{5} - 2508347904706 T^{6} + 199773729396042 T^{7} + 30834150746767545 T^{8} + 2625732276260869232 T^{9} + \)\(21\!\cdots\!55\)\( T^{10} + \)\(93\!\cdots\!02\)\( T^{11} - \)\(80\!\cdots\!74\)\( T^{12} - \)\(16\!\cdots\!08\)\( T^{13} - \)\(99\!\cdots\!59\)\( T^{14} - \)\(50\!\cdots\!70\)\( T^{15} + \)\(33\!\cdots\!17\)\( T^{16} + \)\(18\!\cdots\!12\)\( T^{17} + \)\(33\!\cdots\!39\)\( T^{18} \)
$23$ \( 1 - 198 T - 705 T^{2} - 91301 T^{3} + 438671529 T^{4} + 2726772345 T^{5} - 4473792236496 T^{6} - 529950580493355 T^{7} + 16379374840441131 T^{8} + 14761633655243648819 T^{9} - \)\(56\!\cdots\!61\)\( T^{10} - \)\(80\!\cdots\!90\)\( T^{11} - \)\(80\!\cdots\!40\)\( T^{12} + \)\(18\!\cdots\!88\)\( T^{13} + \)\(11\!\cdots\!81\)\( T^{14} - \)\(18\!\cdots\!33\)\( T^{15} - \)\(23\!\cdots\!57\)\( T^{16} - \)\(27\!\cdots\!01\)\( T^{17} + \)\(55\!\cdots\!22\)\( T^{18} - \)\(33\!\cdots\!67\)\( T^{19} - \)\(34\!\cdots\!73\)\( T^{20} - \)\(32\!\cdots\!79\)\( T^{21} + \)\(25\!\cdots\!01\)\( T^{22} + \)\(49\!\cdots\!16\)\( T^{23} - \)\(26\!\cdots\!60\)\( T^{24} - \)\(31\!\cdots\!70\)\( T^{25} - \)\(26\!\cdots\!01\)\( T^{26} + \)\(86\!\cdots\!93\)\( T^{27} + \)\(11\!\cdots\!19\)\( T^{28} - \)\(45\!\cdots\!65\)\( T^{29} - \)\(47\!\cdots\!56\)\( T^{30} + \)\(34\!\cdots\!15\)\( T^{31} + \)\(68\!\cdots\!41\)\( T^{32} - \)\(17\!\cdots\!43\)\( T^{33} - \)\(16\!\cdots\!05\)\( T^{34} - \)\(55\!\cdots\!46\)\( T^{35} + \)\(34\!\cdots\!09\)\( T^{36} \)
$29$ \( 1 - 669 T + 215391 T^{2} - 38499082 T^{3} + 2308041732 T^{4} + 721346289873 T^{5} - 230277676776537 T^{6} + 29270701823941662 T^{7} - 256611486308977863 T^{8} - \)\(65\!\cdots\!33\)\( T^{9} + \)\(14\!\cdots\!05\)\( T^{10} - \)\(15\!\cdots\!87\)\( T^{11} - \)\(16\!\cdots\!43\)\( T^{12} + \)\(46\!\cdots\!60\)\( T^{13} - \)\(10\!\cdots\!28\)\( T^{14} + \)\(12\!\cdots\!28\)\( T^{15} + \)\(33\!\cdots\!71\)\( T^{16} - \)\(43\!\cdots\!60\)\( T^{17} + \)\(93\!\cdots\!58\)\( T^{18} - \)\(10\!\cdots\!40\)\( T^{19} + \)\(19\!\cdots\!91\)\( T^{20} + \)\(17\!\cdots\!32\)\( T^{21} - \)\(38\!\cdots\!48\)\( T^{22} + \)\(40\!\cdots\!40\)\( T^{23} - \)\(34\!\cdots\!23\)\( T^{24} - \)\(79\!\cdots\!23\)\( T^{25} + \)\(18\!\cdots\!05\)\( T^{26} - \)\(20\!\cdots\!97\)\( T^{27} - \)\(19\!\cdots\!63\)\( T^{28} + \)\(53\!\cdots\!18\)\( T^{29} - \)\(10\!\cdots\!77\)\( T^{30} + \)\(77\!\cdots\!37\)\( T^{31} + \)\(60\!\cdots\!12\)\( T^{32} - \)\(24\!\cdots\!18\)\( T^{33} + \)\(33\!\cdots\!51\)\( T^{34} - \)\(25\!\cdots\!01\)\( T^{35} + \)\(93\!\cdots\!81\)\( T^{36} \)
$31$ \( 1 + 42 T - 94569 T^{2} - 127700 T^{3} + 5864595627 T^{4} + 35375058546 T^{5} - 164526482997895 T^{6} + 1524651579146895 T^{7} + 1463730669606344745 T^{8} + \)\(18\!\cdots\!63\)\( T^{9} + \)\(17\!\cdots\!97\)\( T^{10} - \)\(78\!\cdots\!41\)\( T^{11} - \)\(73\!\cdots\!09\)\( T^{12} + \)\(56\!\cdots\!93\)\( T^{13} + \)\(16\!\cdots\!00\)\( T^{14} - \)\(10\!\cdots\!81\)\( T^{15} + \)\(20\!\cdots\!48\)\( T^{16} + \)\(25\!\cdots\!41\)\( T^{17} - \)\(11\!\cdots\!30\)\( T^{18} + \)\(75\!\cdots\!31\)\( T^{19} + \)\(18\!\cdots\!88\)\( T^{20} - \)\(27\!\cdots\!51\)\( T^{21} + \)\(12\!\cdots\!00\)\( T^{22} + \)\(13\!\cdots\!43\)\( T^{23} - \)\(51\!\cdots\!69\)\( T^{24} - \)\(16\!\cdots\!71\)\( T^{25} + \)\(11\!\cdots\!37\)\( T^{26} + \)\(35\!\cdots\!93\)\( T^{27} + \)\(80\!\cdots\!45\)\( T^{28} + \)\(25\!\cdots\!45\)\( T^{29} - \)\(80\!\cdots\!95\)\( T^{30} + \)\(51\!\cdots\!66\)\( T^{31} + \)\(25\!\cdots\!47\)\( T^{32} - \)\(16\!\cdots\!00\)\( T^{33} - \)\(36\!\cdots\!29\)\( T^{34} + \)\(48\!\cdots\!02\)\( T^{35} + \)\(34\!\cdots\!21\)\( T^{36} \)
$37$ \( ( 1 + 528 T + 331644 T^{2} + 146244143 T^{3} + 58414086261 T^{4} + 19852757182098 T^{5} + 6310834206147419 T^{6} + 1748373264723073161 T^{7} + \)\(45\!\cdots\!51\)\( T^{8} + \)\(10\!\cdots\!80\)\( T^{9} + \)\(23\!\cdots\!03\)\( T^{10} + \)\(44\!\cdots\!49\)\( T^{11} + \)\(82\!\cdots\!63\)\( T^{12} + \)\(13\!\cdots\!38\)\( T^{13} + \)\(19\!\cdots\!73\)\( T^{14} + \)\(24\!\cdots\!47\)\( T^{15} + \)\(28\!\cdots\!28\)\( T^{16} + \)\(22\!\cdots\!08\)\( T^{17} + \)\(21\!\cdots\!33\)\( T^{18} )^{2} \)
$41$ \( 1 + 210 T + 104286 T^{2} + 22997855 T^{3} + 9934534284 T^{4} + 1831929309408 T^{5} + 826578393532683 T^{6} + 95912877955086318 T^{7} + 53991387444209097630 T^{8} + \)\(49\!\cdots\!04\)\( T^{9} + \)\(27\!\cdots\!06\)\( T^{10} + \)\(19\!\cdots\!38\)\( T^{11} + \)\(18\!\cdots\!77\)\( T^{12} + \)\(56\!\cdots\!12\)\( T^{13} + \)\(97\!\cdots\!84\)\( T^{14} - \)\(65\!\cdots\!35\)\( T^{15} + \)\(32\!\cdots\!66\)\( T^{16} - \)\(73\!\cdots\!62\)\( T^{17} + \)\(24\!\cdots\!06\)\( T^{18} - \)\(50\!\cdots\!02\)\( T^{19} + \)\(15\!\cdots\!06\)\( T^{20} - \)\(21\!\cdots\!35\)\( T^{21} + \)\(22\!\cdots\!04\)\( T^{22} + \)\(87\!\cdots\!12\)\( T^{23} + \)\(19\!\cdots\!17\)\( T^{24} + \)\(14\!\cdots\!58\)\( T^{25} + \)\(13\!\cdots\!66\)\( T^{26} + \)\(17\!\cdots\!24\)\( T^{27} + \)\(13\!\cdots\!30\)\( T^{28} + \)\(15\!\cdots\!78\)\( T^{29} + \)\(94\!\cdots\!03\)\( T^{30} + \)\(14\!\cdots\!88\)\( T^{31} + \)\(54\!\cdots\!04\)\( T^{32} + \)\(86\!\cdots\!55\)\( T^{33} + \)\(27\!\cdots\!06\)\( T^{34} + \)\(37\!\cdots\!10\)\( T^{35} + \)\(12\!\cdots\!61\)\( T^{36} \)
$43$ \( 1 + 399 T - 16899 T^{2} - 29870330 T^{3} - 6681874995 T^{4} - 2945583704673 T^{5} - 279451424238580 T^{6} + 114723718715195925 T^{7} - 22893423556005010389 T^{8} - \)\(50\!\cdots\!26\)\( T^{9} + \)\(49\!\cdots\!25\)\( T^{10} + \)\(83\!\cdots\!89\)\( T^{11} + \)\(32\!\cdots\!32\)\( T^{12} + \)\(19\!\cdots\!11\)\( T^{13} + \)\(15\!\cdots\!39\)\( T^{14} - \)\(77\!\cdots\!98\)\( T^{15} - \)\(13\!\cdots\!01\)\( T^{16} - \)\(57\!\cdots\!89\)\( T^{17} - \)\(27\!\cdots\!34\)\( T^{18} - \)\(45\!\cdots\!23\)\( T^{19} - \)\(87\!\cdots\!49\)\( T^{20} - \)\(38\!\cdots\!14\)\( T^{21} + \)\(60\!\cdots\!39\)\( T^{22} + \)\(61\!\cdots\!77\)\( T^{23} + \)\(81\!\cdots\!68\)\( T^{24} + \)\(16\!\cdots\!27\)\( T^{25} + \)\(78\!\cdots\!25\)\( T^{26} - \)\(64\!\cdots\!82\)\( T^{27} - \)\(23\!\cdots\!61\)\( T^{28} + \)\(92\!\cdots\!75\)\( T^{29} - \)\(17\!\cdots\!80\)\( T^{30} - \)\(14\!\cdots\!11\)\( T^{31} - \)\(26\!\cdots\!55\)\( T^{32} - \)\(95\!\cdots\!90\)\( T^{33} - \)\(43\!\cdots\!99\)\( T^{34} + \)\(80\!\cdots\!93\)\( T^{35} + \)\(16\!\cdots\!49\)\( T^{36} \)
$47$ \( 1 - 1149 T + 878931 T^{2} - 558739520 T^{3} + 314481583065 T^{4} - 156054243768369 T^{5} + 71251581438674772 T^{6} - 30322087008759916809 T^{7} + \)\(12\!\cdots\!45\)\( T^{8} - \)\(46\!\cdots\!22\)\( T^{9} + \)\(16\!\cdots\!03\)\( T^{10} - \)\(58\!\cdots\!25\)\( T^{11} + \)\(19\!\cdots\!00\)\( T^{12} - \)\(62\!\cdots\!97\)\( T^{13} + \)\(19\!\cdots\!99\)\( T^{14} - \)\(61\!\cdots\!00\)\( T^{15} + \)\(19\!\cdots\!97\)\( T^{16} - \)\(59\!\cdots\!05\)\( T^{17} + \)\(18\!\cdots\!14\)\( T^{18} - \)\(61\!\cdots\!15\)\( T^{19} + \)\(20\!\cdots\!13\)\( T^{20} - \)\(68\!\cdots\!00\)\( T^{21} + \)\(22\!\cdots\!59\)\( T^{22} - \)\(75\!\cdots\!71\)\( T^{23} + \)\(24\!\cdots\!00\)\( T^{24} - \)\(75\!\cdots\!75\)\( T^{25} + \)\(22\!\cdots\!43\)\( T^{26} - \)\(64\!\cdots\!86\)\( T^{27} + \)\(17\!\cdots\!05\)\( T^{28} - \)\(45\!\cdots\!43\)\( T^{29} + \)\(11\!\cdots\!12\)\( T^{30} - \)\(25\!\cdots\!27\)\( T^{31} + \)\(53\!\cdots\!85\)\( T^{32} - \)\(98\!\cdots\!40\)\( T^{33} + \)\(16\!\cdots\!91\)\( T^{34} - \)\(21\!\cdots\!47\)\( T^{35} + \)\(19\!\cdots\!69\)\( T^{36} \)
$53$ \( 1 + 633 T + 255945 T^{2} + 126268180 T^{3} - 12487303869 T^{4} - 27394679707125 T^{5} - 11925746432672124 T^{6} - 7608133460276670363 T^{7} - \)\(19\!\cdots\!93\)\( T^{8} - \)\(65\!\cdots\!24\)\( T^{9} + \)\(31\!\cdots\!37\)\( T^{10} + \)\(13\!\cdots\!07\)\( T^{11} + \)\(68\!\cdots\!80\)\( T^{12} + \)\(20\!\cdots\!69\)\( T^{13} + \)\(75\!\cdots\!69\)\( T^{14} + \)\(59\!\cdots\!68\)\( T^{15} - \)\(67\!\cdots\!89\)\( T^{16} - \)\(44\!\cdots\!41\)\( T^{17} - \)\(22\!\cdots\!34\)\( T^{18} - \)\(66\!\cdots\!57\)\( T^{19} - \)\(14\!\cdots\!81\)\( T^{20} + \)\(19\!\cdots\!44\)\( T^{21} + \)\(36\!\cdots\!29\)\( T^{22} + \)\(14\!\cdots\!33\)\( T^{23} + \)\(74\!\cdots\!20\)\( T^{24} + \)\(21\!\cdots\!71\)\( T^{25} + \)\(76\!\cdots\!97\)\( T^{26} - \)\(23\!\cdots\!88\)\( T^{27} - \)\(10\!\cdots\!57\)\( T^{28} - \)\(60\!\cdots\!99\)\( T^{29} - \)\(14\!\cdots\!04\)\( T^{30} - \)\(48\!\cdots\!25\)\( T^{31} - \)\(32\!\cdots\!21\)\( T^{32} + \)\(49\!\cdots\!40\)\( T^{33} + \)\(14\!\cdots\!45\)\( T^{34} + \)\(54\!\cdots\!01\)\( T^{35} + \)\(12\!\cdots\!69\)\( T^{36} \)
$59$ \( 1 - 51 T - 451647 T^{2} - 21382794 T^{3} + 80110363617 T^{4} + 50594271665385 T^{5} - 1977453238201620 T^{6} - 16700186133425135661 T^{7} - \)\(47\!\cdots\!09\)\( T^{8} + \)\(17\!\cdots\!90\)\( T^{9} + \)\(22\!\cdots\!41\)\( T^{10} + \)\(40\!\cdots\!47\)\( T^{11} - \)\(43\!\cdots\!08\)\( T^{12} - \)\(23\!\cdots\!67\)\( T^{13} + \)\(27\!\cdots\!51\)\( T^{14} + \)\(60\!\cdots\!82\)\( T^{15} + \)\(18\!\cdots\!03\)\( T^{16} - \)\(60\!\cdots\!27\)\( T^{17} - \)\(51\!\cdots\!02\)\( T^{18} - \)\(12\!\cdots\!33\)\( T^{19} + \)\(77\!\cdots\!23\)\( T^{20} + \)\(52\!\cdots\!98\)\( T^{21} + \)\(49\!\cdots\!31\)\( T^{22} - \)\(85\!\cdots\!33\)\( T^{23} - \)\(32\!\cdots\!68\)\( T^{24} + \)\(61\!\cdots\!73\)\( T^{25} + \)\(72\!\cdots\!01\)\( T^{26} + \)\(11\!\cdots\!10\)\( T^{27} - \)\(63\!\cdots\!09\)\( T^{28} - \)\(45\!\cdots\!19\)\( T^{29} - \)\(11\!\cdots\!20\)\( T^{30} + \)\(58\!\cdots\!15\)\( T^{31} + \)\(19\!\cdots\!77\)\( T^{32} - \)\(10\!\cdots\!06\)\( T^{33} - \)\(45\!\cdots\!87\)\( T^{34} - \)\(10\!\cdots\!09\)\( T^{35} + \)\(42\!\cdots\!61\)\( T^{36} \)
$61$ \( 1 + 4104 T + 8721243 T^{2} + 12845942331 T^{3} + 14723429394375 T^{4} + 13952021159336931 T^{5} + 11344408872753650538 T^{6} + \)\(81\!\cdots\!39\)\( T^{7} + \)\(52\!\cdots\!41\)\( T^{8} + \)\(30\!\cdots\!55\)\( T^{9} + \)\(16\!\cdots\!01\)\( T^{10} + \)\(86\!\cdots\!10\)\( T^{11} + \)\(43\!\cdots\!06\)\( T^{12} + \)\(21\!\cdots\!18\)\( T^{13} + \)\(10\!\cdots\!71\)\( T^{14} + \)\(54\!\cdots\!31\)\( T^{15} + \)\(27\!\cdots\!47\)\( T^{16} + \)\(13\!\cdots\!21\)\( T^{17} + \)\(65\!\cdots\!78\)\( T^{18} + \)\(30\!\cdots\!01\)\( T^{19} + \)\(14\!\cdots\!67\)\( T^{20} + \)\(63\!\cdots\!71\)\( T^{21} + \)\(28\!\cdots\!91\)\( T^{22} + \)\(13\!\cdots\!18\)\( T^{23} + \)\(59\!\cdots\!86\)\( T^{24} + \)\(26\!\cdots\!10\)\( T^{25} + \)\(11\!\cdots\!41\)\( T^{26} + \)\(48\!\cdots\!55\)\( T^{27} + \)\(18\!\cdots\!41\)\( T^{28} + \)\(66\!\cdots\!59\)\( T^{29} + \)\(21\!\cdots\!18\)\( T^{30} + \)\(59\!\cdots\!71\)\( T^{31} + \)\(14\!\cdots\!75\)\( T^{32} + \)\(28\!\cdots\!31\)\( T^{33} + \)\(43\!\cdots\!83\)\( T^{34} + \)\(46\!\cdots\!44\)\( T^{35} + \)\(25\!\cdots\!41\)\( T^{36} \)
$67$ \( 1 + 675 T + 659343 T^{2} + 100752002 T^{3} + 218221755063 T^{4} + 72082004612133 T^{5} + 110077824496142386 T^{6} + 12369574887734108319 T^{7} + \)\(32\!\cdots\!89\)\( T^{8} + \)\(94\!\cdots\!86\)\( T^{9} + \)\(13\!\cdots\!51\)\( T^{10} + \)\(12\!\cdots\!51\)\( T^{11} + \)\(25\!\cdots\!06\)\( T^{12} + \)\(65\!\cdots\!97\)\( T^{13} + \)\(11\!\cdots\!85\)\( T^{14} + \)\(15\!\cdots\!54\)\( T^{15} + \)\(19\!\cdots\!93\)\( T^{16} + \)\(72\!\cdots\!19\)\( T^{17} + \)\(11\!\cdots\!06\)\( T^{18} + \)\(21\!\cdots\!97\)\( T^{19} + \)\(17\!\cdots\!17\)\( T^{20} + \)\(41\!\cdots\!38\)\( T^{21} + \)\(95\!\cdots\!85\)\( T^{22} + \)\(16\!\cdots\!71\)\( T^{23} + \)\(18\!\cdots\!54\)\( T^{24} + \)\(28\!\cdots\!17\)\( T^{25} + \)\(93\!\cdots\!71\)\( T^{26} + \)\(19\!\cdots\!78\)\( T^{27} + \)\(19\!\cdots\!61\)\( T^{28} + \)\(22\!\cdots\!53\)\( T^{29} + \)\(60\!\cdots\!66\)\( T^{30} + \)\(11\!\cdots\!99\)\( T^{31} + \)\(10\!\cdots\!07\)\( T^{32} + \)\(15\!\cdots\!14\)\( T^{33} + \)\(29\!\cdots\!63\)\( T^{34} + \)\(91\!\cdots\!25\)\( T^{35} + \)\(40\!\cdots\!29\)\( T^{36} \)
$71$ \( 1 - 2964 T + 5181987 T^{2} - 6692184605 T^{3} + 7021800717027 T^{4} - 6149705076747273 T^{5} + 4509578307489711810 T^{6} - \)\(26\!\cdots\!71\)\( T^{7} + \)\(11\!\cdots\!39\)\( T^{8} - \)\(10\!\cdots\!51\)\( T^{9} - \)\(38\!\cdots\!05\)\( T^{10} + \)\(48\!\cdots\!64\)\( T^{11} - \)\(38\!\cdots\!26\)\( T^{12} + \)\(22\!\cdots\!36\)\( T^{13} - \)\(89\!\cdots\!81\)\( T^{14} + \)\(69\!\cdots\!55\)\( T^{15} + \)\(26\!\cdots\!55\)\( T^{16} - \)\(30\!\cdots\!81\)\( T^{17} + \)\(21\!\cdots\!62\)\( T^{18} - \)\(10\!\cdots\!91\)\( T^{19} + \)\(34\!\cdots\!55\)\( T^{20} + \)\(31\!\cdots\!05\)\( T^{21} - \)\(14\!\cdots\!21\)\( T^{22} + \)\(13\!\cdots\!36\)\( T^{23} - \)\(81\!\cdots\!86\)\( T^{24} + \)\(36\!\cdots\!44\)\( T^{25} - \)\(10\!\cdots\!05\)\( T^{26} - \)\(10\!\cdots\!41\)\( T^{27} + \)\(38\!\cdots\!39\)\( T^{28} - \)\(33\!\cdots\!81\)\( T^{29} + \)\(19\!\cdots\!10\)\( T^{30} - \)\(97\!\cdots\!63\)\( T^{31} + \)\(39\!\cdots\!07\)\( T^{32} - \)\(13\!\cdots\!55\)\( T^{33} + \)\(37\!\cdots\!07\)\( T^{34} - \)\(76\!\cdots\!44\)\( T^{35} + \)\(92\!\cdots\!81\)\( T^{36} \)
$73$ \( 1 + 2004 T + 2000946 T^{2} + 455386946 T^{3} - 1347367876605 T^{4} - 2017261870614000 T^{5} - 1209003634116669326 T^{6} + \)\(10\!\cdots\!82\)\( T^{7} + \)\(84\!\cdots\!51\)\( T^{8} + \)\(72\!\cdots\!80\)\( T^{9} + \)\(20\!\cdots\!52\)\( T^{10} - \)\(16\!\cdots\!78\)\( T^{11} - \)\(22\!\cdots\!56\)\( T^{12} - \)\(98\!\cdots\!40\)\( T^{13} + \)\(13\!\cdots\!24\)\( T^{14} + \)\(41\!\cdots\!42\)\( T^{15} + \)\(18\!\cdots\!49\)\( T^{16} - \)\(36\!\cdots\!00\)\( T^{17} - \)\(74\!\cdots\!40\)\( T^{18} - \)\(14\!\cdots\!00\)\( T^{19} + \)\(28\!\cdots\!61\)\( T^{20} + \)\(24\!\cdots\!46\)\( T^{21} + \)\(31\!\cdots\!04\)\( T^{22} - \)\(87\!\cdots\!80\)\( T^{23} - \)\(77\!\cdots\!64\)\( T^{24} - \)\(22\!\cdots\!94\)\( T^{25} + \)\(10\!\cdots\!32\)\( T^{26} + \)\(14\!\cdots\!60\)\( T^{27} + \)\(67\!\cdots\!99\)\( T^{28} + \)\(31\!\cdots\!06\)\( T^{29} - \)\(14\!\cdots\!86\)\( T^{30} - \)\(94\!\cdots\!00\)\( T^{31} - \)\(24\!\cdots\!45\)\( T^{32} + \)\(32\!\cdots\!78\)\( T^{33} + \)\(55\!\cdots\!26\)\( T^{34} + \)\(21\!\cdots\!08\)\( T^{35} + \)\(41\!\cdots\!09\)\( T^{36} \)
$79$ \( 1 - 543 T + 237567 T^{2} - 999160254 T^{3} + 523827589830 T^{4} - 96194492457153 T^{5} + 539738595627010821 T^{6} - \)\(40\!\cdots\!30\)\( T^{7} + \)\(94\!\cdots\!63\)\( T^{8} - \)\(22\!\cdots\!17\)\( T^{9} + \)\(23\!\cdots\!31\)\( T^{10} - \)\(59\!\cdots\!77\)\( T^{11} + \)\(54\!\cdots\!91\)\( T^{12} - \)\(95\!\cdots\!60\)\( T^{13} + \)\(34\!\cdots\!30\)\( T^{14} - \)\(76\!\cdots\!74\)\( T^{15} + \)\(33\!\cdots\!23\)\( T^{16} - \)\(18\!\cdots\!82\)\( T^{17} + \)\(10\!\cdots\!22\)\( T^{18} - \)\(93\!\cdots\!98\)\( T^{19} + \)\(81\!\cdots\!83\)\( T^{20} - \)\(91\!\cdots\!06\)\( T^{21} + \)\(20\!\cdots\!30\)\( T^{22} - \)\(27\!\cdots\!40\)\( T^{23} + \)\(77\!\cdots\!51\)\( T^{24} - \)\(42\!\cdots\!83\)\( T^{25} + \)\(81\!\cdots\!11\)\( T^{26} - \)\(39\!\cdots\!03\)\( T^{27} + \)\(80\!\cdots\!63\)\( T^{28} - \)\(16\!\cdots\!70\)\( T^{29} + \)\(11\!\cdots\!41\)\( T^{30} - \)\(97\!\cdots\!07\)\( T^{31} + \)\(26\!\cdots\!30\)\( T^{32} - \)\(24\!\cdots\!46\)\( T^{33} + \)\(28\!\cdots\!87\)\( T^{34} - \)\(32\!\cdots\!97\)\( T^{35} + \)\(29\!\cdots\!81\)\( T^{36} \)
$83$ \( 1 - 381 T - 2900673 T^{2} + 447885504 T^{3} + 4876458766569 T^{4} + 264353813817459 T^{5} - 5416917118351160406 T^{6} - \)\(17\!\cdots\!41\)\( T^{7} + \)\(42\!\cdots\!03\)\( T^{8} + \)\(29\!\cdots\!18\)\( T^{9} - \)\(20\!\cdots\!59\)\( T^{10} - \)\(32\!\cdots\!17\)\( T^{11} + \)\(83\!\cdots\!18\)\( T^{12} + \)\(23\!\cdots\!99\)\( T^{13} + \)\(98\!\cdots\!71\)\( T^{14} - \)\(11\!\cdots\!64\)\( T^{15} - \)\(11\!\cdots\!55\)\( T^{16} + \)\(27\!\cdots\!23\)\( T^{17} + \)\(77\!\cdots\!90\)\( T^{18} + \)\(15\!\cdots\!01\)\( T^{19} - \)\(36\!\cdots\!95\)\( T^{20} - \)\(22\!\cdots\!92\)\( T^{21} + \)\(10\!\cdots\!31\)\( T^{22} + \)\(14\!\cdots\!93\)\( T^{23} + \)\(29\!\cdots\!62\)\( T^{24} - \)\(64\!\cdots\!11\)\( T^{25} - \)\(23\!\cdots\!39\)\( T^{26} + \)\(19\!\cdots\!86\)\( T^{27} + \)\(15\!\cdots\!47\)\( T^{28} - \)\(37\!\cdots\!83\)\( T^{29} - \)\(66\!\cdots\!86\)\( T^{30} + \)\(18\!\cdots\!73\)\( T^{31} + \)\(19\!\cdots\!41\)\( T^{32} + \)\(10\!\cdots\!72\)\( T^{33} - \)\(37\!\cdots\!93\)\( T^{34} - \)\(28\!\cdots\!27\)\( T^{35} + \)\(42\!\cdots\!29\)\( T^{36} \)
$89$ \( 1 - 4386 T + 8417457 T^{2} - 8148247434 T^{3} + 1704831378354 T^{4} + 6057030791519313 T^{5} - 8758244485320704556 T^{6} + \)\(47\!\cdots\!75\)\( T^{7} + \)\(23\!\cdots\!85\)\( T^{8} - \)\(79\!\cdots\!84\)\( T^{9} + \)\(88\!\cdots\!08\)\( T^{10} - \)\(56\!\cdots\!43\)\( T^{11} + \)\(59\!\cdots\!62\)\( T^{12} + \)\(36\!\cdots\!15\)\( T^{13} - \)\(49\!\cdots\!97\)\( T^{14} + \)\(31\!\cdots\!54\)\( T^{15} + \)\(36\!\cdots\!84\)\( T^{16} - \)\(30\!\cdots\!90\)\( T^{17} + \)\(35\!\cdots\!28\)\( T^{18} - \)\(21\!\cdots\!10\)\( T^{19} + \)\(18\!\cdots\!24\)\( T^{20} + \)\(10\!\cdots\!86\)\( T^{21} - \)\(12\!\cdots\!37\)\( T^{22} + \)\(62\!\cdots\!35\)\( T^{23} + \)\(73\!\cdots\!22\)\( T^{24} - \)\(49\!\cdots\!27\)\( T^{25} + \)\(54\!\cdots\!28\)\( T^{26} - \)\(34\!\cdots\!36\)\( T^{27} + \)\(72\!\cdots\!85\)\( T^{28} + \)\(10\!\cdots\!75\)\( T^{29} - \)\(13\!\cdots\!16\)\( T^{30} + \)\(64\!\cdots\!17\)\( T^{31} + \)\(12\!\cdots\!34\)\( T^{32} - \)\(43\!\cdots\!66\)\( T^{33} + \)\(31\!\cdots\!17\)\( T^{34} - \)\(11\!\cdots\!54\)\( T^{35} + \)\(18\!\cdots\!41\)\( T^{36} \)
$97$ \( 1 - 7599 T + 26158659 T^{2} - 55340804582 T^{3} + 85085612025453 T^{4} - 109382795406678699 T^{5} + \)\(12\!\cdots\!66\)\( T^{6} - \)\(13\!\cdots\!89\)\( T^{7} + \)\(14\!\cdots\!97\)\( T^{8} - \)\(16\!\cdots\!24\)\( T^{9} + \)\(18\!\cdots\!71\)\( T^{10} - \)\(18\!\cdots\!19\)\( T^{11} + \)\(16\!\cdots\!70\)\( T^{12} - \)\(14\!\cdots\!97\)\( T^{13} + \)\(12\!\cdots\!23\)\( T^{14} - \)\(10\!\cdots\!14\)\( T^{15} + \)\(10\!\cdots\!05\)\( T^{16} - \)\(10\!\cdots\!21\)\( T^{17} + \)\(10\!\cdots\!22\)\( T^{18} - \)\(96\!\cdots\!33\)\( T^{19} + \)\(88\!\cdots\!45\)\( T^{20} - \)\(83\!\cdots\!38\)\( T^{21} + \)\(86\!\cdots\!43\)\( T^{22} - \)\(93\!\cdots\!21\)\( T^{23} + \)\(97\!\cdots\!30\)\( T^{24} - \)\(95\!\cdots\!43\)\( T^{25} + \)\(87\!\cdots\!51\)\( T^{26} - \)\(72\!\cdots\!12\)\( T^{27} + \)\(59\!\cdots\!53\)\( T^{28} - \)\(51\!\cdots\!53\)\( T^{29} + \)\(42\!\cdots\!86\)\( T^{30} - \)\(33\!\cdots\!67\)\( T^{31} + \)\(23\!\cdots\!77\)\( T^{32} - \)\(14\!\cdots\!74\)\( T^{33} + \)\(60\!\cdots\!99\)\( T^{34} - \)\(16\!\cdots\!47\)\( T^{35} + \)\(19\!\cdots\!69\)\( T^{36} \)
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