Properties

Label 13.5.f.a
Level 13
Weight 5
Character orbit 13.f
Analytic conductor 1.344
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 13.f (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.34380952009\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2}\cdot 13^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{12} q^{2} + ( -1 - \beta_{8} + \beta_{9} + \beta_{12} - \beta_{14} ) q^{3} + ( 8 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{11} + \beta_{12} ) q^{4} + ( 4 - 8 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{5} + ( -9 + 14 \beta_{1} + 9 \beta_{2} - 13 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{6} + ( 8 - 7 \beta_{1} + 7 \beta_{2} - 6 \beta_{3} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{7} + ( 22 \beta_{1} - 14 \beta_{2} - 4 \beta_{3} - 8 \beta_{4} + \beta_{5} + 4 \beta_{7} + 4 \beta_{8} + 4 \beta_{10} + 5 \beta_{12} - \beta_{13} + 12 \beta_{14} - \beta_{15} ) q^{8} + ( -3 - 21 \beta_{1} - 18 \beta_{2} + 11 \beta_{3} + 3 \beta_{4} + 12 \beta_{5} - 6 \beta_{7} - 4 \beta_{9} - 3 \beta_{10} - \beta_{11} - 6 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} ) q^{9} +O(q^{10})\) \( q -\beta_{12} q^{2} + ( -1 - \beta_{8} + \beta_{9} + \beta_{12} - \beta_{14} ) q^{3} + ( 8 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{11} + \beta_{12} ) q^{4} + ( 4 - 8 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{5} + ( -9 + 14 \beta_{1} + 9 \beta_{2} - 13 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{6} + ( 8 - 7 \beta_{1} + 7 \beta_{2} - 6 \beta_{3} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{7} + ( 22 \beta_{1} - 14 \beta_{2} - 4 \beta_{3} - 8 \beta_{4} + \beta_{5} + 4 \beta_{7} + 4 \beta_{8} + 4 \beta_{10} + 5 \beta_{12} - \beta_{13} + 12 \beta_{14} - \beta_{15} ) q^{8} + ( -3 - 21 \beta_{1} - 18 \beta_{2} + 11 \beta_{3} + 3 \beta_{4} + 12 \beta_{5} - 6 \beta_{7} - 4 \beta_{9} - 3 \beta_{10} - \beta_{11} - 6 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} ) q^{9} + ( -23 - 20 \beta_{1} + 21 \beta_{2} + 9 \beta_{4} - 20 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + 14 \beta_{12} + \beta_{13} - 18 \beta_{14} + 2 \beta_{15} ) q^{10} + ( -26 + 29 \beta_{1} + 33 \beta_{2} - 8 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} + \beta_{6} - 4 \beta_{7} + 3 \beta_{8} + 4 \beta_{9} - 7 \beta_{10} - 2 \beta_{12} + \beta_{13} - 4 \beta_{14} - \beta_{15} ) q^{11} + ( -2 - 34 \beta_{2} + 30 \beta_{3} + 19 \beta_{4} + 19 \beta_{5} - \beta_{6} + 2 \beta_{7} - 4 \beta_{8} + 8 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} - 9 \beta_{12} - 3 \beta_{13} - 17 \beta_{14} + 2 \beta_{15} ) q^{12} + ( 37 - 43 \beta_{1} + 17 \beta_{2} + 31 \beta_{3} - 28 \beta_{4} + 3 \beta_{5} + 3 \beta_{7} - 11 \beta_{8} + 3 \beta_{9} + \beta_{10} - 3 \beta_{11} + 9 \beta_{12} + 12 \beta_{14} - 2 \beta_{15} ) q^{13} + ( 60 - 28 \beta_{2} - 29 \beta_{3} + 18 \beta_{4} - 18 \beta_{5} - 4 \beta_{6} - 7 \beta_{7} - \beta_{8} + 7 \beta_{10} - \beta_{11} - 7 \beta_{12} + \beta_{13} + 7 \beta_{14} ) q^{14} + ( -2 + 83 \beta_{1} - 76 \beta_{2} + 68 \beta_{3} - 21 \beta_{4} - 7 \beta_{5} + \beta_{6} + 7 \beta_{7} - \beta_{8} + 7 \beta_{9} + 6 \beta_{10} + 2 \beta_{11} - 25 \beta_{12} + \beta_{13} + 35 \beta_{14} + \beta_{15} ) q^{15} + ( 16 - 28 \beta_{1} + 74 \beta_{2} - 148 \beta_{3} + 3 \beta_{4} + 25 \beta_{5} - 4 \beta_{7} - 4 \beta_{8} + 4 \beta_{9} - 8 \beta_{10} + 2 \beta_{11} - 26 \beta_{12} + \beta_{13} - 33 \beta_{14} ) q^{16} + ( 69 - 33 \beta_{1} - 3 \beta_{2} + 81 \beta_{3} + 25 \beta_{4} - 2 \beta_{5} + 6 \beta_{6} + 6 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} + 9 \beta_{11} + 16 \beta_{12} - 17 \beta_{14} - 3 \beta_{15} ) q^{17} + ( 134 + 32 \beta_{1} + 152 \beta_{2} - 109 \beta_{3} - 16 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 11 \beta_{8} - 14 \beta_{9} - 11 \beta_{10} + 2 \beta_{11} - 14 \beta_{12} + 7 \beta_{13} - 27 \beta_{14} - 5 \beta_{15} ) q^{18} + ( -40 - 83 \beta_{1} + 41 \beta_{2} + 94 \beta_{3} + 2 \beta_{4} - 13 \beta_{5} - 8 \beta_{6} + \beta_{7} + \beta_{8} - 10 \beta_{9} + 10 \beta_{10} - 12 \beta_{11} - 8 \beta_{13} + 5 \beta_{14} + 12 \beta_{15} ) q^{19} + ( -294 + 80 \beta_{1} - 300 \beta_{2} + 88 \beta_{3} + 2 \beta_{4} - 36 \beta_{5} + 13 \beta_{6} - 6 \beta_{7} + 6 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 4 \beta_{11} + 72 \beta_{12} - 13 \beta_{13} + 61 \beta_{14} - 4 \beta_{15} ) q^{20} + ( -96 + 33 \beta_{1} + 56 \beta_{2} - 89 \beta_{3} + 12 \beta_{4} + 37 \beta_{5} - 2 \beta_{6} - 7 \beta_{7} - 7 \beta_{9} - 2 \beta_{11} + 23 \beta_{12} + 5 \beta_{13} - 12 \beta_{14} + 7 \beta_{15} ) q^{21} + ( 7 - 168 \beta_{1} - 93 \beta_{2} + 43 \beta_{3} - \beta_{4} + 37 \beta_{5} + 14 \beta_{7} + 7 \beta_{9} + 7 \beta_{10} + 6 \beta_{11} + 8 \beta_{12} + 12 \beta_{13} + 52 \beta_{14} - \beta_{15} ) q^{22} + ( -45 - 56 \beta_{1} - 52 \beta_{2} - 24 \beta_{4} - 34 \beta_{5} - 14 \beta_{6} + 2 \beta_{7} + 11 \beta_{8} + 11 \beta_{9} + 47 \beta_{12} - 4 \beta_{13} - 21 \beta_{14} - 14 \beta_{15} ) q^{23} + ( -328 + 378 \beta_{1} + 376 \beta_{2} - 60 \beta_{3} - 55 \beta_{4} - 32 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 14 \beta_{8} - 2 \beta_{9} + 16 \beta_{10} + 31 \beta_{12} - 3 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} ) q^{24} + ( -96 + 182 \beta_{1} + 82 \beta_{2} - 74 \beta_{3} + 75 \beta_{4} + 75 \beta_{5} + 6 \beta_{6} - 13 \beta_{7} + 8 \beta_{8} - 16 \beta_{9} - 13 \beta_{10} + 21 \beta_{11} - 87 \beta_{12} + 21 \beta_{13} - 71 \beta_{14} - 12 \beta_{15} ) q^{25} + ( 362 - 598 \beta_{1} + 114 \beta_{2} - 113 \beta_{3} - 68 \beta_{4} + 7 \beta_{5} - 2 \beta_{6} - 5 \beta_{7} + 27 \beta_{8} - 8 \beta_{9} + 9 \beta_{10} + 16 \beta_{11} - 67 \beta_{12} - 3 \beta_{13} + 65 \beta_{14} + 17 \beta_{15} ) q^{26} + ( 213 - 305 \beta_{2} - 310 \beta_{3} + 73 \beta_{4} - 73 \beta_{5} + 27 \beta_{6} + 22 \beta_{7} - 5 \beta_{8} - 22 \beta_{10} + 5 \beta_{11} + 6 \beta_{12} - 5 \beta_{13} - 6 \beta_{14} ) q^{27} + ( 116 + 478 \beta_{1} - 498 \beta_{2} + 636 \beta_{3} + 16 \beta_{4} + 20 \beta_{5} - 4 \beta_{6} - 20 \beta_{7} - 2 \beta_{8} - 20 \beta_{9} - 22 \beta_{10} - 14 \beta_{11} - 54 \beta_{12} - 10 \beta_{13} - 12 \beta_{14} - 10 \beta_{15} ) q^{28} + ( 146 - 99 \beta_{1} + 247 \beta_{2} - 494 \beta_{3} - 31 \beta_{4} - 23 \beta_{5} - 2 \beta_{6} + 11 \beta_{7} + 25 \beta_{8} - 25 \beta_{9} + 22 \beta_{10} - 16 \beta_{11} - 11 \beta_{12} - 8 \beta_{13} + 59 \beta_{14} + 2 \beta_{15} ) q^{29} + ( 764 - 392 \beta_{1} + 14 \beta_{2} + 762 \beta_{3} - 38 \beta_{4} + 45 \beta_{5} - 40 \beta_{6} - 28 \beta_{8} + 14 \beta_{9} + 8 \beta_{10} - 30 \beta_{11} - 2 \beta_{12} - 35 \beta_{14} + 20 \beta_{15} ) q^{30} + ( 181 + 102 \beta_{1} + 355 \beta_{2} - 308 \beta_{3} - 34 \beta_{4} + 4 \beta_{5} + 15 \beta_{6} - 17 \beta_{7} - 55 \beta_{8} + 72 \beta_{9} + 55 \beta_{10} - 15 \beta_{11} + 51 \beta_{12} - 19 \beta_{13} + 21 \beta_{14} + 4 \beta_{15} ) q^{31} + ( -306 - 102 \beta_{1} + 306 \beta_{2} + 42 \beta_{3} + 120 \beta_{4} - 11 \beta_{5} + 21 \beta_{6} + 60 \beta_{9} - 60 \beta_{10} + 15 \beta_{11} + 21 \beta_{13} - 90 \beta_{14} - 15 \beta_{15} ) q^{32} + ( -506 + 334 \beta_{1} - 457 \beta_{2} + 269 \beta_{3} - 16 \beta_{4} - 13 \beta_{5} - 25 \beta_{6} + 49 \beta_{7} - 49 \beta_{8} + 16 \beta_{9} + 16 \beta_{10} - 4 \beta_{11} + 26 \beta_{12} + 25 \beta_{13} - 63 \beta_{14} - 4 \beta_{15} ) q^{33} + ( -456 + 468 \beta_{1} - 78 \beta_{2} - 438 \beta_{3} - 3 \beta_{4} - 16 \beta_{5} + 17 \beta_{6} - 18 \beta_{7} - 48 \beta_{8} + 30 \beta_{9} - 48 \beta_{10} + 17 \beta_{11} - 4 \beta_{12} + \beta_{13} - 45 \beta_{14} - 16 \beta_{15} ) q^{34} + ( 1 - 651 \beta_{1} - 465 \beta_{2} + 222 \beta_{3} - 5 \beta_{4} - 108 \beta_{5} + 2 \beta_{7} + 21 \beta_{9} + \beta_{10} - 8 \beta_{11} + 6 \beta_{12} - 16 \beta_{13} - 101 \beta_{14} + 6 \beta_{15} ) q^{35} + ( -672 - 658 \beta_{1} + 262 \beta_{2} + 81 \beta_{4} + 113 \beta_{5} + 28 \beta_{6} - 92 \beta_{7} - 14 \beta_{8} - 14 \beta_{9} - 180 \beta_{12} - 3 \beta_{13} + 7 \beta_{14} + 28 \beta_{15} ) q^{36} + ( -472 + 860 \beta_{1} + 801 \beta_{2} - 245 \beta_{3} - \beta_{4} + 31 \beta_{5} - 16 \beta_{6} + 59 \beta_{7} + 25 \beta_{8} - 59 \beta_{9} + 34 \beta_{10} + 2 \beta_{11} - 11 \beta_{12} - 14 \beta_{13} + 59 \beta_{14} + 14 \beta_{15} ) q^{37} + ( -20 + 192 \beta_{1} - 416 \beta_{2} + 447 \beta_{3} - 185 \beta_{4} - 185 \beta_{5} - 5 \beta_{6} + 45 \beta_{7} + 31 \beta_{8} - 62 \beta_{9} + 45 \beta_{10} - 42 \beta_{11} + 188 \beta_{12} - 42 \beta_{13} + 250 \beta_{14} + 10 \beta_{15} ) q^{38} + ( 805 - 790 \beta_{1} + 94 \beta_{2} + 284 \beta_{3} + 209 \beta_{4} + 91 \beta_{5} + 17 \beta_{6} - 28 \beta_{7} + 55 \beta_{8} - 9 \beta_{9} - 74 \beta_{10} - 7 \beta_{11} - 32 \beta_{12} + 19 \beta_{13} - 100 \beta_{14} - 52 \beta_{15} ) q^{39} + ( 1146 - 512 \beta_{2} - 454 \beta_{3} - 183 \beta_{4} + 183 \beta_{5} - 53 \beta_{6} + 10 \beta_{7} + 58 \beta_{8} - 10 \beta_{10} + 24 \beta_{12} - 24 \beta_{14} ) q^{40} + ( -7 + 484 \beta_{1} - 514 \beta_{2} + 499 \beta_{3} + 30 \beta_{4} + 30 \beta_{5} - 8 \beta_{6} - 30 \beta_{7} + 38 \beta_{8} - 30 \beta_{9} + 8 \beta_{10} + 30 \beta_{11} + 110 \beta_{12} + 38 \beta_{13} - 30 \beta_{14} + 38 \beta_{15} ) q^{41} + ( -161 + 198 \beta_{1} + 763 \beta_{2} - 1526 \beta_{3} + 33 \beta_{4} - 45 \beta_{5} + 12 \beta_{6} + 21 \beta_{7} - 5 \beta_{8} + 5 \beta_{9} + 42 \beta_{10} + 46 \beta_{11} + 125 \beta_{12} + 23 \beta_{13} + 61 \beta_{14} - 12 \beta_{15} ) q^{42} + ( -44 + 50 \beta_{1} - 21 \beta_{2} + 924 \beta_{3} + 34 \beta_{4} - 181 \beta_{5} + 70 \beta_{6} + 42 \beta_{8} - 21 \beta_{9} + 14 \beta_{10} + 35 \beta_{11} + 6 \beta_{12} + 189 \beta_{14} - 35 \beta_{15} ) q^{43} + ( 776 - 130 \beta_{1} + 596 \beta_{2} - 712 \beta_{3} + 62 \beta_{4} - 72 \beta_{5} - 36 \beta_{6} + 36 \beta_{7} + 14 \beta_{8} - 50 \beta_{9} - 14 \beta_{10} + 36 \beta_{11} + 58 \beta_{12} + 28 \beta_{13} + 48 \beta_{14} + 8 \beta_{15} ) q^{44} + ( -606 + 365 \beta_{1} + 583 \beta_{2} - 319 \beta_{3} - 375 \beta_{4} + 134 \beta_{5} - 5 \beta_{6} - 23 \beta_{7} - 23 \beta_{8} - 69 \beta_{9} + 69 \beta_{10} + 38 \beta_{11} - 5 \beta_{13} + 199 \beta_{14} - 38 \beta_{15} ) q^{45} + ( -819 - 310 \beta_{1} - 909 \beta_{2} - 161 \beta_{3} + 59 \beta_{4} + 212 \beta_{5} + \beta_{6} - 90 \beta_{7} + 90 \beta_{8} - 59 \beta_{9} - 59 \beta_{10} + 26 \beta_{11} - 424 \beta_{12} - \beta_{13} - 199 \beta_{14} + 26 \beta_{15} ) q^{46} + ( 85 + 56 \beta_{1} - 149 \beta_{2} + 94 \beta_{3} + 130 \beta_{4} - 54 \beta_{5} - 49 \beta_{6} - 9 \beta_{7} + \beta_{8} - 10 \beta_{9} + \beta_{10} - 49 \beta_{11} - 73 \beta_{12} - 43 \beta_{13} - 129 \beta_{14} + 6 \beta_{15} ) q^{47} + ( 48 - 342 \beta_{1} - 904 \beta_{2} + 454 \beta_{3} - 36 \beta_{4} - 75 \beta_{5} + 96 \beta_{7} - 4 \beta_{9} + 48 \beta_{10} - 17 \beta_{11} + 84 \beta_{12} - 34 \beta_{13} + 57 \beta_{14} - 3 \beta_{15} ) q^{48} + ( -406 - 448 \beta_{1} - 62 \beta_{2} - 178 \beta_{4} + 129 \beta_{5} + 16 \beta_{6} + 133 \beta_{7} + 42 \beta_{8} + 42 \beta_{9} + 7 \beta_{12} + 19 \beta_{13} + 304 \beta_{14} + 16 \beta_{15} ) q^{49} + ( -1553 + 1274 \beta_{1} + 1377 \beta_{2} + 34 \beta_{3} + 346 \beta_{4} + 131 \beta_{5} + 79 \beta_{6} - 103 \beta_{7} - 39 \beta_{8} + 103 \beta_{9} - 64 \beta_{10} - 15 \beta_{11} - 78 \beta_{12} + 64 \beta_{13} - 103 \beta_{14} - 64 \beta_{15} ) q^{50} + ( -103 - 168 \beta_{1} - 463 \beta_{2} + 390 \beta_{3} - 156 \beta_{4} - 156 \beta_{5} - 36 \beta_{6} - 114 \beta_{7} - 73 \beta_{8} + 146 \beta_{9} - 114 \beta_{10} - 24 \beta_{11} + 31 \beta_{12} - 24 \beta_{13} - 115 \beta_{14} + 72 \beta_{15} ) q^{51} + ( 576 - 906 \beta_{1} + 714 \beta_{2} + 342 \beta_{3} + 252 \beta_{4} - 452 \beta_{5} - 50 \beta_{6} + 54 \beta_{7} - 120 \beta_{8} + 18 \beta_{9} + 146 \beta_{10} - 91 \beta_{11} + 383 \beta_{12} - 23 \beta_{13} - 233 \beta_{14} + 50 \beta_{15} ) q^{52} + ( 388 - 61 \beta_{2} - 215 \beta_{3} - 144 \beta_{4} + 144 \beta_{5} - 13 \beta_{6} - 35 \beta_{7} - 154 \beta_{8} + 35 \beta_{10} - 36 \beta_{11} + 36 \beta_{12} + 36 \beta_{13} - 36 \beta_{14} ) q^{53} + ( 55 + 1166 \beta_{1} - 1061 \beta_{2} + 1112 \beta_{3} + 197 \beta_{4} - 105 \beta_{5} + 55 \beta_{6} + 105 \beta_{7} - 101 \beta_{8} + 105 \beta_{9} + 4 \beta_{10} - \beta_{11} + 365 \beta_{12} - 56 \beta_{13} - 499 \beta_{14} - 56 \beta_{15} ) q^{54} + ( 854 - 1035 \beta_{1} - 46 \beta_{2} + 92 \beta_{3} - 8 \beta_{4} - 147 \beta_{5} - 12 \beta_{6} - 63 \beta_{7} - 55 \beta_{8} + 55 \beta_{9} - 126 \beta_{10} - 48 \beta_{11} + 68 \beta_{12} - 24 \beta_{13} + 29 \beta_{14} + 12 \beta_{15} ) q^{55} + ( 610 - 352 \beta_{1} + 34 \beta_{2} - 174 \beta_{3} - 328 \beta_{4} + 292 \beta_{5} + 68 \beta_{6} - 68 \beta_{8} + 34 \beta_{9} - 26 \beta_{10} + 34 \beta_{11} - 286 \beta_{12} - 32 \beta_{14} - 34 \beta_{15} ) q^{56} + ( 936 - 69 \beta_{1} + 770 \beta_{2} - 695 \beta_{3} + 330 \beta_{4} + 293 \beta_{5} + 8 \beta_{6} - 47 \beta_{7} + 144 \beta_{8} - 97 \beta_{9} - 144 \beta_{10} - 8 \beta_{11} - 437 \beta_{12} - 19 \beta_{13} + 186 \beta_{14} + 11 \beta_{15} ) q^{57} + ( 642 - 766 \beta_{1} - 600 \beta_{2} + 812 \beta_{3} + 340 \beta_{4} - 83 \beta_{5} - 70 \beta_{6} + 42 \beta_{7} + 42 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} - 92 \beta_{11} - 70 \beta_{13} - 126 \beta_{14} + 92 \beta_{15} ) q^{58} + ( -386 + 555 \beta_{1} - 331 \beta_{2} + 366 \beta_{3} - 134 \beta_{4} + 126 \beta_{5} + 26 \beta_{6} + 55 \beta_{7} - 55 \beta_{8} + 134 \beta_{9} + 134 \beta_{10} + 29 \beta_{11} - 252 \beta_{12} - 26 \beta_{13} - 94 \beta_{14} + 29 \beta_{15} ) q^{59} + ( -388 + 968 \beta_{1} - 228 \beta_{2} - 518 \beta_{3} - 441 \beta_{4} - 415 \beta_{5} + 25 \beta_{6} + 130 \beta_{7} + 222 \beta_{8} - 92 \beta_{9} + 222 \beta_{10} + 25 \beta_{11} - 377 \beta_{12} + 53 \beta_{13} + 663 \beta_{14} + 28 \beta_{15} ) q^{60} + ( -160 - 225 \beta_{1} + 830 \beta_{2} - 360 \beta_{3} - 56 \beta_{4} - 127 \beta_{5} - 320 \beta_{7} - 110 \beta_{9} - 160 \beta_{10} + 53 \beta_{11} - 104 \beta_{12} + 106 \beta_{13} - 391 \beta_{14} - 55 \beta_{15} ) q^{61} + ( 876 + 964 \beta_{1} + 632 \beta_{2} + 59 \beta_{4} + 96 \beta_{5} - 110 \beta_{6} + 98 \beta_{7} - 88 \beta_{8} - 88 \beta_{9} - 67 \beta_{12} + 38 \beta_{13} + 106 \beta_{14} - 110 \beta_{15} ) q^{62} + ( 1074 - 674 \beta_{1} - 652 \beta_{2} - 388 \beta_{3} + 210 \beta_{4} + 10 \beta_{5} - 64 \beta_{6} - 22 \beta_{7} + 56 \beta_{8} + 22 \beta_{9} - 78 \beta_{10} + 32 \beta_{11} - 72 \beta_{12} - 32 \beta_{13} - 22 \beta_{14} + 32 \beta_{15} ) q^{63} + ( 108 + 40 \beta_{1} + 1286 \beta_{2} - 1306 \beta_{3} - 134 \beta_{4} - 134 \beta_{5} + 74 \beta_{6} + 148 \beta_{7} - 20 \beta_{8} + 40 \beta_{9} + 148 \beta_{10} + 169 \beta_{11} + 399 \beta_{12} + 169 \beta_{13} + 359 \beta_{14} - 148 \beta_{15} ) q^{64} + ( -621 + 27 \beta_{1} - 1645 \beta_{2} - 55 \beta_{3} - 38 \beta_{4} + 478 \beta_{5} + 35 \beta_{6} + 38 \beta_{7} - 187 \beta_{8} + 19 \beta_{9} - 44 \beta_{10} + 114 \beta_{11} + 49 \beta_{12} - 84 \beta_{13} - 237 \beta_{14} + 67 \beta_{15} ) q^{65} + ( -1532 + 146 \beta_{2} + 295 \beta_{3} - 272 \beta_{4} + 272 \beta_{5} + 129 \beta_{6} - 149 \beta_{7} + 149 \beta_{8} + 149 \beta_{10} + 46 \beta_{11} - 115 \beta_{12} - 46 \beta_{13} + 115 \beta_{14} ) q^{66} + ( -494 - 489 \beta_{1} + 545 \beta_{2} - 1164 \beta_{3} - 407 \beta_{4} - 56 \beta_{5} - 34 \beta_{6} + 56 \beta_{7} + 69 \beta_{8} + 56 \beta_{9} + 125 \beta_{10} - 62 \beta_{11} + 325 \beta_{12} - 28 \beta_{13} + 758 \beta_{14} - 28 \beta_{15} ) q^{67} + ( -980 + 858 \beta_{1} + 144 \beta_{2} - 288 \beta_{3} + 354 \beta_{4} + 369 \beta_{5} - 51 \beta_{6} - 82 \beta_{7} + 42 \beta_{8} - 42 \beta_{9} - 164 \beta_{10} - 8 \beta_{11} - 221 \beta_{12} - 4 \beta_{13} - 409 \beta_{14} + 51 \beta_{15} ) q^{68} + ( 327 - 102 \beta_{1} - 114 \beta_{2} + 693 \beta_{3} + 453 \beta_{4} - 108 \beta_{5} - 264 \beta_{6} + 228 \beta_{8} - 114 \beta_{9} - 105 \beta_{10} - 123 \beta_{11} + 120 \beta_{12} - 117 \beta_{14} + 132 \beta_{15} ) q^{69} + ( -2406 + 228 \beta_{1} - 2240 \beta_{2} + 2454 \beta_{3} - 190 \beta_{4} - 792 \beta_{5} + 81 \beta_{6} + 76 \beta_{7} - 14 \beta_{8} - 62 \beta_{9} + 14 \beta_{10} - 81 \beta_{11} + 806 \beta_{12} - 67 \beta_{13} - 176 \beta_{14} - 14 \beta_{15} ) q^{70} + ( -504 + 437 \beta_{1} + 595 \beta_{2} - 220 \beta_{3} - 506 \beta_{4} - 163 \beta_{5} + 98 \beta_{6} + 91 \beta_{7} + 91 \beta_{8} - 126 \beta_{9} + 126 \beta_{10} + 2 \beta_{11} + 98 \beta_{13} + 407 \beta_{14} - 2 \beta_{15} ) q^{71} + ( 1004 - 222 \beta_{1} + 976 \beta_{2} - 12 \beta_{3} + 182 \beta_{4} - 455 \beta_{5} + 9 \beta_{6} - 28 \beta_{7} + 28 \beta_{8} - 182 \beta_{9} - 182 \beta_{10} - 141 \beta_{11} + 910 \beta_{12} - 9 \beta_{13} + 350 \beta_{14} - 141 \beta_{15} ) q^{72} + ( -356 - 1826 \beta_{1} + 2129 \beta_{2} - 334 \beta_{3} + 751 \beta_{4} + 330 \beta_{5} + 139 \beta_{6} - 22 \beta_{7} - 31 \beta_{8} + 9 \beta_{9} - 31 \beta_{10} + 139 \beta_{11} + 317 \beta_{12} + 94 \beta_{13} - 782 \beta_{14} - 45 \beta_{15} ) q^{73} + ( -42 - 198 \beta_{1} + 576 \beta_{2} - 270 \beta_{3} + 439 \beta_{4} + 714 \beta_{5} - 84 \beta_{7} - 36 \beta_{9} - 42 \beta_{10} - 48 \beta_{11} - 481 \beta_{12} - 96 \beta_{13} + 191 \beta_{14} + 144 \beta_{15} ) q^{74} + ( 113 + 159 \beta_{1} - 1444 \beta_{2} - 135 \beta_{4} - 373 \beta_{5} + 53 \beta_{6} - 65 \beta_{7} - 46 \beta_{8} - 46 \beta_{9} + 554 \beta_{12} - 99 \beta_{13} - 484 \beta_{14} + 53 \beta_{15} ) q^{75} + ( 1990 - 3040 \beta_{1} - 2992 \beta_{2} + 944 \beta_{3} - 785 \beta_{4} + 542 \beta_{5} - 155 \beta_{6} - 48 \beta_{7} - 10 \beta_{8} + 48 \beta_{9} - 38 \beta_{10} + 24 \beta_{11} - 285 \beta_{12} - 131 \beta_{13} - 48 \beta_{14} + 131 \beta_{15} ) q^{76} + ( -1346 + 2958 \beta_{1} - 434 \beta_{2} + 483 \beta_{3} + 208 \beta_{4} + 208 \beta_{5} + 49 \beta_{6} + 84 \beta_{7} + 49 \beta_{8} - 98 \beta_{9} + 84 \beta_{10} - 114 \beta_{11} - 395 \beta_{12} - 114 \beta_{13} - 297 \beta_{14} - 98 \beta_{15} ) q^{77} + ( -2239 + 3512 \beta_{1} + 2215 \beta_{2} - 2809 \beta_{3} - 668 \beta_{4} - 254 \beta_{5} + 100 \beta_{6} + 8 \beta_{7} + 196 \beta_{8} + 171 \beta_{9} - 67 \beta_{10} + 170 \beta_{11} - 353 \beta_{12} + 202 \beta_{13} + 813 \beta_{14} - 173 \beta_{15} ) q^{78} + ( -774 - 86 \beta_{2} - 90 \beta_{3} + 744 \beta_{4} - 744 \beta_{5} - 68 \beta_{6} + 164 \beta_{7} - 4 \beta_{8} - 164 \beta_{10} + 34 \beta_{11} - 232 \beta_{12} - 34 \beta_{13} + 232 \beta_{14} ) q^{79} + ( 1932 - 2994 \beta_{1} + 2818 \beta_{2} - 810 \beta_{3} + 263 \beta_{4} + 176 \beta_{5} - 158 \beta_{6} - 176 \beta_{7} + 100 \beta_{8} - 176 \beta_{9} - 76 \beta_{10} + 29 \beta_{11} - 1420 \beta_{12} + 187 \beta_{13} - 350 \beta_{14} + 187 \beta_{15} ) q^{80} + ( 381 - 141 \beta_{1} - 2410 \beta_{2} + 4820 \beta_{3} - 550 \beta_{4} - 326 \beta_{5} + 78 \beta_{6} + 248 \beta_{7} - 256 \beta_{8} + 256 \beta_{9} + 496 \beta_{10} + 44 \beta_{11} + 528 \beta_{12} + 22 \beta_{13} + 318 \beta_{14} - 78 \beta_{15} ) q^{81} + ( -508 + 184 \beta_{1} + 128 \beta_{2} - 4428 \beta_{3} - 271 \beta_{4} - 44 \beta_{5} - 16 \beta_{6} - 256 \beta_{8} + 128 \beta_{9} + 116 \beta_{10} + 64 \beta_{11} + 101 \beta_{12} + 59 \beta_{14} + 8 \beta_{15} ) q^{82} + ( 573 + 546 \beta_{1} + 1335 \beta_{2} - 912 \beta_{3} - 269 \beta_{4} + 779 \beta_{5} - 60 \beta_{6} - 93 \beta_{7} - 123 \beta_{8} + 216 \beta_{9} + 123 \beta_{10} + 60 \beta_{11} - 656 \beta_{12} + 216 \beta_{13} - 146 \beta_{14} - 156 \beta_{15} ) q^{83} + ( 1328 - 404 \beta_{1} - 1666 \beta_{2} - 90 \beta_{3} + 888 \beta_{4} + 248 \beta_{5} + 30 \beta_{6} - 338 \beta_{7} - 338 \beta_{8} + 156 \beta_{9} - 156 \beta_{10} + 54 \beta_{11} + 30 \beta_{13} - 860 \beta_{14} - 54 \beta_{15} ) q^{84} + ( 2188 + 432 \beta_{1} + 2111 \beta_{2} + 456 \beta_{3} - 53 \beta_{4} + 224 \beta_{5} + 73 \beta_{6} - 77 \beta_{7} + 77 \beta_{8} + 53 \beta_{9} + 53 \beta_{10} - 44 \beta_{11} - 448 \beta_{12} - 73 \beta_{13} + 553 \beta_{14} - 44 \beta_{15} ) q^{85} + ( 3752 + 74 \beta_{1} - 4134 \beta_{2} + 3745 \beta_{3} - 908 \beta_{4} + 420 \beta_{5} - 224 \beta_{6} + 7 \beta_{7} - 315 \beta_{8} + 322 \beta_{9} - 315 \beta_{10} - 224 \beta_{11} + 749 \beta_{12} - 195 \beta_{13} + 593 \beta_{14} + 29 \beta_{15} ) q^{86} + ( 356 + 3972 \beta_{1} + 2259 \beta_{2} - 1220 \beta_{3} - 188 \beta_{4} - 670 \beta_{5} + 712 \beta_{7} + 181 \beta_{9} + 356 \beta_{10} + 22 \beta_{11} + 544 \beta_{12} + 44 \beta_{13} + 230 \beta_{14} - 54 \beta_{15} ) q^{87} + ( 1658 + 1528 \beta_{1} + 406 \beta_{2} + 396 \beta_{4} - 334 \beta_{5} + 134 \beta_{6} - 166 \beta_{7} + 130 \beta_{8} + 130 \beta_{9} - 192 \beta_{12} - 160 \beta_{13} - 370 \beta_{14} + 134 \beta_{15} ) q^{88} + ( 2804 - 2459 \beta_{1} - 2685 \beta_{2} + 116 \beta_{3} - 181 \beta_{4} - 1570 \beta_{5} + 221 \beta_{6} + 226 \beta_{7} + 9 \beta_{8} - 226 \beta_{9} + 217 \beta_{10} - 171 \beta_{11} + 889 \beta_{12} + 50 \beta_{13} + 226 \beta_{14} - 50 \beta_{15} ) q^{89} + ( 3702 - 8288 \beta_{1} + 730 \beta_{2} - 687 \beta_{3} + 1205 \beta_{4} + 1205 \beta_{5} - 199 \beta_{6} - 485 \beta_{7} + 43 \beta_{8} - 86 \beta_{9} - 485 \beta_{10} - 96 \beta_{11} - 1395 \beta_{12} - 96 \beta_{13} - 1309 \beta_{14} + 398 \beta_{15} ) q^{90} + ( -442 + 2431 \beta_{1} - 2587 \beta_{2} + 2834 \beta_{3} - 286 \beta_{4} + 676 \beta_{5} - 156 \beta_{6} - 273 \beta_{7} + 377 \beta_{8} - 442 \beta_{9} - 78 \beta_{10} - 195 \beta_{11} - 1118 \beta_{12} + 78 \beta_{13} - 208 \beta_{14} + 91 \beta_{15} ) q^{91} + ( -3362 + 4450 \beta_{2} + 4298 \beta_{3} + 881 \beta_{4} - 881 \beta_{5} + 73 \beta_{6} + 270 \beta_{7} - 152 \beta_{8} - 270 \beta_{10} - 19 \beta_{11} + 845 \beta_{12} + 19 \beta_{13} - 845 \beta_{14} ) q^{92} + ( -3276 - 3731 \beta_{1} + 3640 \beta_{2} - 6558 \beta_{3} - 248 \beta_{4} + 91 \beta_{5} + 183 \beta_{6} - 91 \beta_{7} - 267 \beta_{8} - 91 \beta_{9} - 358 \beta_{10} + 24 \beta_{11} - 244 \beta_{12} - 159 \beta_{13} + 587 \beta_{14} - 159 \beta_{15} ) q^{93} + ( -3472 + 3876 \beta_{1} - 1976 \beta_{2} + 3952 \beta_{3} - 329 \beta_{4} + 164 \beta_{5} + 128 \beta_{6} + 14 \beta_{7} + 376 \beta_{8} - 376 \beta_{9} + 28 \beta_{10} - 148 \beta_{11} - 841 \beta_{12} - 74 \beta_{13} + 226 \beta_{14} - 128 \beta_{15} ) q^{94} + ( -8673 + 4349 \beta_{1} + 129 \beta_{2} - 2670 \beta_{3} + 685 \beta_{4} - 359 \beta_{5} + 274 \beta_{6} - 258 \beta_{8} + 129 \beta_{9} + 283 \beta_{10} + 67 \beta_{11} + 1226 \beta_{12} - 584 \beta_{14} - 137 \beta_{15} ) q^{95} + ( -5602 - 1072 \beta_{1} - 6458 \beta_{2} + 5198 \beta_{3} - 593 \beta_{4} + 397 \beta_{5} - 53 \beta_{6} - 28 \beta_{7} - 188 \beta_{8} + 216 \beta_{9} + 188 \beta_{10} + 53 \beta_{11} - 209 \beta_{12} - 83 \beta_{13} - 405 \beta_{14} + 136 \beta_{15} ) q^{96} + ( 1876 + 2537 \beta_{1} - 1629 \beta_{2} - 2656 \beta_{3} + 354 \beta_{4} - 125 \beta_{5} - 113 \beta_{6} + 247 \beta_{7} + 247 \beta_{8} + 366 \beta_{9} - 366 \beta_{10} + 54 \beta_{11} - 113 \beta_{13} - 113 \beta_{14} - 54 \beta_{15} ) q^{97} + ( 6143 - 4934 \beta_{1} + 6355 \beta_{2} - 5461 \beta_{3} - 315 \beta_{4} - 851 \beta_{5} - 255 \beta_{6} + 212 \beta_{7} - 212 \beta_{8} + 315 \beta_{9} + 315 \beta_{10} + 311 \beta_{11} + 1702 \beta_{12} + 255 \beta_{13} + 67 \beta_{14} + 311 \beta_{15} ) q^{98} + ( 608 - 4754 \beta_{1} + 3656 \beta_{2} + 986 \beta_{3} + 142 \beta_{4} + 816 \beta_{5} - 40 \beta_{6} - 378 \beta_{7} - 112 \beta_{8} - 266 \beta_{9} - 112 \beta_{10} - 40 \beta_{11} + 172 \beta_{12} - 26 \beta_{13} - 254 \beta_{14} + 14 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 4q^{2} - 2q^{3} - 6q^{4} + 8q^{5} - 38q^{6} + 56q^{7} + 90q^{8} - 164q^{9} + O(q^{10}) \) \( 16q - 4q^{2} - 2q^{3} - 6q^{4} + 8q^{5} - 38q^{6} + 56q^{7} + 90q^{8} - 164q^{9} - 486q^{10} - 100q^{11} + 294q^{13} + 808q^{14} + 346q^{15} + 230q^{16} + 984q^{17} + 2434q^{18} - 1498q^{19} - 3962q^{20} - 1076q^{21} - 1524q^{22} - 1014q^{23} - 2142q^{24} + 614q^{26} + 3352q^{27} + 5764q^{28} + 814q^{29} + 9162q^{30} + 4060q^{31} - 4996q^{32} - 5636q^{33} - 2502q^{34} - 4892q^{35} - 15750q^{36} - 1790q^{37} + 6982q^{39} + 18816q^{40} + 4280q^{41} - 1204q^{42} - 1368q^{43} + 10736q^{44} - 6806q^{45} - 15246q^{46} + 1484q^{47} - 3002q^{48} - 11820q^{49} - 13574q^{50} + 1432q^{52} + 7204q^{53} + 13240q^{54} + 6936q^{55} + 8124q^{56} + 12736q^{57} + 3030q^{58} - 2380q^{59} - 6472q^{60} - 162q^{61} + 19614q^{62} + 12004q^{63} - 5248q^{65} - 23872q^{66} - 14854q^{67} - 6444q^{68} + 2412q^{69} - 34524q^{70} - 8050q^{71} + 15420q^{72} - 15448q^{73} - 2882q^{74} + 8280q^{75} + 10622q^{76} - 11672q^{78} - 17064q^{79} + 2564q^{80} + 2128q^{81} - 5346q^{82} + 12788q^{83} + 25948q^{84} + 35382q^{85} + 67260q^{86} + 29342q^{87} + 40836q^{88} + 20492q^{89} + 8996q^{91} - 49884q^{92} - 78920q^{93} - 30606q^{94} - 98574q^{95} - 94664q^{96} + 50944q^{97} + 61484q^{98} - 21632q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 152 x^{14} + 9190 x^{12} + 285720 x^{10} + 4862025 x^{8} + 43573680 x^{6} + 169417008 x^{4} + 100636992 x^{2} + 3779136\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-403105 \nu^{15} - 60681659 \nu^{13} - 3635180779 \nu^{11} - 112263516645 \nu^{9} - 1906294858020 \nu^{7} - 17198065117872 \nu^{5} - 68934581954784 \nu^{3} - 50621014857024 \nu + 5545371476736\)\()/ 11090742953472 \)
\(\beta_{2}\)\(=\)\((\)\(-450212 \nu^{15} - 12268161 \nu^{14} - 40141192 \nu^{13} - 1670382423 \nu^{12} - 336194096 \nu^{11} - 86530231647 \nu^{10} + 64569965208 \nu^{9} - 2167037463465 \nu^{8} + 2514078351540 \nu^{7} - 26905461929136 \nu^{6} + 35960915423664 \nu^{5} - 144684110994768 \nu^{4} + 189141832996704 \nu^{3} - 182218604617728 \nu^{2} + 90475406179200 \nu - 35916509215296\)\()/ 33272228860416 \)
\(\beta_{3}\)\(=\)\((\)\(450212 \nu^{15} - 12268161 \nu^{14} + 40141192 \nu^{13} - 1670382423 \nu^{12} + 336194096 \nu^{11} - 86530231647 \nu^{10} - 64569965208 \nu^{9} - 2167037463465 \nu^{8} - 2514078351540 \nu^{7} - 26905461929136 \nu^{6} - 35960915423664 \nu^{5} - 144684110994768 \nu^{4} - 189141832996704 \nu^{3} - 182218604617728 \nu^{2} - 90475406179200 \nu - 35916509215296\)\()/ 33272228860416 \)
\(\beta_{4}\)\(=\)\((\)\(-53905 \nu^{15} - 3031335 \nu^{14} - 1946495 \nu^{13} - 399803697 \nu^{12} + 403691033 \nu^{11} - 19722145833 \nu^{10} + 33967214799 \nu^{9} - 456891538383 \nu^{8} + 1036401623208 \nu^{7} - 4928117078304 \nu^{6} + 14113769934816 \nu^{5} - 18429203735856 \nu^{4} + 73741688677440 \nu^{3} + 20576677484160 \nu^{2} + 32386072836672 \nu + 22901572977984\)\()/ 3696914317824 \)
\(\beta_{5}\)\(=\)\((\)\(-53905 \nu^{15} + 3031335 \nu^{14} - 1946495 \nu^{13} + 399803697 \nu^{12} + 403691033 \nu^{11} + 19722145833 \nu^{10} + 33967214799 \nu^{9} + 456891538383 \nu^{8} + 1036401623208 \nu^{7} + 4928117078304 \nu^{6} + 14113769934816 \nu^{5} + 18429203735856 \nu^{4} + 73741688677440 \nu^{3} - 20576677484160 \nu^{2} + 32386072836672 \nu - 22901572977984\)\()/ 3696914317824 \)
\(\beta_{6}\)\(=\)\((\)\(958933 \nu^{14} + 127064675 \nu^{12} + 6269679835 \nu^{10} + 142169221341 \nu^{8} + 1357351279200 \nu^{6} + 1111509749520 \nu^{4} - 41761008777600 \nu^{2} - 17722730943936\)\()/ 616152386304 \)
\(\beta_{7}\)\(=\)\((\)\(1611011 \nu^{15} + 20386710 \nu^{14} + 235643173 \nu^{13} + 2770864650 \nu^{12} + 13401654797 \nu^{11} + 141965775450 \nu^{10} + 377367280059 \nu^{9} + 3436503751926 \nu^{8} + 5435904315744 \nu^{7} + 38802395601648 \nu^{6} + 35965207159632 \nu^{5} + 147897464997600 \nu^{4} + 69173551698048 \nu^{3} - 236124309230208 \nu^{2} - 38119690075968 \nu - 127810286301312\)\()/ 11090742953472 \)
\(\beta_{8}\)\(=\)\((\)\(-64316 \nu^{15} + 10690839 \nu^{14} - 5734456 \nu^{13} + 1454826609 \nu^{12} - 48027728 \nu^{11} + 75309720489 \nu^{10} + 9224280744 \nu^{9} + 1881613092303 \nu^{8} + 359154050220 \nu^{7} + 23121020302128 \nu^{6} + 5137273631952 \nu^{5} + 118652121636336 \nu^{4} + 27020261856672 \nu^{3} + 95067845558784 \nu^{2} + 12925058025600 \nu - 46660774261824\)\()/ 4753175551488 \)
\(\beta_{9}\)\(=\)\((\)\(8483506 \nu^{15} - 22455 \nu^{14} + 1257390830 \nu^{13} - 34398657 \nu^{12} + 73776312574 \nu^{11} - 3085996905 \nu^{10} + 2217207089202 \nu^{9} - 83390031711 \nu^{8} + 36294499625328 \nu^{7} - 575259720336 \nu^{6} + 310555374841728 \nu^{5} + 2193532745904 \nu^{4} + 1153914652058112 \nu^{3} - 10688655293568 \nu^{2} + 921694995453312 \nu - 121083795836352\)\()/ 33272228860416 \)
\(\beta_{10}\)\(=\)\((\)\(1611011 \nu^{15} - 20386710 \nu^{14} + 235643173 \nu^{13} - 2770864650 \nu^{12} + 13401654797 \nu^{11} - 141965775450 \nu^{10} + 377367280059 \nu^{9} - 3436503751926 \nu^{8} + 5435904315744 \nu^{7} - 38802395601648 \nu^{6} + 35965207159632 \nu^{5} - 147897464997600 \nu^{4} + 69173551698048 \nu^{3} + 236124309230208 \nu^{2} - 38119690075968 \nu + 116719543347840\)\()/ 11090742953472 \)
\(\beta_{11}\)\(=\)\((\)\(3060949 \nu^{15} + 30779505 \nu^{14} + 538883915 \nu^{13} + 4066826247 \nu^{12} + 37885091971 \nu^{11} + 202355357535 \nu^{10} + 1360934568357 \nu^{9} + 4811620057209 \nu^{8} + 26434213763832 \nu^{7} + 55871559620352 \nu^{6} + 266224281763680 \nu^{5} + 274948280891280 \nu^{4} + 1144577610929088 \nu^{3} + 313787054673792 \nu^{2} + 765185781986496 \nu + 262228572037440\)\()/ 11090742953472 \)
\(\beta_{12}\)\(=\)\((\)\(1668943 \nu^{15} + 3478479 \nu^{14} + 244791977 \nu^{13} + 476788953 \nu^{12} + 14182744225 \nu^{11} + 24822766977 \nu^{10} + 420766469319 \nu^{9} + 620613013671 \nu^{8} + 6825957638976 \nu^{7} + 7560677757024 \nu^{6} + 58482708936192 \nu^{5} + 37860759720432 \nu^{4} + 218257879045248 \nu^{3} + 28035201382272 \nu^{2} + 118564929748800 \nu - 4839006028608\)\()/ 3696914317824 \)
\(\beta_{13}\)\(=\)\((\)\(3060949 \nu^{15} - 30779505 \nu^{14} + 538883915 \nu^{13} - 4066826247 \nu^{12} + 37885091971 \nu^{11} - 202355357535 \nu^{10} + 1360934568357 \nu^{9} - 4811620057209 \nu^{8} + 26434213763832 \nu^{7} - 55871559620352 \nu^{6} + 266224281763680 \nu^{5} - 274948280891280 \nu^{4} + 1144577610929088 \nu^{3} - 313787054673792 \nu^{2} + 765185781986496 \nu - 262228572037440\)\()/ 11090742953472 \)
\(\beta_{14}\)\(=\)\((\)\(1668943 \nu^{15} - 3478479 \nu^{14} + 244791977 \nu^{13} - 476788953 \nu^{12} + 14182744225 \nu^{11} - 24822766977 \nu^{10} + 420766469319 \nu^{9} - 620613013671 \nu^{8} + 6825957638976 \nu^{7} - 7560677757024 \nu^{6} + 58482708936192 \nu^{5} - 37860759720432 \nu^{4} + 218257879045248 \nu^{3} - 28035201382272 \nu^{2} + 118564929748800 \nu + 4839006028608\)\()/ 3696914317824 \)
\(\beta_{15}\)\(=\)\((\)\(7320853 \nu^{15} + 2876799 \nu^{14} + 1089833795 \nu^{13} + 381194025 \nu^{12} + 64086192235 \nu^{11} + 18809039505 \nu^{10} + 1921681816845 \nu^{9} + 426507664023 \nu^{8} + 31200573373200 \nu^{7} + 4072053837600 \nu^{6} + 262878864690816 \nu^{5} + 3334529248560 \nu^{4} + 938832132172032 \nu^{3} - 125283026332800 \nu^{2} + 452794700306880 \nu - 53168192831808\)\()/ 3696914317824 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-6 \beta_{15} + 8 \beta_{14} + \beta_{13} + 12 \beta_{12} + \beta_{11} + 8 \beta_{10} + 4 \beta_{9} - 2 \beta_{8} + 8 \beta_{7} + 3 \beta_{6} - 4 \beta_{5} - 4 \beta_{4} - 18 \beta_{3} + 16 \beta_{2} + 20 \beta_{1} - 4\)\()/26\)
\(\nu^{2}\)\(=\)\((\)\(7 \beta_{14} - 4 \beta_{13} - 7 \beta_{12} + 4 \beta_{11} - 7 \beta_{10} + 2 \beta_{8} + 7 \beta_{7} - 3 \beta_{6} - 13 \beta_{5} + 13 \beta_{4} + 19 \beta_{3} + 17 \beta_{2} - 242\)\()/13\)
\(\nu^{3}\)\(=\)\((\)\(92 \beta_{15} - 114 \beta_{14} + 15 \beta_{13} - 184 \beta_{12} + 15 \beta_{11} - 140 \beta_{10} - 70 \beta_{9} + 35 \beta_{8} - 140 \beta_{7} - 46 \beta_{6} + 83 \beta_{5} + 83 \beta_{4} + 588 \beta_{3} - 553 \beta_{2} + 274 \beta_{1} - 242\)\()/13\)
\(\nu^{4}\)\(=\)\((\)\(-407 \beta_{14} + 175 \beta_{13} + 407 \beta_{12} - 175 \beta_{11} + 381 \beta_{10} - 380 \beta_{8} - 381 \beta_{7} + 258 \beta_{6} + 507 \beta_{5} - 507 \beta_{4} - 1855 \beta_{3} - 1475 \beta_{2} + 7318\)\()/13\)
\(\nu^{5}\)\(=\)\((\)\(-3476 \beta_{15} + 3430 \beta_{14} - 1479 \beta_{13} + 7212 \beta_{12} - 1479 \beta_{11} + 5588 \beta_{10} + 3782 \beta_{9} - 1891 \beta_{8} + 5588 \beta_{7} + 1738 \beta_{6} - 3977 \beta_{5} - 3977 \beta_{4} - 32190 \beta_{3} + 30299 \beta_{2} - 27838 \beta_{1} + 17616\)\()/13\)
\(\nu^{6}\)\(=\)\((\)\(17913 \beta_{14} - 8533 \beta_{13} - 17913 \beta_{12} + 8533 \beta_{11} - 18797 \beta_{10} + 24644 \beta_{8} + 18797 \beta_{7} - 14814 \beta_{6} - 22607 \beta_{5} + 22607 \beta_{4} + 118171 \beta_{3} + 93527 \beta_{2} - 271710\)\()/13\)
\(\nu^{7}\)\(=\)\(11468 \beta_{15} - 8888 \beta_{14} + 7017 \beta_{13} - 24998 \beta_{12} + 7017 \beta_{11} - 19004 \beta_{10} - 16110 \beta_{9} + 8055 \beta_{8} - 19004 \beta_{7} - 5734 \beta_{6} + 15173 \beta_{5} + 15173 \beta_{4} + 130698 \beta_{3} - 122643 \beta_{2} + 124726 \beta_{1} - 73312\)
\(\nu^{8}\)\(=\)\((\)\(-772903 \beta_{14} + 430867 \beta_{13} + 772903 \beta_{12} - 430867 \beta_{11} + 918997 \beta_{10} - 1362716 \beta_{8} - 918997 \beta_{7} + 772518 \beta_{6} + 1088347 \beta_{5} - 1088347 \beta_{4} - 6639151 \beta_{3} - 5276435 \beta_{2} + 11600462\)\()/13\)
\(\nu^{9}\)\(=\)\((\)\(-6901292 \beta_{15} + 4408926 \beta_{14} - 5008851 \beta_{13} + 15652900 \beta_{12} - 5008851 \beta_{11} + 11583236 \beta_{10} + 11243974 \beta_{9} - 5621987 \beta_{8} + 11583236 \beta_{7} + 3450646 \beta_{6} - 9941777 \beta_{5} - 9941777 \beta_{4} - 88036518 \beta_{3} + 82414531 \beta_{2} - 84818014 \beta_{1} + 48370256\)\()/13\)
\(\nu^{10}\)\(=\)\((\)\(34597169 \beta_{14} - 21896869 \beta_{13} - 34597169 \beta_{12} + 21896869 \beta_{11} - 45236733 \beta_{10} + 71658740 \beta_{8} + 45236733 \beta_{7} - 39176646 \beta_{6} - 54100527 \beta_{5} + 54100527 \beta_{4} + 353441755 \beta_{3} + 281783015 \beta_{2} - 536617966\)\()/13\)
\(\nu^{11}\)\(=\)\((\)\(332946236 \beta_{15} - 187007200 \beta_{14} + 263030493 \beta_{13} - 775259574 \beta_{12} + 263030493 \beta_{11} - 561086828 \beta_{10} - 588252374 \beta_{9} + 294126187 \beta_{8} - 561086828 \beta_{7} - 166473118 \beta_{6} + 503828801 \beta_{5} + 503828801 \beta_{4} + 4512195906 \beta_{3} - 4218069719 \beta_{2} + 4307663710 \beta_{1} - 2420792496\)\()/13\)
\(\nu^{12}\)\(=\)\((\)\(-1614876399 \beta_{14} + 1111784059 \beta_{13} + 1614876399 \beta_{12} - 1111784059 \beta_{11} + 2244399605 \beta_{10} - 3691353404 \beta_{8} - 2244399605 \beta_{7} + 1971703926 \beta_{6} + 2719026011 \beta_{5} - 2719026011 \beta_{4} - 18332662303 \beta_{3} - 14641308899 \beta_{2} + 25885701678\)\()/13\)
\(\nu^{13}\)\(=\)\((\)\(-16408659212 \beta_{15} + 8537243750 \beta_{14} - 13536977019 \beta_{13} + 38821589996 \beta_{12} - 13536977019 \beta_{11} + 27670764260 \beta_{10} + 30284346246 \beta_{9} - 15142173123 \beta_{8} + 27670764260 \beta_{7} + 8204329606 \beta_{6} - 25547886401 \beta_{5} - 25547886401 \beta_{4} - 229745363670 \beta_{3} + 214603190547 \beta_{2} - 217025181214 \beta_{1} + 121041181744\)\()/13\)
\(\nu^{14}\)\(=\)\(5983739437 \beta_{14} - 4333135633 \beta_{13} - 5983739437 \beta_{12} + 4333135633 \beta_{11} - 8616856273 \beta_{10} + 14483635364 \beta_{8} + 8616856273 \beta_{7} - 7623329934 \beta_{6} - 10544524843 \beta_{5} + 10544524843 \beta_{4} + 72180049879 \beta_{3} + 57696414515 \beta_{2} - 98119308806\)
\(\nu^{15}\)\(=\)\((\)\(817505384348 \beta_{15} - 407544050424 \beta_{14} + 689836505013 \beta_{13} - 1952234100238 \beta_{12} + 689836505013 \beta_{11} - 1378130038796 \beta_{10} - 1544690049814 \beta_{9} + 772345024907 \beta_{8} - 1378130038796 \beta_{7} - 408752692174 \beta_{6} + 1294054470737 \beta_{5} + 1294054470737 \beta_{4} + 11650966452498 \beta_{3} - 10878621427591 \beta_{2} + 10916665870750 \beta_{1} - 6064117949264\)\()/13\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
4.89748i
5.33868i
0.816521i
3.98977i
4.34562i
7.09996i
3.68702i
0.200628i
4.89748i
5.33868i
0.816521i
3.98977i
4.34562i
7.09996i
3.68702i
0.200628i
−6.34141 1.69918i 7.28678 + 12.6211i 23.4699 + 13.5504i −20.2793 + 20.2793i −24.7631 92.4170i 21.3318 5.71584i −51.5321 51.5321i −65.6944 + 113.786i 163.058 94.1415i
2.2 −3.41998 0.916381i −5.16500 8.94604i −2.99988 1.73198i 8.65321 8.65321i 9.46622 + 35.3284i −2.94871 + 0.790103i 48.7300 + 48.7300i −12.8545 + 22.2646i −37.5235 + 21.6642i
2.3 2.38644 + 0.639444i 4.25709 + 7.37349i −8.57021 4.94801i 16.9748 16.9748i 5.44434 + 20.3185i −50.2192 + 13.4562i −45.2402 45.2402i 4.25442 7.36887i 51.3637 29.6548i
2.4 5.50893 + 1.47611i −4.28079 7.41455i 14.3130 + 8.26362i −29.3294 + 29.3294i −12.6379 47.1652i 45.8361 12.2817i 2.12627 + 2.12627i 3.84960 6.66770i −204.867 + 118.280i
6.1 −1.88469 7.03375i −0.939567 + 1.62738i −32.0652 + 18.5128i 26.6717 26.6717i 13.2174 + 3.54158i 7.85376 29.3106i 108.263 + 108.263i 38.7344 + 67.0900i −237.870 137.334i
6.2 −0.387376 1.44571i 3.29308 5.70378i 11.9164 6.87993i −10.5663 + 10.5663i −9.52166 2.55132i −14.3010 + 53.3722i −31.4958 31.4958i 18.8113 + 32.5821i 19.3688 + 11.1826i
6.3 0.540374 + 2.01670i −8.21171 + 14.2231i 10.0813 5.82045i 17.2508 17.2508i −33.1212 8.87479i −2.95016 + 11.0101i 40.8071 + 40.8071i −94.3643 163.444i 44.1117 + 25.4679i
6.4 1.59772 + 5.96275i 2.76012 4.78067i −19.1453 + 11.0536i −5.37551 + 5.37551i 32.9158 + 8.81977i 23.3974 87.3205i −26.6579 26.6579i 25.2635 + 43.7576i −40.6414 23.4643i
7.1 −6.34141 + 1.69918i 7.28678 12.6211i 23.4699 13.5504i −20.2793 20.2793i −24.7631 + 92.4170i 21.3318 + 5.71584i −51.5321 + 51.5321i −65.6944 113.786i 163.058 + 94.1415i
7.2 −3.41998 + 0.916381i −5.16500 + 8.94604i −2.99988 + 1.73198i 8.65321 + 8.65321i 9.46622 35.3284i −2.94871 0.790103i 48.7300 48.7300i −12.8545 22.2646i −37.5235 21.6642i
7.3 2.38644 0.639444i 4.25709 7.37349i −8.57021 + 4.94801i 16.9748 + 16.9748i 5.44434 20.3185i −50.2192 13.4562i −45.2402 + 45.2402i 4.25442 + 7.36887i 51.3637 + 29.6548i
7.4 5.50893 1.47611i −4.28079 + 7.41455i 14.3130 8.26362i −29.3294 29.3294i −12.6379 + 47.1652i 45.8361 + 12.2817i 2.12627 2.12627i 3.84960 + 6.66770i −204.867 118.280i
11.1 −1.88469 + 7.03375i −0.939567 1.62738i −32.0652 18.5128i 26.6717 + 26.6717i 13.2174 3.54158i 7.85376 + 29.3106i 108.263 108.263i 38.7344 67.0900i −237.870 + 137.334i
11.2 −0.387376 + 1.44571i 3.29308 + 5.70378i 11.9164 + 6.87993i −10.5663 10.5663i −9.52166 + 2.55132i −14.3010 53.3722i −31.4958 + 31.4958i 18.8113 32.5821i 19.3688 11.1826i
11.3 0.540374 2.01670i −8.21171 14.2231i 10.0813 + 5.82045i 17.2508 + 17.2508i −33.1212 + 8.87479i −2.95016 11.0101i 40.8071 40.8071i −94.3643 + 163.444i 44.1117 25.4679i
11.4 1.59772 5.96275i 2.76012 + 4.78067i −19.1453 11.0536i −5.37551 5.37551i 32.9158 8.81977i 23.3974 + 87.3205i −26.6579 + 26.6579i 25.2635 43.7576i −40.6414 + 23.4643i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.5.f.a 16
3.b odd 2 1 117.5.bd.c 16
13.c even 3 1 169.5.d.d 16
13.e even 6 1 169.5.d.c 16
13.f odd 12 1 inner 13.5.f.a 16
13.f odd 12 1 169.5.d.c 16
13.f odd 12 1 169.5.d.d 16
39.k even 12 1 117.5.bd.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.5.f.a 16 1.a even 1 1 trivial
13.5.f.a 16 13.f odd 12 1 inner
117.5.bd.c 16 3.b odd 2 1
117.5.bd.c 16 39.k even 12 1
169.5.d.c 16 13.e even 6 1
169.5.d.c 16 13.f odd 12 1
169.5.d.d 16 13.c even 3 1
169.5.d.d 16 13.f odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 11 T^{2} - 18 T^{3} + 75 T^{4} + 502 T^{5} + 2554 T^{6} + 9868 T^{7} + 22616 T^{8} + 29288 T^{9} - 27048 T^{10} - 898544 T^{11} - 17177376 T^{12} - 108914720 T^{13} - 426373536 T^{14} - 382913728 T^{15} - 329617344 T^{16} - 6126619648 T^{17} - 109151625216 T^{18} - 446114693120 T^{19} - 1125736513536 T^{20} - 942191673344 T^{21} - 453790138368 T^{22} + 7861937635328 T^{23} + 97134980366336 T^{24} + 678123796430848 T^{25} + 2808152697339904 T^{26} + 8831277394296832 T^{27} + 21110623253299200 T^{28} - 81064793292668928 T^{29} + 792633534417207296 T^{30} + 4611686018427387904 T^{31} + 18446744073709551616 T^{32} \)
$3$ \( 1 + 2 T - 240 T^{2} - 1492 T^{3} + 22927 T^{4} + 271668 T^{5} - 890196 T^{6} - 21609198 T^{7} - 25045623 T^{8} + 1172175192 T^{9} + 5748348060 T^{10} - 93679542360 T^{11} - 1026060864642 T^{12} + 8900356926348 T^{13} + 169584054967116 T^{14} - 370754464597920 T^{15} - 17423631492664230 T^{16} - 30031111632431520 T^{17} + 1112640984639248076 T^{18} + 4730014585295307468 T^{19} - 44168555769262938882 T^{20} - \)\(32\!\cdots\!60\)\( T^{21} + \)\(16\!\cdots\!60\)\( T^{22} + \)\(26\!\cdots\!12\)\( T^{23} - \)\(46\!\cdots\!43\)\( T^{24} - \)\(32\!\cdots\!58\)\( T^{25} - \)\(10\!\cdots\!96\)\( T^{26} + \)\(26\!\cdots\!08\)\( T^{27} + \)\(18\!\cdots\!47\)\( T^{28} - \)\(96\!\cdots\!72\)\( T^{29} - \)\(12\!\cdots\!40\)\( T^{30} + \)\(84\!\cdots\!02\)\( T^{31} + \)\(34\!\cdots\!81\)\( T^{32} \)
$5$ \( 1 - 8 T + 32 T^{2} - 12096 T^{3} - 527442 T^{4} + 6817816 T^{5} + 35492224 T^{6} + 3499118776 T^{7} + 214598054801 T^{8} - 1770375811288 T^{9} + 6931925138112 T^{10} - 204666718110056 T^{11} - 91059199232161746 T^{12} - 449348487028373168 T^{13} - 924223306284337248 T^{14} - \)\(13\!\cdots\!88\)\( T^{15} + \)\(53\!\cdots\!36\)\( T^{16} - \)\(84\!\cdots\!00\)\( T^{17} - \)\(36\!\cdots\!00\)\( T^{18} - \)\(10\!\cdots\!00\)\( T^{19} - \)\(13\!\cdots\!50\)\( T^{20} - \)\(19\!\cdots\!00\)\( T^{21} + \)\(41\!\cdots\!00\)\( T^{22} - \)\(65\!\cdots\!00\)\( T^{23} + \)\(49\!\cdots\!25\)\( T^{24} + \)\(50\!\cdots\!00\)\( T^{25} + \)\(32\!\cdots\!00\)\( T^{26} + \)\(38\!\cdots\!00\)\( T^{27} - \)\(18\!\cdots\!50\)\( T^{28} - \)\(26\!\cdots\!00\)\( T^{29} + \)\(44\!\cdots\!00\)\( T^{30} - \)\(69\!\cdots\!00\)\( T^{31} + \)\(54\!\cdots\!25\)\( T^{32} \)
$7$ \( 1 - 56 T + 7478 T^{2} - 202660 T^{3} + 15425611 T^{4} + 147757036 T^{5} + 351071986 T^{6} + 612647989816 T^{7} + 16654656981341 T^{8} - 3342938469481784 T^{9} + 73292174911510116 T^{10} - 4996161548264166712 T^{11} - \)\(32\!\cdots\!86\)\( T^{12} + \)\(46\!\cdots\!48\)\( T^{13} - \)\(17\!\cdots\!68\)\( T^{14} - \)\(47\!\cdots\!88\)\( T^{15} + \)\(27\!\cdots\!66\)\( T^{16} - \)\(11\!\cdots\!88\)\( T^{17} - \)\(10\!\cdots\!68\)\( T^{18} + \)\(64\!\cdots\!48\)\( T^{19} - \)\(10\!\cdots\!86\)\( T^{20} - \)\(39\!\cdots\!12\)\( T^{21} + \)\(14\!\cdots\!16\)\( T^{22} - \)\(15\!\cdots\!84\)\( T^{23} + \)\(18\!\cdots\!41\)\( T^{24} + \)\(16\!\cdots\!16\)\( T^{25} + \)\(22\!\cdots\!86\)\( T^{26} + \)\(22\!\cdots\!36\)\( T^{27} + \)\(56\!\cdots\!11\)\( T^{28} - \)\(17\!\cdots\!60\)\( T^{29} + \)\(15\!\cdots\!78\)\( T^{30} - \)\(28\!\cdots\!56\)\( T^{31} + \)\(12\!\cdots\!01\)\( T^{32} \)
$11$ \( 1 + 100 T + 27170 T^{2} - 106032 T^{3} + 1057947 T^{4} - 44543342240 T^{5} - 2123313764498 T^{6} - 578703081018404 T^{7} - 42234317976348187 T^{8} - 16300704176801189728 T^{9} - \)\(18\!\cdots\!52\)\( T^{10} - \)\(15\!\cdots\!56\)\( T^{11} + \)\(32\!\cdots\!42\)\( T^{12} + \)\(33\!\cdots\!12\)\( T^{13} + \)\(38\!\cdots\!24\)\( T^{14} + \)\(42\!\cdots\!68\)\( T^{15} + \)\(31\!\cdots\!70\)\( T^{16} + \)\(62\!\cdots\!88\)\( T^{17} + \)\(81\!\cdots\!44\)\( T^{18} + \)\(10\!\cdots\!52\)\( T^{19} + \)\(14\!\cdots\!62\)\( T^{20} - \)\(10\!\cdots\!56\)\( T^{21} - \)\(18\!\cdots\!32\)\( T^{22} - \)\(23\!\cdots\!68\)\( T^{23} - \)\(89\!\cdots\!27\)\( T^{24} - \)\(17\!\cdots\!44\)\( T^{25} - \)\(96\!\cdots\!98\)\( T^{26} - \)\(29\!\cdots\!40\)\( T^{27} + \)\(10\!\cdots\!07\)\( T^{28} - \)\(15\!\cdots\!72\)\( T^{29} + \)\(56\!\cdots\!70\)\( T^{30} + \)\(30\!\cdots\!00\)\( T^{31} + \)\(44\!\cdots\!41\)\( T^{32} \)
$13$ \( 1 - 294 T - 66586 T^{2} + 42455504 T^{3} - 2316760667 T^{4} - 2128791278900 T^{5} + 410094472234278 T^{6} + 33452215036870058 T^{7} - 17726001609919307172 T^{8} + \)\(95\!\cdots\!38\)\( T^{9} + \)\(33\!\cdots\!38\)\( T^{10} - \)\(49\!\cdots\!00\)\( T^{11} - \)\(15\!\cdots\!47\)\( T^{12} + \)\(80\!\cdots\!04\)\( T^{13} - \)\(36\!\cdots\!46\)\( T^{14} - \)\(45\!\cdots\!74\)\( T^{15} + \)\(44\!\cdots\!81\)\( T^{16} \)
$17$ \( 1 - 984 T + 934532 T^{2} - 601991520 T^{3} + 366227956458 T^{4} - 186786892356648 T^{5} + 90290052706451368 T^{6} - 39430381111636884696 T^{7} + \)\(16\!\cdots\!09\)\( T^{8} - \)\(64\!\cdots\!32\)\( T^{9} + \)\(23\!\cdots\!64\)\( T^{10} - \)\(85\!\cdots\!84\)\( T^{11} + \)\(29\!\cdots\!86\)\( T^{12} - \)\(95\!\cdots\!24\)\( T^{13} + \)\(30\!\cdots\!88\)\( T^{14} - \)\(92\!\cdots\!00\)\( T^{15} + \)\(27\!\cdots\!68\)\( T^{16} - \)\(77\!\cdots\!00\)\( T^{17} + \)\(21\!\cdots\!08\)\( T^{18} - \)\(55\!\cdots\!64\)\( T^{19} + \)\(14\!\cdots\!66\)\( T^{20} - \)\(34\!\cdots\!84\)\( T^{21} + \)\(81\!\cdots\!44\)\( T^{22} - \)\(18\!\cdots\!12\)\( T^{23} + \)\(38\!\cdots\!49\)\( T^{24} - \)\(77\!\cdots\!76\)\( T^{25} + \)\(14\!\cdots\!68\)\( T^{26} - \)\(25\!\cdots\!08\)\( T^{27} + \)\(42\!\cdots\!78\)\( T^{28} - \)\(57\!\cdots\!20\)\( T^{29} + \)\(75\!\cdots\!92\)\( T^{30} - \)\(66\!\cdots\!84\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \)
$19$ \( 1 + 1498 T + 1242140 T^{2} + 605628956 T^{3} + 144809054215 T^{4} - 30840378908480 T^{5} - 42820390632894416 T^{6} - 17457142236019195214 T^{7} - \)\(21\!\cdots\!35\)\( T^{8} + \)\(14\!\cdots\!12\)\( T^{9} + \)\(88\!\cdots\!60\)\( T^{10} + \)\(16\!\cdots\!32\)\( T^{11} - \)\(48\!\cdots\!18\)\( T^{12} - \)\(39\!\cdots\!12\)\( T^{13} - \)\(82\!\cdots\!76\)\( T^{14} + \)\(18\!\cdots\!60\)\( T^{15} + \)\(15\!\cdots\!38\)\( T^{16} + \)\(23\!\cdots\!60\)\( T^{17} - \)\(14\!\cdots\!16\)\( T^{18} - \)\(87\!\cdots\!32\)\( T^{19} - \)\(14\!\cdots\!58\)\( T^{20} + \)\(62\!\cdots\!32\)\( T^{21} + \)\(43\!\cdots\!60\)\( T^{22} + \)\(92\!\cdots\!92\)\( T^{23} - \)\(17\!\cdots\!35\)\( T^{24} - \)\(18\!\cdots\!34\)\( T^{25} - \)\(60\!\cdots\!16\)\( T^{26} - \)\(56\!\cdots\!80\)\( T^{27} + \)\(34\!\cdots\!15\)\( T^{28} + \)\(18\!\cdots\!16\)\( T^{29} + \)\(50\!\cdots\!40\)\( T^{30} + \)\(79\!\cdots\!98\)\( T^{31} + \)\(69\!\cdots\!21\)\( T^{32} \)
$23$ \( 1 + 1014 T + 1541624 T^{2} + 1215676488 T^{3} + 1070749656711 T^{4} + 761258748687096 T^{5} + 523855300773466804 T^{6} + \)\(37\!\cdots\!10\)\( T^{7} + \)\(23\!\cdots\!53\)\( T^{8} + \)\(16\!\cdots\!08\)\( T^{9} + \)\(96\!\cdots\!40\)\( T^{10} + \)\(61\!\cdots\!96\)\( T^{11} + \)\(35\!\cdots\!06\)\( T^{12} + \)\(20\!\cdots\!24\)\( T^{13} + \)\(11\!\cdots\!00\)\( T^{14} + \)\(60\!\cdots\!24\)\( T^{15} + \)\(34\!\cdots\!06\)\( T^{16} + \)\(16\!\cdots\!84\)\( T^{17} + \)\(90\!\cdots\!00\)\( T^{18} + \)\(44\!\cdots\!04\)\( T^{19} + \)\(21\!\cdots\!66\)\( T^{20} + \)\(10\!\cdots\!96\)\( T^{21} + \)\(46\!\cdots\!40\)\( T^{22} + \)\(21\!\cdots\!48\)\( T^{23} + \)\(87\!\cdots\!13\)\( T^{24} + \)\(39\!\cdots\!10\)\( T^{25} + \)\(15\!\cdots\!04\)\( T^{26} + \)\(62\!\cdots\!36\)\( T^{27} + \)\(24\!\cdots\!91\)\( T^{28} + \)\(78\!\cdots\!48\)\( T^{29} + \)\(27\!\cdots\!64\)\( T^{30} + \)\(51\!\cdots\!14\)\( T^{31} + \)\(14\!\cdots\!41\)\( T^{32} \)
$29$ \( 1 - 814 T - 3744612 T^{2} + 2089659432 T^{3} + 8177655703406 T^{4} - 2736264841720610 T^{5} - 12804081872369783884 T^{6} + \)\(25\!\cdots\!10\)\( T^{7} + \)\(15\!\cdots\!13\)\( T^{8} - \)\(18\!\cdots\!86\)\( T^{9} - \)\(16\!\cdots\!00\)\( T^{10} + \)\(13\!\cdots\!06\)\( T^{11} + \)\(14\!\cdots\!38\)\( T^{12} - \)\(78\!\cdots\!60\)\( T^{13} - \)\(12\!\cdots\!92\)\( T^{14} + \)\(21\!\cdots\!10\)\( T^{15} + \)\(93\!\cdots\!80\)\( T^{16} + \)\(15\!\cdots\!10\)\( T^{17} - \)\(62\!\cdots\!12\)\( T^{18} - \)\(27\!\cdots\!60\)\( T^{19} + \)\(37\!\cdots\!98\)\( T^{20} + \)\(23\!\cdots\!06\)\( T^{21} - \)\(20\!\cdots\!00\)\( T^{22} - \)\(16\!\cdots\!46\)\( T^{23} + \)\(97\!\cdots\!33\)\( T^{24} + \)\(11\!\cdots\!10\)\( T^{25} - \)\(40\!\cdots\!84\)\( T^{26} - \)\(60\!\cdots\!10\)\( T^{27} + \)\(12\!\cdots\!66\)\( T^{28} + \)\(23\!\cdots\!12\)\( T^{29} - \)\(29\!\cdots\!52\)\( T^{30} - \)\(45\!\cdots\!14\)\( T^{31} + \)\(39\!\cdots\!81\)\( T^{32} \)
$31$ \( 1 - 4060 T + 8241800 T^{2} - 11888153764 T^{3} + 12102195192780 T^{4} - 6844355067784460 T^{5} - 1291690806395612552 T^{6} + \)\(91\!\cdots\!16\)\( T^{7} - \)\(12\!\cdots\!64\)\( T^{8} + \)\(80\!\cdots\!00\)\( T^{9} + \)\(12\!\cdots\!36\)\( T^{10} - \)\(10\!\cdots\!52\)\( T^{11} + \)\(15\!\cdots\!04\)\( T^{12} - \)\(11\!\cdots\!48\)\( T^{13} + \)\(36\!\cdots\!28\)\( T^{14} + \)\(41\!\cdots\!28\)\( T^{15} - \)\(74\!\cdots\!42\)\( T^{16} + \)\(38\!\cdots\!88\)\( T^{17} + \)\(31\!\cdots\!48\)\( T^{18} - \)\(93\!\cdots\!28\)\( T^{19} + \)\(11\!\cdots\!24\)\( T^{20} - \)\(72\!\cdots\!52\)\( T^{21} + \)\(78\!\cdots\!56\)\( T^{22} + \)\(46\!\cdots\!00\)\( T^{23} - \)\(66\!\cdots\!04\)\( T^{24} + \)\(44\!\cdots\!96\)\( T^{25} - \)\(58\!\cdots\!52\)\( T^{26} - \)\(28\!\cdots\!60\)\( T^{27} + \)\(46\!\cdots\!80\)\( T^{28} - \)\(42\!\cdots\!04\)\( T^{29} + \)\(27\!\cdots\!00\)\( T^{30} - \)\(12\!\cdots\!60\)\( T^{31} + \)\(27\!\cdots\!21\)\( T^{32} \)
$37$ \( 1 + 1790 T + 8613908 T^{2} + 6266080760 T^{3} + 17905874153442 T^{4} - 23706724104957758 T^{5} - 22615506964230414788 T^{6} - \)\(14\!\cdots\!38\)\( T^{7} - \)\(62\!\cdots\!27\)\( T^{8} - \)\(13\!\cdots\!82\)\( T^{9} + \)\(30\!\cdots\!92\)\( T^{10} + \)\(33\!\cdots\!18\)\( T^{11} + \)\(99\!\cdots\!54\)\( T^{12} + \)\(24\!\cdots\!76\)\( T^{13} + \)\(15\!\cdots\!00\)\( T^{14} - \)\(24\!\cdots\!14\)\( T^{15} - \)\(23\!\cdots\!64\)\( T^{16} - \)\(45\!\cdots\!54\)\( T^{17} + \)\(53\!\cdots\!00\)\( T^{18} + \)\(16\!\cdots\!56\)\( T^{19} + \)\(12\!\cdots\!14\)\( T^{20} + \)\(77\!\cdots\!18\)\( T^{21} + \)\(13\!\cdots\!12\)\( T^{22} - \)\(11\!\cdots\!22\)\( T^{23} - \)\(94\!\cdots\!87\)\( T^{24} - \)\(40\!\cdots\!58\)\( T^{25} - \)\(12\!\cdots\!88\)\( T^{26} - \)\(23\!\cdots\!38\)\( T^{27} + \)\(33\!\cdots\!82\)\( T^{28} + \)\(22\!\cdots\!60\)\( T^{29} + \)\(56\!\cdots\!28\)\( T^{30} + \)\(22\!\cdots\!90\)\( T^{31} + \)\(23\!\cdots\!61\)\( T^{32} \)
$41$ \( 1 - 4280 T + 16206692 T^{2} - 41605604352 T^{3} + 89007507297078 T^{4} - 156507905015338472 T^{5} + \)\(22\!\cdots\!52\)\( T^{6} - \)\(29\!\cdots\!84\)\( T^{7} + \)\(29\!\cdots\!57\)\( T^{8} - \)\(44\!\cdots\!84\)\( T^{9} + \)\(61\!\cdots\!76\)\( T^{10} - \)\(12\!\cdots\!68\)\( T^{11} + \)\(17\!\cdots\!90\)\( T^{12} - \)\(89\!\cdots\!44\)\( T^{13} - \)\(29\!\cdots\!48\)\( T^{14} + \)\(15\!\cdots\!64\)\( T^{15} - \)\(27\!\cdots\!92\)\( T^{16} + \)\(44\!\cdots\!04\)\( T^{17} - \)\(23\!\cdots\!08\)\( T^{18} - \)\(20\!\cdots\!64\)\( T^{19} + \)\(11\!\cdots\!90\)\( T^{20} - \)\(23\!\cdots\!68\)\( T^{21} + \)\(31\!\cdots\!36\)\( T^{22} - \)\(64\!\cdots\!64\)\( T^{23} + \)\(12\!\cdots\!17\)\( T^{24} - \)\(34\!\cdots\!44\)\( T^{25} + \)\(72\!\cdots\!52\)\( T^{26} - \)\(14\!\cdots\!92\)\( T^{27} + \)\(23\!\cdots\!38\)\( T^{28} - \)\(30\!\cdots\!12\)\( T^{29} + \)\(33\!\cdots\!72\)\( T^{30} - \)\(25\!\cdots\!80\)\( T^{31} + \)\(16\!\cdots\!61\)\( T^{32} \)
$43$ \( 1 + 1368 T + 13386518 T^{2} + 17459387280 T^{3} + 89338317662175 T^{4} + 129060665859360744 T^{5} + \)\(40\!\cdots\!42\)\( T^{6} + \)\(73\!\cdots\!56\)\( T^{7} + \)\(14\!\cdots\!61\)\( T^{8} + \)\(35\!\cdots\!04\)\( T^{9} + \)\(55\!\cdots\!44\)\( T^{10} + \)\(14\!\cdots\!84\)\( T^{11} + \)\(24\!\cdots\!50\)\( T^{12} + \)\(52\!\cdots\!76\)\( T^{13} + \)\(10\!\cdots\!68\)\( T^{14} + \)\(17\!\cdots\!08\)\( T^{15} + \)\(39\!\cdots\!58\)\( T^{16} + \)\(61\!\cdots\!08\)\( T^{17} + \)\(12\!\cdots\!68\)\( T^{18} + \)\(21\!\cdots\!76\)\( T^{19} + \)\(33\!\cdots\!50\)\( T^{20} + \)\(67\!\cdots\!84\)\( T^{21} + \)\(89\!\cdots\!44\)\( T^{22} + \)\(19\!\cdots\!04\)\( T^{23} + \)\(27\!\cdots\!61\)\( T^{24} + \)\(46\!\cdots\!56\)\( T^{25} + \)\(87\!\cdots\!42\)\( T^{26} + \)\(96\!\cdots\!44\)\( T^{27} + \)\(22\!\cdots\!75\)\( T^{28} + \)\(15\!\cdots\!80\)\( T^{29} + \)\(39\!\cdots\!18\)\( T^{30} + \)\(13\!\cdots\!68\)\( T^{31} + \)\(34\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 - 1484 T + 1101128 T^{2} + 8571170172 T^{3} + 67218351180012 T^{4} - 197396893555353260 T^{5} + \)\(25\!\cdots\!96\)\( T^{6} + \)\(27\!\cdots\!20\)\( T^{7} + \)\(18\!\cdots\!48\)\( T^{8} - \)\(96\!\cdots\!72\)\( T^{9} + \)\(16\!\cdots\!92\)\( T^{10} - \)\(79\!\cdots\!80\)\( T^{11} + \)\(29\!\cdots\!72\)\( T^{12} - \)\(27\!\cdots\!40\)\( T^{13} + \)\(63\!\cdots\!92\)\( T^{14} - \)\(86\!\cdots\!68\)\( T^{15} + \)\(34\!\cdots\!58\)\( T^{16} - \)\(42\!\cdots\!08\)\( T^{17} + \)\(15\!\cdots\!12\)\( T^{18} - \)\(31\!\cdots\!40\)\( T^{19} + \)\(16\!\cdots\!12\)\( T^{20} - \)\(21\!\cdots\!80\)\( T^{21} + \)\(22\!\cdots\!52\)\( T^{22} - \)\(63\!\cdots\!92\)\( T^{23} + \)\(59\!\cdots\!68\)\( T^{24} + \)\(42\!\cdots\!20\)\( T^{25} + \)\(19\!\cdots\!96\)\( T^{26} - \)\(73\!\cdots\!60\)\( T^{27} + \)\(12\!\cdots\!32\)\( T^{28} + \)\(76\!\cdots\!52\)\( T^{29} + \)\(47\!\cdots\!88\)\( T^{30} - \)\(31\!\cdots\!84\)\( T^{31} + \)\(10\!\cdots\!81\)\( T^{32} \)
$53$ \( ( 1 - 3602 T + 54510142 T^{2} - 169091095000 T^{3} + 1347824194996053 T^{4} - 3593979789946993116 T^{5} + \)\(19\!\cdots\!86\)\( T^{6} - \)\(44\!\cdots\!26\)\( T^{7} + \)\(19\!\cdots\!96\)\( T^{8} - \)\(35\!\cdots\!06\)\( T^{9} + \)\(12\!\cdots\!46\)\( T^{10} - \)\(17\!\cdots\!56\)\( T^{11} + \)\(52\!\cdots\!13\)\( T^{12} - \)\(51\!\cdots\!00\)\( T^{13} + \)\(13\!\cdots\!02\)\( T^{14} - \)\(68\!\cdots\!22\)\( T^{15} + \)\(15\!\cdots\!41\)\( T^{16} )^{2} \)
$59$ \( 1 + 2380 T + 26675390 T^{2} + 58910752356 T^{3} + 184713311551875 T^{4} - 411423908773382660 T^{5} - \)\(31\!\cdots\!82\)\( T^{6} - \)\(24\!\cdots\!80\)\( T^{7} - \)\(72\!\cdots\!87\)\( T^{8} - \)\(10\!\cdots\!96\)\( T^{9} - \)\(55\!\cdots\!12\)\( T^{10} + \)\(30\!\cdots\!12\)\( T^{11} + \)\(10\!\cdots\!74\)\( T^{12} + \)\(31\!\cdots\!40\)\( T^{13} + \)\(19\!\cdots\!96\)\( T^{14} - \)\(82\!\cdots\!24\)\( T^{15} - \)\(14\!\cdots\!10\)\( T^{16} - \)\(99\!\cdots\!64\)\( T^{17} + \)\(27\!\cdots\!16\)\( T^{18} + \)\(55\!\cdots\!40\)\( T^{19} + \)\(23\!\cdots\!34\)\( T^{20} + \)\(78\!\cdots\!12\)\( T^{21} - \)\(17\!\cdots\!32\)\( T^{22} - \)\(41\!\cdots\!16\)\( T^{23} - \)\(33\!\cdots\!47\)\( T^{24} - \)\(13\!\cdots\!80\)\( T^{25} - \)\(21\!\cdots\!82\)\( T^{26} - \)\(34\!\cdots\!60\)\( T^{27} + \)\(18\!\cdots\!75\)\( T^{28} + \)\(71\!\cdots\!36\)\( T^{29} + \)\(39\!\cdots\!90\)\( T^{30} + \)\(42\!\cdots\!80\)\( T^{31} + \)\(21\!\cdots\!61\)\( T^{32} \)
$61$ \( 1 + 162 T - 50302892 T^{2} - 60770764376 T^{3} + 1076406400246734 T^{4} + 2212979358132074374 T^{5} - \)\(12\!\cdots\!00\)\( T^{6} - \)\(23\!\cdots\!38\)\( T^{7} + \)\(12\!\cdots\!45\)\( T^{8} - \)\(19\!\cdots\!06\)\( T^{9} - \)\(19\!\cdots\!64\)\( T^{10} + \)\(60\!\cdots\!86\)\( T^{11} + \)\(31\!\cdots\!06\)\( T^{12} - \)\(33\!\cdots\!96\)\( T^{13} - \)\(25\!\cdots\!92\)\( T^{14} - \)\(37\!\cdots\!34\)\( T^{15} + \)\(15\!\cdots\!84\)\( T^{16} - \)\(52\!\cdots\!94\)\( T^{17} - \)\(49\!\cdots\!52\)\( T^{18} - \)\(89\!\cdots\!16\)\( T^{19} + \)\(11\!\cdots\!66\)\( T^{20} + \)\(30\!\cdots\!86\)\( T^{21} - \)\(14\!\cdots\!24\)\( T^{22} - \)\(19\!\cdots\!86\)\( T^{23} + \)\(16\!\cdots\!45\)\( T^{24} - \)\(43\!\cdots\!18\)\( T^{25} - \)\(32\!\cdots\!00\)\( T^{26} + \)\(79\!\cdots\!34\)\( T^{27} + \)\(53\!\cdots\!54\)\( T^{28} - \)\(41\!\cdots\!96\)\( T^{29} - \)\(47\!\cdots\!12\)\( T^{30} + \)\(21\!\cdots\!62\)\( T^{31} + \)\(18\!\cdots\!41\)\( T^{32} \)
$67$ \( 1 + 14854 T + 152246780 T^{2} + 1156949917028 T^{3} + 7271671748877823 T^{4} + 42419416555472380768 T^{5} + \)\(23\!\cdots\!44\)\( T^{6} + \)\(13\!\cdots\!62\)\( T^{7} + \)\(73\!\cdots\!33\)\( T^{8} + \)\(39\!\cdots\!52\)\( T^{9} + \)\(21\!\cdots\!32\)\( T^{10} + \)\(10\!\cdots\!60\)\( T^{11} + \)\(53\!\cdots\!70\)\( T^{12} + \)\(25\!\cdots\!60\)\( T^{13} + \)\(12\!\cdots\!72\)\( T^{14} + \)\(55\!\cdots\!96\)\( T^{15} + \)\(24\!\cdots\!74\)\( T^{16} + \)\(11\!\cdots\!16\)\( T^{17} + \)\(49\!\cdots\!52\)\( T^{18} + \)\(21\!\cdots\!60\)\( T^{19} + \)\(88\!\cdots\!70\)\( T^{20} + \)\(35\!\cdots\!60\)\( T^{21} + \)\(14\!\cdots\!72\)\( T^{22} + \)\(53\!\cdots\!32\)\( T^{23} + \)\(20\!\cdots\!13\)\( T^{24} + \)\(72\!\cdots\!22\)\( T^{25} + \)\(25\!\cdots\!44\)\( T^{26} + \)\(94\!\cdots\!28\)\( T^{27} + \)\(32\!\cdots\!43\)\( T^{28} + \)\(10\!\cdots\!08\)\( T^{29} + \)\(27\!\cdots\!80\)\( T^{30} + \)\(54\!\cdots\!54\)\( T^{31} + \)\(73\!\cdots\!21\)\( T^{32} \)
$71$ \( 1 + 8050 T + 23381036 T^{2} + 11907331644 T^{3} - 640051132997169 T^{4} - 10630217014363370528 T^{5} - \)\(61\!\cdots\!68\)\( T^{6} - \)\(22\!\cdots\!54\)\( T^{7} - \)\(28\!\cdots\!59\)\( T^{8} + \)\(44\!\cdots\!28\)\( T^{9} + \)\(49\!\cdots\!12\)\( T^{10} + \)\(30\!\cdots\!36\)\( T^{11} + \)\(12\!\cdots\!98\)\( T^{12} + \)\(11\!\cdots\!48\)\( T^{13} - \)\(13\!\cdots\!32\)\( T^{14} - \)\(16\!\cdots\!32\)\( T^{15} - \)\(11\!\cdots\!78\)\( T^{16} - \)\(43\!\cdots\!92\)\( T^{17} - \)\(87\!\cdots\!52\)\( T^{18} + \)\(19\!\cdots\!68\)\( T^{19} + \)\(52\!\cdots\!58\)\( T^{20} + \)\(32\!\cdots\!36\)\( T^{21} + \)\(13\!\cdots\!72\)\( T^{22} + \)\(30\!\cdots\!08\)\( T^{23} - \)\(49\!\cdots\!19\)\( T^{24} - \)\(10\!\cdots\!34\)\( T^{25} - \)\(68\!\cdots\!68\)\( T^{26} - \)\(30\!\cdots\!68\)\( T^{27} - \)\(46\!\cdots\!09\)\( T^{28} + \)\(21\!\cdots\!04\)\( T^{29} + \)\(10\!\cdots\!56\)\( T^{30} + \)\(95\!\cdots\!50\)\( T^{31} + \)\(30\!\cdots\!81\)\( T^{32} \)
$73$ \( 1 + 15448 T + 119320352 T^{2} + 673174693736 T^{3} + 2488100657363446 T^{4} + 439794287857885936 T^{5} - \)\(63\!\cdots\!16\)\( T^{6} - \)\(62\!\cdots\!20\)\( T^{7} - \)\(47\!\cdots\!31\)\( T^{8} - \)\(27\!\cdots\!12\)\( T^{9} - \)\(13\!\cdots\!00\)\( T^{10} - \)\(44\!\cdots\!16\)\( T^{11} - \)\(23\!\cdots\!62\)\( T^{12} + \)\(97\!\cdots\!12\)\( T^{13} + \)\(86\!\cdots\!96\)\( T^{14} + \)\(55\!\cdots\!96\)\( T^{15} + \)\(31\!\cdots\!12\)\( T^{16} + \)\(15\!\cdots\!36\)\( T^{17} + \)\(70\!\cdots\!76\)\( T^{18} + \)\(22\!\cdots\!52\)\( T^{19} - \)\(15\!\cdots\!82\)\( T^{20} - \)\(82\!\cdots\!16\)\( T^{21} - \)\(68\!\cdots\!00\)\( T^{22} - \)\(41\!\cdots\!72\)\( T^{23} - \)\(19\!\cdots\!51\)\( T^{24} - \)\(75\!\cdots\!20\)\( T^{25} - \)\(21\!\cdots\!16\)\( T^{26} + \)\(42\!\cdots\!76\)\( T^{27} + \)\(68\!\cdots\!26\)\( T^{28} + \)\(52\!\cdots\!56\)\( T^{29} + \)\(26\!\cdots\!72\)\( T^{30} + \)\(97\!\cdots\!48\)\( T^{31} + \)\(17\!\cdots\!41\)\( T^{32} \)
$79$ \( ( 1 + 8532 T + 181789616 T^{2} + 1264697206412 T^{3} + 17222120260071052 T^{4} + 98573552246045188900 T^{5} + \)\(10\!\cdots\!52\)\( T^{6} + \)\(51\!\cdots\!68\)\( T^{7} + \)\(46\!\cdots\!78\)\( T^{8} + \)\(20\!\cdots\!08\)\( T^{9} + \)\(15\!\cdots\!72\)\( T^{10} + \)\(58\!\cdots\!00\)\( T^{11} + \)\(39\!\cdots\!92\)\( T^{12} + \)\(11\!\cdots\!12\)\( T^{13} + \)\(63\!\cdots\!96\)\( T^{14} + \)\(11\!\cdots\!52\)\( T^{15} + \)\(52\!\cdots\!41\)\( T^{16} )^{2} \)
$83$ \( 1 - 12788 T + 81766472 T^{2} - 219288483636 T^{3} + 2030465335614048 T^{4} - 45830993951572452908 T^{5} + \)\(44\!\cdots\!96\)\( T^{6} - \)\(25\!\cdots\!88\)\( T^{7} + \)\(16\!\cdots\!16\)\( T^{8} - \)\(16\!\cdots\!32\)\( T^{9} + \)\(14\!\cdots\!56\)\( T^{10} - \)\(91\!\cdots\!60\)\( T^{11} + \)\(55\!\cdots\!64\)\( T^{12} - \)\(46\!\cdots\!64\)\( T^{13} + \)\(38\!\cdots\!88\)\( T^{14} - \)\(25\!\cdots\!16\)\( T^{15} + \)\(16\!\cdots\!58\)\( T^{16} - \)\(12\!\cdots\!36\)\( T^{17} + \)\(87\!\cdots\!08\)\( T^{18} - \)\(49\!\cdots\!04\)\( T^{19} + \)\(28\!\cdots\!84\)\( T^{20} - \)\(21\!\cdots\!60\)\( T^{21} + \)\(16\!\cdots\!76\)\( T^{22} - \)\(88\!\cdots\!12\)\( T^{23} + \)\(41\!\cdots\!76\)\( T^{24} - \)\(31\!\cdots\!28\)\( T^{25} + \)\(25\!\cdots\!96\)\( T^{26} - \)\(12\!\cdots\!68\)\( T^{27} + \)\(26\!\cdots\!68\)\( T^{28} - \)\(13\!\cdots\!96\)\( T^{29} + \)\(24\!\cdots\!32\)\( T^{30} - \)\(17\!\cdots\!88\)\( T^{31} + \)\(66\!\cdots\!21\)\( T^{32} \)
$89$ \( 1 - 20492 T + 140290034 T^{2} + 294316631388 T^{3} - 7791818672057541 T^{4} + 30934354648294655428 T^{5} - \)\(17\!\cdots\!22\)\( T^{6} + \)\(68\!\cdots\!36\)\( T^{7} - \)\(79\!\cdots\!23\)\( T^{8} + \)\(35\!\cdots\!60\)\( T^{9} + \)\(95\!\cdots\!64\)\( T^{10} - \)\(14\!\cdots\!56\)\( T^{11} + \)\(75\!\cdots\!86\)\( T^{12} - \)\(44\!\cdots\!80\)\( T^{13} + \)\(74\!\cdots\!96\)\( T^{14} - \)\(60\!\cdots\!12\)\( T^{15} + \)\(48\!\cdots\!30\)\( T^{16} - \)\(37\!\cdots\!92\)\( T^{17} + \)\(29\!\cdots\!76\)\( T^{18} - \)\(10\!\cdots\!80\)\( T^{19} + \)\(11\!\cdots\!46\)\( T^{20} - \)\(13\!\cdots\!56\)\( T^{21} + \)\(58\!\cdots\!24\)\( T^{22} + \)\(13\!\cdots\!60\)\( T^{23} - \)\(19\!\cdots\!83\)\( T^{24} + \)\(10\!\cdots\!96\)\( T^{25} - \)\(16\!\cdots\!22\)\( T^{26} + \)\(18\!\cdots\!48\)\( T^{27} - \)\(28\!\cdots\!21\)\( T^{28} + \)\(68\!\cdots\!48\)\( T^{29} + \)\(20\!\cdots\!74\)\( T^{30} - \)\(18\!\cdots\!92\)\( T^{31} + \)\(57\!\cdots\!41\)\( T^{32} \)
$97$ \( 1 - 50944 T + 891925202 T^{2} + 15221697404 T^{3} - 236517605790318237 T^{4} + \)\(33\!\cdots\!96\)\( T^{5} - \)\(12\!\cdots\!26\)\( T^{6} - \)\(46\!\cdots\!40\)\( T^{7} + \)\(51\!\cdots\!85\)\( T^{8} - \)\(82\!\cdots\!12\)\( T^{9} - \)\(48\!\cdots\!08\)\( T^{10} + \)\(44\!\cdots\!76\)\( T^{11} - \)\(17\!\cdots\!98\)\( T^{12} - \)\(33\!\cdots\!24\)\( T^{13} + \)\(25\!\cdots\!04\)\( T^{14} + \)\(77\!\cdots\!12\)\( T^{15} - \)\(23\!\cdots\!26\)\( T^{16} + \)\(68\!\cdots\!72\)\( T^{17} + \)\(19\!\cdots\!44\)\( T^{18} - \)\(23\!\cdots\!84\)\( T^{19} - \)\(10\!\cdots\!58\)\( T^{20} + \)\(24\!\cdots\!76\)\( T^{21} - \)\(23\!\cdots\!48\)\( T^{22} - \)\(35\!\cdots\!32\)\( T^{23} + \)\(19\!\cdots\!85\)\( T^{24} - \)\(15\!\cdots\!40\)\( T^{25} - \)\(37\!\cdots\!26\)\( T^{26} + \)\(87\!\cdots\!76\)\( T^{27} - \)\(54\!\cdots\!57\)\( T^{28} + \)\(31\!\cdots\!64\)\( T^{29} + \)\(16\!\cdots\!42\)\( T^{30} - \)\(81\!\cdots\!44\)\( T^{31} + \)\(14\!\cdots\!81\)\( T^{32} \)
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