Properties

Label 11.6.c.a
Level 11
Weight 6
Character orbit 11.c
Analytic conductor 1.764
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 11.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.76422201794\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{2} + ( -3 \beta_{6} + 3 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{3} + ( -3 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 9 \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{14} ) q^{4} + ( 3 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} + 9 \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{5} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 5 \beta_{6} - 17 \beta_{7} + 7 \beta_{8} + 2 \beta_{11} + 2 \beta_{14} + \beta_{15} ) q^{6} + ( 5 + \beta_{2} - \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 34 \beta_{6} + 5 \beta_{7} + \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} + \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{7} + ( 41 + 16 \beta_{1} - 19 \beta_{3} - 19 \beta_{4} - 5 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} - 41 \beta_{8} + 4 \beta_{10} - 4 \beta_{11} - 3 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{8} + ( -42 + 3 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 4 \beta_{5} + 37 \beta_{6} - 5 \beta_{7} - 21 \beta_{8} + 10 \beta_{9} + 10 \beta_{10} + 5 \beta_{12} + \beta_{13} - 4 \beta_{15} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{2} + ( -3 \beta_{6} + 3 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{3} + ( -3 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 9 \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{14} ) q^{4} + ( 3 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} + 9 \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{5} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 5 \beta_{6} - 17 \beta_{7} + 7 \beta_{8} + 2 \beta_{11} + 2 \beta_{14} + \beta_{15} ) q^{6} + ( 5 + \beta_{2} - \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 34 \beta_{6} + 5 \beta_{7} + \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} + \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{7} + ( 41 + 16 \beta_{1} - 19 \beta_{3} - 19 \beta_{4} - 5 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} - 41 \beta_{8} + 4 \beta_{10} - 4 \beta_{11} - 3 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{8} + ( -42 + 3 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 4 \beta_{5} + 37 \beta_{6} - 5 \beta_{7} - 21 \beta_{8} + 10 \beta_{9} + 10 \beta_{10} + 5 \beta_{12} + \beta_{13} - 4 \beta_{15} ) q^{9} + ( -125 - 2 \beta_{1} + 31 \beta_{2} + 2 \beta_{4} + 5 \beta_{5} - 139 \beta_{7} + 139 \beta_{8} - 4 \beta_{9} - 10 \beta_{11} + 10 \beta_{12} + 4 \beta_{13} - 4 \beta_{14} ) q^{10} + ( -23 + 32 \beta_{1} + 16 \beta_{2} + 2 \beta_{3} - 35 \beta_{4} - 2 \beta_{5} - 35 \beta_{6} + 97 \beta_{7} - 69 \beta_{8} + 8 \beta_{9} - 7 \beta_{10} + 5 \beta_{11} + 5 \beta_{12} - 3 \beta_{13} - 4 \beta_{14} + \beta_{15} ) q^{11} + ( -52 - 33 \beta_{1} - 40 \beta_{2} + 33 \beta_{4} + 3 \beta_{5} + 9 \beta_{7} - 9 \beta_{8} - 20 \beta_{9} + 12 \beta_{11} - 12 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{12} + ( 105 + 20 \beta_{1} - 15 \beta_{2} - 20 \beta_{3} + 15 \beta_{4} + 3 \beta_{5} - 90 \beta_{6} + 15 \beta_{7} + 27 \beta_{8} - 5 \beta_{9} - 5 \beta_{10} - 30 \beta_{12} - 5 \beta_{13} - 3 \beta_{15} ) q^{13} + ( 164 - 104 \beta_{1} + 17 \beta_{3} + 17 \beta_{4} - 2 \beta_{5} - 112 \beta_{6} + 112 \beta_{7} - 164 \beta_{8} - 14 \beta_{9} - 24 \beta_{10} + 24 \beta_{11} - 14 \beta_{12} - 3 \beta_{13} - \beta_{14} - \beta_{15} ) q^{14} + ( 333 - 91 \beta_{2} + 91 \beta_{3} + 136 \beta_{4} - 5 \beta_{5} - 222 \beta_{6} + 333 \beta_{7} - 91 \beta_{8} - 10 \beta_{9} - 25 \beta_{10} + 10 \beta_{11} + 10 \beta_{13} + 5 \beta_{14} + 10 \beta_{15} ) q^{15} + ( 109 - 109 \beta_{1} - 109 \beta_{2} + 148 \beta_{3} + 630 \beta_{6} - 186 \beta_{7} + 521 \beta_{8} + 12 \beta_{10} - 4 \beta_{11} + 12 \beta_{12} + 10 \beta_{14} + 3 \beta_{15} ) q^{16} + ( -36 + 36 \beta_{1} + 36 \beta_{2} + 41 \beta_{3} - 223 \beta_{6} - 372 \beta_{7} - 187 \beta_{8} + 20 \beta_{10} + \beta_{11} + 20 \beta_{12} - 13 \beta_{14} - 10 \beta_{15} ) q^{17} + ( -49 + 138 \beta_{2} - 138 \beta_{3} + 11 \beta_{4} + 17 \beta_{5} + 137 \beta_{6} - 49 \beta_{7} + 138 \beta_{8} + 32 \beta_{9} + 18 \beta_{10} - 32 \beta_{11} - 14 \beta_{13} - 17 \beta_{14} - 14 \beta_{15} ) q^{18} + ( 151 + 65 \beta_{1} - 176 \beta_{3} - 176 \beta_{4} + 25 \beta_{5} - 664 \beta_{6} + 664 \beta_{7} - 151 \beta_{8} + 10 \beta_{9} + 16 \beta_{10} - 16 \beta_{11} + 10 \beta_{12} + 18 \beta_{13} - 7 \beta_{14} - 7 \beta_{15} ) q^{19} + ( -156 + 214 \beta_{1} + 74 \beta_{2} - 214 \beta_{3} - 74 \beta_{4} - 30 \beta_{5} + 82 \beta_{6} - 74 \beta_{7} - 1166 \beta_{8} + 4 \beta_{12} + 30 \beta_{15} ) q^{20} + ( -1153 + 137 \beta_{1} + 221 \beta_{2} - 137 \beta_{4} - 38 \beta_{5} - 931 \beta_{7} + 931 \beta_{8} + 2 \beta_{9} + 7 \beta_{11} - 7 \beta_{12} - 31 \beta_{13} + 31 \beta_{14} ) q^{21} + ( -1747 + 90 \beta_{1} + 67 \beta_{2} + 95 \beta_{3} - 150 \beta_{4} + 15 \beta_{5} + 1610 \beta_{6} - 458 \beta_{7} + 1117 \beta_{8} - 16 \beta_{9} - 8 \beta_{10} - 10 \beta_{11} - 10 \beta_{12} + 28 \beta_{13} + 30 \beta_{14} - 2 \beta_{15} ) q^{22} + ( -270 - 114 \beta_{1} - 84 \beta_{2} + 114 \beta_{4} - 18 \beta_{5} + 594 \beta_{7} - 594 \beta_{8} + 70 \beta_{9} - 22 \beta_{11} + 22 \beta_{12} - 18 \beta_{13} + 18 \beta_{14} ) q^{23} + ( 1257 + 33 \beta_{1} - 60 \beta_{2} - 33 \beta_{3} + 60 \beta_{4} - 24 \beta_{5} - 1197 \beta_{6} + 60 \beta_{7} - 288 \beta_{8} + 20 \beta_{9} + 20 \beta_{10} + 100 \beta_{12} + 33 \beta_{13} + 24 \beta_{15} ) q^{24} + ( 1129 - 215 \beta_{1} + 25 \beta_{3} + 25 \beta_{4} + 25 \beta_{5} + 411 \beta_{6} - 411 \beta_{7} - 1129 \beta_{8} + 35 \beta_{9} + 90 \beta_{10} - 90 \beta_{11} + 35 \beta_{12} + 25 \beta_{13} ) q^{25} + ( 1856 - 200 \beta_{2} + 200 \beta_{3} + 119 \beta_{4} - 2 \beta_{5} - 1036 \beta_{6} + 1856 \beta_{7} - 200 \beta_{8} + 22 \beta_{9} + 102 \beta_{10} - 22 \beta_{11} - 77 \beta_{13} + 2 \beta_{14} - 77 \beta_{15} ) q^{26} + ( 141 - 141 \beta_{1} - 141 \beta_{2} - 30 \beta_{3} + 2430 \beta_{6} - 834 \beta_{7} + 2289 \beta_{8} - 54 \beta_{10} + 26 \beta_{11} - 54 \beta_{12} - 45 \beta_{14} + 15 \beta_{15} ) q^{27} + ( -64 + 64 \beta_{1} + 64 \beta_{2} + 34 \beta_{3} - 376 \beta_{6} - 2926 \beta_{7} - 312 \beta_{8} - 100 \beta_{10} - 36 \beta_{11} - 100 \beta_{12} + 12 \beta_{14} + 40 \beta_{15} ) q^{28} + ( 1063 - 55 \beta_{2} + 55 \beta_{3} + 252 \beta_{4} - 7 \beta_{5} + 502 \beta_{6} + 1063 \beta_{7} - 55 \beta_{8} - 203 \beta_{9} - 94 \beta_{10} + 203 \beta_{11} + 70 \beta_{13} + 7 \beta_{14} + 70 \beta_{15} ) q^{29} + ( 1458 - 51 \beta_{1} + 87 \beta_{3} + 87 \beta_{4} + 21 \beta_{5} - 5010 \beta_{6} + 5010 \beta_{7} - 1458 \beta_{8} - 70 \beta_{9} - 150 \beta_{10} + 150 \beta_{11} - 70 \beta_{12} - 15 \beta_{13} - 36 \beta_{14} - 36 \beta_{15} ) q^{30} + ( -577 - 287 \beta_{1} + 3 \beta_{2} + 287 \beta_{3} - 3 \beta_{4} + 72 \beta_{5} + 574 \beta_{6} - 3 \beta_{7} - 1923 \beta_{8} - 155 \beta_{9} - 155 \beta_{10} - 92 \beta_{12} - 43 \beta_{13} - 72 \beta_{15} ) q^{31} + ( -5242 - 347 \beta_{1} - 183 \beta_{2} + 347 \beta_{4} + 97 \beta_{5} - 3779 \beta_{7} + 3779 \beta_{8} + 56 \beta_{9} + 156 \beta_{11} - 156 \beta_{12} + 75 \beta_{13} - 75 \beta_{14} ) q^{32} + ( -4212 - 87 \beta_{1} - 225 \beta_{2} - 13 \beta_{3} + 156 \beta_{4} - 42 \beta_{5} + 3379 \beta_{6} - 779 \beta_{7} + 1279 \beta_{8} - 74 \beta_{9} + 183 \beta_{10} - 16 \beta_{11} - 49 \beta_{12} - 96 \beta_{13} - 62 \beta_{14} - 23 \beta_{15} ) q^{33} + ( -849 + 218 \beta_{1} - 108 \beta_{2} - 218 \beta_{4} + 13 \beta_{5} + 1845 \beta_{7} - 1845 \beta_{8} - 10 \beta_{9} - 132 \beta_{11} + 132 \beta_{12} + 57 \beta_{13} - 57 \beta_{14} ) q^{34} + ( 2966 - 187 \beta_{1} + 22 \beta_{2} + 187 \beta_{3} - 22 \beta_{4} + 59 \beta_{5} - 2988 \beta_{6} - 22 \beta_{7} + 1912 \beta_{8} - 16 \beta_{9} - 16 \beta_{10} + 45 \beta_{12} - 40 \beta_{13} - 59 \beta_{15} ) q^{35} + ( 7016 + 599 \beta_{1} - 63 \beta_{3} - 63 \beta_{4} - 129 \beta_{5} - 2971 \beta_{6} + 2971 \beta_{7} - 7016 \beta_{8} + 188 \beta_{9} + 56 \beta_{10} - 56 \beta_{11} + 188 \beta_{12} - 74 \beta_{13} + 55 \beta_{14} + 55 \beta_{15} ) q^{36} + ( -553 + 879 \beta_{2} - 879 \beta_{3} - 650 \beta_{4} + 43 \beta_{5} + 544 \beta_{6} - 553 \beta_{7} + 879 \beta_{8} + 143 \beta_{9} - 20 \beta_{10} - 143 \beta_{11} + 176 \beta_{13} - 43 \beta_{14} + 176 \beta_{15} ) q^{37} + ( -849 + 849 \beta_{1} + 849 \beta_{2} - 349 \beta_{3} + 6769 \beta_{6} - 3721 \beta_{7} + 7618 \beta_{8} - 32 \beta_{10} - 16 \beta_{11} - 32 \beta_{12} + 120 \beta_{14} - 62 \beta_{15} ) q^{38} + ( 211 - 211 \beta_{1} - 211 \beta_{2} - 263 \beta_{3} - 596 \beta_{6} - 6633 \beta_{7} - 807 \beta_{8} + 120 \beta_{10} + 63 \beta_{11} + 120 \beta_{12} + 100 \beta_{14} - 81 \beta_{15} ) q^{39} + ( 2612 - 514 \beta_{2} + 514 \beta_{3} - 1032 \beta_{4} - 120 \beta_{5} + 4722 \beta_{6} + 2612 \beta_{7} - 514 \beta_{8} + 200 \beta_{9} + 320 \beta_{10} - 200 \beta_{11} - 130 \beta_{13} + 120 \beta_{14} - 130 \beta_{15} ) q^{40} + ( 3724 + 51 \beta_{1} + 500 \beta_{3} + 500 \beta_{4} - 311 \beta_{5} - 7531 \beta_{6} + 7531 \beta_{7} - 3724 \beta_{8} + 217 \beta_{9} + 205 \beta_{10} - 205 \beta_{11} + 217 \beta_{12} - 94 \beta_{13} + 217 \beta_{14} + 217 \beta_{15} ) q^{41} + ( -3539 - 281 \beta_{1} - 465 \beta_{2} + 281 \beta_{3} + 465 \beta_{4} - 30 \beta_{5} + 4004 \beta_{6} + 465 \beta_{7} - 5403 \beta_{8} + 110 \beta_{9} + 110 \beta_{10} + 24 \beta_{12} + 143 \beta_{13} + 30 \beta_{15} ) q^{42} + ( -3091 + 313 \beta_{1} - 1131 \beta_{2} - 313 \beta_{4} - 82 \beta_{5} - 3836 \beta_{7} + 3836 \beta_{8} - 42 \beta_{9} - 301 \beta_{11} + 301 \beta_{12} - 9 \beta_{13} + 9 \beta_{14} ) q^{43} + ( -8758 - 651 \beta_{1} + 439 \beta_{2} - 1030 \beta_{3} + 733 \beta_{4} + 73 \beta_{5} + 3934 \beta_{6} - 2303 \beta_{7} + 4757 \beta_{8} + 192 \beta_{9} - 256 \beta_{10} - 144 \beta_{11} + 164 \beta_{12} + 115 \beta_{13} - 63 \beta_{14} + 68 \beta_{15} ) q^{44} + ( -7451 + 486 \beta_{1} + 1729 \beta_{2} - 486 \beta_{4} + 111 \beta_{5} - 447 \beta_{7} + 447 \beta_{8} + 35 \beta_{9} + 320 \beta_{11} - 320 \beta_{12} - 44 \beta_{13} + 44 \beta_{14} ) q^{45} + ( 3736 + 118 \beta_{1} + 754 \beta_{2} - 118 \beta_{3} - 754 \beta_{4} + 14 \beta_{5} - 4490 \beta_{6} - 754 \beta_{7} - 996 \beta_{8} + 28 \beta_{9} + 28 \beta_{10} - 140 \beta_{12} - 170 \beta_{13} - 14 \beta_{15} ) q^{46} + ( 3786 + 1610 \beta_{1} - 277 \beta_{3} - 277 \beta_{4} + 328 \beta_{5} - 1580 \beta_{6} + 1580 \beta_{7} - 3786 \beta_{8} - 531 \beta_{9} - 212 \beta_{10} + 212 \beta_{11} - 531 \beta_{12} + 75 \beta_{13} - 253 \beta_{14} - 253 \beta_{15} ) q^{47} + ( 3426 + 233 \beta_{2} - 233 \beta_{3} - 812 \beta_{4} - 20 \beta_{5} - 4143 \beta_{6} + 3426 \beta_{7} + 233 \beta_{8} - 572 \beta_{9} - 264 \beta_{10} + 572 \beta_{11} + 55 \beta_{13} + 20 \beta_{14} + 55 \beta_{15} ) q^{48} + ( -173 + 173 \beta_{1} + 173 \beta_{2} - 1186 \beta_{3} + 3544 \beta_{6} - 2368 \beta_{7} + 3717 \beta_{8} + 199 \beta_{10} + 131 \beta_{11} + 199 \beta_{12} - 177 \beta_{14} - \beta_{15} ) q^{49} + ( 1025 - 1025 \beta_{1} - 1025 \beta_{2} - 11 \beta_{3} - 860 \beta_{6} - 4676 \beta_{7} - 1885 \beta_{8} + 10 \beta_{10} - 80 \beta_{11} + 10 \beta_{12} - 285 \beta_{14} + 95 \beta_{15} ) q^{50} + ( 7781 - 675 \beta_{2} + 675 \beta_{3} + 983 \beta_{4} + 197 \beta_{5} - 1915 \beta_{6} + 7781 \beta_{7} - 675 \beta_{8} + 282 \beta_{9} - 616 \beta_{10} - 282 \beta_{11} - 83 \beta_{13} - 197 \beta_{14} - 83 \beta_{15} ) q^{51} + ( -3818 - 1702 \beta_{1} + 2146 \beta_{3} + 2146 \beta_{4} + 400 \beta_{5} - 3208 \beta_{6} + 3208 \beta_{7} + 3818 \beta_{8} - 492 \beta_{9} + 12 \beta_{10} - 12 \beta_{11} - 492 \beta_{12} + 132 \beta_{13} - 268 \beta_{14} - 268 \beta_{15} ) q^{52} + ( 1703 - 1830 \beta_{1} - 631 \beta_{2} + 1830 \beta_{3} + 631 \beta_{4} - 75 \beta_{5} - 1072 \beta_{6} + 631 \beta_{7} - 6015 \beta_{8} + 461 \beta_{9} + 461 \beta_{10} + 432 \beta_{12} + 23 \beta_{13} + 75 \beta_{15} ) q^{53} + ( -284 - 1184 \beta_{1} - 941 \beta_{2} + 1184 \beta_{4} + 62 \beta_{5} - 8537 \beta_{7} + 8537 \beta_{8} - 236 \beta_{9} - 12 \beta_{11} + 12 \beta_{12} - 170 \beta_{13} + 170 \beta_{14} ) q^{54} + ( 1638 - 364 \beta_{1} - 2008 \beta_{2} + 1685 \beta_{3} + 1560 \beta_{4} - 156 \beta_{5} + 3980 \beta_{6} + 2748 \beta_{7} - 1730 \beta_{8} - 80 \beta_{9} - 535 \beta_{10} + 841 \beta_{11} - 325 \beta_{12} + 96 \beta_{13} + 381 \beta_{14} + 122 \beta_{15} ) q^{55} + ( 6932 + 506 \beta_{1} - 1122 \beta_{2} - 506 \beta_{4} - 198 \beta_{5} + 4606 \beta_{7} - 4606 \beta_{8} - 672 \beta_{9} + 88 \beta_{11} - 88 \beta_{12} - 154 \beta_{13} + 154 \beta_{14} ) q^{56} + ( -3312 - 225 \beta_{1} - 750 \beta_{2} + 225 \beta_{3} + 750 \beta_{4} - 285 \beta_{5} + 4062 \beta_{6} + 750 \beta_{7} + 2121 \beta_{8} - 141 \beta_{9} - 141 \beta_{10} - 923 \beta_{12} + 420 \beta_{13} + 285 \beta_{15} ) q^{57} + ( 5448 - 1086 \beta_{1} + 101 \beta_{3} + 101 \beta_{4} - 292 \beta_{5} - 6236 \beta_{6} + 6236 \beta_{7} - 5448 \beta_{8} - 62 \beta_{9} - 776 \beta_{10} + 776 \beta_{11} - 62 \beta_{12} + 37 \beta_{13} + 329 \beta_{14} + 329 \beta_{15} ) q^{58} + ( 4001 + 412 \beta_{2} - 412 \beta_{3} - 1975 \beta_{4} - 182 \beta_{5} + 2204 \beta_{6} + 4001 \beta_{7} + 412 \beta_{8} + 550 \beta_{9} - 92 \beta_{10} - 550 \beta_{11} - 671 \beta_{13} + 182 \beta_{14} - 671 \beta_{15} ) q^{59} + ( -1240 + 1240 \beta_{1} + 1240 \beta_{2} - 20 \beta_{3} + 2862 \beta_{6} - 5588 \beta_{7} + 4102 \beta_{8} + 364 \beta_{10} - 848 \beta_{11} + 364 \beta_{12} - 12 \beta_{14} + 142 \beta_{15} ) q^{60} + ( -2315 + 2315 \beta_{1} + 2315 \beta_{2} - 1762 \beta_{3} - 3800 \beta_{6} - 1343 \beta_{7} - 1485 \beta_{8} + 367 \beta_{10} + 141 \beta_{11} + 367 \beta_{12} + 81 \beta_{14} - 79 \beta_{15} ) q^{61} + ( -15176 + 722 \beta_{2} - 722 \beta_{3} - 699 \beta_{4} + 98 \beta_{5} + 5278 \beta_{6} - 15176 \beta_{7} + 722 \beta_{8} + 150 \beta_{9} + 506 \beta_{10} - 150 \beta_{11} + 643 \beta_{13} - 98 \beta_{14} + 643 \beta_{15} ) q^{62} + ( -2540 + 309 \beta_{1} - 1826 \beta_{3} - 1826 \beta_{4} + 649 \beta_{5} + 2892 \beta_{6} - 2892 \beta_{7} + 2540 \beta_{8} + 968 \beta_{9} + 561 \beta_{10} - 561 \beta_{11} + 968 \beta_{12} + 308 \beta_{13} - 341 \beta_{14} - 341 \beta_{15} ) q^{63} + ( -2065 + 4092 \beta_{1} + 2167 \beta_{2} - 4092 \beta_{3} - 2167 \beta_{4} + 11 \beta_{5} - 102 \beta_{6} - 2167 \beta_{7} + 3752 \beta_{8} + 484 \beta_{9} + 484 \beta_{10} + 56 \beta_{12} - 682 \beta_{13} - 11 \beta_{15} ) q^{64} + ( 458 + 2897 \beta_{1} + 2746 \beta_{2} - 2897 \beta_{4} - 265 \beta_{5} + 12574 \beta_{7} - 12574 \beta_{8} + 49 \beta_{9} - 215 \beta_{11} + 215 \beta_{12} - 9 \beta_{13} + 9 \beta_{14} ) q^{65} + ( 2583 + 537 \beta_{1} + 2320 \beta_{2} - 1547 \beta_{3} - 3502 \beta_{4} + 128 \beta_{5} - 8177 \beta_{6} - 719 \beta_{7} + 1226 \beta_{8} + 302 \beta_{9} + 1064 \beta_{10} - 1090 \beta_{11} + 1022 \beta_{12} - 325 \beta_{13} - 283 \beta_{14} - 570 \beta_{15} ) q^{66} + ( -6743 - 859 \beta_{1} + 122 \beta_{2} + 859 \beta_{4} - 131 \beta_{5} - 6259 \beta_{7} + 6259 \beta_{8} + 419 \beta_{9} - 275 \beta_{11} + 275 \beta_{12} + 353 \beta_{13} - 353 \beta_{14} ) q^{67} + ( -7704 + 481 \beta_{1} + 1119 \beta_{2} - 481 \beta_{3} - 1119 \beta_{4} + 359 \beta_{5} + 6585 \beta_{6} - 1119 \beta_{7} - 3914 \beta_{8} - 28 \beta_{9} - 28 \beta_{10} + 868 \beta_{12} + 161 \beta_{13} - 359 \beta_{15} ) q^{68} + ( -17746 - 226 \beta_{1} - 96 \beta_{3} - 96 \beta_{4} - 558 \beta_{5} + 18854 \beta_{6} - 18854 \beta_{7} + 17746 \beta_{8} + 474 \beta_{9} + 1046 \beta_{10} - 1046 \beta_{11} + 474 \beta_{12} - 140 \beta_{13} + 418 \beta_{14} + 418 \beta_{15} ) q^{69} + ( -3503 - 2663 \beta_{2} + 2663 \beta_{3} + 6474 \beta_{4} + 129 \beta_{5} - 2482 \beta_{6} - 3503 \beta_{7} - 2663 \beta_{8} + 364 \beta_{9} + 434 \beta_{10} - 364 \beta_{11} + 406 \beta_{13} - 129 \beta_{14} + 406 \beta_{15} ) q^{70} + ( 4156 - 4156 \beta_{1} - 4156 \beta_{2} + 1235 \beta_{3} - 17648 \beta_{6} + 11038 \beta_{7} - 21804 \beta_{8} - 901 \beta_{10} + 1105 \beta_{11} - 901 \beta_{12} + 571 \beta_{14} + 246 \beta_{15} ) q^{71} + ( 1931 - 1931 \beta_{1} - 1931 \beta_{2} + 5555 \beta_{3} + 9935 \beta_{6} + 29916 \beta_{7} + 8004 \beta_{8} - 1212 \beta_{10} + 608 \beta_{11} - 1212 \beta_{12} + 605 \beta_{14} - 108 \beta_{15} ) q^{72} + ( 4477 + 2066 \beta_{2} - 2066 \beta_{3} + 2323 \beta_{4} - 92 \beta_{5} - 14787 \beta_{6} + 4477 \beta_{7} + 2066 \beta_{8} - 1066 \beta_{9} + 261 \beta_{10} + 1066 \beta_{11} - 551 \beta_{13} + 92 \beta_{14} - 551 \beta_{15} ) q^{73} + ( 6876 + 4972 \beta_{1} - 4341 \beta_{3} - 4341 \beta_{4} - 1390 \beta_{5} + 24518 \beta_{6} - 24518 \beta_{7} - 6876 \beta_{8} - 1030 \beta_{9} - 1276 \beta_{10} + 1276 \beta_{11} - 1030 \beta_{12} - 571 \beta_{13} + 819 \beta_{14} + 819 \beta_{15} ) q^{74} + ( 15668 - 535 \beta_{1} - 385 \beta_{2} + 535 \beta_{3} + 385 \beta_{4} - 200 \beta_{5} - 15283 \beta_{6} + 385 \beta_{7} + 15675 \beta_{8} - 2245 \beta_{9} - 2245 \beta_{10} - 1529 \beta_{12} + 485 \beta_{13} + 200 \beta_{15} ) q^{75} + ( 19530 - 3886 \beta_{1} + 465 \beta_{2} + 3886 \beta_{4} - 170 \beta_{5} + 13000 \beta_{7} - 13000 \beta_{8} + 704 \beta_{9} + 488 \beta_{11} - 488 \beta_{12} + 393 \beta_{13} - 393 \beta_{14} ) q^{76} + ( 27579 + 1036 \beta_{1} + 595 \beta_{2} + 3857 \beta_{3} - 1470 \beta_{4} + 763 \beta_{5} - 9588 \beta_{6} + 2325 \beta_{7} - 10741 \beta_{8} - 401 \beta_{9} + 344 \beta_{10} - 175 \beta_{11} - 1264 \beta_{12} + 116 \beta_{13} - 487 \beta_{14} + 64 \beta_{15} ) q^{77} + ( 27326 - 2679 \beta_{1} - 5248 \beta_{2} + 2679 \beta_{4} + 273 \beta_{5} + 798 \beta_{7} - 798 \beta_{8} + 766 \beta_{9} - 164 \beta_{11} + 164 \beta_{12} - 25 \beta_{13} + 25 \beta_{14} ) q^{78} + ( 5135 - 2207 \beta_{1} - 5497 \beta_{2} + 2207 \beta_{3} + 5497 \beta_{4} + 172 \beta_{5} + 362 \beta_{6} + 5497 \beta_{7} + 8841 \beta_{8} + 201 \beta_{9} + 201 \beta_{10} + 2108 \beta_{12} - 855 \beta_{13} - 172 \beta_{15} ) q^{79} + ( -25860 - 8222 \beta_{1} + 3154 \beta_{3} + 3154 \beta_{4} + 1790 \beta_{5} - 4074 \beta_{6} + 4074 \beta_{7} + 25860 \beta_{8} + 888 \beta_{9} + 1328 \beta_{10} - 1328 \beta_{11} + 888 \beta_{12} + 316 \beta_{13} - 1474 \beta_{14} - 1474 \beta_{15} ) q^{80} + ( -25242 - 1520 \beta_{2} + 1520 \beta_{3} + 2414 \beta_{4} + 524 \beta_{5} + 17100 \beta_{6} - 25242 \beta_{7} - 1520 \beta_{8} - 1264 \beta_{9} + 634 \beta_{10} + 1264 \beta_{11} + 626 \beta_{13} - 524 \beta_{14} + 626 \beta_{15} ) q^{81} + ( 1536 - 1536 \beta_{1} - 1536 \beta_{2} + 9063 \beta_{3} - 28957 \beta_{6} + 32573 \beta_{7} - 30493 \beta_{8} - 516 \beta_{10} + 950 \beta_{11} - 516 \beta_{12} - 804 \beta_{14} - 555 \beta_{15} ) q^{82} + ( -2611 + 2611 \beta_{1} + 2611 \beta_{2} - 3517 \beta_{3} - 1005 \beta_{6} + 5528 \beta_{7} + 1606 \beta_{8} + 628 \beta_{10} - 1568 \beta_{11} + 628 \beta_{12} - 780 \beta_{14} + 841 \beta_{15} ) q^{83} + ( -19194 + 2412 \beta_{2} - 2412 \beta_{3} - 3258 \beta_{4} - 402 \beta_{5} - 19182 \beta_{6} - 19194 \beta_{7} + 2412 \beta_{8} - 696 \beta_{9} - 1380 \beta_{10} + 696 \beta_{11} - 720 \beta_{13} + 402 \beta_{14} - 720 \beta_{15} ) q^{84} + ( -13242 - 747 \beta_{1} - 3087 \beta_{3} - 3087 \beta_{4} - 147 \beta_{5} + 28246 \beta_{6} - 28246 \beta_{7} + 13242 \beta_{8} - 711 \beta_{9} - 748 \beta_{10} + 748 \beta_{11} - 711 \beta_{12} - 219 \beta_{13} - 72 \beta_{14} - 72 \beta_{15} ) q^{85} + ( -13889 + 1239 \beta_{1} - 1322 \beta_{2} - 1239 \beta_{3} + 1322 \beta_{4} + 882 \beta_{5} + 15211 \beta_{6} + 1322 \beta_{7} + 51810 \beta_{8} - 858 \beta_{9} - 858 \beta_{10} + 376 \beta_{12} + 1309 \beta_{13} - 882 \beta_{15} ) q^{86} + ( 10815 + 4700 \beta_{1} + 7797 \beta_{2} - 4700 \beta_{4} + 1735 \beta_{5} + 40463 \beta_{7} - 40463 \beta_{8} - 166 \beta_{9} + 1861 \beta_{11} - 1861 \beta_{12} + 318 \beta_{13} - 318 \beta_{14} ) q^{87} + ( 31295 + 8998 \beta_{1} + 2299 \beta_{2} - 6578 \beta_{3} - 3421 \beta_{4} - 1771 \beta_{5} - 45144 \beta_{6} + 12067 \beta_{7} - 27808 \beta_{8} - 1540 \beta_{9} - 1276 \beta_{10} + 1056 \beta_{11} - 1848 \beta_{12} - 110 \beta_{13} + 682 \beta_{14} + 1331 \beta_{15} ) q^{88} + ( -3047 - 3719 \beta_{1} + 2311 \beta_{2} + 3719 \beta_{4} + 56 \beta_{5} - 27170 \beta_{7} + 27170 \beta_{8} + 1222 \beta_{9} - 1419 \beta_{11} + 1419 \beta_{12} - 659 \beta_{13} + 659 \beta_{14} ) q^{89} + ( -18980 + 7093 \beta_{1} + 5452 \beta_{2} - 7093 \beta_{3} - 5452 \beta_{4} - 679 \beta_{5} + 13528 \beta_{6} - 5452 \beta_{7} - 47790 \beta_{8} + 1436 \beta_{9} + 1436 \beta_{10} - 690 \beta_{12} - 500 \beta_{13} + 679 \beta_{15} ) q^{90} + ( -44946 + 1339 \beta_{1} - 5297 \beta_{3} - 5297 \beta_{4} - 1499 \beta_{5} + 33942 \beta_{6} - 33942 \beta_{7} + 44946 \beta_{8} + 51 \beta_{9} - 597 \beta_{10} + 597 \beta_{11} + 51 \beta_{12} - 587 \beta_{13} + 912 \beta_{14} + 912 \beta_{15} ) q^{91} + ( -578 - 4064 \beta_{2} + 4064 \beta_{3} + 1326 \beta_{4} - 654 \beta_{5} - 4580 \beta_{6} - 578 \beta_{7} - 4064 \beta_{8} + 1496 \beta_{9} + 216 \beta_{10} - 1496 \beta_{11} + 286 \beta_{13} + 654 \beta_{14} + 286 \beta_{15} ) q^{92} + ( 4438 - 4438 \beta_{1} - 4438 \beta_{2} - 1853 \beta_{3} - 35318 \beta_{6} + 23244 \beta_{7} - 39756 \beta_{8} + 892 \beta_{10} - 2717 \beta_{11} + 892 \beta_{12} + 175 \beta_{14} - 960 \beta_{15} ) q^{93} + ( -947 + 947 \beta_{1} + 947 \beta_{2} + 1240 \beta_{3} + 18948 \beta_{6} + 54143 \beta_{7} + 19895 \beta_{8} + 1350 \beta_{10} - 1300 \beta_{11} + 1350 \beta_{12} + 307 \beta_{14} - 1389 \beta_{15} ) q^{94} + ( -2947 - 3165 \beta_{2} + 3165 \beta_{3} + 4478 \beta_{4} - 919 \beta_{5} + 18336 \beta_{6} - 2947 \beta_{7} - 3165 \beta_{8} + 1020 \beta_{9} + 2217 \beta_{10} - 1020 \beta_{11} + 908 \beta_{13} + 919 \beta_{14} + 908 \beta_{15} ) q^{95} + ( -17748 - 157 \beta_{1} + 1969 \beta_{3} + 1969 \beta_{4} + 355 \beta_{5} + 67967 \beta_{6} - 67967 \beta_{7} + 17748 \beta_{8} + 3596 \beta_{9} + 1512 \beta_{10} - 1512 \beta_{11} + 3596 \beta_{12} + 612 \beta_{13} + 257 \beta_{14} + 257 \beta_{15} ) q^{96} + ( 33313 - 1188 \beta_{1} + 5586 \beta_{2} + 1188 \beta_{3} - 5586 \beta_{4} - 198 \beta_{5} - 38899 \beta_{6} - 5586 \beta_{7} + 3126 \beta_{8} + 3046 \beta_{9} + 3046 \beta_{10} + 1596 \beta_{12} - 978 \beta_{13} + 198 \beta_{15} ) q^{97} + ( 49597 + 262 \beta_{1} - 1180 \beta_{2} - 262 \beta_{4} - 1403 \beta_{5} + 5758 \beta_{7} - 5758 \beta_{8} - 48 \beta_{9} - 1110 \beta_{11} + 1110 \beta_{12} - 1294 \beta_{13} + 1294 \beta_{14} ) q^{98} + ( 54518 - 6917 \beta_{1} - 3882 \beta_{2} + 6794 \beta_{3} + 4789 \beta_{4} - 293 \beta_{5} - 6321 \beta_{6} - 8675 \beta_{7} - 25899 \beta_{8} + 1535 \beta_{9} + 1444 \beta_{10} - 1517 \beta_{11} + 5006 \beta_{12} + 919 \beta_{13} + 107 \beta_{14} - 244 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - q^{2} - 24q^{3} - 73q^{4} - 10q^{5} + 121q^{6} + 196q^{7} + 527q^{8} - 530q^{9} + O(q^{10}) \) \( 16q - q^{2} - 24q^{3} - 73q^{4} - 10q^{5} + 121q^{6} + 196q^{7} + 527q^{8} - 530q^{9} - 672q^{10} - 692q^{11} - 1562q^{12} + 1162q^{13} + 560q^{14} + 1796q^{15} + 6399q^{16} - 22q^{17} + 834q^{18} - 3236q^{19} - 5514q^{20} - 7772q^{21} - 13059q^{22} - 10848q^{23} + 13233q^{24} + 15686q^{25} + 16216q^{26} + 22500q^{27} + 8838q^{28} + 13070q^{29} - 22830q^{30} - 14764q^{31} - 58812q^{32} - 48544q^{33} - 26966q^{34} + 43368q^{35} + 63696q^{36} + 4638q^{37} + 68144q^{38} + 20300q^{39} + 52284q^{40} - 14806q^{41} - 69922q^{42} - 24376q^{43} - 104960q^{44} - 97480q^{45} + 50452q^{46} + 40364q^{47} + 30236q^{48} + 32246q^{49} + 10839q^{50} + 75564q^{51} - 80654q^{52} - 11654q^{53} + 43796q^{54} + 7052q^{55} + 70632q^{56} - 40020q^{57} + 10276q^{58} + 70804q^{59} + 46116q^{60} - 31446q^{61} - 153388q^{62} - 7064q^{63} + 17695q^{64} - 41284q^{65} + 48006q^{66} - 64200q^{67} - 94114q^{68} - 62412q^{69} - 103768q^{70} - 184380q^{71} - 14729q^{72} - 1750q^{73} + 306048q^{74} + 246596q^{75} + 174806q^{76} + 384646q^{77} + 360916q^{78} + 24324q^{79} - 386444q^{80} - 259412q^{81} - 316255q^{82} - 46028q^{83} - 271956q^{84} + 63914q^{85} + 22931q^{86} - 47592q^{87} + 211871q^{88} + 148364q^{89} - 347186q^{90} - 258448q^{91} - 58658q^{92} - 387478q^{93} - 55370q^{94} - 4716q^{95} + 328396q^{96} + 484296q^{97} + 743692q^{98} + 724964q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 5 x^{15} + 86 x^{14} - 146 x^{13} + 7205 x^{12} - 23732 x^{11} + 774165 x^{10} - 1228996 x^{9} + 44649817 x^{8} - 92720004 x^{7} + 943915592 x^{6} - 1892495035 x^{5} + 8611100939 x^{4} - 17760110698 x^{3} + 18348896492 x^{2} - 4224271656 x + 393784336\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(25\!\cdots\!53\)\( \nu^{15} - \)\(10\!\cdots\!45\)\( \nu^{14} + \)\(21\!\cdots\!82\)\( \nu^{13} - \)\(21\!\cdots\!94\)\( \nu^{12} + \)\(18\!\cdots\!97\)\( \nu^{11} - \)\(46\!\cdots\!28\)\( \nu^{10} + \)\(19\!\cdots\!45\)\( \nu^{9} - \)\(16\!\cdots\!76\)\( \nu^{8} + \)\(11\!\cdots\!17\)\( \nu^{7} - \)\(14\!\cdots\!44\)\( \nu^{6} + \)\(23\!\cdots\!48\)\( \nu^{5} - \)\(27\!\cdots\!55\)\( \nu^{4} + \)\(17\!\cdots\!07\)\( \nu^{3} - \)\(24\!\cdots\!30\)\( \nu^{2} + \)\(57\!\cdots\!92\)\( \nu + \)\(31\!\cdots\!52\)\(\)\()/ \)\(22\!\cdots\!92\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(19\!\cdots\!81\)\( \nu^{15} + \)\(78\!\cdots\!11\)\( \nu^{14} - \)\(15\!\cdots\!68\)\( \nu^{13} + \)\(14\!\cdots\!06\)\( \nu^{12} - \)\(13\!\cdots\!13\)\( \nu^{11} + \)\(33\!\cdots\!06\)\( \nu^{10} - \)\(14\!\cdots\!09\)\( \nu^{9} + \)\(10\!\cdots\!54\)\( \nu^{8} - \)\(85\!\cdots\!69\)\( \nu^{7} + \)\(10\!\cdots\!06\)\( \nu^{6} - \)\(17\!\cdots\!68\)\( \nu^{5} + \)\(21\!\cdots\!31\)\( \nu^{4} - \)\(16\!\cdots\!73\)\( \nu^{3} + \)\(22\!\cdots\!08\)\( \nu^{2} - \)\(25\!\cdots\!84\)\( \nu + \)\(38\!\cdots\!44\)\(\)\()/ \)\(12\!\cdots\!56\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(21\!\cdots\!19\)\( \nu^{15} - \)\(81\!\cdots\!75\)\( \nu^{14} + \)\(17\!\cdots\!26\)\( \nu^{13} - \)\(89\!\cdots\!78\)\( \nu^{12} + \)\(15\!\cdots\!15\)\( \nu^{11} - \)\(32\!\cdots\!60\)\( \nu^{10} + \)\(16\!\cdots\!99\)\( \nu^{9} - \)\(58\!\cdots\!76\)\( \nu^{8} + \)\(94\!\cdots\!51\)\( \nu^{7} - \)\(83\!\cdots\!24\)\( \nu^{6} + \)\(18\!\cdots\!40\)\( \nu^{5} - \)\(18\!\cdots\!97\)\( \nu^{4} + \)\(15\!\cdots\!89\)\( \nu^{3} - \)\(20\!\cdots\!82\)\( \nu^{2} + \)\(17\!\cdots\!88\)\( \nu - \)\(37\!\cdots\!20\)\(\)\()/ \)\(12\!\cdots\!56\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(35\!\cdots\!61\)\( \nu^{15} - \)\(24\!\cdots\!13\)\( \nu^{14} + \)\(30\!\cdots\!94\)\( \nu^{13} - \)\(87\!\cdots\!26\)\( \nu^{12} + \)\(23\!\cdots\!01\)\( \nu^{11} - \)\(12\!\cdots\!56\)\( \nu^{10} + \)\(26\!\cdots\!97\)\( \nu^{9} - \)\(82\!\cdots\!28\)\( \nu^{8} + \)\(14\!\cdots\!61\)\( \nu^{7} - \)\(54\!\cdots\!44\)\( \nu^{6} + \)\(25\!\cdots\!52\)\( \nu^{5} - \)\(78\!\cdots\!51\)\( \nu^{4} + \)\(18\!\cdots\!19\)\( \nu^{3} - \)\(18\!\cdots\!62\)\( \nu^{2} + \)\(42\!\cdots\!60\)\( \nu + \)\(32\!\cdots\!44\)\(\)\()/ \)\(12\!\cdots\!56\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(48\!\cdots\!69\)\( \nu^{15} + \)\(23\!\cdots\!05\)\( \nu^{14} - \)\(41\!\cdots\!70\)\( \nu^{13} + \)\(62\!\cdots\!58\)\( \nu^{12} - \)\(34\!\cdots\!05\)\( \nu^{11} + \)\(10\!\cdots\!80\)\( \nu^{10} - \)\(37\!\cdots\!37\)\( \nu^{9} + \)\(51\!\cdots\!52\)\( \nu^{8} - \)\(21\!\cdots\!25\)\( \nu^{7} + \)\(40\!\cdots\!84\)\( \nu^{6} - \)\(45\!\cdots\!80\)\( \nu^{5} + \)\(82\!\cdots\!39\)\( \nu^{4} - \)\(40\!\cdots\!95\)\( \nu^{3} + \)\(78\!\cdots\!74\)\( \nu^{2} - \)\(78\!\cdots\!04\)\( \nu + \)\(18\!\cdots\!36\)\(\)\()/ \)\(99\!\cdots\!92\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(78\!\cdots\!15\)\( \nu^{15} + \)\(37\!\cdots\!17\)\( \nu^{14} - \)\(67\!\cdots\!40\)\( \nu^{13} + \)\(10\!\cdots\!58\)\( \nu^{12} - \)\(56\!\cdots\!79\)\( \nu^{11} + \)\(17\!\cdots\!50\)\( \nu^{10} - \)\(60\!\cdots\!55\)\( \nu^{9} + \)\(83\!\cdots\!22\)\( \nu^{8} - \)\(35\!\cdots\!23\)\( \nu^{7} + \)\(65\!\cdots\!78\)\( \nu^{6} - \)\(73\!\cdots\!12\)\( \nu^{5} + \)\(13\!\cdots\!45\)\( \nu^{4} - \)\(66\!\cdots\!31\)\( \nu^{3} + \)\(12\!\cdots\!72\)\( \nu^{2} - \)\(12\!\cdots\!56\)\( \nu + \)\(19\!\cdots\!24\)\(\)\()/ \)\(99\!\cdots\!92\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(39\!\cdots\!39\)\( \nu^{15} + \)\(19\!\cdots\!74\)\( \nu^{14} - \)\(33\!\cdots\!03\)\( \nu^{13} + \)\(51\!\cdots\!06\)\( \nu^{12} - \)\(28\!\cdots\!49\)\( \nu^{11} + \)\(88\!\cdots\!15\)\( \nu^{10} - \)\(30\!\cdots\!89\)\( \nu^{9} + \)\(42\!\cdots\!75\)\( \nu^{8} - \)\(17\!\cdots\!49\)\( \nu^{7} + \)\(33\!\cdots\!27\)\( \nu^{6} - \)\(37\!\cdots\!42\)\( \nu^{5} + \)\(67\!\cdots\!77\)\( \nu^{4} - \)\(33\!\cdots\!50\)\( \nu^{3} + \)\(63\!\cdots\!29\)\( \nu^{2} - \)\(63\!\cdots\!60\)\( \nu + \)\(65\!\cdots\!40\)\(\)\()/ \)\(49\!\cdots\!96\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(47\!\cdots\!26\)\( \nu^{15} + \)\(18\!\cdots\!92\)\( \nu^{14} - \)\(38\!\cdots\!80\)\( \nu^{13} + \)\(23\!\cdots\!61\)\( \nu^{12} - \)\(33\!\cdots\!19\)\( \nu^{11} + \)\(74\!\cdots\!53\)\( \nu^{10} - \)\(35\!\cdots\!87\)\( \nu^{9} + \)\(16\!\cdots\!15\)\( \nu^{8} - \)\(20\!\cdots\!25\)\( \nu^{7} + \)\(20\!\cdots\!57\)\( \nu^{6} - \)\(40\!\cdots\!17\)\( \nu^{5} + \)\(42\!\cdots\!45\)\( \nu^{4} - \)\(32\!\cdots\!49\)\( \nu^{3} + \)\(45\!\cdots\!90\)\( \nu^{2} - \)\(10\!\cdots\!24\)\( \nu - \)\(20\!\cdots\!72\)\(\)\()/ \)\(12\!\cdots\!56\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(28\!\cdots\!59\)\( \nu^{15} + \)\(15\!\cdots\!16\)\( \nu^{14} - \)\(24\!\cdots\!81\)\( \nu^{13} + \)\(53\!\cdots\!12\)\( \nu^{12} - \)\(20\!\cdots\!98\)\( \nu^{11} + \)\(78\!\cdots\!54\)\( \nu^{10} - \)\(22\!\cdots\!48\)\( \nu^{9} + \)\(46\!\cdots\!06\)\( \nu^{8} - \)\(12\!\cdots\!58\)\( \nu^{7} + \)\(33\!\cdots\!88\)\( \nu^{6} - \)\(27\!\cdots\!13\)\( \nu^{5} + \)\(67\!\cdots\!78\)\( \nu^{4} - \)\(25\!\cdots\!39\)\( \nu^{3} + \)\(61\!\cdots\!16\)\( \nu^{2} - \)\(67\!\cdots\!28\)\( \nu + \)\(15\!\cdots\!04\)\(\)\()/ \)\(49\!\cdots\!96\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(64\!\cdots\!06\)\( \nu^{15} + \)\(32\!\cdots\!06\)\( \nu^{14} - \)\(55\!\cdots\!37\)\( \nu^{13} + \)\(98\!\cdots\!99\)\( \nu^{12} - \)\(46\!\cdots\!36\)\( \nu^{11} + \)\(15\!\cdots\!65\)\( \nu^{10} - \)\(50\!\cdots\!34\)\( \nu^{9} + \)\(83\!\cdots\!07\)\( \nu^{8} - \)\(28\!\cdots\!54\)\( \nu^{7} + \)\(62\!\cdots\!65\)\( \nu^{6} - \)\(60\!\cdots\!50\)\( \nu^{5} + \)\(12\!\cdots\!73\)\( \nu^{4} - \)\(55\!\cdots\!29\)\( \nu^{3} + \)\(11\!\cdots\!32\)\( \nu^{2} - \)\(11\!\cdots\!44\)\( \nu + \)\(17\!\cdots\!80\)\(\)\()/ \)\(49\!\cdots\!96\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(76\!\cdots\!69\)\( \nu^{15} + \)\(37\!\cdots\!46\)\( \nu^{14} - \)\(65\!\cdots\!41\)\( \nu^{13} + \)\(10\!\cdots\!29\)\( \nu^{12} - \)\(55\!\cdots\!68\)\( \nu^{11} + \)\(17\!\cdots\!87\)\( \nu^{10} - \)\(59\!\cdots\!38\)\( \nu^{9} + \)\(89\!\cdots\!93\)\( \nu^{8} - \)\(34\!\cdots\!52\)\( \nu^{7} + \)\(68\!\cdots\!17\)\( \nu^{6} - \)\(71\!\cdots\!25\)\( \nu^{5} + \)\(13\!\cdots\!33\)\( \nu^{4} - \)\(63\!\cdots\!43\)\( \nu^{3} + \)\(12\!\cdots\!24\)\( \nu^{2} - \)\(11\!\cdots\!32\)\( \nu + \)\(11\!\cdots\!76\)\(\)\()/ \)\(49\!\cdots\!96\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(10\!\cdots\!27\)\( \nu^{15} - \)\(50\!\cdots\!23\)\( \nu^{14} + \)\(90\!\cdots\!94\)\( \nu^{13} - \)\(13\!\cdots\!70\)\( \nu^{12} + \)\(76\!\cdots\!71\)\( \nu^{11} - \)\(23\!\cdots\!84\)\( \nu^{10} + \)\(81\!\cdots\!07\)\( \nu^{9} - \)\(11\!\cdots\!44\)\( \nu^{8} + \)\(47\!\cdots\!31\)\( \nu^{7} - \)\(88\!\cdots\!64\)\( \nu^{6} + \)\(98\!\cdots\!92\)\( \nu^{5} - \)\(17\!\cdots\!53\)\( \nu^{4} + \)\(88\!\cdots\!33\)\( \nu^{3} - \)\(17\!\cdots\!58\)\( \nu^{2} + \)\(17\!\cdots\!84\)\( \nu - \)\(17\!\cdots\!96\)\(\)\()/ \)\(49\!\cdots\!96\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(26\!\cdots\!26\)\( \nu^{15} - \)\(12\!\cdots\!10\)\( \nu^{14} + \)\(22\!\cdots\!01\)\( \nu^{13} - \)\(34\!\cdots\!20\)\( \nu^{12} + \)\(19\!\cdots\!44\)\( \nu^{11} - \)\(59\!\cdots\!15\)\( \nu^{10} + \)\(20\!\cdots\!54\)\( \nu^{9} - \)\(28\!\cdots\!26\)\( \nu^{8} + \)\(11\!\cdots\!30\)\( \nu^{7} - \)\(22\!\cdots\!41\)\( \nu^{6} + \)\(24\!\cdots\!29\)\( \nu^{5} - \)\(45\!\cdots\!58\)\( \nu^{4} + \)\(22\!\cdots\!92\)\( \nu^{3} - \)\(42\!\cdots\!36\)\( \nu^{2} + \)\(42\!\cdots\!31\)\( \nu - \)\(64\!\cdots\!38\)\(\)\()/ \)\(12\!\cdots\!99\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(27\!\cdots\!65\)\( \nu^{15} + \)\(13\!\cdots\!43\)\( \nu^{14} - \)\(23\!\cdots\!00\)\( \nu^{13} + \)\(35\!\cdots\!02\)\( \nu^{12} - \)\(20\!\cdots\!93\)\( \nu^{11} + \)\(62\!\cdots\!50\)\( \nu^{10} - \)\(21\!\cdots\!61\)\( \nu^{9} + \)\(29\!\cdots\!86\)\( \nu^{8} - \)\(12\!\cdots\!57\)\( \nu^{7} + \)\(23\!\cdots\!74\)\( \nu^{6} - \)\(26\!\cdots\!24\)\( \nu^{5} + \)\(47\!\cdots\!51\)\( \nu^{4} - \)\(23\!\cdots\!61\)\( \nu^{3} + \)\(45\!\cdots\!76\)\( \nu^{2} - \)\(45\!\cdots\!28\)\( \nu + \)\(68\!\cdots\!96\)\(\)\()/ \)\(81\!\cdots\!36\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{15} - 43 \beta_{8} + \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-2 \beta_{14} - 4 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} - 12 \beta_{8} + 34 \beta_{7} - 28 \beta_{6} - 77 \beta_{3} + 16 \beta_{2} + 16 \beta_{1} - 16\)
\(\nu^{4}\)\(=\)\(2 \beta_{15} - 87 \beta_{14} + 2 \beta_{13} + 4 \beta_{11} - 12 \beta_{10} - 4 \beta_{9} + 135 \beta_{8} - 833 \beta_{7} - 2576 \beta_{6} + 87 \beta_{5} - 107 \beta_{4} - 135 \beta_{3} + 135 \beta_{2} - 833\)
\(\nu^{5}\)\(=\)\(-14 \beta_{14} + 14 \beta_{13} + 336 \beta_{12} - 336 \beta_{11} + 16 \beta_{9} + 9007 \beta_{8} - 9007 \beta_{7} + 262 \beta_{5} + 1934 \beta_{4} + 4483 \beta_{2} - 1934 \beta_{1} - 3705\)
\(\nu^{6}\)\(=\)\(-724 \beta_{15} - 724 \beta_{14} + 7321 \beta_{13} + 1008 \beta_{12} - 1608 \beta_{11} + 1608 \beta_{10} + 1008 \beta_{9} + 272598 \beta_{8} - 191468 \beta_{7} + 191468 \beta_{6} + 8045 \beta_{5} + 14617 \beta_{4} + 14617 \beta_{3} - 5736 \beta_{1} - 272598\)
\(\nu^{7}\)\(=\)\(-4252 \beta_{15} + 25030 \beta_{13} + 3904 \beta_{12} + 25788 \beta_{10} + 25788 \beta_{9} - 282911 \beta_{8} + 353785 \beta_{7} + 518476 \beta_{6} + 4252 \beta_{5} + 353785 \beta_{4} + 549413 \beta_{3} - 353785 \beta_{2} - 549413 \beta_{1} - 164691\)
\(\nu^{8}\)\(=\)\(-622115 \beta_{15} + 105294 \beta_{14} - 57324 \beta_{12} + 161348 \beta_{11} - 57324 \beta_{10} + 6672519 \beta_{8} + 16558939 \beta_{7} + 8139610 \beta_{6} + 824020 \beta_{3} - 1467091 \beta_{2} - 1467091 \beta_{1} + 1467091\)
\(\nu^{9}\)\(=\)\(-2363932 \beta_{15} + 710642 \beta_{14} - 2363932 \beta_{13} + 2009960 \beta_{11} - 1484760 \beta_{10} - 2009960 \beta_{9} - 18818226 \beta_{8} + 72285022 \beta_{7} - 33616894 \beta_{6} - 710642 \beta_{5} - 28866391 \beta_{4} + 18818226 \beta_{3} - 18818226 \beta_{2} + 72285022\)
\(\nu^{10}\)\(=\)\(53543269 \beta_{14} - 53543269 \beta_{13} + 4603200 \beta_{12} - 4603200 \beta_{11} - 9980928 \beta_{9} - 738647557 \beta_{8} + 738647557 \beta_{7} - 65728725 \beta_{5} - 142452645 \beta_{4} + 40578149 \beta_{2} + 142452645 \beta_{1} + 1965217305\)
\(\nu^{11}\)\(=\)\(217581914 \beta_{15} + 217581914 \beta_{14} - 94872624 \beta_{13} - 58722752 \beta_{12} - 102105300 \beta_{11} + 102105300 \beta_{10} - 58722752 \beta_{9} - 2995735314 \beta_{8} - 2280079802 \beta_{7} + 2280079802 \beta_{6} - 312454538 \beta_{5} - 1774094784 \beta_{4} - 1774094784 \beta_{3} + 4176840669 \beta_{1} + 2995735314\)
\(\nu^{12}\)\(=\)\(4657355935 \beta_{15} - 1286346842 \beta_{13} - 929050408 \beta_{12} + 330009108 \beta_{10} + 330009108 \beta_{9} - 176806829869 \beta_{8} + 2019861251 \beta_{7} - 73142895322 \beta_{6} - 4657355935 \beta_{5} + 2019861251 \beta_{4} - 11596224332 \beta_{3} - 2019861251 \beta_{2} + 11596224332 \beta_{1} + 75162756573\)
\(\nu^{13}\)\(=\)\(11293502550 \beta_{15} - 19835222816 \beta_{14} - 13154027264 \beta_{12} + 7079589488 \beta_{11} - 13154027264 \beta_{10} - 344191010582 \beta_{8} - 30939301040 \beta_{7} - 509884005868 \beta_{6} - 368505500737 \beta_{3} + 165692995286 \beta_{2} + 165692995286 \beta_{1} - 165692995286\)
\(\nu^{14}\)\(=\)\(129838068780 \beta_{15} - 408574340305 \beta_{14} + 129838068780 \beta_{13} + 21012853800 \beta_{11} - 106428182840 \beta_{10} - 21012853800 \beta_{9} + 1291350775377 \beta_{8} - 8094359895355 \beta_{7} - 8541029006908 \beta_{6} + 408574340305 \beta_{5} - 37642698937 \beta_{4} - 1291350775377 \beta_{3} + 1291350775377 \beta_{2} - 8094359895355\)
\(\nu^{15}\)\(=\)\(-1256105108580 \beta_{14} + 1256105108580 \beta_{13} + 1093932232300 \beta_{12} - 1093932232300 \beta_{11} + 604767604160 \beta_{9} + 66067478443437 \beta_{8} - 66067478443437 \beta_{7} + 3060061374962 \beta_{5} + 15404294453876 \beta_{4} + 17305426183777 \beta_{2} - 15404294453876 \beta_{1} - 60157769831531\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
2.73971 + 8.43195i
1.27432 + 3.92196i
−1.19154 3.66717i
−2.13151 6.56011i
2.73971 8.43195i
1.27432 3.92196i
−1.19154 + 3.66717i
−2.13151 + 6.56011i
7.81150 + 5.67539i
1.13051 + 0.821361i
0.136182 + 0.0989419i
−7.26917 5.28136i
7.81150 5.67539i
1.13051 0.821361i
0.136182 0.0989419i
−7.26917 + 5.28136i
−6.36363 + 4.62345i −5.98507 18.4202i 9.23097 28.4100i −59.3249 43.1020i 123.251 + 89.5474i −51.6143 + 158.852i −5.17243 15.9191i −106.890 + 77.6601i 576.801
3.2 −2.52720 + 1.83612i 4.03595 + 12.4214i −6.87313 + 21.1533i 23.3906 + 16.9942i −33.0068 23.9808i 27.3683 84.2310i −52.3600 161.147i 58.5894 42.5677i −90.3160
3.3 3.92850 2.85422i −7.68717 23.6587i −2.60202 + 8.00819i 68.6567 + 49.8820i −97.7261 71.0022i 19.5777 60.2541i 60.6528 + 186.670i −304.049 + 220.905i 412.092
3.4 6.38938 4.64216i 3.63629 + 11.1914i 9.38602 28.8872i −51.9930 37.7751i 75.1856 + 54.6256i −33.5385 + 103.221i 3.96879 + 12.2147i 84.5674 61.4418i −507.561
4.1 −6.36363 4.62345i −5.98507 + 18.4202i 9.23097 + 28.4100i −59.3249 + 43.1020i 123.251 89.5474i −51.6143 158.852i −5.17243 + 15.9191i −106.890 77.6601i 576.801
4.2 −2.52720 1.83612i 4.03595 12.4214i −6.87313 21.1533i 23.3906 16.9942i −33.0068 + 23.9808i 27.3683 + 84.2310i −52.3600 + 161.147i 58.5894 + 42.5677i −90.3160
4.3 3.92850 + 2.85422i −7.68717 + 23.6587i −2.60202 8.00819i 68.6567 49.8820i −97.7261 + 71.0022i 19.5777 + 60.2541i 60.6528 186.670i −304.049 220.905i 412.092
4.4 6.38938 + 4.64216i 3.63629 11.1914i 9.38602 + 28.8872i −51.9930 + 37.7751i 75.1856 54.6256i −33.5385 103.221i 3.96879 12.2147i 84.5674 + 61.4418i −507.561
5.1 −3.29275 10.1340i 2.26410 + 1.64496i −65.9678 + 47.9284i 20.4388 62.9043i 9.21501 28.3609i 58.3278 42.3777i 427.066 + 310.282i −72.6709 223.658i −704.774
5.2 −0.740832 2.28005i −21.7558 15.8065i 21.2388 15.4309i −5.23817 + 16.1214i −19.9222 + 61.3143i 87.7560 63.7585i −112.982 82.0864i 148.378 + 456.662i 40.6382
5.3 −0.361034 1.11115i 12.7716 + 9.27912i 24.7842 18.0068i 1.31139 4.03604i 5.69949 17.5412i −146.551 + 106.476i −59.2025 43.0132i 1.92087 + 5.91183i −4.95810
5.4 2.46756 + 7.59437i 0.720120 + 0.523198i −25.6970 + 18.6700i −2.24156 + 6.89880i −2.19642 + 6.75988i 136.674 99.2995i 1.52919 + 1.11102i −74.8463 230.353i −57.9232
9.1 −3.29275 + 10.1340i 2.26410 1.64496i −65.9678 47.9284i 20.4388 + 62.9043i 9.21501 + 28.3609i 58.3278 + 42.3777i 427.066 310.282i −72.6709 + 223.658i −704.774
9.2 −0.740832 + 2.28005i −21.7558 + 15.8065i 21.2388 + 15.4309i −5.23817 16.1214i −19.9222 61.3143i 87.7560 + 63.7585i −112.982 + 82.0864i 148.378 456.662i 40.6382
9.3 −0.361034 + 1.11115i 12.7716 9.27912i 24.7842 + 18.0068i 1.31139 + 4.03604i 5.69949 + 17.5412i −146.551 106.476i −59.2025 + 43.0132i 1.92087 5.91183i −4.95810
9.4 2.46756 7.59437i 0.720120 0.523198i −25.6970 18.6700i −2.24156 6.89880i −2.19642 6.75988i 136.674 + 99.2995i 1.52919 1.11102i −74.8463 + 230.353i −57.9232
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.6.c.a 16
3.b odd 2 1 99.6.f.a 16
11.c even 5 1 inner 11.6.c.a 16
11.c even 5 1 121.6.a.g 8
11.d odd 10 1 121.6.a.i 8
33.f even 10 1 1089.6.a.bb 8
33.h odd 10 1 99.6.f.a 16
33.h odd 10 1 1089.6.a.bg 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.c.a 16 1.a even 1 1 trivial
11.6.c.a 16 11.c even 5 1 inner
99.6.f.a 16 3.b odd 2 1
99.6.f.a 16 33.h odd 10 1
121.6.a.g 8 11.c even 5 1
121.6.a.i 8 11.d odd 10 1
1089.6.a.bb 8 33.f even 10 1
1089.6.a.bg 8 33.h odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(11, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T - 27 T^{2} - 107 T^{3} - 1419 T^{4} + 8820 T^{5} + 87252 T^{6} + 263536 T^{7} - 340032 T^{8} - 22378176 T^{9} - 80633472 T^{10} - 24058624 T^{11} + 3920778752 T^{12} + 27633161216 T^{13} + 33187021824 T^{14} - 453607768064 T^{15} - 5295115431936 T^{16} - 14515448578048 T^{17} + 33983510347776 T^{18} + 905483426725888 T^{19} + 4111234500657152 T^{20} - 807273463021568 T^{21} - 86579531300732928 T^{22} - 768908272513056768 T^{23} - 373869137815928832 T^{24} + 9272348682802429952 T^{25} + 98237018671832629248 T^{26} + \)\(31\!\cdots\!60\)\( T^{27} - \)\(16\!\cdots\!44\)\( T^{28} - \)\(39\!\cdots\!24\)\( T^{29} - \)\(31\!\cdots\!48\)\( T^{30} + \)\(37\!\cdots\!68\)\( T^{31} + \)\(12\!\cdots\!76\)\( T^{32} \)
$3$ \( 1 + 24 T + 67 T^{2} - 7584 T^{3} - 59024 T^{4} + 902472 T^{5} - 5098832 T^{6} - 823238520 T^{7} - 7438218224 T^{8} + 149244259392 T^{9} + 2932904949129 T^{10} + 100988556408 T^{11} - 211038704936325 T^{12} + 5168526851861856 T^{13} + 100693217419097472 T^{14} - 1083983071988425824 T^{15} - 45946119917427802848 T^{16} - \)\(26\!\cdots\!32\)\( T^{17} + \)\(59\!\cdots\!28\)\( T^{18} + \)\(74\!\cdots\!92\)\( T^{19} - \)\(73\!\cdots\!25\)\( T^{20} + \)\(85\!\cdots\!44\)\( T^{21} + \)\(60\!\cdots\!21\)\( T^{22} + \)\(74\!\cdots\!44\)\( T^{23} - \)\(90\!\cdots\!24\)\( T^{24} - \)\(24\!\cdots\!60\)\( T^{25} - \)\(36\!\cdots\!68\)\( T^{26} + \)\(15\!\cdots\!04\)\( T^{27} - \)\(25\!\cdots\!24\)\( T^{28} - \)\(78\!\cdots\!12\)\( T^{29} + \)\(16\!\cdots\!83\)\( T^{30} + \)\(14\!\cdots\!68\)\( T^{31} + \)\(14\!\cdots\!01\)\( T^{32} \)
$5$ \( 1 + 10 T - 14043 T^{2} - 180020 T^{3} + 75055968 T^{4} + 1848164610 T^{5} - 100342386816 T^{6} - 10842868406750 T^{7} - 789417562199820 T^{8} + 23704907416159500 T^{9} + 4246287590163916179 T^{10} + \)\(10\!\cdots\!90\)\( T^{11} - \)\(60\!\cdots\!77\)\( T^{12} - \)\(80\!\cdots\!80\)\( T^{13} - \)\(20\!\cdots\!88\)\( T^{14} + \)\(14\!\cdots\!40\)\( T^{15} + \)\(11\!\cdots\!96\)\( T^{16} + \)\(44\!\cdots\!00\)\( T^{17} - \)\(19\!\cdots\!00\)\( T^{18} - \)\(24\!\cdots\!00\)\( T^{19} - \)\(57\!\cdots\!25\)\( T^{20} + \)\(30\!\cdots\!50\)\( T^{21} + \)\(39\!\cdots\!75\)\( T^{22} + \)\(68\!\cdots\!00\)\( T^{23} - \)\(71\!\cdots\!00\)\( T^{24} - \)\(30\!\cdots\!50\)\( T^{25} - \)\(89\!\cdots\!00\)\( T^{26} + \)\(51\!\cdots\!50\)\( T^{27} + \)\(65\!\cdots\!00\)\( T^{28} - \)\(48\!\cdots\!00\)\( T^{29} - \)\(11\!\cdots\!75\)\( T^{30} + \)\(26\!\cdots\!50\)\( T^{31} + \)\(82\!\cdots\!25\)\( T^{32} \)
$7$ \( 1 - 196 T - 30529 T^{2} + 7524960 T^{3} + 181603532 T^{4} - 65266983916 T^{5} + 1646601296492 T^{6} - 2264192257484276 T^{7} + 153203956518588668 T^{8} + 71959842929557303168 T^{9} - \)\(69\!\cdots\!11\)\( T^{10} - \)\(58\!\cdots\!36\)\( T^{11} + \)\(70\!\cdots\!87\)\( T^{12} - \)\(70\!\cdots\!88\)\( T^{13} + \)\(17\!\cdots\!96\)\( T^{14} + \)\(10\!\cdots\!92\)\( T^{15} - \)\(58\!\cdots\!88\)\( T^{16} + \)\(18\!\cdots\!44\)\( T^{17} + \)\(50\!\cdots\!04\)\( T^{18} - \)\(33\!\cdots\!84\)\( T^{19} + \)\(55\!\cdots\!87\)\( T^{20} - \)\(78\!\cdots\!52\)\( T^{21} - \)\(15\!\cdots\!39\)\( T^{22} + \)\(27\!\cdots\!24\)\( T^{23} + \)\(97\!\cdots\!68\)\( T^{24} - \)\(24\!\cdots\!32\)\( T^{25} + \)\(29\!\cdots\!08\)\( T^{26} - \)\(19\!\cdots\!88\)\( T^{27} + \)\(92\!\cdots\!32\)\( T^{28} + \)\(64\!\cdots\!20\)\( T^{29} - \)\(43\!\cdots\!21\)\( T^{30} - \)\(47\!\cdots\!28\)\( T^{31} + \)\(40\!\cdots\!01\)\( T^{32} \)
$11$ \( 1 + 692 T - 42911 T^{2} - 151678824 T^{3} - 24749320761 T^{4} + 8945461252640 T^{5} + 2054880279217667 T^{6} + 655572739411977772 T^{7} + \)\(54\!\cdots\!48\)\( T^{8} + \)\(10\!\cdots\!72\)\( T^{9} + \)\(53\!\cdots\!67\)\( T^{10} + \)\(37\!\cdots\!40\)\( T^{11} - \)\(16\!\cdots\!61\)\( T^{12} - \)\(16\!\cdots\!24\)\( T^{13} - \)\(74\!\cdots\!11\)\( T^{14} + \)\(19\!\cdots\!92\)\( T^{15} + \)\(45\!\cdots\!01\)\( T^{16} \)
$13$ \( 1 - 1162 T - 94551 T^{2} + 579290788 T^{3} + 146427987596 T^{4} - 600870003540266 T^{5} + 125052452572816140 T^{6} + \)\(30\!\cdots\!46\)\( T^{7} - \)\(11\!\cdots\!28\)\( T^{8} - \)\(13\!\cdots\!64\)\( T^{9} + \)\(84\!\cdots\!75\)\( T^{10} + \)\(52\!\cdots\!34\)\( T^{11} - \)\(50\!\cdots\!93\)\( T^{12} - \)\(13\!\cdots\!68\)\( T^{13} + \)\(22\!\cdots\!84\)\( T^{14} + \)\(19\!\cdots\!00\)\( T^{15} - \)\(91\!\cdots\!84\)\( T^{16} + \)\(73\!\cdots\!00\)\( T^{17} + \)\(31\!\cdots\!16\)\( T^{18} - \)\(69\!\cdots\!76\)\( T^{19} - \)\(95\!\cdots\!93\)\( T^{20} + \)\(37\!\cdots\!62\)\( T^{21} + \)\(22\!\cdots\!75\)\( T^{22} - \)\(13\!\cdots\!48\)\( T^{23} - \)\(42\!\cdots\!28\)\( T^{24} + \)\(40\!\cdots\!78\)\( T^{25} + \)\(62\!\cdots\!60\)\( T^{26} - \)\(11\!\cdots\!62\)\( T^{27} + \)\(10\!\cdots\!96\)\( T^{28} + \)\(14\!\cdots\!84\)\( T^{29} - \)\(89\!\cdots\!99\)\( T^{30} - \)\(40\!\cdots\!34\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \)
$17$ \( 1 + 22 T - 4363527 T^{2} + 2994827912 T^{3} + 8715016018952 T^{4} - 7089763996838630 T^{5} - 2533784024596431108 T^{6} + \)\(31\!\cdots\!86\)\( T^{7} - \)\(40\!\cdots\!24\)\( T^{8} + \)\(31\!\cdots\!52\)\( T^{9} - \)\(84\!\cdots\!97\)\( T^{10} - \)\(30\!\cdots\!66\)\( T^{11} + \)\(27\!\cdots\!07\)\( T^{12} - \)\(24\!\cdots\!92\)\( T^{13} + \)\(63\!\cdots\!68\)\( T^{14} + \)\(53\!\cdots\!56\)\( T^{15} - \)\(17\!\cdots\!84\)\( T^{16} + \)\(75\!\cdots\!92\)\( T^{17} + \)\(12\!\cdots\!32\)\( T^{18} - \)\(69\!\cdots\!56\)\( T^{19} + \)\(11\!\cdots\!07\)\( T^{20} - \)\(17\!\cdots\!62\)\( T^{21} - \)\(69\!\cdots\!53\)\( T^{22} + \)\(36\!\cdots\!36\)\( T^{23} - \)\(66\!\cdots\!24\)\( T^{24} + \)\(74\!\cdots\!02\)\( T^{25} - \)\(84\!\cdots\!92\)\( T^{26} - \)\(33\!\cdots\!90\)\( T^{27} + \)\(58\!\cdots\!52\)\( T^{28} + \)\(28\!\cdots\!84\)\( T^{29} - \)\(59\!\cdots\!23\)\( T^{30} + \)\(42\!\cdots\!46\)\( T^{31} + \)\(27\!\cdots\!01\)\( T^{32} \)
$19$ \( 1 + 3236 T + 2242450 T^{2} - 5377355164 T^{3} - 6608264663933 T^{4} + 17695395326312452 T^{5} + 42919893614145313944 T^{6} - \)\(12\!\cdots\!12\)\( T^{7} - \)\(79\!\cdots\!53\)\( T^{8} + \)\(63\!\cdots\!04\)\( T^{9} + \)\(19\!\cdots\!08\)\( T^{10} - \)\(18\!\cdots\!68\)\( T^{11} - \)\(43\!\cdots\!59\)\( T^{12} + \)\(10\!\cdots\!68\)\( T^{13} + \)\(31\!\cdots\!86\)\( T^{14} + \)\(74\!\cdots\!36\)\( T^{15} - \)\(65\!\cdots\!88\)\( T^{16} + \)\(18\!\cdots\!64\)\( T^{17} + \)\(19\!\cdots\!86\)\( T^{18} + \)\(15\!\cdots\!32\)\( T^{19} - \)\(16\!\cdots\!59\)\( T^{20} - \)\(16\!\cdots\!32\)\( T^{21} + \)\(45\!\cdots\!08\)\( T^{22} + \)\(36\!\cdots\!96\)\( T^{23} - \)\(11\!\cdots\!53\)\( T^{24} - \)\(43\!\cdots\!88\)\( T^{25} + \)\(37\!\cdots\!44\)\( T^{26} + \)\(37\!\cdots\!48\)\( T^{27} - \)\(35\!\cdots\!33\)\( T^{28} - \)\(70\!\cdots\!36\)\( T^{29} + \)\(73\!\cdots\!50\)\( T^{30} + \)\(26\!\cdots\!64\)\( T^{31} + \)\(19\!\cdots\!01\)\( T^{32} \)
$23$ \( ( 1 + 5424 T + 55274044 T^{2} + 223055086704 T^{3} + 1278967597796756 T^{4} + 4059196611960971568 T^{5} + \)\(16\!\cdots\!28\)\( T^{6} + \)\(42\!\cdots\!04\)\( T^{7} + \)\(13\!\cdots\!98\)\( T^{8} + \)\(27\!\cdots\!72\)\( T^{9} + \)\(68\!\cdots\!72\)\( T^{10} + \)\(10\!\cdots\!76\)\( T^{11} + \)\(21\!\cdots\!56\)\( T^{12} + \)\(24\!\cdots\!72\)\( T^{13} + \)\(39\!\cdots\!56\)\( T^{14} + \)\(24\!\cdots\!68\)\( T^{15} + \)\(29\!\cdots\!01\)\( T^{16} )^{2} \)
$29$ \( 1 - 13070 T + 47857613 T^{2} + 326036461272 T^{3} - 3615673984576960 T^{4} + 10457250674731621870 T^{5} + \)\(18\!\cdots\!00\)\( T^{6} - \)\(17\!\cdots\!94\)\( T^{7} + \)\(53\!\cdots\!04\)\( T^{8} - \)\(39\!\cdots\!36\)\( T^{9} + \)\(22\!\cdots\!23\)\( T^{10} + \)\(76\!\cdots\!14\)\( T^{11} - \)\(39\!\cdots\!21\)\( T^{12} + \)\(17\!\cdots\!72\)\( T^{13} + \)\(58\!\cdots\!92\)\( T^{14} + \)\(19\!\cdots\!56\)\( T^{15} - \)\(28\!\cdots\!24\)\( T^{16} + \)\(39\!\cdots\!44\)\( T^{17} + \)\(24\!\cdots\!92\)\( T^{18} + \)\(14\!\cdots\!28\)\( T^{19} - \)\(70\!\cdots\!21\)\( T^{20} + \)\(27\!\cdots\!86\)\( T^{21} + \)\(16\!\cdots\!23\)\( T^{22} - \)\(60\!\cdots\!64\)\( T^{23} + \)\(16\!\cdots\!04\)\( T^{24} - \)\(11\!\cdots\!06\)\( T^{25} + \)\(24\!\cdots\!00\)\( T^{26} + \)\(28\!\cdots\!30\)\( T^{27} - \)\(20\!\cdots\!60\)\( T^{28} + \)\(37\!\cdots\!28\)\( T^{29} + \)\(11\!\cdots\!13\)\( T^{30} - \)\(62\!\cdots\!30\)\( T^{31} + \)\(98\!\cdots\!01\)\( T^{32} \)
$31$ \( 1 + 14764 T + 23868303 T^{2} - 736989794128 T^{3} - 4108642278550732 T^{4} + 13702752669168442004 T^{5} + \)\(19\!\cdots\!64\)\( T^{6} + \)\(41\!\cdots\!60\)\( T^{7} - \)\(23\!\cdots\!76\)\( T^{8} - \)\(16\!\cdots\!24\)\( T^{9} - \)\(56\!\cdots\!07\)\( T^{10} - \)\(25\!\cdots\!32\)\( T^{11} + \)\(90\!\cdots\!75\)\( T^{12} + \)\(25\!\cdots\!48\)\( T^{13} + \)\(11\!\cdots\!12\)\( T^{14} - \)\(43\!\cdots\!36\)\( T^{15} - \)\(59\!\cdots\!92\)\( T^{16} - \)\(12\!\cdots\!36\)\( T^{17} + \)\(95\!\cdots\!12\)\( T^{18} + \)\(60\!\cdots\!48\)\( T^{19} + \)\(60\!\cdots\!75\)\( T^{20} - \)\(48\!\cdots\!32\)\( T^{21} - \)\(31\!\cdots\!07\)\( T^{22} - \)\(25\!\cdots\!24\)\( T^{23} - \)\(10\!\cdots\!76\)\( T^{24} + \)\(53\!\cdots\!60\)\( T^{25} + \)\(71\!\cdots\!64\)\( T^{26} + \)\(14\!\cdots\!04\)\( T^{27} - \)\(12\!\cdots\!32\)\( T^{28} - \)\(63\!\cdots\!28\)\( T^{29} + \)\(59\!\cdots\!03\)\( T^{30} + \)\(10\!\cdots\!64\)\( T^{31} + \)\(20\!\cdots\!01\)\( T^{32} \)
$37$ \( 1 - 4638 T - 119292171 T^{2} + 271516931476 T^{3} + 9768318205415568 T^{4} + 23159902741520564482 T^{5} - \)\(10\!\cdots\!96\)\( T^{6} - \)\(28\!\cdots\!78\)\( T^{7} + \)\(98\!\cdots\!12\)\( T^{8} + \)\(81\!\cdots\!96\)\( T^{9} - \)\(70\!\cdots\!13\)\( T^{10} + \)\(96\!\cdots\!26\)\( T^{11} + \)\(44\!\cdots\!51\)\( T^{12} + \)\(67\!\cdots\!72\)\( T^{13} - \)\(38\!\cdots\!24\)\( T^{14} - \)\(44\!\cdots\!16\)\( T^{15} + \)\(33\!\cdots\!44\)\( T^{16} - \)\(30\!\cdots\!12\)\( T^{17} - \)\(18\!\cdots\!76\)\( T^{18} + \)\(22\!\cdots\!96\)\( T^{19} + \)\(10\!\cdots\!51\)\( T^{20} + \)\(15\!\cdots\!82\)\( T^{21} - \)\(78\!\cdots\!37\)\( T^{22} + \)\(62\!\cdots\!28\)\( T^{23} + \)\(52\!\cdots\!12\)\( T^{24} - \)\(10\!\cdots\!46\)\( T^{25} - \)\(27\!\cdots\!04\)\( T^{26} + \)\(41\!\cdots\!26\)\( T^{27} + \)\(12\!\cdots\!68\)\( T^{28} + \)\(23\!\cdots\!32\)\( T^{29} - \)\(70\!\cdots\!79\)\( T^{30} - \)\(19\!\cdots\!34\)\( T^{31} + \)\(28\!\cdots\!01\)\( T^{32} \)
$41$ \( 1 + 14806 T + 116461329 T^{2} + 2057638142512 T^{3} + 26494339964805888 T^{4} - 71982306131587525038 T^{5} - \)\(91\!\cdots\!00\)\( T^{6} - \)\(10\!\cdots\!42\)\( T^{7} - \)\(45\!\cdots\!68\)\( T^{8} - \)\(37\!\cdots\!52\)\( T^{9} + \)\(16\!\cdots\!83\)\( T^{10} - \)\(24\!\cdots\!54\)\( T^{11} - \)\(11\!\cdots\!93\)\( T^{12} + \)\(39\!\cdots\!24\)\( T^{13} - \)\(13\!\cdots\!04\)\( T^{14} - \)\(46\!\cdots\!40\)\( T^{15} + \)\(21\!\cdots\!96\)\( T^{16} - \)\(53\!\cdots\!40\)\( T^{17} - \)\(18\!\cdots\!04\)\( T^{18} + \)\(61\!\cdots\!24\)\( T^{19} - \)\(21\!\cdots\!93\)\( T^{20} - \)\(51\!\cdots\!54\)\( T^{21} + \)\(40\!\cdots\!83\)\( T^{22} - \)\(10\!\cdots\!52\)\( T^{23} - \)\(14\!\cdots\!68\)\( T^{24} - \)\(41\!\cdots\!42\)\( T^{25} - \)\(39\!\cdots\!00\)\( T^{26} - \)\(36\!\cdots\!38\)\( T^{27} + \)\(15\!\cdots\!88\)\( T^{28} + \)\(13\!\cdots\!12\)\( T^{29} + \)\(91\!\cdots\!29\)\( T^{30} + \)\(13\!\cdots\!06\)\( T^{31} + \)\(10\!\cdots\!01\)\( T^{32} \)
$43$ \( ( 1 + 12188 T + 699074717 T^{2} + 6483619268984 T^{3} + 212300340532873551 T^{4} + \)\(15\!\cdots\!48\)\( T^{5} + \)\(40\!\cdots\!95\)\( T^{6} + \)\(26\!\cdots\!52\)\( T^{7} + \)\(63\!\cdots\!24\)\( T^{8} + \)\(38\!\cdots\!36\)\( T^{9} + \)\(88\!\cdots\!55\)\( T^{10} + \)\(49\!\cdots\!36\)\( T^{11} + \)\(99\!\cdots\!51\)\( T^{12} + \)\(44\!\cdots\!12\)\( T^{13} + \)\(70\!\cdots\!33\)\( T^{14} + \)\(18\!\cdots\!16\)\( T^{15} + \)\(21\!\cdots\!01\)\( T^{16} )^{2} \)
$47$ \( 1 - 40364 T - 192282453 T^{2} + 29504805032296 T^{3} - 228545046034285440 T^{4} - \)\(90\!\cdots\!64\)\( T^{5} + \)\(14\!\cdots\!72\)\( T^{6} + \)\(11\!\cdots\!56\)\( T^{7} - \)\(35\!\cdots\!88\)\( T^{8} + \)\(65\!\cdots\!12\)\( T^{9} + \)\(41\!\cdots\!01\)\( T^{10} - \)\(83\!\cdots\!92\)\( T^{11} + \)\(84\!\cdots\!75\)\( T^{12} + \)\(26\!\cdots\!80\)\( T^{13} - \)\(69\!\cdots\!16\)\( T^{14} - \)\(31\!\cdots\!28\)\( T^{15} + \)\(22\!\cdots\!32\)\( T^{16} - \)\(73\!\cdots\!96\)\( T^{17} - \)\(36\!\cdots\!84\)\( T^{18} + \)\(32\!\cdots\!40\)\( T^{19} + \)\(23\!\cdots\!75\)\( T^{20} - \)\(52\!\cdots\!44\)\( T^{21} + \)\(59\!\cdots\!49\)\( T^{22} + \)\(21\!\cdots\!16\)\( T^{23} - \)\(27\!\cdots\!88\)\( T^{24} + \)\(20\!\cdots\!92\)\( T^{25} + \)\(57\!\cdots\!28\)\( T^{26} - \)\(83\!\cdots\!52\)\( T^{27} - \)\(48\!\cdots\!40\)\( T^{28} + \)\(14\!\cdots\!72\)\( T^{29} - \)\(21\!\cdots\!97\)\( T^{30} - \)\(10\!\cdots\!52\)\( T^{31} + \)\(58\!\cdots\!01\)\( T^{32} \)
$53$ \( 1 + 11654 T - 311337711 T^{2} - 419332324384 T^{3} + 580390322613555156 T^{4} + \)\(59\!\cdots\!86\)\( T^{5} - \)\(26\!\cdots\!80\)\( T^{6} - \)\(24\!\cdots\!54\)\( T^{7} + \)\(19\!\cdots\!08\)\( T^{8} + \)\(12\!\cdots\!92\)\( T^{9} - \)\(99\!\cdots\!13\)\( T^{10} - \)\(17\!\cdots\!30\)\( T^{11} + \)\(52\!\cdots\!27\)\( T^{12} + \)\(61\!\cdots\!28\)\( T^{13} - \)\(24\!\cdots\!24\)\( T^{14} - \)\(41\!\cdots\!84\)\( T^{15} + \)\(10\!\cdots\!04\)\( T^{16} - \)\(17\!\cdots\!12\)\( T^{17} - \)\(42\!\cdots\!76\)\( T^{18} + \)\(45\!\cdots\!96\)\( T^{19} + \)\(15\!\cdots\!27\)\( T^{20} - \)\(21\!\cdots\!90\)\( T^{21} - \)\(53\!\cdots\!37\)\( T^{22} + \)\(28\!\cdots\!44\)\( T^{23} + \)\(18\!\cdots\!08\)\( T^{24} - \)\(96\!\cdots\!22\)\( T^{25} - \)\(42\!\cdots\!20\)\( T^{26} + \)\(40\!\cdots\!02\)\( T^{27} + \)\(16\!\cdots\!56\)\( T^{28} - \)\(50\!\cdots\!12\)\( T^{29} - \)\(15\!\cdots\!39\)\( T^{30} + \)\(24\!\cdots\!78\)\( T^{31} + \)\(87\!\cdots\!01\)\( T^{32} \)
$59$ \( 1 - 70804 T + 650956378 T^{2} + 77911949681268 T^{3} - 2676199163934843125 T^{4} + \)\(30\!\cdots\!04\)\( T^{5} - \)\(16\!\cdots\!64\)\( T^{6} - \)\(34\!\cdots\!60\)\( T^{7} + \)\(27\!\cdots\!19\)\( T^{8} - \)\(65\!\cdots\!44\)\( T^{9} - \)\(22\!\cdots\!08\)\( T^{10} + \)\(48\!\cdots\!08\)\( T^{11} - \)\(15\!\cdots\!95\)\( T^{12} + \)\(35\!\cdots\!28\)\( T^{13} - \)\(16\!\cdots\!18\)\( T^{14} - \)\(27\!\cdots\!84\)\( T^{15} + \)\(11\!\cdots\!04\)\( T^{16} - \)\(19\!\cdots\!16\)\( T^{17} - \)\(83\!\cdots\!18\)\( T^{18} + \)\(13\!\cdots\!72\)\( T^{19} - \)\(41\!\cdots\!95\)\( T^{20} + \)\(89\!\cdots\!92\)\( T^{21} - \)\(29\!\cdots\!08\)\( T^{22} - \)\(62\!\cdots\!56\)\( T^{23} + \)\(18\!\cdots\!19\)\( T^{24} - \)\(16\!\cdots\!40\)\( T^{25} - \)\(58\!\cdots\!64\)\( T^{26} + \)\(76\!\cdots\!96\)\( T^{27} - \)\(47\!\cdots\!25\)\( T^{28} + \)\(99\!\cdots\!32\)\( T^{29} + \)\(59\!\cdots\!78\)\( T^{30} - \)\(46\!\cdots\!96\)\( T^{31} + \)\(46\!\cdots\!01\)\( T^{32} \)
$61$ \( 1 + 31446 T - 822013127 T^{2} - 8661690172160 T^{3} + 2725823414585587092 T^{4} + \)\(42\!\cdots\!82\)\( T^{5} - \)\(26\!\cdots\!64\)\( T^{6} - \)\(18\!\cdots\!62\)\( T^{7} + \)\(38\!\cdots\!68\)\( T^{8} + \)\(25\!\cdots\!48\)\( T^{9} - \)\(35\!\cdots\!81\)\( T^{10} - \)\(41\!\cdots\!46\)\( T^{11} + \)\(39\!\cdots\!67\)\( T^{12} + \)\(15\!\cdots\!48\)\( T^{13} - \)\(29\!\cdots\!56\)\( T^{14} + \)\(10\!\cdots\!60\)\( T^{15} + \)\(32\!\cdots\!68\)\( T^{16} + \)\(87\!\cdots\!60\)\( T^{17} - \)\(21\!\cdots\!56\)\( T^{18} + \)\(94\!\cdots\!48\)\( T^{19} + \)\(20\!\cdots\!67\)\( T^{20} - \)\(17\!\cdots\!46\)\( T^{21} - \)\(12\!\cdots\!81\)\( T^{22} + \)\(76\!\cdots\!48\)\( T^{23} + \)\(98\!\cdots\!68\)\( T^{24} - \)\(40\!\cdots\!62\)\( T^{25} - \)\(49\!\cdots\!64\)\( T^{26} + \)\(67\!\cdots\!82\)\( T^{27} + \)\(35\!\cdots\!92\)\( T^{28} - \)\(96\!\cdots\!60\)\( T^{29} - \)\(77\!\cdots\!27\)\( T^{30} + \)\(24\!\cdots\!46\)\( T^{31} + \)\(67\!\cdots\!01\)\( T^{32} \)
$67$ \( ( 1 + 32100 T + 9230620865 T^{2} + 244904972239352 T^{3} + 38707469935726110055 T^{4} + \)\(85\!\cdots\!84\)\( T^{5} + \)\(97\!\cdots\!71\)\( T^{6} + \)\(18\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!56\)\( T^{8} + \)\(24\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!79\)\( T^{10} + \)\(21\!\cdots\!12\)\( T^{11} + \)\(12\!\cdots\!55\)\( T^{12} + \)\(10\!\cdots\!64\)\( T^{13} + \)\(55\!\cdots\!85\)\( T^{14} + \)\(26\!\cdots\!00\)\( T^{15} + \)\(11\!\cdots\!01\)\( T^{16} )^{2} \)
$71$ \( 1 + 184380 T + 9105023403 T^{2} - 197336579903880 T^{3} - 24287550568222943480 T^{4} - \)\(40\!\cdots\!60\)\( T^{5} - \)\(21\!\cdots\!24\)\( T^{6} - \)\(83\!\cdots\!20\)\( T^{7} + \)\(75\!\cdots\!36\)\( T^{8} + \)\(77\!\cdots\!20\)\( T^{9} - \)\(13\!\cdots\!67\)\( T^{10} + \)\(52\!\cdots\!20\)\( T^{11} + \)\(10\!\cdots\!63\)\( T^{12} - \)\(14\!\cdots\!40\)\( T^{13} + \)\(35\!\cdots\!44\)\( T^{14} + \)\(16\!\cdots\!80\)\( T^{15} - \)\(86\!\cdots\!52\)\( T^{16} + \)\(29\!\cdots\!80\)\( T^{17} + \)\(11\!\cdots\!44\)\( T^{18} - \)\(84\!\cdots\!40\)\( T^{19} + \)\(11\!\cdots\!63\)\( T^{20} + \)\(99\!\cdots\!20\)\( T^{21} - \)\(45\!\cdots\!67\)\( T^{22} + \)\(48\!\cdots\!20\)\( T^{23} + \)\(84\!\cdots\!36\)\( T^{24} - \)\(16\!\cdots\!20\)\( T^{25} - \)\(78\!\cdots\!24\)\( T^{26} - \)\(26\!\cdots\!60\)\( T^{27} - \)\(28\!\cdots\!80\)\( T^{28} - \)\(42\!\cdots\!80\)\( T^{29} + \)\(35\!\cdots\!03\)\( T^{30} + \)\(12\!\cdots\!80\)\( T^{31} + \)\(12\!\cdots\!01\)\( T^{32} \)
$73$ \( 1 + 1750 T - 9685718607 T^{2} - 212498950812584 T^{3} + 43813058702423151592 T^{4} + \)\(15\!\cdots\!34\)\( T^{5} - \)\(11\!\cdots\!52\)\( T^{6} - \)\(51\!\cdots\!14\)\( T^{7} + \)\(20\!\cdots\!12\)\( T^{8} + \)\(86\!\cdots\!36\)\( T^{9} - \)\(48\!\cdots\!49\)\( T^{10} - \)\(52\!\cdots\!86\)\( T^{11} + \)\(16\!\cdots\!67\)\( T^{12} - \)\(52\!\cdots\!80\)\( T^{13} - \)\(49\!\cdots\!88\)\( T^{14} + \)\(79\!\cdots\!72\)\( T^{15} + \)\(11\!\cdots\!64\)\( T^{16} + \)\(16\!\cdots\!96\)\( T^{17} - \)\(21\!\cdots\!12\)\( T^{18} - \)\(47\!\cdots\!60\)\( T^{19} + \)\(30\!\cdots\!67\)\( T^{20} - \)\(20\!\cdots\!98\)\( T^{21} - \)\(38\!\cdots\!01\)\( T^{22} + \)\(14\!\cdots\!52\)\( T^{23} + \)\(70\!\cdots\!12\)\( T^{24} - \)\(36\!\cdots\!02\)\( T^{25} - \)\(16\!\cdots\!48\)\( T^{26} + \)\(48\!\cdots\!38\)\( T^{27} + \)\(27\!\cdots\!92\)\( T^{28} - \)\(27\!\cdots\!12\)\( T^{29} - \)\(26\!\cdots\!43\)\( T^{30} + \)\(98\!\cdots\!50\)\( T^{31} + \)\(11\!\cdots\!01\)\( T^{32} \)
$79$ \( 1 - 24324 T - 12402045505 T^{2} + 275159567550064 T^{3} + 65785434916137516948 T^{4} - \)\(60\!\cdots\!48\)\( T^{5} - \)\(18\!\cdots\!32\)\( T^{6} - \)\(63\!\cdots\!40\)\( T^{7} + \)\(35\!\cdots\!12\)\( T^{8} + \)\(52\!\cdots\!80\)\( T^{9} - \)\(57\!\cdots\!31\)\( T^{10} - \)\(17\!\cdots\!40\)\( T^{11} - \)\(19\!\cdots\!49\)\( T^{12} + \)\(31\!\cdots\!76\)\( T^{13} + \)\(27\!\cdots\!16\)\( T^{14} - \)\(24\!\cdots\!92\)\( T^{15} - \)\(12\!\cdots\!40\)\( T^{16} - \)\(75\!\cdots\!08\)\( T^{17} + \)\(26\!\cdots\!16\)\( T^{18} + \)\(91\!\cdots\!24\)\( T^{19} - \)\(17\!\cdots\!49\)\( T^{20} - \)\(48\!\cdots\!60\)\( T^{21} - \)\(48\!\cdots\!31\)\( T^{22} + \)\(13\!\cdots\!20\)\( T^{23} + \)\(28\!\cdots\!12\)\( T^{24} - \)\(15\!\cdots\!60\)\( T^{25} - \)\(14\!\cdots\!32\)\( T^{26} - \)\(14\!\cdots\!52\)\( T^{27} + \)\(47\!\cdots\!48\)\( T^{28} + \)\(61\!\cdots\!36\)\( T^{29} - \)\(84\!\cdots\!05\)\( T^{30} - \)\(51\!\cdots\!76\)\( T^{31} + \)\(64\!\cdots\!01\)\( T^{32} \)
$83$ \( 1 + 46028 T - 1440977818 T^{2} - 472279336812396 T^{3} + 7484936123526321167 T^{4} + \)\(23\!\cdots\!36\)\( T^{5} + \)\(69\!\cdots\!52\)\( T^{6} - \)\(37\!\cdots\!04\)\( T^{7} + \)\(61\!\cdots\!31\)\( T^{8} - \)\(34\!\cdots\!88\)\( T^{9} - \)\(12\!\cdots\!48\)\( T^{10} - \)\(99\!\cdots\!88\)\( T^{11} + \)\(13\!\cdots\!97\)\( T^{12} + \)\(60\!\cdots\!04\)\( T^{13} - \)\(15\!\cdots\!10\)\( T^{14} - \)\(16\!\cdots\!32\)\( T^{15} + \)\(13\!\cdots\!48\)\( T^{16} - \)\(63\!\cdots\!76\)\( T^{17} - \)\(24\!\cdots\!90\)\( T^{18} + \)\(36\!\cdots\!28\)\( T^{19} + \)\(32\!\cdots\!97\)\( T^{20} - \)\(94\!\cdots\!84\)\( T^{21} - \)\(47\!\cdots\!52\)\( T^{22} - \)\(51\!\cdots\!16\)\( T^{23} + \)\(35\!\cdots\!31\)\( T^{24} - \)\(86\!\cdots\!72\)\( T^{25} + \)\(62\!\cdots\!48\)\( T^{26} + \)\(83\!\cdots\!52\)\( T^{27} + \)\(10\!\cdots\!67\)\( T^{28} - \)\(25\!\cdots\!28\)\( T^{29} - \)\(31\!\cdots\!82\)\( T^{30} + \)\(39\!\cdots\!96\)\( T^{31} + \)\(33\!\cdots\!01\)\( T^{32} \)
$89$ \( ( 1 - 74182 T + 30909376975 T^{2} - 1846418512823988 T^{3} + \)\(45\!\cdots\!91\)\( T^{4} - \)\(22\!\cdots\!92\)\( T^{5} + \)\(42\!\cdots\!45\)\( T^{6} - \)\(18\!\cdots\!10\)\( T^{7} + \)\(27\!\cdots\!76\)\( T^{8} - \)\(10\!\cdots\!90\)\( T^{9} + \)\(13\!\cdots\!45\)\( T^{10} - \)\(39\!\cdots\!08\)\( T^{11} + \)\(43\!\cdots\!91\)\( T^{12} - \)\(10\!\cdots\!12\)\( T^{13} + \)\(93\!\cdots\!75\)\( T^{14} - \)\(12\!\cdots\!18\)\( T^{15} + \)\(94\!\cdots\!01\)\( T^{16} )^{2} \)
$97$ \( 1 - 484296 T + 85937368298 T^{2} - 5038221560960800 T^{3} - \)\(39\!\cdots\!13\)\( T^{4} + \)\(77\!\cdots\!88\)\( T^{5} - \)\(26\!\cdots\!88\)\( T^{6} - \)\(51\!\cdots\!72\)\( T^{7} + \)\(68\!\cdots\!99\)\( T^{8} + \)\(11\!\cdots\!72\)\( T^{9} - \)\(88\!\cdots\!96\)\( T^{10} + \)\(59\!\cdots\!56\)\( T^{11} + \)\(18\!\cdots\!01\)\( T^{12} - \)\(59\!\cdots\!72\)\( T^{13} + \)\(41\!\cdots\!14\)\( T^{14} + \)\(17\!\cdots\!44\)\( T^{15} - \)\(49\!\cdots\!68\)\( T^{16} + \)\(15\!\cdots\!08\)\( T^{17} + \)\(30\!\cdots\!86\)\( T^{18} - \)\(37\!\cdots\!96\)\( T^{19} + \)\(10\!\cdots\!01\)\( T^{20} + \)\(27\!\cdots\!92\)\( T^{21} - \)\(35\!\cdots\!04\)\( T^{22} + \)\(40\!\cdots\!96\)\( T^{23} + \)\(20\!\cdots\!99\)\( T^{24} - \)\(13\!\cdots\!04\)\( T^{25} - \)\(58\!\cdots\!12\)\( T^{26} + \)\(14\!\cdots\!84\)\( T^{27} - \)\(64\!\cdots\!13\)\( T^{28} - \)\(69\!\cdots\!00\)\( T^{29} + \)\(10\!\cdots\!02\)\( T^{30} - \)\(49\!\cdots\!28\)\( T^{31} + \)\(87\!\cdots\!01\)\( T^{32} \)
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