Properties

Label 33.6.e.a
Level $33$
Weight $6$
Character orbit 33.e
Analytic conductor $5.293$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 33.e (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.29266605383\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 3 x^{19} + 142 x^{18} - 389 x^{17} + 14927 x^{16} - 6599 x^{15} + 1399353 x^{14} + 1969563 x^{13} + 129076450 x^{12} + 14028509 x^{11} + 4158720649 x^{10} - 7877307064 x^{9} + 107078602608 x^{8} - 53496436514 x^{7} + 2527225278875 x^{6} - 6173773919508 x^{5} + 63231264058241 x^{4} - 118816044595930 x^{3} + 398041877081860 x^{2} - 161017247606600 x + 25735126080400\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 11^{2} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{7} ) q^{2} -9 \beta_{9} q^{3} + ( -\beta_{6} + 2 \beta_{7} - 20 \beta_{8} - 2 \beta_{9} - \beta_{15} ) q^{4} + ( -3 + \beta_{1} - 2 \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + 3 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{5} + ( -9 + 9 \beta_{1} - 9 \beta_{3} + 9 \beta_{5} + 9 \beta_{6} - 9 \beta_{7} + 9 \beta_{8} + 9 \beta_{9} ) q^{6} + ( -1 - 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - \beta_{6} + 8 \beta_{7} - 50 \beta_{8} - 7 \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} - 2 \beta_{15} + 2 \beta_{19} ) q^{7} + ( -19 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - 18 \beta_{5} - 19 \beta_{7} - 27 \beta_{9} - \beta_{10} - 3 \beta_{11} + 2 \beta_{13} + \beta_{14} + 4 \beta_{15} + 3 \beta_{16} + 3 \beta_{17} - \beta_{18} ) q^{8} + 81 \beta_{7} q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{7} ) q^{2} -9 \beta_{9} q^{3} + ( -\beta_{6} + 2 \beta_{7} - 20 \beta_{8} - 2 \beta_{9} - \beta_{15} ) q^{4} + ( -3 + \beta_{1} - 2 \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + 3 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{5} + ( -9 + 9 \beta_{1} - 9 \beta_{3} + 9 \beta_{5} + 9 \beta_{6} - 9 \beta_{7} + 9 \beta_{8} + 9 \beta_{9} ) q^{6} + ( -1 - 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - \beta_{6} + 8 \beta_{7} - 50 \beta_{8} - 7 \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} - 2 \beta_{15} + 2 \beta_{19} ) q^{7} + ( -19 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - 18 \beta_{5} - 19 \beta_{7} - 27 \beta_{9} - \beta_{10} - 3 \beta_{11} + 2 \beta_{13} + \beta_{14} + 4 \beta_{15} + 3 \beta_{16} + 3 \beta_{17} - \beta_{18} ) q^{8} + 81 \beta_{7} q^{9} + ( 95 + 3 \beta_{1} - 10 \beta_{2} - 10 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 4 \beta_{7} - 5 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - 4 \beta_{12} - 4 \beta_{13} - 5 \beta_{14} + \beta_{15} + 4 \beta_{17} - 3 \beta_{18} ) q^{10} + ( -2 + 6 \beta_{2} + 5 \beta_{3} + \beta_{4} + 22 \beta_{5} + 7 \beta_{6} + 64 \beta_{7} - 71 \beta_{8} - 50 \beta_{9} + \beta_{10} + 8 \beta_{11} + \beta_{12} + 5 \beta_{13} + 7 \beta_{14} - 4 \beta_{16} - 7 \beta_{17} - 3 \beta_{18} + 2 \beta_{19} ) q^{11} + ( 162 - 9 \beta_{2} + 9 \beta_{3} + 18 \beta_{8} + 18 \beta_{9} ) q^{12} + ( 21 + 32 \beta_{1} - 14 \beta_{2} + 10 \beta_{3} - 7 \beta_{4} - 5 \beta_{5} - 9 \beta_{6} + 236 \beta_{7} - 25 \beta_{8} - 3 \beta_{9} - \beta_{10} - 3 \beta_{11} - 7 \beta_{13} - 6 \beta_{14} - 3 \beta_{15} - 2 \beta_{16} + 3 \beta_{17} + 2 \beta_{18} - 7 \beta_{19} ) q^{13} + ( -207 - 12 \beta_{1} + 18 \beta_{2} + 18 \beta_{3} + 5 \beta_{4} - 74 \beta_{5} - 201 \beta_{7} + 4 \beta_{8} + 44 \beta_{9} - 3 \beta_{10} + 11 \beta_{11} + 4 \beta_{12} + 14 \beta_{13} + 12 \beta_{14} - 3 \beta_{15} + 5 \beta_{16} - 5 \beta_{17} - 3 \beta_{18} + 4 \beta_{19} ) q^{14} + ( -18 \beta_{6} - 27 \beta_{7} + 27 \beta_{9} + 9 \beta_{18} + 9 \beta_{19} ) q^{15} + ( -270 + 106 \beta_{1} - 4 \beta_{2} - 24 \beta_{3} + 17 \beta_{5} + 109 \beta_{6} - 265 \beta_{7} + 265 \beta_{8} + 270 \beta_{9} + 3 \beta_{10} - 14 \beta_{11} + 11 \beta_{12} + 4 \beta_{13} - 13 \beta_{14} + 11 \beta_{15} + 7 \beta_{16} - 3 \beta_{17} + 14 \beta_{18} + 3 \beta_{19} ) q^{16} + ( -379 - 21 \beta_{1} - 2 \beta_{2} - 37 \beta_{3} + 29 \beta_{5} - 15 \beta_{6} - 255 \beta_{7} + 255 \beta_{8} + 379 \beta_{9} + 6 \beta_{10} - 3 \beta_{12} - 27 \beta_{13} - 12 \beta_{14} - 19 \beta_{15} - 8 \beta_{16} + 10 \beta_{17} - 16 \beta_{18} - 10 \beta_{19} ) q^{17} + ( -81 \beta_{6} - 81 \beta_{8} ) q^{18} + ( 151 - 29 \beta_{1} - 28 \beta_{2} + 22 \beta_{3} - 22 \beta_{4} + 53 \beta_{5} + 144 \beta_{7} - 19 \beta_{8} - 631 \beta_{9} - 5 \beta_{10} - 29 \beta_{11} - 19 \beta_{12} - 9 \beta_{13} - 32 \beta_{14} + 16 \beta_{15} - 4 \beta_{16} + 22 \beta_{17} - 5 \beta_{18} - 19 \beta_{19} ) q^{19} + ( 131 - 205 \beta_{1} + 47 \beta_{2} + 4 \beta_{3} + 24 \beta_{4} + 21 \beta_{5} - 9 \beta_{6} - 147 \beta_{7} - 124 \beta_{8} + 17 \beta_{9} + 5 \beta_{10} + 17 \beta_{11} + 10 \beta_{13} + 19 \beta_{14} + 7 \beta_{15} + 3 \beta_{16} - 11 \beta_{17} - 4 \beta_{18} + 24 \beta_{19} ) q^{20} + ( 378 + 9 \beta_{1} + 9 \beta_{2} - 27 \beta_{3} + 18 \beta_{4} + 54 \beta_{5} + 36 \beta_{6} + 72 \beta_{8} + 90 \beta_{9} + 18 \beta_{11} + 18 \beta_{13} + 9 \beta_{14} - 9 \beta_{15} - 18 \beta_{16} - 9 \beta_{18} ) q^{21} + ( 728 + 42 \beta_{1} + 10 \beta_{2} - 179 \beta_{3} + \beta_{4} + 77 \beta_{5} + 36 \beta_{6} + 1063 \beta_{7} - 828 \beta_{8} - 545 \beta_{9} + 9 \beta_{10} - 21 \beta_{11} + 2 \beta_{12} - 38 \beta_{13} + 24 \beta_{14} + \beta_{15} + 15 \beta_{16} - \beta_{17} + 9 \beta_{18} - 16 \beta_{19} ) q^{22} + ( 554 - 26 \beta_{1} + 61 \beta_{2} + 206 \beta_{3} + 32 \beta_{4} - 142 \beta_{5} - 143 \beta_{6} - 32 \beta_{7} - 843 \beta_{8} - 863 \beta_{9} - 21 \beta_{10} - 20 \beta_{11} + 32 \beta_{12} + 71 \beta_{13} + 76 \beta_{14} - 6 \beta_{15} - \beta_{16} - 32 \beta_{17} + 26 \beta_{18} ) q^{23} + ( 180 - 135 \beta_{1} + 18 \beta_{2} + 9 \beta_{3} + 27 \beta_{4} + 9 \beta_{5} + 387 \beta_{7} - 171 \beta_{8} + 18 \beta_{9} - 9 \beta_{10} + 18 \beta_{11} - 9 \beta_{13} + 36 \beta_{14} - 27 \beta_{15} + 18 \beta_{16} - 27 \beta_{17} + 9 \beta_{18} + 27 \beta_{19} ) q^{24} + ( -395 - 97 \beta_{1} - 60 \beta_{2} + 118 \beta_{3} - 5 \beta_{4} - 33 \beta_{5} - 374 \beta_{7} - 36 \beta_{8} - 697 \beta_{9} - 11 \beta_{10} + 16 \beta_{11} - 36 \beta_{12} - 24 \beta_{13} - 59 \beta_{14} - 39 \beta_{15} - 10 \beta_{16} + 5 \beta_{17} - 11 \beta_{18} - 36 \beta_{19} ) q^{25} + ( 27 - 125 \beta_{1} + 5 \beta_{2} + 54 \beta_{3} - 32 \beta_{4} - 152 \beta_{5} - 330 \beta_{6} + 368 \beta_{7} + 1350 \beta_{8} - 395 \beta_{9} - 54 \beta_{10} - 27 \beta_{11} - 32 \beta_{12} + 27 \beta_{13} + 32 \beta_{14} + 117 \beta_{15} + 31 \beta_{16} + 14 \beta_{18} - 9 \beta_{19} ) q^{26} + ( -729 - 729 \beta_{7} + 729 \beta_{8} + 729 \beta_{9} ) q^{27} + ( -2728 + 307 \beta_{1} - 17 \beta_{2} - 40 \beta_{3} + 6 \beta_{5} + 324 \beta_{6} - 1993 \beta_{7} + 1993 \beta_{8} + 2728 \beta_{9} + 17 \beta_{10} - 28 \beta_{11} + 6 \beta_{12} - 45 \beta_{13} - 136 \beta_{14} - 11 \beta_{15} - 6 \beta_{16} + 23 \beta_{17} - 12 \beta_{18} - 23 \beta_{19} ) q^{28} + ( -16 + 179 \beta_{1} - 17 \beta_{2} - 32 \beta_{3} + 55 \beta_{4} + 195 \beta_{5} - 27 \beta_{6} - 872 \beta_{7} - 665 \beta_{8} + 888 \beta_{9} + 32 \beta_{10} + 16 \beta_{11} + 55 \beta_{12} - 16 \beta_{13} - 33 \beta_{14} - 59 \beta_{15} - 50 \beta_{16} + 35 \beta_{18} + 17 \beta_{19} ) q^{29} + ( -9 + 9 \beta_{1} + 126 \beta_{2} + 45 \beta_{4} + 108 \beta_{5} + 9 \beta_{8} - 819 \beta_{9} - 27 \beta_{10} + 54 \beta_{11} + 9 \beta_{12} + 117 \beta_{13} + 90 \beta_{14} + 27 \beta_{15} - 27 \beta_{16} - 45 \beta_{17} - 27 \beta_{18} + 9 \beta_{19} ) q^{30} + ( 694 - 142 \beta_{1} - 61 \beta_{2} + 193 \beta_{3} - 37 \beta_{4} + 28 \beta_{5} - 209 \beta_{6} + 284 \beta_{7} - 785 \beta_{8} + 54 \beta_{9} + 16 \beta_{10} + 54 \beta_{11} - 45 \beta_{13} - 53 \beta_{14} - 36 \beta_{15} - 65 \beta_{16} + 4 \beta_{17} + 26 \beta_{18} - 37 \beta_{19} ) q^{31} + ( 1526 + 22 \beta_{1} - 129 \beta_{2} + 129 \beta_{3} + 10 \beta_{4} + 312 \beta_{5} + 300 \beta_{6} + 34 \beta_{7} + 3746 \beta_{8} + 3756 \beta_{9} - 2 \beta_{10} + 10 \beta_{11} - 34 \beta_{12} - 58 \beta_{13} - 82 \beta_{14} + 12 \beta_{15} - 12 \beta_{16} + 34 \beta_{17} - 22 \beta_{18} ) q^{32} + ( 63 + 108 \beta_{1} + 36 \beta_{2} - 18 \beta_{3} + 9 \beta_{4} - 99 \beta_{5} - 18 \beta_{6} - 108 \beta_{7} + 639 \beta_{8} + 540 \beta_{9} - 27 \beta_{10} - 45 \beta_{11} + 54 \beta_{12} - 9 \beta_{13} + 36 \beta_{14} + 45 \beta_{15} + 27 \beta_{16} + 9 \beta_{17} + 18 \beta_{18} ) q^{33} + ( 1025 - 7 \beta_{1} - 61 \beta_{2} + 573 \beta_{3} - 179 \beta_{4} - 816 \beta_{5} - 724 \beta_{6} + 93 \beta_{7} - 2495 \beta_{8} - 2602 \beta_{9} - 15 \beta_{10} - 107 \beta_{11} - 93 \beta_{12} - 8 \beta_{13} - 16 \beta_{14} + 100 \beta_{15} + 92 \beta_{16} + 93 \beta_{17} + 7 \beta_{18} ) q^{34} + ( -1860 + 237 \beta_{1} - 45 \beta_{2} - 426 \beta_{3} + 35 \beta_{4} + 31 \beta_{5} + 450 \beta_{6} + 1176 \beta_{7} + 1815 \beta_{8} + 80 \beta_{9} - 24 \beta_{10} + 80 \beta_{11} - \beta_{13} + 59 \beta_{14} - 135 \beta_{15} + 4 \beta_{16} - 79 \beta_{17} + 49 \beta_{18} + 35 \beta_{19} ) q^{35} + ( -162 + 81 \beta_{2} - 81 \beta_{5} - 162 \beta_{7} - 1458 \beta_{9} + 81 \beta_{13} + 81 \beta_{14} + 81 \beta_{15} ) q^{36} + ( -19 - 702 \beta_{1} - 41 \beta_{2} - 38 \beta_{3} + 49 \beta_{4} - 683 \beta_{5} + 24 \beta_{6} + 446 \beta_{7} + 101 \beta_{8} - 427 \beta_{9} + 38 \beta_{10} + 19 \beta_{11} + 49 \beta_{12} - 19 \beta_{13} - 60 \beta_{14} + 13 \beta_{15} - 32 \beta_{16} - 52 \beta_{18} - 46 \beta_{19} ) q^{37} + ( 2813 + 509 \beta_{1} + 64 \beta_{2} - 463 \beta_{3} + 536 \beta_{5} + 500 \beta_{6} + 4287 \beta_{7} - 4287 \beta_{8} - 2813 \beta_{9} - 9 \beta_{10} + 108 \beta_{11} - 71 \beta_{12} + 161 \beta_{13} + 121 \beta_{14} + 88 \beta_{15} - 35 \beta_{16} - 29 \beta_{17} - 70 \beta_{18} + 29 \beta_{19} ) q^{38} + ( -1998 - 288 \beta_{1} + 72 \beta_{2} + 333 \beta_{3} - 279 \beta_{5} - 270 \beta_{6} - 2160 \beta_{7} + 2160 \beta_{8} + 1998 \beta_{9} + 18 \beta_{10} + 63 \beta_{11} + 36 \beta_{12} + 45 \beta_{13} + 45 \beta_{14} - 9 \beta_{15} - 9 \beta_{16} - 63 \beta_{17} - 18 \beta_{18} + 63 \beta_{19} ) q^{39} + ( -38 + 301 \beta_{1} + 3 \beta_{2} - 76 \beta_{3} - 62 \beta_{4} + 339 \beta_{5} - 93 \beta_{6} + 1574 \beta_{7} - 7494 \beta_{8} - 1536 \beta_{9} + 76 \beta_{10} + 38 \beta_{11} - 62 \beta_{12} - 38 \beta_{13} - 35 \beta_{14} - 118 \beta_{15} + 52 \beta_{16} - 94 \beta_{18} + 34 \beta_{19} ) q^{40} + ( -109 + 471 \beta_{1} - 176 \beta_{2} - 447 \beta_{3} - 42 \beta_{4} + 981 \beta_{5} - 85 \beta_{7} + 11 \beta_{8} - 1631 \beta_{9} + 4 \beta_{10} - 18 \beta_{11} + 11 \beta_{12} - 187 \beta_{13} - 98 \beta_{14} - 91 \beta_{15} + 77 \beta_{16} + 42 \beta_{17} + 4 \beta_{18} + 11 \beta_{19} ) q^{41} + ( 1890 - 630 \beta_{1} - 63 \beta_{2} - 108 \beta_{3} - 9 \beta_{4} - 99 \beta_{5} + 135 \beta_{6} + 1512 \beta_{7} - 1773 \beta_{8} - 126 \beta_{9} - 27 \beta_{10} - 126 \beta_{11} - 99 \beta_{13} + 18 \beta_{14} + 18 \beta_{15} + 90 \beta_{16} + 45 \beta_{17} - 27 \beta_{18} - 9 \beta_{19} ) q^{42} + ( 1430 - 143 \beta_{1} + 41 \beta_{2} + 770 \beta_{3} - 87 \beta_{4} + 436 \beta_{5} + 569 \beta_{6} - 122 \beta_{7} + 3316 \beta_{8} + 3152 \beta_{9} - 31 \beta_{10} - 164 \beta_{11} + 122 \beta_{12} + 36 \beta_{13} + 148 \beta_{14} + 21 \beta_{15} + 133 \beta_{16} - 122 \beta_{17} + 143 \beta_{18} ) q^{43} + ( 5170 + 663 \beta_{1} + 31 \beta_{2} - 1103 \beta_{3} - 58 \beta_{4} - 55 \beta_{5} - 763 \beta_{6} - 1597 \beta_{7} - 5587 \beta_{8} - 4668 \beta_{9} + 117 \beta_{10} + 14 \beta_{11} - 54 \beta_{12} + 110 \beta_{13} + 47 \beta_{14} + 37 \beta_{15} - 44 \beta_{16} + 111 \beta_{17} - 42 \beta_{18} + 87 \beta_{19} ) q^{44} + ( 162 + 81 \beta_{1} + 81 \beta_{2} + 81 \beta_{4} + 162 \beta_{5} + 81 \beta_{6} - 243 \beta_{8} - 81 \beta_{9} + 81 \beta_{10} + 162 \beta_{11} - 81 \beta_{15} - 81 \beta_{16} - 81 \beta_{18} ) q^{45} + ( -6065 + 1168 \beta_{1} - 152 \beta_{2} - 2509 \beta_{3} - 142 \beta_{4} - 83 \beta_{5} + 2364 \beta_{6} + 1134 \beta_{7} + 6214 \beta_{8} - 291 \beta_{9} + 145 \beta_{10} - 291 \beta_{11} - 250 \beta_{13} - 287 \beta_{14} + 218 \beta_{15} - 59 \beta_{16} + 373 \beta_{17} - 208 \beta_{18} - 142 \beta_{19} ) q^{46} + ( -2974 - 958 \beta_{1} + 122 \beta_{2} + 799 \beta_{3} + 36 \beta_{4} - 514 \beta_{5} - 3133 \beta_{7} + 156 \beta_{8} + 512 \beta_{9} + 149 \beta_{10} - 123 \beta_{11} + 156 \beta_{12} - 34 \beta_{13} - 170 \beta_{14} - 203 \beta_{15} - 39 \beta_{16} - 36 \beta_{17} + 149 \beta_{18} + 156 \beta_{19} ) q^{47} + ( -63 - 702 \beta_{1} + 135 \beta_{2} - 126 \beta_{3} + 27 \beta_{4} - 639 \beta_{5} - 981 \beta_{6} - 198 \beta_{7} - 2259 \beta_{8} + 261 \beta_{9} + 126 \beta_{10} + 63 \beta_{11} + 27 \beta_{12} - 63 \beta_{13} + 72 \beta_{14} - 207 \beta_{15} - 54 \beta_{16} + 27 \beta_{18} + 99 \beta_{19} ) q^{48} + ( 1871 + 156 \beta_{1} - 53 \beta_{2} + 315 \beta_{3} - 333 \beta_{5} + 121 \beta_{6} + 3148 \beta_{7} - 3148 \beta_{8} - 1871 \beta_{9} - 35 \beta_{10} - 31 \beta_{11} + 71 \beta_{12} - 117 \beta_{13} - 115 \beta_{14} - 99 \beta_{15} + 13 \beta_{16} + 40 \beta_{17} + 26 \beta_{18} - 40 \beta_{19} ) q^{49} + ( -2745 + 37 \beta_{1} + 5 \beta_{2} + 798 \beta_{3} - 709 \beta_{5} - 47 \beta_{6} + 3236 \beta_{7} - 3236 \beta_{8} + 2745 \beta_{9} - 84 \beta_{10} + 80 \beta_{11} - 25 \beta_{12} - 100 \beta_{13} + 164 \beta_{14} - 189 \beta_{15} + 9 \beta_{16} - 14 \beta_{17} + 18 \beta_{18} + 14 \beta_{19} ) q^{50} + ( 72 + 396 \beta_{1} - 270 \beta_{2} + 144 \beta_{3} - 90 \beta_{4} + 324 \beta_{5} + 135 \beta_{6} - 1026 \beta_{7} - 2412 \beta_{8} + 954 \beta_{9} - 144 \beta_{10} - 72 \beta_{11} - 90 \beta_{12} + 72 \beta_{13} - 198 \beta_{14} + 54 \beta_{15} + 63 \beta_{16} + 54 \beta_{18} - 27 \beta_{19} ) q^{51} + ( -5130 + 1012 \beta_{1} + 375 \beta_{2} - 1069 \beta_{3} + 183 \beta_{4} + 3004 \beta_{5} - 5187 \beta_{7} + 141 \beta_{8} - 1700 \beta_{9} + 195 \beta_{10} + 126 \beta_{11} + 141 \beta_{12} + 234 \beta_{13} + 275 \beta_{14} + 8 \beta_{15} - 99 \beta_{16} - 183 \beta_{17} + 195 \beta_{18} + 141 \beta_{19} ) q^{52} + ( -1921 - 1448 \beta_{1} - 17 \beta_{2} - 545 \beta_{3} - 202 \beta_{4} - 428 \beta_{5} + 699 \beta_{6} - 843 \beta_{7} + 2160 \beta_{8} - 441 \beta_{9} - 154 \beta_{10} - 441 \beta_{11} + 149 \beta_{13} - 48 \beta_{14} + 459 \beta_{15} + 226 \beta_{16} + 120 \beta_{17} - 13 \beta_{18} - 202 \beta_{19} ) q^{53} + ( 729 + 729 \beta_{3} ) q^{54} + ( -1187 + 1371 \beta_{1} - 81 \beta_{2} - 1258 \beta_{3} + 152 \beta_{4} + 913 \beta_{5} - 671 \beta_{6} - 7975 \beta_{7} + 6504 \beta_{8} + 2619 \beta_{9} - 250 \beta_{10} + 399 \beta_{11} - 176 \beta_{12} + 69 \beta_{13} - 200 \beta_{14} - 262 \beta_{15} - 108 \beta_{16} - 205 \beta_{17} - 69 \beta_{18} - 62 \beta_{19} ) q^{55} + ( 3880 + 76 \beta_{1} - 266 \beta_{2} + 3539 \beta_{3} + 244 \beta_{4} - 873 \beta_{5} - 1009 \beta_{6} - 186 \beta_{7} + 9664 \beta_{8} + 10002 \beta_{9} + 202 \beta_{10} + 338 \beta_{11} + 186 \beta_{12} - 303 \beta_{13} - 177 \beta_{14} - 262 \beta_{15} - 136 \beta_{16} - 186 \beta_{17} - 76 \beta_{18} ) q^{56} + ( -1314 + 891 \beta_{1} - 9 \beta_{2} - 108 \beta_{3} + 27 \beta_{4} + 108 \beta_{5} + 153 \beta_{6} + 4122 \beta_{7} + 1125 \beta_{8} + 216 \beta_{9} - 45 \beta_{10} + 216 \beta_{11} + 126 \beta_{13} + 72 \beta_{14} - 189 \beta_{15} - 81 \beta_{16} - 198 \beta_{17} + 108 \beta_{18} + 27 \beta_{19} ) q^{57} + ( 9899 - 1550 \beta_{1} - 120 \beta_{2} + 1588 \beta_{3} - 19 \beta_{4} - 1637 \beta_{5} + 9937 \beta_{7} - 40 \beta_{8} - 10953 \beta_{9} - 275 \beta_{10} + 19 \beta_{11} - 40 \beta_{12} - 80 \beta_{13} + 326 \beta_{14} + 347 \beta_{15} + 17 \beta_{16} + 19 \beta_{17} - 275 \beta_{18} - 40 \beta_{19} ) q^{58} + ( -59 - 2463 \beta_{1} - 48 \beta_{2} - 118 \beta_{3} - 14 \beta_{4} - 2404 \beta_{5} - 2560 \beta_{6} - 10108 \beta_{7} + 596 \beta_{8} + 10167 \beta_{9} + 118 \beta_{10} + 59 \beta_{11} - 14 \beta_{12} - 59 \beta_{13} - 107 \beta_{14} - 128 \beta_{15} - 93 \beta_{16} - 152 \beta_{18} - 127 \beta_{19} ) q^{59} + ( 2718 + 1899 \beta_{1} - 207 \beta_{2} - 1746 \beta_{3} + 1575 \beta_{5} + 1863 \beta_{6} + 1440 \beta_{7} - 1440 \beta_{8} - 2718 \beta_{9} - 36 \beta_{10} - 162 \beta_{11} - 117 \beta_{12} - 171 \beta_{13} - 297 \beta_{14} - 9 \beta_{16} + 216 \beta_{17} - 18 \beta_{18} - 216 \beta_{19} ) q^{60} + ( -2762 + 50 \beta_{1} - 109 \beta_{2} + 1507 \beta_{3} - 1335 \beta_{5} - 231 \beta_{6} - 9410 \beta_{7} + 9410 \beta_{8} + 2762 \beta_{9} - 281 \beta_{10} - 125 \beta_{11} - 183 \beta_{12} + 717 \beta_{13} + 637 \beta_{14} + 545 \beta_{15} + 297 \beta_{16} - 188 \beta_{17} + 594 \beta_{18} + 188 \beta_{19} ) q^{61} + ( 331 - 460 \beta_{1} + 311 \beta_{2} + 662 \beta_{3} - 244 \beta_{4} - 791 \beta_{5} - 1272 \beta_{6} + 12625 \beta_{7} - 9266 \beta_{8} - 12956 \beta_{9} - 662 \beta_{10} - 331 \beta_{11} - 244 \beta_{12} + 331 \beta_{13} + 642 \beta_{14} + 1025 \beta_{15} + 215 \beta_{16} + 62 \beta_{18} - 385 \beta_{19} ) q^{62} + ( -567 + 324 \beta_{1} - 162 \beta_{2} - 243 \beta_{3} - 162 \beta_{4} - 486 \beta_{7} - 162 \beta_{8} - 3645 \beta_{9} - 81 \beta_{10} - 81 \beta_{11} - 162 \beta_{12} + 243 \beta_{15} + 81 \beta_{16} + 162 \beta_{17} - 81 \beta_{18} - 162 \beta_{19} ) q^{63} + ( 20122 - 4565 \beta_{1} + 410 \beta_{2} + 2411 \beta_{3} + 563 \beta_{4} + 847 \beta_{5} - 2648 \beta_{6} + 22165 \beta_{7} - 20423 \beta_{8} + 864 \beta_{9} + 237 \beta_{10} + 864 \beta_{11} + 267 \beta_{13} + 326 \beta_{14} - 763 \beta_{15} - 284 \beta_{16} - 373 \beta_{17} + 17 \beta_{18} + 563 \beta_{19} ) q^{64} + ( -1547 + 314 \beta_{1} + 900 \beta_{2} - 400 \beta_{3} + 299 \beta_{4} + 2838 \beta_{5} + 2479 \beta_{6} + 97 \beta_{7} - 245 \beta_{8} + 286 \beta_{9} + 172 \beta_{10} + 531 \beta_{11} - 97 \beta_{12} + 67 \beta_{13} - 75 \beta_{14} - 217 \beta_{15} - 359 \beta_{16} + 97 \beta_{17} - 314 \beta_{18} ) q^{65} + ( -2052 + 387 \beta_{1} - 405 \beta_{2} + 90 \beta_{3} - 153 \beta_{4} + 1188 \beta_{5} - 495 \beta_{6} - 4851 \beta_{7} + 9585 \beta_{8} + 2997 \beta_{9} + 81 \beta_{10} - 216 \beta_{11} - 315 \beta_{13} - 648 \beta_{14} + 54 \beta_{15} - 144 \beta_{16} + 9 \beta_{17} + 117 \beta_{18} + 9 \beta_{19} ) q^{66} + ( -5109 + 85 \beta_{1} + 248 \beta_{2} + 1304 \beta_{3} + 412 \beta_{4} - 2248 \beta_{5} - 2625 \beta_{6} + 131 \beta_{7} + 13652 \beta_{8} + 13691 \beta_{9} - 338 \beta_{10} + 39 \beta_{11} - 131 \beta_{12} + 28 \beta_{13} - 395 \beta_{14} + 46 \beta_{15} - 377 \beta_{16} + 131 \beta_{17} - 85 \beta_{18} ) q^{67} + ( -35342 + 2388 \beta_{1} + 728 \beta_{2} - 2379 \beta_{3} - 187 \beta_{4} - 443 \beta_{5} + 2620 \beta_{6} - 22237 \beta_{7} + 35523 \beta_{8} - 368 \beta_{9} - 241 \beta_{10} - 368 \beta_{11} + 840 \beta_{13} + 54 \beta_{14} + 1117 \beta_{15} + 256 \beta_{16} - 39 \beta_{17} + 75 \beta_{18} - 187 \beta_{19} ) q^{68} + ( 7920 - 1098 \beta_{1} - 783 \beta_{2} + 1053 \beta_{3} - 288 \beta_{4} - 1908 \beta_{5} + 7875 \beta_{7} - 5139 \beta_{9} + 234 \beta_{10} - 333 \beta_{11} - 783 \beta_{13} - 144 \beta_{14} + 189 \beta_{15} + 243 \beta_{16} + 288 \beta_{17} + 234 \beta_{18} ) q^{69} + ( 106 - 747 \beta_{1} + 95 \beta_{2} + 212 \beta_{3} + 41 \beta_{4} - 853 \beta_{5} - 2791 \beta_{6} - 25389 \beta_{7} + 8960 \beta_{8} + 25283 \beta_{9} - 212 \beta_{10} - 106 \beta_{11} + 41 \beta_{12} + 106 \beta_{13} + 201 \beta_{14} - 470 \beta_{15} + 215 \beta_{16} + 73 \beta_{18} + 76 \beta_{19} ) q^{70} + ( 25748 + 1910 \beta_{1} + 54 \beta_{2} - 513 \beta_{3} + 290 \beta_{5} + 2187 \beta_{6} + 21869 \beta_{7} - 21869 \beta_{8} - 25748 \beta_{9} + 277 \beta_{10} - 211 \beta_{11} + 114 \beta_{12} - 564 \beta_{13} - 142 \beta_{14} - 341 \beta_{15} - 12 \beta_{16} - 42 \beta_{17} - 24 \beta_{18} + 42 \beta_{19} ) q^{71} + ( -1620 + 1134 \beta_{1} - 243 \beta_{2} - 1296 \beta_{3} + 972 \beta_{5} + 1215 \beta_{6} - 3483 \beta_{7} + 3483 \beta_{8} + 1620 \beta_{9} + 81 \beta_{10} - 324 \beta_{11} - 324 \beta_{14} + 324 \beta_{15} + 243 \beta_{17} - 243 \beta_{19} ) q^{72} + ( -264 + 1797 \beta_{1} + 751 \beta_{2} - 528 \beta_{3} + 643 \beta_{4} + 2061 \beta_{5} - 1139 \beta_{6} - 3130 \beta_{7} - 19039 \beta_{8} + 3394 \beta_{9} + 528 \beta_{10} + 264 \beta_{11} + 643 \beta_{12} - 264 \beta_{13} + 487 \beta_{14} - 225 \beta_{15} - 296 \beta_{16} - 241 \beta_{18} - 9 \beta_{19} ) q^{73} + ( -33832 - 1246 \beta_{1} - 1008 \beta_{2} + 1251 \beta_{3} + 59 \beta_{4} + 1500 \beta_{5} - 33827 \beta_{7} + 53 \beta_{8} + 34539 \beta_{9} + 103 \beta_{10} + 64 \beta_{11} + 53 \beta_{12} - 1061 \beta_{13} - 1128 \beta_{14} - 1245 \beta_{15} - \beta_{16} - 59 \beta_{17} + 103 \beta_{18} + 53 \beta_{19} ) q^{74} + ( 3654 + 522 \beta_{1} - 270 \beta_{2} - 864 \beta_{3} - 279 \beta_{4} - 153 \beta_{5} + 963 \beta_{6} + 9783 \beta_{7} - 3690 \beta_{8} - 243 \beta_{9} - 99 \beta_{10} - 243 \beta_{11} + 315 \beta_{13} - 180 \beta_{14} + 63 \beta_{15} - 126 \beta_{16} - 45 \beta_{17} - 90 \beta_{18} - 279 \beta_{19} ) q^{75} + ( 767 - 402 \beta_{1} - 329 \beta_{2} - 3357 \beta_{3} - 200 \beta_{4} + 1428 \beta_{5} + 1878 \beta_{6} - 95 \beta_{7} + 5054 \beta_{8} + 4345 \beta_{9} - 259 \beta_{10} - 709 \beta_{11} + 95 \beta_{12} + 203 \beta_{13} + 346 \beta_{14} + 307 \beta_{15} + 450 \beta_{16} - 95 \beta_{17} + 402 \beta_{18} ) q^{76} + ( 4202 + 1821 \beta_{1} - 433 \beta_{2} - 633 \beta_{3} - 237 \beta_{4} + 1408 \beta_{5} - 866 \beta_{6} - 18044 \beta_{7} - 1967 \beta_{8} - 21161 \beta_{9} - 338 \beta_{10} - 599 \beta_{11} - 20 \beta_{12} + 110 \beta_{13} + 402 \beta_{14} - 80 \beta_{15} + 407 \beta_{16} + 200 \beta_{17} + 169 \beta_{18} - 222 \beta_{19} ) q^{77} + ( -15597 - 117 \beta_{1} + 684 \beta_{2} + 1836 \beta_{3} + 207 \beta_{4} + 936 \beta_{5} + 1008 \beta_{6} - 288 \beta_{7} + 3312 \beta_{8} + 3366 \beta_{9} + 126 \beta_{10} + 54 \beta_{11} + 288 \beta_{12} - 252 \beta_{13} - 9 \beta_{14} - 171 \beta_{15} + 72 \beta_{16} - 288 \beta_{17} + 117 \beta_{18} ) q^{78} + ( -15985 - 2369 \beta_{1} + 924 \beta_{2} - 1459 \beta_{3} + 627 \beta_{4} + 1107 \beta_{5} + 1185 \beta_{6} + 13571 \beta_{7} + 15691 \beta_{8} + 921 \beta_{9} + 274 \beta_{10} + 921 \beta_{11} + 444 \beta_{13} + 353 \beta_{14} - 536 \beta_{15} - 480 \beta_{16} - 559 \beta_{17} - 186 \beta_{18} + 627 \beta_{19} ) q^{79} + ( 13173 + 20 \beta_{1} + 695 \beta_{2} + 267 \beta_{3} + 230 \beta_{4} - 4019 \beta_{5} + 13460 \beta_{7} - 179 \beta_{8} - 33709 \beta_{9} + 60 \beta_{10} + 517 \beta_{11} - 179 \beta_{12} + 874 \beta_{13} - 21 \beta_{14} - 359 \beta_{15} - 122 \beta_{16} - 230 \beta_{17} + 60 \beta_{18} - 179 \beta_{19} ) q^{80} -6561 \beta_{8} q^{81} + ( 46005 - 1952 \beta_{1} + 305 \beta_{2} + 2000 \beta_{3} - 1389 \beta_{5} - 2258 \beta_{6} + 26715 \beta_{7} - 26715 \beta_{8} - 46005 \beta_{9} - 306 \beta_{10} + 328 \beta_{11} + 41 \beta_{12} + 468 \beta_{13} + 1246 \beta_{14} - 143 \beta_{15} + 283 \beta_{16} - 588 \beta_{17} + 566 \beta_{18} + 588 \beta_{19} ) q^{82} + ( 3435 - 1169 \beta_{1} + 182 \beta_{2} - 1114 \beta_{3} + 700 \beta_{5} - 573 \beta_{6} + 31002 \beta_{7} - 31002 \beta_{8} - 3435 \beta_{9} + 596 \beta_{10} - 123 \beta_{11} + 297 \beta_{12} - 656 \beta_{13} + 127 \beta_{14} - 242 \beta_{15} - 291 \beta_{16} + 109 \beta_{17} - 582 \beta_{18} - 109 \beta_{19} ) q^{83} + ( 54 - 2610 \beta_{1} - 351 \beta_{2} + 108 \beta_{3} - 207 \beta_{4} - 2664 \beta_{5} - 2916 \beta_{6} - 6660 \beta_{7} - 18090 \beta_{8} + 6606 \beta_{9} - 108 \beta_{10} - 54 \beta_{11} - 207 \beta_{12} + 54 \beta_{13} - 297 \beta_{14} - 963 \beta_{15} + 9 \beta_{16} + 153 \beta_{18} + 54 \beta_{19} ) q^{84} + ( -31998 - 4457 \beta_{1} + 134 \beta_{2} + 4176 \beta_{3} - 81 \beta_{4} + 139 \beta_{5} - 32279 \beta_{7} + 328 \beta_{8} + 4301 \beta_{9} - 215 \beta_{10} - 362 \beta_{11} + 328 \beta_{12} - 194 \beta_{13} + 697 \beta_{14} + 731 \beta_{15} + 128 \beta_{16} + 81 \beta_{17} - 215 \beta_{18} + 328 \beta_{19} ) q^{85} + ( 38313 - 2364 \beta_{1} - 1102 \beta_{2} + 195 \beta_{3} - 346 \beta_{4} + 197 \beta_{5} - 564 \beta_{6} + 65548 \beta_{7} - 38732 \beta_{8} + 73 \beta_{9} + 369 \beta_{10} + 73 \beta_{11} - 1252 \beta_{13} - 715 \beta_{14} - 584 \beta_{15} - 543 \beta_{16} + 541 \beta_{17} - 124 \beta_{18} - 346 \beta_{19} ) q^{86} + ( 13563 + 459 \beta_{1} - 81 \beta_{2} + 1080 \beta_{3} - 342 \beta_{4} - 1044 \beta_{5} - 1152 \beta_{6} + 495 \beta_{7} - 7848 \beta_{8} - 7425 \beta_{9} + 315 \beta_{10} + 423 \beta_{11} - 495 \beta_{12} - 9 \beta_{13} - 153 \beta_{14} + 36 \beta_{15} - 108 \beta_{16} + 495 \beta_{17} - 459 \beta_{18} ) q^{87} + ( -5609 + 2186 \beta_{1} + 967 \beta_{2} - 7705 \beta_{3} + 316 \beta_{4} + 7750 \beta_{6} - 33635 \beta_{7} + 44304 \beta_{8} - 26041 \beta_{9} + 823 \beta_{10} + 279 \beta_{11} + 576 \beta_{12} + 718 \beta_{13} + 230 \beta_{14} + 579 \beta_{15} + 101 \beta_{16} + 63 \beta_{17} - 512 \beta_{18} + 88 \beta_{19} ) q^{88} + ( -3210 - 128 \beta_{1} - 1873 \beta_{2} - 867 \beta_{3} - 1634 \beta_{4} - 1012 \beta_{5} - 101 \beta_{6} + 763 \beta_{7} + 23711 \beta_{8} + 22692 \beta_{9} - 108 \beta_{10} - 1019 \beta_{11} - 763 \beta_{12} + 130 \beta_{13} + 150 \beta_{14} + 891 \beta_{15} + 911 \beta_{16} + 763 \beta_{17} + 128 \beta_{18} ) q^{89} + ( 324 + 243 \beta_{1} - 648 \beta_{2} + 486 \beta_{3} - 324 \beta_{4} - 891 \beta_{5} - 243 \beta_{6} + 7857 \beta_{7} + 81 \beta_{8} - 729 \beta_{9} - 243 \beta_{10} - 729 \beta_{11} - 810 \beta_{13} - 81 \beta_{14} + 324 \beta_{15} + 567 \beta_{16} + 405 \beta_{17} + 162 \beta_{18} - 324 \beta_{19} ) q^{90} + ( 22221 + 5670 \beta_{1} - 551 \beta_{2} - 5568 \beta_{3} - 609 \beta_{4} + 7406 \beta_{5} + 22323 \beta_{7} - 681 \beta_{8} - 10731 \beta_{9} - 327 \beta_{10} - 507 \beta_{11} - 681 \beta_{12} + 130 \beta_{13} - 455 \beta_{14} + 733 \beta_{15} + 30 \beta_{16} + 609 \beta_{17} - 327 \beta_{18} - 681 \beta_{19} ) q^{91} + ( -59 + 4445 \beta_{1} - 2039 \beta_{2} - 118 \beta_{3} + 96 \beta_{4} + 4504 \beta_{5} + 1462 \beta_{6} - 90688 \beta_{7} + 57201 \beta_{8} + 90747 \beta_{9} + 118 \beta_{10} + 59 \beta_{11} + 96 \beta_{12} - 59 \beta_{13} - 2098 \beta_{14} - 78 \beta_{15} - 203 \beta_{16} + 612 \beta_{18} + 527 \beta_{19} ) q^{92} + ( 3357 + 1800 \beta_{1} + 711 \beta_{2} - 387 \beta_{3} + 864 \beta_{5} + 2034 \beta_{6} - 2853 \beta_{7} + 2853 \beta_{8} - 3357 \beta_{9} + 234 \beta_{10} + 855 \beta_{11} + 297 \beta_{12} - 18 \beta_{13} - 90 \beta_{14} - 495 \beta_{15} - 378 \beta_{16} - 333 \beta_{17} - 756 \beta_{18} + 333 \beta_{19} ) q^{93} + ( -25124 + 5191 \beta_{1} + 629 \beta_{2} + 3603 \beta_{3} - 3013 \beta_{5} + 5230 \beta_{6} + 15744 \beta_{7} - 15744 \beta_{8} + 25124 \beta_{9} + 39 \beta_{10} + 1420 \beta_{11} + 332 \beta_{12} + 115 \beta_{13} - 391 \beta_{14} - 475 \beta_{15} - 830 \beta_{16} + 201 \beta_{17} - 1660 \beta_{18} - 201 \beta_{19} ) q^{94} + ( -659 - 465 \beta_{1} + 408 \beta_{2} - 1318 \beta_{3} - 134 \beta_{4} + 194 \beta_{5} + 2489 \beta_{6} + 27567 \beta_{7} - 47788 \beta_{8} - 26908 \beta_{9} + 1318 \beta_{10} + 659 \beta_{11} - 134 \beta_{12} - 659 \beta_{13} - 251 \beta_{14} + 1016 \beta_{15} - 62 \beta_{16} + 202 \beta_{18} + 1458 \beta_{19} ) q^{95} + ( -33840 + 2718 \beta_{1} + 963 \beta_{2} - 2502 \beta_{3} - 90 \beta_{4} - 1449 \beta_{5} - 33624 \beta_{7} - 396 \beta_{8} - 13698 \beta_{9} - 198 \beta_{10} + 126 \beta_{11} - 396 \beta_{12} + 1359 \beta_{13} + 441 \beta_{14} + 711 \beta_{15} - 90 \beta_{16} + 90 \beta_{17} - 198 \beta_{18} - 396 \beta_{19} ) q^{96} + ( 760 + 5660 \beta_{1} + 972 \beta_{2} + 944 \beta_{3} + 395 \beta_{4} - 533 \beta_{5} - 595 \beta_{6} + 27053 \beta_{7} - 362 \beta_{8} - 3 \beta_{9} - 349 \beta_{10} - 3 \beta_{11} + 971 \beta_{13} + 744 \beta_{14} + 761 \beta_{15} + 928 \beta_{16} - 165 \beta_{17} + 530 \beta_{18} + 395 \beta_{19} ) q^{97} + ( -17699 - 204 \beta_{1} - 685 \beta_{2} + 2220 \beta_{3} - 500 \beta_{4} - 4163 \beta_{5} - 4183 \beta_{6} - 55 \beta_{7} + 22412 \beta_{8} + 22059 \beta_{9} - 373 \beta_{10} - 353 \beta_{11} + 55 \beta_{12} - 938 \beta_{13} - 1107 \beta_{14} + 149 \beta_{15} - 20 \beta_{16} - 55 \beta_{17} + 204 \beta_{18} ) q^{98} + ( -4617 - 1620 \beta_{1} + 81 \beta_{2} + 1134 \beta_{3} - 81 \beta_{4} - 891 \beta_{5} - 1134 \beta_{6} - 4131 \beta_{7} - 648 \beta_{8} - 1296 \beta_{9} + 162 \beta_{10} - 81 \beta_{11} - 162 \beta_{12} - 486 \beta_{13} - 162 \beta_{14} - 81 \beta_{15} + 567 \beta_{16} + 81 \beta_{17} + 162 \beta_{18} + 405 \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 2q^{2} - 45q^{3} - 124q^{4} - 33q^{5} - 18q^{6} - 335q^{7} - 472q^{8} - 405q^{9} + O(q^{10}) \) \( 20q - 2q^{2} - 45q^{3} - 124q^{4} - 33q^{5} - 18q^{6} - 335q^{7} - 472q^{8} - 405q^{9} + 1952q^{10} - 835q^{11} + 3474q^{12} - 959q^{13} - 3020q^{14} + 198q^{15} - 1428q^{16} - 3144q^{17} - 567q^{18} - 930q^{19} + 3177q^{20} + 8460q^{21} + 2713q^{22} + 1400q^{23} + 1332q^{24} - 9388q^{25} + 3022q^{26} - 3645q^{27} - 21753q^{28} + 4494q^{29} - 4302q^{30} + 9208q^{31} + 69114q^{32} + 6615q^{33} - 9084q^{34} - 31371q^{35} - 10044q^{36} - 3153q^{37} + 1852q^{38} - 8631q^{39} - 53123q^{40} - 7497q^{41} + 22320q^{42} + 63624q^{43} + 58299q^{44} + 1782q^{45} - 94681q^{46} - 43031q^{47} - 12357q^{48} - 5640q^{49} - 76590q^{50} + 459q^{51} - 80149q^{52} - 20452q^{53} + 13122q^{54} + 62974q^{55} + 169638q^{56} - 39375q^{57} + 93145q^{58} + 101730q^{59} + 28593q^{60} + 50745q^{61} - 171957q^{62} - 27135q^{63} + 199638q^{64} - 28210q^{65} + 46107q^{66} + 10730q^{67} - 420141q^{68} + 94365q^{69} + 292664q^{70} + 164895q^{71} + 11988q^{72} - 77236q^{73} - 328228q^{74} + 9063q^{75} + 79212q^{76} + 56087q^{77} - 280062q^{78} - 291931q^{79} + 18026q^{80} - 32805q^{81} + 424659q^{82} - 255451q^{83} - 26127q^{84} - 452271q^{85} + 252843q^{86} + 187056q^{87} + 171694q^{88} + 181632q^{89} - 40338q^{90} + 284615q^{91} + 1199691q^{92} + 82872q^{93} - 550105q^{94} - 511091q^{95} - 587574q^{96} - 145083q^{97} - 150570q^{98} - 83430q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 3 x^{19} + 142 x^{18} - 389 x^{17} + 14927 x^{16} - 6599 x^{15} + 1399353 x^{14} + 1969563 x^{13} + 129076450 x^{12} + 14028509 x^{11} + 4158720649 x^{10} - 7877307064 x^{9} + 107078602608 x^{8} - 53496436514 x^{7} + 2527225278875 x^{6} - 6173773919508 x^{5} + 63231264058241 x^{4} - 118816044595930 x^{3} + 398041877081860 x^{2} - 161017247606600 x + 25735126080400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(43\!\cdots\!67\)\( \nu^{19} + \)\(18\!\cdots\!85\)\( \nu^{18} - \)\(99\!\cdots\!74\)\( \nu^{17} + \)\(25\!\cdots\!03\)\( \nu^{16} - \)\(70\!\cdots\!89\)\( \nu^{15} + \)\(24\!\cdots\!13\)\( \nu^{14} + \)\(36\!\cdots\!13\)\( \nu^{13} + \)\(21\!\cdots\!99\)\( \nu^{12} + \)\(52\!\cdots\!70\)\( \nu^{11} + \)\(22\!\cdots\!57\)\( \nu^{10} + \)\(56\!\cdots\!17\)\( \nu^{9} + \)\(90\!\cdots\!68\)\( \nu^{8} + \)\(40\!\cdots\!84\)\( \nu^{7} + \)\(24\!\cdots\!78\)\( \nu^{6} + \)\(15\!\cdots\!35\)\( \nu^{5} + \)\(39\!\cdots\!36\)\( \nu^{4} + \)\(24\!\cdots\!49\)\( \nu^{3} + \)\(13\!\cdots\!50\)\( \nu^{2} + \)\(72\!\cdots\!40\)\( \nu - \)\(33\!\cdots\!00\)\(\)\()/ \)\(78\!\cdots\!80\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(11\!\cdots\!83\)\( \nu^{19} - \)\(65\!\cdots\!93\)\( \nu^{18} + \)\(17\!\cdots\!90\)\( \nu^{17} - \)\(40\!\cdots\!31\)\( \nu^{16} + \)\(17\!\cdots\!69\)\( \nu^{15} + \)\(12\!\cdots\!39\)\( \nu^{14} + \)\(15\!\cdots\!15\)\( \nu^{13} + \)\(48\!\cdots\!81\)\( \nu^{12} + \)\(14\!\cdots\!94\)\( \nu^{11} + \)\(25\!\cdots\!87\)\( \nu^{10} + \)\(63\!\cdots\!59\)\( \nu^{9} + \)\(38\!\cdots\!00\)\( \nu^{8} + \)\(21\!\cdots\!52\)\( \nu^{7} + \)\(14\!\cdots\!22\)\( \nu^{6} + \)\(11\!\cdots\!13\)\( \nu^{5} + \)\(24\!\cdots\!36\)\( \nu^{4} + \)\(11\!\cdots\!91\)\( \nu^{3} + \)\(67\!\cdots\!70\)\( \nu^{2} - \)\(30\!\cdots\!60\)\( \nu + \)\(41\!\cdots\!00\)\(\)\()/ \)\(86\!\cdots\!80\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(45\!\cdots\!18\)\( \nu^{19} - \)\(83\!\cdots\!99\)\( \nu^{18} - \)\(58\!\cdots\!93\)\( \nu^{17} - \)\(13\!\cdots\!49\)\( \nu^{16} - \)\(60\!\cdots\!63\)\( \nu^{15} - \)\(28\!\cdots\!93\)\( \nu^{14} - \)\(63\!\cdots\!58\)\( \nu^{13} - \)\(39\!\cdots\!12\)\( \nu^{12} - \)\(63\!\cdots\!30\)\( \nu^{11} - \)\(28\!\cdots\!46\)\( \nu^{10} - \)\(20\!\cdots\!16\)\( \nu^{9} - \)\(51\!\cdots\!69\)\( \nu^{8} - \)\(31\!\cdots\!57\)\( \nu^{7} - \)\(16\!\cdots\!99\)\( \nu^{6} - \)\(76\!\cdots\!60\)\( \nu^{5} - \)\(18\!\cdots\!39\)\( \nu^{4} - \)\(15\!\cdots\!08\)\( \nu^{3} - \)\(72\!\cdots\!20\)\( \nu^{2} + \)\(33\!\cdots\!80\)\( \nu - \)\(64\!\cdots\!50\)\(\)\()/ \)\(21\!\cdots\!95\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(52\!\cdots\!85\)\( \nu^{19} - \)\(55\!\cdots\!27\)\( \nu^{18} - \)\(47\!\cdots\!94\)\( \nu^{17} - \)\(24\!\cdots\!11\)\( \nu^{16} - \)\(35\!\cdots\!63\)\( \nu^{15} - \)\(32\!\cdots\!91\)\( \nu^{14} - \)\(50\!\cdots\!07\)\( \nu^{13} - \)\(35\!\cdots\!17\)\( \nu^{12} - \)\(44\!\cdots\!94\)\( \nu^{11} - \)\(14\!\cdots\!35\)\( \nu^{10} + \)\(25\!\cdots\!83\)\( \nu^{9} + \)\(50\!\cdots\!88\)\( \nu^{8} + \)\(29\!\cdots\!32\)\( \nu^{7} + \)\(12\!\cdots\!96\)\( \nu^{6} + \)\(11\!\cdots\!77\)\( \nu^{5} + \)\(18\!\cdots\!80\)\( \nu^{4} + \)\(18\!\cdots\!53\)\( \nu^{3} + \)\(85\!\cdots\!30\)\( \nu^{2} - \)\(39\!\cdots\!80\)\( \nu + \)\(76\!\cdots\!80\)\(\)\()/ \)\(17\!\cdots\!60\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(24\!\cdots\!56\)\( \nu^{19} - \)\(97\!\cdots\!79\)\( \nu^{18} + \)\(34\!\cdots\!15\)\( \nu^{17} - \)\(12\!\cdots\!70\)\( \nu^{16} + \)\(35\!\cdots\!05\)\( \nu^{15} - \)\(52\!\cdots\!99\)\( \nu^{14} + \)\(32\!\cdots\!51\)\( \nu^{13} + \)\(10\!\cdots\!35\)\( \nu^{12} + \)\(29\!\cdots\!85\)\( \nu^{11} - \)\(33\!\cdots\!30\)\( \nu^{10} + \)\(86\!\cdots\!25\)\( \nu^{9} - \)\(32\!\cdots\!35\)\( \nu^{8} + \)\(22\!\cdots\!40\)\( \nu^{7} - \)\(39\!\cdots\!60\)\( \nu^{6} + \)\(54\!\cdots\!00\)\( \nu^{5} - \)\(20\!\cdots\!91\)\( \nu^{4} + \)\(15\!\cdots\!74\)\( \nu^{3} - \)\(44\!\cdots\!25\)\( \nu^{2} + \)\(87\!\cdots\!80\)\( \nu - \)\(14\!\cdots\!00\)\(\)\()/ \)\(43\!\cdots\!90\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(65\!\cdots\!47\)\( \nu^{19} + \)\(14\!\cdots\!27\)\( \nu^{18} - \)\(90\!\cdots\!88\)\( \nu^{17} + \)\(17\!\cdots\!11\)\( \nu^{16} - \)\(94\!\cdots\!83\)\( \nu^{15} - \)\(36\!\cdots\!17\)\( \nu^{14} - \)\(90\!\cdots\!77\)\( \nu^{13} - \)\(20\!\cdots\!07\)\( \nu^{12} - \)\(84\!\cdots\!60\)\( \nu^{11} - \)\(72\!\cdots\!51\)\( \nu^{10} - \)\(26\!\cdots\!01\)\( \nu^{9} + \)\(33\!\cdots\!46\)\( \nu^{8} - \)\(61\!\cdots\!32\)\( \nu^{7} - \)\(12\!\cdots\!34\)\( \nu^{6} - \)\(15\!\cdots\!25\)\( \nu^{5} + \)\(29\!\cdots\!30\)\( \nu^{4} - \)\(39\!\cdots\!19\)\( \nu^{3} + \)\(45\!\cdots\!20\)\( \nu^{2} - \)\(15\!\cdots\!00\)\( \nu + \)\(39\!\cdots\!00\)\(\)\()/ \)\(86\!\cdots\!80\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(46\!\cdots\!17\)\( \nu^{19} - \)\(14\!\cdots\!47\)\( \nu^{18} + \)\(65\!\cdots\!86\)\( \nu^{17} - \)\(19\!\cdots\!09\)\( \nu^{16} + \)\(68\!\cdots\!31\)\( \nu^{15} - \)\(41\!\cdots\!19\)\( \nu^{14} + \)\(63\!\cdots\!05\)\( \nu^{13} + \)\(79\!\cdots\!95\)\( \nu^{12} + \)\(58\!\cdots\!86\)\( \nu^{11} - \)\(53\!\cdots\!07\)\( \nu^{10} + \)\(18\!\cdots\!21\)\( \nu^{9} - \)\(40\!\cdots\!40\)\( \nu^{8} + \)\(48\!\cdots\!68\)\( \nu^{7} - \)\(30\!\cdots\!42\)\( \nu^{6} + \)\(11\!\cdots\!47\)\( \nu^{5} - \)\(29\!\cdots\!56\)\( \nu^{4} + \)\(28\!\cdots\!89\)\( \nu^{3} - \)\(57\!\cdots\!86\)\( \nu^{2} + \)\(16\!\cdots\!80\)\( \nu - \)\(67\!\cdots\!40\)\(\)\()/ \)\(39\!\cdots\!40\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(59\!\cdots\!67\)\( \nu^{19} - \)\(17\!\cdots\!07\)\( \nu^{18} + \)\(85\!\cdots\!44\)\( \nu^{17} - \)\(23\!\cdots\!95\)\( \nu^{16} + \)\(89\!\cdots\!63\)\( \nu^{15} - \)\(39\!\cdots\!35\)\( \nu^{14} + \)\(83\!\cdots\!85\)\( \nu^{13} + \)\(11\!\cdots\!55\)\( \nu^{12} + \)\(77\!\cdots\!32\)\( \nu^{11} + \)\(86\!\cdots\!63\)\( \nu^{10} + \)\(24\!\cdots\!09\)\( \nu^{9} - \)\(46\!\cdots\!82\)\( \nu^{8} + \)\(64\!\cdots\!60\)\( \nu^{7} - \)\(30\!\cdots\!26\)\( \nu^{6} + \)\(15\!\cdots\!29\)\( \nu^{5} - \)\(36\!\cdots\!06\)\( \nu^{4} + \)\(37\!\cdots\!95\)\( \nu^{3} - \)\(69\!\cdots\!28\)\( \nu^{2} + \)\(24\!\cdots\!20\)\( \nu - \)\(62\!\cdots\!80\)\(\)\()/ \)\(36\!\cdots\!40\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(35\!\cdots\!52\)\( \nu^{19} + \)\(92\!\cdots\!83\)\( \nu^{18} - \)\(50\!\cdots\!91\)\( \nu^{17} + \)\(11\!\cdots\!36\)\( \nu^{16} - \)\(52\!\cdots\!55\)\( \nu^{15} + \)\(18\!\cdots\!51\)\( \nu^{14} - \)\(50\!\cdots\!59\)\( \nu^{13} - \)\(91\!\cdots\!19\)\( \nu^{12} - \)\(46\!\cdots\!13\)\( \nu^{11} - \)\(24\!\cdots\!08\)\( \nu^{10} - \)\(15\!\cdots\!57\)\( \nu^{9} + \)\(22\!\cdots\!69\)\( \nu^{8} - \)\(37\!\cdots\!02\)\( \nu^{7} + \)\(49\!\cdots\!40\)\( \nu^{6} - \)\(90\!\cdots\!06\)\( \nu^{5} + \)\(18\!\cdots\!41\)\( \nu^{4} - \)\(21\!\cdots\!62\)\( \nu^{3} + \)\(33\!\cdots\!39\)\( \nu^{2} - \)\(13\!\cdots\!40\)\( \nu + \)\(21\!\cdots\!00\)\(\)\()/ \)\(19\!\cdots\!20\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(18\!\cdots\!63\)\( \nu^{19} + \)\(25\!\cdots\!17\)\( \nu^{18} + \)\(12\!\cdots\!18\)\( \nu^{17} + \)\(33\!\cdots\!85\)\( \nu^{16} + \)\(10\!\cdots\!51\)\( \nu^{15} + \)\(40\!\cdots\!37\)\( \nu^{14} + \)\(18\!\cdots\!63\)\( \nu^{13} + \)\(40\!\cdots\!33\)\( \nu^{12} + \)\(23\!\cdots\!74\)\( \nu^{11} + \)\(33\!\cdots\!23\)\( \nu^{10} + \)\(85\!\cdots\!07\)\( \nu^{9} + \)\(50\!\cdots\!22\)\( \nu^{8} - \)\(28\!\cdots\!44\)\( \nu^{7} + \)\(20\!\cdots\!46\)\( \nu^{6} + \)\(23\!\cdots\!91\)\( \nu^{5} + \)\(36\!\cdots\!22\)\( \nu^{4} - \)\(22\!\cdots\!01\)\( \nu^{3} + \)\(98\!\cdots\!82\)\( \nu^{2} - \)\(31\!\cdots\!80\)\( \nu - \)\(73\!\cdots\!40\)\(\)\()/ \)\(36\!\cdots\!40\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(62\!\cdots\!25\)\( \nu^{19} + \)\(41\!\cdots\!01\)\( \nu^{18} - \)\(88\!\cdots\!77\)\( \nu^{17} + \)\(53\!\cdots\!63\)\( \nu^{16} - \)\(92\!\cdots\!49\)\( \nu^{15} + \)\(33\!\cdots\!94\)\( \nu^{14} - \)\(79\!\cdots\!90\)\( \nu^{13} + \)\(17\!\cdots\!65\)\( \nu^{12} - \)\(67\!\cdots\!80\)\( \nu^{11} + \)\(27\!\cdots\!53\)\( \nu^{10} - \)\(17\!\cdots\!21\)\( \nu^{9} + \)\(12\!\cdots\!76\)\( \nu^{8} - \)\(67\!\cdots\!93\)\( \nu^{7} + \)\(16\!\cdots\!62\)\( \nu^{6} - \)\(12\!\cdots\!41\)\( \nu^{5} + \)\(89\!\cdots\!83\)\( \nu^{4} - \)\(41\!\cdots\!52\)\( \nu^{3} + \)\(14\!\cdots\!52\)\( \nu^{2} - \)\(20\!\cdots\!90\)\( \nu + \)\(12\!\cdots\!00\)\(\)\()/ \)\(19\!\cdots\!20\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(10\!\cdots\!65\)\( \nu^{19} + \)\(27\!\cdots\!03\)\( \nu^{18} - \)\(14\!\cdots\!53\)\( \nu^{17} + \)\(35\!\cdots\!84\)\( \nu^{16} - \)\(15\!\cdots\!48\)\( \nu^{15} + \)\(86\!\cdots\!64\)\( \nu^{14} - \)\(14\!\cdots\!02\)\( \nu^{13} - \)\(26\!\cdots\!41\)\( \nu^{12} - \)\(13\!\cdots\!54\)\( \nu^{11} - \)\(61\!\cdots\!90\)\( \nu^{10} - \)\(42\!\cdots\!57\)\( \nu^{9} + \)\(68\!\cdots\!23\)\( \nu^{8} - \)\(10\!\cdots\!73\)\( \nu^{7} + \)\(11\!\cdots\!46\)\( \nu^{6} - \)\(26\!\cdots\!18\)\( \nu^{5} + \)\(56\!\cdots\!50\)\( \nu^{4} - \)\(61\!\cdots\!72\)\( \nu^{3} + \)\(94\!\cdots\!59\)\( \nu^{2} - \)\(35\!\cdots\!00\)\( \nu + \)\(49\!\cdots\!80\)\(\)\()/ \)\(19\!\cdots\!20\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(21\!\cdots\!75\)\( \nu^{19} - \)\(69\!\cdots\!65\)\( \nu^{18} + \)\(31\!\cdots\!70\)\( \nu^{17} - \)\(90\!\cdots\!27\)\( \nu^{16} + \)\(32\!\cdots\!13\)\( \nu^{15} - \)\(19\!\cdots\!73\)\( \nu^{14} + \)\(30\!\cdots\!31\)\( \nu^{13} + \)\(37\!\cdots\!57\)\( \nu^{12} + \)\(27\!\cdots\!38\)\( \nu^{11} - \)\(22\!\cdots\!21\)\( \nu^{10} + \)\(88\!\cdots\!23\)\( \nu^{9} - \)\(18\!\cdots\!80\)\( \nu^{8} + \)\(23\!\cdots\!24\)\( \nu^{7} - \)\(14\!\cdots\!26\)\( \nu^{6} + \)\(54\!\cdots\!61\)\( \nu^{5} - \)\(14\!\cdots\!52\)\( \nu^{4} + \)\(13\!\cdots\!31\)\( \nu^{3} - \)\(27\!\cdots\!10\)\( \nu^{2} + \)\(81\!\cdots\!60\)\( \nu - \)\(32\!\cdots\!00\)\(\)\()/ \)\(39\!\cdots\!40\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(61\!\cdots\!65\)\( \nu^{19} + \)\(15\!\cdots\!92\)\( \nu^{18} - \)\(86\!\cdots\!62\)\( \nu^{17} + \)\(20\!\cdots\!34\)\( \nu^{16} - \)\(90\!\cdots\!82\)\( \nu^{15} + \)\(16\!\cdots\!05\)\( \nu^{14} - \)\(85\!\cdots\!09\)\( \nu^{13} - \)\(15\!\cdots\!73\)\( \nu^{12} - \)\(79\!\cdots\!40\)\( \nu^{11} - \)\(40\!\cdots\!84\)\( \nu^{10} - \)\(25\!\cdots\!74\)\( \nu^{9} + \)\(38\!\cdots\!84\)\( \nu^{8} - \)\(62\!\cdots\!33\)\( \nu^{7} + \)\(68\!\cdots\!74\)\( \nu^{6} - \)\(15\!\cdots\!85\)\( \nu^{5} + \)\(31\!\cdots\!87\)\( \nu^{4} - \)\(36\!\cdots\!50\)\( \nu^{3} + \)\(56\!\cdots\!45\)\( \nu^{2} - \)\(19\!\cdots\!05\)\( \nu + \)\(29\!\cdots\!00\)\(\)\()/ \)\(99\!\cdots\!10\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(31\!\cdots\!87\)\( \nu^{19} + \)\(94\!\cdots\!55\)\( \nu^{18} - \)\(45\!\cdots\!28\)\( \nu^{17} + \)\(12\!\cdots\!79\)\( \nu^{16} - \)\(47\!\cdots\!35\)\( \nu^{15} + \)\(20\!\cdots\!75\)\( \nu^{14} - \)\(44\!\cdots\!73\)\( \nu^{13} - \)\(63\!\cdots\!55\)\( \nu^{12} - \)\(41\!\cdots\!32\)\( \nu^{11} - \)\(55\!\cdots\!27\)\( \nu^{10} - \)\(13\!\cdots\!33\)\( \nu^{9} + \)\(24\!\cdots\!86\)\( \nu^{8} - \)\(34\!\cdots\!28\)\( \nu^{7} + \)\(15\!\cdots\!30\)\( \nu^{6} - \)\(80\!\cdots\!49\)\( \nu^{5} + \)\(19\!\cdots\!38\)\( \nu^{4} - \)\(20\!\cdots\!31\)\( \nu^{3} + \)\(36\!\cdots\!60\)\( \nu^{2} - \)\(12\!\cdots\!00\)\( \nu + \)\(33\!\cdots\!00\)\(\)\()/ \)\(39\!\cdots\!40\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(34\!\cdots\!31\)\( \nu^{19} - \)\(47\!\cdots\!27\)\( \nu^{18} + \)\(48\!\cdots\!18\)\( \nu^{17} - \)\(60\!\cdots\!93\)\( \nu^{16} + \)\(50\!\cdots\!09\)\( \nu^{15} + \)\(54\!\cdots\!63\)\( \nu^{14} + \)\(49\!\cdots\!03\)\( \nu^{13} + \)\(14\!\cdots\!07\)\( \nu^{12} + \)\(46\!\cdots\!00\)\( \nu^{11} + \)\(73\!\cdots\!21\)\( \nu^{10} + \)\(15\!\cdots\!09\)\( \nu^{9} - \)\(99\!\cdots\!84\)\( \nu^{8} + \)\(33\!\cdots\!90\)\( \nu^{7} + \)\(20\!\cdots\!74\)\( \nu^{6} + \)\(91\!\cdots\!07\)\( \nu^{5} - \)\(10\!\cdots\!58\)\( \nu^{4} + \)\(19\!\cdots\!53\)\( \nu^{3} - \)\(19\!\cdots\!22\)\( \nu^{2} + \)\(10\!\cdots\!00\)\( \nu - \)\(22\!\cdots\!60\)\(\)\()/ \)\(39\!\cdots\!40\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(18\!\cdots\!43\)\( \nu^{19} + \)\(49\!\cdots\!70\)\( \nu^{18} - \)\(24\!\cdots\!96\)\( \nu^{17} + \)\(63\!\cdots\!66\)\( \nu^{16} - \)\(26\!\cdots\!99\)\( \nu^{15} + \)\(44\!\cdots\!24\)\( \nu^{14} - \)\(24\!\cdots\!11\)\( \nu^{13} - \)\(40\!\cdots\!09\)\( \nu^{12} - \)\(22\!\cdots\!74\)\( \nu^{11} - \)\(56\!\cdots\!80\)\( \nu^{10} - \)\(67\!\cdots\!85\)\( \nu^{9} + \)\(14\!\cdots\!86\)\( \nu^{8} - \)\(17\!\cdots\!56\)\( \nu^{7} + \)\(37\!\cdots\!06\)\( \nu^{6} - \)\(41\!\cdots\!05\)\( \nu^{5} + \)\(11\!\cdots\!52\)\( \nu^{4} - \)\(98\!\cdots\!13\)\( \nu^{3} + \)\(17\!\cdots\!69\)\( \nu^{2} - \)\(49\!\cdots\!70\)\( \nu + \)\(22\!\cdots\!40\)\(\)\()/ \)\(19\!\cdots\!20\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(18\!\cdots\!89\)\( \nu^{19} + \)\(58\!\cdots\!03\)\( \nu^{18} - \)\(25\!\cdots\!45\)\( \nu^{17} + \)\(75\!\cdots\!75\)\( \nu^{16} - \)\(26\!\cdots\!46\)\( \nu^{15} + \)\(15\!\cdots\!41\)\( \nu^{14} - \)\(24\!\cdots\!07\)\( \nu^{13} - \)\(30\!\cdots\!01\)\( \nu^{12} - \)\(22\!\cdots\!09\)\( \nu^{11} + \)\(47\!\cdots\!38\)\( \nu^{10} - \)\(66\!\cdots\!90\)\( \nu^{9} + \)\(17\!\cdots\!51\)\( \nu^{8} - \)\(17\!\cdots\!98\)\( \nu^{7} + \)\(79\!\cdots\!43\)\( \nu^{6} - \)\(41\!\cdots\!17\)\( \nu^{5} + \)\(12\!\cdots\!73\)\( \nu^{4} - \)\(10\!\cdots\!84\)\( \nu^{3} + \)\(19\!\cdots\!19\)\( \nu^{2} - \)\(48\!\cdots\!50\)\( \nu + \)\(14\!\cdots\!20\)\(\)\()/ \)\(19\!\cdots\!20\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(40\!\cdots\!63\)\( \nu^{19} + \)\(73\!\cdots\!55\)\( \nu^{18} - \)\(57\!\cdots\!36\)\( \nu^{17} + \)\(95\!\cdots\!45\)\( \nu^{16} - \)\(60\!\cdots\!65\)\( \nu^{15} - \)\(39\!\cdots\!25\)\( \nu^{14} - \)\(58\!\cdots\!13\)\( \nu^{13} - \)\(14\!\cdots\!63\)\( \nu^{12} - \)\(54\!\cdots\!80\)\( \nu^{11} - \)\(67\!\cdots\!53\)\( \nu^{10} - \)\(18\!\cdots\!21\)\( \nu^{9} + \)\(15\!\cdots\!96\)\( \nu^{8} - \)\(41\!\cdots\!84\)\( \nu^{7} - \)\(11\!\cdots\!02\)\( \nu^{6} - \)\(10\!\cdots\!07\)\( \nu^{5} + \)\(14\!\cdots\!96\)\( \nu^{4} - \)\(24\!\cdots\!95\)\( \nu^{3} + \)\(31\!\cdots\!14\)\( \nu^{2} - \)\(14\!\cdots\!80\)\( \nu + \)\(33\!\cdots\!80\)\(\)\()/ \)\(39\!\cdots\!40\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{6} + \beta_{5} - \beta_{3} + \beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{15} + \beta_{14} + \beta_{13} - 49 \beta_{9} - 2 \beta_{7} + \beta_{5} + \beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{18} - \beta_{16} + 2 \beta_{15} - 3 \beta_{12} - 16 \beta_{9} + 44 \beta_{8} + 16 \beta_{7} - 79 \beta_{6} + \beta_{5} - 3 \beta_{4} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(-15 \beta_{19} - \beta_{18} + 4 \beta_{17} + \beta_{16} + 7 \beta_{15} - 9 \beta_{14} - 118 \beta_{13} - 17 \beta_{11} - 6 \beta_{10} - 17 \beta_{9} - 109 \beta_{8} + 4103 \beta_{7} - 88 \beta_{6} - 16 \beta_{5} - 15 \beta_{4} + 94 \beta_{3} - 18 \beta_{2} + 95 \beta_{1} + 111\)
\(\nu^{5}\)\(=\)\(-110 \beta_{18} - 395 \beta_{17} - 494 \beta_{16} - 505 \beta_{15} - 150 \beta_{14} - 161 \beta_{13} + 395 \beta_{12} + 615 \beta_{11} + 121 \beta_{10} + 6527 \beta_{9} + 5912 \beta_{8} - 395 \beta_{7} - 140 \beta_{6} + 354 \beta_{5} + 384 \beta_{4} + 7246 \beta_{3} + 323 \beta_{2} + 110 \beta_{1} - 10803\)
\(\nu^{6}\)\(=\)\(2122 \beta_{19} + 1654 \beta_{18} - 2122 \beta_{17} + 827 \beta_{16} - 1053 \beta_{15} - 9744 \beta_{14} + 476 \beta_{13} + 1471 \beta_{12} + 702 \beta_{11} - 234 \beta_{10} + 378939 \beta_{9} + 379025 \beta_{8} - 379025 \beta_{7} - 11569 \beta_{6} - 26522 \beta_{5} + 28051 \beta_{3} + 1295 \beta_{2} - 11335 \beta_{1} - 378939\)
\(\nu^{7}\)\(=\)\(3254 \beta_{19} - 9950 \beta_{18} + 40006 \beta_{17} + 40970 \beta_{16} - 10342 \beta_{15} - 49384 \beta_{14} - 28754 \beta_{13} + 3254 \beta_{12} - 42296 \beta_{11} - 9950 \beta_{10} + 1368252 \beta_{9} + 3254 \beta_{8} - 924934 \beta_{7} - 772403 \beta_{5} - 40006 \beta_{4} + 48194 \beta_{3} - 25500 \beta_{2} - 50484 \beta_{1} - 922644\)
\(\nu^{8}\)\(=\)\(177348 \beta_{19} - 31076 \beta_{18} + 50584 \beta_{16} - 1303965 \beta_{15} + 124144 \beta_{14} - 78920 \beta_{13} + 241280 \beta_{12} + 78920 \beta_{11} + 157840 \beta_{10} + 1595710 \beta_{9} - 36213063 \beta_{8} - 1516790 \beta_{7} + 1288533 \beta_{6} - 2149096 \beta_{5} + 241280 \beta_{4} - 157840 \beta_{3} + 203064 \beta_{2} - 2228016 \beta_{1} - 78920\)
\(\nu^{9}\)\(=\)\(4524927 \beta_{19} + 1170881 \beta_{18} - 3886103 \beta_{17} + 2336386 \beta_{16} - 6312007 \beta_{15} + 5373708 \beta_{14} + 4247382 \beta_{13} + 3359422 \beta_{11} - 848781 \beta_{10} + 3359422 \beta_{9} - 124946721 \beta_{8} - 54855695 \beta_{7} + 6993388 \beta_{6} + 2188541 \beta_{5} + 4524927 \beta_{4} - 6144607 \beta_{3} + 1250242 \beta_{2} - 68093391 \beta_{1} + 126112226\)
\(\nu^{10}\)\(=\)\(-3689918 \beta_{18} + 25938091 \beta_{17} + 15102918 \beta_{16} + 22248173 \beta_{15} - 1763270 \beta_{14} + 5381985 \beta_{13} - 25938091 \beta_{12} - 18558255 \beta_{11} - 3455337 \beta_{10} - 339395649 \beta_{9} - 320837394 \beta_{8} + 25938091 \beta_{7} + 289170332 \beta_{6} + 274067414 \beta_{5} - 4743924 \beta_{4} - 432696864 \beta_{3} - 125708178 \beta_{2} + 3689918 \beta_{1} + 3935325780\)
\(\nu^{11}\)\(=\)\(-467464551 \beta_{19} - 77745526 \beta_{18} + 467464551 \beta_{17} - 38872763 \beta_{16} + 851165385 \beta_{15} + 298877648 \beta_{14} + 312846500 \beta_{13} - 102547439 \beta_{12} - 499446122 \beta_{11} + 109727097 \beta_{10} - 22616158914 \beta_{9} - 6504545063 \beta_{8} + 6504545063 \beta_{7} + 6882773099 \beta_{6} + 7540331207 \beta_{5} - 8078650092 \beta_{3} - 428591788 \beta_{2} + 6773046002 \beta_{1} + 22616158914\)
\(\nu^{12}\)\(=\)\(-2521311739 \beta_{19} - 285290813 \beta_{18} - 230635103 \beta_{17} - 2111953519 \beta_{16} + 13233551923 \beta_{15} + 11582868610 \beta_{14} + 12924941839 \beta_{13} - 2521311739 \beta_{12} + 870628426 \beta_{11} - 285290813 \beta_{10} - 411859343235 \beta_{9} - 2521311739 \beta_{8} + 52094969207 \beta_{7} + 51576033857 \beta_{5} + 230635103 \beta_{4} - 34801024927 \beta_{3} + 10403630100 \beta_{2} + 35441018250 \beta_{1} + 51454975884\)
\(\nu^{13}\)\(=\)\(-14716314606 \beta_{19} + 10210150996 \beta_{18} - 16215654966 \beta_{16} + 94634925546 \beta_{15} + 31751667616 \beta_{14} + 4355405318 \beta_{13} - 48249591124 \beta_{12} - 4355405318 \beta_{11} - 8710810636 \beta_{10} - 1926949231874 \beta_{9} + 704687282694 \beta_{8} + 1922593826556 \beta_{7} - 696597054827 \beta_{6} + 156042572200 \beta_{5} - 48249591124 \beta_{4} + 8710810636 \beta_{3} + 27396262298 \beta_{2} + 160397977518 \beta_{1} + 4355405318\)
\(\nu^{14}\)\(=\)\(-292352283784 \beta_{19} - 34854111888 \beta_{18} - 5557400036 \beta_{17} - 252545818884 \beta_{16} + 213143858856 \beta_{15} - 269670351316 \beta_{14} - 1276170297737 \beta_{13} - 74660576788 \beta_{11} - 22681932468 \beta_{10} - 74660576788 \beta_{9} + 7326882251822 \beta_{8} + 36497822712087 \beta_{7} - 4149926973608 \beta_{6} - 39806464900 \beta_{5} - 292352283784 \beta_{4} + 4172608906076 \beta_{3} - 96332957360 \beta_{2} + 1831752828189 \beta_{1} - 7544573958818\)
\(\nu^{15}\)\(=\)\(-479735928081 \beta_{18} - 4998525075371 \beta_{17} - 5009638076492 \beta_{16} - 5478261003452 \beta_{15} - 5833470453945 \beta_{14} - 6302093380905 \beta_{13} + 4998525075371 \beta_{12} + 5957996931533 \beta_{11} + 948358855041 \beta_{10} + 230996984179853 \beta_{9} + 225038987248320 \beta_{8} - 4998525075371 \beta_{7} - 20815879578792 \beta_{6} - 15806241502300 \beta_{5} + 3024703924323 \beta_{4} + 91011257319678 \beta_{3} + 5134831490538 \beta_{2} + 479735928081 \beta_{1} - 316478871362913\)
\(\nu^{16}\)\(=\)\(31257017635655 \beta_{19} + 10444761992334 \beta_{18} - 31257017635655 \beta_{17} + 5222380996167 \beta_{16} - 54798838289509 \beta_{15} - 123780821569211 \beta_{14} - 25434676952452 \beta_{13} + 34946349192051 \beta_{12} + 24141780340890 \beta_{11} - 3329524697569 \beta_{10} + 4794010431468943 \beta_{9} + 3761379846385786 \beta_{8} - 3761379846385786 \beta_{7} - 224364841258877 \beta_{6} - 681629133294317 \beta_{5} + 710993294631374 \beta_{3} + 26034636639488 \beta_{2} - 221035316561308 \beta_{1} - 4794010431468943\)
\(\nu^{17}\)\(=\)\(253472038415379 \beta_{19} - 38850817478382 \beta_{18} + 267287994097648 \beta_{17} + 471516705285384 \beta_{16} - 941051393639587 \beta_{15} - 1004110676549499 \beta_{14} - 512520647815253 \beta_{13} + 253472038415379 \beta_{12} - 316531321325291 \beta_{11} - 38850817478382 \beta_{10} + 35797604647510521 \beta_{9} + 253472038415379 \beta_{8} - 25735185625742184 \beta_{7} - 10244286260390596 \beta_{5} - 267287994097648 \beta_{4} + 2739023730421602 \beta_{3} - 259048609399874 \beta_{2} - 2788267057649245 \beta_{1} - 25685942298514541\)
\(\nu^{18}\)\(=\)\(4072836091727351 \beta_{19} - 311297496607090 \beta_{18} + 3424330209203155 \beta_{16} - 17349138802577429 \beta_{15} - 27004552600256 \beta_{14} - 479901689565643 \beta_{13} + 3368374317400506 \beta_{12} + 479901689565643 \beta_{11} + 959803379131286 \beta_{10} + 136466835017997473 \beta_{9} - 380165483354979040 \beta_{8} - 135986933328431830 \beta_{7} + 25860116896127953 \beta_{6} - 55080114063016684 \beta_{5} + 3368374317400506 \beta_{4} - 959803379131286 \beta_{3} + 452897136965387 \beta_{2} - 55560015752582327 \beta_{1} - 479901689565643\)
\(\nu^{19}\)\(=\)\(54604404403695164 \beta_{19} + 8186613684898552 \beta_{18} - 22999220571188550 \beta_{17} + 37775541561213170 \beta_{16} - 106882899856484164 \beta_{15} + 57689583268048442 \beta_{14} + 59702677649205624 \beta_{13} + 25015476527380546 \beta_{11} - 3085178864353278 \beta_{10} + 25015476527380546 \beta_{9} - 2925276882208096176 \beta_{8} - 1214189184287924294 \beta_{7} + 340815369890122060 \beta_{6} + 16828862842481994 \beta_{5} + 54604404403695164 \beta_{4} - 337730191025768782 \beta_{3} - 32364453745953728 \beta_{2} - 750073753709315399 \beta_{1} + 2954865810084410794\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(-\beta_{8}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
−7.12695 + 5.17803i
−4.29115 + 3.11770i
0.218287 0.158595i
3.88934 2.82577i
8.61948 6.26242i
−3.00337 + 9.24343i
−2.05193 + 6.31518i
0.992840 3.05565i
1.17535 3.61737i
3.07809 9.47338i
−7.12695 5.17803i
−4.29115 3.11770i
0.218287 + 0.158595i
3.88934 + 2.82577i
8.61948 + 6.26242i
−3.00337 9.24343i
−2.05193 6.31518i
0.992840 + 3.05565i
1.17535 + 3.61737i
3.07809 + 9.47338i
−7.93597 5.76582i 2.78115 8.55951i 19.8464 + 61.0809i −37.2750 + 27.0819i −71.4237 + 51.8924i 37.2644 + 114.688i 97.6803 300.629i −65.5304 47.6106i 451.963
4.2 −5.10017 3.70549i 2.78115 8.55951i 2.39251 + 7.36340i 54.7018 39.7432i −45.9015 + 33.3494i −37.8504 116.492i −47.2561 + 145.439i −65.5304 47.6106i −426.256
4.3 −0.590730 0.429190i 2.78115 8.55951i −9.72379 29.9267i −77.5329 + 56.3310i −5.31657 + 3.86271i 29.5747 + 91.0216i −14.3206 + 44.0742i −65.5304 47.6106i 69.9777
4.4 3.08033 + 2.23799i 2.78115 8.55951i −5.40872 16.6463i 20.5684 14.9438i 27.7229 20.1419i −26.9084 82.8155i 58.2442 179.257i −65.5304 47.6106i 96.8014
4.5 7.81047 + 5.67464i 2.78115 8.55951i 18.9134 + 58.2093i 25.1385 18.2642i 70.2942 51.0717i 56.7190 + 174.563i −87.1280 + 268.152i −65.5304 47.6106i 299.987
16.1 −2.69436 8.29237i −7.28115 5.29007i −35.6154 + 25.8761i −23.0548 + 70.9555i −24.2492 + 74.6314i 46.3463 33.6726i 84.8093 + 61.6176i 25.0304 + 77.0356i 650.507
16.2 −1.74291 5.36413i −7.28115 5.29007i 0.152432 0.110748i 21.0541 64.7978i −15.6862 + 48.2771i −106.538 + 77.4041i −146.876 106.711i 25.0304 + 77.0356i −384.279
16.3 1.30186 + 4.00670i −7.28115 5.29007i 11.5297 8.37681i −26.9724 + 83.0125i 11.7167 36.0603i −114.007 + 82.8310i 157.639 + 114.532i 25.0304 + 77.0356i −367.721
16.4 1.48437 + 4.56842i −7.28115 5.29007i 7.22140 5.24666i 18.4760 56.8633i 13.3593 41.1158i 58.2735 42.3382i 159.044 + 115.553i 25.0304 + 77.0356i 287.201
16.5 3.38711 + 10.4244i −7.28115 5.29007i −71.3079 + 51.8082i 8.39633 25.8412i 30.4839 93.8199i −110.374 + 80.1917i −497.837 361.700i 25.0304 + 77.0356i 297.820
25.1 −7.93597 + 5.76582i 2.78115 + 8.55951i 19.8464 61.0809i −37.2750 27.0819i −71.4237 51.8924i 37.2644 114.688i 97.6803 + 300.629i −65.5304 + 47.6106i 451.963
25.2 −5.10017 + 3.70549i 2.78115 + 8.55951i 2.39251 7.36340i 54.7018 + 39.7432i −45.9015 33.3494i −37.8504 + 116.492i −47.2561 145.439i −65.5304 + 47.6106i −426.256
25.3 −0.590730 + 0.429190i 2.78115 + 8.55951i −9.72379 + 29.9267i −77.5329 56.3310i −5.31657 3.86271i 29.5747 91.0216i −14.3206 44.0742i −65.5304 + 47.6106i 69.9777
25.4 3.08033 2.23799i 2.78115 + 8.55951i −5.40872 + 16.6463i 20.5684 + 14.9438i 27.7229 + 20.1419i −26.9084 + 82.8155i 58.2442 + 179.257i −65.5304 + 47.6106i 96.8014
25.5 7.81047 5.67464i 2.78115 + 8.55951i 18.9134 58.2093i 25.1385 + 18.2642i 70.2942 + 51.0717i 56.7190 174.563i −87.1280 268.152i −65.5304 + 47.6106i 299.987
31.1 −2.69436 + 8.29237i −7.28115 + 5.29007i −35.6154 25.8761i −23.0548 70.9555i −24.2492 74.6314i 46.3463 + 33.6726i 84.8093 61.6176i 25.0304 77.0356i 650.507
31.2 −1.74291 + 5.36413i −7.28115 + 5.29007i 0.152432 + 0.110748i 21.0541 + 64.7978i −15.6862 48.2771i −106.538 77.4041i −146.876 + 106.711i 25.0304 77.0356i −384.279
31.3 1.30186 4.00670i −7.28115 + 5.29007i 11.5297 + 8.37681i −26.9724 83.0125i 11.7167 + 36.0603i −114.007 82.8310i 157.639 114.532i 25.0304 77.0356i −367.721
31.4 1.48437 4.56842i −7.28115 + 5.29007i 7.22140 + 5.24666i 18.4760 + 56.8633i 13.3593 + 41.1158i 58.2735 + 42.3382i 159.044 115.553i 25.0304 77.0356i 287.201
31.5 3.38711 10.4244i −7.28115 + 5.29007i −71.3079 51.8082i 8.39633 + 25.8412i 30.4839 + 93.8199i −110.374 80.1917i −497.837 + 361.700i 25.0304 77.0356i 297.820
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.5
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 33.6.e.a 20
3.b odd 2 1 99.6.f.c 20
11.c even 5 1 inner 33.6.e.a 20
11.c even 5 1 363.6.a.u 10
11.d odd 10 1 363.6.a.q 10
33.f even 10 1 1089.6.a.bl 10
33.h odd 10 1 99.6.f.c 20
33.h odd 10 1 1089.6.a.bh 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.e.a 20 1.a even 1 1 trivial
33.6.e.a 20 11.c even 5 1 inner
99.6.f.c 20 3.b odd 2 1
99.6.f.c 20 33.h odd 10 1
363.6.a.q 10 11.d odd 10 1
363.6.a.u 10 11.c even 5 1
1089.6.a.bh 10 33.h odd 10 1
1089.6.a.bl 10 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{20} + \cdots\) acting on \(S_{6}^{\mathrm{new}}(33, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T - 16 T^{2} + 48 T^{3} + 97 T^{4} - 11088 T^{5} - 57229 T^{6} - 65569 T^{7} - 928679 T^{8} - 1815992 T^{9} + 48355944 T^{10} + 343339400 T^{11} + 1961223104 T^{12} + 10055712064 T^{13} + 57349720768 T^{14} + 180538204672 T^{15} - 800270554624 T^{16} - 8469852554752 T^{17} - 8738588782592 T^{18} - 282455771086848 T^{19} - 3521355536494592 T^{20} - 9038584674779136 T^{21} - 8948314913374208 T^{22} - 277540128514113536 T^{23} - 839144497085415424 T^{24} + 6057856912068706304 T^{25} + 61578793783323000832 T^{26} + \)\(34\!\cdots\!52\)\( T^{27} + \)\(21\!\cdots\!04\)\( T^{28} + \)\(12\!\cdots\!00\)\( T^{29} + \)\(54\!\cdots\!56\)\( T^{30} - \)\(65\!\cdots\!56\)\( T^{31} - \)\(10\!\cdots\!04\)\( T^{32} - \)\(24\!\cdots\!08\)\( T^{33} - \)\(67\!\cdots\!96\)\( T^{34} - \)\(41\!\cdots\!84\)\( T^{35} + \)\(11\!\cdots\!72\)\( T^{36} + \)\(18\!\cdots\!36\)\( T^{37} - \)\(19\!\cdots\!84\)\( T^{38} + \)\(79\!\cdots\!36\)\( T^{39} + \)\(12\!\cdots\!76\)\( T^{40} \)
$3$ \( ( 1 + 9 T + 81 T^{2} + 729 T^{3} + 6561 T^{4} )^{5} \)
$5$ \( 1 + 33 T - 2574 T^{2} - 442362 T^{3} - 3649743 T^{4} + 859431888 T^{5} + 68263423063 T^{6} - 613695163242 T^{7} - 8224692529806 T^{8} - 3916916706538077 T^{9} - 259205059944140452 T^{10} - 18234184730469093483 T^{11} + \)\(13\!\cdots\!54\)\( T^{12} + \)\(10\!\cdots\!62\)\( T^{13} + \)\(30\!\cdots\!73\)\( T^{14} - \)\(21\!\cdots\!88\)\( T^{15} - \)\(11\!\cdots\!33\)\( T^{16} - \)\(41\!\cdots\!58\)\( T^{17} - \)\(29\!\cdots\!14\)\( T^{18} + \)\(24\!\cdots\!27\)\( T^{19} + \)\(19\!\cdots\!06\)\( T^{20} + \)\(76\!\cdots\!75\)\( T^{21} - \)\(28\!\cdots\!50\)\( T^{22} - \)\(12\!\cdots\!50\)\( T^{23} - \)\(10\!\cdots\!25\)\( T^{24} - \)\(64\!\cdots\!00\)\( T^{25} + \)\(28\!\cdots\!25\)\( T^{26} + \)\(30\!\cdots\!50\)\( T^{27} + \)\(11\!\cdots\!50\)\( T^{28} - \)\(51\!\cdots\!75\)\( T^{29} - \)\(23\!\cdots\!00\)\( T^{30} - \)\(10\!\cdots\!25\)\( T^{31} - \)\(71\!\cdots\!50\)\( T^{32} - \)\(16\!\cdots\!50\)\( T^{33} + \)\(57\!\cdots\!75\)\( T^{34} + \)\(22\!\cdots\!00\)\( T^{35} - \)\(30\!\cdots\!75\)\( T^{36} - \)\(11\!\cdots\!50\)\( T^{37} - \)\(20\!\cdots\!50\)\( T^{38} + \)\(83\!\cdots\!25\)\( T^{39} + \)\(78\!\cdots\!25\)\( T^{40} \)
$7$ \( 1 + 335 T + 16915 T^{2} - 7182176 T^{3} - 1051965620 T^{4} + 72793993411 T^{5} + 35141188593091 T^{6} + 2739326802417471 T^{7} - 473045475333609542 T^{8} - \)\(12\!\cdots\!56\)\( T^{9} - \)\(77\!\cdots\!78\)\( T^{10} + \)\(16\!\cdots\!28\)\( T^{11} + \)\(37\!\cdots\!83\)\( T^{12} + \)\(67\!\cdots\!79\)\( T^{13} - \)\(73\!\cdots\!23\)\( T^{14} - \)\(94\!\cdots\!99\)\( T^{15} + \)\(33\!\cdots\!51\)\( T^{16} + \)\(20\!\cdots\!36\)\( T^{17} + \)\(17\!\cdots\!95\)\( T^{18} - \)\(14\!\cdots\!09\)\( T^{19} - \)\(44\!\cdots\!35\)\( T^{20} - \)\(24\!\cdots\!63\)\( T^{21} + \)\(49\!\cdots\!55\)\( T^{22} + \)\(97\!\cdots\!48\)\( T^{23} + \)\(27\!\cdots\!51\)\( T^{24} - \)\(12\!\cdots\!93\)\( T^{25} - \)\(16\!\cdots\!27\)\( T^{26} + \)\(25\!\cdots\!97\)\( T^{27} + \)\(23\!\cdots\!83\)\( T^{28} + \)\(17\!\cdots\!96\)\( T^{29} - \)\(13\!\cdots\!22\)\( T^{30} - \)\(38\!\cdots\!08\)\( T^{31} - \)\(24\!\cdots\!42\)\( T^{32} + \)\(23\!\cdots\!97\)\( T^{33} + \)\(50\!\cdots\!59\)\( T^{34} + \)\(17\!\cdots\!73\)\( T^{35} - \)\(42\!\cdots\!20\)\( T^{36} - \)\(48\!\cdots\!32\)\( T^{37} + \)\(19\!\cdots\!35\)\( T^{38} + \)\(64\!\cdots\!05\)\( T^{39} + \)\(32\!\cdots\!01\)\( T^{40} \)
$11$ \( 1 + 835 T + 411576 T^{2} + 135113924 T^{3} + 51276137451 T^{4} + 21291035814554 T^{5} + 14177742943869417 T^{6} + 7062016859740932364 T^{7} + \)\(25\!\cdots\!52\)\( T^{8} + \)\(69\!\cdots\!43\)\( T^{9} + \)\(25\!\cdots\!46\)\( T^{10} + \)\(11\!\cdots\!93\)\( T^{11} + \)\(66\!\cdots\!52\)\( T^{12} + \)\(29\!\cdots\!64\)\( T^{13} + \)\(95\!\cdots\!17\)\( T^{14} + \)\(23\!\cdots\!54\)\( T^{15} + \)\(89\!\cdots\!51\)\( T^{16} + \)\(37\!\cdots\!24\)\( T^{17} + \)\(18\!\cdots\!76\)\( T^{18} + \)\(60\!\cdots\!85\)\( T^{19} + \)\(11\!\cdots\!01\)\( T^{20} \)
$13$ \( 1 + 959 T - 957789 T^{2} - 1383202614 T^{3} - 13508261034 T^{4} + 834097485595449 T^{5} + 599161941806513193 T^{6} - 91772052083258869065 T^{7} - \)\(43\!\cdots\!68\)\( T^{8} - \)\(20\!\cdots\!70\)\( T^{9} + \)\(96\!\cdots\!94\)\( T^{10} + \)\(14\!\cdots\!66\)\( T^{11} + \)\(51\!\cdots\!11\)\( T^{12} - \)\(33\!\cdots\!31\)\( T^{13} - \)\(43\!\cdots\!29\)\( T^{14} - \)\(11\!\cdots\!87\)\( T^{15} + \)\(97\!\cdots\!99\)\( T^{16} + \)\(10\!\cdots\!82\)\( T^{17} + \)\(30\!\cdots\!91\)\( T^{18} - \)\(19\!\cdots\!83\)\( T^{19} - \)\(24\!\cdots\!17\)\( T^{20} - \)\(73\!\cdots\!19\)\( T^{21} + \)\(41\!\cdots\!59\)\( T^{22} + \)\(51\!\cdots\!74\)\( T^{23} + \)\(18\!\cdots\!99\)\( T^{24} - \)\(82\!\cdots\!91\)\( T^{25} - \)\(11\!\cdots\!21\)\( T^{26} - \)\(32\!\cdots\!67\)\( T^{27} + \)\(18\!\cdots\!11\)\( T^{28} + \)\(19\!\cdots\!38\)\( T^{29} + \)\(48\!\cdots\!06\)\( T^{30} - \)\(37\!\cdots\!90\)\( T^{31} - \)\(30\!\cdots\!68\)\( T^{32} - \)\(23\!\cdots\!45\)\( T^{33} + \)\(56\!\cdots\!57\)\( T^{34} + \)\(29\!\cdots\!93\)\( T^{35} - \)\(17\!\cdots\!34\)\( T^{36} - \)\(66\!\cdots\!02\)\( T^{37} - \)\(17\!\cdots\!61\)\( T^{38} + \)\(64\!\cdots\!63\)\( T^{39} + \)\(24\!\cdots\!01\)\( T^{40} \)
$17$ \( 1 + 3144 T + 5250373 T^{2} + 7642103232 T^{3} + 11637656778967 T^{4} + 14893166309468376 T^{5} + 19425847958633989495 T^{6} + \)\(28\!\cdots\!20\)\( T^{7} + \)\(40\!\cdots\!14\)\( T^{8} + \)\(53\!\cdots\!92\)\( T^{9} + \)\(68\!\cdots\!49\)\( T^{10} + \)\(82\!\cdots\!20\)\( T^{11} + \)\(10\!\cdots\!49\)\( T^{12} + \)\(13\!\cdots\!32\)\( T^{13} + \)\(17\!\cdots\!75\)\( T^{14} + \)\(21\!\cdots\!08\)\( T^{15} + \)\(26\!\cdots\!16\)\( T^{16} + \)\(32\!\cdots\!88\)\( T^{17} + \)\(37\!\cdots\!13\)\( T^{18} + \)\(43\!\cdots\!24\)\( T^{19} + \)\(53\!\cdots\!21\)\( T^{20} + \)\(62\!\cdots\!68\)\( T^{21} + \)\(75\!\cdots\!37\)\( T^{22} + \)\(94\!\cdots\!84\)\( T^{23} + \)\(10\!\cdots\!16\)\( T^{24} + \)\(12\!\cdots\!56\)\( T^{25} + \)\(14\!\cdots\!75\)\( T^{26} + \)\(15\!\cdots\!76\)\( T^{27} + \)\(16\!\cdots\!49\)\( T^{28} + \)\(19\!\cdots\!40\)\( T^{29} + \)\(22\!\cdots\!01\)\( T^{30} + \)\(25\!\cdots\!56\)\( T^{31} + \)\(27\!\cdots\!14\)\( T^{32} + \)\(27\!\cdots\!40\)\( T^{33} + \)\(26\!\cdots\!55\)\( T^{34} + \)\(28\!\cdots\!68\)\( T^{35} + \)\(31\!\cdots\!67\)\( T^{36} + \)\(29\!\cdots\!24\)\( T^{37} + \)\(28\!\cdots\!77\)\( T^{38} + \)\(24\!\cdots\!92\)\( T^{39} + \)\(11\!\cdots\!01\)\( T^{40} \)
$19$ \( 1 + 930 T - 8334375 T^{2} - 15923705314 T^{3} + 36437156271477 T^{4} + 111140628302011558 T^{5} - \)\(10\!\cdots\!01\)\( T^{6} - \)\(48\!\cdots\!10\)\( T^{7} + \)\(16\!\cdots\!30\)\( T^{8} + \)\(17\!\cdots\!94\)\( T^{9} - \)\(16\!\cdots\!55\)\( T^{10} - \)\(56\!\cdots\!02\)\( T^{11} - \)\(17\!\cdots\!21\)\( T^{12} + \)\(15\!\cdots\!78\)\( T^{13} + \)\(11\!\cdots\!67\)\( T^{14} - \)\(35\!\cdots\!06\)\( T^{15} - \)\(46\!\cdots\!72\)\( T^{16} + \)\(59\!\cdots\!14\)\( T^{17} + \)\(14\!\cdots\!97\)\( T^{18} - \)\(48\!\cdots\!26\)\( T^{19} - \)\(37\!\cdots\!41\)\( T^{20} - \)\(11\!\cdots\!74\)\( T^{21} + \)\(87\!\cdots\!97\)\( T^{22} + \)\(90\!\cdots\!86\)\( T^{23} - \)\(17\!\cdots\!72\)\( T^{24} - \)\(33\!\cdots\!94\)\( T^{25} + \)\(27\!\cdots\!67\)\( T^{26} + \)\(88\!\cdots\!22\)\( T^{27} - \)\(25\!\cdots\!21\)\( T^{28} - \)\(19\!\cdots\!98\)\( T^{29} - \)\(14\!\cdots\!55\)\( T^{30} + \)\(37\!\cdots\!06\)\( T^{31} + \)\(85\!\cdots\!30\)\( T^{32} - \)\(63\!\cdots\!90\)\( T^{33} - \)\(33\!\cdots\!01\)\( T^{34} + \)\(89\!\cdots\!42\)\( T^{35} + \)\(72\!\cdots\!77\)\( T^{36} - \)\(78\!\cdots\!86\)\( T^{37} - \)\(10\!\cdots\!75\)\( T^{38} + \)\(28\!\cdots\!70\)\( T^{39} + \)\(75\!\cdots\!01\)\( T^{40} \)
$23$ \( ( 1 - 700 T + 9710719 T^{2} - 9119650776 T^{3} + 130079109788580 T^{4} - 95004700444084768 T^{5} + \)\(83\!\cdots\!69\)\( T^{6} - \)\(92\!\cdots\!84\)\( T^{7} + \)\(68\!\cdots\!71\)\( T^{8} - \)\(70\!\cdots\!40\)\( T^{9} + \)\(39\!\cdots\!92\)\( T^{10} - \)\(45\!\cdots\!20\)\( T^{11} + \)\(28\!\cdots\!79\)\( T^{12} - \)\(24\!\cdots\!88\)\( T^{13} + \)\(14\!\cdots\!69\)\( T^{14} - \)\(10\!\cdots\!24\)\( T^{15} + \)\(92\!\cdots\!20\)\( T^{16} - \)\(41\!\cdots\!32\)\( T^{17} + \)\(28\!\cdots\!19\)\( T^{18} - \)\(13\!\cdots\!00\)\( T^{19} + \)\(12\!\cdots\!49\)\( T^{20} )^{2} \)
$29$ \( 1 - 4494 T - 31128298 T^{2} + 208818981018 T^{3} + 208510993266398 T^{4} + 2116509401480725632 T^{5} - \)\(19\!\cdots\!38\)\( T^{6} - \)\(13\!\cdots\!02\)\( T^{7} + \)\(93\!\cdots\!74\)\( T^{8} + \)\(15\!\cdots\!06\)\( T^{9} - \)\(10\!\cdots\!36\)\( T^{10} - \)\(64\!\cdots\!14\)\( T^{11} - \)\(18\!\cdots\!70\)\( T^{12} + \)\(23\!\cdots\!90\)\( T^{13} + \)\(23\!\cdots\!22\)\( T^{14} - \)\(12\!\cdots\!16\)\( T^{15} - \)\(13\!\cdots\!66\)\( T^{16} + \)\(24\!\cdots\!06\)\( T^{17} + \)\(35\!\cdots\!62\)\( T^{18} - \)\(48\!\cdots\!98\)\( T^{19} - \)\(28\!\cdots\!22\)\( T^{20} - \)\(10\!\cdots\!02\)\( T^{21} + \)\(14\!\cdots\!62\)\( T^{22} + \)\(21\!\cdots\!94\)\( T^{23} - \)\(23\!\cdots\!66\)\( T^{24} - \)\(46\!\cdots\!84\)\( T^{25} + \)\(17\!\cdots\!22\)\( T^{26} + \)\(35\!\cdots\!10\)\( T^{27} - \)\(59\!\cdots\!70\)\( T^{28} - \)\(41\!\cdots\!86\)\( T^{29} - \)\(13\!\cdots\!36\)\( T^{30} + \)\(42\!\cdots\!94\)\( T^{31} + \)\(51\!\cdots\!74\)\( T^{32} - \)\(15\!\cdots\!98\)\( T^{33} - \)\(44\!\cdots\!38\)\( T^{34} + \)\(10\!\cdots\!68\)\( T^{35} + \)\(20\!\cdots\!98\)\( T^{36} + \)\(42\!\cdots\!82\)\( T^{37} - \)\(12\!\cdots\!98\)\( T^{38} - \)\(38\!\cdots\!06\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$31$ \( 1 - 9208 T - 100967666 T^{2} + 1256420919406 T^{3} + 2526682313975648 T^{4} - 58366787411029429012 T^{5} + \)\(11\!\cdots\!08\)\( T^{6} + \)\(63\!\cdots\!66\)\( T^{7} + \)\(41\!\cdots\!14\)\( T^{8} + \)\(28\!\cdots\!72\)\( T^{9} - \)\(40\!\cdots\!56\)\( T^{10} - \)\(10\!\cdots\!44\)\( T^{11} + \)\(97\!\cdots\!82\)\( T^{12} + \)\(39\!\cdots\!78\)\( T^{13} + \)\(64\!\cdots\!04\)\( T^{14} - \)\(23\!\cdots\!36\)\( T^{15} - \)\(79\!\cdots\!36\)\( T^{16} + \)\(77\!\cdots\!38\)\( T^{17} + \)\(24\!\cdots\!62\)\( T^{18} - \)\(93\!\cdots\!64\)\( T^{19} - \)\(71\!\cdots\!14\)\( T^{20} - \)\(26\!\cdots\!64\)\( T^{21} + \)\(20\!\cdots\!62\)\( T^{22} + \)\(18\!\cdots\!38\)\( T^{23} - \)\(53\!\cdots\!36\)\( T^{24} - \)\(45\!\cdots\!36\)\( T^{25} + \)\(35\!\cdots\!04\)\( T^{26} + \)\(61\!\cdots\!78\)\( T^{27} + \)\(44\!\cdots\!82\)\( T^{28} - \)\(13\!\cdots\!44\)\( T^{29} - \)\(15\!\cdots\!56\)\( T^{30} + \)\(30\!\cdots\!72\)\( T^{31} + \)\(12\!\cdots\!14\)\( T^{32} + \)\(55\!\cdots\!66\)\( T^{33} + \)\(27\!\cdots\!08\)\( T^{34} - \)\(41\!\cdots\!12\)\( T^{35} + \)\(51\!\cdots\!48\)\( T^{36} + \)\(73\!\cdots\!06\)\( T^{37} - \)\(16\!\cdots\!66\)\( T^{38} - \)\(44\!\cdots\!08\)\( T^{39} + \)\(13\!\cdots\!01\)\( T^{40} \)
$37$ \( 1 + 3153 T - 189334617 T^{2} - 648063671706 T^{3} + 15007180081744662 T^{4} + \)\(14\!\cdots\!63\)\( T^{5} - \)\(63\!\cdots\!67\)\( T^{6} - \)\(16\!\cdots\!27\)\( T^{7} - \)\(76\!\cdots\!68\)\( T^{8} + \)\(11\!\cdots\!02\)\( T^{9} + \)\(68\!\cdots\!42\)\( T^{10} - \)\(13\!\cdots\!10\)\( T^{11} - \)\(31\!\cdots\!09\)\( T^{12} - \)\(30\!\cdots\!53\)\( T^{13} + \)\(16\!\cdots\!47\)\( T^{14} + \)\(98\!\cdots\!59\)\( T^{15} - \)\(10\!\cdots\!33\)\( T^{16} + \)\(10\!\cdots\!22\)\( T^{17} + \)\(27\!\cdots\!39\)\( T^{18} - \)\(65\!\cdots\!17\)\( T^{19} - \)\(43\!\cdots\!05\)\( T^{20} - \)\(45\!\cdots\!69\)\( T^{21} + \)\(13\!\cdots\!11\)\( T^{22} + \)\(36\!\cdots\!46\)\( T^{23} - \)\(24\!\cdots\!33\)\( T^{24} + \)\(15\!\cdots\!63\)\( T^{25} + \)\(18\!\cdots\!03\)\( T^{26} - \)\(23\!\cdots\!29\)\( T^{27} - \)\(17\!\cdots\!09\)\( T^{28} - \)\(50\!\cdots\!70\)\( T^{29} + \)\(17\!\cdots\!58\)\( T^{30} + \)\(19\!\cdots\!86\)\( T^{31} - \)\(94\!\cdots\!68\)\( T^{32} - \)\(14\!\cdots\!39\)\( T^{33} - \)\(38\!\cdots\!83\)\( T^{34} + \)\(58\!\cdots\!59\)\( T^{35} + \)\(42\!\cdots\!62\)\( T^{36} - \)\(12\!\cdots\!42\)\( T^{37} - \)\(26\!\cdots\!33\)\( T^{38} + \)\(30\!\cdots\!29\)\( T^{39} + \)\(66\!\cdots\!01\)\( T^{40} \)
$41$ \( 1 + 7497 T - 230704809 T^{2} - 1973403635478 T^{3} + 38370662638350894 T^{4} + \)\(49\!\cdots\!71\)\( T^{5} - \)\(34\!\cdots\!15\)\( T^{6} - \)\(91\!\cdots\!31\)\( T^{7} - \)\(44\!\cdots\!36\)\( T^{8} + \)\(99\!\cdots\!78\)\( T^{9} + \)\(49\!\cdots\!90\)\( T^{10} - \)\(59\!\cdots\!42\)\( T^{11} - \)\(10\!\cdots\!73\)\( T^{12} - \)\(65\!\cdots\!73\)\( T^{13} + \)\(73\!\cdots\!71\)\( T^{14} + \)\(18\!\cdots\!39\)\( T^{15} + \)\(58\!\cdots\!87\)\( T^{16} - \)\(20\!\cdots\!26\)\( T^{17} - \)\(24\!\cdots\!25\)\( T^{18} + \)\(11\!\cdots\!39\)\( T^{19} + \)\(40\!\cdots\!03\)\( T^{20} + \)\(13\!\cdots\!39\)\( T^{21} - \)\(32\!\cdots\!25\)\( T^{22} - \)\(31\!\cdots\!26\)\( T^{23} + \)\(10\!\cdots\!87\)\( T^{24} + \)\(37\!\cdots\!39\)\( T^{25} + \)\(17\!\cdots\!71\)\( T^{26} - \)\(18\!\cdots\!73\)\( T^{27} - \)\(33\!\cdots\!73\)\( T^{28} - \)\(22\!\cdots\!42\)\( T^{29} + \)\(21\!\cdots\!90\)\( T^{30} + \)\(50\!\cdots\!78\)\( T^{31} - \)\(25\!\cdots\!36\)\( T^{32} - \)\(61\!\cdots\!31\)\( T^{33} - \)\(27\!\cdots\!15\)\( T^{34} + \)\(44\!\cdots\!71\)\( T^{35} + \)\(40\!\cdots\!94\)\( T^{36} - \)\(24\!\cdots\!78\)\( T^{37} - \)\(32\!\cdots\!09\)\( T^{38} + \)\(12\!\cdots\!97\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$43$ \( ( 1 - 31812 T + 1038663587 T^{2} - 19467514122584 T^{3} + 372119581798977172 T^{4} - \)\(48\!\cdots\!16\)\( T^{5} + \)\(67\!\cdots\!53\)\( T^{6} - \)\(66\!\cdots\!48\)\( T^{7} + \)\(79\!\cdots\!67\)\( T^{8} - \)\(66\!\cdots\!80\)\( T^{9} + \)\(92\!\cdots\!08\)\( T^{10} - \)\(98\!\cdots\!40\)\( T^{11} + \)\(17\!\cdots\!83\)\( T^{12} - \)\(21\!\cdots\!36\)\( T^{13} + \)\(31\!\cdots\!53\)\( T^{14} - \)\(33\!\cdots\!88\)\( T^{15} + \)\(37\!\cdots\!28\)\( T^{16} - \)\(28\!\cdots\!88\)\( T^{17} + \)\(22\!\cdots\!87\)\( T^{18} - \)\(10\!\cdots\!16\)\( T^{19} + \)\(47\!\cdots\!49\)\( T^{20} )^{2} \)
$47$ \( 1 + 43031 T + 1013626010 T^{2} + 25690964156472 T^{3} + 573036087382146913 T^{4} + \)\(86\!\cdots\!54\)\( T^{5} + \)\(12\!\cdots\!19\)\( T^{6} + \)\(20\!\cdots\!88\)\( T^{7} + \)\(16\!\cdots\!46\)\( T^{8} - \)\(23\!\cdots\!25\)\( T^{9} - \)\(97\!\cdots\!24\)\( T^{10} - \)\(47\!\cdots\!21\)\( T^{11} - \)\(15\!\cdots\!86\)\( T^{12} - \)\(13\!\cdots\!08\)\( T^{13} - \)\(13\!\cdots\!79\)\( T^{14} - \)\(42\!\cdots\!50\)\( T^{15} - \)\(13\!\cdots\!81\)\( T^{16} + \)\(51\!\cdots\!08\)\( T^{17} - \)\(26\!\cdots\!74\)\( T^{18} + \)\(36\!\cdots\!55\)\( T^{19} + \)\(34\!\cdots\!22\)\( T^{20} + \)\(83\!\cdots\!85\)\( T^{21} - \)\(13\!\cdots\!26\)\( T^{22} + \)\(61\!\cdots\!44\)\( T^{23} - \)\(36\!\cdots\!81\)\( T^{24} - \)\(27\!\cdots\!50\)\( T^{25} - \)\(19\!\cdots\!71\)\( T^{26} - \)\(44\!\cdots\!44\)\( T^{27} - \)\(12\!\cdots\!86\)\( T^{28} - \)\(83\!\cdots\!47\)\( T^{29} - \)\(39\!\cdots\!76\)\( T^{30} - \)\(21\!\cdots\!75\)\( T^{31} + \)\(34\!\cdots\!46\)\( T^{32} + \)\(10\!\cdots\!16\)\( T^{33} + \)\(14\!\cdots\!31\)\( T^{34} + \)\(22\!\cdots\!22\)\( T^{35} + \)\(33\!\cdots\!13\)\( T^{36} + \)\(34\!\cdots\!04\)\( T^{37} + \)\(31\!\cdots\!90\)\( T^{38} + \)\(30\!\cdots\!33\)\( T^{39} + \)\(16\!\cdots\!01\)\( T^{40} \)
$53$ \( 1 + 20452 T - 1381076488 T^{2} - 30178549123020 T^{3} + 1051734263644423168 T^{4} + \)\(30\!\cdots\!92\)\( T^{5} - \)\(41\!\cdots\!04\)\( T^{6} - \)\(21\!\cdots\!32\)\( T^{7} + \)\(12\!\cdots\!08\)\( T^{8} + \)\(10\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!32\)\( T^{10} - \)\(30\!\cdots\!96\)\( T^{11} - \)\(79\!\cdots\!00\)\( T^{12} - \)\(33\!\cdots\!80\)\( T^{13} + \)\(28\!\cdots\!08\)\( T^{14} + \)\(53\!\cdots\!36\)\( T^{15} - \)\(14\!\cdots\!40\)\( T^{16} - \)\(28\!\cdots\!56\)\( T^{17} - \)\(48\!\cdots\!28\)\( T^{18} + \)\(51\!\cdots\!68\)\( T^{19} + \)\(30\!\cdots\!34\)\( T^{20} + \)\(21\!\cdots\!24\)\( T^{21} - \)\(85\!\cdots\!72\)\( T^{22} - \)\(20\!\cdots\!92\)\( T^{23} - \)\(45\!\cdots\!40\)\( T^{24} + \)\(68\!\cdots\!48\)\( T^{25} + \)\(15\!\cdots\!92\)\( T^{26} - \)\(74\!\cdots\!60\)\( T^{27} - \)\(74\!\cdots\!00\)\( T^{28} - \)\(11\!\cdots\!28\)\( T^{29} + \)\(17\!\cdots\!68\)\( T^{30} + \)\(70\!\cdots\!00\)\( T^{31} + \)\(36\!\cdots\!08\)\( T^{32} - \)\(25\!\cdots\!76\)\( T^{33} - \)\(20\!\cdots\!96\)\( T^{34} + \)\(63\!\cdots\!44\)\( T^{35} + \)\(92\!\cdots\!68\)\( T^{36} - \)\(11\!\cdots\!60\)\( T^{37} - \)\(21\!\cdots\!12\)\( T^{38} + \)\(13\!\cdots\!64\)\( T^{39} + \)\(26\!\cdots\!01\)\( T^{40} \)
$59$ \( 1 - 101730 T + 1604806505 T^{2} + 134091168397710 T^{3} - 3128765010982782815 T^{4} - \)\(13\!\cdots\!94\)\( T^{5} + \)\(43\!\cdots\!55\)\( T^{6} + \)\(42\!\cdots\!70\)\( T^{7} - \)\(27\!\cdots\!10\)\( T^{8} + \)\(67\!\cdots\!90\)\( T^{9} + \)\(29\!\cdots\!97\)\( T^{10} - \)\(61\!\cdots\!70\)\( T^{11} - \)\(18\!\cdots\!05\)\( T^{12} + \)\(58\!\cdots\!90\)\( T^{13} + \)\(10\!\cdots\!15\)\( T^{14} - \)\(53\!\cdots\!42\)\( T^{15} + \)\(39\!\cdots\!00\)\( T^{16} + \)\(18\!\cdots\!50\)\( T^{17} - \)\(56\!\cdots\!75\)\( T^{18} - \)\(73\!\cdots\!50\)\( T^{19} + \)\(64\!\cdots\!95\)\( T^{20} - \)\(52\!\cdots\!50\)\( T^{21} - \)\(28\!\cdots\!75\)\( T^{22} + \)\(65\!\cdots\!50\)\( T^{23} + \)\(10\!\cdots\!00\)\( T^{24} - \)\(99\!\cdots\!58\)\( T^{25} + \)\(14\!\cdots\!15\)\( T^{26} + \)\(55\!\cdots\!10\)\( T^{27} - \)\(12\!\cdots\!05\)\( T^{28} - \)\(29\!\cdots\!30\)\( T^{29} + \)\(10\!\cdots\!97\)\( T^{30} + \)\(16\!\cdots\!10\)\( T^{31} - \)\(48\!\cdots\!10\)\( T^{32} + \)\(53\!\cdots\!30\)\( T^{33} + \)\(39\!\cdots\!55\)\( T^{34} - \)\(90\!\cdots\!06\)\( T^{35} - \)\(14\!\cdots\!15\)\( T^{36} + \)\(44\!\cdots\!90\)\( T^{37} + \)\(38\!\cdots\!05\)\( T^{38} - \)\(17\!\cdots\!70\)\( T^{39} + \)\(12\!\cdots\!01\)\( T^{40} \)
$61$ \( 1 - 50745 T - 56399280 T^{2} + 21422082501262 T^{3} - 684714878530830057 T^{4} + \)\(12\!\cdots\!00\)\( T^{5} - \)\(35\!\cdots\!43\)\( T^{6} - \)\(37\!\cdots\!26\)\( T^{7} + \)\(12\!\cdots\!04\)\( T^{8} - \)\(51\!\cdots\!15\)\( T^{9} + \)\(60\!\cdots\!76\)\( T^{10} - \)\(10\!\cdots\!29\)\( T^{11} - \)\(17\!\cdots\!92\)\( T^{12} + \)\(20\!\cdots\!46\)\( T^{13} - \)\(20\!\cdots\!93\)\( T^{14} + \)\(18\!\cdots\!00\)\( T^{15} - \)\(22\!\cdots\!03\)\( T^{16} - \)\(34\!\cdots\!82\)\( T^{17} - \)\(22\!\cdots\!32\)\( T^{18} - \)\(52\!\cdots\!07\)\( T^{19} + \)\(40\!\cdots\!46\)\( T^{20} - \)\(43\!\cdots\!07\)\( T^{21} - \)\(16\!\cdots\!32\)\( T^{22} - \)\(20\!\cdots\!82\)\( T^{23} - \)\(11\!\cdots\!03\)\( T^{24} + \)\(77\!\cdots\!00\)\( T^{25} - \)\(74\!\cdots\!93\)\( T^{26} + \)\(62\!\cdots\!46\)\( T^{27} - \)\(46\!\cdots\!92\)\( T^{28} - \)\(23\!\cdots\!29\)\( T^{29} + \)\(11\!\cdots\!76\)\( T^{30} - \)\(81\!\cdots\!15\)\( T^{31} + \)\(15\!\cdots\!04\)\( T^{32} - \)\(41\!\cdots\!26\)\( T^{33} - \)\(33\!\cdots\!43\)\( T^{34} + \)\(97\!\cdots\!00\)\( T^{35} - \)\(45\!\cdots\!57\)\( T^{36} + \)\(12\!\cdots\!62\)\( T^{37} - \)\(26\!\cdots\!80\)\( T^{38} - \)\(20\!\cdots\!45\)\( T^{39} + \)\(34\!\cdots\!01\)\( T^{40} \)
$67$ \( ( 1 - 5365 T + 2064927002 T^{2} + 94838601342392 T^{3} + 1894255487996497761 T^{4} + \)\(25\!\cdots\!86\)\( T^{5} + \)\(74\!\cdots\!51\)\( T^{6} + \)\(29\!\cdots\!72\)\( T^{7} + \)\(16\!\cdots\!70\)\( T^{8} + \)\(48\!\cdots\!03\)\( T^{9} + \)\(22\!\cdots\!82\)\( T^{10} + \)\(66\!\cdots\!21\)\( T^{11} + \)\(30\!\cdots\!30\)\( T^{12} + \)\(71\!\cdots\!96\)\( T^{13} + \)\(24\!\cdots\!51\)\( T^{14} + \)\(11\!\cdots\!02\)\( T^{15} + \)\(11\!\cdots\!89\)\( T^{16} + \)\(77\!\cdots\!56\)\( T^{17} + \)\(22\!\cdots\!02\)\( T^{18} - \)\(79\!\cdots\!55\)\( T^{19} + \)\(20\!\cdots\!49\)\( T^{20} )^{2} \)
$71$ \( 1 - 164895 T + 8885582890 T^{2} + 28899802515492 T^{3} - 19721275394629928803 T^{4} + \)\(61\!\cdots\!34\)\( T^{5} - \)\(21\!\cdots\!49\)\( T^{6} + \)\(24\!\cdots\!80\)\( T^{7} - \)\(11\!\cdots\!10\)\( T^{8} - \)\(57\!\cdots\!23\)\( T^{9} + \)\(16\!\cdots\!20\)\( T^{10} - \)\(38\!\cdots\!79\)\( T^{11} + \)\(13\!\cdots\!74\)\( T^{12} - \)\(20\!\cdots\!36\)\( T^{13} + \)\(12\!\cdots\!73\)\( T^{14} - \)\(26\!\cdots\!82\)\( T^{15} - \)\(36\!\cdots\!57\)\( T^{16} + \)\(81\!\cdots\!52\)\( T^{17} + \)\(77\!\cdots\!58\)\( T^{18} + \)\(14\!\cdots\!33\)\( T^{19} - \)\(11\!\cdots\!46\)\( T^{20} + \)\(26\!\cdots\!83\)\( T^{21} + \)\(25\!\cdots\!58\)\( T^{22} + \)\(47\!\cdots\!52\)\( T^{23} - \)\(38\!\cdots\!57\)\( T^{24} - \)\(50\!\cdots\!82\)\( T^{25} + \)\(42\!\cdots\!73\)\( T^{26} - \)\(12\!\cdots\!36\)\( T^{27} + \)\(14\!\cdots\!74\)\( T^{28} - \)\(78\!\cdots\!29\)\( T^{29} + \)\(62\!\cdots\!20\)\( T^{30} - \)\(37\!\cdots\!73\)\( T^{31} - \)\(13\!\cdots\!10\)\( T^{32} + \)\(53\!\cdots\!80\)\( T^{33} - \)\(81\!\cdots\!49\)\( T^{34} + \)\(43\!\cdots\!34\)\( T^{35} - \)\(24\!\cdots\!03\)\( T^{36} + \)\(65\!\cdots\!92\)\( T^{37} + \)\(36\!\cdots\!90\)\( T^{38} - \)\(12\!\cdots\!45\)\( T^{39} + \)\(13\!\cdots\!01\)\( T^{40} \)
$73$ \( 1 + 77236 T - 2734137636 T^{2} - 439400254875574 T^{3} - 6136008583422538516 T^{4} + \)\(89\!\cdots\!20\)\( T^{5} + \)\(34\!\cdots\!16\)\( T^{6} - \)\(82\!\cdots\!90\)\( T^{7} - \)\(34\!\cdots\!12\)\( T^{8} - \)\(11\!\cdots\!92\)\( T^{9} - \)\(19\!\cdots\!52\)\( T^{10} - \)\(55\!\cdots\!84\)\( T^{11} + \)\(30\!\cdots\!52\)\( T^{12} + \)\(20\!\cdots\!78\)\( T^{13} + \)\(26\!\cdots\!04\)\( T^{14} - \)\(18\!\cdots\!96\)\( T^{15} - \)\(36\!\cdots\!56\)\( T^{16} - \)\(81\!\cdots\!30\)\( T^{17} - \)\(41\!\cdots\!16\)\( T^{18} + \)\(10\!\cdots\!12\)\( T^{19} + \)\(14\!\cdots\!30\)\( T^{20} + \)\(22\!\cdots\!16\)\( T^{21} - \)\(17\!\cdots\!84\)\( T^{22} - \)\(72\!\cdots\!10\)\( T^{23} - \)\(67\!\cdots\!56\)\( T^{24} - \)\(70\!\cdots\!28\)\( T^{25} + \)\(21\!\cdots\!96\)\( T^{26} + \)\(33\!\cdots\!46\)\( T^{27} + \)\(10\!\cdots\!52\)\( T^{28} - \)\(39\!\cdots\!12\)\( T^{29} - \)\(28\!\cdots\!48\)\( T^{30} - \)\(34\!\cdots\!44\)\( T^{31} - \)\(21\!\cdots\!12\)\( T^{32} - \)\(10\!\cdots\!70\)\( T^{33} + \)\(92\!\cdots\!84\)\( T^{34} + \)\(50\!\cdots\!40\)\( T^{35} - \)\(71\!\cdots\!16\)\( T^{36} - \)\(10\!\cdots\!82\)\( T^{37} - \)\(13\!\cdots\!64\)\( T^{38} + \)\(80\!\cdots\!52\)\( T^{39} + \)\(21\!\cdots\!01\)\( T^{40} \)
$79$ \( 1 + 291931 T + 30414943043 T^{2} + 646593104785424 T^{3} - \)\(12\!\cdots\!32\)\( T^{4} - \)\(98\!\cdots\!49\)\( T^{5} - \)\(74\!\cdots\!89\)\( T^{6} + \)\(12\!\cdots\!11\)\( T^{7} - \)\(71\!\cdots\!66\)\( T^{8} - \)\(11\!\cdots\!36\)\( T^{9} - \)\(28\!\cdots\!62\)\( T^{10} + \)\(25\!\cdots\!40\)\( T^{11} + \)\(27\!\cdots\!99\)\( T^{12} + \)\(12\!\cdots\!87\)\( T^{13} + \)\(29\!\cdots\!17\)\( T^{14} + \)\(48\!\cdots\!13\)\( T^{15} + \)\(60\!\cdots\!79\)\( T^{16} + \)\(32\!\cdots\!00\)\( T^{17} - \)\(21\!\cdots\!01\)\( T^{18} - \)\(40\!\cdots\!93\)\( T^{19} - \)\(28\!\cdots\!03\)\( T^{20} - \)\(12\!\cdots\!07\)\( T^{21} - \)\(20\!\cdots\!01\)\( T^{22} + \)\(95\!\cdots\!00\)\( T^{23} + \)\(53\!\cdots\!79\)\( T^{24} + \)\(13\!\cdots\!87\)\( T^{25} + \)\(25\!\cdots\!17\)\( T^{26} + \)\(33\!\cdots\!13\)\( T^{27} + \)\(21\!\cdots\!99\)\( T^{28} + \)\(64\!\cdots\!60\)\( T^{29} - \)\(21\!\cdots\!62\)\( T^{30} - \)\(26\!\cdots\!64\)\( T^{31} - \)\(51\!\cdots\!66\)\( T^{32} + \)\(27\!\cdots\!89\)\( T^{33} - \)\(50\!\cdots\!89\)\( T^{34} - \)\(20\!\cdots\!51\)\( T^{35} - \)\(81\!\cdots\!32\)\( T^{36} + \)\(12\!\cdots\!76\)\( T^{37} + \)\(18\!\cdots\!43\)\( T^{38} + \)\(54\!\cdots\!69\)\( T^{39} + \)\(57\!\cdots\!01\)\( T^{40} \)
$83$ \( 1 + 255451 T + 34519672899 T^{2} + 3575139843671212 T^{3} + \)\(37\!\cdots\!76\)\( T^{4} + \)\(35\!\cdots\!75\)\( T^{5} + \)\(25\!\cdots\!59\)\( T^{6} + \)\(14\!\cdots\!07\)\( T^{7} + \)\(82\!\cdots\!10\)\( T^{8} + \)\(30\!\cdots\!72\)\( T^{9} - \)\(12\!\cdots\!50\)\( T^{10} - \)\(29\!\cdots\!64\)\( T^{11} - \)\(24\!\cdots\!05\)\( T^{12} - \)\(18\!\cdots\!21\)\( T^{13} - \)\(13\!\cdots\!51\)\( T^{14} - \)\(61\!\cdots\!79\)\( T^{15} - \)\(41\!\cdots\!85\)\( T^{16} + \)\(14\!\cdots\!88\)\( T^{17} + \)\(16\!\cdots\!11\)\( T^{18} + \)\(17\!\cdots\!39\)\( T^{19} + \)\(13\!\cdots\!57\)\( T^{20} + \)\(67\!\cdots\!77\)\( T^{21} + \)\(26\!\cdots\!39\)\( T^{22} + \)\(86\!\cdots\!16\)\( T^{23} - \)\(98\!\cdots\!85\)\( T^{24} - \)\(57\!\cdots\!97\)\( T^{25} - \)\(49\!\cdots\!99\)\( T^{26} - \)\(27\!\cdots\!47\)\( T^{27} - \)\(14\!\cdots\!05\)\( T^{28} - \)\(67\!\cdots\!52\)\( T^{29} - \)\(11\!\cdots\!50\)\( T^{30} + \)\(10\!\cdots\!04\)\( T^{31} + \)\(11\!\cdots\!10\)\( T^{32} + \)\(81\!\cdots\!01\)\( T^{33} + \)\(54\!\cdots\!91\)\( T^{34} + \)\(29\!\cdots\!25\)\( T^{35} + \)\(12\!\cdots\!76\)\( T^{36} + \)\(47\!\cdots\!16\)\( T^{37} + \)\(17\!\cdots\!51\)\( T^{38} + \)\(52\!\cdots\!57\)\( T^{39} + \)\(80\!\cdots\!01\)\( T^{40} \)
$89$ \( ( 1 - 90816 T + 23172579003 T^{2} - 1801726174975080 T^{3} + \)\(27\!\cdots\!36\)\( T^{4} - \)\(16\!\cdots\!08\)\( T^{5} + \)\(19\!\cdots\!73\)\( T^{6} - \)\(94\!\cdots\!52\)\( T^{7} + \)\(10\!\cdots\!47\)\( T^{8} - \)\(41\!\cdots\!60\)\( T^{9} + \)\(54\!\cdots\!44\)\( T^{10} - \)\(23\!\cdots\!40\)\( T^{11} + \)\(32\!\cdots\!47\)\( T^{12} - \)\(16\!\cdots\!48\)\( T^{13} + \)\(19\!\cdots\!73\)\( T^{14} - \)\(91\!\cdots\!92\)\( T^{15} + \)\(81\!\cdots\!36\)\( T^{16} - \)\(30\!\cdots\!20\)\( T^{17} + \)\(21\!\cdots\!03\)\( T^{18} - \)\(47\!\cdots\!84\)\( T^{19} + \)\(29\!\cdots\!01\)\( T^{20} )^{2} \)
$97$ \( 1 + 145083 T - 17142285784 T^{2} - 3965082155917146 T^{3} + 78196752542732073907 T^{4} + \)\(53\!\cdots\!28\)\( T^{5} + \)\(10\!\cdots\!81\)\( T^{6} - \)\(42\!\cdots\!58\)\( T^{7} - \)\(17\!\cdots\!52\)\( T^{8} + \)\(21\!\cdots\!73\)\( T^{9} + \)\(14\!\cdots\!80\)\( T^{10} - \)\(20\!\cdots\!81\)\( T^{11} - \)\(68\!\cdots\!20\)\( T^{12} - \)\(11\!\cdots\!06\)\( T^{13} - \)\(17\!\cdots\!25\)\( T^{14} + \)\(15\!\cdots\!12\)\( T^{15} + \)\(88\!\cdots\!65\)\( T^{16} - \)\(10\!\cdots\!42\)\( T^{17} - \)\(99\!\cdots\!76\)\( T^{18} + \)\(33\!\cdots\!41\)\( T^{19} + \)\(90\!\cdots\!26\)\( T^{20} + \)\(28\!\cdots\!37\)\( T^{21} - \)\(73\!\cdots\!24\)\( T^{22} - \)\(65\!\cdots\!06\)\( T^{23} + \)\(48\!\cdots\!65\)\( T^{24} + \)\(72\!\cdots\!84\)\( T^{25} - \)\(68\!\cdots\!25\)\( T^{26} - \)\(39\!\cdots\!58\)\( T^{27} - \)\(20\!\cdots\!20\)\( T^{28} - \)\(52\!\cdots\!17\)\( T^{29} + \)\(31\!\cdots\!20\)\( T^{30} + \)\(40\!\cdots\!89\)\( T^{31} - \)\(28\!\cdots\!52\)\( T^{32} - \)\(58\!\cdots\!06\)\( T^{33} + \)\(12\!\cdots\!69\)\( T^{34} + \)\(53\!\cdots\!04\)\( T^{35} + \)\(68\!\cdots\!07\)\( T^{36} - \)\(29\!\cdots\!22\)\( T^{37} - \)\(11\!\cdots\!16\)\( T^{38} + \)\(80\!\cdots\!19\)\( T^{39} + \)\(47\!\cdots\!01\)\( T^{40} \)
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