Properties

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

Related objects

Downloads

Learn more about

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: \(x^{20} - x^{19} + 78 x^{18} + 79 x^{17} + 10573 x^{16} - 33409 x^{15} + 1262953 x^{14} - 1581925 x^{13} + 89291182 x^{12} - 100741271 x^{11} + 2277268901 x^{10} - 1144062486 x^{9} + 70677624924 x^{8} + 93641098134 x^{7} + 1732079176577 x^{6} + 1113900934906 x^{5} + 27749747552381 x^{4} + 4154923377746 x^{3} + 57625758470132 x^{2} + 60036938210920 x + 25599187870096\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{5} + \beta_{6} ) q^{2} + 9 \beta_{7} q^{3} + ( -12 + \beta_{1} - 12 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 18 \beta_{6} + 18 \beta_{7} + \beta_{12} + \beta_{16} ) q^{4} + ( -3 - 5 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{7} + \beta_{9} + \beta_{11} - \beta_{14} + \beta_{17} ) q^{5} + ( -9 \beta_{1} + 9 \beta_{2} ) q^{6} + ( -13 + 2 \beta_{1} - 13 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{11} + 2 \beta_{13} + \beta_{14} + \beta_{19} ) q^{7} + ( -25 - 2 \beta_{1} + 10 \beta_{3} + 17 \beta_{4} - 12 \beta_{5} + 25 \beta_{6} + 56 \beta_{7} - 2 \beta_{9} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{8} -81 \beta_{6} q^{9} +O(q^{10})\) \( q + ( -\beta_{5} + \beta_{6} ) q^{2} + 9 \beta_{7} q^{3} + ( -12 + \beta_{1} - 12 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 18 \beta_{6} + 18 \beta_{7} + \beta_{12} + \beta_{16} ) q^{4} + ( -3 - 5 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{7} + \beta_{9} + \beta_{11} - \beta_{14} + \beta_{17} ) q^{5} + ( -9 \beta_{1} + 9 \beta_{2} ) q^{6} + ( -13 + 2 \beta_{1} - 13 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{11} + 2 \beta_{13} + \beta_{14} + \beta_{19} ) q^{7} + ( -25 - 2 \beta_{1} + 10 \beta_{3} + 17 \beta_{4} - 12 \beta_{5} + 25 \beta_{6} + 56 \beta_{7} - 2 \beta_{9} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{8} -81 \beta_{6} q^{9} + ( 41 - 2 \beta_{1} + 111 \beta_{2} - 26 \beta_{3} - \beta_{5} - 111 \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} + 5 \beta_{11} - 3 \beta_{13} + \beta_{16} + 3 \beta_{17} + \beta_{18} - \beta_{19} ) q^{10} + ( 91 + 18 \beta_{1} - 57 \beta_{2} + 8 \beta_{3} + 18 \beta_{4} - 26 \beta_{5} + 5 \beta_{6} + 25 \beta_{7} + \beta_{8} - 3 \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{16} + 2 \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{11} + ( -108 + 54 \beta_{2} + 9 \beta_{3} - 54 \beta_{6} - 9 \beta_{8} ) q^{12} + ( -28 \beta_{1} + 119 \beta_{2} - 6 \beta_{3} - 22 \beta_{4} - 14 \beta_{5} + 56 \beta_{6} - 119 \beta_{7} - 3 \beta_{8} - 4 \beta_{10} + 6 \beta_{11} - 3 \beta_{12} - 6 \beta_{13} - 8 \beta_{14} + 3 \beta_{15} + 2 \beta_{16} + 8 \beta_{17} - 3 \beta_{18} + 4 \beta_{19} ) q^{13} + ( 241 + 4 \beta_{1} - 41 \beta_{3} + 5 \beta_{4} + 45 \beta_{5} - 241 \beta_{6} - 253 \beta_{7} + 2 \beta_{8} + 13 \beta_{9} - 13 \beta_{12} + 4 \beta_{13} + 4 \beta_{14} - 4 \beta_{15} - 15 \beta_{16} + 15 \beta_{17} + 5 \beta_{18} - 4 \beta_{19} ) q^{14} + ( -45 - 45 \beta_{2} - 18 \beta_{4} + 18 \beta_{5} + 18 \beta_{6} + 18 \beta_{7} + 9 \beta_{11} + 9 \beta_{12} - 9 \beta_{14} + 9 \beta_{16} ) q^{15} + ( -489 + 21 \beta_{1} - 26 \beta_{2} + 50 \beta_{3} + 50 \beta_{4} + 489 \beta_{7} - 15 \beta_{9} - 3 \beta_{10} - 17 \beta_{11} + 19 \beta_{12} + \beta_{14} + \beta_{15} - 17 \beta_{17} - 3 \beta_{18} ) q^{16} + ( 121 + 24 \beta_{1} - 105 \beta_{2} + 8 \beta_{3} + 8 \beta_{4} - 121 \beta_{7} + 9 \beta_{9} + 7 \beta_{10} - 16 \beta_{11} - 4 \beta_{12} + 7 \beta_{14} - 5 \beta_{15} - 16 \beta_{17} + 7 \beta_{18} ) q^{17} + ( 81 - 81 \beta_{1} + 81 \beta_{2} - 81 \beta_{3} - 81 \beta_{4} + 81 \beta_{5} - 81 \beta_{6} - 81 \beta_{7} ) q^{18} + ( 336 + 28 \beta_{1} + 110 \beta_{3} + 82 \beta_{4} - 82 \beta_{5} - 336 \beta_{6} - 464 \beta_{7} + 8 \beta_{8} + 13 \beta_{9} - 13 \beta_{12} + 28 \beta_{13} + 28 \beta_{14} - 21 \beta_{16} - 17 \beta_{17} - 8 \beta_{18} ) q^{19} + ( -74 \beta_{1} - 210 \beta_{2} - 19 \beta_{3} - 55 \beta_{4} + 32 \beta_{5} + 1127 \beta_{6} + 210 \beta_{7} + 4 \beta_{8} + \beta_{10} + 19 \beta_{11} + 4 \beta_{12} - 19 \beta_{13} - 20 \beta_{14} + 6 \beta_{15} + 37 \beta_{16} + 20 \beta_{17} - 6 \beta_{18} - \beta_{19} ) q^{20} + ( -117 + 27 \beta_{1} - 81 \beta_{2} - 9 \beta_{3} - 36 \beta_{5} + 81 \beta_{6} - 18 \beta_{8} + 18 \beta_{9} + 9 \beta_{10} + 18 \beta_{11} - 9 \beta_{13} + 18 \beta_{16} + 9 \beta_{17} ) q^{21} + ( -286 + 90 \beta_{1} - 270 \beta_{2} + 72 \beta_{3} + 20 \beta_{4} - 179 \beta_{5} + 1033 \beta_{6} + 1248 \beta_{7} + 34 \beta_{8} - 49 \beta_{9} - 6 \beta_{10} - 18 \beta_{11} + 13 \beta_{12} + 6 \beta_{13} + 24 \beta_{14} + 8 \beta_{15} + 5 \beta_{16} - 11 \beta_{17} - 13 \beta_{18} + 2 \beta_{19} ) q^{22} + ( 442 + 58 \beta_{1} + 9 \beta_{2} - 38 \beta_{3} - 30 \beta_{5} - 9 \beta_{6} + 16 \beta_{8} - 36 \beta_{9} + 13 \beta_{10} + 5 \beta_{11} + 28 \beta_{13} - 36 \beta_{16} - 28 \beta_{17} + 3 \beta_{18} - 3 \beta_{19} ) q^{23} + ( -81 \beta_{1} + 225 \beta_{2} + 27 \beta_{3} - 108 \beta_{4} + 153 \beta_{5} - 504 \beta_{6} - 225 \beta_{7} - 18 \beta_{8} + 9 \beta_{10} - 27 \beta_{11} - 18 \beta_{12} + 27 \beta_{13} + 9 \beta_{14} - 9 \beta_{15} - 18 \beta_{16} - 9 \beta_{17} + 9 \beta_{18} - 9 \beta_{19} ) q^{24} + ( -928 - 32 \beta_{1} - 86 \beta_{3} - 22 \beta_{4} + 54 \beta_{5} + 928 \beta_{6} + 258 \beta_{7} + 53 \beta_{8} - 22 \beta_{9} + 22 \beta_{12} - 32 \beta_{13} - 32 \beta_{14} + 13 \beta_{15} - 31 \beta_{16} + 6 \beta_{17} - 19 \beta_{18} + 13 \beta_{19} ) q^{25} + ( -1374 + 253 \beta_{1} - 1374 \beta_{2} + 253 \beta_{3} - 26 \beta_{4} + 26 \beta_{5} + 510 \beta_{6} + 510 \beta_{7} - 11 \beta_{8} + 11 \beta_{9} + 11 \beta_{10} + 13 \beta_{11} + 46 \beta_{12} + 11 \beta_{13} - 13 \beta_{14} - 11 \beta_{15} + 46 \beta_{16} - 11 \beta_{19} ) q^{26} -729 \beta_{2} q^{27} + ( 2399 + 22 \beta_{1} + 1601 \beta_{2} - 345 \beta_{3} - 345 \beta_{4} - 2399 \beta_{7} + 96 \beta_{9} - 22 \beta_{10} + 8 \beta_{11} - 107 \beta_{12} + 81 \beta_{14} + 7 \beta_{15} + 8 \beta_{17} - 22 \beta_{18} ) q^{28} + ( 143 - 46 \beta_{1} + 143 \beta_{2} - 46 \beta_{3} + 18 \beta_{4} - 18 \beta_{5} - 2209 \beta_{6} - 2209 \beta_{7} - 9 \beta_{8} + 9 \beta_{9} + 3 \beta_{10} + 6 \beta_{11} - 43 \beta_{12} - 50 \beta_{13} - 6 \beta_{14} - 3 \beta_{15} - 43 \beta_{16} - 7 \beta_{19} ) q^{29} + ( 999 - 27 \beta_{1} - 18 \beta_{3} + 216 \beta_{4} - 9 \beta_{5} - 999 \beta_{6} - 630 \beta_{7} + 9 \beta_{8} - 18 \beta_{9} + 18 \beta_{12} - 27 \beta_{13} - 27 \beta_{14} - 9 \beta_{15} + 9 \beta_{16} - 18 \beta_{17} + 18 \beta_{18} - 9 \beta_{19} ) q^{30} + ( 221 \beta_{1} + 51 \beta_{2} + 79 \beta_{3} + 142 \beta_{4} + 482 \beta_{5} + 1372 \beta_{6} - 51 \beta_{7} + 21 \beta_{8} - 3 \beta_{10} - 79 \beta_{11} + 21 \beta_{12} + 79 \beta_{13} + 66 \beta_{14} - 7 \beta_{15} - 33 \beta_{16} - 66 \beta_{17} + 7 \beta_{18} + 3 \beta_{19} ) q^{31} + ( 1064 - 482 \beta_{1} + 2250 \beta_{2} - 259 \beta_{3} + 540 \beta_{5} - 2250 \beta_{6} - 101 \beta_{8} - 38 \beta_{9} - 22 \beta_{10} - 74 \beta_{11} + 58 \beta_{13} - 38 \beta_{16} - 58 \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{32} + ( -513 - 63 \beta_{1} - 468 \beta_{2} + 171 \beta_{3} + 90 \beta_{4} - 18 \beta_{5} + 288 \beta_{6} + 1332 \beta_{7} + 27 \beta_{8} - 36 \beta_{9} + 27 \beta_{10} - 18 \beta_{12} - 9 \beta_{13} - 18 \beta_{14} - 18 \beta_{15} - 27 \beta_{16} + 36 \beta_{17} + 27 \beta_{18} - 18 \beta_{19} ) q^{33} + ( 826 + 432 \beta_{1} - 234 \beta_{2} - 418 \beta_{3} - 484 \beta_{5} + 234 \beta_{6} + 88 \beta_{8} - 109 \beta_{9} - 41 \beta_{10} - 31 \beta_{11} - 52 \beta_{13} - 109 \beta_{16} + 52 \beta_{17} - 12 \beta_{18} + 12 \beta_{19} ) q^{34} + ( -815 \beta_{1} + 2696 \beta_{2} - 69 \beta_{3} - 746 \beta_{4} + 288 \beta_{5} - 1795 \beta_{6} - 2696 \beta_{7} - 99 \beta_{8} + 8 \beta_{10} + 69 \beta_{11} - 99 \beta_{12} - 69 \beta_{13} - 32 \beta_{14} + 6 \beta_{15} + 54 \beta_{16} + 32 \beta_{17} - 6 \beta_{18} - 8 \beta_{19} ) q^{35} + ( 486 - 81 \beta_{4} - 486 \beta_{6} - 1458 \beta_{7} - 81 \beta_{8} + 81 \beta_{9} - 81 \beta_{12} ) q^{36} + ( -2687 + 854 \beta_{1} - 2687 \beta_{2} + 854 \beta_{3} + 530 \beta_{4} - 530 \beta_{5} + 1046 \beta_{6} + 1046 \beta_{7} + 73 \beta_{8} - 73 \beta_{9} - 29 \beta_{10} - 6 \beta_{11} + 88 \beta_{12} - 14 \beta_{13} + 6 \beta_{14} + 29 \beta_{15} + 88 \beta_{16} + 34 \beta_{19} ) q^{37} + ( -3874 - 260 \beta_{1} - 736 \beta_{2} - 736 \beta_{3} - 736 \beta_{4} + 3874 \beta_{7} + 149 \beta_{9} + 31 \beta_{10} + 121 \beta_{11} + 64 \beta_{12} - 7 \beta_{14} - 25 \beta_{15} + 121 \beta_{17} + 31 \beta_{18} ) q^{38} + ( 1071 - 324 \beta_{1} + 1575 \beta_{2} - 198 \beta_{3} - 198 \beta_{4} - 1071 \beta_{7} - 18 \beta_{9} + 9 \beta_{10} + 18 \beta_{11} - 27 \beta_{12} - 72 \beta_{14} + 27 \beta_{15} + 18 \beta_{17} + 9 \beta_{18} ) q^{39} + ( -184 + 969 \beta_{1} - 184 \beta_{2} + 969 \beta_{3} + 1390 \beta_{4} - 1390 \beta_{5} + 2236 \beta_{6} + 2236 \beta_{7} - 39 \beta_{8} + 39 \beta_{9} - 26 \beta_{10} + 86 \beta_{11} + 58 \beta_{12} - 86 \beta_{14} + 26 \beta_{15} + 58 \beta_{16} + 8 \beta_{19} ) q^{40} + ( 3928 + 65 \beta_{1} - 23 \beta_{3} + 796 \beta_{4} + 88 \beta_{5} - 3928 \beta_{6} - 105 \beta_{7} - 8 \beta_{8} - 43 \beta_{9} + 43 \beta_{12} + 65 \beta_{13} + 65 \beta_{14} + 38 \beta_{15} + 51 \beta_{16} + 26 \beta_{17} - 7 \beta_{18} + 38 \beta_{19} ) q^{41} + ( 234 \beta_{1} - 2169 \beta_{2} - 171 \beta_{3} + 405 \beta_{4} + 45 \beta_{5} + 2277 \beta_{6} + 2169 \beta_{7} + 135 \beta_{8} + 9 \beta_{10} + 171 \beta_{11} + 135 \beta_{12} - 171 \beta_{13} - 135 \beta_{14} - 45 \beta_{15} + 117 \beta_{16} + 135 \beta_{17} + 45 \beta_{18} - 9 \beta_{19} ) q^{42} + ( -108 - 603 \beta_{1} + 1658 \beta_{2} - 473 \beta_{3} + 556 \beta_{5} - 1658 \beta_{6} - 41 \beta_{8} + 112 \beta_{9} + 16 \beta_{10} + 42 \beta_{11} - 47 \beta_{13} + 112 \beta_{16} + 47 \beta_{17} - 26 \beta_{18} + 26 \beta_{19} ) q^{43} + ( -2513 - 385 \beta_{1} - 3971 \beta_{2} + 1559 \beta_{3} + 1015 \beta_{4} - 981 \beta_{5} + 2562 \beta_{6} + 6971 \beta_{7} - 68 \beta_{8} + 83 \beta_{9} - 6 \beta_{10} - 44 \beta_{11} + 133 \beta_{12} - 24 \beta_{13} - 69 \beta_{14} - 11 \beta_{15} + 185 \beta_{16} - 56 \beta_{17} - 26 \beta_{18} + 24 \beta_{19} ) q^{44} + ( -405 + 243 \beta_{1} - 243 \beta_{2} + 81 \beta_{3} - 162 \beta_{5} + 243 \beta_{6} - 81 \beta_{8} + 81 \beta_{13} - 81 \beta_{17} ) q^{45} + ( -813 \beta_{1} - 2146 \beta_{2} + 91 \beta_{3} - 904 \beta_{4} - 46 \beta_{5} + 4028 \beta_{6} + 2146 \beta_{7} + 55 \beta_{8} + 3 \beta_{10} - 91 \beta_{11} + 55 \beta_{12} + 91 \beta_{13} - 12 \beta_{14} - 28 \beta_{15} - 120 \beta_{16} + 12 \beta_{17} + 28 \beta_{18} - 3 \beta_{19} ) q^{46} + ( -4726 + 85 \beta_{1} - 569 \beta_{3} - 1158 \beta_{4} + 654 \beta_{5} + 4726 \beta_{6} + 4413 \beta_{7} - 72 \beta_{8} - 80 \beta_{9} + 80 \beta_{12} + 85 \beta_{13} + 85 \beta_{14} - 24 \beta_{15} + 152 \beta_{16} - 103 \beta_{17} + 85 \beta_{18} - 24 \beta_{19} ) q^{47} + ( -234 + 333 \beta_{1} - 234 \beta_{2} + 333 \beta_{3} - 261 \beta_{4} + 261 \beta_{5} - 4167 \beta_{6} - 4167 \beta_{7} - 171 \beta_{8} + 171 \beta_{9} - 27 \beta_{10} - 153 \beta_{11} - 135 \beta_{12} + 144 \beta_{13} + 153 \beta_{14} + 27 \beta_{15} - 135 \beta_{16} + 18 \beta_{19} ) q^{48} + ( -2735 - 2494 \beta_{1} + 6981 \beta_{2} - 270 \beta_{3} - 270 \beta_{4} + 2735 \beta_{7} - 61 \beta_{9} - 14 \beta_{10} - 100 \beta_{11} - 146 \beta_{12} + 106 \beta_{14} + 45 \beta_{15} - 100 \beta_{17} - 14 \beta_{18} ) q^{49} + ( 1689 - 343 \beta_{1} + 911 \beta_{2} - 133 \beta_{3} - 133 \beta_{4} - 1689 \beta_{7} - 24 \beta_{9} - 19 \beta_{10} + 51 \beta_{11} + 323 \beta_{12} - 278 \beta_{14} - 18 \beta_{15} + 51 \beta_{17} - 19 \beta_{18} ) q^{50} + ( -945 + 297 \beta_{1} - 945 \beta_{2} + 297 \beta_{3} + 144 \beta_{4} - 144 \beta_{5} + 2034 \beta_{6} + 2034 \beta_{7} + 36 \beta_{8} - 36 \beta_{9} + 63 \beta_{10} - 144 \beta_{11} + 81 \beta_{12} + 81 \beta_{13} + 144 \beta_{14} - 63 \beta_{15} + 81 \beta_{16} - 18 \beta_{19} ) q^{51} + ( 8646 - 213 \beta_{1} - 101 \beta_{3} + 1690 \beta_{4} - 112 \beta_{5} - 8646 \beta_{6} - 2889 \beta_{7} + 84 \beta_{8} + 71 \beta_{9} - 71 \beta_{12} - 213 \beta_{13} - 213 \beta_{14} - 45 \beta_{15} - 155 \beta_{16} + 252 \beta_{17} + 18 \beta_{18} - 45 \beta_{19} ) q^{52} + ( 737 \beta_{1} + 6304 \beta_{2} + 9 \beta_{3} + 728 \beta_{4} + 1890 \beta_{5} - 1172 \beta_{6} - 6304 \beta_{7} - 164 \beta_{8} - 63 \beta_{10} - 9 \beta_{11} - 164 \beta_{12} + 9 \beta_{13} - 17 \beta_{14} + 62 \beta_{15} - 168 \beta_{16} + 17 \beta_{17} - 62 \beta_{18} + 63 \beta_{19} ) q^{53} + ( 729 - 729 \beta_{3} ) q^{54} + ( -4850 + 1470 \beta_{1} - 5317 \beta_{2} + 1582 \beta_{3} + 996 \beta_{4} - 1422 \beta_{5} - 2874 \beta_{6} + 5696 \beta_{7} - 106 \beta_{8} + 220 \beta_{9} - 5 \beta_{10} + 398 \beta_{11} - 82 \beta_{12} + 70 \beta_{13} - 28 \beta_{14} - 18 \beta_{15} + 5 \beta_{16} - 21 \beta_{17} + 47 \beta_{18} - 30 \beta_{19} ) q^{55} + ( 388 + 2857 \beta_{1} - 15088 \beta_{2} - 266 \beta_{3} - 3039 \beta_{5} + 15088 \beta_{6} + 200 \beta_{8} + 399 \beta_{9} + 42 \beta_{10} + 138 \beta_{11} - 182 \beta_{13} + 399 \beta_{16} + 182 \beta_{17} + 30 \beta_{18} - 30 \beta_{19} ) q^{56} + ( -837 \beta_{1} - 3024 \beta_{2} - 99 \beta_{3} - 738 \beta_{4} + 738 \beta_{5} + 4176 \beta_{6} + 3024 \beta_{7} + 189 \beta_{8} - 72 \beta_{10} + 99 \beta_{11} + 189 \beta_{12} - 99 \beta_{13} + 153 \beta_{14} + 72 \beta_{15} + 117 \beta_{16} - 153 \beta_{17} - 72 \beta_{18} + 72 \beta_{19} ) q^{57} + ( 1221 - 20 \beta_{1} - 2953 \beta_{3} - 4056 \beta_{4} + 2933 \beta_{5} - 1221 \beta_{6} - 2392 \beta_{7} + 52 \beta_{8} - 151 \beta_{9} + 151 \beta_{12} - 20 \beta_{13} - 20 \beta_{14} + 99 \beta_{16} - 35 \beta_{17} - 77 \beta_{18} ) q^{58} + ( -1782 + 1211 \beta_{1} - 1782 \beta_{2} + 1211 \beta_{3} + 634 \beta_{4} - 634 \beta_{5} - 8752 \beta_{6} - 8752 \beta_{7} + 396 \beta_{8} - 396 \beta_{9} + 55 \beta_{10} - 25 \beta_{11} - 387 \beta_{12} - 297 \beta_{13} + 25 \beta_{14} - 55 \beta_{15} - 387 \beta_{16} - 121 \beta_{19} ) q^{59} + ( -1890 - 207 \beta_{1} + 8253 \beta_{2} - 495 \beta_{3} - 495 \beta_{4} + 1890 \beta_{7} - 333 \beta_{9} - 63 \beta_{10} + 9 \beta_{11} + 36 \beta_{12} - 180 \beta_{14} + 54 \beta_{15} + 9 \beta_{17} - 63 \beta_{18} ) q^{60} + ( 1528 - 58 \beta_{1} + 343 \beta_{2} - 486 \beta_{3} - 486 \beta_{4} - 1528 \beta_{7} - 567 \beta_{9} + 106 \beta_{10} - 50 \beta_{11} + 170 \beta_{12} - 36 \beta_{14} - 55 \beta_{15} - 50 \beta_{17} + 106 \beta_{18} ) q^{61} + ( 23530 + 783 \beta_{1} + 23530 \beta_{2} + 783 \beta_{3} + 1347 \beta_{4} - 1347 \beta_{5} - 21669 \beta_{6} - 21669 \beta_{7} - 93 \beta_{8} + 93 \beta_{9} - 45 \beta_{10} - 307 \beta_{11} - 870 \beta_{12} + 311 \beta_{13} + 307 \beta_{14} + 45 \beta_{15} - 870 \beta_{16} + 13 \beta_{19} ) q^{62} + ( -729 - 81 \beta_{1} + 243 \beta_{3} + 324 \beta_{4} - 324 \beta_{5} + 729 \beta_{6} - 324 \beta_{7} - 162 \beta_{8} - 81 \beta_{13} - 81 \beta_{14} + 162 \beta_{16} - 81 \beta_{17} - 81 \beta_{18} ) q^{63} + ( -1281 \beta_{1} + 13399 \beta_{2} + 479 \beta_{3} - 1760 \beta_{4} + 3843 \beta_{5} - 4606 \beta_{6} - 13399 \beta_{7} - 608 \beta_{8} + 145 \beta_{10} - 479 \beta_{11} - 608 \beta_{12} + 479 \beta_{13} + 587 \beta_{14} + \beta_{15} - 284 \beta_{16} - 587 \beta_{17} - \beta_{18} - 145 \beta_{19} ) q^{64} + ( -7804 - 1089 \beta_{1} - 5003 \beta_{2} - 3609 \beta_{3} + 996 \beta_{5} + 5003 \beta_{6} + 865 \beta_{8} - 707 \beta_{9} - 37 \beta_{10} - 124 \beta_{11} - 93 \beta_{13} - 707 \beta_{16} + 93 \beta_{17} + 3 \beta_{18} - 3 \beta_{19} ) q^{65} + ( -2430 - 972 \beta_{1} + 6867 \beta_{2} + 639 \beta_{3} + 162 \beta_{4} - 576 \beta_{5} - 8802 \beta_{6} - 144 \beta_{7} + 189 \beta_{8} - 234 \beta_{9} - 99 \beta_{10} - 45 \beta_{11} - 135 \beta_{12} - 117 \beta_{13} + 99 \beta_{14} + 117 \beta_{15} - 441 \beta_{16} + 63 \beta_{17} - 63 \beta_{18} + 45 \beta_{19} ) q^{66} + ( -2678 - 660 \beta_{1} - 8546 \beta_{2} - 2154 \beta_{3} + 550 \beta_{5} + 8546 \beta_{6} - 530 \beta_{8} + 319 \beta_{9} + 132 \beta_{10} - 187 \beta_{11} - 110 \beta_{13} + 319 \beta_{16} + 110 \beta_{17} + 44 \beta_{18} - 44 \beta_{19} ) q^{67} + ( -2343 \beta_{1} - 19073 \beta_{2} + 233 \beta_{3} - 2576 \beta_{4} + 1694 \beta_{5} + 37240 \beta_{6} + 19073 \beta_{7} + 280 \beta_{8} + 3 \beta_{10} - 233 \beta_{11} + 280 \beta_{12} + 233 \beta_{13} + 587 \beta_{14} + 33 \beta_{15} + 529 \beta_{16} - 587 \beta_{17} - 33 \beta_{18} - 3 \beta_{19} ) q^{68} + ( 81 + 252 \beta_{1} + 522 \beta_{3} + 864 \beta_{4} - 270 \beta_{5} - 81 \beta_{6} + 3897 \beta_{7} + 144 \beta_{8} + 180 \beta_{9} - 180 \beta_{12} + 252 \beta_{13} + 252 \beta_{14} - 27 \beta_{15} - 324 \beta_{16} - 297 \beta_{17} - 90 \beta_{18} - 27 \beta_{19} ) q^{69} + ( -16093 + 3488 \beta_{1} - 16093 \beta_{2} + 3488 \beta_{3} - 1579 \beta_{4} + 1579 \beta_{5} - 20147 \beta_{6} - 20147 \beta_{7} - 415 \beta_{8} + 415 \beta_{9} + 140 \beta_{10} + 300 \beta_{11} - 9 \beta_{12} - 259 \beta_{13} - 300 \beta_{14} - 140 \beta_{15} - 9 \beta_{16} + 35 \beta_{19} ) q^{70} + ( 13221 + 806 \beta_{1} + 12581 \beta_{2} + 1588 \beta_{3} + 1588 \beta_{4} - 13221 \beta_{7} + 290 \beta_{9} - 39 \beta_{10} - 379 \beta_{11} - 314 \beta_{12} - 38 \beta_{14} - 97 \beta_{15} - 379 \beta_{17} - 39 \beta_{18} ) q^{71} + ( 2025 + 405 \beta_{1} - 2511 \beta_{2} - 972 \beta_{3} - 972 \beta_{4} - 2025 \beta_{7} + 162 \beta_{9} + 162 \beta_{11} - 162 \beta_{12} + 81 \beta_{14} - 81 \beta_{15} + 162 \beta_{17} ) q^{72} + ( 17469 - 114 \beta_{1} + 17469 \beta_{2} - 114 \beta_{3} + 210 \beta_{4} - 210 \beta_{5} - 9935 \beta_{6} - 9935 \beta_{7} + 81 \beta_{8} - 81 \beta_{9} + 67 \beta_{10} + 466 \beta_{11} + 1123 \beta_{12} + 152 \beta_{13} - 466 \beta_{14} - 67 \beta_{15} + 1123 \beta_{16} + 15 \beta_{19} ) q^{73} + ( 11110 - 83 \beta_{1} - 37 \beta_{3} + 3326 \beta_{4} - 46 \beta_{5} - 11110 \beta_{6} + 27134 \beta_{7} + 887 \beta_{8} - 832 \beta_{9} + 832 \beta_{12} - 83 \beta_{13} - 83 \beta_{14} + 35 \beta_{15} - 55 \beta_{16} - 206 \beta_{17} - 82 \beta_{18} + 35 \beta_{19} ) q^{74} + ( 720 \beta_{1} + 8352 \beta_{2} + 234 \beta_{3} + 486 \beta_{4} - 198 \beta_{5} - 2322 \beta_{6} - 8352 \beta_{7} + 279 \beta_{8} - 54 \beta_{10} - 234 \beta_{11} + 279 \beta_{12} + 234 \beta_{13} - 54 \beta_{14} + 171 \beta_{15} - 198 \beta_{16} + 54 \beta_{17} - 171 \beta_{18} + 54 \beta_{19} ) q^{75} + ( -6680 - 2795 \beta_{1} - 20419 \beta_{2} - 4467 \beta_{3} + 2816 \beta_{5} + 20419 \beta_{6} + 497 \beta_{8} + 553 \beta_{9} + 122 \beta_{10} + 500 \beta_{11} + 21 \beta_{13} + 553 \beta_{16} - 21 \beta_{17} + 229 \beta_{18} - 229 \beta_{19} ) q^{76} + ( -30025 + 3955 \beta_{1} - 14853 \beta_{2} + 4295 \beta_{3} - 742 \beta_{4} - 2970 \beta_{5} + 9411 \beta_{6} + 14444 \beta_{7} - 695 \beta_{8} + 1384 \beta_{9} + \beta_{10} - 403 \beta_{11} - 668 \beta_{12} + 329 \beta_{13} + 200 \beta_{14} + 39 \beta_{15} - 114 \beta_{16} + 364 \beta_{17} - 33 \beta_{18} + 50 \beta_{19} ) q^{77} + ( -12366 + 2529 \beta_{1} - 7776 \beta_{2} + 2295 \beta_{3} - 2412 \beta_{5} + 7776 \beta_{6} - 513 \beta_{8} + 99 \beta_{9} - 99 \beta_{10} + 99 \beta_{11} + 117 \beta_{13} + 99 \beta_{16} - 117 \beta_{17} - 99 \beta_{18} + 99 \beta_{19} ) q^{78} + ( -2656 \beta_{1} - 28602 \beta_{2} - 334 \beta_{3} - 2322 \beta_{4} + 44 \beta_{5} + 16473 \beta_{6} + 28602 \beta_{7} + 143 \beta_{8} + 71 \beta_{10} + 334 \beta_{11} + 143 \beta_{12} - 334 \beta_{13} - 1169 \beta_{14} - 80 \beta_{15} - 994 \beta_{16} + 1169 \beta_{17} + 80 \beta_{18} - 71 \beta_{19} ) q^{79} + ( -20245 + 41 \beta_{1} + 2694 \beta_{3} + 3750 \beta_{4} - 2653 \beta_{5} + 20245 \beta_{6} + 38699 \beta_{7} - 462 \beta_{8} - 336 \beta_{9} + 336 \beta_{12} + 41 \beta_{13} + 41 \beta_{14} + 127 \beta_{15} + 798 \beta_{16} + 277 \beta_{17} - 123 \beta_{18} + 127 \beta_{19} ) q^{80} + ( -6561 - 6561 \beta_{2} + 6561 \beta_{6} + 6561 \beta_{7} ) q^{81} + ( 2186 - 4422 \beta_{1} + 41036 \beta_{2} - 2088 \beta_{3} - 2088 \beta_{4} - 2186 \beta_{7} - 340 \beta_{9} + 99 \beta_{10} + 397 \beta_{11} - 121 \beta_{12} - 122 \beta_{14} - 110 \beta_{15} + 397 \beta_{17} + 99 \beta_{18} ) q^{82} + ( 15185 - 608 \beta_{1} - 11089 \beta_{2} - 4626 \beta_{3} - 4626 \beta_{4} - 15185 \beta_{7} - 303 \beta_{9} - 160 \beta_{10} + 114 \beta_{11} + 1035 \beta_{12} + 469 \beta_{14} + 146 \beta_{15} + 114 \beta_{17} - 160 \beta_{18} ) q^{83} + ( 14409 - 603 \beta_{1} + 14409 \beta_{2} - 603 \beta_{3} + 3303 \beta_{4} - 3303 \beta_{5} + 7182 \beta_{6} + 7182 \beta_{7} + 963 \beta_{8} - 963 \beta_{9} - 198 \beta_{10} + 72 \beta_{11} + 864 \beta_{12} - 801 \beta_{13} - 72 \beta_{14} + 198 \beta_{15} + 864 \beta_{16} + 135 \beta_{19} ) q^{84} + ( 19727 + 628 \beta_{1} + 7894 \beta_{3} + 2358 \beta_{4} - 7266 \beta_{5} - 19727 \beta_{6} + 6952 \beta_{7} - 2123 \beta_{8} + 1518 \beta_{9} - 1518 \beta_{12} + 628 \beta_{13} + 628 \beta_{14} - 25 \beta_{15} + 605 \beta_{16} - 208 \beta_{17} + 249 \beta_{18} - 25 \beta_{19} ) q^{85} + ( 33 \beta_{1} + 24672 \beta_{2} - 847 \beta_{3} + 880 \beta_{4} - 354 \beta_{5} - 8116 \beta_{6} - 24672 \beta_{7} + 55 \beta_{8} + 33 \beta_{10} + 847 \beta_{11} + 55 \beta_{12} - 847 \beta_{13} - 770 \beta_{14} - 198 \beta_{15} + 222 \beta_{16} + 770 \beta_{17} + 198 \beta_{18} - 33 \beta_{19} ) q^{86} + ( 1287 - 72 \beta_{1} - 18594 \beta_{2} + 90 \beta_{3} + 126 \beta_{5} + 18594 \beta_{6} + 306 \beta_{8} + 81 \beta_{9} - 63 \beta_{10} - 450 \beta_{11} + 54 \beta_{13} + 81 \beta_{16} - 54 \beta_{17} - 27 \beta_{18} + 27 \beta_{19} ) q^{87} + ( -32842 - 225 \beta_{1} + 8480 \beta_{2} - 661 \beta_{3} + 3589 \beta_{4} - 844 \beta_{5} - 29483 \beta_{6} + 26416 \beta_{7} - 146 \beta_{8} - 1507 \beta_{9} + 127 \beta_{10} - 759 \beta_{11} + 594 \beta_{12} - 33 \beta_{13} - 58 \beta_{14} - 191 \beta_{16} - 830 \beta_{17} + 24 \beta_{18} - 19 \beta_{19} ) q^{88} + ( 11099 + 1435 \beta_{1} + 1896 \beta_{2} - 185 \beta_{3} - 848 \beta_{5} - 1896 \beta_{6} + 548 \beta_{8} + 157 \beta_{9} - 277 \beta_{10} + 479 \beta_{11} + 587 \beta_{13} + 157 \beta_{16} - 587 \beta_{17} - 102 \beta_{18} + 102 \beta_{19} ) q^{89} + ( 324 \beta_{1} - 8991 \beta_{2} + 405 \beta_{3} - 81 \beta_{4} + 1944 \beta_{5} + 5670 \beta_{6} + 8991 \beta_{7} - 81 \beta_{8} + 81 \beta_{10} - 405 \beta_{11} - 81 \beta_{12} + 405 \beta_{13} + 162 \beta_{14} - 162 \beta_{15} - 162 \beta_{16} - 162 \beta_{17} + 162 \beta_{18} - 81 \beta_{19} ) q^{90} + ( 15404 - 1101 \beta_{1} - 5253 \beta_{3} - 1138 \beta_{4} + 4152 \beta_{5} - 15404 \beta_{6} + 37524 \beta_{7} + 485 \beta_{8} + 37 \beta_{9} - 37 \beta_{12} - 1101 \beta_{13} - 1101 \beta_{14} - 126 \beta_{15} - 522 \beta_{16} + 825 \beta_{17} + 570 \beta_{18} - 126 \beta_{19} ) q^{91} + ( -20143 - 2393 \beta_{1} - 20143 \beta_{2} - 2393 \beta_{3} - 602 \beta_{4} + 602 \beta_{5} - 16215 \beta_{6} - 16215 \beta_{7} - 105 \beta_{8} + 105 \beta_{9} - 313 \beta_{10} + 167 \beta_{11} - 165 \beta_{12} + 793 \beta_{13} - 167 \beta_{14} + 313 \beta_{15} - 165 \beta_{16} + 315 \beta_{19} ) q^{92} + ( 459 + 5616 \beta_{1} + 12807 \beta_{2} + 1278 \beta_{3} + 1278 \beta_{4} - 459 \beta_{7} + 297 \beta_{9} + 90 \beta_{10} + 117 \beta_{11} + 189 \beta_{12} + 594 \beta_{14} - 63 \beta_{15} + 117 \beta_{17} + 90 \beta_{18} ) q^{93} + ( 26448 + 1518 \beta_{1} - 27852 \beta_{2} + 8936 \beta_{3} + 8936 \beta_{4} - 26448 \beta_{7} + 447 \beta_{9} + 6 \beta_{10} - 686 \beta_{11} - 1331 \beta_{12} + 757 \beta_{14} + 67 \beta_{15} - 686 \beta_{17} + 6 \beta_{18} ) q^{94} + ( 63827 - 4812 \beta_{1} + 63827 \beta_{2} - 4812 \beta_{3} - 2730 \beta_{4} + 2730 \beta_{5} - 67395 \beta_{6} - 67395 \beta_{7} - 682 \beta_{8} + 682 \beta_{9} - 56 \beta_{10} + 165 \beta_{11} + 454 \beta_{12} - 1236 \beta_{13} - 165 \beta_{14} + 56 \beta_{15} + 454 \beta_{16} - 57 \beta_{19} ) q^{95} + ( 20250 + 522 \beta_{1} - 4338 \beta_{3} - 2007 \beta_{4} + 4860 \beta_{5} - 20250 \beta_{6} - 10674 \beta_{7} - 909 \beta_{8} + 1251 \beta_{9} - 1251 \beta_{12} + 522 \beta_{13} + 522 \beta_{14} - 18 \beta_{15} - 342 \beta_{16} + 144 \beta_{17} + 216 \beta_{18} - 18 \beta_{19} ) q^{96} + ( 570 \beta_{1} + 29218 \beta_{2} - 1380 \beta_{3} + 1950 \beta_{4} - 5070 \beta_{5} - 2182 \beta_{6} - 29218 \beta_{7} + 1471 \beta_{8} - 394 \beta_{10} + 1380 \beta_{11} + 1471 \beta_{12} - 1380 \beta_{13} - 872 \beta_{14} - 219 \beta_{15} + 864 \beta_{16} + 872 \beta_{17} + 219 \beta_{18} + 394 \beta_{19} ) q^{97} + ( -116760 + 1010 \beta_{1} + 1371 \beta_{2} + 10433 \beta_{3} - 713 \beta_{5} - 1371 \beta_{6} - 2813 \beta_{8} + 274 \beta_{9} + 78 \beta_{10} + 186 \beta_{11} + 297 \beta_{13} + 274 \beta_{16} - 297 \beta_{17} - 367 \beta_{18} + 367 \beta_{19} ) q^{98} + ( -4212 - 810 \beta_{1} - 1620 \beta_{2} - 648 \beta_{3} - 2106 \beta_{4} + 1296 \beta_{5} - 7776 \beta_{6} - 405 \beta_{7} + 405 \beta_{8} - 162 \beta_{9} + 81 \beta_{10} + 243 \beta_{11} - 81 \beta_{12} - 162 \beta_{13} - 324 \beta_{14} - 243 \beta_{15} - 324 \beta_{16} + 324 \beta_{17} - 81 \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{2} + 45q^{3} + 8q^{4} - 11q^{5} - 54q^{6} - 139q^{7} - 76q^{8} - 405q^{9} + O(q^{10}) \) \( 20q + 6q^{2} + 45q^{3} + 8q^{4} - 11q^{5} - 54q^{6} - 139q^{7} - 76q^{8} - 405q^{9} - 424q^{10} + 2289q^{11} - 2682q^{12} - 847q^{13} + 2022q^{14} - 396q^{15} - 7148q^{16} + 2482q^{17} + 81q^{18} + 2958q^{19} + 8037q^{20} - 1404q^{21} + 7441q^{22} + 8140q^{23} - 4626q^{24} - 13120q^{25} - 13508q^{26} + 3645q^{27} + 27819q^{28} - 20210q^{29} + 11106q^{30} + 5540q^{31} - 4626q^{32} + 639q^{33} + 16540q^{34} - 34101q^{35} + 648q^{36} - 25173q^{37} - 55878q^{38} + 7623q^{39} + 22689q^{40} + 54349q^{41} + 32292q^{42} - 21688q^{43} + 24587q^{44} - 5346q^{45} + 44155q^{46} - 48387q^{47} - 43083q^{48} - 78880q^{49} + 19956q^{50} + 7317q^{51} + 109051q^{52} - 74382q^{53} + 10206q^{54} - 47630q^{55} + 168894q^{56} + 52983q^{57} + 1501q^{58} - 108412q^{59} - 72333q^{60} + 16737q^{61} + 132685q^{62} - 11259q^{63} - 157030q^{64} - 134194q^{65} - 127539q^{66} + 20178q^{67} + 387013q^{68} + 19305q^{69} - 411988q^{70} + 143157q^{71} + 41634q^{72} + 164980q^{73} + 286114q^{74} - 96255q^{75} + 46868q^{76} - 367461q^{77} - 152658q^{78} + 369613q^{79} - 107448q^{80} - 32805q^{81} - 184129q^{82} + 267741q^{83} + 279819q^{84} + 379937q^{85} - 290531q^{86} + 214920q^{87} - 740846q^{88} + 205492q^{89} + 117126q^{90} + 394559q^{91} - 483181q^{92} - 49860q^{93} + 561327q^{94} + 267881q^{95} + 228564q^{96} - 300567q^{97} - 2296174q^{98} - 115506q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - x^{19} + 78 x^{18} + 79 x^{17} + 10573 x^{16} - 33409 x^{15} + 1262953 x^{14} - 1581925 x^{13} + 89291182 x^{12} - 100741271 x^{11} + 2277268901 x^{10} - 1144062486 x^{9} + 70677624924 x^{8} + 93641098134 x^{7} + 1732079176577 x^{6} + 1113900934906 x^{5} + 27749747552381 x^{4} + 4154923377746 x^{3} + 57625758470132 x^{2} + 60036938210920 x + 25599187870096\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(24\!\cdots\!63\)\( \nu^{19} - \)\(65\!\cdots\!25\)\( \nu^{18} + \)\(20\!\cdots\!83\)\( \nu^{17} - \)\(14\!\cdots\!56\)\( \nu^{16} + \)\(26\!\cdots\!59\)\( \nu^{15} - \)\(12\!\cdots\!42\)\( \nu^{14} + \)\(33\!\cdots\!30\)\( \nu^{13} - \)\(94\!\cdots\!46\)\( \nu^{12} + \)\(24\!\cdots\!89\)\( \nu^{11} - \)\(63\!\cdots\!26\)\( \nu^{10} + \)\(70\!\cdots\!47\)\( \nu^{9} - \)\(13\!\cdots\!45\)\( \nu^{8} + \)\(20\!\cdots\!11\)\( \nu^{7} - \)\(52\!\cdots\!16\)\( \nu^{6} + \)\(46\!\cdots\!51\)\( \nu^{5} - \)\(35\!\cdots\!72\)\( \nu^{4} + \)\(81\!\cdots\!00\)\( \nu^{3} - \)\(94\!\cdots\!68\)\( \nu^{2} + \)\(42\!\cdots\!05\)\( \nu - \)\(65\!\cdots\!64\)\(\)\()/ \)\(34\!\cdots\!02\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(98\!\cdots\!84\)\( \nu^{19} - \)\(26\!\cdots\!19\)\( \nu^{18} - \)\(73\!\cdots\!63\)\( \nu^{17} - \)\(17\!\cdots\!83\)\( \nu^{16} - \)\(10\!\cdots\!27\)\( \nu^{15} + \)\(20\!\cdots\!65\)\( \nu^{14} - \)\(11\!\cdots\!14\)\( \nu^{13} - \)\(72\!\cdots\!26\)\( \nu^{12} - \)\(84\!\cdots\!46\)\( \nu^{11} - \)\(93\!\cdots\!04\)\( \nu^{10} - \)\(19\!\cdots\!30\)\( \nu^{9} - \)\(17\!\cdots\!67\)\( \nu^{8} - \)\(65\!\cdots\!67\)\( \nu^{7} - \)\(18\!\cdots\!67\)\( \nu^{6} - \)\(15\!\cdots\!72\)\( \nu^{5} - \)\(30\!\cdots\!19\)\( \nu^{4} - \)\(26\!\cdots\!60\)\( \nu^{3} - \)\(37\!\cdots\!44\)\( \nu^{2} - \)\(21\!\cdots\!40\)\( \nu + \)\(78\!\cdots\!30\)\(\)\()/ \)\(13\!\cdots\!11\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(32\!\cdots\!82\)\( \nu^{19} + \)\(11\!\cdots\!59\)\( \nu^{18} - \)\(26\!\cdots\!63\)\( \nu^{17} + \)\(41\!\cdots\!60\)\( \nu^{16} - \)\(33\!\cdots\!25\)\( \nu^{15} + \)\(19\!\cdots\!01\)\( \nu^{14} - \)\(43\!\cdots\!81\)\( \nu^{13} + \)\(16\!\cdots\!53\)\( \nu^{12} - \)\(30\!\cdots\!83\)\( \nu^{11} + \)\(10\!\cdots\!24\)\( \nu^{10} - \)\(83\!\cdots\!97\)\( \nu^{9} + \)\(22\!\cdots\!89\)\( \nu^{8} - \)\(22\!\cdots\!38\)\( \nu^{7} + \)\(27\!\cdots\!00\)\( \nu^{6} - \)\(49\!\cdots\!50\)\( \nu^{5} + \)\(10\!\cdots\!67\)\( \nu^{4} - \)\(82\!\cdots\!26\)\( \nu^{3} + \)\(22\!\cdots\!79\)\( \nu^{2} - \)\(16\!\cdots\!64\)\( \nu - \)\(50\!\cdots\!28\)\(\)\()/ \)\(27\!\cdots\!22\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(68\!\cdots\!67\)\( \nu^{19} + \)\(31\!\cdots\!53\)\( \nu^{18} + \)\(50\!\cdots\!18\)\( \nu^{17} + \)\(13\!\cdots\!33\)\( \nu^{16} + \)\(72\!\cdots\!47\)\( \nu^{15} - \)\(12\!\cdots\!23\)\( \nu^{14} + \)\(82\!\cdots\!47\)\( \nu^{13} + \)\(20\!\cdots\!85\)\( \nu^{12} + \)\(58\!\cdots\!50\)\( \nu^{11} + \)\(17\!\cdots\!47\)\( \nu^{10} + \)\(13\!\cdots\!99\)\( \nu^{9} + \)\(13\!\cdots\!10\)\( \nu^{8} + \)\(48\!\cdots\!36\)\( \nu^{7} + \)\(12\!\cdots\!74\)\( \nu^{6} + \)\(12\!\cdots\!31\)\( \nu^{5} + \)\(20\!\cdots\!62\)\( \nu^{4} + \)\(20\!\cdots\!79\)\( \nu^{3} + \)\(27\!\cdots\!02\)\( \nu^{2} + \)\(70\!\cdots\!00\)\( \nu + \)\(56\!\cdots\!00\)\(\)\()/ \)\(54\!\cdots\!44\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(30\!\cdots\!65\)\( \nu^{19} - \)\(20\!\cdots\!13\)\( \nu^{18} + \)\(24\!\cdots\!02\)\( \nu^{17} + \)\(31\!\cdots\!99\)\( \nu^{16} + \)\(32\!\cdots\!69\)\( \nu^{15} - \)\(92\!\cdots\!29\)\( \nu^{14} + \)\(38\!\cdots\!25\)\( \nu^{13} - \)\(37\!\cdots\!33\)\( \nu^{12} + \)\(27\!\cdots\!58\)\( \nu^{11} - \)\(22\!\cdots\!27\)\( \nu^{10} + \)\(70\!\cdots\!77\)\( \nu^{9} - \)\(15\!\cdots\!50\)\( \nu^{8} + \)\(22\!\cdots\!36\)\( \nu^{7} + \)\(35\!\cdots\!86\)\( \nu^{6} + \)\(55\!\cdots\!81\)\( \nu^{5} + \)\(49\!\cdots\!06\)\( \nu^{4} + \)\(89\!\cdots\!97\)\( \nu^{3} + \)\(40\!\cdots\!70\)\( \nu^{2} + \)\(21\!\cdots\!12\)\( \nu + \)\(20\!\cdots\!20\)\(\)\()/ \)\(13\!\cdots\!08\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(17\!\cdots\!06\)\( \nu^{19} - \)\(39\!\cdots\!03\)\( \nu^{18} + \)\(13\!\cdots\!67\)\( \nu^{17} - \)\(30\!\cdots\!62\)\( \nu^{16} + \)\(18\!\cdots\!43\)\( \nu^{15} - \)\(81\!\cdots\!09\)\( \nu^{14} + \)\(22\!\cdots\!53\)\( \nu^{13} - \)\(55\!\cdots\!65\)\( \nu^{12} + \)\(16\!\cdots\!39\)\( \nu^{11} - \)\(36\!\cdots\!26\)\( \nu^{10} + \)\(42\!\cdots\!41\)\( \nu^{9} - \)\(68\!\cdots\!99\)\( \nu^{8} + \)\(12\!\cdots\!44\)\( \nu^{7} + \)\(13\!\cdots\!24\)\( \nu^{6} + \)\(28\!\cdots\!40\)\( \nu^{5} - \)\(16\!\cdots\!93\)\( \nu^{4} + \)\(47\!\cdots\!22\)\( \nu^{3} - \)\(52\!\cdots\!85\)\( \nu^{2} + \)\(10\!\cdots\!72\)\( \nu + \)\(32\!\cdots\!16\)\(\)\()/ \)\(69\!\cdots\!04\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(56\!\cdots\!13\)\( \nu^{19} + \)\(18\!\cdots\!13\)\( \nu^{18} - \)\(42\!\cdots\!98\)\( \nu^{17} - \)\(80\!\cdots\!83\)\( \nu^{16} - \)\(58\!\cdots\!61\)\( \nu^{15} + \)\(14\!\cdots\!57\)\( \nu^{14} - \)\(68\!\cdots\!69\)\( \nu^{13} + \)\(32\!\cdots\!65\)\( \nu^{12} - \)\(47\!\cdots\!86\)\( \nu^{11} + \)\(14\!\cdots\!43\)\( \nu^{10} - \)\(11\!\cdots\!89\)\( \nu^{9} - \)\(76\!\cdots\!50\)\( \nu^{8} - \)\(36\!\cdots\!64\)\( \nu^{7} - \)\(89\!\cdots\!54\)\( \nu^{6} - \)\(94\!\cdots\!13\)\( \nu^{5} - \)\(14\!\cdots\!78\)\( \nu^{4} - \)\(14\!\cdots\!81\)\( \nu^{3} - \)\(20\!\cdots\!66\)\( \nu^{2} - \)\(11\!\cdots\!36\)\( \nu - \)\(29\!\cdots\!88\)\(\)\()/ \)\(54\!\cdots\!44\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(16\!\cdots\!73\)\( \nu^{19} - \)\(40\!\cdots\!82\)\( \nu^{18} + \)\(13\!\cdots\!24\)\( \nu^{17} - \)\(69\!\cdots\!04\)\( \nu^{16} + \)\(17\!\cdots\!28\)\( \nu^{15} - \)\(79\!\cdots\!49\)\( \nu^{14} + \)\(21\!\cdots\!81\)\( \nu^{13} - \)\(58\!\cdots\!17\)\( \nu^{12} + \)\(15\!\cdots\!20\)\( \nu^{11} - \)\(39\!\cdots\!24\)\( \nu^{10} + \)\(43\!\cdots\!06\)\( \nu^{9} - \)\(79\!\cdots\!50\)\( \nu^{8} + \)\(12\!\cdots\!07\)\( \nu^{7} - \)\(23\!\cdots\!88\)\( \nu^{6} + \)\(28\!\cdots\!43\)\( \nu^{5} - \)\(20\!\cdots\!83\)\( \nu^{4} + \)\(50\!\cdots\!26\)\( \nu^{3} - \)\(57\!\cdots\!15\)\( \nu^{2} + \)\(21\!\cdots\!13\)\( \nu - \)\(39\!\cdots\!64\)\(\)\()/ \)\(34\!\cdots\!02\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(81\!\cdots\!62\)\( \nu^{19} - \)\(29\!\cdots\!41\)\( \nu^{18} + \)\(60\!\cdots\!06\)\( \nu^{17} + \)\(11\!\cdots\!49\)\( \nu^{16} + \)\(84\!\cdots\!73\)\( \nu^{15} - \)\(21\!\cdots\!58\)\( \nu^{14} + \)\(98\!\cdots\!75\)\( \nu^{13} - \)\(49\!\cdots\!12\)\( \nu^{12} + \)\(69\!\cdots\!13\)\( \nu^{11} - \)\(25\!\cdots\!52\)\( \nu^{10} + \)\(16\!\cdots\!11\)\( \nu^{9} + \)\(10\!\cdots\!33\)\( \nu^{8} + \)\(52\!\cdots\!19\)\( \nu^{7} + \)\(13\!\cdots\!91\)\( \nu^{6} + \)\(12\!\cdots\!96\)\( \nu^{5} + \)\(22\!\cdots\!70\)\( \nu^{4} + \)\(21\!\cdots\!34\)\( \nu^{3} + \)\(29\!\cdots\!56\)\( \nu^{2} + \)\(16\!\cdots\!20\)\( \nu + \)\(18\!\cdots\!84\)\(\)\()/ \)\(15\!\cdots\!76\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(24\!\cdots\!41\)\( \nu^{19} + \)\(13\!\cdots\!69\)\( \nu^{18} + \)\(18\!\cdots\!94\)\( \nu^{17} + \)\(47\!\cdots\!37\)\( \nu^{16} + \)\(25\!\cdots\!75\)\( \nu^{15} - \)\(41\!\cdots\!93\)\( \nu^{14} + \)\(28\!\cdots\!55\)\( \nu^{13} + \)\(89\!\cdots\!71\)\( \nu^{12} + \)\(20\!\cdots\!24\)\( \nu^{11} + \)\(81\!\cdots\!49\)\( \nu^{10} + \)\(48\!\cdots\!51\)\( \nu^{9} + \)\(49\!\cdots\!00\)\( \nu^{8} + \)\(15\!\cdots\!26\)\( \nu^{7} + \)\(47\!\cdots\!52\)\( \nu^{6} + \)\(43\!\cdots\!65\)\( \nu^{5} + \)\(81\!\cdots\!02\)\( \nu^{4} + \)\(64\!\cdots\!93\)\( \nu^{3} + \)\(90\!\cdots\!66\)\( \nu^{2} + \)\(52\!\cdots\!40\)\( \nu + \)\(16\!\cdots\!84\)\(\)\()/ \)\(31\!\cdots\!52\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(56\!\cdots\!84\)\( \nu^{19} + \)\(11\!\cdots\!77\)\( \nu^{18} - \)\(44\!\cdots\!89\)\( \nu^{17} + \)\(25\!\cdots\!06\)\( \nu^{16} - \)\(58\!\cdots\!57\)\( \nu^{15} + \)\(25\!\cdots\!25\)\( \nu^{14} - \)\(72\!\cdots\!73\)\( \nu^{13} + \)\(16\!\cdots\!81\)\( \nu^{12} - \)\(50\!\cdots\!25\)\( \nu^{11} + \)\(11\!\cdots\!62\)\( \nu^{10} - \)\(12\!\cdots\!83\)\( \nu^{9} + \)\(19\!\cdots\!49\)\( \nu^{8} - \)\(39\!\cdots\!42\)\( \nu^{7} - \)\(98\!\cdots\!96\)\( \nu^{6} - \)\(87\!\cdots\!34\)\( \nu^{5} + \)\(48\!\cdots\!65\)\( \nu^{4} - \)\(14\!\cdots\!78\)\( \nu^{3} + \)\(15\!\cdots\!05\)\( \nu^{2} - \)\(13\!\cdots\!10\)\( \nu + \)\(10\!\cdots\!64\)\(\)\()/ \)\(69\!\cdots\!04\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(40\!\cdots\!71\)\( \nu^{19} + \)\(81\!\cdots\!37\)\( \nu^{18} - \)\(32\!\cdots\!84\)\( \nu^{17} - \)\(14\!\cdots\!43\)\( \nu^{16} - \)\(43\!\cdots\!65\)\( \nu^{15} + \)\(17\!\cdots\!11\)\( \nu^{14} - \)\(53\!\cdots\!55\)\( \nu^{13} + \)\(11\!\cdots\!81\)\( \nu^{12} - \)\(37\!\cdots\!78\)\( \nu^{11} + \)\(76\!\cdots\!29\)\( \nu^{10} - \)\(10\!\cdots\!91\)\( \nu^{9} + \)\(13\!\cdots\!92\)\( \nu^{8} - \)\(30\!\cdots\!98\)\( \nu^{7} - \)\(10\!\cdots\!88\)\( \nu^{6} - \)\(70\!\cdots\!29\)\( \nu^{5} + \)\(10\!\cdots\!66\)\( \nu^{4} - \)\(12\!\cdots\!31\)\( \nu^{3} + \)\(70\!\cdots\!08\)\( \nu^{2} - \)\(37\!\cdots\!08\)\( \nu - \)\(24\!\cdots\!12\)\(\)\()/ \)\(40\!\cdots\!32\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(21\!\cdots\!25\)\( \nu^{19} + \)\(43\!\cdots\!99\)\( \nu^{18} - \)\(16\!\cdots\!68\)\( \nu^{17} + \)\(62\!\cdots\!03\)\( \nu^{16} - \)\(22\!\cdots\!67\)\( \nu^{15} + \)\(94\!\cdots\!84\)\( \nu^{14} - \)\(27\!\cdots\!19\)\( \nu^{13} + \)\(60\!\cdots\!66\)\( \nu^{12} - \)\(18\!\cdots\!41\)\( \nu^{11} + \)\(41\!\cdots\!69\)\( \nu^{10} - \)\(48\!\cdots\!48\)\( \nu^{9} + \)\(75\!\cdots\!10\)\( \nu^{8} - \)\(14\!\cdots\!17\)\( \nu^{7} - \)\(40\!\cdots\!70\)\( \nu^{6} - \)\(33\!\cdots\!74\)\( \nu^{5} + \)\(19\!\cdots\!29\)\( \nu^{4} - \)\(52\!\cdots\!62\)\( \nu^{3} + \)\(58\!\cdots\!71\)\( \nu^{2} - \)\(69\!\cdots\!04\)\( \nu + \)\(40\!\cdots\!80\)\(\)\()/ \)\(20\!\cdots\!16\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(23\!\cdots\!85\)\( \nu^{19} + \)\(59\!\cdots\!66\)\( \nu^{18} - \)\(19\!\cdots\!03\)\( \nu^{17} + \)\(10\!\cdots\!85\)\( \nu^{16} - \)\(25\!\cdots\!16\)\( \nu^{15} + \)\(11\!\cdots\!15\)\( \nu^{14} - \)\(31\!\cdots\!66\)\( \nu^{13} + \)\(84\!\cdots\!79\)\( \nu^{12} - \)\(22\!\cdots\!02\)\( \nu^{11} + \)\(56\!\cdots\!83\)\( \nu^{10} - \)\(63\!\cdots\!61\)\( \nu^{9} + \)\(11\!\cdots\!73\)\( \nu^{8} - \)\(18\!\cdots\!43\)\( \nu^{7} + \)\(49\!\cdots\!38\)\( \nu^{6} - \)\(42\!\cdots\!58\)\( \nu^{5} + \)\(30\!\cdots\!88\)\( \nu^{4} - \)\(73\!\cdots\!96\)\( \nu^{3} + \)\(83\!\cdots\!92\)\( \nu^{2} - \)\(24\!\cdots\!16\)\( \nu + \)\(58\!\cdots\!24\)\(\)\()/ \)\(20\!\cdots\!16\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(16\!\cdots\!35\)\( \nu^{19} + \)\(13\!\cdots\!87\)\( \nu^{18} - \)\(13\!\cdots\!50\)\( \nu^{17} - \)\(15\!\cdots\!05\)\( \nu^{16} - \)\(17\!\cdots\!11\)\( \nu^{15} + \)\(52\!\cdots\!59\)\( \nu^{14} - \)\(21\!\cdots\!51\)\( \nu^{13} + \)\(22\!\cdots\!31\)\( \nu^{12} - \)\(14\!\cdots\!78\)\( \nu^{11} + \)\(14\!\cdots\!85\)\( \nu^{10} - \)\(38\!\cdots\!39\)\( \nu^{9} + \)\(12\!\cdots\!34\)\( \nu^{8} - \)\(11\!\cdots\!28\)\( \nu^{7} - \)\(17\!\cdots\!74\)\( \nu^{6} - \)\(29\!\cdots\!31\)\( \nu^{5} - \)\(23\!\cdots\!98\)\( \nu^{4} - \)\(47\!\cdots\!11\)\( \nu^{3} - \)\(15\!\cdots\!06\)\( \nu^{2} - \)\(10\!\cdots\!04\)\( \nu - \)\(10\!\cdots\!84\)\(\)\()/ \)\(13\!\cdots\!08\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(61\!\cdots\!69\)\( \nu^{19} + \)\(59\!\cdots\!75\)\( \nu^{18} - \)\(48\!\cdots\!16\)\( \nu^{17} - \)\(48\!\cdots\!71\)\( \nu^{16} - \)\(65\!\cdots\!05\)\( \nu^{15} + \)\(20\!\cdots\!51\)\( \nu^{14} - \)\(77\!\cdots\!37\)\( \nu^{13} + \)\(97\!\cdots\!89\)\( \nu^{12} - \)\(55\!\cdots\!16\)\( \nu^{11} + \)\(62\!\cdots\!09\)\( \nu^{10} - \)\(14\!\cdots\!31\)\( \nu^{9} + \)\(81\!\cdots\!14\)\( \nu^{8} - \)\(43\!\cdots\!00\)\( \nu^{7} - \)\(55\!\cdots\!48\)\( \nu^{6} - \)\(10\!\cdots\!19\)\( \nu^{5} - \)\(67\!\cdots\!42\)\( \nu^{4} - \)\(17\!\cdots\!07\)\( \nu^{3} - \)\(77\!\cdots\!84\)\( \nu^{2} - \)\(41\!\cdots\!52\)\( \nu - \)\(13\!\cdots\!12\)\(\)\()/ \)\(40\!\cdots\!32\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(62\!\cdots\!69\)\( \nu^{19} + \)\(11\!\cdots\!97\)\( \nu^{18} - \)\(48\!\cdots\!76\)\( \nu^{17} - \)\(43\!\cdots\!13\)\( \nu^{16} - \)\(65\!\cdots\!52\)\( \nu^{15} + \)\(26\!\cdots\!81\)\( \nu^{14} - \)\(80\!\cdots\!76\)\( \nu^{13} + \)\(17\!\cdots\!22\)\( \nu^{12} - \)\(56\!\cdots\!72\)\( \nu^{11} + \)\(11\!\cdots\!82\)\( \nu^{10} - \)\(14\!\cdots\!39\)\( \nu^{9} + \)\(20\!\cdots\!05\)\( \nu^{8} - \)\(45\!\cdots\!68\)\( \nu^{7} - \)\(17\!\cdots\!97\)\( \nu^{6} - \)\(10\!\cdots\!80\)\( \nu^{5} + \)\(28\!\cdots\!21\)\( \nu^{4} - \)\(16\!\cdots\!12\)\( \nu^{3} + \)\(12\!\cdots\!40\)\( \nu^{2} - \)\(38\!\cdots\!08\)\( \nu - \)\(11\!\cdots\!44\)\(\)\()/ \)\(20\!\cdots\!16\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(80\!\cdots\!34\)\( \nu^{19} + \)\(11\!\cdots\!68\)\( \nu^{18} - \)\(63\!\cdots\!74\)\( \nu^{17} - \)\(40\!\cdots\!49\)\( \nu^{16} - \)\(85\!\cdots\!38\)\( \nu^{15} + \)\(30\!\cdots\!11\)\( \nu^{14} - \)\(10\!\cdots\!93\)\( \nu^{13} + \)\(16\!\cdots\!08\)\( \nu^{12} - \)\(72\!\cdots\!05\)\( \nu^{11} + \)\(10\!\cdots\!96\)\( \nu^{10} - \)\(18\!\cdots\!47\)\( \nu^{9} + \)\(16\!\cdots\!04\)\( \nu^{8} - \)\(57\!\cdots\!63\)\( \nu^{7} - \)\(53\!\cdots\!75\)\( \nu^{6} - \)\(13\!\cdots\!63\)\( \nu^{5} - \)\(35\!\cdots\!99\)\( \nu^{4} - \)\(21\!\cdots\!52\)\( \nu^{3} + \)\(54\!\cdots\!24\)\( \nu^{2} - \)\(42\!\cdots\!20\)\( \nu - \)\(26\!\cdots\!28\)\(\)\()/ \)\(20\!\cdots\!16\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{12} + \beta_{9} - \beta_{8} - 49 \beta_{7} - 6 \beta_{6} + \beta_{4} + 6\)
\(\nu^{3}\)\(=\)\(-\beta_{19} + \beta_{16} + 2 \beta_{14} + 3 \beta_{13} + \beta_{12} - 2 \beta_{11} + 2 \beta_{9} - 2 \beta_{8} + 28 \beta_{7} + 28 \beta_{6} + 81 \beta_{5} - 81 \beta_{4} - 66 \beta_{3} - 35 \beta_{2} - 66 \beta_{1} - 35\)
\(\nu^{4}\)\(=\)\(6 \beta_{19} - 3 \beta_{18} + 5 \beta_{17} - 109 \beta_{16} + 3 \beta_{15} - 5 \beta_{14} + 4 \beta_{13} - 11 \beta_{12} - 4 \beta_{11} - 6 \beta_{10} - 11 \beta_{8} - 1001 \beta_{7} - 4056 \beta_{6} - 197 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 1001 \beta_{2} + 2 \beta_{1}\)
\(\nu^{5}\)\(=\)\(-17 \beta_{19} + 17 \beta_{18} - 249 \beta_{17} - 219 \beta_{16} + 249 \beta_{13} - 408 \beta_{11} + 126 \beta_{10} - 219 \beta_{9} + 369 \beta_{8} - 315 \beta_{6} + 1946 \beta_{5} + 6002 \beta_{3} + 315 \beta_{2} - 1697 \beta_{1} + 7332\)
\(\nu^{6}\)\(=\)\(-391 \beta_{18} - 943 \beta_{17} - 470 \beta_{15} + 472 \beta_{14} + 2129 \beta_{12} - 943 \beta_{11} - 391 \beta_{10} - 11170 \beta_{9} + 129800 \beta_{7} - 2186 \beta_{4} - 2186 \beta_{3} + 259700 \beta_{2} + 24965 \beta_{1} - 129800\)
\(\nu^{7}\)\(=\)\(3378 \beta_{19} + 10216 \beta_{18} - 18712 \beta_{17} + 18726 \beta_{16} + 3378 \beta_{15} - 26738 \beta_{14} - 26738 \beta_{13} - 21000 \beta_{12} + 21000 \beta_{9} - 39726 \beta_{8} - 1059378 \beta_{7} - 101768 \beta_{6} - 255964 \beta_{5} + 779871 \beta_{4} + 229226 \beta_{3} - 26738 \beta_{1} + 101768\)
\(\nu^{8}\)\(=\)\(-103772 \beta_{19} + 1140633 \beta_{16} - 45416 \beta_{15} - 74628 \beta_{14} - 33804 \beta_{13} + 1140633 \beta_{12} + 74628 \beta_{11} + 45416 \beta_{10} - 298572 \beta_{9} + 298572 \beta_{8} + 39621781 \beta_{7} + 39621781 \beta_{6} + 3383445 \beta_{5} - 3383445 \beta_{4} - 2817041 \beta_{3} - 24013831 \beta_{2} - 2817041 \beta_{1} - 24013831\)
\(\nu^{9}\)\(=\)\(1450621 \beta_{19} - 960777 \beta_{18} + 2035773 \beta_{17} - 2777337 \beta_{16} + 960777 \beta_{15} - 2035773 \beta_{14} - 4826063 \beta_{13} + 1337594 \beta_{12} + 4826063 \beta_{11} - 1450621 \beta_{10} + 1337594 \beta_{8} - 29236885 \beta_{7} - 139458096 \beta_{6} - 81253177 \beta_{5} + 31230388 \beta_{4} - 4826063 \beta_{3} + 29236885 \beta_{2} + 26404325 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-5314715 \beta_{19} + 5314715 \beta_{18} + 4888469 \beta_{17} + 37443407 \beta_{16} - 4888469 \beta_{13} + 891068 \beta_{11} + 11974174 \beta_{10} + 37443407 \beta_{9} + 79924742 \beta_{8} + 1811301461 \beta_{6} + 107288446 \beta_{5} + 293855786 \beta_{3} - 1811301461 \beta_{2} - 112176915 \beta_{1} + 2321917415\)
\(\nu^{11}\)\(=\)\(-63162745 \beta_{18} + 291186469 \beta_{17} - 93206829 \beta_{15} + 215238939 \beta_{14} - 68191175 \beta_{12} + 291186469 \beta_{11} - 63162745 \beta_{10} - 352872962 \beta_{9} + 5229729401 \beta_{7} - 3676709050 \beta_{4} - 3676709050 \beta_{3} + 12160673160 \beta_{2} + 4919241041 \beta_{1} - 5229729401\)
\(\nu^{12}\)\(=\)\(628941271 \beta_{19} + 732584482 \beta_{18} - 444888904 \beta_{17} - 4464778965 \beta_{16} + 628941271 \beta_{15} + 212777339 \beta_{14} + 212777339 \beta_{13} - 12190834454 \beta_{12} + 12190834454 \beta_{9} - 7726055489 \beta_{8} - 437142897208 \beta_{7} - 206326475808 \beta_{6} - 16155268842 \beta_{5} + 47901253935 \beta_{4} + 16368046181 \beta_{3} + 212777339 \beta_{1} + 206326475808\)
\(\nu^{13}\)\(=\)\(-17032821338 \beta_{19} + 43657317140 \beta_{16} - 7707637614 \beta_{15} + 30587060194 \beta_{14} + 53107249298 \beta_{13} + 43657317140 \beta_{12} - 30587060194 \beta_{11} + 7707637614 \beta_{10} - 564855954 \beta_{9} + 564855954 \beta_{8} + 2111203895678 \beta_{7} + 2111203895678 \beta_{6} + 918668035351 \beta_{5} - 918668035351 \beta_{4} - 440934661377 \beta_{3} - 1324846024046 \beta_{2} - 440934661377 \beta_{1} - 1324846024046\)
\(\nu^{14}\)\(=\)\(153953121956 \beta_{19} - 79410379408 \beta_{18} + 53176977104 \beta_{17} - 1277679603561 \beta_{16} + 79410379408 \beta_{15} - 53176977104 \beta_{14} - 60819880868 \beta_{13} - 518644733732 \beta_{12} + 60819880868 \beta_{11} - 153953121956 \beta_{10} - 518644733732 \beta_{8} - 23268015977126 \beta_{7} - 46668114553869 \beta_{6} - 5598937791885 \beta_{5} + 2220877241464 \beta_{4} - 60819880868 \beta_{3} + 23268015977126 \beta_{2} + 2160057360596 \beta_{1}\)
\(\nu^{15}\)\(=\)\(-913604260816 \beta_{19} + 913604260816 \beta_{18} - 3235932238730 \beta_{17} + 863704227014 \beta_{16} + 3235932238730 \beta_{13} - 5587446334931 \beta_{11} + 1870033955985 \beta_{10} + 863704227014 \beta_{9} + 4440614608619 \beta_{8} + 108028556736337 \beta_{6} + 48501985560908 \beta_{5} + 53677152610279 \beta_{3} - 108028556736337 \beta_{2} - 45266053322178 \beta_{1} + 144241411965627\)
\(\nu^{16}\)\(=\)\(-8796134072883 \beta_{18} + 3823389968623 \beta_{17} - 8576511792099 \beta_{15} + 6519642754429 \beta_{14} + 59362074981075 \beta_{12} + 3823389968623 \beta_{11} - 8796134072883 \beta_{10} - 135011957705357 \beta_{9} + 2609849489416065 \beta_{7} - 288891397343934 \beta_{4} - 288891397343934 \beta_{3} + 2408804192105871 \beta_{2} + 361665423162631 \beta_{1} - 2609849489416065\)
\(\nu^{17}\)\(=\)\(106507268471761 \beta_{19} + 99981415905261 \beta_{18} - 246103968451527 \beta_{17} - 173761853717853 \beta_{16} + 106507268471761 \beta_{15} - 344543328079569 \beta_{14} - 344543328079569 \beta_{13} - 636134042095086 \beta_{12} + 636134042095086 \beta_{9} - 462372188377233 \beta_{8} - 29823212853601257 \beta_{7} - 14053547135257677 \beta_{6} - 5503388066028218 \beta_{5} + 10724421940459171 \beta_{4} + 5158844737948649 \beta_{3} - 344543328079569 \beta_{1} + 14053547135257677\)
\(\nu^{18}\)\(=\)\(-1959280512795509 \beta_{19} + 14373005898818122 \beta_{16} - 1031263966456727 \beta_{15} + 710341976432377 \beta_{14} + 1511443569723505 \beta_{13} + 14373005898818122 \beta_{12} - 710341976432377 \beta_{11} + 1031263966456727 \beta_{10} - 6734284022345609 \beta_{9} + 6734284022345609 \beta_{8} + 543015139771997908 \beta_{7} + 543015139771997908 \beta_{6} + 75264211365490351 \beta_{5} - 75264211365490351 \beta_{4} - 37513050346101844 \beta_{3} - 251105127525787820 \beta_{2} - 37513050346101844 \beta_{1} - 251105127525787820\)
\(\nu^{19}\)\(=\)\(22896802393502826 \beta_{19} - 10605598822795040 \beta_{18} + 25872979532316616 \beta_{17} - 75540542807022976 \beta_{16} + 10605598822795040 \beta_{15} - 25872979532316616 \beta_{14} - 62765186339032202 \beta_{13} - 26826430903030538 \beta_{12} + 62765186339032202 \beta_{11} - 22896802393502826 \beta_{10} - 26826430903030538 \beta_{8} - 1763986712176702248 \beta_{7} - 3497975938847469514 \beta_{6} - 1168721828218833167 \beta_{5} + 621899392402054124 \beta_{4} - 62765186339032202 \beta_{3} + 1763986712176702248 \beta_{2} + 559134206063021922 \beta_{1}\)

Character values

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

\(n\) \(13\) \(23\)
\(\chi(n)\) \(-1 - \beta_{2} + \beta_{6} + \beta_{7}\) \(1\)

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
−8.59311 + 6.24326i
−3.89647 + 2.83095i
−0.490010 + 0.356013i
4.70404 3.41769i
7.96653 5.78803i
−2.34585 + 7.21978i
−1.64776 + 5.07128i
0.527694 1.62407i
1.34436 4.13753i
2.93057 9.01936i
−8.59311 6.24326i
−3.89647 2.83095i
−0.490010 0.356013i
4.70404 + 3.41769i
7.96653 + 5.78803i
−2.34585 7.21978i
−1.64776 5.07128i
0.527694 + 1.62407i
1.34436 + 4.13753i
2.93057 + 9.01936i
−7.78409 5.65548i −2.78115 + 8.55951i 18.7192 + 57.6117i 48.1408 34.9763i 70.0568 50.8993i 9.43666 + 29.0431i 84.9654 261.497i −65.5304 47.6106i −572.540
4.2 −3.08746 2.24317i −2.78115 + 8.55951i −5.38796 16.5824i 3.77724 2.74432i 27.7871 20.1885i 49.4196 + 152.098i −58.2998 + 179.428i −65.5304 47.6106i −17.8180
4.3 0.319007 + 0.231772i −2.78115 + 8.55951i −9.84050 30.2859i −17.3054 + 12.5731i −2.87106 + 2.08595i −52.3175 161.017i 7.77944 23.9427i −65.5304 47.6106i −8.43465
4.4 5.51306 + 4.00547i −2.78115 + 8.55951i 4.46148 + 13.7310i −62.5986 + 45.4806i −49.6176 + 36.0493i 59.3828 + 182.761i 36.9828 113.821i −65.5304 47.6106i −527.281
4.5 8.77555 + 6.37581i −2.78115 + 8.55951i 26.4708 + 81.4687i 31.3852 22.8027i −78.9800 + 57.3823i −67.6896 208.327i −179.871 + 553.584i −65.5304 47.6106i 420.808
16.1 −2.65487 8.17084i 7.28115 + 5.29007i −33.8257 + 24.5758i −21.1802 + 65.1858i 23.8938 73.5376i −196.384 + 142.681i 68.1911 + 49.5438i 25.0304 + 77.0356i 588.853
16.2 −1.95678 6.02234i 7.28115 + 5.29007i −6.55106 + 4.75963i 8.65323 26.6319i 17.6110 54.2011i 121.539 88.3034i −122.450 88.9651i 25.0304 + 77.0356i −177.319
16.3 0.218677 + 0.673018i 7.28115 + 5.29007i 25.4834 18.5148i −21.0138 + 64.6739i −1.96809 + 6.05716i 98.3565 71.4602i 36.3535 + 26.4124i 25.0304 + 77.0356i −48.1219
16.4 1.03535 + 3.18647i 7.28115 + 5.29007i 16.8069 12.2109i 32.9083 101.281i −9.31813 + 28.6783i −32.7853 + 23.8199i 143.049 + 103.931i 25.0304 + 77.0356i 356.801
16.5 2.62155 + 8.06830i 7.28115 + 5.29007i −32.3365 + 23.4938i −8.26670 + 25.4423i −23.5940 + 72.6147i −58.4588 + 42.4728i −54.7010 39.7426i 25.0304 + 77.0356i −226.948
25.1 −7.78409 + 5.65548i −2.78115 8.55951i 18.7192 57.6117i 48.1408 + 34.9763i 70.0568 + 50.8993i 9.43666 29.0431i 84.9654 + 261.497i −65.5304 + 47.6106i −572.540
25.2 −3.08746 + 2.24317i −2.78115 8.55951i −5.38796 + 16.5824i 3.77724 + 2.74432i 27.7871 + 20.1885i 49.4196 152.098i −58.2998 179.428i −65.5304 + 47.6106i −17.8180
25.3 0.319007 0.231772i −2.78115 8.55951i −9.84050 + 30.2859i −17.3054 12.5731i −2.87106 2.08595i −52.3175 + 161.017i 7.77944 + 23.9427i −65.5304 + 47.6106i −8.43465
25.4 5.51306 4.00547i −2.78115 8.55951i 4.46148 13.7310i −62.5986 45.4806i −49.6176 36.0493i 59.3828 182.761i 36.9828 + 113.821i −65.5304 + 47.6106i −527.281
25.5 8.77555 6.37581i −2.78115 8.55951i 26.4708 81.4687i 31.3852 + 22.8027i −78.9800 57.3823i −67.6896 + 208.327i −179.871 553.584i −65.5304 + 47.6106i 420.808
31.1 −2.65487 + 8.17084i 7.28115 5.29007i −33.8257 24.5758i −21.1802 65.1858i 23.8938 + 73.5376i −196.384 142.681i 68.1911 49.5438i 25.0304 77.0356i 588.853
31.2 −1.95678 + 6.02234i 7.28115 5.29007i −6.55106 4.75963i 8.65323 + 26.6319i 17.6110 + 54.2011i 121.539 + 88.3034i −122.450 + 88.9651i 25.0304 77.0356i −177.319
31.3 0.218677 0.673018i 7.28115 5.29007i 25.4834 + 18.5148i −21.0138 64.6739i −1.96809 6.05716i 98.3565 + 71.4602i 36.3535 26.4124i 25.0304 77.0356i −48.1219
31.4 1.03535 3.18647i 7.28115 5.29007i 16.8069 + 12.2109i 32.9083 + 101.281i −9.31813 28.6783i −32.7853 23.8199i 143.049 103.931i 25.0304 77.0356i 356.801
31.5 2.62155 8.06830i 7.28115 5.29007i −32.3365 23.4938i −8.26670 25.4423i −23.5940 72.6147i −58.4588 42.4728i −54.7010 + 39.7426i 25.0304 77.0356i −226.948
\(n\): 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.b 20
3.b odd 2 1 99.6.f.b 20
11.c even 5 1 inner 33.6.e.b 20
11.c even 5 1 363.6.a.r 10
11.d odd 10 1 363.6.a.t 10
33.f even 10 1 1089.6.a.bi 10
33.h odd 10 1 99.6.f.b 20
33.h odd 10 1 1089.6.a.bk 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.e.b 20 1.a even 1 1 trivial
33.6.e.b 20 11.c even 5 1 inner
99.6.f.b 20 3.b odd 2 1
99.6.f.b 20 33.h odd 10 1
363.6.a.r 10 11.c even 5 1
363.6.a.t 10 11.d odd 10 1
1089.6.a.bi 10 33.f even 10 1
1089.6.a.bk 10 33.h odd 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

$p$ $F_p(T)$
$2$ \( 1 - 6 T - 66 T^{2} + 504 T^{3} + 2385 T^{4} - 28452 T^{5} - 41165 T^{6} + 1071615 T^{7} + 917613 T^{8} - 35227644 T^{9} - 27914152 T^{10} + 1271584920 T^{11} - 348543616 T^{12} - 47976238272 T^{13} + 97406507712 T^{14} + 1344142785024 T^{15} - 5381764079104 T^{16} - 22758073680384 T^{17} + 186607195168768 T^{18} + 192925698600960 T^{19} - 5970303143669760 T^{20} + 6173622355230720 T^{21} + 191085767852818432 T^{22} - 745736558358822912 T^{23} - 5643188651010555904 T^{24} + 45101947678378426368 T^{25} + \)\(10\!\cdots\!88\)\( T^{26} - \)\(16\!\cdots\!96\)\( T^{27} - \)\(38\!\cdots\!16\)\( T^{28} + \)\(44\!\cdots\!40\)\( T^{29} - \)\(31\!\cdots\!48\)\( T^{30} - \)\(12\!\cdots\!92\)\( T^{31} + \)\(10\!\cdots\!88\)\( T^{32} + \)\(39\!\cdots\!80\)\( T^{33} - \)\(48\!\cdots\!60\)\( T^{34} - \)\(10\!\cdots\!36\)\( T^{35} + \)\(28\!\cdots\!60\)\( T^{36} + \)\(19\!\cdots\!28\)\( T^{37} - \)\(81\!\cdots\!84\)\( T^{38} - \)\(23\!\cdots\!08\)\( T^{39} + \)\(12\!\cdots\!76\)\( T^{40} \)
$3$ \( ( 1 - 9 T + 81 T^{2} - 729 T^{3} + 6561 T^{4} )^{5} \)
$5$ \( 1 + 11 T - 1192 T^{2} + 391578 T^{3} + 7985791 T^{4} - 7835820 T^{5} + 76020239189 T^{6} - 252536679958 T^{7} + 42068789311252 T^{8} + 17769917591746189 T^{9} - 184077767376804972 T^{10} - 17686850814152975101 T^{11} + \)\(18\!\cdots\!32\)\( T^{12} - \)\(31\!\cdots\!18\)\( T^{13} - \)\(30\!\cdots\!01\)\( T^{14} - \)\(26\!\cdots\!20\)\( T^{15} - \)\(16\!\cdots\!99\)\( T^{16} - \)\(76\!\cdots\!82\)\( T^{17} + \)\(12\!\cdots\!88\)\( T^{18} - \)\(18\!\cdots\!79\)\( T^{19} - \)\(26\!\cdots\!14\)\( T^{20} - \)\(59\!\cdots\!75\)\( T^{21} + \)\(11\!\cdots\!00\)\( T^{22} - \)\(23\!\cdots\!50\)\( T^{23} - \)\(15\!\cdots\!75\)\( T^{24} - \)\(79\!\cdots\!00\)\( T^{25} - \)\(28\!\cdots\!25\)\( T^{26} - \)\(90\!\cdots\!50\)\( T^{27} + \)\(16\!\cdots\!00\)\( T^{28} - \)\(50\!\cdots\!25\)\( T^{29} - \)\(16\!\cdots\!00\)\( T^{30} + \)\(49\!\cdots\!25\)\( T^{31} + \)\(36\!\cdots\!00\)\( T^{32} - \)\(68\!\cdots\!50\)\( T^{33} + \)\(64\!\cdots\!25\)\( T^{34} - \)\(20\!\cdots\!00\)\( T^{35} + \)\(66\!\cdots\!75\)\( T^{36} + \)\(10\!\cdots\!50\)\( T^{37} - \)\(96\!\cdots\!00\)\( T^{38} + \)\(27\!\cdots\!75\)\( T^{39} + \)\(78\!\cdots\!25\)\( T^{40} \)
$7$ \( 1 + 139 T + 7083 T^{2} - 3092128 T^{3} - 175291068 T^{4} - 33053179529 T^{5} + 2979421835355 T^{6} + 482586576948395 T^{7} + 305219033995671954 T^{8} + 18888116552559950812 T^{9} + \)\(17\!\cdots\!94\)\( T^{10} - \)\(52\!\cdots\!20\)\( T^{11} - \)\(42\!\cdots\!21\)\( T^{12} - \)\(98\!\cdots\!33\)\( T^{13} + \)\(78\!\cdots\!85\)\( T^{14} + \)\(10\!\cdots\!13\)\( T^{15} + \)\(37\!\cdots\!51\)\( T^{16} + \)\(17\!\cdots\!88\)\( T^{17} + \)\(28\!\cdots\!43\)\( T^{18} - \)\(58\!\cdots\!41\)\( T^{19} - \)\(56\!\cdots\!19\)\( T^{20} - \)\(98\!\cdots\!87\)\( T^{21} + \)\(79\!\cdots\!07\)\( T^{22} + \)\(83\!\cdots\!84\)\( T^{23} + \)\(29\!\cdots\!51\)\( T^{24} + \)\(13\!\cdots\!91\)\( T^{25} + \)\(17\!\cdots\!65\)\( T^{26} - \)\(37\!\cdots\!19\)\( T^{27} - \)\(27\!\cdots\!21\)\( T^{28} - \)\(56\!\cdots\!40\)\( T^{29} + \)\(31\!\cdots\!06\)\( T^{30} + \)\(57\!\cdots\!16\)\( T^{31} + \)\(15\!\cdots\!54\)\( T^{32} + \)\(41\!\cdots\!65\)\( T^{33} + \)\(42\!\cdots\!95\)\( T^{34} - \)\(79\!\cdots\!47\)\( T^{35} - \)\(71\!\cdots\!68\)\( T^{36} - \)\(21\!\cdots\!96\)\( T^{37} + \)\(81\!\cdots\!67\)\( T^{38} + \)\(26\!\cdots\!77\)\( T^{39} + \)\(32\!\cdots\!01\)\( T^{40} \)
$11$ \( 1 - 2289 T + 2971338 T^{2} - 2736985632 T^{3} + 1953224328341 T^{4} - 1123060363206210 T^{5} + 530782182037499579 T^{6} - \)\(20\!\cdots\!88\)\( T^{7} + \)\(69\!\cdots\!82\)\( T^{8} - \)\(21\!\cdots\!81\)\( T^{9} + \)\(72\!\cdots\!18\)\( T^{10} - \)\(33\!\cdots\!31\)\( T^{11} + \)\(17\!\cdots\!82\)\( T^{12} - \)\(86\!\cdots\!88\)\( T^{13} + \)\(35\!\cdots\!79\)\( T^{14} - \)\(12\!\cdots\!10\)\( T^{15} + \)\(34\!\cdots\!41\)\( T^{16} - \)\(76\!\cdots\!32\)\( T^{17} + \)\(13\!\cdots\!38\)\( T^{18} - \)\(16\!\cdots\!39\)\( T^{19} + \)\(11\!\cdots\!01\)\( T^{20} \)
$13$ \( 1 + 847 T - 381353 T^{2} - 23203690 T^{3} + 614927875370 T^{4} - 84069398831999 T^{5} - 376117754446506463 T^{6} + \)\(14\!\cdots\!51\)\( T^{7} + \)\(15\!\cdots\!08\)\( T^{8} - \)\(13\!\cdots\!18\)\( T^{9} - \)\(45\!\cdots\!94\)\( T^{10} + \)\(84\!\cdots\!82\)\( T^{11} + \)\(27\!\cdots\!07\)\( T^{12} - \)\(38\!\cdots\!19\)\( T^{13} + \)\(94\!\cdots\!27\)\( T^{14} + \)\(14\!\cdots\!61\)\( T^{15} - \)\(82\!\cdots\!33\)\( T^{16} - \)\(43\!\cdots\!30\)\( T^{17} + \)\(41\!\cdots\!79\)\( T^{18} + \)\(56\!\cdots\!77\)\( T^{19} - \)\(17\!\cdots\!13\)\( T^{20} + \)\(20\!\cdots\!61\)\( T^{21} + \)\(57\!\cdots\!71\)\( T^{22} - \)\(22\!\cdots\!10\)\( T^{23} - \)\(15\!\cdots\!33\)\( T^{24} + \)\(10\!\cdots\!73\)\( T^{25} + \)\(24\!\cdots\!23\)\( T^{26} - \)\(37\!\cdots\!83\)\( T^{27} + \)\(98\!\cdots\!07\)\( T^{28} + \)\(11\!\cdots\!26\)\( T^{29} - \)\(22\!\cdots\!06\)\( T^{30} - \)\(24\!\cdots\!26\)\( T^{31} + \)\(10\!\cdots\!08\)\( T^{32} + \)\(37\!\cdots\!43\)\( T^{33} - \)\(35\!\cdots\!87\)\( T^{34} - \)\(29\!\cdots\!43\)\( T^{35} + \)\(80\!\cdots\!70\)\( T^{36} - \)\(11\!\cdots\!70\)\( T^{37} - \)\(68\!\cdots\!97\)\( T^{38} + \)\(56\!\cdots\!79\)\( T^{39} + \)\(24\!\cdots\!01\)\( T^{40} \)
$17$ \( 1 - 2482 T - 586109 T^{2} + 3633625872 T^{3} + 3530566411959 T^{4} - 8503032341651140 T^{5} - 833559953449917351 T^{6} + \)\(11\!\cdots\!90\)\( T^{7} - \)\(42\!\cdots\!02\)\( T^{8} - \)\(16\!\cdots\!86\)\( T^{9} + \)\(31\!\cdots\!27\)\( T^{10} + \)\(44\!\cdots\!52\)\( T^{11} - \)\(19\!\cdots\!03\)\( T^{12} - \)\(44\!\cdots\!76\)\( T^{13} + \)\(16\!\cdots\!81\)\( T^{14} + \)\(57\!\cdots\!98\)\( T^{15} - \)\(45\!\cdots\!36\)\( T^{16} - \)\(47\!\cdots\!90\)\( T^{17} + \)\(12\!\cdots\!51\)\( T^{18} - \)\(28\!\cdots\!76\)\( T^{19} - \)\(11\!\cdots\!15\)\( T^{20} - \)\(40\!\cdots\!32\)\( T^{21} + \)\(25\!\cdots\!99\)\( T^{22} - \)\(13\!\cdots\!70\)\( T^{23} - \)\(18\!\cdots\!36\)\( T^{24} + \)\(32\!\cdots\!86\)\( T^{25} + \)\(13\!\cdots\!69\)\( T^{26} - \)\(51\!\cdots\!68\)\( T^{27} - \)\(31\!\cdots\!03\)\( T^{28} + \)\(10\!\cdots\!64\)\( T^{29} + \)\(10\!\cdots\!23\)\( T^{30} - \)\(78\!\cdots\!98\)\( T^{31} - \)\(28\!\cdots\!02\)\( T^{32} + \)\(11\!\cdots\!30\)\( T^{33} - \)\(11\!\cdots\!99\)\( T^{34} - \)\(16\!\cdots\!20\)\( T^{35} + \)\(96\!\cdots\!59\)\( T^{36} + \)\(14\!\cdots\!04\)\( T^{37} - \)\(32\!\cdots\!41\)\( T^{38} - \)\(19\!\cdots\!26\)\( T^{39} + \)\(11\!\cdots\!01\)\( T^{40} \)
$19$ \( 1 - 2958 T + 2646201 T^{2} - 3660731050 T^{3} + 2792721703941 T^{4} + 3451918392922414 T^{5} + 18067528085611636271 T^{6} - \)\(25\!\cdots\!26\)\( T^{7} - \)\(41\!\cdots\!58\)\( T^{8} + \)\(17\!\cdots\!10\)\( T^{9} - \)\(10\!\cdots\!47\)\( T^{10} + \)\(12\!\cdots\!50\)\( T^{11} - \)\(13\!\cdots\!41\)\( T^{12} + \)\(19\!\cdots\!10\)\( T^{13} - \)\(42\!\cdots\!33\)\( T^{14} + \)\(11\!\cdots\!26\)\( T^{15} - \)\(89\!\cdots\!08\)\( T^{16} + \)\(23\!\cdots\!06\)\( T^{17} - \)\(89\!\cdots\!51\)\( T^{18} - \)\(45\!\cdots\!18\)\( T^{19} + \)\(24\!\cdots\!03\)\( T^{20} - \)\(11\!\cdots\!82\)\( T^{21} - \)\(54\!\cdots\!51\)\( T^{22} + \)\(35\!\cdots\!94\)\( T^{23} - \)\(33\!\cdots\!08\)\( T^{24} + \)\(10\!\cdots\!74\)\( T^{25} - \)\(97\!\cdots\!33\)\( T^{26} + \)\(10\!\cdots\!90\)\( T^{27} - \)\(19\!\cdots\!41\)\( T^{28} + \)\(43\!\cdots\!50\)\( T^{29} - \)\(86\!\cdots\!47\)\( T^{30} + \)\(36\!\cdots\!90\)\( T^{31} - \)\(22\!\cdots\!58\)\( T^{32} - \)\(34\!\cdots\!74\)\( T^{33} + \)\(58\!\cdots\!71\)\( T^{34} + \)\(27\!\cdots\!86\)\( T^{35} + \)\(55\!\cdots\!41\)\( T^{36} - \)\(18\!\cdots\!50\)\( T^{37} + \)\(32\!\cdots\!01\)\( T^{38} - \)\(89\!\cdots\!42\)\( T^{39} + \)\(75\!\cdots\!01\)\( T^{40} \)
$23$ \( ( 1 - 4070 T + 45637381 T^{2} - 146013507066 T^{3} + 948326619574152 T^{4} - 2478949772209263686 T^{5} + \)\(12\!\cdots\!67\)\( T^{6} - \)\(26\!\cdots\!42\)\( T^{7} + \)\(11\!\cdots\!39\)\( T^{8} - \)\(21\!\cdots\!76\)\( T^{9} + \)\(80\!\cdots\!96\)\( T^{10} - \)\(13\!\cdots\!68\)\( T^{11} + \)\(46\!\cdots\!11\)\( T^{12} - \)\(71\!\cdots\!94\)\( T^{13} + \)\(20\!\cdots\!67\)\( T^{14} - \)\(27\!\cdots\!98\)\( T^{15} + \)\(67\!\cdots\!48\)\( T^{16} - \)\(66\!\cdots\!62\)\( T^{17} + \)\(13\!\cdots\!81\)\( T^{18} - \)\(77\!\cdots\!10\)\( T^{19} + \)\(12\!\cdots\!49\)\( T^{20} )^{2} \)
$29$ \( 1 + 20210 T + 202559070 T^{2} + 1543626359990 T^{3} + 11921455751952854 T^{4} + 86898022076080579600 T^{5} + \)\(52\!\cdots\!66\)\( T^{6} + \)\(27\!\cdots\!10\)\( T^{7} + \)\(14\!\cdots\!66\)\( T^{8} + \)\(75\!\cdots\!50\)\( T^{9} + \)\(30\!\cdots\!64\)\( T^{10} + \)\(10\!\cdots\!70\)\( T^{11} + \)\(38\!\cdots\!10\)\( T^{12} + \)\(84\!\cdots\!50\)\( T^{13} - \)\(34\!\cdots\!14\)\( T^{14} - \)\(40\!\cdots\!80\)\( T^{15} - \)\(23\!\cdots\!70\)\( T^{16} - \)\(15\!\cdots\!30\)\( T^{17} - \)\(92\!\cdots\!50\)\( T^{18} - \)\(42\!\cdots\!10\)\( T^{19} - \)\(18\!\cdots\!14\)\( T^{20} - \)\(87\!\cdots\!90\)\( T^{21} - \)\(38\!\cdots\!50\)\( T^{22} - \)\(13\!\cdots\!70\)\( T^{23} - \)\(41\!\cdots\!70\)\( T^{24} - \)\(14\!\cdots\!20\)\( T^{25} - \)\(25\!\cdots\!14\)\( T^{26} + \)\(12\!\cdots\!50\)\( T^{27} + \)\(11\!\cdots\!10\)\( T^{28} + \)\(67\!\cdots\!30\)\( T^{29} + \)\(40\!\cdots\!64\)\( T^{30} + \)\(20\!\cdots\!50\)\( T^{31} + \)\(83\!\cdots\!66\)\( T^{32} + \)\(31\!\cdots\!90\)\( T^{33} + \)\(12\!\cdots\!66\)\( T^{34} + \)\(41\!\cdots\!00\)\( T^{35} + \)\(11\!\cdots\!54\)\( T^{36} + \)\(31\!\cdots\!10\)\( T^{37} + \)\(83\!\cdots\!70\)\( T^{38} + \)\(17\!\cdots\!90\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$31$ \( 1 - 5540 T - 37687062 T^{2} + 228873818242 T^{3} + 55200276680500 T^{4} - 4073041469087644628 T^{5} + \)\(58\!\cdots\!28\)\( T^{6} - \)\(23\!\cdots\!30\)\( T^{7} - \)\(22\!\cdots\!46\)\( T^{8} + \)\(21\!\cdots\!36\)\( T^{9} + \)\(48\!\cdots\!04\)\( T^{10} - \)\(19\!\cdots\!36\)\( T^{11} + \)\(12\!\cdots\!42\)\( T^{12} + \)\(55\!\cdots\!06\)\( T^{13} - \)\(25\!\cdots\!24\)\( T^{14} - \)\(15\!\cdots\!44\)\( T^{15} + \)\(90\!\cdots\!72\)\( T^{16} + \)\(34\!\cdots\!46\)\( T^{17} - \)\(15\!\cdots\!98\)\( T^{18} - \)\(71\!\cdots\!36\)\( T^{19} + \)\(55\!\cdots\!34\)\( T^{20} - \)\(20\!\cdots\!36\)\( T^{21} - \)\(12\!\cdots\!98\)\( T^{22} + \)\(81\!\cdots\!46\)\( T^{23} + \)\(60\!\cdots\!72\)\( T^{24} - \)\(28\!\cdots\!44\)\( T^{25} - \)\(14\!\cdots\!24\)\( T^{26} + \)\(88\!\cdots\!06\)\( T^{27} + \)\(57\!\cdots\!42\)\( T^{28} - \)\(25\!\cdots\!36\)\( T^{29} + \)\(18\!\cdots\!04\)\( T^{30} + \)\(22\!\cdots\!36\)\( T^{31} - \)\(67\!\cdots\!46\)\( T^{32} - \)\(20\!\cdots\!30\)\( T^{33} + \)\(14\!\cdots\!28\)\( T^{34} - \)\(28\!\cdots\!28\)\( T^{35} + \)\(11\!\cdots\!00\)\( T^{36} + \)\(13\!\cdots\!42\)\( T^{37} - \)\(62\!\cdots\!62\)\( T^{38} - \)\(26\!\cdots\!40\)\( T^{39} + \)\(13\!\cdots\!01\)\( T^{40} \)
$37$ \( 1 + 25173 T + 101240475 T^{2} - 3137677574014 T^{3} - 38583783979797222 T^{4} + 17854665244699157015 T^{5} + \)\(30\!\cdots\!13\)\( T^{6} + \)\(17\!\cdots\!17\)\( T^{7} - \)\(49\!\cdots\!64\)\( T^{8} - \)\(71\!\cdots\!22\)\( T^{9} + \)\(21\!\cdots\!58\)\( T^{10} + \)\(22\!\cdots\!30\)\( T^{11} - \)\(72\!\cdots\!29\)\( T^{12} - \)\(96\!\cdots\!73\)\( T^{13} + \)\(65\!\cdots\!39\)\( T^{14} + \)\(89\!\cdots\!67\)\( T^{15} + \)\(53\!\cdots\!07\)\( T^{16} - \)\(15\!\cdots\!54\)\( T^{17} - \)\(28\!\cdots\!17\)\( T^{18} - \)\(62\!\cdots\!13\)\( T^{19} + \)\(50\!\cdots\!27\)\( T^{20} - \)\(43\!\cdots\!41\)\( T^{21} - \)\(13\!\cdots\!33\)\( T^{22} - \)\(51\!\cdots\!22\)\( T^{23} + \)\(12\!\cdots\!07\)\( T^{24} + \)\(14\!\cdots\!19\)\( T^{25} + \)\(72\!\cdots\!11\)\( T^{26} - \)\(74\!\cdots\!89\)\( T^{27} - \)\(38\!\cdots\!29\)\( T^{28} + \)\(81\!\cdots\!10\)\( T^{29} + \)\(55\!\cdots\!42\)\( T^{30} - \)\(12\!\cdots\!46\)\( T^{31} - \)\(60\!\cdots\!64\)\( T^{32} + \)\(15\!\cdots\!69\)\( T^{33} + \)\(18\!\cdots\!37\)\( T^{34} + \)\(73\!\cdots\!95\)\( T^{35} - \)\(11\!\cdots\!22\)\( T^{36} - \)\(62\!\cdots\!98\)\( T^{37} + \)\(13\!\cdots\!75\)\( T^{38} + \)\(23\!\cdots\!89\)\( T^{39} + \)\(66\!\cdots\!01\)\( T^{40} \)
$41$ \( 1 - 54349 T + 1240096769 T^{2} - 14082883973844 T^{3} + 61786091684525404 T^{4} + \)\(16\!\cdots\!81\)\( T^{5} - \)\(36\!\cdots\!71\)\( T^{6} - \)\(27\!\cdots\!41\)\( T^{7} + \)\(17\!\cdots\!70\)\( T^{8} - \)\(43\!\cdots\!20\)\( T^{9} + \)\(79\!\cdots\!72\)\( T^{10} + \)\(17\!\cdots\!42\)\( T^{11} - \)\(12\!\cdots\!03\)\( T^{12} - \)\(39\!\cdots\!49\)\( T^{13} + \)\(18\!\cdots\!39\)\( T^{14} - \)\(48\!\cdots\!41\)\( T^{15} - \)\(21\!\cdots\!63\)\( T^{16} + \)\(14\!\cdots\!14\)\( T^{17} + \)\(16\!\cdots\!09\)\( T^{18} - \)\(30\!\cdots\!57\)\( T^{19} + \)\(29\!\cdots\!99\)\( T^{20} - \)\(35\!\cdots\!57\)\( T^{21} + \)\(22\!\cdots\!09\)\( T^{22} + \)\(22\!\cdots\!14\)\( T^{23} - \)\(38\!\cdots\!63\)\( T^{24} - \)\(10\!\cdots\!41\)\( T^{25} + \)\(44\!\cdots\!39\)\( T^{26} - \)\(11\!\cdots\!49\)\( T^{27} - \)\(40\!\cdots\!03\)\( T^{28} + \)\(64\!\cdots\!42\)\( T^{29} + \)\(34\!\cdots\!72\)\( T^{30} - \)\(22\!\cdots\!20\)\( T^{31} + \)\(10\!\cdots\!70\)\( T^{32} - \)\(18\!\cdots\!41\)\( T^{33} - \)\(28\!\cdots\!71\)\( T^{34} + \)\(14\!\cdots\!81\)\( T^{35} + \)\(65\!\cdots\!04\)\( T^{36} - \)\(17\!\cdots\!44\)\( T^{37} + \)\(17\!\cdots\!69\)\( T^{38} - \)\(89\!\cdots\!49\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$43$ \( ( 1 + 10844 T + 1196564203 T^{2} + 10995563504928 T^{3} + 667098098771873060 T^{4} + \)\(52\!\cdots\!40\)\( T^{5} + \)\(23\!\cdots\!21\)\( T^{6} + \)\(15\!\cdots\!32\)\( T^{7} + \)\(54\!\cdots\!03\)\( T^{8} + \)\(32\!\cdots\!04\)\( T^{9} + \)\(93\!\cdots\!20\)\( T^{10} + \)\(47\!\cdots\!72\)\( T^{11} + \)\(11\!\cdots\!47\)\( T^{12} + \)\(49\!\cdots\!24\)\( T^{13} + \)\(10\!\cdots\!21\)\( T^{14} + \)\(36\!\cdots\!20\)\( T^{15} + \)\(67\!\cdots\!40\)\( T^{16} + \)\(16\!\cdots\!96\)\( T^{17} + \)\(26\!\cdots\!03\)\( T^{18} + \)\(34\!\cdots\!92\)\( T^{19} + \)\(47\!\cdots\!49\)\( T^{20} )^{2} \)
$47$ \( 1 + 48387 T + 710746584 T^{2} - 3158063032284 T^{3} - 262897144295380353 T^{4} - \)\(63\!\cdots\!86\)\( T^{5} - \)\(10\!\cdots\!91\)\( T^{6} - \)\(65\!\cdots\!36\)\( T^{7} + \)\(17\!\cdots\!96\)\( T^{8} + \)\(53\!\cdots\!07\)\( T^{9} + \)\(85\!\cdots\!00\)\( T^{10} + \)\(96\!\cdots\!47\)\( T^{11} + \)\(54\!\cdots\!36\)\( T^{12} - \)\(10\!\cdots\!76\)\( T^{13} - \)\(48\!\cdots\!57\)\( T^{14} - \)\(10\!\cdots\!38\)\( T^{15} - \)\(13\!\cdots\!67\)\( T^{16} - \)\(10\!\cdots\!72\)\( T^{17} + \)\(81\!\cdots\!96\)\( T^{18} + \)\(55\!\cdots\!75\)\( T^{19} + \)\(11\!\cdots\!02\)\( T^{20} + \)\(12\!\cdots\!25\)\( T^{21} + \)\(43\!\cdots\!04\)\( T^{22} - \)\(12\!\cdots\!96\)\( T^{23} - \)\(37\!\cdots\!67\)\( T^{24} - \)\(65\!\cdots\!66\)\( T^{25} - \)\(71\!\cdots\!93\)\( T^{26} - \)\(35\!\cdots\!68\)\( T^{27} + \)\(41\!\cdots\!36\)\( T^{28} + \)\(16\!\cdots\!29\)\( T^{29} + \)\(34\!\cdots\!00\)\( T^{30} + \)\(49\!\cdots\!01\)\( T^{31} + \)\(37\!\cdots\!96\)\( T^{32} - \)\(31\!\cdots\!52\)\( T^{33} - \)\(12\!\cdots\!59\)\( T^{34} - \)\(16\!\cdots\!98\)\( T^{35} - \)\(15\!\cdots\!53\)\( T^{36} - \)\(42\!\cdots\!88\)\( T^{37} + \)\(21\!\cdots\!16\)\( T^{38} + \)\(34\!\cdots\!41\)\( T^{39} + \)\(16\!\cdots\!01\)\( T^{40} \)
$53$ \( 1 + 74382 T + 2332170246 T^{2} + 27820741138818 T^{3} - 310764907217742870 T^{4} - \)\(17\!\cdots\!96\)\( T^{5} - \)\(41\!\cdots\!70\)\( T^{6} - \)\(10\!\cdots\!70\)\( T^{7} - \)\(17\!\cdots\!82\)\( T^{8} + \)\(10\!\cdots\!78\)\( T^{9} + \)\(13\!\cdots\!92\)\( T^{10} + \)\(36\!\cdots\!70\)\( T^{11} + \)\(45\!\cdots\!34\)\( T^{12} - \)\(14\!\cdots\!66\)\( T^{13} - \)\(20\!\cdots\!82\)\( T^{14} - \)\(52\!\cdots\!92\)\( T^{15} - \)\(94\!\cdots\!14\)\( T^{16} - \)\(14\!\cdots\!02\)\( T^{17} + \)\(54\!\cdots\!02\)\( T^{18} + \)\(10\!\cdots\!90\)\( T^{19} + \)\(32\!\cdots\!70\)\( T^{20} + \)\(43\!\cdots\!70\)\( T^{21} + \)\(95\!\cdots\!98\)\( T^{22} - \)\(10\!\cdots\!14\)\( T^{23} - \)\(28\!\cdots\!14\)\( T^{24} - \)\(67\!\cdots\!56\)\( T^{25} - \)\(11\!\cdots\!18\)\( T^{26} - \)\(32\!\cdots\!62\)\( T^{27} + \)\(42\!\cdots\!34\)\( T^{28} + \)\(14\!\cdots\!10\)\( T^{29} + \)\(22\!\cdots\!08\)\( T^{30} + \)\(73\!\cdots\!46\)\( T^{31} - \)\(48\!\cdots\!82\)\( T^{32} - \)\(12\!\cdots\!10\)\( T^{33} - \)\(20\!\cdots\!30\)\( T^{34} - \)\(35\!\cdots\!72\)\( T^{35} - \)\(27\!\cdots\!70\)\( T^{36} + \)\(10\!\cdots\!74\)\( T^{37} + \)\(35\!\cdots\!54\)\( T^{38} + \)\(47\!\cdots\!74\)\( T^{39} + \)\(26\!\cdots\!01\)\( T^{40} \)
$59$ \( 1 + 108412 T + 3177982803 T^{2} - 14395521095770 T^{3} + 228820294084344289 T^{4} + \)\(12\!\cdots\!38\)\( T^{5} + \)\(23\!\cdots\!45\)\( T^{6} - \)\(30\!\cdots\!96\)\( T^{7} - \)\(33\!\cdots\!06\)\( T^{8} + \)\(14\!\cdots\!32\)\( T^{9} + \)\(54\!\cdots\!75\)\( T^{10} + \)\(12\!\cdots\!74\)\( T^{11} - \)\(90\!\cdots\!89\)\( T^{12} - \)\(22\!\cdots\!90\)\( T^{13} + \)\(64\!\cdots\!05\)\( T^{14} + \)\(11\!\cdots\!16\)\( T^{15} - \)\(20\!\cdots\!08\)\( T^{16} + \)\(34\!\cdots\!76\)\( T^{17} + \)\(16\!\cdots\!83\)\( T^{18} + \)\(46\!\cdots\!74\)\( T^{19} + \)\(22\!\cdots\!59\)\( T^{20} + \)\(33\!\cdots\!26\)\( T^{21} + \)\(82\!\cdots\!83\)\( T^{22} + \)\(12\!\cdots\!24\)\( T^{23} - \)\(53\!\cdots\!08\)\( T^{24} + \)\(20\!\cdots\!84\)\( T^{25} + \)\(86\!\cdots\!05\)\( T^{26} - \)\(21\!\cdots\!10\)\( T^{27} - \)\(61\!\cdots\!89\)\( T^{28} + \)\(58\!\cdots\!26\)\( T^{29} + \)\(18\!\cdots\!75\)\( T^{30} + \)\(35\!\cdots\!68\)\( T^{31} - \)\(60\!\cdots\!06\)\( T^{32} - \)\(38\!\cdots\!04\)\( T^{33} + \)\(21\!\cdots\!45\)\( T^{34} + \)\(83\!\cdots\!62\)\( T^{35} + \)\(10\!\cdots\!89\)\( T^{36} - \)\(47\!\cdots\!30\)\( T^{37} + \)\(75\!\cdots\!03\)\( T^{38} + \)\(18\!\cdots\!88\)\( T^{39} + \)\(12\!\cdots\!01\)\( T^{40} \)
$61$ \( 1 - 16737 T - 4257807684 T^{2} + 74454312318022 T^{3} + 8195809786068315819 T^{4} - \)\(18\!\cdots\!36\)\( T^{5} - \)\(81\!\cdots\!39\)\( T^{6} + \)\(32\!\cdots\!78\)\( T^{7} + \)\(18\!\cdots\!48\)\( T^{8} - \)\(39\!\cdots\!35\)\( T^{9} + \)\(71\!\cdots\!52\)\( T^{10} + \)\(30\!\cdots\!67\)\( T^{11} - \)\(12\!\cdots\!36\)\( T^{12} - \)\(10\!\cdots\!70\)\( T^{13} + \)\(10\!\cdots\!63\)\( T^{14} - \)\(89\!\cdots\!00\)\( T^{15} - \)\(50\!\cdots\!19\)\( T^{16} + \)\(14\!\cdots\!66\)\( T^{17} - \)\(24\!\cdots\!24\)\( T^{18} - \)\(59\!\cdots\!19\)\( T^{19} + \)\(19\!\cdots\!66\)\( T^{20} - \)\(50\!\cdots\!19\)\( T^{21} - \)\(17\!\cdots\!24\)\( T^{22} + \)\(86\!\cdots\!66\)\( T^{23} - \)\(25\!\cdots\!19\)\( T^{24} - \)\(38\!\cdots\!00\)\( T^{25} + \)\(39\!\cdots\!63\)\( T^{26} - \)\(31\!\cdots\!70\)\( T^{27} - \)\(32\!\cdots\!36\)\( T^{28} + \)\(67\!\cdots\!67\)\( T^{29} + \)\(13\!\cdots\!52\)\( T^{30} - \)\(60\!\cdots\!35\)\( T^{31} + \)\(24\!\cdots\!48\)\( T^{32} + \)\(35\!\cdots\!78\)\( T^{33} - \)\(77\!\cdots\!39\)\( T^{34} - \)\(14\!\cdots\!36\)\( T^{35} + \)\(54\!\cdots\!19\)\( T^{36} + \)\(42\!\cdots\!22\)\( T^{37} - \)\(20\!\cdots\!84\)\( T^{38} - \)\(67\!\cdots\!37\)\( T^{39} + \)\(34\!\cdots\!01\)\( T^{40} \)
$67$ \( ( 1 - 10089 T + 9200644318 T^{2} - 30582343541560 T^{3} + 40026382528996867101 T^{4} + \)\(76\!\cdots\!62\)\( T^{5} + \)\(11\!\cdots\!19\)\( T^{6} + \)\(60\!\cdots\!56\)\( T^{7} + \)\(22\!\cdots\!74\)\( T^{8} + \)\(15\!\cdots\!43\)\( T^{9} + \)\(33\!\cdots\!74\)\( T^{10} + \)\(20\!\cdots\!01\)\( T^{11} + \)\(40\!\cdots\!26\)\( T^{12} + \)\(14\!\cdots\!08\)\( T^{13} + \)\(36\!\cdots\!19\)\( T^{14} + \)\(34\!\cdots\!34\)\( T^{15} + \)\(24\!\cdots\!49\)\( T^{16} - \)\(25\!\cdots\!80\)\( T^{17} + \)\(10\!\cdots\!18\)\( T^{18} - \)\(15\!\cdots\!23\)\( T^{19} + \)\(20\!\cdots\!49\)\( T^{20} )^{2} \)
$71$ \( 1 - 143157 T + 1051422668 T^{2} + 559993621841664 T^{3} + 567598711388826911 T^{4} - \)\(20\!\cdots\!10\)\( T^{5} - \)\(98\!\cdots\!19\)\( T^{6} + \)\(48\!\cdots\!08\)\( T^{7} + \)\(86\!\cdots\!32\)\( T^{8} - \)\(92\!\cdots\!57\)\( T^{9} - \)\(23\!\cdots\!20\)\( T^{10} + \)\(10\!\cdots\!43\)\( T^{11} + \)\(56\!\cdots\!28\)\( T^{12} - \)\(46\!\cdots\!32\)\( T^{13} - \)\(10\!\cdots\!69\)\( T^{14} - \)\(12\!\cdots\!70\)\( T^{15} + \)\(15\!\cdots\!65\)\( T^{16} + \)\(40\!\cdots\!24\)\( T^{17} - \)\(20\!\cdots\!92\)\( T^{18} - \)\(27\!\cdots\!17\)\( T^{19} + \)\(23\!\cdots\!98\)\( T^{20} - \)\(50\!\cdots\!67\)\( T^{21} - \)\(65\!\cdots\!92\)\( T^{22} + \)\(23\!\cdots\!24\)\( T^{23} + \)\(16\!\cdots\!65\)\( T^{24} - \)\(24\!\cdots\!70\)\( T^{25} - \)\(36\!\cdots\!69\)\( T^{26} - \)\(28\!\cdots\!32\)\( T^{27} + \)\(62\!\cdots\!28\)\( T^{28} + \)\(21\!\cdots\!93\)\( T^{29} - \)\(87\!\cdots\!20\)\( T^{30} - \)\(60\!\cdots\!07\)\( T^{31} + \)\(10\!\cdots\!32\)\( T^{32} + \)\(10\!\cdots\!08\)\( T^{33} - \)\(38\!\cdots\!19\)\( T^{34} - \)\(14\!\cdots\!10\)\( T^{35} + \)\(71\!\cdots\!11\)\( T^{36} + \)\(12\!\cdots\!64\)\( T^{37} + \)\(43\!\cdots\!68\)\( T^{38} - \)\(10\!\cdots\!07\)\( T^{39} + \)\(13\!\cdots\!01\)\( T^{40} \)
$73$ \( 1 - 164980 T + 10183330756 T^{2} - 64891522430754 T^{3} - 24454208090589510900 T^{4} + \)\(67\!\cdots\!92\)\( T^{5} + \)\(62\!\cdots\!92\)\( T^{6} - \)\(31\!\cdots\!30\)\( T^{7} - \)\(15\!\cdots\!52\)\( T^{8} + \)\(14\!\cdots\!56\)\( T^{9} + \)\(34\!\cdots\!48\)\( T^{10} - \)\(36\!\cdots\!80\)\( T^{11} + \)\(54\!\cdots\!20\)\( T^{12} + \)\(79\!\cdots\!86\)\( T^{13} - \)\(24\!\cdots\!68\)\( T^{14} - \)\(16\!\cdots\!72\)\( T^{15} + \)\(83\!\cdots\!28\)\( T^{16} + \)\(24\!\cdots\!98\)\( T^{17} - \)\(23\!\cdots\!04\)\( T^{18} - \)\(13\!\cdots\!68\)\( T^{19} + \)\(48\!\cdots\!46\)\( T^{20} - \)\(27\!\cdots\!24\)\( T^{21} - \)\(99\!\cdots\!96\)\( T^{22} + \)\(21\!\cdots\!86\)\( T^{23} + \)\(15\!\cdots\!28\)\( T^{24} - \)\(61\!\cdots\!96\)\( T^{25} - \)\(19\!\cdots\!32\)\( T^{26} + \)\(13\!\cdots\!02\)\( T^{27} + \)\(18\!\cdots\!20\)\( T^{28} - \)\(25\!\cdots\!40\)\( T^{29} + \)\(50\!\cdots\!52\)\( T^{30} + \)\(44\!\cdots\!92\)\( T^{31} - \)\(96\!\cdots\!52\)\( T^{32} - \)\(41\!\cdots\!90\)\( T^{33} + \)\(16\!\cdots\!08\)\( T^{34} + \)\(37\!\cdots\!44\)\( T^{35} - \)\(28\!\cdots\!00\)\( T^{36} - \)\(15\!\cdots\!22\)\( T^{37} + \)\(50\!\cdots\!44\)\( T^{38} - \)\(17\!\cdots\!60\)\( T^{39} + \)\(21\!\cdots\!01\)\( T^{40} \)
$79$ \( 1 - 369613 T + 72619945775 T^{2} - 9956180775758220 T^{3} + \)\(10\!\cdots\!40\)\( T^{4} - \)\(93\!\cdots\!89\)\( T^{5} + \)\(68\!\cdots\!63\)\( T^{6} - \)\(40\!\cdots\!93\)\( T^{7} + \)\(19\!\cdots\!50\)\( T^{8} - \)\(57\!\cdots\!80\)\( T^{9} - \)\(63\!\cdots\!38\)\( T^{10} + \)\(24\!\cdots\!60\)\( T^{11} - \)\(21\!\cdots\!57\)\( T^{12} + \)\(12\!\cdots\!67\)\( T^{13} - \)\(56\!\cdots\!35\)\( T^{14} + \)\(14\!\cdots\!93\)\( T^{15} + \)\(26\!\cdots\!03\)\( T^{16} - \)\(68\!\cdots\!00\)\( T^{17} + \)\(58\!\cdots\!03\)\( T^{18} - \)\(39\!\cdots\!33\)\( T^{19} + \)\(22\!\cdots\!45\)\( T^{20} - \)\(12\!\cdots\!67\)\( T^{21} + \)\(55\!\cdots\!03\)\( T^{22} - \)\(19\!\cdots\!00\)\( T^{23} + \)\(23\!\cdots\!03\)\( T^{24} + \)\(40\!\cdots\!07\)\( T^{25} - \)\(48\!\cdots\!35\)\( T^{26} + \)\(33\!\cdots\!33\)\( T^{27} - \)\(17\!\cdots\!57\)\( T^{28} + \)\(61\!\cdots\!40\)\( T^{29} - \)\(48\!\cdots\!38\)\( T^{30} - \)\(13\!\cdots\!20\)\( T^{31} + \)\(13\!\cdots\!50\)\( T^{32} - \)\(90\!\cdots\!07\)\( T^{33} + \)\(46\!\cdots\!63\)\( T^{34} - \)\(19\!\cdots\!11\)\( T^{35} + \)\(68\!\cdots\!40\)\( T^{36} - \)\(19\!\cdots\!80\)\( T^{37} + \)\(44\!\cdots\!75\)\( T^{38} - \)\(69\!\cdots\!87\)\( T^{39} + \)\(57\!\cdots\!01\)\( T^{40} \)
$83$ \( 1 - 267741 T + 18435478109 T^{2} + 672321419443590 T^{3} - 76827219835151070874 T^{4} - \)\(48\!\cdots\!05\)\( T^{5} + \)\(49\!\cdots\!67\)\( T^{6} - \)\(13\!\cdots\!53\)\( T^{7} + \)\(27\!\cdots\!40\)\( T^{8} + \)\(10\!\cdots\!18\)\( T^{9} - \)\(35\!\cdots\!40\)\( T^{10} - \)\(79\!\cdots\!76\)\( T^{11} + \)\(20\!\cdots\!93\)\( T^{12} + \)\(12\!\cdots\!93\)\( T^{13} + \)\(20\!\cdots\!33\)\( T^{14} + \)\(58\!\cdots\!63\)\( T^{15} - \)\(77\!\cdots\!27\)\( T^{16} - \)\(58\!\cdots\!92\)\( T^{17} - \)\(12\!\cdots\!23\)\( T^{18} + \)\(92\!\cdots\!33\)\( T^{19} + \)\(54\!\cdots\!97\)\( T^{20} + \)\(36\!\cdots\!19\)\( T^{21} - \)\(20\!\cdots\!27\)\( T^{22} - \)\(35\!\cdots\!44\)\( T^{23} - \)\(18\!\cdots\!27\)\( T^{24} + \)\(55\!\cdots\!09\)\( T^{25} + \)\(77\!\cdots\!17\)\( T^{26} + \)\(17\!\cdots\!51\)\( T^{27} + \)\(11\!\cdots\!93\)\( T^{28} - \)\(18\!\cdots\!68\)\( T^{29} - \)\(31\!\cdots\!60\)\( T^{30} + \)\(38\!\cdots\!26\)\( T^{31} + \)\(39\!\cdots\!40\)\( T^{32} - \)\(76\!\cdots\!79\)\( T^{33} + \)\(10\!\cdots\!83\)\( T^{34} - \)\(41\!\cdots\!35\)\( T^{35} - \)\(25\!\cdots\!74\)\( T^{36} + \)\(88\!\cdots\!70\)\( T^{37} + \)\(96\!\cdots\!41\)\( T^{38} - \)\(54\!\cdots\!87\)\( T^{39} + \)\(80\!\cdots\!01\)\( T^{40} \)
$89$ \( ( 1 - 102746 T + 45851660001 T^{2} - 3724467972360430 T^{3} + \)\(94\!\cdots\!60\)\( T^{4} - \)\(62\!\cdots\!70\)\( T^{5} + \)\(11\!\cdots\!59\)\( T^{6} - \)\(66\!\cdots\!02\)\( T^{7} + \)\(10\!\cdots\!79\)\( T^{8} - \)\(50\!\cdots\!80\)\( T^{9} + \)\(66\!\cdots\!80\)\( T^{10} - \)\(28\!\cdots\!20\)\( T^{11} + \)\(32\!\cdots\!79\)\( T^{12} - \)\(11\!\cdots\!98\)\( T^{13} + \)\(11\!\cdots\!59\)\( T^{14} - \)\(34\!\cdots\!30\)\( T^{15} + \)\(28\!\cdots\!60\)\( T^{16} - \)\(63\!\cdots\!70\)\( T^{17} + \)\(43\!\cdots\!01\)\( T^{18} - \)\(54\!\cdots\!54\)\( T^{19} + \)\(29\!\cdots\!01\)\( T^{20} )^{2} \)
$97$ \( 1 + 300567 T + 8991582556 T^{2} - 5546439472556530 T^{3} - \)\(64\!\cdots\!73\)\( T^{4} - \)\(98\!\cdots\!20\)\( T^{5} + \)\(54\!\cdots\!85\)\( T^{6} + \)\(10\!\cdots\!62\)\( T^{7} + \)\(49\!\cdots\!00\)\( T^{8} - \)\(91\!\cdots\!31\)\( T^{9} - \)\(99\!\cdots\!40\)\( T^{10} - \)\(97\!\cdots\!73\)\( T^{11} + \)\(33\!\cdots\!40\)\( T^{12} + \)\(58\!\cdots\!54\)\( T^{13} + \)\(25\!\cdots\!39\)\( T^{14} - \)\(52\!\cdots\!96\)\( T^{15} - \)\(20\!\cdots\!35\)\( T^{16} + \)\(43\!\cdots\!70\)\( T^{17} + \)\(10\!\cdots\!24\)\( T^{18} - \)\(19\!\cdots\!59\)\( T^{19} - \)\(10\!\cdots\!78\)\( T^{20} - \)\(16\!\cdots\!63\)\( T^{21} + \)\(75\!\cdots\!76\)\( T^{22} + \)\(27\!\cdots\!10\)\( T^{23} - \)\(11\!\cdots\!35\)\( T^{24} - \)\(24\!\cdots\!72\)\( T^{25} + \)\(10\!\cdots\!11\)\( T^{26} + \)\(20\!\cdots\!22\)\( T^{27} + \)\(98\!\cdots\!40\)\( T^{28} - \)\(24\!\cdots\!61\)\( T^{29} - \)\(21\!\cdots\!60\)\( T^{30} - \)\(17\!\cdots\!83\)\( T^{31} + \)\(79\!\cdots\!00\)\( T^{32} + \)\(14\!\cdots\!34\)\( T^{33} + \)\(65\!\cdots\!65\)\( T^{34} - \)\(99\!\cdots\!60\)\( T^{35} - \)\(56\!\cdots\!73\)\( T^{36} - \)\(41\!\cdots\!10\)\( T^{37} + \)\(57\!\cdots\!44\)\( T^{38} + \)\(16\!\cdots\!31\)\( T^{39} + \)\(47\!\cdots\!01\)\( T^{40} \)
show more
show less