Properties

Label 43.10.a.a
Level 43
Weight 10
Character orbit 43.a
Self dual yes
Analytic conductor 22.147
Analytic rank 1
Dimension 15
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.1465409550\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 - \beta_{1} ) q^{2} + ( -21 - \beta_{2} ) q^{3} + ( 216 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} + ( -315 + 4 \beta_{1} - \beta_{2} + \beta_{8} ) q^{5} + ( 39 + 39 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{8} + \beta_{9} ) q^{6} + ( -642 - 37 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} + \beta_{4} - \beta_{8} - 2 \beta_{9} ) q^{7} + ( -1333 - 242 \beta_{1} + 31 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} + \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{8} + ( 4618 + 38 \beta_{1} + 44 \beta_{2} - 12 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - 6 \beta_{8} - \beta_{9} + \beta_{10} - 3 \beta_{11} - \beta_{12} - \beta_{14} ) q^{9} +O(q^{10})\) \( q + ( -2 - \beta_{1} ) q^{2} + ( -21 - \beta_{2} ) q^{3} + ( 216 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} + ( -315 + 4 \beta_{1} - \beta_{2} + \beta_{8} ) q^{5} + ( 39 + 39 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{8} + \beta_{9} ) q^{6} + ( -642 - 37 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} + \beta_{4} - \beta_{8} - 2 \beta_{9} ) q^{7} + ( -1333 - 242 \beta_{1} + 31 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} + \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{8} + ( 4618 + 38 \beta_{1} + 44 \beta_{2} - 12 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - 6 \beta_{8} - \beta_{9} + \beta_{10} - 3 \beta_{11} - \beta_{12} - \beta_{14} ) q^{9} + ( -2498 + 511 \beta_{1} + 104 \beta_{2} - 15 \beta_{3} + 5 \beta_{4} - \beta_{5} - 5 \beta_{6} - 4 \beta_{7} - 9 \beta_{8} + \beta_{9} - 4 \beta_{10} - 3 \beta_{11} + 4 \beta_{12} + 4 \beta_{14} ) q^{10} + ( -7072 + 687 \beta_{1} + 38 \beta_{2} - 10 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} + 6 \beta_{6} + 12 \beta_{8} + 10 \beta_{9} - \beta_{10} + 9 \beta_{11} + \beta_{13} ) q^{11} + ( -17920 + 1067 \beta_{1} - 123 \beta_{2} - 53 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} + \beta_{6} + 11 \beta_{7} + 17 \beta_{8} + \beta_{9} + 18 \beta_{10} + 8 \beta_{11} - 11 \beta_{12} - 5 \beta_{13} - 10 \beta_{14} ) q^{12} + ( -7911 + 1267 \beta_{1} + 100 \beta_{2} + 26 \beta_{3} + 8 \beta_{4} - 16 \beta_{5} - \beta_{6} - 11 \beta_{8} + 6 \beta_{9} - 14 \beta_{10} - 4 \beta_{11} - \beta_{12} + \beta_{13} - 7 \beta_{14} ) q^{13} + ( 27336 + 2956 \beta_{1} + 205 \beta_{2} + 69 \beta_{3} - 4 \beta_{4} + 14 \beta_{5} + 14 \beta_{6} + 2 \beta_{7} + 51 \beta_{8} + 7 \beta_{9} - 4 \beta_{10} - 14 \beta_{11} + 5 \beta_{12} + 25 \beta_{13} + 11 \beta_{14} ) q^{14} + ( 27150 + 3562 \beta_{1} + 882 \beta_{2} + 124 \beta_{3} - 20 \beta_{4} + 16 \beta_{5} - 43 \beta_{6} - 30 \beta_{7} - 7 \beta_{8} + 16 \beta_{9} + 19 \beta_{10} - 6 \beta_{11} + 19 \beta_{12} - 15 \beta_{13} + 41 \beta_{14} ) q^{15} + ( 66198 + 2617 \beta_{1} + 458 \beta_{2} + 351 \beta_{3} - \beta_{4} + 10 \beta_{5} + 18 \beta_{6} - 27 \beta_{7} - 8 \beta_{8} - 29 \beta_{9} - 65 \beta_{10} - 13 \beta_{11} + 3 \beta_{12} - 51 \beta_{13} - 25 \beta_{14} ) q^{16} + ( -59323 + 2978 \beta_{1} - 44 \beta_{2} - 2 \beta_{3} - 37 \beta_{4} - 23 \beta_{5} - 39 \beta_{6} + 37 \beta_{7} - 6 \beta_{8} - 15 \beta_{9} + 33 \beta_{10} + 8 \beta_{11} + 34 \beta_{12} + 6 \beta_{13} - 13 \beta_{14} ) q^{17} + ( -38873 - 1427 \beta_{1} - 1078 \beta_{2} + 144 \beta_{3} + 21 \beta_{4} - 29 \beta_{5} + 119 \beta_{6} + 37 \beta_{7} + 109 \beta_{8} - 99 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} - 26 \beta_{12} + 54 \beta_{13} - 26 \beta_{14} ) q^{18} + ( -46688 + 6066 \beta_{1} - 520 \beta_{2} + 73 \beta_{3} + \beta_{4} - 11 \beta_{5} - 61 \beta_{6} + 20 \beta_{7} - 96 \beta_{8} - 16 \beta_{9} + 46 \beta_{10} + 11 \beta_{11} - 128 \beta_{12} + 49 \beta_{13} + 27 \beta_{14} ) q^{19} + ( -205890 + 5736 \beta_{1} - 2665 \beta_{2} - 572 \beta_{3} + 11 \beta_{4} + 27 \beta_{5} - \beta_{6} + 40 \beta_{7} + 163 \beta_{8} - 56 \beta_{9} + 207 \beta_{10} + 45 \beta_{11} - 68 \beta_{12} + 60 \beta_{13} - 17 \beta_{14} ) q^{20} + ( -138638 + 7394 \beta_{1} - 474 \beta_{2} + 399 \beta_{3} - 10 \beta_{4} + 24 \beta_{5} + 41 \beta_{6} - 93 \beta_{7} - 271 \beta_{8} + 111 \beta_{9} - 256 \beta_{10} + 83 \beta_{11} + 102 \beta_{12} - 51 \beta_{13} - 65 \beta_{14} ) q^{21} + ( -486716 + 12702 \beta_{1} - 1727 \beta_{2} - 604 \beta_{3} + 173 \beta_{4} - 35 \beta_{5} - 127 \beta_{6} + 7 \beta_{7} - 342 \beta_{8} - 49 \beta_{9} - 19 \beta_{10} - 5 \beta_{11} + 135 \beta_{12} - 269 \beta_{13} + 21 \beta_{14} ) q^{22} + ( -167207 + 782 \beta_{1} - 578 \beta_{2} - 144 \beta_{3} - 69 \beta_{4} + 29 \beta_{5} + 321 \beta_{6} + 17 \beta_{7} - 495 \beta_{8} + 269 \beta_{9} - 9 \beta_{10} + 34 \beta_{11} + 180 \beta_{12} - 150 \beta_{13} - 23 \beta_{14} ) q^{23} + ( -772426 + 28059 \beta_{1} - 969 \beta_{2} - 551 \beta_{3} + 116 \beta_{4} + 63 \beta_{5} - 485 \beta_{6} - 109 \beta_{7} + 197 \beta_{8} + 41 \beta_{9} - 408 \beta_{10} - 170 \beta_{11} + 115 \beta_{12} + 225 \beta_{13} + 258 \beta_{14} ) q^{24} + ( 86713 + 3275 \beta_{1} - 110 \beta_{2} - 624 \beta_{3} + 27 \beta_{4} + 69 \beta_{5} + 166 \beta_{6} - 141 \beta_{7} - 495 \beta_{8} + 291 \beta_{9} + 377 \beta_{10} - 206 \beta_{11} - 295 \beta_{12} + 229 \beta_{13} + 136 \beta_{14} ) q^{25} + ( -889024 - 6816 \beta_{1} - 5621 \beta_{2} - 2060 \beta_{3} - 297 \beta_{4} + 119 \beta_{5} - 317 \beta_{6} - 99 \beta_{7} + 990 \beta_{8} + 93 \beta_{9} + 599 \beta_{10} + 9 \beta_{11} - 491 \beta_{12} + 241 \beta_{13} - 33 \beta_{14} ) q^{26} + ( -831176 - 35781 \beta_{1} - 884 \beta_{2} + 596 \beta_{3} - 40 \beta_{4} - 284 \beta_{5} + 370 \beta_{6} + 232 \beta_{7} + 140 \beta_{8} - 90 \beta_{9} - 503 \beta_{10} - 142 \beta_{11} + 89 \beta_{12} + 287 \beta_{13} - 286 \beta_{14} ) q^{27} + ( -1863686 - 54068 \beta_{1} - 4722 \beta_{2} - 2022 \beta_{3} + 258 \beta_{4} - 540 \beta_{5} + 372 \beta_{6} + 296 \beta_{7} + 46 \beta_{8} - 376 \beta_{9} + 124 \beta_{10} - 166 \beta_{11} + 476 \beta_{12} - 300 \beta_{13} - 168 \beta_{14} ) q^{28} + ( -1224855 - 22936 \beta_{1} + 7955 \beta_{2} + 1939 \beta_{3} - 468 \beta_{4} + 400 \beta_{5} - 116 \beta_{6} + 543 \beta_{7} + 625 \beta_{8} + 81 \beta_{9} - 87 \beta_{10} + 375 \beta_{11} - 255 \beta_{12} - 564 \beta_{13} + 130 \beta_{14} ) q^{29} + ( -2619272 - 100911 \beta_{1} + 455 \beta_{2} + 1838 \beta_{3} - 301 \beta_{4} - 99 \beta_{5} + 289 \beta_{6} - 111 \beta_{7} + 1932 \beta_{8} - 667 \beta_{9} - 717 \beta_{10} + 701 \beta_{11} + 391 \beta_{12} - 681 \beta_{13} - 335 \beta_{14} ) q^{30} + ( -800503 - 10806 \beta_{1} + 4784 \beta_{2} - 193 \beta_{3} + 265 \beta_{4} - 89 \beta_{5} - 1179 \beta_{6} - 304 \beta_{7} - 659 \beta_{8} - 230 \beta_{9} + 308 \beta_{10} - 135 \beta_{11} - 109 \beta_{12} + 32 \beta_{13} - 1007 \beta_{14} ) q^{31} + ( -1248646 - 122091 \beta_{1} + 4192 \beta_{2} + 1211 \beta_{3} - 551 \beta_{4} + 338 \beta_{5} + 1682 \beta_{6} - 333 \beta_{7} + 2012 \beta_{8} + 185 \beta_{9} + 633 \beta_{10} + 253 \beta_{11} - 391 \beta_{12} + 639 \beta_{13} + 701 \beta_{14} ) q^{32} + ( -941356 - 59317 \beta_{1} + 27195 \beta_{2} + 2757 \beta_{3} + 363 \beta_{4} + 902 \beta_{5} - 696 \beta_{6} - 888 \beta_{7} + 1467 \beta_{8} - 592 \beta_{9} + 1808 \beta_{10} + 448 \beta_{11} - 332 \beta_{12} + 444 \beta_{13} + 1500 \beta_{14} ) q^{33} + ( -2032566 + 56292 \beta_{1} + 6401 \beta_{2} + 614 \beta_{3} + 415 \beta_{4} + 359 \beta_{5} - 1605 \beta_{6} - 94 \beta_{7} - 998 \beta_{8} + 132 \beta_{9} - 774 \beta_{10} - 283 \beta_{11} - 229 \beta_{12} + 323 \beta_{13} + 315 \beta_{14} ) q^{34} + ( -1849286 - 50676 \beta_{1} + 19084 \beta_{2} + 5617 \beta_{3} + 588 \beta_{4} - 894 \beta_{5} - 35 \beta_{6} + 1163 \beta_{7} - 2685 \beta_{8} - 271 \beta_{9} - 2406 \beta_{10} - 749 \beta_{11} + 666 \beta_{12} - 195 \beta_{13} - 817 \beta_{14} ) q^{35} + ( -1218140 - 49427 \beta_{1} + 16676 \beta_{2} + 4513 \beta_{3} + 277 \beta_{4} + 285 \beta_{5} + 1121 \beta_{6} - 1230 \beta_{7} - 941 \beta_{8} + 514 \beta_{9} - 1357 \beta_{10} - 365 \beta_{11} + 1150 \beta_{12} + 522 \beta_{13} + 935 \beta_{14} ) q^{36} + ( -577015 - 63537 \beta_{1} + 28794 \beta_{2} + 1034 \beta_{3} + 2291 \beta_{4} - 746 \beta_{5} + 1550 \beta_{6} + 735 \beta_{7} - 5538 \beta_{8} - 439 \beta_{9} + 1501 \beta_{10} - 1335 \beta_{11} + 825 \beta_{12} + 418 \beta_{13} + 1638 \beta_{14} ) q^{37} + ( -4262208 + 12543 \beta_{1} - 33819 \beta_{2} - 6326 \beta_{3} - 1485 \beta_{4} - 829 \beta_{5} - 1481 \beta_{6} + 752 \beta_{7} - 2790 \beta_{8} + 1598 \beta_{9} - 188 \beta_{10} + 697 \beta_{11} - 949 \beta_{12} - 1501 \beta_{13} - 3549 \beta_{14} ) q^{38} + ( -2041884 - 183703 \beta_{1} + 20997 \beta_{2} + 1737 \beta_{3} + 653 \beta_{4} - 1124 \beta_{5} + 162 \beta_{6} + 140 \beta_{7} - 495 \beta_{8} - 1702 \beta_{9} + 2486 \beta_{10} - 558 \beta_{11} - 500 \beta_{12} - 210 \beta_{13} - 60 \beta_{14} ) q^{39} + ( -2675192 + 261864 \beta_{1} - 32415 \beta_{2} - 3488 \beta_{3} + 543 \beta_{4} - 213 \beta_{5} + 239 \beta_{6} + 1686 \beta_{7} - 6881 \beta_{8} + 2714 \beta_{9} - 3421 \beta_{10} - 1143 \beta_{11} + 464 \beta_{12} + 804 \beta_{13} - 1107 \beta_{14} ) q^{40} + ( -1235816 - 83615 \beta_{1} + 18918 \beta_{2} + 5275 \beta_{3} - 2411 \beta_{4} + 1540 \beta_{5} - 347 \beta_{6} - 3240 \beta_{7} + 972 \beta_{8} + 2546 \beta_{9} - 411 \beta_{10} + 1600 \beta_{11} - 1042 \beta_{12} - 1440 \beta_{13} - 1139 \beta_{14} ) q^{41} + ( -4873037 - 67775 \beta_{1} - 38787 \beta_{2} - 8558 \beta_{3} - 2872 \beta_{4} + 1186 \beta_{5} + 202 \beta_{6} + 795 \beta_{7} + 5681 \beta_{8} - 759 \beta_{9} + 6603 \beta_{10} + 2194 \beta_{11} - 2993 \beta_{12} - 1225 \beta_{13} + 619 \beta_{14} ) q^{42} -3418801 q^{43} + ( -4650026 + 482962 \beta_{1} - 44379 \beta_{2} - 20196 \beta_{3} - 1843 \beta_{4} + 1956 \beta_{5} - 2604 \beta_{6} - 197 \beta_{7} + 2088 \beta_{8} + 4637 \beta_{9} + 2345 \beta_{10} - 2015 \beta_{11} - 487 \beta_{12} + 4303 \beta_{13} + 4709 \beta_{14} ) q^{44} + ( -14443123 - 81295 \beta_{1} - 18923 \beta_{2} + 11950 \beta_{3} - 257 \beta_{4} - 333 \beta_{5} - 2147 \beta_{6} + 2367 \beta_{7} + 14292 \beta_{8} - 5139 \beta_{9} - 8624 \beta_{10} + 3964 \beta_{11} + 2656 \beta_{12} - 2090 \beta_{13} - 2125 \beta_{14} ) q^{45} + ( -149539 + 106300 \beta_{1} - 58369 \beta_{2} + 2358 \beta_{3} - 252 \beta_{4} + 1938 \beta_{5} + 9730 \beta_{6} + 642 \beta_{7} - 1257 \beta_{8} + 648 \beta_{9} + 1266 \beta_{10} + 1914 \beta_{11} + 603 \beta_{12} - 1761 \beta_{13} + 2787 \beta_{14} ) q^{46} + ( -6941909 - 413295 \beta_{1} - 23822 \beta_{2} - 7375 \beta_{3} - 641 \beta_{4} - 4314 \beta_{5} + 5183 \beta_{6} - 1306 \beta_{7} + 7919 \beta_{8} - 1628 \beta_{9} - 2615 \beta_{10} - 2710 \beta_{11} + 3673 \beta_{12} + 2471 \beta_{13} - 113 \beta_{14} ) q^{47} + ( -9770268 + 649637 \beta_{1} - 107807 \beta_{2} - 32971 \beta_{3} + 4512 \beta_{4} - 2505 \beta_{5} - 9437 \beta_{6} - 43 \beta_{7} + 3519 \beta_{8} - 5141 \beta_{9} + 5554 \beta_{10} - 1150 \beta_{11} + 2925 \beta_{12} - 289 \beta_{13} - 1816 \beta_{14} ) q^{48} + ( 6108319 + 540200 \beta_{1} + 8578 \beta_{2} + 77 \beta_{3} - 278 \beta_{4} + 6962 \beta_{5} - 6751 \beta_{6} - 8177 \beta_{7} + 10451 \beta_{8} + 905 \beta_{9} + 1766 \beta_{10} + 273 \beta_{11} - 4750 \beta_{12} + 7663 \beta_{13} + 3677 \beta_{14} ) q^{49} + ( -2809962 - 15193 \beta_{1} - 175262 \beta_{2} - 13070 \beta_{3} + 2850 \beta_{4} - 6858 \beta_{5} + 10070 \beta_{6} + 3065 \beta_{7} - 4042 \beta_{8} - 4133 \beta_{9} - 3337 \beta_{10} - 642 \beta_{11} + 430 \beta_{12} - 1718 \beta_{13} - 8028 \beta_{14} ) q^{50} + ( 2480386 + 130627 \beta_{1} + 90199 \beta_{2} + 38038 \beta_{3} + 268 \beta_{4} + 1317 \beta_{5} - 1505 \beta_{6} + 2362 \beta_{7} + 13289 \beta_{8} - 5628 \beta_{9} - 8486 \beta_{10} - 1789 \beta_{11} - 3108 \beta_{12} - 4359 \beta_{13} + 2445 \beta_{14} ) q^{51} + ( 9764810 + 1529876 \beta_{1} - 26015 \beta_{2} + 6726 \beta_{3} + 7951 \beta_{4} - 1340 \beta_{5} - 11468 \beta_{6} - 3127 \beta_{7} - 29576 \beta_{8} + 1327 \beta_{9} - 11429 \beta_{10} - 3941 \beta_{11} + 4243 \beta_{12} - 3675 \beta_{13} - 1969 \beta_{14} ) q^{52} + ( -14437909 + 196345 \beta_{1} + 10610 \beta_{2} - 34069 \beta_{3} + 3140 \beta_{4} - 1040 \beta_{5} - 2452 \beta_{6} + 5553 \beta_{7} - 2900 \beta_{8} + 1465 \beta_{9} + 2718 \beta_{10} + 3653 \beta_{11} + 8383 \beta_{12} - 9990 \beta_{13} + 3490 \beta_{14} ) q^{53} + ( 27876511 + 515176 \beta_{1} + 11804 \beta_{2} + 15974 \beta_{3} + 1581 \beta_{4} + 11 \beta_{5} - 249 \beta_{6} + 506 \beta_{7} + 2677 \beta_{8} - 9832 \beta_{9} + 14376 \beta_{10} - 5031 \beta_{11} - 7984 \beta_{12} + 3348 \beta_{13} - 514 \beta_{14} ) q^{54} + ( 25655099 - 374031 \beta_{1} - 20537 \beta_{2} - 12603 \beta_{3} - 6833 \beta_{4} + 1127 \beta_{5} + 10715 \beta_{6} + 4866 \beta_{7} - 3626 \beta_{8} + 4752 \beta_{9} + 21093 \beta_{10} + 481 \beta_{11} - 15401 \beta_{12} + 5838 \beta_{13} - 3389 \beta_{14} ) q^{55} + ( 28489776 + 1469346 \beta_{1} + 94528 \beta_{2} + 11226 \beta_{3} + 3226 \beta_{4} + 6056 \beta_{5} - 1432 \beta_{6} - 10858 \beta_{7} - 18408 \beta_{8} + 15002 \beta_{9} - 1786 \beta_{10} - 2310 \beta_{11} - 3270 \beta_{12} + 4390 \beta_{13} + 12038 \beta_{14} ) q^{56} + ( 17404849 - 1183401 \beta_{1} + 9750 \beta_{2} + 1398 \beta_{3} - 5362 \beta_{4} - 5923 \beta_{5} + 15591 \beta_{6} + 6176 \beta_{7} + 12441 \beta_{8} - 2436 \beta_{9} + 3156 \beta_{10} + 83 \beta_{11} + 1524 \beta_{12} + 11409 \beta_{13} - 11967 \beta_{14} ) q^{57} + ( 19686251 + 457083 \beta_{1} + 111780 \beta_{2} + 66389 \beta_{3} - 5282 \beta_{4} + 220 \beta_{5} - 660 \beta_{6} + 3882 \beta_{7} - 25634 \beta_{8} - 695 \beta_{9} - 8160 \beta_{10} + 15202 \beta_{11} + 10370 \beta_{12} - 20482 \beta_{13} - 6116 \beta_{14} ) q^{58} + ( 12277796 + 521718 \beta_{1} + 211084 \beta_{2} - 43251 \beta_{3} - 9324 \beta_{4} + 4802 \beta_{5} - 15937 \beta_{6} + 2801 \beta_{7} - 11591 \beta_{8} + 18693 \beta_{9} + 3376 \beta_{10} - 5983 \beta_{11} + 11018 \beta_{12} - 4153 \beta_{13} + 6447 \beta_{14} ) q^{59} + ( 64811226 + 624661 \beta_{1} + 290144 \beta_{2} + 130593 \beta_{3} - 3861 \beta_{4} + 8554 \beta_{5} + 15906 \beta_{6} - 5593 \beta_{7} - 1670 \beta_{8} + 6041 \beta_{9} - 21337 \beta_{10} - 437 \beta_{11} - 4547 \beta_{12} + 2971 \beta_{13} - 85 \beta_{14} ) q^{60} + ( 16677889 - 1267897 \beta_{1} + 55313 \beta_{2} + 24719 \beta_{3} - 7677 \beta_{4} - 689 \beta_{5} + 16615 \beta_{6} - 306 \beta_{7} - 21614 \beta_{8} + 14710 \beta_{9} - 16593 \beta_{10} + 14441 \beta_{11} + 9311 \beta_{12} + 834 \beta_{13} - 8983 \beta_{14} ) q^{61} + ( 9233364 + 690422 \beta_{1} + 24823 \beta_{2} - 26298 \beta_{3} - 4305 \beta_{4} - 3999 \beta_{5} - 38643 \beta_{6} + 3178 \beta_{7} + 41860 \beta_{8} - 11798 \beta_{9} + 2544 \beta_{10} - 5599 \beta_{11} - 18375 \beta_{12} + 39957 \beta_{13} + 16459 \beta_{14} ) q^{62} + ( 27062359 - 738503 \beta_{1} + 243907 \beta_{2} - 36471 \beta_{3} - 1333 \beta_{4} + 4731 \beta_{5} - 16689 \beta_{6} - 3332 \beta_{7} + 18426 \beta_{8} + 6994 \beta_{9} + 34661 \beta_{10} - 387 \beta_{11} - 16469 \beta_{12} - 2006 \beta_{13} + 6197 \beta_{14} ) q^{63} + ( 56661316 - 735489 \beta_{1} + 193758 \beta_{2} + 51379 \beta_{3} + 11025 \beta_{4} - 13866 \beta_{5} + 31894 \beta_{6} + 101 \beta_{7} - 52242 \beta_{8} - 1605 \beta_{9} - 3441 \beta_{10} - 7067 \beta_{11} + 26479 \beta_{12} - 11095 \beta_{13} + 827 \beta_{14} ) q^{64} + ( 6582039 - 2503589 \beta_{1} - 5495 \beta_{2} - 88955 \beta_{3} + 6989 \beta_{4} + 2349 \beta_{5} + 4745 \beta_{6} - 12982 \beta_{7} - 19678 \beta_{8} - 346 \beta_{9} + 8063 \beta_{10} + 5891 \beta_{11} + 4775 \beta_{12} - 20576 \beta_{13} - 2661 \beta_{14} ) q^{65} + ( 44617794 - 738116 \beta_{1} + 519725 \beta_{2} + 185875 \beta_{3} + 296 \beta_{4} - 6642 \beta_{5} + 2094 \beta_{6} - 5248 \beta_{7} + 16939 \beta_{8} - 17801 \beta_{9} - 56926 \beta_{10} - 2954 \beta_{11} + 41305 \beta_{12} - 21587 \beta_{13} - 7737 \beta_{14} ) q^{66} + ( 31474008 - 2031243 \beta_{1} + 157164 \beta_{2} + 41092 \beta_{3} - 1458 \beta_{4} - 9897 \beta_{5} - 24470 \beta_{6} - 8188 \beta_{7} - 22008 \beta_{8} - 7644 \beta_{9} - 21447 \beta_{10} - 16709 \beta_{11} - 21014 \beta_{12} + 3113 \beta_{13} + 4634 \beta_{14} ) q^{67} + ( -5905276 + 195013 \beta_{1} - 536050 \beta_{2} - 144623 \beta_{3} + 20627 \beta_{4} - 325 \beta_{5} - 9425 \beta_{6} + 5638 \beta_{7} + 48625 \beta_{8} - 9550 \beta_{9} + 22349 \beta_{10} + 14413 \beta_{11} - 30850 \beta_{12} - 3998 \beta_{13} - 15611 \beta_{14} ) q^{68} + ( 11183061 - 1980065 \beta_{1} - 160635 \beta_{2} - 70896 \beta_{3} + 31713 \beta_{4} - 7917 \beta_{5} + 7385 \beta_{6} + 19973 \beta_{7} + 8164 \beta_{8} - 20849 \beta_{9} + 5040 \beta_{10} - 16864 \beta_{11} - 14296 \beta_{12} + 15446 \beta_{13} + 869 \beta_{14} ) q^{69} + ( 43367055 - 1562805 \beta_{1} - 409719 \beta_{2} - 69940 \beta_{3} - 27526 \beta_{4} + 12272 \beta_{5} - 4944 \beta_{6} + 22909 \beta_{7} + 110371 \beta_{8} - 34875 \beta_{9} + 87493 \beta_{10} + 19036 \beta_{11} - 42209 \beta_{12} + 27191 \beta_{13} - 1773 \beta_{14} ) q^{70} + ( -424137 - 891494 \beta_{1} - 25534 \beta_{2} - 62584 \beta_{3} + 14308 \beta_{4} + 15225 \beta_{5} - 8823 \beta_{6} - 30932 \beta_{7} - 17091 \beta_{8} + 21428 \beta_{9} - 54755 \beta_{10} - 16825 \beta_{11} + 42649 \beta_{12} - 22558 \beta_{13} + 4779 \beta_{14} ) q^{71} + ( 59358455 - 969930 \beta_{1} + 502002 \beta_{2} + 7327 \beta_{3} - 15670 \beta_{4} + 14229 \beta_{5} - 7967 \beta_{6} - 16150 \beta_{7} + 28598 \beta_{8} + 2588 \beta_{9} + 36405 \beta_{10} + 5860 \beta_{11} + 16216 \beta_{12} - 45804 \beta_{13} + 8499 \beta_{14} ) q^{72} + ( -47436012 - 1913733 \beta_{1} - 392065 \beta_{2} - 146775 \beta_{3} + 26405 \beta_{4} - 6468 \beta_{5} - 26456 \beta_{6} + 13416 \beta_{7} - 30649 \beta_{8} - 26636 \beta_{9} + 8084 \beta_{10} - 5796 \beta_{11} + 11354 \beta_{12} + 21498 \beta_{13} + 40602 \beta_{14} ) q^{73} + ( 48651705 - 2236729 \beta_{1} - 550609 \beta_{2} - 44918 \beta_{3} - 15246 \beta_{4} + 19410 \beta_{5} + 83346 \beta_{6} + 21568 \beta_{7} + 123743 \beta_{8} - 11416 \beta_{9} + 16092 \beta_{10} - 552 \beta_{11} + 13293 \beta_{12} + 27501 \beta_{13} - 3407 \beta_{14} ) q^{74} + ( -389719 - 5602972 \beta_{1} - 1075609 \beta_{2} - 64171 \beta_{3} + 285 \beta_{4} - 25931 \beta_{5} + 97937 \beta_{6} + 40654 \beta_{7} + 46900 \beta_{8} + 9882 \beta_{9} - 7934 \beta_{10} + 5715 \beta_{11} - 33830 \beta_{12} + 50747 \beta_{13} - 60281 \beta_{14} ) q^{75} + ( 22434992 + 5048134 \beta_{1} - 321935 \beta_{2} - 84268 \beta_{3} - 17265 \beta_{4} + 7101 \beta_{5} - 57463 \beta_{6} - 11392 \beta_{7} - 11665 \beta_{8} + 65080 \beta_{9} - 6315 \beta_{10} + 5357 \beta_{11} - 6308 \beta_{12} + 21252 \beta_{13} + 5853 \beta_{14} ) q^{76} + ( -81653897 - 5097981 \beta_{1} - 788469 \beta_{2} - 88213 \beta_{3} + 9855 \beta_{4} - 17103 \beta_{5} - 17761 \beta_{6} - 5624 \beta_{7} - 81016 \beta_{8} - 33680 \beta_{9} - 66129 \beta_{10} + 8375 \beta_{11} + 5439 \beta_{12} - 41832 \beta_{13} - 23753 \beta_{14} ) q^{77} + ( 137202276 + 493738 \beta_{1} + 862835 \beta_{2} + 308371 \beta_{3} - 33680 \beta_{4} + 17906 \beta_{5} + 16626 \beta_{6} - 28488 \beta_{7} + 49565 \beta_{8} + 16991 \beta_{9} - 64914 \beta_{10} - 23686 \beta_{11} + 4995 \beta_{12} + 45791 \beta_{13} + 19229 \beta_{14} ) q^{78} + ( 37253847 + 1469461 \beta_{1} + 210372 \beta_{2} + 174610 \beta_{3} - 36295 \beta_{4} + 24166 \beta_{5} - 64683 \beta_{6} - 37899 \beta_{7} - 115787 \beta_{8} + 31219 \beta_{9} - 26528 \beta_{10} - 2581 \beta_{11} + 45144 \beta_{12} - 88833 \beta_{13} - 12757 \beta_{14} ) q^{79} + ( -77262124 + 510888 \beta_{1} - 1234475 \beta_{2} - 437680 \beta_{3} + 21111 \beta_{4} - 35053 \beta_{5} - 7065 \beta_{6} + 59738 \beta_{7} - 66289 \beta_{8} + 142 \beta_{9} + 65435 \beta_{10} + 15009 \beta_{11} - 34160 \beta_{12} - 23060 \beta_{13} - 32071 \beta_{14} ) q^{80} + ( -50187992 - 344047 \beta_{1} - 76210 \beta_{2} - 176165 \beta_{3} + 19358 \beta_{4} - 7009 \beta_{5} - 43518 \beta_{6} - 13911 \beta_{7} - 125856 \beta_{8} + 46515 \beta_{9} + 110662 \beta_{10} + 23862 \beta_{11} - 36986 \beta_{12} - 1614 \beta_{13} + 48484 \beta_{14} ) q^{81} + ( 62992568 - 877529 \beta_{1} + 449734 \beta_{2} + 433624 \beta_{3} - 41762 \beta_{4} - 20818 \beta_{5} - 2178 \beta_{6} - 32403 \beta_{7} + 3610 \beta_{8} - 24063 \beta_{9} - 42945 \beta_{10} + 26050 \beta_{11} - 12614 \beta_{12} - 43098 \beta_{13} + 1672 \beta_{14} ) q^{82} + ( -95779038 - 2750841 \beta_{1} - 826494 \beta_{2} - 95733 \beta_{3} + 25832 \beta_{4} + 24903 \beta_{5} + 66643 \beta_{6} - 45317 \beta_{7} + 79257 \beta_{8} + 17317 \beta_{9} + 33255 \beta_{10} + 27420 \beta_{11} - 3218 \beta_{12} + 70158 \beta_{13} + 65569 \beta_{14} ) q^{83} + ( 125308612 + 7040210 \beta_{1} + 1814554 \beta_{2} + 272842 \beta_{3} + 23980 \beta_{4} - 19754 \beta_{5} - 20666 \beta_{6} - 6594 \beta_{7} - 226570 \beta_{8} + 64822 \beta_{9} - 114240 \beta_{10} - 65712 \beta_{11} + 39022 \beta_{12} - 40550 \beta_{13} + 26972 \beta_{14} ) q^{84} + ( 46642644 + 1575075 \beta_{1} - 565115 \beta_{2} + 258692 \beta_{3} - 30735 \beta_{4} + 22030 \beta_{5} + 29574 \beta_{6} + 12277 \beta_{7} - 114961 \beta_{8} - 28413 \beta_{9} - 46369 \beta_{10} - 14471 \beta_{11} + 21315 \beta_{12} + 40792 \beta_{13} - 19728 \beta_{14} ) q^{85} + ( 6837602 + 3418801 \beta_{1} ) q^{86} + ( -140275743 + 3886177 \beta_{1} + 1317924 \beta_{2} + 19248 \beta_{3} - 35243 \beta_{4} + 198 \beta_{5} - 101579 \beta_{6} - 9863 \beta_{7} + 59809 \beta_{8} + 42729 \beta_{9} + 50676 \beta_{10} - 4417 \beta_{11} + 132790 \beta_{12} + 34181 \beta_{13} + 68025 \beta_{14} ) q^{87} + ( -98599791 + 8455177 \beta_{1} - 2180293 \beta_{2} - 376372 \beta_{3} + 107388 \beta_{4} - 89234 \beta_{5} + 46782 \beta_{6} + 68273 \beta_{7} - 33623 \beta_{8} - 15863 \beta_{9} + 17735 \beta_{10} + 11764 \beta_{11} - 23557 \beta_{12} + 10173 \beta_{13} - 150901 \beta_{14} ) q^{88} + ( -26222191 - 67477 \beta_{1} - 1618843 \beta_{2} + 20813 \beta_{3} - 9373 \beta_{4} + 25405 \beta_{5} + 17133 \beta_{6} + 49610 \beta_{7} + 58560 \beta_{8} - 104674 \beta_{9} + 61983 \beta_{10} - 25855 \beta_{11} - 107279 \beta_{12} + 37656 \beta_{13} - 89595 \beta_{14} ) q^{89} + ( 91860711 + 14175630 \beta_{1} + 2909871 \beta_{2} + 66707 \beta_{3} - 22365 \beta_{4} + 78345 \beta_{5} - 198875 \beta_{6} - 42534 \beta_{7} - 45722 \beta_{8} + 49411 \beta_{9} + 93614 \beta_{10} + 2865 \beta_{11} + 2433 \beta_{12} - 27491 \beta_{13} + 85677 \beta_{14} ) q^{90} + ( -218557543 + 560803 \beta_{1} - 1028035 \beta_{2} + 191729 \beta_{3} - 69001 \beta_{4} - 29077 \beta_{5} + 90597 \beta_{6} + 139820 \beta_{7} + 301916 \beta_{8} - 47566 \beta_{9} + 126661 \beta_{10} + 37865 \beta_{11} - 53217 \beta_{12} + 18890 \beta_{13} - 2925 \beta_{14} ) q^{91} + ( 10551674 - 2169178 \beta_{1} + 1043252 \beta_{2} + 57990 \beta_{3} + 15224 \beta_{4} + 25585 \beta_{5} + 130853 \beta_{6} - 30581 \beta_{7} - 47621 \beta_{8} - 49311 \beta_{9} - 93134 \beta_{10} - 13966 \beta_{11} + 39405 \beta_{12} - 55045 \beta_{13} + 23542 \beta_{14} ) q^{92} + ( -94993684 + 1095516 \beta_{1} + 926457 \beta_{2} + 642566 \beta_{3} + 49870 \beta_{4} + 29435 \beta_{5} + 71023 \beta_{6} - 43978 \beta_{7} - 44628 \beta_{8} - 25306 \beta_{9} - 134013 \beta_{10} + 78529 \beta_{11} - 92447 \beta_{12} - 90600 \beta_{13} - 55337 \beta_{14} ) q^{93} + ( 309506090 + 11774278 \beta_{1} + 1766086 \beta_{2} + 612092 \beta_{3} + 61954 \beta_{4} + 32792 \beta_{5} + 92720 \beta_{6} - 109289 \beta_{7} + 73758 \beta_{8} - 24251 \beta_{9} + 14751 \beta_{10} - 121898 \beta_{11} - 24918 \beta_{12} + 42554 \beta_{13} + 58230 \beta_{14} ) q^{94} + ( -256861319 - 374965 \beta_{1} + 232400 \beta_{2} + 306388 \beta_{3} - 68239 \beta_{4} + 84808 \beta_{5} + 41664 \beta_{6} + 11293 \beta_{7} + 256534 \beta_{8} - 37491 \beta_{9} + 1531 \beta_{10} + 87245 \beta_{11} - 4785 \beta_{12} - 68968 \beta_{13} - 1034 \beta_{14} ) q^{95} + ( -66004652 + 14554589 \beta_{1} + 1025189 \beta_{2} - 915643 \beta_{3} + 46028 \beta_{4} + 17855 \beta_{5} - 42725 \beta_{6} + 19537 \beta_{7} - 73897 \beta_{8} + 201399 \beta_{9} + 123310 \beta_{10} - 17714 \beta_{11} - 71439 \beta_{12} + 156539 \beta_{13} + 38900 \beta_{14} ) q^{96} + ( -204121880 + 1038679 \beta_{1} + 742450 \beta_{2} + 498290 \beta_{3} + 62061 \beta_{4} - 110556 \beta_{5} + 15464 \beta_{6} - 13215 \beta_{7} - 30771 \beta_{8} - 200193 \beta_{9} - 172109 \beta_{10} - 159331 \beta_{11} + 146140 \beta_{12} + 57059 \beta_{13} + 59510 \beta_{14} ) q^{97} + ( -409938059 - 5504792 \beta_{1} - 868617 \beta_{2} - 596368 \beta_{3} + 138284 \beta_{4} - 196562 \beta_{5} - 125434 \beta_{6} + 63193 \beta_{7} - 81565 \beta_{8} - 223351 \beta_{9} - 61599 \beta_{10} - 25886 \beta_{11} + 77461 \beta_{12} - 110811 \beta_{13} - 143911 \beta_{14} ) q^{98} + ( -438998692 + 5425129 \beta_{1} + 1355980 \beta_{2} + 560931 \beta_{3} + 24816 \beta_{4} - 22309 \beta_{5} - 41729 \beta_{6} - 13397 \beta_{7} + 299077 \beta_{8} + 37485 \beta_{9} - 236095 \beta_{10} + 109832 \beta_{11} + 32132 \beta_{12} - 63224 \beta_{13} - 113291 \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q - 32q^{2} - 317q^{3} + 3242q^{4} - 4717q^{5} + 687q^{6} - 9680q^{7} - 20394q^{8} + 69516q^{9} + O(q^{10}) \) \( 15q - 32q^{2} - 317q^{3} + 3242q^{4} - 4717q^{5} + 687q^{6} - 9680q^{7} - 20394q^{8} + 69516q^{9} - 36237q^{10} - 104484q^{11} - 266395q^{12} - 116174q^{13} + 416064q^{14} + 415388q^{15} + 996762q^{16} - 884265q^{17} - 588735q^{18} - 689535q^{19} - 3077879q^{20} - 2070198q^{21} - 7276218q^{22} - 2504077q^{23} - 11534895q^{24} + 1315350q^{25} - 13343414q^{26} - 12546986q^{27} - 28059568q^{28} - 18406221q^{29} - 39503820q^{30} - 12033699q^{31} - 18952630q^{32} - 14197716q^{33} - 30383125q^{34} - 27855546q^{35} - 18372959q^{36} - 8722847q^{37} - 63941843q^{38} - 30955510q^{39} - 39665611q^{40} - 18689389q^{41} - 73185310q^{42} - 51282015q^{43} - 68723220q^{44} - 216992888q^{45} - 2067521q^{46} - 104960741q^{47} - 145362479q^{48} + 92663095q^{49} - 42446347q^{50} + 37433407q^{51} + 149226080q^{52} - 215907800q^{53} + 419158122q^{54} + 384379852q^{55} + 430441344q^{56} + 258744488q^{57} + 295963139q^{58} + 185924544q^{59} + 973236172q^{60} + 247538102q^{61} + 139798853q^{62} + 405429926q^{63} + 848556290q^{64} + 94294394q^{65} + 667230492q^{66} + 467904656q^{67} - 88234341q^{68} + 163914994q^{69} + 647526126q^{70} - 8252944q^{71} + 889796745q^{72} - 715627902q^{73} + 725122989q^{74} - 18301762q^{75} + 346300359q^{76} - 1236779964q^{77} + 2058642146q^{78} + 560681783q^{79} - 1157214179q^{80} - 752010645q^{81} + 941346367q^{82} - 1442854698q^{83} + 1895248718q^{84} + 699302088q^{85} + 109401632q^{86} - 2094576907q^{87} - 1464507256q^{88} - 396710008q^{89} + 1411356270q^{90} - 3278076852q^{91} + 155864647q^{92} - 1424759183q^{93} + 4666638949q^{94} - 3854114395q^{95} - 952489551q^{96} - 3063837815q^{97} - 6161086984q^{98} - 6576160348q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15} - 2 x^{14} - 5425 x^{13} + 14888 x^{12} + 11288030 x^{11} - 37600244 x^{10} - 11474166224 x^{9} + 47465836576 x^{8} + 5986976782464 x^{7} - 32493903147264 x^{6} - 1516975415483904 x^{5} + 10892588268404224 x^{4} + 139803541742443008 x^{3} - 1349125586394823680 x^{2} + 2103623681144094720 x + 529838441422848000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(19\!\cdots\!31\)\( \nu^{14} + \)\(19\!\cdots\!96\)\( \nu^{13} - \)\(99\!\cdots\!55\)\( \nu^{12} - \)\(92\!\cdots\!54\)\( \nu^{11} + \)\(19\!\cdots\!26\)\( \nu^{10} + \)\(16\!\cdots\!32\)\( \nu^{9} - \)\(18\!\cdots\!24\)\( \nu^{8} - \)\(11\!\cdots\!36\)\( \nu^{7} + \)\(89\!\cdots\!60\)\( \nu^{6} + \)\(27\!\cdots\!04\)\( \nu^{5} - \)\(20\!\cdots\!00\)\( \nu^{4} + \)\(37\!\cdots\!44\)\( \nu^{3} + \)\(18\!\cdots\!40\)\( \nu^{2} - \)\(13\!\cdots\!24\)\( \nu + \)\(11\!\cdots\!12\)\(\)\()/ \)\(77\!\cdots\!72\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(19\!\cdots\!31\)\( \nu^{14} - \)\(19\!\cdots\!96\)\( \nu^{13} + \)\(99\!\cdots\!55\)\( \nu^{12} + \)\(92\!\cdots\!54\)\( \nu^{11} - \)\(19\!\cdots\!26\)\( \nu^{10} - \)\(16\!\cdots\!32\)\( \nu^{9} + \)\(18\!\cdots\!24\)\( \nu^{8} + \)\(11\!\cdots\!36\)\( \nu^{7} - \)\(89\!\cdots\!60\)\( \nu^{6} - \)\(27\!\cdots\!04\)\( \nu^{5} + \)\(20\!\cdots\!00\)\( \nu^{4} - \)\(37\!\cdots\!44\)\( \nu^{3} - \)\(17\!\cdots\!68\)\( \nu^{2} + \)\(13\!\cdots\!96\)\( \nu - \)\(67\!\cdots\!40\)\(\)\()/ \)\(77\!\cdots\!72\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(22\!\cdots\!81\)\( \nu^{14} + \)\(53\!\cdots\!70\)\( \nu^{13} + \)\(12\!\cdots\!97\)\( \nu^{12} - \)\(27\!\cdots\!28\)\( \nu^{11} - \)\(27\!\cdots\!02\)\( \nu^{10} + \)\(52\!\cdots\!28\)\( \nu^{9} + \)\(28\!\cdots\!96\)\( \nu^{8} - \)\(45\!\cdots\!36\)\( \nu^{7} - \)\(16\!\cdots\!72\)\( \nu^{6} + \)\(18\!\cdots\!72\)\( \nu^{5} + \)\(44\!\cdots\!52\)\( \nu^{4} - \)\(28\!\cdots\!48\)\( \nu^{3} - \)\(48\!\cdots\!36\)\( \nu^{2} + \)\(51\!\cdots\!64\)\( \nu + \)\(54\!\cdots\!28\)\(\)\()/ \)\(24\!\cdots\!96\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(17\!\cdots\!29\)\( \nu^{14} + \)\(88\!\cdots\!48\)\( \nu^{13} + \)\(10\!\cdots\!25\)\( \nu^{12} - \)\(45\!\cdots\!02\)\( \nu^{11} - \)\(25\!\cdots\!70\)\( \nu^{10} + \)\(84\!\cdots\!96\)\( \nu^{9} + \)\(31\!\cdots\!16\)\( \nu^{8} - \)\(73\!\cdots\!04\)\( \nu^{7} - \)\(19\!\cdots\!56\)\( \nu^{6} + \)\(31\!\cdots\!16\)\( \nu^{5} + \)\(61\!\cdots\!16\)\( \nu^{4} - \)\(65\!\cdots\!16\)\( \nu^{3} - \)\(74\!\cdots\!52\)\( \nu^{2} + \)\(57\!\cdots\!80\)\( \nu + \)\(27\!\cdots\!80\)\(\)\()/ \)\(12\!\cdots\!20\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(10\!\cdots\!57\)\( \nu^{14} - \)\(22\!\cdots\!76\)\( \nu^{13} + \)\(57\!\cdots\!85\)\( \nu^{12} + \)\(10\!\cdots\!94\)\( \nu^{11} - \)\(11\!\cdots\!90\)\( \nu^{10} - \)\(20\!\cdots\!32\)\( \nu^{9} + \)\(10\!\cdots\!28\)\( \nu^{8} + \)\(17\!\cdots\!88\)\( \nu^{7} - \)\(48\!\cdots\!08\)\( \nu^{6} - \)\(63\!\cdots\!72\)\( \nu^{5} + \)\(10\!\cdots\!28\)\( \nu^{4} + \)\(83\!\cdots\!72\)\( \nu^{3} - \)\(96\!\cdots\!56\)\( \nu^{2} - \)\(65\!\cdots\!00\)\( \nu + \)\(39\!\cdots\!60\)\(\)\()/ \)\(38\!\cdots\!60\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(12\!\cdots\!19\)\( \nu^{14} - \)\(19\!\cdots\!08\)\( \nu^{13} - \)\(73\!\cdots\!15\)\( \nu^{12} + \)\(10\!\cdots\!22\)\( \nu^{11} + \)\(16\!\cdots\!70\)\( \nu^{10} - \)\(22\!\cdots\!96\)\( \nu^{9} - \)\(18\!\cdots\!16\)\( \nu^{8} + \)\(22\!\cdots\!04\)\( \nu^{7} + \)\(10\!\cdots\!76\)\( \nu^{6} - \)\(11\!\cdots\!56\)\( \nu^{5} - \)\(30\!\cdots\!96\)\( \nu^{4} + \)\(28\!\cdots\!96\)\( \nu^{3} + \)\(31\!\cdots\!32\)\( \nu^{2} - \)\(26\!\cdots\!40\)\( \nu + \)\(13\!\cdots\!20\)\(\)\()/ \)\(38\!\cdots\!60\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(14\!\cdots\!49\)\( \nu^{14} - \)\(24\!\cdots\!52\)\( \nu^{13} + \)\(75\!\cdots\!25\)\( \nu^{12} + \)\(11\!\cdots\!78\)\( \nu^{11} - \)\(14\!\cdots\!90\)\( \nu^{10} - \)\(22\!\cdots\!64\)\( \nu^{9} + \)\(12\!\cdots\!56\)\( \nu^{8} + \)\(18\!\cdots\!16\)\( \nu^{7} - \)\(52\!\cdots\!36\)\( \nu^{6} - \)\(70\!\cdots\!64\)\( \nu^{5} + \)\(94\!\cdots\!16\)\( \nu^{4} + \)\(99\!\cdots\!64\)\( \nu^{3} - \)\(57\!\cdots\!12\)\( \nu^{2} - \)\(27\!\cdots\!80\)\( \nu + \)\(47\!\cdots\!80\)\(\)\()/ \)\(32\!\cdots\!80\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(26\!\cdots\!51\)\( \nu^{14} - \)\(42\!\cdots\!88\)\( \nu^{13} + \)\(13\!\cdots\!15\)\( \nu^{12} + \)\(21\!\cdots\!62\)\( \nu^{11} - \)\(25\!\cdots\!70\)\( \nu^{10} - \)\(39\!\cdots\!16\)\( \nu^{9} + \)\(22\!\cdots\!44\)\( \nu^{8} + \)\(34\!\cdots\!04\)\( \nu^{7} - \)\(94\!\cdots\!24\)\( \nu^{6} - \)\(13\!\cdots\!76\)\( \nu^{5} + \)\(17\!\cdots\!84\)\( \nu^{4} + \)\(19\!\cdots\!36\)\( \nu^{3} - \)\(10\!\cdots\!28\)\( \nu^{2} - \)\(60\!\cdots\!00\)\( \nu + \)\(74\!\cdots\!00\)\(\)\()/ \)\(38\!\cdots\!60\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(23\!\cdots\!61\)\( \nu^{14} + \)\(38\!\cdots\!28\)\( \nu^{13} - \)\(11\!\cdots\!25\)\( \nu^{12} - \)\(18\!\cdots\!02\)\( \nu^{11} + \)\(22\!\cdots\!70\)\( \nu^{10} + \)\(34\!\cdots\!16\)\( \nu^{9} - \)\(20\!\cdots\!44\)\( \nu^{8} - \)\(28\!\cdots\!44\)\( \nu^{7} + \)\(90\!\cdots\!24\)\( \nu^{6} + \)\(10\!\cdots\!96\)\( \nu^{5} - \)\(18\!\cdots\!24\)\( \nu^{4} - \)\(12\!\cdots\!76\)\( \nu^{3} + \)\(14\!\cdots\!48\)\( \nu^{2} + \)\(14\!\cdots\!80\)\( \nu - \)\(97\!\cdots\!40\)\(\)\()/ \)\(19\!\cdots\!80\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(40\!\cdots\!87\)\( \nu^{14} - \)\(33\!\cdots\!16\)\( \nu^{13} + \)\(20\!\cdots\!95\)\( \nu^{12} + \)\(15\!\cdots\!34\)\( \nu^{11} - \)\(41\!\cdots\!50\)\( \nu^{10} - \)\(28\!\cdots\!92\)\( \nu^{9} + \)\(38\!\cdots\!08\)\( \nu^{8} + \)\(22\!\cdots\!08\)\( \nu^{7} - \)\(18\!\cdots\!88\)\( \nu^{6} - \)\(70\!\cdots\!72\)\( \nu^{5} + \)\(40\!\cdots\!68\)\( \nu^{4} + \)\(29\!\cdots\!92\)\( \nu^{3} - \)\(35\!\cdots\!76\)\( \nu^{2} + \)\(11\!\cdots\!20\)\( \nu + \)\(50\!\cdots\!00\)\(\)\()/ \)\(19\!\cdots\!80\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(16\!\cdots\!99\)\( \nu^{14} - \)\(17\!\cdots\!36\)\( \nu^{13} + \)\(88\!\cdots\!19\)\( \nu^{12} + \)\(84\!\cdots\!78\)\( \nu^{11} - \)\(17\!\cdots\!86\)\( \nu^{10} - \)\(15\!\cdots\!60\)\( \nu^{9} + \)\(16\!\cdots\!52\)\( \nu^{8} + \)\(12\!\cdots\!76\)\( \nu^{7} - \)\(76\!\cdots\!36\)\( \nu^{6} - \)\(38\!\cdots\!88\)\( \nu^{5} + \)\(16\!\cdots\!68\)\( \nu^{4} + \)\(22\!\cdots\!88\)\( \nu^{3} - \)\(13\!\cdots\!68\)\( \nu^{2} + \)\(53\!\cdots\!28\)\( \nu - \)\(52\!\cdots\!04\)\(\)\()/ \)\(77\!\cdots\!72\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(86\!\cdots\!09\)\( \nu^{14} - \)\(11\!\cdots\!72\)\( \nu^{13} + \)\(45\!\cdots\!45\)\( \nu^{12} + \)\(55\!\cdots\!38\)\( \nu^{11} - \)\(91\!\cdots\!50\)\( \nu^{10} - \)\(99\!\cdots\!44\)\( \nu^{9} + \)\(87\!\cdots\!56\)\( \nu^{8} + \)\(79\!\cdots\!56\)\( \nu^{7} - \)\(41\!\cdots\!76\)\( \nu^{6} - \)\(25\!\cdots\!64\)\( \nu^{5} + \)\(97\!\cdots\!96\)\( \nu^{4} + \)\(17\!\cdots\!84\)\( \nu^{3} - \)\(87\!\cdots\!12\)\( \nu^{2} + \)\(29\!\cdots\!80\)\( \nu - \)\(15\!\cdots\!80\)\(\)\()/ \)\(38\!\cdots\!60\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(20\!\cdots\!97\)\( \nu^{14} + \)\(27\!\cdots\!12\)\( \nu^{13} - \)\(11\!\cdots\!85\)\( \nu^{12} - \)\(13\!\cdots\!06\)\( \nu^{11} + \)\(22\!\cdots\!26\)\( \nu^{10} + \)\(24\!\cdots\!44\)\( \nu^{9} - \)\(21\!\cdots\!28\)\( \nu^{8} - \)\(19\!\cdots\!80\)\( \nu^{7} + \)\(10\!\cdots\!96\)\( \nu^{6} + \)\(66\!\cdots\!80\)\( \nu^{5} - \)\(22\!\cdots\!28\)\( \nu^{4} - \)\(56\!\cdots\!36\)\( \nu^{3} + \)\(20\!\cdots\!52\)\( \nu^{2} - \)\(57\!\cdots\!00\)\( \nu + \)\(74\!\cdots\!08\)\(\)\()/ \)\(77\!\cdots\!72\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} - \beta_{1} + 724\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + 2 \beta_{9} - \beta_{8} + 3 \beta_{4} - \beta_{3} - 37 \beta_{2} + 1260 \beta_{1} - 971\)
\(\nu^{4}\)\(=\)\(-25 \beta_{14} - 51 \beta_{13} + 3 \beta_{12} - 5 \beta_{11} - 65 \beta_{10} - 45 \beta_{9} - 27 \beta_{7} + 18 \beta_{6} + 10 \beta_{5} - 25 \beta_{4} + 1871 \beta_{3} + 2266 \beta_{2} - 2863 \beta_{1} + 912638\)
\(\nu^{5}\)\(=\)\(-451 \beta_{14} - 129 \beta_{13} + 361 \beta_{12} - 2211 \beta_{11} + 17 \beta_{10} + 4281 \beta_{9} - 4020 \beta_{8} + 603 \beta_{7} - 1862 \beta_{6} - 438 \beta_{5} + 6825 \beta_{4} - 9721 \beta_{3} - 88940 \beta_{2} + 1906657 \beta_{1} - 2545422\)
\(\nu^{6}\)\(=\)\(-56261 \beta_{14} - 137047 \beta_{13} + 29647 \beta_{12} - 13355 \beta_{11} - 166145 \beta_{10} - 124837 \beta_{9} - 24322 \beta_{8} - 74635 \beta_{7} + 99238 \beta_{6} + 16390 \beta_{5} - 72415 \beta_{4} + 3313547 \beta_{3} + 4662534 \beta_{2} - 9867737 \beta_{1} + 1382593092\)
\(\nu^{7}\)\(=\)\(-1476377 \beta_{14} - 101683 \beta_{13} + 624443 \beta_{12} - 3891795 \beta_{11} + 499011 \beta_{10} + 8280255 \beta_{9} - 9578646 \beta_{8} + 2022465 \beta_{7} - 6641514 \beta_{6} - 1574810 \beta_{5} + 13341257 \beta_{4} - 32063849 \beta_{3} - 185691482 \beta_{2} + 3159193055 \beta_{1} - 7998183896\)
\(\nu^{8}\)\(=\)\(-99010825 \beta_{14} - 289235299 \beta_{13} + 92490507 \beta_{12} - 24459623 \beta_{11} - 343317101 \beta_{10} - 293492905 \beta_{9} - 55318490 \beta_{8} - 161180207 \beta_{7} + 304193382 \beta_{6} + 28019062 \beta_{5} - 172377627 \beta_{4} + 5940685531 \beta_{3} + 9358786346 \beta_{2} - 28988601017 \beta_{1} + 2294173232028\)
\(\nu^{9}\)\(=\)\(-3505734621 \beta_{14} + 504553057 \beta_{13} + 400597703 \beta_{12} - 6498434995 \beta_{11} + 2257945439 \beta_{10} + 16121418211 \beta_{9} - 19514435778 \beta_{8} + 5036973621 \beta_{7} - 17240539538 \beta_{6} - 3950278306 \beta_{5} + 25209696809 \beta_{4} - 84361806929 \beta_{3} - 382404555110 \beta_{2} + 5517998843179 \beta_{1} - 22538241354596\)
\(\nu^{10}\)\(=\)\(-159606282881 \beta_{14} - 567035896587 \beta_{13} + 222487096275 \beta_{12} - 36914590119 \beta_{11} - 667007471669 \beta_{10} - 649593397041 \beta_{9} - 62970416226 \beta_{8} - 326638116615 \beta_{7} + 761822926566 \beta_{6} + 58080805974 \beta_{5} - 394538691451 \beta_{4} + 10877476562787 \beta_{3} + 18715850805194 \beta_{2} - 75178497365145 \beta_{1} + 4013632938170404\)
\(\nu^{11}\)\(=\)\(-7366410638861 \beta_{14} + 2712863734673 \beta_{13} - 1119132304553 \beta_{12} - 10759881290443 \beta_{11} + 7161703054575 \beta_{10} + 31937223329091 \beta_{9} - 37451118980778 \beta_{8} + 11373902175909 \beta_{7} - 39697676628498 \beta_{6} - 8660440017570 \beta_{5} + 47419492364081 \beta_{4} - 202075889447449 \beta_{3} - 787012630239486 \beta_{2} + 9982495692381739 \beta_{1} - 57282897571561212\)
\(\nu^{12}\)\(=\)\(-245248710611449 \beta_{14} - 1083049183240819 \beta_{13} + 481111511396539 \beta_{12} - 44325492776047 \beta_{11} - 1270622059987117 \beta_{10} - 1390238797362409 \beta_{9} + 21040233310126 \beta_{8} - 651653446777327 \beta_{7} + 1740214024685526 \beta_{6} + 133596085914118 \beta_{5} - 888233290791267 \beta_{4} + 20314753348708091 \beta_{3} + 37512834772097354 \beta_{2} - 180468425483910945 \beta_{1} + 7273162571598589668\)
\(\nu^{13}\)\(=\)\(-14571618375601765 \beta_{14} + 8905237902902281 \beta_{13} - 5718447457129825 \beta_{12} - 17961495859399203 \beta_{11} + 19293763112871287 \beta_{10} + 64070228644553227 \beta_{9} - 70251027192678154 \beta_{8} + 24602114347149757 \beta_{7} - 86357267464454082 \beta_{6} - 17865031729309778 \beta_{5} + 89692877074905401 \beta_{4} - 460231065324273793 \beta_{3} - 1617767787839936766 \beta_{2} + 18525240229224447251 \beta_{1} - 135987911070271592364\)
\(\nu^{14}\)\(=\)\(-361913292449488609 \beta_{14} - 2053445921854982699 \beta_{13} + 989174116651597619 \beta_{12} - 28336960724061799 \beta_{11} - 2413405825177909877 \beta_{10} - 2914782746773674961 \beta_{9} + 354219391839938942 \beta_{8} - 1298720351038661703 \beta_{7} + 3789566164596080582 \beta_{6} + 312873342030878966 \beta_{5} - 1970212909654523387 \beta_{4} + 38579726935375072371 \beta_{3} + 75440756904593150170 \beta_{2} - 412635637684278462185 \beta_{1} + 13519253316696560922340\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
40.4171
39.2075
32.6728
21.7142
16.4876
15.9205
7.62988
2.38595
−0.220103
−17.0644
−22.4329
−25.8680
−28.3075
−35.4490
−45.0936
−42.4171 107.942 1287.21 267.050 −4578.58 1084.23 −32882.2 −8031.56 −11327.5
1.2 −41.2075 −186.815 1186.05 −1998.59 7698.15 −10690.1 −27776.1 15216.7 82356.9
1.3 −34.6728 96.4953 690.205 1855.36 −3345.76 −11539.9 −6178.86 −10371.7 −64330.7
1.4 −23.7142 2.00552 50.3616 −764.097 −47.5592 4079.63 10947.4 −19679.0 18119.9
1.5 −18.4876 −262.112 −170.210 −2363.43 4845.81 7025.62 12612.4 49019.5 43694.1
1.6 −17.9205 −172.986 −190.855 238.484 3099.99 −4524.82 12595.5 10241.0 −4273.75
1.7 −9.62988 189.920 −419.265 636.384 −1828.91 −3288.51 8967.97 16386.6 −6128.30
1.8 −4.38595 117.920 −492.763 −1237.15 −517.190 12549.3 4406.84 −5777.96 5426.06
1.9 −1.77990 −203.939 −508.832 1139.29 362.990 4324.96 1816.98 21908.1 −2027.82
1.10 15.0644 34.0074 −285.063 1876.98 512.302 −7041.18 −12007.3 −18526.5 28275.7
1.11 20.4329 231.971 −94.4979 −2583.58 4739.83 −1769.99 −12392.5 34127.5 −52790.0
1.12 23.8680 64.8300 57.6838 −578.837 1547.37 −413.035 −10843.6 −15480.1 −13815.7
1.13 26.3075 −164.924 180.084 879.044 −4338.73 8431.97 −8731.88 7516.88 23125.4
1.14 33.4490 8.30450 606.833 −752.925 277.777 −7164.62 3172.07 −19614.0 −25184.6
1.15 43.0936 −179.621 1345.05 −1330.99 −7740.49 −743.545 35899.3 12580.5 −57356.9
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.10.a.a 15
3.b odd 2 1 387.10.a.c 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.10.a.a 15 1.a even 1 1 trivial
387.10.a.c 15 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(43\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{15} + \cdots\) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(43))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 32 T + 2731 T^{2} + 77806 T^{3} + 3850358 T^{4} + 101839448 T^{5} + 3733988120 T^{6} + 89284085280 T^{7} + 2722683702592 T^{8} + 57999472061696 T^{9} + 1597215028061696 T^{10} + 30230480697280512 T^{11} + 801326892628557824 T^{12} + 13973639350763585536 T^{13} + \)\(38\!\cdots\!16\)\( T^{14} + \)\(67\!\cdots\!40\)\( T^{15} + \)\(19\!\cdots\!92\)\( T^{16} + \)\(36\!\cdots\!84\)\( T^{17} + \)\(10\!\cdots\!72\)\( T^{18} + \)\(20\!\cdots\!32\)\( T^{19} + \)\(56\!\cdots\!72\)\( T^{20} + \)\(10\!\cdots\!64\)\( T^{21} + \)\(25\!\cdots\!36\)\( T^{22} + \)\(42\!\cdots\!80\)\( T^{23} + \)\(90\!\cdots\!40\)\( T^{24} + \)\(12\!\cdots\!52\)\( T^{25} + \)\(24\!\cdots\!04\)\( T^{26} + \)\(25\!\cdots\!36\)\( T^{27} + \)\(45\!\cdots\!32\)\( T^{28} + \)\(27\!\cdots\!48\)\( T^{29} + \)\(43\!\cdots\!68\)\( T^{30} \)
$3$ \( 1 + 317 T + 163109 T^{2} + 43189707 T^{3} + 13657333057 T^{4} + 3049240516432 T^{5} + 749507085253976 T^{6} + 147091126597541388 T^{7} + 30515837023086629418 T^{8} + \)\(53\!\cdots\!26\)\( T^{9} + \)\(98\!\cdots\!89\)\( T^{10} + \)\(15\!\cdots\!61\)\( T^{11} + \)\(26\!\cdots\!65\)\( T^{12} + \)\(39\!\cdots\!21\)\( T^{13} + \)\(60\!\cdots\!99\)\( T^{14} + \)\(84\!\cdots\!20\)\( T^{15} + \)\(11\!\cdots\!17\)\( T^{16} + \)\(15\!\cdots\!69\)\( T^{17} + \)\(20\!\cdots\!55\)\( T^{18} + \)\(24\!\cdots\!81\)\( T^{19} + \)\(29\!\cdots\!27\)\( T^{20} + \)\(31\!\cdots\!94\)\( T^{21} + \)\(34\!\cdots\!86\)\( T^{22} + \)\(33\!\cdots\!08\)\( T^{23} + \)\(33\!\cdots\!28\)\( T^{24} + \)\(26\!\cdots\!68\)\( T^{25} + \)\(23\!\cdots\!19\)\( T^{26} + \)\(14\!\cdots\!27\)\( T^{27} + \)\(10\!\cdots\!67\)\( T^{28} + \)\(41\!\cdots\!93\)\( T^{29} + \)\(25\!\cdots\!07\)\( T^{30} \)
$5$ \( 1 + 4717 T + 25115807 T^{2} + 83707229397 T^{3} + 275195357593257 T^{4} + 730256475016765590 T^{5} + \)\(18\!\cdots\!78\)\( T^{6} + \)\(41\!\cdots\!18\)\( T^{7} + \)\(89\!\cdots\!28\)\( T^{8} + \)\(17\!\cdots\!68\)\( T^{9} + \)\(32\!\cdots\!59\)\( T^{10} + \)\(57\!\cdots\!55\)\( T^{11} + \)\(96\!\cdots\!25\)\( T^{12} + \)\(15\!\cdots\!25\)\( T^{13} + \)\(22\!\cdots\!25\)\( T^{14} + \)\(32\!\cdots\!00\)\( T^{15} + \)\(44\!\cdots\!25\)\( T^{16} + \)\(57\!\cdots\!25\)\( T^{17} + \)\(71\!\cdots\!25\)\( T^{18} + \)\(83\!\cdots\!75\)\( T^{19} + \)\(93\!\cdots\!75\)\( T^{20} + \)\(97\!\cdots\!00\)\( T^{21} + \)\(97\!\cdots\!00\)\( T^{22} + \)\(88\!\cdots\!50\)\( T^{23} + \)\(77\!\cdots\!50\)\( T^{24} + \)\(58\!\cdots\!50\)\( T^{25} + \)\(43\!\cdots\!25\)\( T^{26} + \)\(25\!\cdots\!25\)\( T^{27} + \)\(15\!\cdots\!75\)\( T^{28} + \)\(55\!\cdots\!25\)\( T^{29} + \)\(22\!\cdots\!25\)\( T^{30} \)
$7$ \( 1 + 9680 T + 303171705 T^{2} + 2420677168232 T^{3} + 41789850773745109 T^{4} + \)\(27\!\cdots\!60\)\( T^{5} + \)\(34\!\cdots\!17\)\( T^{6} + \)\(18\!\cdots\!84\)\( T^{7} + \)\(18\!\cdots\!61\)\( T^{8} + \)\(63\!\cdots\!36\)\( T^{9} + \)\(58\!\cdots\!33\)\( T^{10} - \)\(81\!\cdots\!12\)\( T^{11} + \)\(53\!\cdots\!53\)\( T^{12} - \)\(14\!\cdots\!36\)\( T^{13} - \)\(44\!\cdots\!71\)\( T^{14} - \)\(84\!\cdots\!16\)\( T^{15} - \)\(17\!\cdots\!97\)\( T^{16} - \)\(23\!\cdots\!64\)\( T^{17} + \)\(35\!\cdots\!79\)\( T^{18} - \)\(21\!\cdots\!12\)\( T^{19} + \)\(62\!\cdots\!31\)\( T^{20} + \)\(27\!\cdots\!64\)\( T^{21} + \)\(31\!\cdots\!23\)\( T^{22} + \)\(12\!\cdots\!84\)\( T^{23} + \)\(97\!\cdots\!19\)\( T^{24} + \)\(31\!\cdots\!40\)\( T^{25} + \)\(19\!\cdots\!87\)\( T^{26} + \)\(45\!\cdots\!32\)\( T^{27} + \)\(22\!\cdots\!35\)\( T^{28} + \)\(29\!\cdots\!20\)\( T^{29} + \)\(12\!\cdots\!43\)\( T^{30} \)
$11$ \( 1 + 104484 T + 21671047097 T^{2} + 1826154580412208 T^{3} + \)\(22\!\cdots\!79\)\( T^{4} + \)\(16\!\cdots\!04\)\( T^{5} + \)\(15\!\cdots\!71\)\( T^{6} + \)\(10\!\cdots\!88\)\( T^{7} + \)\(81\!\cdots\!64\)\( T^{8} + \)\(46\!\cdots\!00\)\( T^{9} + \)\(32\!\cdots\!04\)\( T^{10} + \)\(17\!\cdots\!28\)\( T^{11} + \)\(10\!\cdots\!38\)\( T^{12} + \)\(52\!\cdots\!68\)\( T^{13} + \)\(30\!\cdots\!42\)\( T^{14} + \)\(13\!\cdots\!92\)\( T^{15} + \)\(72\!\cdots\!22\)\( T^{16} + \)\(29\!\cdots\!08\)\( T^{17} + \)\(14\!\cdots\!98\)\( T^{18} + \)\(53\!\cdots\!08\)\( T^{19} + \)\(24\!\cdots\!04\)\( T^{20} + \)\(79\!\cdots\!00\)\( T^{21} + \)\(32\!\cdots\!84\)\( T^{22} + \)\(96\!\cdots\!48\)\( T^{23} + \)\(35\!\cdots\!81\)\( T^{24} + \)\(87\!\cdots\!04\)\( T^{25} + \)\(28\!\cdots\!89\)\( T^{26} + \)\(53\!\cdots\!48\)\( T^{27} + \)\(15\!\cdots\!87\)\( T^{28} + \)\(17\!\cdots\!24\)\( T^{29} + \)\(38\!\cdots\!51\)\( T^{30} \)
$13$ \( 1 + 116174 T + 93110646219 T^{2} + 9939551633003756 T^{3} + \)\(43\!\cdots\!83\)\( T^{4} + \)\(42\!\cdots\!54\)\( T^{5} + \)\(13\!\cdots\!41\)\( T^{6} + \)\(12\!\cdots\!40\)\( T^{7} + \)\(31\!\cdots\!88\)\( T^{8} + \)\(25\!\cdots\!00\)\( T^{9} + \)\(57\!\cdots\!96\)\( T^{10} + \)\(42\!\cdots\!04\)\( T^{11} + \)\(85\!\cdots\!78\)\( T^{12} + \)\(58\!\cdots\!52\)\( T^{13} + \)\(10\!\cdots\!26\)\( T^{14} + \)\(67\!\cdots\!12\)\( T^{15} + \)\(11\!\cdots\!98\)\( T^{16} + \)\(65\!\cdots\!08\)\( T^{17} + \)\(10\!\cdots\!26\)\( T^{18} + \)\(53\!\cdots\!64\)\( T^{19} + \)\(76\!\cdots\!28\)\( T^{20} + \)\(36\!\cdots\!00\)\( T^{21} + \)\(47\!\cdots\!36\)\( T^{22} + \)\(19\!\cdots\!40\)\( T^{23} + \)\(23\!\cdots\!33\)\( T^{24} + \)\(76\!\cdots\!46\)\( T^{25} + \)\(83\!\cdots\!91\)\( T^{26} + \)\(20\!\cdots\!76\)\( T^{27} + \)\(19\!\cdots\!27\)\( T^{28} + \)\(26\!\cdots\!66\)\( T^{29} + \)\(24\!\cdots\!57\)\( T^{30} \)
$17$ \( 1 + 884265 T + 1384046763853 T^{2} + 1008991515705553461 T^{3} + \)\(92\!\cdots\!03\)\( T^{4} + \)\(56\!\cdots\!24\)\( T^{5} + \)\(38\!\cdots\!52\)\( T^{6} + \)\(20\!\cdots\!80\)\( T^{7} + \)\(11\!\cdots\!15\)\( T^{8} + \)\(54\!\cdots\!13\)\( T^{9} + \)\(26\!\cdots\!40\)\( T^{10} + \)\(11\!\cdots\!12\)\( T^{11} + \)\(47\!\cdots\!38\)\( T^{12} + \)\(17\!\cdots\!77\)\( T^{13} + \)\(69\!\cdots\!95\)\( T^{14} + \)\(23\!\cdots\!62\)\( T^{15} + \)\(81\!\cdots\!15\)\( T^{16} + \)\(25\!\cdots\!93\)\( T^{17} + \)\(79\!\cdots\!74\)\( T^{18} + \)\(21\!\cdots\!72\)\( T^{19} + \)\(62\!\cdots\!80\)\( T^{20} + \)\(15\!\cdots\!77\)\( T^{21} + \)\(38\!\cdots\!95\)\( T^{22} + \)\(80\!\cdots\!80\)\( T^{23} + \)\(18\!\cdots\!84\)\( T^{24} + \)\(31\!\cdots\!76\)\( T^{25} + \)\(60\!\cdots\!59\)\( T^{26} + \)\(78\!\cdots\!01\)\( T^{27} + \)\(12\!\cdots\!81\)\( T^{28} + \)\(96\!\cdots\!85\)\( T^{29} + \)\(12\!\cdots\!93\)\( T^{30} \)
$19$ \( 1 + 689535 T + 2651913177639 T^{2} + 1907898195520184901 T^{3} + \)\(36\!\cdots\!69\)\( T^{4} + \)\(26\!\cdots\!68\)\( T^{5} + \)\(34\!\cdots\!48\)\( T^{6} + \)\(24\!\cdots\!24\)\( T^{7} + \)\(24\!\cdots\!78\)\( T^{8} + \)\(17\!\cdots\!10\)\( T^{9} + \)\(14\!\cdots\!57\)\( T^{10} + \)\(92\!\cdots\!71\)\( T^{11} + \)\(66\!\cdots\!35\)\( T^{12} + \)\(40\!\cdots\!15\)\( T^{13} + \)\(25\!\cdots\!19\)\( T^{14} + \)\(14\!\cdots\!88\)\( T^{15} + \)\(82\!\cdots\!01\)\( T^{16} + \)\(41\!\cdots\!15\)\( T^{17} + \)\(22\!\cdots\!65\)\( T^{18} + \)\(99\!\cdots\!51\)\( T^{19} + \)\(49\!\cdots\!43\)\( T^{20} + \)\(19\!\cdots\!10\)\( T^{21} + \)\(91\!\cdots\!02\)\( T^{22} + \)\(29\!\cdots\!64\)\( T^{23} + \)\(13\!\cdots\!12\)\( T^{24} + \)\(32\!\cdots\!68\)\( T^{25} + \)\(14\!\cdots\!51\)\( T^{26} + \)\(24\!\cdots\!41\)\( T^{27} + \)\(10\!\cdots\!21\)\( T^{28} + \)\(91\!\cdots\!35\)\( T^{29} + \)\(42\!\cdots\!99\)\( T^{30} \)
$23$ \( 1 + 2504077 T + 19837371032721 T^{2} + 44904598844137902367 T^{3} + \)\(19\!\cdots\!31\)\( T^{4} + \)\(39\!\cdots\!74\)\( T^{5} + \)\(12\!\cdots\!34\)\( T^{6} + \)\(22\!\cdots\!78\)\( T^{7} + \)\(54\!\cdots\!85\)\( T^{8} + \)\(92\!\cdots\!11\)\( T^{9} + \)\(19\!\cdots\!50\)\( T^{10} + \)\(29\!\cdots\!96\)\( T^{11} + \)\(52\!\cdots\!24\)\( T^{12} + \)\(73\!\cdots\!07\)\( T^{13} + \)\(11\!\cdots\!35\)\( T^{14} + \)\(14\!\cdots\!00\)\( T^{15} + \)\(20\!\cdots\!05\)\( T^{16} + \)\(23\!\cdots\!83\)\( T^{17} + \)\(30\!\cdots\!28\)\( T^{18} + \)\(30\!\cdots\!56\)\( T^{19} + \)\(36\!\cdots\!50\)\( T^{20} + \)\(31\!\cdots\!99\)\( T^{21} + \)\(33\!\cdots\!95\)\( T^{22} + \)\(24\!\cdots\!38\)\( T^{23} + \)\(23\!\cdots\!82\)\( T^{24} + \)\(14\!\cdots\!26\)\( T^{25} + \)\(12\!\cdots\!97\)\( T^{26} + \)\(52\!\cdots\!27\)\( T^{27} + \)\(41\!\cdots\!63\)\( T^{28} + \)\(94\!\cdots\!53\)\( T^{29} + \)\(68\!\cdots\!07\)\( T^{30} \)
$29$ \( 1 + 18406221 T + 231322256139583 T^{2} + \)\(21\!\cdots\!69\)\( T^{3} + \)\(16\!\cdots\!45\)\( T^{4} + \)\(10\!\cdots\!86\)\( T^{5} + \)\(64\!\cdots\!34\)\( T^{6} + \)\(34\!\cdots\!58\)\( T^{7} + \)\(16\!\cdots\!04\)\( T^{8} + \)\(73\!\cdots\!72\)\( T^{9} + \)\(30\!\cdots\!31\)\( T^{10} + \)\(11\!\cdots\!11\)\( T^{11} + \)\(44\!\cdots\!21\)\( T^{12} + \)\(16\!\cdots\!53\)\( T^{13} + \)\(60\!\cdots\!37\)\( T^{14} + \)\(22\!\cdots\!12\)\( T^{15} + \)\(87\!\cdots\!53\)\( T^{16} + \)\(34\!\cdots\!33\)\( T^{17} + \)\(13\!\cdots\!89\)\( T^{18} + \)\(52\!\cdots\!31\)\( T^{19} + \)\(19\!\cdots\!19\)\( T^{20} + \)\(68\!\cdots\!32\)\( T^{21} + \)\(22\!\cdots\!56\)\( T^{22} + \)\(66\!\cdots\!78\)\( T^{23} + \)\(18\!\cdots\!86\)\( T^{24} + \)\(45\!\cdots\!86\)\( T^{25} + \)\(99\!\cdots\!05\)\( T^{26} + \)\(18\!\cdots\!09\)\( T^{27} + \)\(29\!\cdots\!47\)\( T^{28} + \)\(33\!\cdots\!41\)\( T^{29} + \)\(26\!\cdots\!49\)\( T^{30} \)
$31$ \( 1 + 12033699 T + 157221027222111 T^{2} + \)\(11\!\cdots\!53\)\( T^{3} + \)\(11\!\cdots\!51\)\( T^{4} + \)\(78\!\cdots\!70\)\( T^{5} + \)\(65\!\cdots\!70\)\( T^{6} + \)\(42\!\cdots\!02\)\( T^{7} + \)\(29\!\cdots\!41\)\( T^{8} + \)\(18\!\cdots\!53\)\( T^{9} + \)\(11\!\cdots\!80\)\( T^{10} + \)\(68\!\cdots\!12\)\( T^{11} + \)\(39\!\cdots\!74\)\( T^{12} + \)\(21\!\cdots\!69\)\( T^{13} + \)\(11\!\cdots\!43\)\( T^{14} + \)\(62\!\cdots\!76\)\( T^{15} + \)\(31\!\cdots\!53\)\( T^{16} + \)\(15\!\cdots\!29\)\( T^{17} + \)\(73\!\cdots\!14\)\( T^{18} + \)\(33\!\cdots\!72\)\( T^{19} + \)\(15\!\cdots\!80\)\( T^{20} + \)\(61\!\cdots\!13\)\( T^{21} + \)\(26\!\cdots\!31\)\( T^{22} + \)\(10\!\cdots\!22\)\( T^{23} + \)\(41\!\cdots\!70\)\( T^{24} + \)\(13\!\cdots\!70\)\( T^{25} + \)\(48\!\cdots\!21\)\( T^{26} + \)\(13\!\cdots\!73\)\( T^{27} + \)\(48\!\cdots\!21\)\( T^{28} + \)\(98\!\cdots\!19\)\( T^{29} + \)\(21\!\cdots\!51\)\( T^{30} \)
$37$ \( 1 + 8722847 T + 568554278402447 T^{2} + \)\(60\!\cdots\!91\)\( T^{3} + \)\(19\!\cdots\!13\)\( T^{4} + \)\(18\!\cdots\!10\)\( T^{5} + \)\(47\!\cdots\!42\)\( T^{6} + \)\(39\!\cdots\!30\)\( T^{7} + \)\(80\!\cdots\!88\)\( T^{8} + \)\(52\!\cdots\!84\)\( T^{9} + \)\(99\!\cdots\!43\)\( T^{10} + \)\(39\!\cdots\!97\)\( T^{11} + \)\(88\!\cdots\!13\)\( T^{12} - \)\(31\!\cdots\!73\)\( T^{13} + \)\(63\!\cdots\!65\)\( T^{14} - \)\(36\!\cdots\!60\)\( T^{15} + \)\(82\!\cdots\!05\)\( T^{16} - \)\(52\!\cdots\!17\)\( T^{17} + \)\(19\!\cdots\!29\)\( T^{18} + \)\(11\!\cdots\!77\)\( T^{19} + \)\(36\!\cdots\!51\)\( T^{20} + \)\(25\!\cdots\!76\)\( T^{21} + \)\(50\!\cdots\!64\)\( T^{22} + \)\(31\!\cdots\!30\)\( T^{23} + \)\(50\!\cdots\!54\)\( T^{24} + \)\(26\!\cdots\!90\)\( T^{25} + \)\(35\!\cdots\!49\)\( T^{26} + \)\(13\!\cdots\!11\)\( T^{27} + \)\(17\!\cdots\!99\)\( T^{28} + \)\(34\!\cdots\!23\)\( T^{29} + \)\(50\!\cdots\!93\)\( T^{30} \)
$41$ \( 1 + 18689389 T + 2776615386417537 T^{2} + \)\(46\!\cdots\!33\)\( T^{3} + \)\(37\!\cdots\!99\)\( T^{4} + \)\(58\!\cdots\!96\)\( T^{5} + \)\(33\!\cdots\!84\)\( T^{6} + \)\(48\!\cdots\!36\)\( T^{7} + \)\(21\!\cdots\!31\)\( T^{8} + \)\(30\!\cdots\!37\)\( T^{9} + \)\(11\!\cdots\!84\)\( T^{10} + \)\(15\!\cdots\!00\)\( T^{11} + \)\(48\!\cdots\!18\)\( T^{12} + \)\(63\!\cdots\!65\)\( T^{13} + \)\(18\!\cdots\!43\)\( T^{14} + \)\(22\!\cdots\!26\)\( T^{15} + \)\(59\!\cdots\!23\)\( T^{16} + \)\(68\!\cdots\!65\)\( T^{17} + \)\(17\!\cdots\!58\)\( T^{18} + \)\(17\!\cdots\!00\)\( T^{19} + \)\(42\!\cdots\!84\)\( T^{20} + \)\(37\!\cdots\!57\)\( T^{21} + \)\(87\!\cdots\!51\)\( T^{22} + \)\(63\!\cdots\!16\)\( T^{23} + \)\(14\!\cdots\!44\)\( T^{24} + \)\(82\!\cdots\!96\)\( T^{25} + \)\(17\!\cdots\!39\)\( T^{26} + \)\(71\!\cdots\!93\)\( T^{27} + \)\(13\!\cdots\!97\)\( T^{28} + \)\(30\!\cdots\!49\)\( T^{29} + \)\(53\!\cdots\!01\)\( T^{30} \)
$43$ \( ( 1 + 3418801 T )^{15} \)
$47$ \( 1 + 104960741 T + 13310159754260471 T^{2} + \)\(95\!\cdots\!91\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} + \)\(43\!\cdots\!16\)\( T^{5} + \)\(26\!\cdots\!08\)\( T^{6} + \)\(13\!\cdots\!76\)\( T^{7} + \)\(66\!\cdots\!90\)\( T^{8} + \)\(28\!\cdots\!38\)\( T^{9} + \)\(12\!\cdots\!89\)\( T^{10} + \)\(49\!\cdots\!97\)\( T^{11} + \)\(20\!\cdots\!91\)\( T^{12} + \)\(70\!\cdots\!13\)\( T^{13} + \)\(26\!\cdots\!11\)\( T^{14} + \)\(85\!\cdots\!24\)\( T^{15} + \)\(29\!\cdots\!37\)\( T^{16} + \)\(88\!\cdots\!57\)\( T^{17} + \)\(28\!\cdots\!33\)\( T^{18} + \)\(77\!\cdots\!37\)\( T^{19} + \)\(22\!\cdots\!23\)\( T^{20} + \)\(56\!\cdots\!22\)\( T^{21} + \)\(14\!\cdots\!70\)\( T^{22} + \)\(31\!\cdots\!16\)\( T^{23} + \)\(73\!\cdots\!76\)\( T^{24} + \)\(13\!\cdots\!84\)\( T^{25} + \)\(26\!\cdots\!43\)\( T^{26} + \)\(36\!\cdots\!51\)\( T^{27} + \)\(57\!\cdots\!77\)\( T^{28} + \)\(50\!\cdots\!89\)\( T^{29} + \)\(54\!\cdots\!43\)\( T^{30} \)
$53$ \( 1 + 215907800 T + 45601589020862355 T^{2} + \)\(63\!\cdots\!44\)\( T^{3} + \)\(82\!\cdots\!07\)\( T^{4} + \)\(85\!\cdots\!20\)\( T^{5} + \)\(82\!\cdots\!01\)\( T^{6} + \)\(67\!\cdots\!72\)\( T^{7} + \)\(51\!\cdots\!60\)\( T^{8} + \)\(34\!\cdots\!76\)\( T^{9} + \)\(21\!\cdots\!24\)\( T^{10} + \)\(11\!\cdots\!24\)\( T^{11} + \)\(58\!\cdots\!90\)\( T^{12} + \)\(26\!\cdots\!88\)\( T^{13} + \)\(12\!\cdots\!62\)\( T^{14} + \)\(61\!\cdots\!84\)\( T^{15} + \)\(40\!\cdots\!46\)\( T^{16} + \)\(28\!\cdots\!32\)\( T^{17} + \)\(20\!\cdots\!30\)\( T^{18} + \)\(13\!\cdots\!04\)\( T^{19} + \)\(83\!\cdots\!32\)\( T^{20} + \)\(44\!\cdots\!44\)\( T^{21} + \)\(22\!\cdots\!20\)\( T^{22} + \)\(95\!\cdots\!52\)\( T^{23} + \)\(38\!\cdots\!53\)\( T^{24} + \)\(13\!\cdots\!80\)\( T^{25} + \)\(41\!\cdots\!19\)\( T^{26} + \)\(10\!\cdots\!84\)\( T^{27} + \)\(25\!\cdots\!15\)\( T^{28} + \)\(39\!\cdots\!00\)\( T^{29} + \)\(59\!\cdots\!57\)\( T^{30} \)
$59$ \( 1 - 185924544 T + 72641643327616985 T^{2} - \)\(11\!\cdots\!56\)\( T^{3} + \)\(26\!\cdots\!25\)\( T^{4} - \)\(37\!\cdots\!76\)\( T^{5} + \)\(67\!\cdots\!65\)\( T^{6} - \)\(85\!\cdots\!64\)\( T^{7} + \)\(12\!\cdots\!49\)\( T^{8} - \)\(14\!\cdots\!84\)\( T^{9} + \)\(19\!\cdots\!45\)\( T^{10} - \)\(20\!\cdots\!16\)\( T^{11} + \)\(24\!\cdots\!49\)\( T^{12} - \)\(23\!\cdots\!92\)\( T^{13} + \)\(25\!\cdots\!69\)\( T^{14} - \)\(22\!\cdots\!84\)\( T^{15} + \)\(21\!\cdots\!91\)\( T^{16} - \)\(17\!\cdots\!32\)\( T^{17} + \)\(15\!\cdots\!31\)\( T^{18} - \)\(11\!\cdots\!56\)\( T^{19} + \)\(95\!\cdots\!55\)\( T^{20} - \)\(63\!\cdots\!24\)\( T^{21} + \)\(47\!\cdots\!71\)\( T^{22} - \)\(27\!\cdots\!84\)\( T^{23} + \)\(18\!\cdots\!35\)\( T^{24} - \)\(89\!\cdots\!76\)\( T^{25} + \)\(55\!\cdots\!75\)\( T^{26} - \)\(20\!\cdots\!76\)\( T^{27} + \)\(11\!\cdots\!15\)\( T^{28} - \)\(24\!\cdots\!04\)\( T^{29} + \)\(11\!\cdots\!99\)\( T^{30} \)
$61$ \( 1 - 247538102 T + 123939924002267915 T^{2} - \)\(25\!\cdots\!76\)\( T^{3} + \)\(72\!\cdots\!53\)\( T^{4} - \)\(12\!\cdots\!30\)\( T^{5} + \)\(27\!\cdots\!23\)\( T^{6} - \)\(42\!\cdots\!04\)\( T^{7} + \)\(73\!\cdots\!05\)\( T^{8} - \)\(10\!\cdots\!70\)\( T^{9} + \)\(15\!\cdots\!63\)\( T^{10} - \)\(19\!\cdots\!64\)\( T^{11} + \)\(26\!\cdots\!33\)\( T^{12} - \)\(30\!\cdots\!38\)\( T^{13} + \)\(37\!\cdots\!95\)\( T^{14} - \)\(39\!\cdots\!08\)\( T^{15} + \)\(43\!\cdots\!95\)\( T^{16} - \)\(41\!\cdots\!78\)\( T^{17} + \)\(42\!\cdots\!93\)\( T^{18} - \)\(36\!\cdots\!04\)\( T^{19} + \)\(34\!\cdots\!63\)\( T^{20} - \)\(26\!\cdots\!70\)\( T^{21} + \)\(22\!\cdots\!05\)\( T^{22} - \)\(14\!\cdots\!84\)\( T^{23} + \)\(11\!\cdots\!03\)\( T^{24} - \)\(60\!\cdots\!30\)\( T^{25} + \)\(40\!\cdots\!73\)\( T^{26} - \)\(16\!\cdots\!56\)\( T^{27} + \)\(94\!\cdots\!15\)\( T^{28} - \)\(22\!\cdots\!22\)\( T^{29} + \)\(10\!\cdots\!01\)\( T^{30} \)
$67$ \( 1 - 467904656 T + 289992155760016797 T^{2} - \)\(96\!\cdots\!32\)\( T^{3} + \)\(36\!\cdots\!55\)\( T^{4} - \)\(99\!\cdots\!68\)\( T^{5} + \)\(29\!\cdots\!91\)\( T^{6} - \)\(69\!\cdots\!20\)\( T^{7} + \)\(17\!\cdots\!04\)\( T^{8} - \)\(36\!\cdots\!76\)\( T^{9} + \)\(81\!\cdots\!48\)\( T^{10} - \)\(15\!\cdots\!04\)\( T^{11} + \)\(31\!\cdots\!82\)\( T^{12} - \)\(54\!\cdots\!16\)\( T^{13} + \)\(10\!\cdots\!46\)\( T^{14} - \)\(16\!\cdots\!44\)\( T^{15} + \)\(27\!\cdots\!62\)\( T^{16} - \)\(40\!\cdots\!44\)\( T^{17} + \)\(63\!\cdots\!86\)\( T^{18} - \)\(85\!\cdots\!24\)\( T^{19} + \)\(12\!\cdots\!36\)\( T^{20} - \)\(14\!\cdots\!04\)\( T^{21} + \)\(19\!\cdots\!52\)\( T^{22} - \)\(20\!\cdots\!20\)\( T^{23} + \)\(24\!\cdots\!97\)\( T^{24} - \)\(22\!\cdots\!32\)\( T^{25} + \)\(22\!\cdots\!65\)\( T^{26} - \)\(15\!\cdots\!12\)\( T^{27} + \)\(12\!\cdots\!19\)\( T^{28} - \)\(56\!\cdots\!64\)\( T^{29} + \)\(33\!\cdots\!43\)\( T^{30} \)
$71$ \( 1 + 8252944 T + 295175699032282861 T^{2} + \)\(11\!\cdots\!80\)\( T^{3} + \)\(45\!\cdots\!81\)\( T^{4} + \)\(31\!\cdots\!80\)\( T^{5} + \)\(48\!\cdots\!57\)\( T^{6} + \)\(47\!\cdots\!60\)\( T^{7} + \)\(40\!\cdots\!77\)\( T^{8} + \)\(47\!\cdots\!80\)\( T^{9} + \)\(28\!\cdots\!97\)\( T^{10} + \)\(35\!\cdots\!60\)\( T^{11} + \)\(17\!\cdots\!41\)\( T^{12} + \)\(20\!\cdots\!76\)\( T^{13} + \)\(94\!\cdots\!13\)\( T^{14} + \)\(10\!\cdots\!84\)\( T^{15} + \)\(43\!\cdots\!03\)\( T^{16} + \)\(43\!\cdots\!36\)\( T^{17} + \)\(17\!\cdots\!31\)\( T^{18} + \)\(15\!\cdots\!60\)\( T^{19} + \)\(58\!\cdots\!47\)\( T^{20} + \)\(43\!\cdots\!80\)\( T^{21} + \)\(17\!\cdots\!47\)\( T^{22} + \)\(92\!\cdots\!60\)\( T^{23} + \)\(43\!\cdots\!47\)\( T^{24} + \)\(13\!\cdots\!80\)\( T^{25} + \)\(85\!\cdots\!11\)\( T^{26} + \)\(98\!\cdots\!80\)\( T^{27} + \)\(11\!\cdots\!51\)\( T^{28} + \)\(14\!\cdots\!24\)\( T^{29} + \)\(83\!\cdots\!51\)\( T^{30} \)
$73$ \( 1 + 715627902 T + 725896042339854143 T^{2} + \)\(39\!\cdots\!04\)\( T^{3} + \)\(23\!\cdots\!09\)\( T^{4} + \)\(10\!\cdots\!42\)\( T^{5} + \)\(48\!\cdots\!43\)\( T^{6} + \)\(18\!\cdots\!64\)\( T^{7} + \)\(69\!\cdots\!09\)\( T^{8} + \)\(23\!\cdots\!46\)\( T^{9} + \)\(77\!\cdots\!47\)\( T^{10} + \)\(23\!\cdots\!12\)\( T^{11} + \)\(68\!\cdots\!33\)\( T^{12} + \)\(18\!\cdots\!86\)\( T^{13} + \)\(49\!\cdots\!39\)\( T^{14} + \)\(11\!\cdots\!52\)\( T^{15} + \)\(28\!\cdots\!07\)\( T^{16} + \)\(63\!\cdots\!34\)\( T^{17} + \)\(13\!\cdots\!01\)\( T^{18} + \)\(27\!\cdots\!32\)\( T^{19} + \)\(54\!\cdots\!71\)\( T^{20} + \)\(96\!\cdots\!14\)\( T^{21} + \)\(17\!\cdots\!53\)\( T^{22} + \)\(26\!\cdots\!44\)\( T^{23} + \)\(40\!\cdots\!39\)\( T^{24} + \)\(52\!\cdots\!58\)\( T^{25} + \)\(69\!\cdots\!33\)\( T^{26} + \)\(68\!\cdots\!24\)\( T^{27} + \)\(74\!\cdots\!79\)\( T^{28} + \)\(42\!\cdots\!78\)\( T^{29} + \)\(35\!\cdots\!57\)\( T^{30} \)
$79$ \( 1 - 560681783 T + 849133708768547829 T^{2} - \)\(32\!\cdots\!85\)\( T^{3} + \)\(33\!\cdots\!49\)\( T^{4} - \)\(10\!\cdots\!12\)\( T^{5} + \)\(90\!\cdots\!32\)\( T^{6} - \)\(23\!\cdots\!28\)\( T^{7} + \)\(19\!\cdots\!98\)\( T^{8} - \)\(43\!\cdots\!74\)\( T^{9} + \)\(34\!\cdots\!01\)\( T^{10} - \)\(67\!\cdots\!51\)\( T^{11} + \)\(53\!\cdots\!45\)\( T^{12} - \)\(93\!\cdots\!63\)\( T^{13} + \)\(72\!\cdots\!51\)\( T^{14} - \)\(11\!\cdots\!88\)\( T^{15} + \)\(86\!\cdots\!69\)\( T^{16} - \)\(13\!\cdots\!43\)\( T^{17} + \)\(91\!\cdots\!55\)\( T^{18} - \)\(13\!\cdots\!71\)\( T^{19} + \)\(85\!\cdots\!99\)\( T^{20} - \)\(12\!\cdots\!94\)\( T^{21} + \)\(69\!\cdots\!22\)\( T^{22} - \)\(10\!\cdots\!48\)\( T^{23} + \)\(46\!\cdots\!28\)\( T^{24} - \)\(62\!\cdots\!12\)\( T^{25} + \)\(24\!\cdots\!31\)\( T^{26} - \)\(28\!\cdots\!85\)\( T^{27} + \)\(89\!\cdots\!11\)\( T^{28} - \)\(70\!\cdots\!43\)\( T^{29} + \)\(15\!\cdots\!99\)\( T^{30} \)
$83$ \( 1 + 1442854698 T + 2517006525957293097 T^{2} + \)\(25\!\cdots\!08\)\( T^{3} + \)\(27\!\cdots\!99\)\( T^{4} + \)\(22\!\cdots\!86\)\( T^{5} + \)\(19\!\cdots\!63\)\( T^{6} + \)\(13\!\cdots\!56\)\( T^{7} + \)\(92\!\cdots\!28\)\( T^{8} + \)\(56\!\cdots\!76\)\( T^{9} + \)\(33\!\cdots\!68\)\( T^{10} + \)\(18\!\cdots\!28\)\( T^{11} + \)\(97\!\cdots\!54\)\( T^{12} + \)\(47\!\cdots\!68\)\( T^{13} + \)\(22\!\cdots\!94\)\( T^{14} + \)\(98\!\cdots\!20\)\( T^{15} + \)\(42\!\cdots\!82\)\( T^{16} + \)\(16\!\cdots\!12\)\( T^{17} + \)\(64\!\cdots\!58\)\( T^{18} + \)\(22\!\cdots\!68\)\( T^{19} + \)\(77\!\cdots\!24\)\( T^{20} + \)\(24\!\cdots\!04\)\( T^{21} + \)\(73\!\cdots\!36\)\( T^{22} + \)\(19\!\cdots\!16\)\( T^{23} + \)\(53\!\cdots\!29\)\( T^{24} + \)\(11\!\cdots\!14\)\( T^{25} + \)\(27\!\cdots\!53\)\( T^{26} + \)\(47\!\cdots\!28\)\( T^{27} + \)\(85\!\cdots\!31\)\( T^{28} + \)\(91\!\cdots\!62\)\( T^{29} + \)\(11\!\cdots\!07\)\( T^{30} \)
$89$ \( 1 + 396710008 T + 2452763342866776295 T^{2} + \)\(86\!\cdots\!68\)\( T^{3} + \)\(29\!\cdots\!85\)\( T^{4} + \)\(87\!\cdots\!92\)\( T^{5} + \)\(22\!\cdots\!31\)\( T^{6} + \)\(52\!\cdots\!68\)\( T^{7} + \)\(12\!\cdots\!05\)\( T^{8} + \)\(17\!\cdots\!32\)\( T^{9} + \)\(51\!\cdots\!71\)\( T^{10} + \)\(14\!\cdots\!24\)\( T^{11} + \)\(17\!\cdots\!37\)\( T^{12} - \)\(18\!\cdots\!80\)\( T^{13} + \)\(54\!\cdots\!11\)\( T^{14} - \)\(10\!\cdots\!56\)\( T^{15} + \)\(19\!\cdots\!99\)\( T^{16} - \)\(23\!\cdots\!80\)\( T^{17} + \)\(74\!\cdots\!73\)\( T^{18} + \)\(21\!\cdots\!64\)\( T^{19} + \)\(27\!\cdots\!79\)\( T^{20} + \)\(32\!\cdots\!12\)\( T^{21} + \)\(79\!\cdots\!45\)\( T^{22} + \)\(11\!\cdots\!28\)\( T^{23} + \)\(17\!\cdots\!59\)\( T^{24} + \)\(24\!\cdots\!92\)\( T^{25} + \)\(28\!\cdots\!65\)\( T^{26} + \)\(29\!\cdots\!08\)\( T^{27} + \)\(29\!\cdots\!55\)\( T^{28} + \)\(16\!\cdots\!88\)\( T^{29} + \)\(14\!\cdots\!49\)\( T^{30} \)
$97$ \( 1 + 3063837815 T + 9536233785462653481 T^{2} + \)\(19\!\cdots\!71\)\( T^{3} + \)\(38\!\cdots\!99\)\( T^{4} + \)\(61\!\cdots\!48\)\( T^{5} + \)\(97\!\cdots\!08\)\( T^{6} + \)\(13\!\cdots\!04\)\( T^{7} + \)\(18\!\cdots\!19\)\( T^{8} + \)\(21\!\cdots\!95\)\( T^{9} + \)\(25\!\cdots\!96\)\( T^{10} + \)\(27\!\cdots\!24\)\( T^{11} + \)\(29\!\cdots\!54\)\( T^{12} + \)\(28\!\cdots\!39\)\( T^{13} + \)\(26\!\cdots\!31\)\( T^{14} + \)\(23\!\cdots\!02\)\( T^{15} + \)\(20\!\cdots\!27\)\( T^{16} + \)\(16\!\cdots\!71\)\( T^{17} + \)\(12\!\cdots\!02\)\( T^{18} + \)\(92\!\cdots\!04\)\( T^{19} + \)\(65\!\cdots\!72\)\( T^{20} + \)\(41\!\cdots\!55\)\( T^{21} + \)\(26\!\cdots\!87\)\( T^{22} + \)\(14\!\cdots\!64\)\( T^{23} + \)\(82\!\cdots\!76\)\( T^{24} + \)\(39\!\cdots\!52\)\( T^{25} + \)\(18\!\cdots\!67\)\( T^{26} + \)\(71\!\cdots\!31\)\( T^{27} + \)\(27\!\cdots\!97\)\( T^{28} + \)\(65\!\cdots\!35\)\( T^{29} + \)\(16\!\cdots\!93\)\( T^{30} \)
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