Properties

Label 72.18.d.b
Level $72$
Weight $18$
Character orbit 72.d
Analytic conductor $131.920$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 72.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(131.919902888\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 8 x^{15} + 109505575668 x^{14} - 766539029536 x^{13} + \)\(45\!\cdots\!58\)\( x^{12} - \)\(27\!\cdots\!44\)\( x^{11} + \)\(92\!\cdots\!84\)\( x^{10} - \)\(46\!\cdots\!88\)\( x^{9} + \)\(97\!\cdots\!41\)\( x^{8} - \)\(39\!\cdots\!08\)\( x^{7} + \)\(54\!\cdots\!12\)\( x^{6} - \)\(16\!\cdots\!68\)\( x^{5} + \)\(15\!\cdots\!52\)\( x^{4} - \)\(31\!\cdots\!24\)\( x^{3} + \)\(18\!\cdots\!52\)\( x^{2} - \)\(18\!\cdots\!92\)\( x + \)\(23\!\cdots\!64\)\(\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{120}\cdot 3^{20}\cdot 7 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -17 - \beta_{1} ) q^{2} + ( -1713 + 17 \beta_{1} + \beta_{4} ) q^{4} + ( 8 + 63 \beta_{1} - \beta_{3} - \beta_{4} ) q^{5} + ( 721562 + 7726 \beta_{1} - \beta_{2} + 11 \beta_{4} + \beta_{5} ) q^{7} + ( -1520694 + 1909 \beta_{1} + 46 \beta_{2} + 5 \beta_{3} - 12 \beta_{4} - \beta_{6} + \beta_{7} ) q^{8} +O(q^{10})\) \( q + ( -17 - \beta_{1} ) q^{2} + ( -1713 + 17 \beta_{1} + \beta_{4} ) q^{4} + ( 8 + 63 \beta_{1} - \beta_{3} - \beta_{4} ) q^{5} + ( 721562 + 7726 \beta_{1} - \beta_{2} + 11 \beta_{4} + \beta_{5} ) q^{7} + ( -1520694 + 1909 \beta_{1} + 46 \beta_{2} + 5 \beta_{3} - 12 \beta_{4} - \beta_{6} + \beta_{7} ) q^{8} + ( 8187569 - 1036 \beta_{1} - 62 \beta_{2} - 18 \beta_{3} - 71 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} - \beta_{7} + \beta_{11} ) q^{10} + ( 13098 + 106505 \beta_{1} - 131 \beta_{2} + 150 \beta_{3} + 714 \beta_{4} - \beta_{5} - 8 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{13} ) q^{11} + ( -147795 - 1179283 \beta_{1} - 434 \beta_{2} - 214 \beta_{3} + 1050 \beta_{4} + 75 \beta_{6} - 12 \beta_{7} - \beta_{8} + \beta_{9} - 5 \beta_{11} + \beta_{12} - \beta_{15} ) q^{13} + ( -1022807807 - 589196 \beta_{1} + 1230 \beta_{2} - 614 \beta_{3} - 7896 \beta_{4} - 96 \beta_{5} - 35 \beta_{6} + 14 \beta_{7} + 2 \beta_{8} + 6 \beta_{9} + \beta_{10} - 5 \beta_{11} - \beta_{12} - 6 \beta_{13} - \beta_{14} + \beta_{15} ) q^{14} + ( 1656444157 + 1312646 \beta_{1} - 14470 \beta_{2} - 150 \beta_{3} - 3207 \beta_{4} - 200 \beta_{5} + 101 \beta_{6} - 16 \beta_{7} + 44 \beta_{8} - 20 \beta_{9} - 2 \beta_{11} - 6 \beta_{12} - 10 \beta_{13} + 4 \beta_{14} ) q^{16} + ( 468338567 + 2074996 \beta_{1} - 3210 \beta_{2} + 306 \beta_{3} - 22216 \beta_{4} + 7 \beta_{5} + 189 \beta_{6} - 274 \beta_{7} + 174 \beta_{8} - \beta_{9} - 6 \beta_{10} - 48 \beta_{11} - 8 \beta_{13} - 4 \beta_{14} + 8 \beta_{15} ) q^{17} + ( -1429238 - 11499141 \beta_{1} - 10257 \beta_{2} - 6734 \beta_{3} - 22574 \beta_{4} + 69 \beta_{5} + 1020 \beta_{6} - 762 \beta_{7} - 605 \beta_{8} + 24 \beta_{9} - 8 \beta_{10} + 40 \beta_{11} - 8 \beta_{12} + 19 \beta_{13} + 32 \beta_{14} + 8 \beta_{15} ) q^{19} + ( 13089106158 - 9921142 \beta_{1} + 100374 \beta_{2} + 39454 \beta_{3} + 20024 \beta_{4} - 3918 \beta_{5} + 16 \beta_{6} - 194 \beta_{7} - 162 \beta_{8} - 128 \beta_{9} + 8 \beta_{10} + 32 \beta_{11} + 16 \beta_{12} + 46 \beta_{13} - 10 \beta_{14} + 30 \beta_{15} ) q^{20} + ( 13945207246 - 1822337 \beta_{1} + 158129 \beta_{2} + 70293 \beta_{3} - 71394 \beta_{4} + 9672 \beta_{5} + 927 \beta_{6} + 536 \beta_{7} + 133 \beta_{8} + 48 \beta_{9} - 224 \beta_{10} - 144 \beta_{11} + 8 \beta_{12} - 160 \beta_{13} + 88 \beta_{14} + 40 \beta_{15} ) q^{22} + ( -46665328280 + 99291338 \beta_{1} - 53315 \beta_{2} + 3055 \beta_{3} - 231317 \beta_{4} + 636 \beta_{5} + 7318 \beta_{6} - 1804 \beta_{7} - 4821 \beta_{8} + 22 \beta_{9} - 55 \beta_{10} - 448 \beta_{11} - 104 \beta_{12} - 92 \beta_{13} - 94 \beta_{14} - 12 \beta_{15} ) q^{23} + ( -113075717159 + 241321164 \beta_{1} + 157544 \beta_{2} - 27938 \beta_{3} + 1879386 \beta_{4} + 2852 \beta_{5} + 14380 \beta_{6} + 604 \beta_{7} - 10818 \beta_{8} + 118 \beta_{9} + 938 \beta_{10} - 944 \beta_{11} - 256 \beta_{12} - 200 \beta_{13} - 100 \beta_{14} - 56 \beta_{15} ) q^{25} + ( -154230823851 + 19572756 \beta_{1} + 126202 \beta_{2} + 552966 \beta_{3} + 1304077 \beta_{4} + 15862 \beta_{5} - 297 \beta_{6} + 435 \beta_{7} - 976 \beta_{8} + 1184 \beta_{9} + 1408 \beta_{10} + 61 \beta_{11} - 16 \beta_{12} + 64 \beta_{13} - 176 \beta_{14} - 336 \beta_{15} ) q^{26} + ( 201408334156 + 999181984 \beta_{1} + 336432 \beta_{2} - 1038296 \beta_{3} + 414148 \beta_{4} + 13368 \beta_{5} - 7388 \beta_{6} - 7792 \beta_{7} + 2840 \beta_{8} - 2512 \beta_{9} - 1744 \beta_{10} + 1560 \beta_{11} - 184 \beta_{12} + 816 \beta_{13} - 600 \beta_{14} - 200 \beta_{15} ) q^{28} + ( -40141752 - 298822087 \beta_{1} - 113712 \beta_{2} + 7729 \beta_{3} + 7475289 \beta_{4} - 3504 \beta_{5} + 23168 \beta_{6} - 18656 \beta_{7} - 22760 \beta_{8} + 664 \beta_{9} - 4688 \beta_{10} + 712 \beta_{11} - 40 \beta_{12} + 928 \beta_{13} + 960 \beta_{14} + 296 \beta_{15} ) q^{29} + ( -19900819522 + 332813772 \beta_{1} + 1750710 \beta_{2} - 233617 \beta_{3} + 15777240 \beta_{4} + 4611 \beta_{5} + 18102 \beta_{6} - 34316 \beta_{7} + 17835 \beta_{8} + 406 \beta_{9} + 7497 \beta_{10} - 5312 \beta_{11} + 152 \beta_{12} - 860 \beta_{13} - 478 \beta_{14} + 1012 \beta_{15} ) q^{31} + ( -91186382286 - 1648580380 \beta_{1} + 13537988 \beta_{2} - 4067124 \beta_{3} - 1010998 \beta_{4} - 134176 \beta_{5} + 65226 \beta_{6} + 5768 \beta_{7} - 12664 \beta_{8} + 1848 \beta_{9} + 10208 \beta_{10} - 340 \beta_{11} + 772 \beta_{12} + 3148 \beta_{13} + 1240 \beta_{14} - 784 \beta_{15} ) q^{32} + ( -278877313966 - 427828826 \beta_{1} - 27175660 \beta_{2} - 4683544 \beta_{3} - 5687898 \beta_{4} + 218620 \beta_{5} + 53648 \beta_{6} - 8612 \beta_{7} + 1816 \beta_{8} + 5404 \beta_{9} - 20042 \beta_{10} - 738 \beta_{11} + 2350 \beta_{12} + 60 \beta_{13} - 2010 \beta_{14} - 1414 \beta_{15} ) q^{34} + ( -621160746 - 4807495824 \beta_{1} + 1936320 \beta_{2} + 392228 \beta_{3} + 53564024 \beta_{4} - 34150 \beta_{5} + 280636 \beta_{6} - 9668 \beta_{7} - 57474 \beta_{8} + 2024 \beta_{9} - 29560 \beta_{10} - 13864 \beta_{11} + 3592 \beta_{12} - 114 \beta_{13} + 480 \beta_{14} - 1544 \beta_{15} ) q^{35} + ( -637029265 - 5296363945 \beta_{1} + 22794010 \beta_{2} - 663018 \beta_{3} - 74608954 \beta_{4} + 31424 \beta_{5} + 449721 \beta_{6} - 39684 \beta_{7} - 226283 \beta_{8} + 2347 \beta_{9} + 39744 \beta_{10} + 809 \beta_{11} + 1067 \beta_{12} + 12160 \beta_{13} + 4352 \beta_{14} + 2005 \beta_{15} ) q^{37} + ( -1504736995718 + 202401845 \beta_{1} - 64867941 \beta_{2} - 4378201 \beta_{3} + 4321578 \beta_{4} + 365912 \beta_{5} - 411659 \beta_{6} + 10696 \beta_{7} - 65417 \beta_{8} + 20112 \beta_{9} + 50016 \beta_{10} + 3024 \beta_{11} + 2328 \beta_{12} + 12832 \beta_{13} + 1032 \beta_{14} + 5496 \beta_{15} ) q^{38} + ( 3787468776428 - 13587845476 \beta_{1} - 170762512 \beta_{2} + 2882092 \beta_{3} - 5083500 \beta_{4} + 215184 \beta_{5} - 394160 \beta_{6} + 39348 \beta_{7} + 42944 \beta_{8} - 9104 \beta_{9} - 97504 \beta_{10} - 19688 \beta_{11} + 8008 \beta_{12} + 20520 \beta_{13} + 12480 \beta_{14} - 592 \beta_{15} ) q^{40} + ( -468502081222 - 7390834884 \beta_{1} - 17033688 \beta_{2} + 2328662 \beta_{3} - 159699070 \beta_{4} + 21956 \beta_{5} - 143636 \beta_{6} - 569428 \beta_{7} - 529674 \beta_{8} - 17346 \beta_{9} - 78062 \beta_{10} + 67216 \beta_{11} + 3584 \beta_{12} + 11800 \beta_{13} - 18676 \beta_{14} - 8216 \beta_{15} ) q^{41} + ( 937769587 + 6848361347 \beta_{1} - 65989517 \beta_{2} - 9885020 \beta_{3} - 202252508 \beta_{4} + 182506 \beta_{5} - 345288 \beta_{6} - 472500 \beta_{7} + 553446 \beta_{8} + 24880 \beta_{9} + 134256 \beta_{10} - 76464 \beta_{11} + 7920 \beta_{12} + 2630 \beta_{13} - 1472 \beta_{14} - 26352 \beta_{15} ) q^{43} + ( -12104895527542 - 12344765601 \beta_{1} - 456061985 \beta_{2} - 36444707 \beta_{3} - 52101555 \beta_{4} - 1607381 \beta_{5} + 2246679 \beta_{6} + 161155 \beta_{7} - 23719 \beta_{8} + 51172 \beta_{9} + 152000 \beta_{10} - 124174 \beta_{11} - 12914 \beta_{12} - 41925 \beta_{13} - 18679 \beta_{14} + 1737 \beta_{15} ) q^{44} + ( -12187595582307 + 48307707140 \beta_{1} + 578238262 \beta_{2} - 50267086 \beta_{3} - 56643816 \beta_{4} + 2595264 \beta_{5} + 1696777 \beta_{6} - 223434 \beta_{7} + 376442 \beta_{8} + 123470 \beta_{9} - 117331 \beta_{10} + 32959 \beta_{11} - 29261 \beta_{12} + 52466 \beta_{13} + 8179 \beta_{14} + 6413 \beta_{15} ) q^{46} + ( 23554874436786 + 88900743824 \beta_{1} - 47541704 \beta_{2} + 3346083 \beta_{3} - 210542166 \beta_{4} + 676397 \beta_{5} + 5286062 \beta_{6} - 279068 \beta_{7} + 326703 \beta_{8} - 163490 \beta_{9} - 121803 \beta_{10} + 7104 \beta_{11} + 3384 \beta_{12} + 1748 \beta_{13} + 3322 \beta_{14} + 1636 \beta_{15} ) q^{47} + ( 8000777495505 + 161250677184 \beta_{1} - 15006384 \beta_{2} - 2947520 \beta_{3} + 233009360 \beta_{4} + 2174856 \beta_{5} + 9345848 \beta_{6} - 1222032 \beta_{7} - 903712 \beta_{8} - 39240 \beta_{9} + 71840 \beta_{10} + 156672 \beta_{11} + 16128 \beta_{12} + 28800 \beta_{13} - 34752 \beta_{14} - 12672 \beta_{15} ) q^{49} + ( -29672857765957 + 117064448419 \beta_{1} + 1400587192 \beta_{2} - 109787728 \beta_{3} - 134349788 \beta_{4} + 4742760 \beta_{5} - 15342336 \beta_{6} + 1708136 \beta_{7} + 715216 \beta_{8} + 191464 \beta_{9} + 15268 \beta_{10} + 59476 \beta_{11} - 85708 \beta_{12} + 71464 \beta_{13} + 37380 \beta_{14} + 11708 \beta_{15} ) q^{50} + ( -16996495557482 + 156448277410 \beta_{1} + 2118762302 \beta_{2} + 149462118 \beta_{3} + 216206712 \beta_{4} + 4749098 \beta_{5} - 14765136 \beta_{6} + 313286 \beta_{7} + 625702 \beta_{8} - 42496 \beta_{9} + 304488 \beta_{10} - 628256 \beta_{11} - 50544 \beta_{12} - 115466 \beta_{13} - 86818 \beta_{14} + 6630 \beta_{15} ) q^{52} + ( 29508838447 + 233821702465 \beta_{1} + 451723082 \beta_{2} - 105883960 \beta_{3} - 974919120 \beta_{4} + 839728 \beta_{5} - 11137575 \beta_{6} - 2815780 \beta_{7} - 4622787 \beta_{8} + 64915 \beta_{9} + 219088 \beta_{10} + 47713 \beta_{11} + 10835 \beta_{12} - 458400 \beta_{13} + 105024 \beta_{14} + 40109 \beta_{15} ) q^{53} + ( 138007751259082 - 461291072042 \beta_{1} - 73760797 \beta_{2} + 24283804 \beta_{3} - 1618026153 \beta_{4} + 3936289 \beta_{5} - 29501240 \beta_{6} - 4914992 \beta_{7} + 6274444 \beta_{8} - 1041704 \beta_{9} - 836924 \beta_{10} - 387968 \beta_{11} + 67808 \beta_{12} - 53360 \beta_{13} - 760 \beta_{14} + 121168 \beta_{15} ) q^{55} + ( 10147460019648 - 203308606808 \beta_{1} + 3010530704 \beta_{2} + 58798824 \beta_{3} - 705215344 \beta_{4} - 12551680 \beta_{5} + 30304840 \beta_{6} + 231656 \beta_{7} + 2650048 \beta_{8} + 445760 \beta_{9} - 1126400 \beta_{10} + 1032736 \beta_{11} + 68192 \beta_{12} - 78048 \beta_{13} + 142912 \beta_{14} + 53376 \beta_{15} ) q^{56} + ( -38952847397289 + 5393468972 \beta_{1} - 3052721458 \beta_{2} - 3960958 \beta_{3} - 2660849 \beta_{4} + 14968562 \beta_{5} + 46081757 \beta_{6} + 9473849 \beta_{7} - 1125504 \beta_{8} + 722176 \beta_{9} + 1259520 \beta_{10} + 84039 \beta_{11} - 2176 \beta_{12} + 365056 \beta_{13} + 90752 \beta_{14} + 179584 \beta_{15} ) q^{58} + ( -134012623541 - 1082661400075 \beta_{1} - 908522203 \beta_{2} + 48141584 \beta_{3} - 3157734260 \beta_{4} + 2437304 \beta_{5} + 64426476 \beta_{6} - 986880 \beta_{7} + 7143008 \beta_{8} - 44312 \beta_{9} + 1954312 \beta_{10} + 606424 \beta_{11} - 208120 \beta_{12} + 8432 \beta_{13} - 53280 \beta_{14} - 8968 \beta_{15} ) q^{59} + ( -234663283339 - 1868535629399 \beta_{1} - 1629236914 \beta_{2} - 138877530 \beta_{3} + 3410728814 \beta_{4} - 378192 \beta_{5} + 108889875 \beta_{6} - 4822860 \beta_{7} + 7357839 \beta_{8} + 41921 \beta_{9} - 2209456 \beta_{10} + 203131 \beta_{11} - 107903 \beta_{12} - 1381792 \beta_{13} - 18368 \beta_{14} - 60289 \beta_{15} ) q^{61} + ( -43477253431004 + 26073579568 \beta_{1} - 2638394920 \beta_{2} - 394804632 \beta_{3} - 657620688 \beta_{4} + 27579168 \beta_{5} - 146165100 \beta_{6} + 15552664 \beta_{7} - 114296 \beta_{8} + 438136 \beta_{9} - 2516684 \beta_{10} - 79076 \beta_{11} + 327212 \beta_{12} - 18488 \beta_{13} - 300948 \beta_{14} - 167788 \beta_{15} ) q^{62} + ( 69505843239596 + 82064080376 \beta_{1} - 2356161896 \beta_{2} + 738198120 \beta_{3} + 1639337660 \beta_{4} + 35109248 \beta_{5} - 201409060 \beta_{6} - 4128944 \beta_{7} + 8234992 \beta_{8} - 249264 \beta_{9} + 2820032 \beta_{10} + 4199048 \beta_{11} + 126360 \beta_{12} + 525192 \beta_{13} + 302160 \beta_{14} - 150816 \beta_{15} ) q^{64} + ( -149595875812628 - 3748109080340 \beta_{1} + 1851000760 \beta_{2} - 126780210 \beta_{3} + 7481288506 \beta_{4} - 85747796 \beta_{5} - 186183388 \beta_{6} - 23223700 \beta_{7} - 16786482 \beta_{8} - 725698 \beta_{9} + 3531354 \beta_{10} + 2713296 \beta_{11} + 316416 \beta_{12} + 504952 \beta_{13} - 632260 \beta_{14} - 188536 \beta_{15} ) q^{65} + ( 575570941366 + 4618932415227 \beta_{1} - 3579248025 \beta_{2} + 468496706 \beta_{3} + 6202726870 \beta_{4} - 174563 \beta_{5} - 289695360 \beta_{6} - 18611802 \beta_{7} + 27474451 \beta_{8} + 758736 \beta_{9} - 2545008 \beta_{10} - 1038928 \beta_{11} - 89584 \beta_{12} + 174515 \beta_{13} + 104896 \beta_{14} - 653840 \beta_{15} ) q^{67} + ( -373778562775126 + 327053896990 \beta_{1} + 1335544032 \beta_{2} + 1455670448 \beta_{3} + 803027542 \beta_{4} - 21095024 \beta_{5} + 318538456 \beta_{6} - 5107168 \beta_{7} - 20639408 \beta_{8} + 1172512 \beta_{9} - 1673568 \beta_{10} + 4578576 \beta_{11} - 476880 \beta_{12} - 1411104 \beta_{13} - 888272 \beta_{14} + 52624 \beta_{15} ) q^{68} + ( -627774583771064 + 81216666016 \beta_{1} - 10254795632 \beta_{2} + 1828263968 \beta_{3} + 4112863384 \beta_{4} + 39277936 \beta_{5} + 369338616 \beta_{6} + 59930448 \beta_{7} + 3222848 \beta_{8} + 3596192 \beta_{9} + 2527168 \beta_{10} + 631200 \beta_{11} - 601104 \beta_{12} + 690496 \beta_{13} - 165040 \beta_{14} - 640080 \beta_{15} ) q^{70} + ( -563339336702048 + 6231671450766 \beta_{1} - 2126246417 \beta_{2} + 64679297 \beta_{3} - 4724211023 \beta_{4} - 58030360 \beta_{5} + 408333738 \beta_{6} + 9508172 \beta_{7} + 44813061 \beta_{8} + 10358730 \beta_{9} - 689849 \beta_{10} - 548672 \beta_{11} + 717032 \beta_{12} + 28060 \beta_{13} + 246782 \beta_{14} + 688972 \beta_{15} ) q^{71} + ( 709190898877569 + 7525726277976 \beta_{1} - 1774134022 \beta_{2} - 44276244 \beta_{3} + 427426778 \beta_{4} - 319755343 \beta_{5} + 615127179 \beta_{6} - 17050938 \beta_{7} + 44227808 \beta_{8} - 943389 \beta_{9} + 755104 \beta_{10} + 5720064 \beta_{11} + 1456896 \beta_{12} + 1196160 \beta_{13} + 155712 \beta_{14} + 260736 \beta_{15} ) q^{73} + ( -693657663895425 + 91837307580 \beta_{1} - 23998401682 \beta_{2} + 204154386 \beta_{3} + 3152694311 \beta_{4} + 144840930 \beta_{5} - 700388475 \beta_{6} - 78923943 \beta_{7} + 2072592 \beta_{8} + 4124384 \beta_{9} + 2382976 \beta_{10} - 444425 \beta_{11} - 501424 \beta_{12} + 645824 \beta_{13} + 1365616 \beta_{14} + 1217424 \beta_{15} ) q^{74} + ( 372651855174254 + 1462965058581 \beta_{1} - 24534548843 \beta_{2} + 427576703 \beta_{3} - 2209039409 \beta_{4} + 189859801 \beta_{5} - 876158339 \beta_{6} - 4642655 \beta_{7} - 87197741 \beta_{8} - 2240308 \beta_{9} - 8918720 \beta_{10} + 2080102 \beta_{11} - 297958 \beta_{12} - 4390263 \beta_{13} - 578973 \beta_{14} + 1818019 \beta_{15} ) q^{76} + ( 2109664147689 + 16918283549310 \beta_{1} - 9346487274 \beta_{2} - 582765497 \beta_{3} + 16048461279 \beta_{4} + 1375184 \beta_{5} - 1040309481 \beta_{6} - 58083996 \beta_{7} + 53355251 \beta_{8} + 1350973 \beta_{9} - 10271696 \beta_{10} + 3428239 \beta_{11} - 1400707 \beta_{12} - 3391840 \beta_{13} + 964032 \beta_{14} - 386941 \beta_{15} ) q^{77} + ( -2833771879260356 - 20240486012964 \beta_{1} + 8593270894 \beta_{2} - 473196876 \beta_{3} + 29330955758 \beta_{4} - 65838146 \beta_{5} - 1215468488 \beta_{6} - 56036368 \beta_{7} - 14503420 \beta_{8} + 16380872 \beta_{9} + 17091948 \beta_{10} - 10968448 \beta_{11} - 271712 \beta_{12} - 1873360 \beta_{13} - 1898792 \beta_{14} + 1601648 \beta_{15} ) q^{79} + ( -84052437882756 - 3781986248024 \beta_{1} - 53694957320 \beta_{2} + 2560489688 \beta_{3} + 8910397772 \beta_{4} + 119340256 \beta_{5} + 1543363068 \beta_{6} + 28060000 \beta_{7} - 183117488 \beta_{8} + 11711888 \beta_{9} + 18221696 \beta_{10} - 4050136 \beta_{11} - 125576 \beta_{12} - 3735032 \beta_{13} + 2127600 \beta_{14} - 388544 \beta_{15} ) q^{80} + ( 977514619271850 + 331784237378 \beta_{1} + 58164739800 \beta_{2} + 8409067760 \beta_{3} + 14799900052 \beta_{4} - 159270712 \beta_{5} + 1452350976 \beta_{6} - 171220792 \beta_{7} + 30981264 \beta_{8} + 12953160 \beta_{9} - 27232236 \beta_{10} + 3896324 \beta_{11} - 295580 \beta_{12} + 9898120 \beta_{13} - 445708 \beta_{14} + 3362252 \beta_{15} ) q^{82} + ( -2892003499089 - 23036387900655 \beta_{1} - 40157164495 \beta_{2} - 7709736352 \beta_{3} + 41942566084 \beta_{4} - 6518176 \beta_{5} + 1318726164 \beta_{6} - 81955376 \beta_{7} + 198091656 \beta_{8} + 487448 \beta_{9} - 24994696 \beta_{10} + 19792424 \beta_{11} - 6081032 \beta_{12} + 1228920 \beta_{13} + 1262112 \beta_{14} + 774664 \beta_{15} ) q^{83} + ( -3773012607499 - 30421053497028 \beta_{1} - 19392554130 \beta_{2} - 3279718039 \beta_{3} - 73973068471 \beta_{4} + 57132000 \beta_{5} + 1818545643 \beta_{6} + 28565556 \beta_{7} + 158368943 \beta_{8} - 2990223 \beta_{9} + 38466080 \beta_{10} + 1657163 \beta_{11} - 1702159 \beta_{12} - 22530112 \beta_{13} - 4714880 \beta_{14} - 1724657 \beta_{15} ) q^{85} + ( 890724081037062 - 128517518583 \beta_{1} + 95352736479 \beta_{2} + 7118067939 \beta_{3} + 4349547870 \beta_{4} + 325325744 \beta_{5} - 1500395307 \beta_{6} - 253518192 \beta_{7} - 74816381 \beta_{8} + 17916704 \beta_{9} + 39398080 \beta_{10} - 1598048 \beta_{11} + 4961072 \beta_{12} - 2437056 \beta_{13} - 5639408 \beta_{14} - 7516176 \beta_{15} ) q^{86} + ( -4183745328663054 + 12657922110886 \beta_{1} + 102587065072 \beta_{2} - 10766469474 \beta_{3} + 19654633878 \beta_{4} + 586925448 \beta_{5} - 1791085868 \beta_{6} - 94385750 \beta_{7} - 424446512 \beta_{8} - 17122872 \beta_{9} - 26092400 \beta_{10} + 45068212 \beta_{11} - 1477348 \beta_{12} + 7676348 \beta_{13} - 2954384 \beta_{14} - 5371208 \beta_{15} ) q^{88} + ( 4364149120033191 - 26565280920616 \beta_{1} - 371616874 \beta_{2} + 1095058636 \beta_{3} - 55678138338 \beta_{4} + 1754984631 \beta_{5} - 2430218067 \beta_{6} - 234868102 \beta_{7} + 258410040 \beta_{8} - 9489827 \beta_{9} - 38533528 \beta_{10} + 50911808 \beta_{11} + 11060992 \beta_{12} + 10328800 \beta_{13} - 2405008 \beta_{14} + 732192 \beta_{15} ) q^{89} + ( 3664556551006 + 29211408027408 \beta_{1} + 35430563712 \beta_{2} - 28143153228 \beta_{3} - 66161970968 \beta_{4} + 17515394 \beta_{5} - 1484883204 \beta_{6} - 138253716 \beta_{7} - 344424938 \beta_{8} + 6971496 \beta_{9} + 14626952 \beta_{10} - 12967080 \beta_{11} + 4608648 \beta_{12} + 4200102 \beta_{13} + 7400928 \beta_{14} + 429432 \beta_{15} ) q^{91} + ( -3581919629112356 + 12686750425760 \beta_{1} + 44235934832 \beta_{2} - 8963379960 \beta_{3} - 46516526092 \beta_{4} + 911847384 \beta_{5} + 1073134420 \beta_{6} - 45627312 \beta_{7} + 686666040 \beta_{8} - 3426704 \beta_{9} + 13916528 \beta_{10} + 42582584 \beta_{11} - 569432 \beta_{12} - 25022096 \beta_{13} - 3989368 \beta_{14} + 10686168 \beta_{15} ) q^{92} + ( -12022645302608358 - 22024703115224 \beta_{1} - 44786531252 \beta_{2} - 5138129564 \beta_{3} - 96164065952 \beta_{4} - 911306656 \beta_{5} + 2170158322 \beta_{6} - 154047604 \beta_{7} - 14808460 \beta_{8} + 34607804 \beta_{9} - 8280598 \beta_{10} - 10833138 \beta_{11} + 4859894 \beta_{12} - 180348 \beta_{13} + 7000374 \beta_{14} - 2776118 \beta_{15} ) q^{94} + ( -5858543697122354 + 26914601278066 \beta_{1} - 5062131439 \beta_{2} - 298573172 \beta_{3} + 15910163725 \beta_{4} - 254398333 \beta_{5} + 1912558392 \beta_{6} - 325471344 \beta_{7} - 147049284 \beta_{8} + 47370104 \beta_{9} + 16579796 \beta_{10} - 27270400 \beta_{11} - 735008 \beta_{12} - 4667568 \beta_{13} - 10630680 \beta_{14} + 3932560 \beta_{15} ) q^{95} + ( 5978985756698985 + 35047586539676 \beta_{1} - 14405517366 \beta_{2} + 1119037974 \beta_{3} - 38524997792 \beta_{4} + 3061352281 \beta_{5} + 700870147 \beta_{6} + 86919746 \beta_{7} + 64301274 \beta_{8} - 6066055 \beta_{9} - 41314594 \beta_{10} + 87548912 \beta_{11} + 8948992 \beta_{12} + 16082984 \beta_{13} + 2241556 \beta_{14} - 7133992 \beta_{15} ) q^{97} + ( -21228188493297481 - 5257821678489 \beta_{1} + 105488565280 \beta_{2} + 21078533184 \beta_{3} - 148962477968 \beta_{4} - 625102752 \beta_{5} - 1446948992 \beta_{6} + 160339296 \beta_{7} + 41269184 \beta_{8} + 32263008 \beta_{9} - 57554960 \beta_{10} - 4296656 \beta_{11} + 975536 \beta_{12} + 5514336 \beta_{13} - 4609168 \beta_{14} + 7959952 \beta_{15} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 270q^{2} - 27436q^{4} + 11529600q^{7} - 24334920q^{8} + O(q^{10}) \) \( 16q - 270q^{2} - 27436q^{4} + 11529600q^{7} - 24334920q^{8} + 131002712q^{10} - 16363788528q^{14} + 26500434192q^{16} + 7489125600q^{17} + 209445719856q^{20} + 223126527100q^{22} - 746845345920q^{23} - 1809682431664q^{25} - 2467726531080q^{26} + 3220542267040q^{28} - 318979758592q^{31} - 1455647316000q^{32} - 4461251980292q^{34} - 24076283913900q^{38} + 60626292962592q^{40} - 7482251536032q^{41} - 193654716236040q^{44} - 195097141003568q^{46} + 376698804821760q^{47} + 127691292101520q^{49} - 474997408872102q^{50} - 272251877663120q^{52} + 2209036687713152q^{55} + 162767516076480q^{56} - 623262610679960q^{58} - 695695648144320q^{62} + 1111931745501248q^{64} - 2385987975356160q^{65} - 5981109959771880q^{68} - 10044559836180288q^{70} - 9025926285576576q^{71} + 11332002046118560q^{73} - 11098735408189464q^{74} + 5959440926938280q^{76} - 45299671392008448q^{79} - 1337342539452480q^{80} + 15639739637081420q^{82} + 14252032276026564q^{86} - 66964872768837680q^{88} + 69879174608766048q^{89} - 57336249810701280q^{92} - 192318922166254176q^{94} - 93790444358203776q^{95} + 95593398602180640q^{97} - 339641261743253790q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{15} + 109505575668 x^{14} - 766539029536 x^{13} + \)\(45\!\cdots\!58\)\( x^{12} - \)\(27\!\cdots\!44\)\( x^{11} + \)\(92\!\cdots\!84\)\( x^{10} - \)\(46\!\cdots\!88\)\( x^{9} + \)\(97\!\cdots\!41\)\( x^{8} - \)\(39\!\cdots\!08\)\( x^{7} + \)\(54\!\cdots\!12\)\( x^{6} - \)\(16\!\cdots\!68\)\( x^{5} + \)\(15\!\cdots\!52\)\( x^{4} - \)\(31\!\cdots\!24\)\( x^{3} + \)\(18\!\cdots\!52\)\( x^{2} - \)\(18\!\cdots\!92\)\( x + \)\(23\!\cdots\!64\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(81\!\cdots\!73\)\( \nu^{15} - \)\(34\!\cdots\!15\)\( \nu^{14} - \)\(83\!\cdots\!59\)\( \nu^{13} - \)\(35\!\cdots\!39\)\( \nu^{12} - \)\(31\!\cdots\!41\)\( \nu^{11} - \)\(13\!\cdots\!21\)\( \nu^{10} - \)\(53\!\cdots\!05\)\( \nu^{9} - \)\(23\!\cdots\!41\)\( \nu^{8} - \)\(41\!\cdots\!26\)\( \nu^{7} - \)\(18\!\cdots\!04\)\( \nu^{6} - \)\(14\!\cdots\!28\)\( \nu^{5} - \)\(66\!\cdots\!00\)\( \nu^{4} - \)\(18\!\cdots\!96\)\( \nu^{3} - \)\(91\!\cdots\!96\)\( \nu^{2} - \)\(42\!\cdots\!44\)\( \nu - \)\(11\!\cdots\!56\)\(\)\()/ \)\(66\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(54\!\cdots\!61\)\( \nu^{15} + \)\(87\!\cdots\!55\)\( \nu^{14} + \)\(56\!\cdots\!63\)\( \nu^{13} + \)\(89\!\cdots\!23\)\( \nu^{12} + \)\(21\!\cdots\!37\)\( \nu^{11} + \)\(33\!\cdots\!97\)\( \nu^{10} + \)\(36\!\cdots\!85\)\( \nu^{9} + \)\(57\!\cdots\!37\)\( \nu^{8} + \)\(28\!\cdots\!82\)\( \nu^{7} + \)\(45\!\cdots\!28\)\( \nu^{6} + \)\(10\!\cdots\!96\)\( \nu^{5} + \)\(16\!\cdots\!00\)\( \nu^{4} + \)\(14\!\cdots\!72\)\( \nu^{3} + \)\(22\!\cdots\!72\)\( \nu^{2} + \)\(11\!\cdots\!08\)\( \nu + \)\(29\!\cdots\!92\)\(\)\()/ \)\(66\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(11\!\cdots\!31\)\( \nu^{15} - \)\(96\!\cdots\!01\)\( \nu^{14} + \)\(12\!\cdots\!15\)\( \nu^{13} - \)\(99\!\cdots\!69\)\( \nu^{12} + \)\(46\!\cdots\!93\)\( \nu^{11} - \)\(37\!\cdots\!07\)\( \nu^{10} + \)\(79\!\cdots\!01\)\( \nu^{9} - \)\(63\!\cdots\!75\)\( \nu^{8} + \)\(63\!\cdots\!44\)\( \nu^{7} - \)\(50\!\cdots\!48\)\( \nu^{6} + \)\(23\!\cdots\!60\)\( \nu^{5} - \)\(17\!\cdots\!44\)\( \nu^{4} + \)\(32\!\cdots\!40\)\( \nu^{3} - \)\(24\!\cdots\!36\)\( \nu^{2} + \)\(54\!\cdots\!32\)\( \nu - \)\(29\!\cdots\!04\)\(\)\()/ \)\(13\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(64\!\cdots\!49\)\( \nu^{15} + \)\(26\!\cdots\!05\)\( \nu^{14} - \)\(66\!\cdots\!67\)\( \nu^{13} + \)\(27\!\cdots\!93\)\( \nu^{12} - \)\(24\!\cdots\!33\)\( \nu^{11} + \)\(10\!\cdots\!27\)\( \nu^{10} - \)\(43\!\cdots\!65\)\( \nu^{9} + \)\(17\!\cdots\!67\)\( \nu^{8} - \)\(34\!\cdots\!38\)\( \nu^{7} + \)\(13\!\cdots\!48\)\( \nu^{6} - \)\(12\!\cdots\!64\)\( \nu^{5} + \)\(47\!\cdots\!00\)\( \nu^{4} - \)\(17\!\cdots\!48\)\( \nu^{3} + \)\(63\!\cdots\!52\)\( \nu^{2} - \)\(22\!\cdots\!72\)\( \nu + \)\(74\!\cdots\!72\)\(\)\()/ \)\(66\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(69\!\cdots\!99\)\( \nu^{15} + \)\(35\!\cdots\!95\)\( \nu^{14} + \)\(71\!\cdots\!67\)\( \nu^{13} + \)\(35\!\cdots\!07\)\( \nu^{12} + \)\(26\!\cdots\!33\)\( \nu^{11} + \)\(13\!\cdots\!73\)\( \nu^{10} + \)\(46\!\cdots\!65\)\( \nu^{9} + \)\(23\!\cdots\!33\)\( \nu^{8} + \)\(36\!\cdots\!88\)\( \nu^{7} + \)\(18\!\cdots\!52\)\( \nu^{6} + \)\(13\!\cdots\!64\)\( \nu^{5} + \)\(65\!\cdots\!00\)\( \nu^{4} + \)\(17\!\cdots\!48\)\( \nu^{3} + \)\(89\!\cdots\!48\)\( \nu^{2} + \)\(14\!\cdots\!72\)\( \nu + \)\(11\!\cdots\!28\)\(\)\()/ \)\(33\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(64\!\cdots\!79\)\( \nu^{15} - \)\(23\!\cdots\!31\)\( \nu^{14} + \)\(65\!\cdots\!89\)\( \nu^{13} - \)\(23\!\cdots\!95\)\( \nu^{12} + \)\(24\!\cdots\!79\)\( \nu^{11} - \)\(89\!\cdots\!41\)\( \nu^{10} + \)\(41\!\cdots\!51\)\( \nu^{9} - \)\(15\!\cdots\!49\)\( \nu^{8} + \)\(32\!\cdots\!10\)\( \nu^{7} - \)\(12\!\cdots\!64\)\( \nu^{6} + \)\(11\!\cdots\!68\)\( \nu^{5} - \)\(44\!\cdots\!24\)\( \nu^{4} + \)\(15\!\cdots\!96\)\( \nu^{3} - \)\(60\!\cdots\!00\)\( \nu^{2} + \)\(55\!\cdots\!56\)\( \nu - \)\(79\!\cdots\!68\)\(\)\()/ \)\(26\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(30\!\cdots\!41\)\( \nu^{15} + \)\(30\!\cdots\!30\)\( \nu^{14} + \)\(31\!\cdots\!28\)\( \nu^{13} + \)\(31\!\cdots\!88\)\( \nu^{12} + \)\(11\!\cdots\!22\)\( \nu^{11} + \)\(11\!\cdots\!32\)\( \nu^{10} + \)\(20\!\cdots\!60\)\( \nu^{9} + \)\(20\!\cdots\!72\)\( \nu^{8} + \)\(16\!\cdots\!17\)\( \nu^{7} + \)\(15\!\cdots\!18\)\( \nu^{6} + \)\(58\!\cdots\!76\)\( \nu^{5} + \)\(56\!\cdots\!00\)\( \nu^{4} + \)\(81\!\cdots\!32\)\( \nu^{3} + \)\(78\!\cdots\!32\)\( \nu^{2} + \)\(13\!\cdots\!48\)\( \nu + \)\(10\!\cdots\!52\)\(\)\()/ \)\(82\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(35\!\cdots\!99\)\( \nu^{15} - \)\(73\!\cdots\!55\)\( \nu^{14} + \)\(36\!\cdots\!17\)\( \nu^{13} - \)\(75\!\cdots\!43\)\( \nu^{12} + \)\(13\!\cdots\!83\)\( \nu^{11} - \)\(28\!\cdots\!77\)\( \nu^{10} + \)\(23\!\cdots\!15\)\( \nu^{9} - \)\(48\!\cdots\!17\)\( \nu^{8} + \)\(18\!\cdots\!38\)\( \nu^{7} - \)\(38\!\cdots\!48\)\( \nu^{6} + \)\(64\!\cdots\!64\)\( \nu^{5} - \)\(13\!\cdots\!00\)\( \nu^{4} + \)\(86\!\cdots\!48\)\( \nu^{3} - \)\(19\!\cdots\!52\)\( \nu^{2} + \)\(10\!\cdots\!72\)\( \nu - \)\(28\!\cdots\!72\)\(\)\()/ \)\(66\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(66\!\cdots\!17\)\( \nu^{15} - \)\(33\!\cdots\!35\)\( \nu^{14} - \)\(67\!\cdots\!11\)\( \nu^{13} - \)\(34\!\cdots\!31\)\( \nu^{12} - \)\(25\!\cdots\!89\)\( \nu^{11} - \)\(12\!\cdots\!09\)\( \nu^{10} - \)\(43\!\cdots\!45\)\( \nu^{9} - \)\(22\!\cdots\!89\)\( \nu^{8} - \)\(34\!\cdots\!54\)\( \nu^{7} - \)\(17\!\cdots\!16\)\( \nu^{6} - \)\(12\!\cdots\!12\)\( \nu^{5} - \)\(61\!\cdots\!00\)\( \nu^{4} - \)\(17\!\cdots\!84\)\( \nu^{3} - \)\(83\!\cdots\!84\)\( \nu^{2} - \)\(28\!\cdots\!76\)\( \nu - \)\(10\!\cdots\!24\)\(\)\()/ \)\(66\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(33\!\cdots\!57\)\( \nu^{15} + \)\(79\!\cdots\!55\)\( \nu^{14} + \)\(34\!\cdots\!91\)\( \nu^{13} + \)\(81\!\cdots\!91\)\( \nu^{12} + \)\(13\!\cdots\!49\)\( \nu^{11} + \)\(30\!\cdots\!69\)\( \nu^{10} + \)\(22\!\cdots\!25\)\( \nu^{9} + \)\(51\!\cdots\!09\)\( \nu^{8} + \)\(17\!\cdots\!94\)\( \nu^{7} + \)\(40\!\cdots\!56\)\( \nu^{6} + \)\(65\!\cdots\!72\)\( \nu^{5} + \)\(14\!\cdots\!80\)\( \nu^{4} + \)\(89\!\cdots\!04\)\( \nu^{3} + \)\(18\!\cdots\!24\)\( \nu^{2} + \)\(11\!\cdots\!16\)\( \nu + \)\(21\!\cdots\!24\)\(\)\()/ \)\(13\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(12\!\cdots\!13\)\( \nu^{15} + \)\(28\!\cdots\!85\)\( \nu^{14} - \)\(12\!\cdots\!79\)\( \nu^{13} + \)\(29\!\cdots\!41\)\( \nu^{12} - \)\(47\!\cdots\!21\)\( \nu^{11} + \)\(11\!\cdots\!99\)\( \nu^{10} - \)\(80\!\cdots\!05\)\( \nu^{9} + \)\(18\!\cdots\!79\)\( \nu^{8} - \)\(64\!\cdots\!06\)\( \nu^{7} + \)\(14\!\cdots\!76\)\( \nu^{6} - \)\(23\!\cdots\!68\)\( \nu^{5} + \)\(52\!\cdots\!00\)\( \nu^{4} - \)\(32\!\cdots\!76\)\( \nu^{3} + \)\(71\!\cdots\!24\)\( \nu^{2} - \)\(41\!\cdots\!64\)\( \nu + \)\(91\!\cdots\!64\)\(\)\()/ \)\(33\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(22\!\cdots\!93\)\( \nu^{15} - \)\(24\!\cdots\!35\)\( \nu^{14} + \)\(23\!\cdots\!69\)\( \nu^{13} - \)\(25\!\cdots\!51\)\( \nu^{12} + \)\(89\!\cdots\!31\)\( \nu^{11} - \)\(94\!\cdots\!89\)\( \nu^{10} + \)\(15\!\cdots\!55\)\( \nu^{9} - \)\(16\!\cdots\!69\)\( \nu^{8} + \)\(13\!\cdots\!16\)\( \nu^{7} - \)\(12\!\cdots\!36\)\( \nu^{6} + \)\(51\!\cdots\!48\)\( \nu^{5} - \)\(46\!\cdots\!00\)\( \nu^{4} + \)\(79\!\cdots\!36\)\( \nu^{3} - \)\(65\!\cdots\!64\)\( \nu^{2} + \)\(17\!\cdots\!04\)\( \nu - \)\(98\!\cdots\!04\)\(\)\()/ \)\(55\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(97\!\cdots\!61\)\( \nu^{15} - \)\(10\!\cdots\!45\)\( \nu^{14} + \)\(10\!\cdots\!63\)\( \nu^{13} - \)\(11\!\cdots\!77\)\( \nu^{12} + \)\(37\!\cdots\!37\)\( \nu^{11} - \)\(41\!\cdots\!03\)\( \nu^{10} + \)\(64\!\cdots\!85\)\( \nu^{9} - \)\(70\!\cdots\!63\)\( \nu^{8} + \)\(50\!\cdots\!82\)\( \nu^{7} - \)\(55\!\cdots\!72\)\( \nu^{6} + \)\(18\!\cdots\!96\)\( \nu^{5} - \)\(20\!\cdots\!00\)\( \nu^{4} + \)\(24\!\cdots\!72\)\( \nu^{3} - \)\(27\!\cdots\!28\)\( \nu^{2} + \)\(21\!\cdots\!08\)\( \nu - \)\(34\!\cdots\!08\)\(\)\()/ \)\(13\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(13\!\cdots\!09\)\( \nu^{15} + \)\(81\!\cdots\!55\)\( \nu^{14} - \)\(14\!\cdots\!97\)\( \nu^{13} + \)\(83\!\cdots\!63\)\( \nu^{12} - \)\(53\!\cdots\!03\)\( \nu^{11} + \)\(31\!\cdots\!57\)\( \nu^{10} - \)\(90\!\cdots\!15\)\( \nu^{9} + \)\(53\!\cdots\!97\)\( \nu^{8} - \)\(71\!\cdots\!08\)\( \nu^{7} + \)\(42\!\cdots\!68\)\( \nu^{6} - \)\(25\!\cdots\!24\)\( \nu^{5} + \)\(15\!\cdots\!00\)\( \nu^{4} - \)\(35\!\cdots\!68\)\( \nu^{3} + \)\(20\!\cdots\!32\)\( \nu^{2} - \)\(52\!\cdots\!52\)\( \nu + \)\(26\!\cdots\!52\)\(\)\()/ \)\(57\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(30\!\cdots\!73\)\( \nu^{15} - \)\(20\!\cdots\!65\)\( \nu^{14} - \)\(31\!\cdots\!09\)\( \nu^{13} - \)\(21\!\cdots\!89\)\( \nu^{12} - \)\(11\!\cdots\!91\)\( \nu^{11} - \)\(79\!\cdots\!71\)\( \nu^{10} - \)\(19\!\cdots\!55\)\( \nu^{9} - \)\(13\!\cdots\!91\)\( \nu^{8} - \)\(15\!\cdots\!76\)\( \nu^{7} - \)\(10\!\cdots\!04\)\( \nu^{6} - \)\(56\!\cdots\!28\)\( \nu^{5} - \)\(38\!\cdots\!00\)\( \nu^{4} - \)\(77\!\cdots\!96\)\( \nu^{3} - \)\(52\!\cdots\!96\)\( \nu^{2} - \)\(98\!\cdots\!44\)\( \nu - \)\(71\!\cdots\!56\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3} - 63 \beta_{1} - 4\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(-28 \beta_{15} - 50 \beta_{14} - 100 \beta_{13} - 128 \beta_{12} - 472 \beta_{11} + 469 \beta_{10} + 59 \beta_{9} - 5409 \beta_{8} + 302 \beta_{7} + 7190 \beta_{6} + 1426 \beta_{5} + 939697 \beta_{4} - 13965 \beta_{3} + 78772 \beta_{2} + 120660330 \beta_{1} - 438007585166\)\()/32\)
\(\nu^{3}\)\(=\)\((\)\(-42474661 \beta_{15} + 20432520 \beta_{14} - 184624480 \beta_{13} - 63610411 \beta_{12} + 105872311 \beta_{11} + 347957908 \beta_{10} + 62908153 \beta_{9} + 2941301207 \beta_{8} - 2617977396 \beta_{7} - 22055714325 \beta_{6} + 912141632 \beta_{5} - 1451220771173 \beta_{4} - 817921184497 \beta_{3} + 408016786774 \beta_{2} + 432824221422466 \beta_{1} + 49133479861965\)\()/256\)
\(\nu^{4}\)\(=\)\((\)\(6048673051210 \beta_{15} + 4859492388390 \beta_{14} + 35378369834860 \beta_{13} + 41427369863530 \beta_{12} + 170845149163190 \beta_{11} - 151850223021735 \beta_{10} - 52624756411900 \beta_{9} + 1396604145331745 \beta_{8} - 498565002554220 \beta_{7} - 2745748443167465 \beta_{6} - 1311732056891285 \beta_{5} - 302521585339437695 \beta_{4} + 4336516824889165 \beta_{3} - 21101825920651510 \beta_{2} - 54983169097626840550 \beta_{1} + 91553904346171859917571\)\()/256\)
\(\nu^{5}\)\(=\)\((\)\(1825482719363627590 \beta_{15} - 1071232144095969610 \beta_{14} + 17928044680434676760 \beta_{13} + 7499923329433990910 \beta_{12} - 18847389400011428750 \beta_{11} - 39396959093493832025 \beta_{10} - 2896972041338345840 \beta_{9} - 264449608778724706965 \beta_{8} + 190330708182173662620 \beta_{7} + 2025943198288988401060 \beta_{6} - 88519901621671572875 \beta_{5} + 125316761168118315984814 \beta_{4} + 51140848805501836582339 \beta_{3} - 75038496113875744821510 \beta_{2} - 39095118292451157957931912 \beta_{1} - 4453735606295800184238956\)\()/512\)
\(\nu^{6}\)\(=\)\((\)\(-188656975235971238906804 \beta_{15} + 94366522190443340063110 \beta_{14} - 1422436224685513835947970 \beta_{13} - 1611111580327591438768584 \beta_{12} - 6923684141386727105210736 \beta_{11} + 5016386600184614452173017 \beta_{10} + 2605198761761624143576062 \beta_{9} - 45496612998520401910481307 \beta_{8} + 29348774358975388641918776 \beta_{7} + 122531223310077234929260000 \beta_{6} + 74082565490679019869489253 \beta_{5} + 9863989881813033577732388346 \beta_{4} - 136204756990876846014239125 \beta_{3} + 470706925498912515779831666 \beta_{2} + 2621679319745666943910097819760 \beta_{1} - 2917901455343091856113477561408258\)\()/256\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(27\!\cdots\!94\)\( \beta_{15} + \)\(39\!\cdots\!50\)\( \beta_{14} - \)\(14\!\cdots\!00\)\( \beta_{13} - \)\(70\!\cdots\!94\)\( \beta_{12} + \)\(20\!\cdots\!74\)\( \beta_{11} + \)\(36\!\cdots\!37\)\( \beta_{10} + \)\(67\!\cdots\!32\)\( \beta_{9} + \)\(21\!\cdots\!93\)\( \beta_{8} - \)\(12\!\cdots\!84\)\( \beta_{7} - \)\(16\!\cdots\!80\)\( \beta_{6} + \)\(74\!\cdots\!63\)\( \beta_{5} - \)\(10\!\cdots\!42\)\( \beta_{4} - \)\(34\!\cdots\!03\)\( \beta_{3} + \)\(79\!\cdots\!46\)\( \beta_{2} + \)\(32\!\cdots\!44\)\( \beta_{1} + \)\(36\!\cdots\!20\)\(\)\()/1024\)
\(\nu^{8}\)\(=\)\((\)\(\)\(15\!\cdots\!80\)\( \beta_{15} - \)\(26\!\cdots\!80\)\( \beta_{14} + \)\(14\!\cdots\!80\)\( \beta_{13} + \)\(15\!\cdots\!40\)\( \beta_{12} + \)\(69\!\cdots\!20\)\( \beta_{11} - \)\(41\!\cdots\!55\)\( \beta_{10} - \)\(28\!\cdots\!00\)\( \beta_{9} + \)\(38\!\cdots\!35\)\( \beta_{8} - \)\(34\!\cdots\!60\)\( \beta_{7} - \)\(12\!\cdots\!45\)\( \beta_{6} - \)\(87\!\cdots\!55\)\( \beta_{5} - \)\(81\!\cdots\!85\)\( \beta_{4} + \)\(10\!\cdots\!95\)\( \beta_{3} - \)\(20\!\cdots\!30\)\( \beta_{2} - \)\(28\!\cdots\!50\)\( \beta_{1} + \)\(25\!\cdots\!59\)\(\)\()/64\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(25\!\cdots\!55\)\( \beta_{15} - \)\(48\!\cdots\!30\)\( \beta_{14} + \)\(72\!\cdots\!80\)\( \beta_{13} + \)\(38\!\cdots\!55\)\( \beta_{12} - \)\(12\!\cdots\!75\)\( \beta_{11} - \)\(19\!\cdots\!25\)\( \beta_{10} + \)\(20\!\cdots\!05\)\( \beta_{9} - \)\(10\!\cdots\!70\)\( \beta_{8} + \)\(52\!\cdots\!60\)\( \beta_{7} + \)\(87\!\cdots\!55\)\( \beta_{6} - \)\(37\!\cdots\!75\)\( \beta_{5} + \)\(53\!\cdots\!41\)\( \beta_{4} + \)\(15\!\cdots\!66\)\( \beta_{3} - \)\(45\!\cdots\!80\)\( \beta_{2} - \)\(16\!\cdots\!98\)\( \beta_{1} - \)\(18\!\cdots\!39\)\(\)\()/128\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(20\!\cdots\!84\)\( \beta_{15} + \)\(58\!\cdots\!70\)\( \beta_{14} - \)\(22\!\cdots\!90\)\( \beta_{13} - \)\(24\!\cdots\!44\)\( \beta_{12} - \)\(11\!\cdots\!96\)\( \beta_{11} + \)\(58\!\cdots\!07\)\( \beta_{10} + \)\(46\!\cdots\!02\)\( \beta_{9} - \)\(55\!\cdots\!17\)\( \beta_{8} + \)\(59\!\cdots\!36\)\( \beta_{7} + \)\(21\!\cdots\!80\)\( \beta_{6} + \)\(15\!\cdots\!23\)\( \beta_{5} + \)\(11\!\cdots\!66\)\( \beta_{4} - \)\(14\!\cdots\!55\)\( \beta_{3} + \)\(47\!\cdots\!06\)\( \beta_{2} + \)\(48\!\cdots\!80\)\( \beta_{1} - \)\(36\!\cdots\!38\)\(\)\()/256\)
\(\nu^{11}\)\(=\)\((\)\(\)\(15\!\cdots\!06\)\( \beta_{15} - \)\(13\!\cdots\!30\)\( \beta_{14} - \)\(22\!\cdots\!80\)\( \beta_{13} - \)\(12\!\cdots\!94\)\( \beta_{12} + \)\(41\!\cdots\!54\)\( \beta_{11} + \)\(65\!\cdots\!67\)\( \beta_{10} - \)\(16\!\cdots\!48\)\( \beta_{9} + \)\(33\!\cdots\!83\)\( \beta_{8} - \)\(14\!\cdots\!84\)\( \beta_{7} - \)\(28\!\cdots\!60\)\( \beta_{6} + \)\(12\!\cdots\!73\)\( \beta_{5} - \)\(17\!\cdots\!42\)\( \beta_{4} - \)\(46\!\cdots\!93\)\( \beta_{3} + \)\(15\!\cdots\!46\)\( \beta_{2} + \)\(54\!\cdots\!24\)\( \beta_{1} + \)\(59\!\cdots\!80\)\(\)\()/1024\)
\(\nu^{12}\)\(=\)\((\)\(\)\(34\!\cdots\!90\)\( \beta_{15} - \)\(13\!\cdots\!40\)\( \beta_{14} + \)\(45\!\cdots\!90\)\( \beta_{13} + \)\(48\!\cdots\!70\)\( \beta_{12} + \)\(22\!\cdots\!10\)\( \beta_{11} - \)\(10\!\cdots\!15\)\( \beta_{10} - \)\(93\!\cdots\!50\)\( \beta_{9} + \)\(10\!\cdots\!55\)\( \beta_{8} - \)\(12\!\cdots\!80\)\( \beta_{7} - \)\(43\!\cdots\!60\)\( \beta_{6} - \)\(32\!\cdots\!65\)\( \beta_{5} - \)\(19\!\cdots\!80\)\( \beta_{4} + \)\(24\!\cdots\!85\)\( \beta_{3} + \)\(24\!\cdots\!10\)\( \beta_{2} - \)\(98\!\cdots\!00\)\( \beta_{1} + \)\(68\!\cdots\!38\)\(\)\()/128\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(40\!\cdots\!30\)\( \beta_{15} + \)\(61\!\cdots\!70\)\( \beta_{14} + \)\(44\!\cdots\!80\)\( \beta_{13} + \)\(26\!\cdots\!30\)\( \beta_{12} - \)\(86\!\cdots\!50\)\( \beta_{11} - \)\(13\!\cdots\!25\)\( \beta_{10} + \)\(46\!\cdots\!80\)\( \beta_{9} - \)\(65\!\cdots\!45\)\( \beta_{8} + \)\(26\!\cdots\!60\)\( \beta_{7} + \)\(57\!\cdots\!80\)\( \beta_{6} - \)\(24\!\cdots\!75\)\( \beta_{5} + \)\(34\!\cdots\!14\)\( \beta_{4} + \)\(87\!\cdots\!39\)\( \beta_{3} - \)\(32\!\cdots\!30\)\( \beta_{2} - \)\(10\!\cdots\!72\)\( \beta_{1} - \)\(11\!\cdots\!56\)\(\)\()/512\)
\(\nu^{14}\)\(=\)\((\)\(-\)\(24\!\cdots\!64\)\( \beta_{15} + \)\(11\!\cdots\!30\)\( \beta_{14} - \)\(36\!\cdots\!10\)\( \beta_{13} - \)\(38\!\cdots\!04\)\( \beta_{12} - \)\(17\!\cdots\!56\)\( \beta_{11} + \)\(77\!\cdots\!97\)\( \beta_{10} + \)\(74\!\cdots\!42\)\( \beta_{9} - \)\(76\!\cdots\!27\)\( \beta_{8} + \)\(10\!\cdots\!96\)\( \beta_{7} + \)\(34\!\cdots\!60\)\( \beta_{6} + \)\(26\!\cdots\!93\)\( \beta_{5} + \)\(14\!\cdots\!86\)\( \beta_{4} - \)\(17\!\cdots\!85\)\( \beta_{3} - \)\(35\!\cdots\!54\)\( \beta_{2} + \)\(79\!\cdots\!00\)\( \beta_{1} - \)\(52\!\cdots\!18\)\(\)\()/256\)
\(\nu^{15}\)\(=\)\((\)\(\)\(36\!\cdots\!06\)\( \beta_{15} - \)\(65\!\cdots\!10\)\( \beta_{14} - \)\(35\!\cdots\!60\)\( \beta_{13} - \)\(21\!\cdots\!94\)\( \beta_{12} + \)\(70\!\cdots\!34\)\( \beta_{11} + \)\(10\!\cdots\!97\)\( \beta_{10} - \)\(43\!\cdots\!28\)\( \beta_{9} + \)\(52\!\cdots\!73\)\( \beta_{8} - \)\(19\!\cdots\!84\)\( \beta_{7} - \)\(45\!\cdots\!40\)\( \beta_{6} + \)\(19\!\cdots\!83\)\( \beta_{5} - \)\(27\!\cdots\!42\)\( \beta_{4} - \)\(67\!\cdots\!83\)\( \beta_{3} + \)\(26\!\cdots\!46\)\( \beta_{2} + \)\(87\!\cdots\!04\)\( \beta_{1} + \)\(94\!\cdots\!40\)\(\)\()/1024\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 + 58156.1i
0.500000 58156.1i
0.500000 + 82996.3i
0.500000 82996.3i
0.500000 151568.i
0.500000 + 151568.i
0.500000 + 159067.i
0.500000 159067.i
0.500000 + 12044.5i
0.500000 12044.5i
0.500000 198996.i
0.500000 + 198996.i
0.500000 + 65608.9i
0.500000 65608.9i
0.500000 + 83132.4i
0.500000 83132.4i
−351.815 85.4288i 0 116476. + 60110.3i 465248.i 0 2.24795e7 −3.58428e7 3.10981e7i 0 −3.97456e7 + 1.63681e8i
37.2 −351.815 + 85.4288i 0 116476. 60110.3i 465248.i 0 2.24795e7 −3.58428e7 + 3.10981e7i 0 −3.97456e7 1.63681e8i
37.3 −328.641 151.878i 0 84938.1 + 99826.8i 663971.i 0 −1.66742e7 −1.27527e7 4.57074e7i 0 −1.00843e8 + 2.18208e8i
37.4 −328.641 + 151.878i 0 84938.1 99826.8i 663971.i 0 −1.66742e7 −1.27527e7 + 4.57074e7i 0 −1.00843e8 2.18208e8i
37.5 −214.125 291.929i 0 −39373.4 + 125018.i 1.21254e6i 0 1.76580e7 4.49273e7 1.52753e7i 0 3.53976e8 2.59635e8i
37.6 −214.125 + 291.929i 0 −39373.4 125018.i 1.21254e6i 0 1.76580e7 4.49273e7 + 1.52753e7i 0 3.53976e8 + 2.59635e8i
37.7 −42.2945 359.560i 0 −127494. + 30414.8i 1.27253e6i 0 5.50569e6 1.63282e7 + 4.45555e7i 0 −4.57551e8 + 5.38211e7i
37.8 −42.2945 + 359.560i 0 −127494. 30414.8i 1.27253e6i 0 5.50569e6 1.63282e7 4.45555e7i 0 −4.57551e8 5.38211e7i
37.9 −18.3339 361.574i 0 −130400. + 13258.2i 96356.3i 0 −1.47728e7 7.18455e6 + 4.69061e7i 0 −3.48399e7 + 1.76659e6i
37.10 −18.3339 + 361.574i 0 −130400. 13258.2i 96356.3i 0 −1.47728e7 7.18455e6 4.69061e7i 0 −3.48399e7 1.76659e6i
37.11 200.394 301.520i 0 −50756.5 120846.i 1.59197e6i 0 −1.66055e7 −4.66086e7 8.91263e6i 0 4.80011e8 + 3.19021e8i
37.12 200.394 + 301.520i 0 −50756.5 + 120846.i 1.59197e6i 0 −1.66055e7 −4.66086e7 + 8.91263e6i 0 4.80011e8 3.19021e8i
37.13 257.790 254.197i 0 1839.67 131059.i 524871.i 0 1.57495e7 −3.28406e7 3.42534e7i 0 −1.33421e8 1.35307e8i
37.14 257.790 + 254.197i 0 1839.67 + 131059.i 524871.i 0 1.57495e7 −3.28406e7 + 3.42534e7i 0 −1.33421e8 + 1.35307e8i
37.15 362.025 3.13586i 0 131052. 2270.52i 665059.i 0 −7.57536e6 4.74371e7 1.23295e6i 0 −2.08553e6 2.40768e8i
37.16 362.025 + 3.13586i 0 131052. + 2270.52i 665059.i 0 −7.57536e6 4.74371e7 + 1.23295e6i 0 −2.08553e6 + 2.40768e8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.18.d.b 16
3.b odd 2 1 8.18.b.a 16
8.b even 2 1 inner 72.18.d.b 16
12.b even 2 1 32.18.b.a 16
24.f even 2 1 32.18.b.a 16
24.h odd 2 1 8.18.b.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.18.b.a 16 3.b odd 2 1
8.18.b.a 16 24.h odd 2 1
32.18.b.a 16 12.b even 2 1
32.18.b.a 16 24.f even 2 1
72.18.d.b 16 1.a even 1 1 trivial
72.18.d.b 16 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{16} + \cdots\) acting on \(S_{18}^{\mathrm{new}}(72, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 270 T + 50168 T^{2} + 15096000 T^{3} - 3619419136 T^{4} - 1008340992000 T^{5} - 342095875276800 T^{6} - 981059695269642240 T^{7} - \)\(42\!\cdots\!80\)\( T^{8} - \)\(12\!\cdots\!80\)\( T^{9} - \)\(58\!\cdots\!00\)\( T^{10} - \)\(22\!\cdots\!00\)\( T^{11} - \)\(10\!\cdots\!16\)\( T^{12} + \)\(58\!\cdots\!00\)\( T^{13} + \)\(25\!\cdots\!72\)\( T^{14} + \)\(17\!\cdots\!60\)\( T^{15} + \)\(87\!\cdots\!36\)\( T^{16} \)
$3$ 1
$5$ \( 1 - 5198674409168 T^{2} + \)\(13\!\cdots\!56\)\( T^{4} - \)\(24\!\cdots\!00\)\( T^{6} + \)\(33\!\cdots\!00\)\( T^{8} - \)\(37\!\cdots\!00\)\( T^{10} + \)\(36\!\cdots\!00\)\( T^{12} - \)\(31\!\cdots\!00\)\( T^{14} + \)\(25\!\cdots\!50\)\( T^{16} - \)\(18\!\cdots\!00\)\( T^{18} + \)\(12\!\cdots\!00\)\( T^{20} - \)\(73\!\cdots\!00\)\( T^{22} + \)\(38\!\cdots\!00\)\( T^{24} - \)\(16\!\cdots\!00\)\( T^{26} + \)\(53\!\cdots\!00\)\( T^{28} - \)\(11\!\cdots\!00\)\( T^{30} + \)\(13\!\cdots\!25\)\( T^{32} \)
$7$ \( ( 1 - 5764800 T + 915215692443448 T^{2} - \)\(65\!\cdots\!20\)\( T^{3} + \)\(50\!\cdots\!24\)\( T^{4} - \)\(34\!\cdots\!80\)\( T^{5} + \)\(18\!\cdots\!60\)\( T^{6} - \)\(12\!\cdots\!60\)\( T^{7} + \)\(50\!\cdots\!90\)\( T^{8} - \)\(27\!\cdots\!20\)\( T^{9} + \)\(10\!\cdots\!40\)\( T^{10} - \)\(44\!\cdots\!40\)\( T^{11} + \)\(14\!\cdots\!24\)\( T^{12} - \)\(44\!\cdots\!40\)\( T^{13} + \)\(14\!\cdots\!52\)\( T^{14} - \)\(21\!\cdots\!00\)\( T^{15} + \)\(85\!\cdots\!01\)\( T^{16} )^{2} \)
$11$ \( 1 - 3877608573076448976 T^{2} + \)\(79\!\cdots\!84\)\( T^{4} - \)\(11\!\cdots\!04\)\( T^{6} + \)\(12\!\cdots\!88\)\( T^{8} - \)\(11\!\cdots\!72\)\( T^{10} + \)\(81\!\cdots\!96\)\( T^{12} - \)\(51\!\cdots\!28\)\( T^{14} + \)\(28\!\cdots\!22\)\( T^{16} - \)\(13\!\cdots\!48\)\( T^{18} + \)\(53\!\cdots\!76\)\( T^{20} - \)\(18\!\cdots\!12\)\( T^{22} + \)\(52\!\cdots\!68\)\( T^{24} - \)\(12\!\cdots\!04\)\( T^{26} + \)\(22\!\cdots\!44\)\( T^{28} - \)\(27\!\cdots\!56\)\( T^{30} + \)\(18\!\cdots\!21\)\( T^{32} \)
$13$ \( 1 - 63371249746529137488 T^{2} + \)\(20\!\cdots\!16\)\( T^{4} - \)\(48\!\cdots\!88\)\( T^{6} + \)\(85\!\cdots\!68\)\( T^{8} - \)\(12\!\cdots\!16\)\( T^{10} + \)\(15\!\cdots\!64\)\( T^{12} - \)\(16\!\cdots\!56\)\( T^{14} + \)\(15\!\cdots\!62\)\( T^{16} - \)\(12\!\cdots\!84\)\( T^{18} + \)\(85\!\cdots\!44\)\( T^{20} - \)\(52\!\cdots\!04\)\( T^{22} + \)\(26\!\cdots\!88\)\( T^{24} - \)\(11\!\cdots\!12\)\( T^{26} + \)\(36\!\cdots\!76\)\( T^{28} - \)\(83\!\cdots\!52\)\( T^{30} + \)\(98\!\cdots\!81\)\( T^{32} \)
$17$ \( ( 1 - 3744562800 T + \)\(34\!\cdots\!44\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!60\)\( T^{4} + \)\(36\!\cdots\!00\)\( T^{5} + \)\(81\!\cdots\!48\)\( T^{6} + \)\(51\!\cdots\!00\)\( T^{7} + \)\(77\!\cdots\!98\)\( T^{8} + \)\(42\!\cdots\!00\)\( T^{9} + \)\(55\!\cdots\!92\)\( T^{10} + \)\(20\!\cdots\!00\)\( T^{11} + \)\(29\!\cdots\!60\)\( T^{12} + \)\(39\!\cdots\!00\)\( T^{13} + \)\(11\!\cdots\!16\)\( T^{14} - \)\(99\!\cdots\!00\)\( T^{15} + \)\(21\!\cdots\!81\)\( T^{16} )^{2} \)
$19$ \( 1 - \)\(50\!\cdots\!16\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{4} - \)\(21\!\cdots\!48\)\( T^{6} + \)\(27\!\cdots\!00\)\( T^{8} - \)\(27\!\cdots\!68\)\( T^{10} + \)\(22\!\cdots\!64\)\( T^{12} - \)\(15\!\cdots\!44\)\( T^{14} + \)\(94\!\cdots\!82\)\( T^{16} - \)\(47\!\cdots\!24\)\( T^{18} + \)\(20\!\cdots\!24\)\( T^{20} - \)\(74\!\cdots\!48\)\( T^{22} + \)\(22\!\cdots\!00\)\( T^{24} - \)\(52\!\cdots\!48\)\( T^{26} + \)\(94\!\cdots\!60\)\( T^{28} - \)\(11\!\cdots\!56\)\( T^{30} + \)\(66\!\cdots\!61\)\( T^{32} \)
$23$ \( ( 1 + 373422672960 T + \)\(64\!\cdots\!76\)\( T^{2} + \)\(17\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!52\)\( T^{4} + \)\(41\!\cdots\!20\)\( T^{5} + \)\(40\!\cdots\!92\)\( T^{6} + \)\(69\!\cdots\!60\)\( T^{7} + \)\(63\!\cdots\!74\)\( T^{8} + \)\(98\!\cdots\!80\)\( T^{9} + \)\(79\!\cdots\!28\)\( T^{10} + \)\(11\!\cdots\!40\)\( T^{11} + \)\(77\!\cdots\!12\)\( T^{12} + \)\(96\!\cdots\!40\)\( T^{13} + \)\(50\!\cdots\!04\)\( T^{14} + \)\(41\!\cdots\!20\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} )^{2} \)
$29$ \( 1 - \)\(62\!\cdots\!20\)\( T^{2} + \)\(20\!\cdots\!80\)\( T^{4} - \)\(44\!\cdots\!20\)\( T^{6} + \)\(75\!\cdots\!16\)\( T^{8} - \)\(10\!\cdots\!80\)\( T^{10} + \)\(11\!\cdots\!20\)\( T^{12} - \)\(10\!\cdots\!80\)\( T^{14} + \)\(81\!\cdots\!06\)\( T^{16} - \)\(54\!\cdots\!80\)\( T^{18} + \)\(30\!\cdots\!20\)\( T^{20} - \)\(14\!\cdots\!80\)\( T^{22} + \)\(57\!\cdots\!36\)\( T^{24} - \)\(18\!\cdots\!20\)\( T^{26} + \)\(43\!\cdots\!80\)\( T^{28} - \)\(70\!\cdots\!20\)\( T^{30} + \)\(59\!\cdots\!41\)\( T^{32} \)
$31$ \( ( 1 + 159489879296 T + \)\(10\!\cdots\!04\)\( T^{2} + \)\(16\!\cdots\!28\)\( T^{3} + \)\(51\!\cdots\!12\)\( T^{4} + \)\(11\!\cdots\!72\)\( T^{5} + \)\(18\!\cdots\!80\)\( T^{6} + \)\(39\!\cdots\!28\)\( T^{7} + \)\(49\!\cdots\!98\)\( T^{8} + \)\(88\!\cdots\!08\)\( T^{9} + \)\(93\!\cdots\!80\)\( T^{10} + \)\(13\!\cdots\!32\)\( T^{11} + \)\(13\!\cdots\!92\)\( T^{12} + \)\(94\!\cdots\!28\)\( T^{13} + \)\(13\!\cdots\!44\)\( T^{14} + \)\(47\!\cdots\!16\)\( T^{15} + \)\(66\!\cdots\!81\)\( T^{16} )^{2} \)
$37$ \( 1 - \)\(39\!\cdots\!44\)\( T^{2} + \)\(80\!\cdots\!76\)\( T^{4} - \)\(10\!\cdots\!72\)\( T^{6} + \)\(11\!\cdots\!64\)\( T^{8} - \)\(90\!\cdots\!04\)\( T^{10} + \)\(60\!\cdots\!24\)\( T^{12} - \)\(34\!\cdots\!60\)\( T^{14} + \)\(17\!\cdots\!90\)\( T^{16} - \)\(72\!\cdots\!40\)\( T^{18} + \)\(26\!\cdots\!04\)\( T^{20} - \)\(81\!\cdots\!76\)\( T^{22} + \)\(20\!\cdots\!24\)\( T^{24} - \)\(42\!\cdots\!28\)\( T^{26} + \)\(65\!\cdots\!36\)\( T^{28} - \)\(67\!\cdots\!76\)\( T^{30} + \)\(35\!\cdots\!81\)\( T^{32} \)
$41$ \( ( 1 + 3741125768016 T + \)\(70\!\cdots\!80\)\( T^{2} + \)\(14\!\cdots\!84\)\( T^{3} + \)\(29\!\cdots\!00\)\( T^{4} - \)\(14\!\cdots\!12\)\( T^{5} + \)\(99\!\cdots\!52\)\( T^{6} - \)\(61\!\cdots\!84\)\( T^{7} + \)\(26\!\cdots\!42\)\( T^{8} - \)\(15\!\cdots\!04\)\( T^{9} + \)\(67\!\cdots\!72\)\( T^{10} - \)\(25\!\cdots\!92\)\( T^{11} + \)\(13\!\cdots\!00\)\( T^{12} + \)\(17\!\cdots\!84\)\( T^{13} + \)\(22\!\cdots\!80\)\( T^{14} + \)\(31\!\cdots\!76\)\( T^{15} + \)\(21\!\cdots\!41\)\( T^{16} )^{2} \)
$43$ \( 1 - \)\(41\!\cdots\!32\)\( T^{2} + \)\(84\!\cdots\!12\)\( T^{4} - \)\(11\!\cdots\!28\)\( T^{6} + \)\(11\!\cdots\!40\)\( T^{8} - \)\(97\!\cdots\!68\)\( T^{10} + \)\(74\!\cdots\!32\)\( T^{12} - \)\(50\!\cdots\!72\)\( T^{14} + \)\(31\!\cdots\!30\)\( T^{16} - \)\(17\!\cdots\!28\)\( T^{18} + \)\(88\!\cdots\!32\)\( T^{20} - \)\(40\!\cdots\!32\)\( T^{22} + \)\(16\!\cdots\!40\)\( T^{24} - \)\(54\!\cdots\!72\)\( T^{26} + \)\(14\!\cdots\!12\)\( T^{28} - \)\(24\!\cdots\!68\)\( T^{30} + \)\(20\!\cdots\!01\)\( T^{32} \)
$47$ \( ( 1 - 188349402410880 T + \)\(19\!\cdots\!72\)\( T^{2} - \)\(31\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!52\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{5} + \)\(87\!\cdots\!84\)\( T^{6} - \)\(99\!\cdots\!60\)\( T^{7} + \)\(28\!\cdots\!54\)\( T^{8} - \)\(26\!\cdots\!20\)\( T^{9} + \)\(62\!\cdots\!96\)\( T^{10} - \)\(43\!\cdots\!00\)\( T^{11} + \)\(85\!\cdots\!72\)\( T^{12} - \)\(42\!\cdots\!00\)\( T^{13} + \)\(69\!\cdots\!48\)\( T^{14} - \)\(17\!\cdots\!40\)\( T^{15} + \)\(25\!\cdots\!21\)\( T^{16} )^{2} \)
$53$ \( 1 - \)\(12\!\cdots\!88\)\( T^{2} + \)\(92\!\cdots\!64\)\( T^{4} - \)\(47\!\cdots\!52\)\( T^{6} + \)\(18\!\cdots\!96\)\( T^{8} - \)\(62\!\cdots\!40\)\( T^{10} + \)\(17\!\cdots\!08\)\( T^{12} - \)\(43\!\cdots\!64\)\( T^{14} + \)\(94\!\cdots\!02\)\( T^{16} - \)\(18\!\cdots\!16\)\( T^{18} + \)\(31\!\cdots\!88\)\( T^{20} - \)\(46\!\cdots\!60\)\( T^{22} + \)\(59\!\cdots\!16\)\( T^{24} - \)\(63\!\cdots\!48\)\( T^{26} + \)\(52\!\cdots\!84\)\( T^{28} - \)\(30\!\cdots\!32\)\( T^{30} + \)\(10\!\cdots\!41\)\( T^{32} \)
$59$ \( 1 - \)\(11\!\cdots\!84\)\( T^{2} + \)\(69\!\cdots\!76\)\( T^{4} - \)\(25\!\cdots\!84\)\( T^{6} + \)\(69\!\cdots\!16\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{10} + \)\(24\!\cdots\!32\)\( T^{12} - \)\(35\!\cdots\!68\)\( T^{14} + \)\(46\!\cdots\!62\)\( T^{16} - \)\(56\!\cdots\!48\)\( T^{18} + \)\(63\!\cdots\!72\)\( T^{20} - \)\(60\!\cdots\!00\)\( T^{22} + \)\(47\!\cdots\!56\)\( T^{24} - \)\(28\!\cdots\!84\)\( T^{26} + \)\(12\!\cdots\!36\)\( T^{28} - \)\(34\!\cdots\!64\)\( T^{30} + \)\(46\!\cdots\!81\)\( T^{32} \)
$61$ \( 1 - \)\(16\!\cdots\!44\)\( T^{2} + \)\(12\!\cdots\!56\)\( T^{4} - \)\(64\!\cdots\!84\)\( T^{6} + \)\(24\!\cdots\!36\)\( T^{8} - \)\(82\!\cdots\!80\)\( T^{10} + \)\(24\!\cdots\!72\)\( T^{12} - \)\(64\!\cdots\!28\)\( T^{14} + \)\(15\!\cdots\!02\)\( T^{16} - \)\(32\!\cdots\!48\)\( T^{18} + \)\(61\!\cdots\!32\)\( T^{20} - \)\(10\!\cdots\!80\)\( T^{22} + \)\(15\!\cdots\!96\)\( T^{24} - \)\(20\!\cdots\!84\)\( T^{26} + \)\(20\!\cdots\!96\)\( T^{28} - \)\(13\!\cdots\!64\)\( T^{30} + \)\(40\!\cdots\!21\)\( T^{32} \)
$67$ \( 1 - \)\(82\!\cdots\!12\)\( T^{2} + \)\(35\!\cdots\!44\)\( T^{4} - \)\(10\!\cdots\!88\)\( T^{6} + \)\(24\!\cdots\!96\)\( T^{8} - \)\(47\!\cdots\!60\)\( T^{10} + \)\(75\!\cdots\!48\)\( T^{12} - \)\(10\!\cdots\!76\)\( T^{14} + \)\(12\!\cdots\!82\)\( T^{16} - \)\(12\!\cdots\!04\)\( T^{18} + \)\(11\!\cdots\!68\)\( T^{20} - \)\(85\!\cdots\!40\)\( T^{22} + \)\(55\!\cdots\!76\)\( T^{24} - \)\(29\!\cdots\!12\)\( T^{26} + \)\(11\!\cdots\!24\)\( T^{28} - \)\(33\!\cdots\!08\)\( T^{30} + \)\(49\!\cdots\!61\)\( T^{32} \)
$71$ \( ( 1 + 4512963142788288 T + \)\(90\!\cdots\!40\)\( T^{2} + \)\(19\!\cdots\!68\)\( T^{3} + \)\(31\!\cdots\!80\)\( T^{4} - \)\(14\!\cdots\!12\)\( T^{5} + \)\(78\!\cdots\!64\)\( T^{6} - \)\(17\!\cdots\!64\)\( T^{7} + \)\(22\!\cdots\!90\)\( T^{8} - \)\(51\!\cdots\!24\)\( T^{9} + \)\(68\!\cdots\!84\)\( T^{10} - \)\(36\!\cdots\!52\)\( T^{11} + \)\(24\!\cdots\!80\)\( T^{12} + \)\(43\!\cdots\!68\)\( T^{13} + \)\(60\!\cdots\!40\)\( T^{14} + \)\(89\!\cdots\!28\)\( T^{15} + \)\(59\!\cdots\!21\)\( T^{16} )^{2} \)
$73$ \( ( 1 - 5666001023059280 T + \)\(17\!\cdots\!28\)\( T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!64\)\( T^{4} - \)\(66\!\cdots\!60\)\( T^{5} + \)\(61\!\cdots\!08\)\( T^{6} - \)\(24\!\cdots\!40\)\( T^{7} + \)\(27\!\cdots\!70\)\( T^{8} - \)\(11\!\cdots\!20\)\( T^{9} + \)\(13\!\cdots\!72\)\( T^{10} - \)\(71\!\cdots\!20\)\( T^{11} + \)\(65\!\cdots\!84\)\( T^{12} - \)\(24\!\cdots\!20\)\( T^{13} + \)\(19\!\cdots\!12\)\( T^{14} - \)\(30\!\cdots\!60\)\( T^{15} + \)\(25\!\cdots\!61\)\( T^{16} )^{2} \)
$79$ \( ( 1 + 22649835696004224 T + \)\(78\!\cdots\!40\)\( T^{2} + \)\(10\!\cdots\!12\)\( T^{3} + \)\(22\!\cdots\!48\)\( T^{4} + \)\(15\!\cdots\!08\)\( T^{5} + \)\(30\!\cdots\!08\)\( T^{6} - \)\(78\!\cdots\!52\)\( T^{7} + \)\(34\!\cdots\!66\)\( T^{8} - \)\(14\!\cdots\!68\)\( T^{9} + \)\(10\!\cdots\!48\)\( T^{10} + \)\(94\!\cdots\!32\)\( T^{11} + \)\(24\!\cdots\!28\)\( T^{12} + \)\(20\!\cdots\!88\)\( T^{13} + \)\(28\!\cdots\!40\)\( T^{14} + \)\(14\!\cdots\!56\)\( T^{15} + \)\(11\!\cdots\!21\)\( T^{16} )^{2} \)
$83$ \( 1 - \)\(26\!\cdots\!92\)\( T^{2} + \)\(34\!\cdots\!32\)\( T^{4} - \)\(29\!\cdots\!68\)\( T^{6} + \)\(21\!\cdots\!00\)\( T^{8} - \)\(13\!\cdots\!28\)\( T^{10} + \)\(71\!\cdots\!12\)\( T^{12} - \)\(34\!\cdots\!12\)\( T^{14} + \)\(15\!\cdots\!30\)\( T^{16} - \)\(61\!\cdots\!48\)\( T^{18} + \)\(22\!\cdots\!92\)\( T^{20} - \)\(73\!\cdots\!92\)\( T^{22} + \)\(21\!\cdots\!00\)\( T^{24} - \)\(52\!\cdots\!32\)\( T^{26} + \)\(10\!\cdots\!72\)\( T^{28} - \)\(14\!\cdots\!28\)\( T^{30} + \)\(97\!\cdots\!61\)\( T^{32} \)
$89$ \( ( 1 - 34939587304383024 T + \)\(39\!\cdots\!00\)\( T^{2} - \)\(75\!\cdots\!32\)\( T^{3} + \)\(62\!\cdots\!48\)\( T^{4} - \)\(60\!\cdots\!48\)\( T^{5} + \)\(64\!\cdots\!08\)\( T^{6} - \)\(76\!\cdots\!28\)\( T^{7} + \)\(64\!\cdots\!66\)\( T^{8} - \)\(10\!\cdots\!12\)\( T^{9} + \)\(12\!\cdots\!28\)\( T^{10} - \)\(15\!\cdots\!72\)\( T^{11} + \)\(22\!\cdots\!88\)\( T^{12} - \)\(37\!\cdots\!68\)\( T^{13} + \)\(27\!\cdots\!00\)\( T^{14} - \)\(33\!\cdots\!16\)\( T^{15} + \)\(13\!\cdots\!61\)\( T^{16} )^{2} \)
$97$ \( ( 1 - 47796699301090320 T + \)\(34\!\cdots\!40\)\( T^{2} - \)\(19\!\cdots\!20\)\( T^{3} + \)\(54\!\cdots\!44\)\( T^{4} - \)\(32\!\cdots\!40\)\( T^{5} + \)\(55\!\cdots\!60\)\( T^{6} - \)\(31\!\cdots\!00\)\( T^{7} + \)\(39\!\cdots\!50\)\( T^{8} - \)\(18\!\cdots\!00\)\( T^{9} + \)\(19\!\cdots\!40\)\( T^{10} - \)\(68\!\cdots\!20\)\( T^{11} + \)\(68\!\cdots\!84\)\( T^{12} - \)\(14\!\cdots\!40\)\( T^{13} + \)\(15\!\cdots\!60\)\( T^{14} - \)\(12\!\cdots\!60\)\( T^{15} + \)\(15\!\cdots\!21\)\( T^{16} )^{2} \)
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