# 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 $2$

# Related objects

## 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
 0.5 + 58156.1i 0.5 − 58156.1i 0.5 + 82996.3i 0.5 − 82996.3i 0.5 − 151568.i 0.5 + 151568.i 0.5 + 159067.i 0.5 − 159067.i 0.5 + 12044.5i 0.5 − 12044.5i 0.5 − 198996.i 0.5 + 198996.i 0.5 + 65608.9i 0.5 − 65608.9i 0.5 + 83132.4i 0.5 − 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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