# Properties

 Label 17.3.e Level 17 Weight 3 Character orbit e Rep. character $$\chi_{17}(3,\cdot)$$ Character field $$\Q(\zeta_{16})$$ Dimension 16 Newform subspaces 2 Sturm bound 4 Trace bound 2

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$17$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 17.e (of order $$16$$ and degree $$8$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$17$$ Character field: $$\Q(\zeta_{16})$$ Newform subspaces: $$2$$ Sturm bound: $$4$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(17, [\chi])$$.

Total New Old
Modular forms 32 32 0
Cusp forms 16 16 0
Eisenstein series 16 16 0

## Trace form

 $$16q - 8q^{2} - 8q^{3} - 8q^{4} - 8q^{5} - 8q^{6} - 8q^{7} - 8q^{8} - 8q^{9} + O(q^{10})$$ $$16q - 8q^{2} - 8q^{3} - 8q^{4} - 8q^{5} - 8q^{6} - 8q^{7} - 8q^{8} - 8q^{9} + 16q^{10} + 32q^{11} + 88q^{12} + 16q^{13} + 24q^{14} + 16q^{15} - 16q^{17} - 80q^{18} - 32q^{19} - 120q^{20} - 128q^{21} - 104q^{22} - 64q^{23} - 56q^{24} + 80q^{25} + 176q^{26} + 136q^{27} + 232q^{28} + 72q^{29} + 184q^{30} + 56q^{31} + 64q^{32} - 72q^{34} - 80q^{35} - 232q^{36} - 136q^{37} - 112q^{38} - 24q^{39} - 136q^{40} + 48q^{41} - 56q^{42} - 136q^{43} + 24q^{44} - 88q^{45} - 88q^{46} + 112q^{47} + 224q^{48} + 24q^{49} - 40q^{51} - 144q^{52} + 64q^{53} + 216q^{54} + 216q^{55} - 48q^{56} + 272q^{57} + 240q^{58} - 40q^{59} + 80q^{60} + 104q^{61} - 304q^{62} + 64q^{63} - 184q^{64} - 128q^{65} + 96q^{68} + 32q^{69} + 144q^{70} + 72q^{71} + 64q^{72} + 72q^{73} + 16q^{74} - 488q^{75} - 264q^{77} - 768q^{78} - 232q^{79} - 216q^{80} - 648q^{81} - 472q^{82} - 352q^{83} + 240q^{85} + 1120q^{86} + 520q^{87} + 440q^{88} + 448q^{89} + 704q^{90} + 296q^{91} + 360q^{92} + 216q^{93} + 24q^{94} + 120q^{95} + 272q^{96} - 296q^{97} - 104q^{98} - 88q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(17, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
17.3.e.a $$8$$ $$0.463$$ $$\Q(\zeta_{16})$$ None $$-8$$ $$-8$$ $$16$$ $$8$$ $$q+(-1+\zeta_{16}+\zeta_{16}^{2}-\zeta_{16}^{4}-\zeta_{16}^{5}+\cdots)q^{2}+\cdots$$
17.3.e.b $$8$$ $$0.463$$ $$\Q(\zeta_{16})$$ None $$0$$ $$0$$ $$-24$$ $$-16$$ $$q+(-\zeta_{16}^{2}+\zeta_{16}^{4}+\zeta_{16}^{7})q^{2}+(2\zeta_{16}^{2}+\cdots)q^{3}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 8 T + 32 T^{2} + 104 T^{3} + 320 T^{4} + 872 T^{5} + 2112 T^{6} + 4776 T^{7} + 10017 T^{8} + 19104 T^{9} + 33792 T^{10} + 55808 T^{11} + 81920 T^{12} + 106496 T^{13} + 131072 T^{14} + 131072 T^{15} + 65536 T^{16}$$)($$1 + 4 T^{2} - 16 T^{3} + 8 T^{4} - 72 T^{5} + 148 T^{6} - 200 T^{7} + 801 T^{8} - 800 T^{9} + 2368 T^{10} - 4608 T^{11} + 2048 T^{12} - 16384 T^{13} + 16384 T^{14} + 65536 T^{16}$$)
$3$ ($$1 + 8 T + 40 T^{2} + 136 T^{3} + 530 T^{4} + 1976 T^{5} + 7064 T^{6} + 19576 T^{7} + 58466 T^{8} + 176184 T^{9} + 572184 T^{10} + 1440504 T^{11} + 3477330 T^{12} + 8030664 T^{13} + 21257640 T^{14} + 38263752 T^{15} + 43046721 T^{16}$$)($$1 - 4 T^{2} + 16 T^{3} + 28 T^{4} - 376 T^{5} + 444 T^{6} + 8 T^{7} - 6760 T^{8} + 72 T^{9} + 35964 T^{10} - 274104 T^{11} + 183708 T^{12} + 944784 T^{13} - 2125764 T^{14} + 43046721 T^{16}$$)
$5$ ($$1 - 16 T + 96 T^{2} - 112 T^{3} - 352 T^{4} - 10432 T^{5} + 95824 T^{6} - 124272 T^{7} - 914688 T^{8} - 3106800 T^{9} + 59890000 T^{10} - 163000000 T^{11} - 137500000 T^{12} - 1093750000 T^{13} + 23437500000 T^{14} - 97656250000 T^{15} + 152587890625 T^{16}$$)($$1 + 24 T + 280 T^{2} + 2168 T^{3} + 13330 T^{4} + 75080 T^{5} + 410024 T^{6} + 2141288 T^{7} + 10795618 T^{8} + 53532200 T^{9} + 256265000 T^{10} + 1173125000 T^{11} + 5207031250 T^{12} + 21171875000 T^{13} + 68359375000 T^{14} + 146484375000 T^{15} + 152587890625 T^{16}$$)
$7$ ($$1 - 8 T + 24 T^{2} - 1128 T^{3} + 8240 T^{4} - 20216 T^{5} + 640360 T^{6} - 4599368 T^{7} + 10019136 T^{8} - 225369032 T^{9} + 1537504360 T^{10} - 2378392184 T^{11} + 47501960240 T^{12} - 318632080872 T^{13} + 332190892824 T^{14} - 5425784582792 T^{15} + 33232930569601 T^{16}$$)($$1 + 16 T + 124 T^{2} + 1296 T^{3} + 10140 T^{4} + 48232 T^{5} + 458172 T^{6} + 3313432 T^{7} + 15111448 T^{8} + 162358168 T^{9} + 1100070972 T^{10} + 5674446568 T^{11} + 58455082140 T^{12} + 366087922704 T^{13} + 1716319612924 T^{14} + 10851569165584 T^{15} + 33232930569601 T^{16}$$)
$11$ ($$1 + 8 T + 12 T^{2} + 536 T^{3} - 10730 T^{4} - 47224 T^{5} + 541212 T^{6} - 224040 T^{7} + 130587298 T^{8} - 27108840 T^{9} + 7923884892 T^{10} - 83660196664 T^{11} - 2300070793130 T^{12} + 13902459586136 T^{13} + 37661140520652 T^{14} + 3037998668665928 T^{15} + 45949729863572161 T^{16}$$)($$1 - 40 T + 700 T^{2} - 7600 T^{3} + 87412 T^{4} - 1438024 T^{5} + 19499716 T^{6} - 176624752 T^{7} + 1537072952 T^{8} - 21371594992 T^{9} + 285495341956 T^{10} - 2547547235464 T^{11} + 18737538505972 T^{12} - 197124426967600 T^{13} + 2196899863704700 T^{14} - 15189993343329640 T^{15} + 45949729863572161 T^{16}$$)
$13$ ($$1 - 16 T + 128 T^{2} - 16 T^{3} - 5116 T^{4} + 117072 T^{5} - 1218176 T^{6} + 73648208 T^{7} - 1288359866 T^{8} + 12446547152 T^{9} - 34792324736 T^{10} + 565084183248 T^{11} - 4173278368636 T^{12} - 2205735869584 T^{13} + 2982154895677568 T^{14} - 62998022171188624 T^{15} + 665416609183179841 T^{16}$$)($$1 + 784 T^{3} + 9888 T^{4} - 237552 T^{5} + 307328 T^{6} - 20917120 T^{7} + 236770498 T^{8} - 3534993280 T^{9} + 8777595008 T^{10} - 1146618131568 T^{11} + 8065945369248 T^{12} + 108081057609616 T^{13} + 665416609183179841 T^{16}$$)
$17$ ($$1 + 6975757441 T^{8}$$)($$1 + 16 T + 240 T^{2} - 784 T^{3} - 11934 T^{4} - 226576 T^{5} + 20045040 T^{6} + 386201104 T^{7} + 6975757441 T^{8}$$)
$19$ ($$1 + 368 T^{2} + 7440 T^{3} + 67712 T^{4} + 1902864 T^{5} + 27896560 T^{6} - 410926464 T^{7} + 18992960962 T^{8} - 148344453504 T^{9} + 3635507595760 T^{10} + 89521913303184 T^{11} + 1149991020632192 T^{12} + 45615132958039440 T^{13} + 814499890216347248 T^{14} +$$$$28\!\cdots\!81$$$$T^{16}$$)($$1 + 32 T + 544 T^{2} + 10608 T^{3} + 192512 T^{4} + 3096432 T^{5} + 96494752 T^{6} + 2593739296 T^{7} + 50479240962 T^{8} + 936339885856 T^{9} + 12575292575392 T^{10} + 145674371396592 T^{11} + 3269539688148992 T^{12} + 65038350862753008 T^{13} + 1204043315971991584 T^{14} + 25568213945052291872 T^{15} +$$$$28\!\cdots\!81$$$$T^{16}$$)
$23$ ($$1 + 56 T + 1032 T^{2} - 18712 T^{3} - 1041904 T^{4} - 8737672 T^{5} + 292851208 T^{6} + 8265977256 T^{7} + 119476335168 T^{8} + 4372701968424 T^{9} + 81951774897928 T^{10} - 1293489042310408 T^{11} - 81592528808215024 T^{12} - 775172877829800088 T^{13} + 22615892413844971272 T^{14} +$$$$64\!\cdots\!04$$$$T^{15} +$$$$61\!\cdots\!61$$$$T^{16}$$)($$1 + 8 T + 732 T^{2} + 3584 T^{3} + 513652 T^{4} + 7272312 T^{5} + 445470244 T^{6} + 2689381328 T^{7} + 220561130936 T^{8} + 1422682722512 T^{9} + 124660838551204 T^{10} + 1076563172005368 T^{11} + 40224594211556212 T^{12} + 148472616189718016 T^{13} + 16041505084238874972 T^{14} + 92742690596309998472 T^{15} +$$$$61\!\cdots\!61$$$$T^{16}$$)
$29$ ($$1 - 48 T + 624 T^{2} + 56960 T^{3} - 3050368 T^{4} + 68290448 T^{5} + 241251280 T^{6} - 64889435200 T^{7} + 2514250952448 T^{8} - 54572015003200 T^{9} + 170632446569680 T^{10} + 40620751071937808 T^{11} - 1525935650211019648 T^{12} + 23963484008779448960 T^{13} +$$$$22\!\cdots\!84$$$$T^{14} -$$$$14\!\cdots\!88$$$$T^{15} +$$$$25\!\cdots\!21$$$$T^{16}$$)($$1 - 24 T + 684 T^{2} - 83736 T^{3} + 1812294 T^{4} - 39076344 T^{5} + 2881128204 T^{6} - 72099991512 T^{7} + 1161408147714 T^{8} - 60636092861592 T^{9} + 2037767237253324 T^{10} - 23243520710618424 T^{11} + 906593572730742534 T^{12} - 35228340887625630936 T^{13} +$$$$24\!\cdots\!44$$$$T^{14} -$$$$71\!\cdots\!44$$$$T^{15} +$$$$25\!\cdots\!21$$$$T^{16}$$)
$31$ ($$1 - 24 T + 232 T^{2} + 33928 T^{3} - 415280 T^{4} - 30121928 T^{5} + 884607752 T^{6} - 20180029736 T^{7} - 595898026688 T^{8} - 19393008576296 T^{9} + 816953835734792 T^{10} - 26733321978816968 T^{11} - 354188590028498480 T^{12} + 27808348520684616328 T^{13} +$$$$18\!\cdots\!52$$$$T^{14} -$$$$18\!\cdots\!04$$$$T^{15} +$$$$72\!\cdots\!81$$$$T^{16}$$)($$1 - 32 T + 1988 T^{2} - 84680 T^{3} + 2224116 T^{4} - 107632640 T^{5} + 3131219084 T^{6} - 96339571608 T^{7} + 3769094500984 T^{8} - 92582328315288 T^{9} + 2891746579674764 T^{10} - 95524364195747840 T^{11} + 1896928602629127156 T^{12} - 69406123341534228680 T^{13} +$$$$15\!\cdots\!68$$$$T^{14} -$$$$24\!\cdots\!72$$$$T^{15} +$$$$72\!\cdots\!81$$$$T^{16}$$)
$37$ ($$1 - 32 T + 1024 T^{2} + 43232 T^{3} - 3988832 T^{4} + 114777552 T^{5} - 1479427536 T^{6} - 166606879104 T^{7} + 7610311129088 T^{8} - 228084817493376 T^{9} - 2772685390297296 T^{10} + 294487796326770768 T^{11} - 14010690445142610272 T^{12} +$$$$20\!\cdots\!68$$$$T^{13} +$$$$67\!\cdots\!44$$$$T^{14} -$$$$28\!\cdots\!48$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16}$$)($$1 + 168 T + 13240 T^{2} + 650184 T^{3} + 19962450 T^{4} + 154153272 T^{5} - 24329640440 T^{6} - 1835917350504 T^{7} - 81344425680670 T^{8} - 2513370852839976 T^{9} - 45597663256670840 T^{10} + 395515121004160248 T^{11} + 70117695474925266450 T^{12} +$$$$31\!\cdots\!16$$$$T^{13} +$$$$87\!\cdots\!40$$$$T^{14} +$$$$15\!\cdots\!52$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16}$$)
$41$ ($$1 - 48 T + 1824 T^{2} + 23984 T^{3} - 1206974 T^{4} + 18567760 T^{5} + 1403379552 T^{6} - 142523024592 T^{7} + 3825436575874 T^{8} - 239581204339152 T^{9} + 3965615206239072 T^{10} + 88198795521870160 T^{11} - 9637597143493089854 T^{12} +$$$$32\!\cdots\!84$$$$T^{13} +$$$$41\!\cdots\!44$$$$T^{14} -$$$$18\!\cdots\!28$$$$T^{15} +$$$$63\!\cdots\!41$$$$T^{16}$$)($$1 + 5396 T^{2} - 54048 T^{3} + 11503750 T^{4} - 349337248 T^{5} + 11972012020 T^{6} - 1013376724544 T^{7} + 10882038609154 T^{8} - 1703486273958464 T^{9} + 33830044657647220 T^{10} - 1659388343264068768 T^{11} + 91856583604500703750 T^{12} -$$$$72\!\cdots\!48$$$$T^{13} +$$$$12\!\cdots\!76$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16}$$)
$43$ ($$1 + 232 T + 27932 T^{2} + 2342968 T^{3} + 154510664 T^{4} + 8541190600 T^{5} + 413867884116 T^{6} + 18437331684504 T^{7} + 795437538561102 T^{8} + 34090626284647896 T^{9} + 1414931936083664916 T^{10} + 53991966653306139400 T^{11} +$$$$18\!\cdots\!64$$$$T^{12} +$$$$50\!\cdots\!32$$$$T^{13} +$$$$11\!\cdots\!32$$$$T^{14} +$$$$17\!\cdots\!68$$$$T^{15} +$$$$13\!\cdots\!01$$$$T^{16}$$)($$1 - 96 T + 6096 T^{2} - 290832 T^{3} + 11553408 T^{4} - 238904592 T^{5} - 4015115952 T^{6} + 725328019488 T^{7} - 38391385271230 T^{8} + 1341131508033312 T^{9} - 13726882431813552 T^{10} - 1510202660105221008 T^{11} +$$$$13\!\cdots\!08$$$$T^{12} -$$$$62\!\cdots\!68$$$$T^{13} +$$$$24\!\cdots\!96$$$$T^{14} -$$$$70\!\cdots\!04$$$$T^{15} +$$$$13\!\cdots\!01$$$$T^{16}$$)
$47$ ($$1 - 192 T + 18432 T^{2} - 1331200 T^{3} + 86363396 T^{4} - 5070960256 T^{5} + 267820974080 T^{6} - 13081254629824 T^{7} + 617534815843590 T^{8} - 28896491477281216 T^{9} + 1306880918619668480 T^{10} - 54660972524224964224 T^{11} +$$$$20\!\cdots\!56$$$$T^{12} -$$$$70\!\cdots\!00$$$$T^{13} +$$$$21\!\cdots\!12$$$$T^{14} -$$$$49\!\cdots\!48$$$$T^{15} +$$$$56\!\cdots\!21$$$$T^{16}$$)($$1 + 80 T + 3200 T^{2} + 228944 T^{3} + 25698900 T^{4} + 1284226160 T^{5} + 46709290368 T^{6} + 3079489449072 T^{7} + 201735188722534 T^{8} + 6802592193000048 T^{9} + 227926436732212608 T^{10} + 13842950309774806640 T^{11} +$$$$61\!\cdots\!00$$$$T^{12} +$$$$12\!\cdots\!56$$$$T^{13} +$$$$37\!\cdots\!00$$$$T^{14} +$$$$20\!\cdots\!20$$$$T^{15} +$$$$56\!\cdots\!21$$$$T^{16}$$)
$53$ ($$1 + 32 T + 2272 T^{2} + 283520 T^{3} + 9589248 T^{4} + 703635840 T^{5} + 40677888352 T^{6} + 1429537588256 T^{7} + 116127856966210 T^{8} + 4015571085411104 T^{9} + 320968105161577312 T^{10} + 15595638861067263360 T^{11} +$$$$59\!\cdots\!28$$$$T^{12} +$$$$49\!\cdots\!80$$$$T^{13} +$$$$11\!\cdots\!52$$$$T^{14} +$$$$44\!\cdots\!08$$$$T^{15} +$$$$38\!\cdots\!21$$$$T^{16}$$)($$1 - 96 T + 3844 T^{2} - 250080 T^{3} + 17203208 T^{4} - 878356128 T^{5} + 61303275468 T^{6} - 2343880946208 T^{7} + 44401903321166 T^{8} - 6583961577898272 T^{9} + 483712330318020108 T^{10} - 19468202420862148512 T^{11} +$$$$10\!\cdots\!88$$$$T^{12} -$$$$43\!\cdots\!20$$$$T^{13} +$$$$18\!\cdots\!04$$$$T^{14} -$$$$13\!\cdots\!24$$$$T^{15} +$$$$38\!\cdots\!21$$$$T^{16}$$)
$59$ ($$1 + 48 T + 5988 T^{2} + 249936 T^{3} + 17455752 T^{4} - 267650832 T^{5} - 2258950740 T^{6} - 4680295454832 T^{7} - 153985341695026 T^{8} - 16292108478270192 T^{9} - 27372521597797140 T^{10} - 11289654923217639312 T^{11} +$$$$25\!\cdots\!92$$$$T^{12} +$$$$12\!\cdots\!36$$$$T^{13} +$$$$10\!\cdots\!28$$$$T^{14} +$$$$29\!\cdots\!28$$$$T^{15} +$$$$21\!\cdots\!41$$$$T^{16}$$)($$1 - 8 T - 1064 T^{2} + 139576 T^{3} - 481440 T^{4} + 260011896 T^{5} + 38349218008 T^{6} - 28053020936 T^{7} + 74913788078050 T^{8} - 97652565878216 T^{9} + 464691318670636888 T^{10} + 10967440526288193336 T^{11} - 70690045880224302240 T^{12} +$$$$71\!\cdots\!76$$$$T^{13} -$$$$18\!\cdots\!84$$$$T^{14} -$$$$49\!\cdots\!88$$$$T^{15} +$$$$21\!\cdots\!41$$$$T^{16}$$)
$61$ ($$1 + 160 T + 8384 T^{2} - 591392 T^{3} - 112289792 T^{4} - 7004799072 T^{5} - 46635351424 T^{6} + 25101045742144 T^{7} + 2320428818600960 T^{8} + 93400991206517824 T^{9} - 645705660795827584 T^{10} -$$$$36\!\cdots\!92$$$$T^{11} -$$$$21\!\cdots\!52$$$$T^{12} -$$$$42\!\cdots\!92$$$$T^{13} +$$$$22\!\cdots\!64$$$$T^{14} +$$$$15\!\cdots\!60$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16}$$)($$1 - 264 T + 33304 T^{2} - 2900248 T^{3} + 200549746 T^{4} - 11914556776 T^{5} + 710898258152 T^{6} - 45325216747576 T^{7} + 2838106764821794 T^{8} - 168655131517730296 T^{9} + 9842984249549545832 T^{10} -$$$$61\!\cdots\!36$$$$T^{11} +$$$$38\!\cdots\!26$$$$T^{12} -$$$$20\!\cdots\!48$$$$T^{13} +$$$$88\!\cdots\!84$$$$T^{14} -$$$$26\!\cdots\!24$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16}$$)
$67$ ($$1 - 14608 T^{2} + 108494176 T^{4} - 545817039504 T^{6} + 2442127301002178 T^{8} - 10998825206906883984 T^{10} +$$$$44\!\cdots\!16$$$$T^{12} -$$$$11\!\cdots\!88$$$$T^{14} +$$$$16\!\cdots\!81$$$$T^{16}$$)($$1 - 30264 T^{2} + 421009100 T^{4} - 3513162425416 T^{6} + 19237575139662822 T^{8} - 70794161127211291336 T^{10} +$$$$17\!\cdots\!00$$$$T^{12} -$$$$24\!\cdots\!04$$$$T^{14} +$$$$16\!\cdots\!81$$$$T^{16}$$)
$71$ ($$1 - 40 T - 12216 T^{2} - 301752 T^{3} + 88296464 T^{4} + 4708166648 T^{5} - 160406696312 T^{6} - 17457547717336 T^{7} - 106016902560192 T^{8} - 88003498043090776 T^{9} - 4076203796944420472 T^{10} +$$$$60\!\cdots\!08$$$$T^{11} +$$$$57\!\cdots\!04$$$$T^{12} -$$$$98\!\cdots\!52$$$$T^{13} -$$$$20\!\cdots\!56$$$$T^{14} -$$$$33\!\cdots\!40$$$$T^{15} +$$$$41\!\cdots\!21$$$$T^{16}$$)($$1 - 32 T - 2196 T^{2} + 376992 T^{3} - 40999012 T^{4} + 1392153480 T^{5} + 50862865468 T^{6} - 10985693543800 T^{7} + 1117214885242200 T^{8} - 55378881154295800 T^{9} + 1292510912018731708 T^{10} +$$$$17\!\cdots\!80$$$$T^{11} -$$$$26\!\cdots\!32$$$$T^{12} +$$$$12\!\cdots\!92$$$$T^{13} -$$$$36\!\cdots\!36$$$$T^{14} -$$$$26\!\cdots\!92$$$$T^{15} +$$$$41\!\cdots\!21$$$$T^{16}$$)
$73$ ($$1 - 48 T - 4096 T^{2} - 616528 T^{3} + 30492930 T^{4} + 4872133264 T^{5} + 204783337600 T^{6} - 9842384223888 T^{7} - 2875677804870654 T^{8} - 52450065529099152 T^{9} + 5815486573949161600 T^{10} +$$$$73\!\cdots\!96$$$$T^{11} +$$$$24\!\cdots\!30$$$$T^{12} -$$$$26\!\cdots\!72$$$$T^{13} -$$$$93\!\cdots\!16$$$$T^{14} -$$$$58\!\cdots\!32$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16}$$)($$1 - 24 T + 3500 T^{2} + 527224 T^{3} - 252666 T^{4} - 799621288 T^{5} + 103501902988 T^{6} + 4216386556104 T^{7} - 1450386823086462 T^{8} + 22469123957478216 T^{9} + 2939271985011844108 T^{10} -$$$$12\!\cdots\!32$$$$T^{11} -$$$$20\!\cdots\!46$$$$T^{12} +$$$$22\!\cdots\!76$$$$T^{13} +$$$$80\!\cdots\!00$$$$T^{14} -$$$$29\!\cdots\!16$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16}$$)
$79$ ($$1 + 136 T + 3800 T^{2} - 1305896 T^{3} - 161585328 T^{4} - 6145771400 T^{5} + 317342984696 T^{6} + 54517171591464 T^{7} + 4138695597908032 T^{8} + 340241667902326824 T^{9} + 12360534958690960376 T^{10} -$$$$14\!\cdots\!00$$$$T^{11} -$$$$24\!\cdots\!08$$$$T^{12} -$$$$12\!\cdots\!96$$$$T^{13} +$$$$22\!\cdots\!00$$$$T^{14} +$$$$50\!\cdots\!16$$$$T^{15} +$$$$23\!\cdots\!21$$$$T^{16}$$)($$1 + 96 T - 11292 T^{2} - 1308856 T^{3} + 36577844 T^{4} + 2295987344 T^{5} - 542602414484 T^{6} + 12994739703176 T^{7} + 6454474657365240 T^{8} + 81100170487521416 T^{9} - 21134407994947373204 T^{10} +$$$$55\!\cdots\!24$$$$T^{11} +$$$$55\!\cdots\!84$$$$T^{12} -$$$$12\!\cdots\!56$$$$T^{13} -$$$$66\!\cdots\!72$$$$T^{14} +$$$$35\!\cdots\!76$$$$T^{15} +$$$$23\!\cdots\!21$$$$T^{16}$$)
$83$ ($$1 + 264 T + 35100 T^{2} + 2811288 T^{3} + 126238536 T^{4} + 5231440488 T^{5} + 1212641715924 T^{6} + 223465365584760 T^{7} + 23797706896032974 T^{8} + 1539452903513411640 T^{9} + 57549939812312003604 T^{10} +$$$$17\!\cdots\!72$$$$T^{11} +$$$$28\!\cdots\!76$$$$T^{12} +$$$$43\!\cdots\!12$$$$T^{13} +$$$$37\!\cdots\!00$$$$T^{14} +$$$$19\!\cdots\!56$$$$T^{15} +$$$$50\!\cdots\!81$$$$T^{16}$$)($$1 + 88 T + 12840 T^{2} - 772808 T^{3} - 70014880 T^{4} - 15998431560 T^{5} + 276363214376 T^{6} + 33752671318936 T^{7} + 12884842311213474 T^{8} + 232522152716150104 T^{9} + 13115734140448022696 T^{10} -$$$$52\!\cdots\!40$$$$T^{11} -$$$$15\!\cdots\!80$$$$T^{12} -$$$$11\!\cdots\!92$$$$T^{13} +$$$$13\!\cdots\!40$$$$T^{14} +$$$$64\!\cdots\!52$$$$T^{15} +$$$$50\!\cdots\!81$$$$T^{16}$$)
$89$ ($$1 - 160 T + 12800 T^{2} - 1203232 T^{3} + 97134272 T^{4} - 4832684576 T^{5} + 253794473472 T^{6} + 8995658901856 T^{7} - 3993981113240190 T^{8} + 71254614161601376 T^{9} + 15923634019048330752 T^{10} -$$$$24\!\cdots\!36$$$$T^{11} +$$$$38\!\cdots\!32$$$$T^{12} -$$$$37\!\cdots\!32$$$$T^{13} +$$$$31\!\cdots\!00$$$$T^{14} -$$$$31\!\cdots\!60$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16}$$)($$1 - 288 T + 41472 T^{2} - 4656976 T^{3} + 519733472 T^{4} - 56560463824 T^{5} + 5578739762816 T^{6} - 499331992822432 T^{7} + 43753371662175234 T^{8} - 3955208715146483872 T^{9} +$$$$35\!\cdots\!56$$$$T^{10} -$$$$28\!\cdots\!64$$$$T^{11} +$$$$20\!\cdots\!32$$$$T^{12} -$$$$14\!\cdots\!76$$$$T^{13} +$$$$10\!\cdots\!12$$$$T^{14} -$$$$56\!\cdots\!08$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16}$$)
$97$ ($$1 - 48 T - 4444 T^{2} - 1737360 T^{3} + 80511142 T^{4} + 14428723088 T^{5} + 1714599118564 T^{6} - 120003485930960 T^{7} - 19840415339669054 T^{8} - 1129112799124402640 T^{9} +$$$$15\!\cdots\!84$$$$T^{10} +$$$$12\!\cdots\!52$$$$T^{11} +$$$$63\!\cdots\!62$$$$T^{12} -$$$$12\!\cdots\!40$$$$T^{13} -$$$$30\!\cdots\!04$$$$T^{14} -$$$$31\!\cdots\!12$$$$T^{15} +$$$$61\!\cdots\!21$$$$T^{16}$$)($$1 + 344 T + 70152 T^{2} + 11174248 T^{3} + 1407887122 T^{4} + 151612370328 T^{5} + 14250743174520 T^{6} + 1250314851633384 T^{7} + 116834146569341026 T^{8} + 11764212439018510056 T^{9} +$$$$12\!\cdots\!20$$$$T^{10} +$$$$12\!\cdots\!12$$$$T^{11} +$$$$11\!\cdots\!42$$$$T^{12} +$$$$82\!\cdots\!52$$$$T^{13} +$$$$48\!\cdots\!32$$$$T^{14} +$$$$22\!\cdots\!36$$$$T^{15} +$$$$61\!\cdots\!21$$$$T^{16}$$)