Properties

Label 224.3.s
Level 224
Weight 3
Character orbit s
Rep. character \(\chi_{224}(33,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 32
Newform subspaces 2
Sturm bound 96
Trace bound 9

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Defining parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.s (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(96\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(224, [\chi])\).

Total New Old
Modular forms 144 32 112
Cusp forms 112 32 80
Eisenstein series 32 0 32

Trace form

\( 32q + 48q^{9} + O(q^{10}) \) \( 32q + 48q^{9} - 80q^{21} + 96q^{25} + 96q^{29} - 144q^{33} + 80q^{37} + 240q^{45} + 16q^{53} - 160q^{57} - 144q^{61} - 176q^{65} - 240q^{73} + 64q^{77} - 352q^{81} - 352q^{85} + 432q^{89} - 400q^{93} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(224, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
224.3.s.a \(16\) \(6.104\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{3}+\beta _{10}q^{5}-\beta _{13}q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
224.3.s.b \(16\) \(6.104\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{3}-\beta _{5}q^{5}+(\beta _{2}+\beta _{12})q^{7}+(-5\beta _{7}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(224, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(224, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 32 T^{2} + 486 T^{4} + 3776 T^{6} + 6841 T^{8} - 164352 T^{10} - 1549098 T^{12} - 1704672 T^{14} + 53230932 T^{16} - 138078432 T^{18} - 10163631978 T^{20} - 87343391232 T^{22} + 294482618361 T^{24} + 13166097898176 T^{26} + 137260754729766 T^{28} + 732057358558752 T^{30} + 1853020188851841 T^{32} \))(\( 1 + 16 T^{2} + 134 T^{4} + 1120 T^{6} + 7897 T^{8} + 73856 T^{10} + 1194422 T^{12} + 13226000 T^{14} + 117489172 T^{16} + 1071306000 T^{18} + 7836602742 T^{20} + 39250106496 T^{22} + 339939955737 T^{24} + 3905198529120 T^{26} + 37845557888454 T^{28} + 366028679279376 T^{30} + 1853020188851841 T^{32} \))
$5$ (\( ( 1 + 46 T^{2} + 1073 T^{4} - 5040 T^{5} + 9678 T^{6} - 250320 T^{7} - 145276 T^{8} - 6258000 T^{9} + 6048750 T^{10} - 78750000 T^{11} + 419140625 T^{12} + 11230468750 T^{14} + 152587890625 T^{16} )^{2} \))(\( ( 1 + 30 T^{2} + 321 T^{4} + 1200 T^{5} - 930 T^{6} + 138960 T^{7} - 137884 T^{8} + 3474000 T^{9} - 581250 T^{10} + 18750000 T^{11} + 125390625 T^{12} + 7324218750 T^{14} + 152587890625 T^{16} )^{2} \))
$7$ (\( 1 + 64 T^{2} + 5084 T^{4} + 338880 T^{6} + 15148742 T^{8} + 813650880 T^{10} + 29308248284 T^{12} + 885842380864 T^{14} + 33232930569601 T^{16} \))(\( 1 - 64 T^{2} + 2268 T^{4} + 3136 T^{6} - 7054138 T^{8} + 7529536 T^{10} + 13074568668 T^{12} - 885842380864 T^{14} + 33232930569601 T^{16} \))
$11$ (\( 1 - 640 T^{2} + 227030 T^{4} - 52241408 T^{6} + 8348702825 T^{8} - 853428477056 T^{10} + 32116185305766 T^{12} + 6409181104515840 T^{14} - 1320087233250529292 T^{16} + 93836820551216413440 T^{18} + \)\(68\!\cdots\!46\)\( T^{20} - \)\(26\!\cdots\!76\)\( T^{22} + \)\(38\!\cdots\!25\)\( T^{24} - \)\(35\!\cdots\!08\)\( T^{26} + \)\(22\!\cdots\!30\)\( T^{28} - \)\(92\!\cdots\!40\)\( T^{30} + \)\(21\!\cdots\!21\)\( T^{32} \))(\( 1 - 432 T^{2} + 84854 T^{4} - 7780896 T^{6} + 22311049 T^{8} + 70438127616 T^{10} + 144760417670 T^{12} - 2441540782042416 T^{14} + 459119675281580212 T^{16} - 35746598589883012656 T^{18} + 31030681144833827270 T^{20} + \)\(22\!\cdots\!36\)\( T^{22} + \)\(10\!\cdots\!89\)\( T^{24} - \)\(52\!\cdots\!96\)\( T^{26} + \)\(83\!\cdots\!14\)\( T^{28} - \)\(62\!\cdots\!92\)\( T^{30} + \)\(21\!\cdots\!21\)\( T^{32} \))
$13$ (\( ( 1 - 192 T^{2} - 16036 T^{4} + 1118400 T^{6} + 1003382406 T^{8} + 31942622400 T^{10} - 13081057841956 T^{12} - 4473232343516352 T^{14} + 665416609183179841 T^{16} )^{2} \))(\( ( 1 - 1056 T^{2} + 510492 T^{4} - 150812640 T^{6} + 30377120006 T^{8} - 4307359811040 T^{10} + 416424007224732 T^{12} - 24602777889339936 T^{14} + 665416609183179841 T^{16} )^{2} \))
$17$ (\( ( 1 - 24 T + 766 T^{2} - 13776 T^{3} + 249905 T^{4} - 2407680 T^{5} - 7157346 T^{6} + 486721512 T^{7} - 15232969756 T^{8} + 140662516968 T^{9} - 597788695266 T^{10} - 58115542129920 T^{11} + 1743276663293105 T^{12} - 27772331972585424 T^{13} + 446288633717996926 T^{14} - 4041067837425622296 T^{15} + 48661191875666868481 T^{16} )^{2} \))(\( ( 1 + 24 T + 1022 T^{2} + 19920 T^{3} + 479985 T^{4} + 8829120 T^{5} + 194168350 T^{6} + 3297192024 T^{7} + 67118725988 T^{8} + 952888494936 T^{9} + 16217134760350 T^{10} + 213113493209280 T^{11} + 3348258935318385 T^{12} + 40158598496944080 T^{13} + 595439926448815742 T^{14} + 4041067837425622296 T^{15} + 48661191875666868481 T^{16} )^{2} \))
$19$ (\( 1 + 528 T^{2} + 24262 T^{4} + 11189472 T^{6} + 3728656729 T^{8} - 6420105050688 T^{10} + 940099701135094 T^{12} + 1121910404491838544 T^{14} + \)\(16\!\cdots\!80\)\( T^{16} + \)\(14\!\cdots\!24\)\( T^{18} + \)\(15\!\cdots\!54\)\( T^{20} - \)\(14\!\cdots\!68\)\( T^{22} + \)\(10\!\cdots\!49\)\( T^{24} + \)\(42\!\cdots\!72\)\( T^{26} + \)\(11\!\cdots\!02\)\( T^{28} + \)\(33\!\cdots\!48\)\( T^{30} + \)\(83\!\cdots\!61\)\( T^{32} \))(\( 1 + 1440 T^{2} + 935910 T^{4} + 438945600 T^{6} + 186482707321 T^{8} + 56682302220480 T^{10} + 7723885307690070 T^{12} - 285644590063174560 T^{14} - \)\(25\!\cdots\!40\)\( T^{16} - \)\(37\!\cdots\!60\)\( T^{18} + \)\(13\!\cdots\!70\)\( T^{20} + \)\(12\!\cdots\!80\)\( T^{22} + \)\(53\!\cdots\!01\)\( T^{24} + \)\(16\!\cdots\!00\)\( T^{26} + \)\(45\!\cdots\!10\)\( T^{28} + \)\(91\!\cdots\!40\)\( T^{30} + \)\(83\!\cdots\!61\)\( T^{32} \))
$23$ (\( 1 - 928 T^{2} - 110362 T^{4} + 76027072 T^{6} + 161836431161 T^{8} - 2090789193920 T^{10} - 54941991557559978 T^{12} + 14303404978544854944 T^{14} - \)\(12\!\cdots\!04\)\( T^{16} + \)\(40\!\cdots\!04\)\( T^{18} - \)\(43\!\cdots\!18\)\( T^{20} - \)\(45\!\cdots\!20\)\( T^{22} + \)\(99\!\cdots\!21\)\( T^{24} + \)\(13\!\cdots\!72\)\( T^{26} - \)\(53\!\cdots\!42\)\( T^{28} - \)\(12\!\cdots\!68\)\( T^{30} + \)\(37\!\cdots\!21\)\( T^{32} \))(\( 1 - 1648 T^{2} + 1017702 T^{4} - 249312032 T^{6} + 42919392953 T^{8} - 81219467468736 T^{10} + 45669118532829142 T^{12} + 12560582613209435984 T^{14} - \)\(19\!\cdots\!76\)\( T^{16} + \)\(35\!\cdots\!44\)\( T^{18} + \)\(35\!\cdots\!02\)\( T^{20} - \)\(17\!\cdots\!56\)\( T^{22} + \)\(26\!\cdots\!33\)\( T^{24} - \)\(42\!\cdots\!32\)\( T^{26} + \)\(48\!\cdots\!82\)\( T^{28} - \)\(22\!\cdots\!88\)\( T^{30} + \)\(37\!\cdots\!21\)\( T^{32} \))
$29$ (\( ( 1 - 28 T + 1376 T^{2} - 41412 T^{3} + 2055086 T^{4} - 34827492 T^{5} + 973218656 T^{6} - 16655052988 T^{7} + 500246412961 T^{8} )^{4} \))(\( ( 1 + 4 T + 2608 T^{2} + 8572 T^{3} + 2977102 T^{4} + 7209052 T^{5} + 1844588848 T^{6} + 2379293284 T^{7} + 500246412961 T^{8} )^{4} \))
$31$ (\( 1 + 4480 T^{2} + 9494774 T^{4} + 14306099840 T^{6} + 19217452630025 T^{8} + 24003699927942080 T^{10} + 26963891194383292806 T^{12} + \)\(27\!\cdots\!40\)\( T^{14} + \)\(26\!\cdots\!04\)\( T^{16} + \)\(25\!\cdots\!40\)\( T^{18} + \)\(22\!\cdots\!46\)\( T^{20} + \)\(18\!\cdots\!80\)\( T^{22} + \)\(13\!\cdots\!25\)\( T^{24} + \)\(96\!\cdots\!40\)\( T^{26} + \)\(58\!\cdots\!54\)\( T^{28} + \)\(25\!\cdots\!80\)\( T^{30} + \)\(52\!\cdots\!61\)\( T^{32} \))(\( 1 + 4944 T^{2} + 12401142 T^{4} + 21706728288 T^{6} + 30278528813833 T^{8} + 35431200182034624 T^{10} + 36428088221729894790 T^{12} + \)\(35\!\cdots\!36\)\( T^{14} + \)\(33\!\cdots\!48\)\( T^{16} + \)\(32\!\cdots\!56\)\( T^{18} + \)\(31\!\cdots\!90\)\( T^{20} + \)\(27\!\cdots\!64\)\( T^{22} + \)\(22\!\cdots\!73\)\( T^{24} + \)\(14\!\cdots\!88\)\( T^{26} + \)\(76\!\cdots\!82\)\( T^{28} + \)\(28\!\cdots\!04\)\( T^{30} + \)\(52\!\cdots\!61\)\( T^{32} \))
$37$ (\( ( 1 - 4 T - 2770 T^{2} + 121912 T^{3} + 3850313 T^{4} - 247260872 T^{5} + 4348488942 T^{6} + 233812310196 T^{7} - 10747661313644 T^{8} + 320089052658324 T^{9} + 8149768384027662 T^{10} - 634403749202768648 T^{11} + 13524145303664927273 T^{12} + \)\(58\!\cdots\!88\)\( T^{13} - \)\(18\!\cdots\!70\)\( T^{14} - \)\(36\!\cdots\!56\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} )^{2} \))(\( ( 1 - 36 T - 2658 T^{2} + 179832 T^{3} + 2782297 T^{4} - 349680552 T^{5} + 4944965886 T^{6} + 248707896180 T^{7} - 11947732536204 T^{8} + 340481109870420 T^{9} + 9267662209871646 T^{10} - 897184626980097768 T^{11} + 9772761047206036537 T^{12} + \)\(86\!\cdots\!68\)\( T^{13} - \)\(17\!\cdots\!98\)\( T^{14} - \)\(32\!\cdots\!04\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} )^{2} \))
$41$ (\( ( 1 - 4384 T^{2} + 14370524 T^{4} - 32104130784 T^{6} + 62557592438726 T^{8} - 90718600708326624 T^{10} + \)\(11\!\cdots\!04\)\( T^{12} - \)\(98\!\cdots\!04\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} )^{2} \))(\( ( 1 - 10816 T^{2} + 53826588 T^{4} - 162742362560 T^{6} + 329722113042758 T^{8} - 459871021169908160 T^{10} + \)\(42\!\cdots\!48\)\( T^{12} - \)\(24\!\cdots\!96\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} )^{2} \))
$43$ (\( ( 1 + 11176 T^{2} + 58293116 T^{4} + 188186249880 T^{6} + 415246366681670 T^{8} + 643371339275993880 T^{10} + \)\(68\!\cdots\!16\)\( T^{12} + \)\(44\!\cdots\!76\)\( T^{14} + \)\(13\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 + 5992 T^{2} + 25433596 T^{4} + 70773684952 T^{6} + 153071479203142 T^{8} + 241961144887582552 T^{10} + \)\(29\!\cdots\!96\)\( T^{12} + \)\(23\!\cdots\!92\)\( T^{14} + \)\(13\!\cdots\!01\)\( T^{16} )^{2} \))
$47$ (\( 1 + 7504 T^{2} + 21182918 T^{4} + 37880838368 T^{6} + 100768600793177 T^{8} + 218689095748226816 T^{10} + \)\(22\!\cdots\!70\)\( T^{12} + \)\(76\!\cdots\!12\)\( T^{14} + \)\(29\!\cdots\!24\)\( T^{16} + \)\(37\!\cdots\!72\)\( T^{18} + \)\(52\!\cdots\!70\)\( T^{20} + \)\(25\!\cdots\!56\)\( T^{22} + \)\(57\!\cdots\!17\)\( T^{24} + \)\(10\!\cdots\!68\)\( T^{26} + \)\(28\!\cdots\!58\)\( T^{28} + \)\(49\!\cdots\!44\)\( T^{30} + \)\(32\!\cdots\!41\)\( T^{32} \))(\( 1 + 7200 T^{2} + 21086790 T^{4} + 20987252160 T^{6} - 28679941197479 T^{8} - 70284530432138880 T^{10} + \)\(21\!\cdots\!30\)\( T^{12} + \)\(13\!\cdots\!80\)\( T^{14} + \)\(38\!\cdots\!20\)\( T^{16} + \)\(66\!\cdots\!80\)\( T^{18} + \)\(50\!\cdots\!30\)\( T^{20} - \)\(81\!\cdots\!80\)\( T^{22} - \)\(16\!\cdots\!59\)\( T^{24} + \)\(58\!\cdots\!60\)\( T^{26} + \)\(28\!\cdots\!90\)\( T^{28} + \)\(47\!\cdots\!00\)\( T^{30} + \)\(32\!\cdots\!41\)\( T^{32} \))
$53$ (\( ( 1 + 12 T - 5314 T^{2} - 123432 T^{3} + 13289689 T^{4} + 412137720 T^{5} + 8001350174 T^{6} - 662204794044 T^{7} - 79868347316108 T^{8} - 1860133266469596 T^{9} + 63134501522293694 T^{10} + 9134769260962685880 T^{11} + \)\(82\!\cdots\!29\)\( T^{12} - \)\(21\!\cdots\!68\)\( T^{13} - \)\(26\!\cdots\!74\)\( T^{14} + \)\(16\!\cdots\!28\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} )^{2} \))(\( ( 1 - 20 T - 6098 T^{2} + 355352 T^{3} + 15866345 T^{4} - 1398217832 T^{5} + 8057716910 T^{6} + 2171403120260 T^{7} - 80650058761388 T^{8} + 6099471364810340 T^{9} + 63579262181733710 T^{10} - 30990604965455452328 T^{11} + \)\(98\!\cdots\!45\)\( T^{12} + \)\(62\!\cdots\!48\)\( T^{13} - \)\(29\!\cdots\!18\)\( T^{14} - \)\(27\!\cdots\!80\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} )^{2} \))
$59$ (\( 1 + 12496 T^{2} + 69591254 T^{4} + 231196127072 T^{6} + 548940902299433 T^{8} + 1224313485336173888 T^{10} + \)\(76\!\cdots\!62\)\( T^{12} - \)\(17\!\cdots\!52\)\( T^{14} - \)\(10\!\cdots\!32\)\( T^{16} - \)\(21\!\cdots\!72\)\( T^{18} + \)\(11\!\cdots\!02\)\( T^{20} + \)\(21\!\cdots\!28\)\( T^{22} + \)\(11\!\cdots\!53\)\( T^{24} + \)\(60\!\cdots\!72\)\( T^{26} + \)\(22\!\cdots\!94\)\( T^{28} + \)\(47\!\cdots\!16\)\( T^{30} + \)\(46\!\cdots\!81\)\( T^{32} \))(\( 1 + 13056 T^{2} + 66899190 T^{4} + 216438091392 T^{6} + 1041428149652041 T^{8} + 6415617182341109568 T^{10} + \)\(27\!\cdots\!66\)\( T^{12} + \)\(85\!\cdots\!44\)\( T^{14} + \)\(25\!\cdots\!80\)\( T^{16} + \)\(10\!\cdots\!84\)\( T^{18} + \)\(41\!\cdots\!86\)\( T^{20} + \)\(11\!\cdots\!08\)\( T^{22} + \)\(22\!\cdots\!81\)\( T^{24} + \)\(56\!\cdots\!92\)\( T^{26} + \)\(21\!\cdots\!90\)\( T^{28} + \)\(50\!\cdots\!76\)\( T^{30} + \)\(46\!\cdots\!81\)\( T^{32} \))
$61$ (\( ( 1 + 180 T + 27726 T^{2} + 3046680 T^{3} + 308297129 T^{4} + 26423205000 T^{5} + 2082233555790 T^{6} + 145145432104140 T^{7} + 9348584465944212 T^{8} + 540086152859504940 T^{9} + 28830274738332969390 T^{10} + \)\(13\!\cdots\!00\)\( T^{11} + \)\(59\!\cdots\!49\)\( T^{12} + \)\(21\!\cdots\!80\)\( T^{13} + \)\(73\!\cdots\!46\)\( T^{14} + \)\(17\!\cdots\!80\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} )^{2} \))(\( ( 1 - 108 T + 15678 T^{2} - 1273320 T^{3} + 112514745 T^{4} - 8281698840 T^{5} + 635104006110 T^{6} - 41916162290292 T^{7} + 2814046950849908 T^{8} - 155970039882176532 T^{9} + 8793549087062088510 T^{10} - \)\(42\!\cdots\!40\)\( T^{11} + \)\(21\!\cdots\!45\)\( T^{12} - \)\(90\!\cdots\!20\)\( T^{13} + \)\(41\!\cdots\!38\)\( T^{14} - \)\(10\!\cdots\!28\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} )^{2} \))
$67$ (\( 1 - 5792 T^{2} - 24227898 T^{4} + 351601426496 T^{6} - 205859799182695 T^{8} - 8707473271916577024 T^{10} + \)\(34\!\cdots\!66\)\( T^{12} + \)\(81\!\cdots\!24\)\( T^{14} - \)\(92\!\cdots\!56\)\( T^{16} + \)\(16\!\cdots\!04\)\( T^{18} + \)\(14\!\cdots\!06\)\( T^{20} - \)\(71\!\cdots\!64\)\( T^{22} - \)\(33\!\cdots\!95\)\( T^{24} + \)\(11\!\cdots\!96\)\( T^{26} - \)\(16\!\cdots\!58\)\( T^{28} - \)\(78\!\cdots\!72\)\( T^{30} + \)\(27\!\cdots\!61\)\( T^{32} \))(\( 1 - 19248 T^{2} + 171825062 T^{4} - 1105279125792 T^{6} + 6322631625605113 T^{8} - 28716244420307792256 T^{10} + \)\(95\!\cdots\!42\)\( T^{12} - \)\(31\!\cdots\!16\)\( T^{14} + \)\(13\!\cdots\!84\)\( T^{16} - \)\(63\!\cdots\!36\)\( T^{18} + \)\(38\!\cdots\!22\)\( T^{20} - \)\(23\!\cdots\!16\)\( T^{22} + \)\(10\!\cdots\!53\)\( T^{24} - \)\(36\!\cdots\!92\)\( T^{26} + \)\(11\!\cdots\!02\)\( T^{28} - \)\(25\!\cdots\!68\)\( T^{30} + \)\(27\!\cdots\!61\)\( T^{32} \))
$71$ (\( ( 1 + 27624 T^{2} + 370534588 T^{4} + 3151580461272 T^{6} + 18747759294515334 T^{8} + 80086957327676918232 T^{10} + \)\(23\!\cdots\!68\)\( T^{12} + \)\(45\!\cdots\!84\)\( T^{14} + \)\(41\!\cdots\!21\)\( T^{16} )^{2} \))(\( ( 1 + 24232 T^{2} + 314425660 T^{4} + 2642217676312 T^{6} + 15735155452974982 T^{8} + 67143192723001800472 T^{10} + \)\(20\!\cdots\!60\)\( T^{12} + \)\(39\!\cdots\!12\)\( T^{14} + \)\(41\!\cdots\!21\)\( T^{16} )^{2} \))
$73$ (\( ( 1 - 36 T + 13374 T^{2} - 465912 T^{3} + 80365289 T^{4} - 3011641896 T^{5} + 480881406846 T^{6} - 17666763587772 T^{7} + 3025613230783092 T^{8} - 94146183159236988 T^{9} + 13656186084031757886 T^{10} - \)\(45\!\cdots\!44\)\( T^{11} + \)\(64\!\cdots\!09\)\( T^{12} - \)\(20\!\cdots\!88\)\( T^{13} + \)\(30\!\cdots\!54\)\( T^{14} - \)\(43\!\cdots\!24\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} )^{2} \))(\( ( 1 + 156 T + 22270 T^{2} + 2208648 T^{3} + 203639977 T^{4} + 17676576216 T^{5} + 1406019151294 T^{6} + 115719264813828 T^{7} + 8350708700464372 T^{8} + 616667962192889412 T^{9} + 39928470709062473854 T^{10} + \)\(26\!\cdots\!24\)\( T^{11} + \)\(16\!\cdots\!37\)\( T^{12} + \)\(94\!\cdots\!52\)\( T^{13} + \)\(51\!\cdots\!70\)\( T^{14} + \)\(19\!\cdots\!04\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} )^{2} \))
$79$ (\( 1 - 10112 T^{2} - 55382202 T^{4} + 565038672512 T^{6} + 4616239476368153 T^{8} - 23803283016229531968 T^{10} - \)\(25\!\cdots\!02\)\( T^{12} + \)\(18\!\cdots\!72\)\( T^{14} + \)\(13\!\cdots\!08\)\( T^{16} + \)\(71\!\cdots\!32\)\( T^{18} - \)\(37\!\cdots\!22\)\( T^{20} - \)\(14\!\cdots\!88\)\( T^{22} + \)\(10\!\cdots\!13\)\( T^{24} + \)\(50\!\cdots\!12\)\( T^{26} - \)\(19\!\cdots\!62\)\( T^{28} - \)\(13\!\cdots\!32\)\( T^{30} + \)\(52\!\cdots\!41\)\( T^{32} \))(\( 1 - 34640 T^{2} + 602988550 T^{4} - 7689396100960 T^{6} + 82922855327705561 T^{8} - \)\(77\!\cdots\!00\)\( T^{10} + \)\(63\!\cdots\!50\)\( T^{12} - \)\(46\!\cdots\!20\)\( T^{14} + \)\(30\!\cdots\!00\)\( T^{16} - \)\(17\!\cdots\!20\)\( T^{18} + \)\(95\!\cdots\!50\)\( T^{20} - \)\(45\!\cdots\!00\)\( T^{22} + \)\(19\!\cdots\!81\)\( T^{24} - \)\(68\!\cdots\!60\)\( T^{26} + \)\(21\!\cdots\!50\)\( T^{28} - \)\(47\!\cdots\!40\)\( T^{30} + \)\(52\!\cdots\!41\)\( T^{32} \))
$83$ (\( ( 1 - 24232 T^{2} + 235912988 T^{4} - 1035732354840 T^{6} + 3398417338650758 T^{8} - 49154118566082623640 T^{10} + \)\(53\!\cdots\!08\)\( T^{12} - \)\(25\!\cdots\!52\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} )^{2} \))(\( ( 1 - 35368 T^{2} + 588987420 T^{4} - 6301011254168 T^{6} + 49538235115611782 T^{8} - \)\(29\!\cdots\!28\)\( T^{10} + \)\(13\!\cdots\!20\)\( T^{12} - \)\(37\!\cdots\!48\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} )^{2} \))
$89$ (\( ( 1 - 204 T + 42982 T^{2} - 5938440 T^{3} + 800796785 T^{4} - 88063687656 T^{5} + 9449600595510 T^{6} - 894257389252164 T^{7} + 83094368751630116 T^{8} - 7083412780266391044 T^{9} + \)\(59\!\cdots\!10\)\( T^{10} - \)\(43\!\cdots\!16\)\( T^{11} + \)\(31\!\cdots\!85\)\( T^{12} - \)\(18\!\cdots\!40\)\( T^{13} + \)\(10\!\cdots\!22\)\( T^{14} - \)\(39\!\cdots\!64\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} )^{2} \))(\( ( 1 - 12 T + 8678 T^{2} - 103560 T^{3} + 5715441 T^{4} + 150778776 T^{5} - 475047477962 T^{6} + 26903544978684 T^{7} - 3925857799931356 T^{8} + 213102979776155964 T^{9} - 29805543348733992842 T^{10} + 74934230745999443736 T^{11} + \)\(22\!\cdots\!21\)\( T^{12} - \)\(32\!\cdots\!60\)\( T^{13} + \)\(21\!\cdots\!38\)\( T^{14} - \)\(23\!\cdots\!92\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} )^{2} \))
$97$ (\( ( 1 - 51968 T^{2} + 1305731356 T^{4} - 20766356196608 T^{6} + 230652504659485894 T^{8} - \)\(18\!\cdots\!48\)\( T^{10} + \)\(10\!\cdots\!16\)\( T^{12} - \)\(36\!\cdots\!88\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} )^{2} \))(\( ( 1 - 39520 T^{2} + 789394140 T^{4} - 11378905321376 T^{6} + 124539045704549702 T^{8} - \)\(10\!\cdots\!56\)\( T^{10} + \)\(61\!\cdots\!40\)\( T^{12} - \)\(27\!\cdots\!20\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} )^{2} \))
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