Properties

Label 450.2.s
Level 450
Weight 2
Character orbit s
Rep. character \(\chi_{450}(17,\cdot)\)
Character field \(\Q(\zeta_{20})\)
Dimension 80
Newform subspaces 4
Sturm bound 180
Trace bound 7

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Defining parameters

Level: \( N \) = \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 450.s (of order \(20\) and degree \(8\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 75 \)
Character field: \(\Q(\zeta_{20})\)
Newform subspaces: \( 4 \)
Sturm bound: \(180\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(450, [\chi])\).

Total New Old
Modular forms 784 80 704
Cusp forms 656 80 576
Eisenstein series 128 0 128

Trace form

\( 80q - 8q^{7} + O(q^{10}) \) \( 80q - 8q^{7} + 8q^{10} + 12q^{13} + 20q^{16} + 80q^{19} + 72q^{22} + 16q^{25} + 32q^{28} + 20q^{34} + 12q^{37} + 4q^{40} - 12q^{52} - 24q^{55} - 4q^{58} - 144q^{67} - 152q^{70} - 164q^{73} - 160q^{79} - 132q^{82} - 172q^{85} - 8q^{88} + 52q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(450, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
450.2.s.a \(16\) \(3.593\) \(\Q(\zeta_{40})\) None \(0\) \(0\) \(0\) \(-12\) \(q+\zeta_{40}^{11}q^{2}-\zeta_{40}^{2}q^{4}+(-\zeta_{40}+2\zeta_{40}^{3}+\cdots)q^{5}+\cdots\)
450.2.s.b \(16\) \(3.593\) \(\Q(\zeta_{40})\) None \(0\) \(0\) \(0\) \(-8\) \(q+\zeta_{40}^{11}q^{2}-\zeta_{40}^{2}q^{4}+(-\zeta_{40}^{3}+\cdots)q^{5}+\cdots\)
450.2.s.c \(16\) \(3.593\) \(\Q(\zeta_{40})\) None \(0\) \(0\) \(0\) \(8\) \(q+\zeta_{40}^{11}q^{2}-\zeta_{40}^{2}q^{4}+(2\zeta_{40}-2\zeta_{40}^{5}+\cdots)q^{5}+\cdots\)
450.2.s.d \(32\) \(3.593\) None \(0\) \(0\) \(0\) \(4\)

Decomposition of \(S_{2}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(450, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T^{4} + T^{8} - T^{12} + T^{16} \))(\( 1 - T^{4} + T^{8} - T^{12} + T^{16} \))(\( 1 - T^{4} + T^{8} - T^{12} + T^{16} \))
$3$ (\( \))(\( \))(\( \))
$5$ (\( 1 + 20 T^{2} + 215 T^{4} + 1600 T^{6} + 9025 T^{8} + 40000 T^{10} + 134375 T^{12} + 312500 T^{14} + 390625 T^{16} \))(\( 1 - 25 T^{4} + 625 T^{8} - 15625 T^{12} + 390625 T^{16} \))(\( 1 - 25 T^{4} + 625 T^{8} - 15625 T^{12} + 390625 T^{16} \))
$7$ (\( ( 1 + 6 T + 18 T^{2} + 42 T^{3} + 51 T^{4} + 6 T^{5} - 84 T^{7} - 391 T^{8} - 588 T^{9} + 2058 T^{11} + 122451 T^{12} + 705894 T^{13} + 2117682 T^{14} + 4941258 T^{15} + 5764801 T^{16} )^{2} \))(\( ( 1 + 4 T + 8 T^{2} + 28 T^{3} + 36 T^{4} - 116 T^{5} - 360 T^{6} - 1836 T^{7} - 8186 T^{8} - 12852 T^{9} - 17640 T^{10} - 39788 T^{11} + 86436 T^{12} + 470596 T^{13} + 941192 T^{14} + 3294172 T^{15} + 5764801 T^{16} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} - 10 T^{3} + 46 T^{4} - 70 T^{5} + 98 T^{6} - 686 T^{7} + 2401 T^{8} )^{4} \))
$11$ (\( 1 - 12 T^{2} + 91 T^{4} - 2688 T^{6} + 17211 T^{8} + 239784 T^{10} - 985351 T^{12} + 34325316 T^{14} - 880731088 T^{16} + 4153363236 T^{18} - 14426523991 T^{20} + 424791982824 T^{22} + 3689330700891 T^{24} - 69719797327488 T^{26} + 285596982281611 T^{28} - 4556998002998892 T^{30} + 45949729863572161 T^{32} \))(\( 1 + 28 T^{2} + 346 T^{4} + 812 T^{6} - 18349 T^{8} - 257236 T^{10} - 1486636 T^{12} - 19318544 T^{14} - 196288523 T^{16} - 2337543824 T^{18} - 21765837676 T^{20} - 455709265396 T^{22} - 3933271107469 T^{24} + 21061188776012 T^{26} + 1085896218345466 T^{28} + 10632995340330748 T^{30} + 45949729863572161 T^{32} \))(\( 1 + 48 T^{2} + 1086 T^{4} + 17112 T^{6} + 223091 T^{8} + 2411184 T^{10} + 21509484 T^{12} + 178999296 T^{14} + 1725584677 T^{16} + 21658914816 T^{18} + 314920355244 T^{20} + 4271559538224 T^{22} + 47821537121171 T^{24} + 443841209772312 T^{26} + 3408333217119006 T^{28} + 18227992011995568 T^{30} + 45949729863572161 T^{32} \))
$13$ (\( ( 1 + 6 T + 18 T^{2} + 18 T^{3} - 397 T^{4} - 1482 T^{5} - 1584 T^{6} + 12384 T^{7} + 106993 T^{8} + 160992 T^{9} - 267696 T^{10} - 3255954 T^{11} - 11338717 T^{12} + 6683274 T^{13} + 86882562 T^{14} + 376491102 T^{15} + 815730721 T^{16} )^{2} \))(\( ( 1 - 26 T + 318 T^{2} - 2478 T^{3} + 14303 T^{4} - 68578 T^{5} + 297696 T^{6} - 1210344 T^{7} + 4559593 T^{8} - 15734472 T^{9} + 50310624 T^{10} - 150665866 T^{11} + 408507983 T^{12} - 920064054 T^{13} + 1534925262 T^{14} - 1631461442 T^{15} + 815730721 T^{16} )^{2} \))(\( ( 1 - 4 T + 18 T^{2} - 92 T^{3} + 303 T^{4} - 1532 T^{5} + 6696 T^{6} - 24136 T^{7} + 110813 T^{8} - 313768 T^{9} + 1131624 T^{10} - 3365804 T^{11} + 8653983 T^{12} - 34158956 T^{13} + 86882562 T^{14} - 250994068 T^{15} + 815730721 T^{16} )^{2} \))
$17$ (\( 1 + 254 T^{4} + 7920 T^{6} + 148035 T^{8} + 2011680 T^{10} + 67255996 T^{12} + 940160160 T^{14} + 18529167749 T^{16} + 271706286240 T^{18} + 5617288041916 T^{20} + 48557064805920 T^{22} + 1032656252778435 T^{24} + 15966671691556080 T^{26} + 147986048256359294 T^{28} + 48661191875666868481 T^{32} \))(\( 1 + 120 T^{2} + 6934 T^{4} + 246280 T^{6} + 5559915 T^{8} + 64132600 T^{10} - 532570324 T^{12} - 41573911040 T^{14} - 955095750571 T^{16} - 12014860290560 T^{18} - 44480806030804 T^{20} + 1548005057649400 T^{22} + 38784618432577515 T^{24} + 496498977802579720 T^{26} + 4039902592951162774 T^{28} + 20205339187128111480 T^{30} + 48661191875666868481 T^{32} \))(\( 1 - 80 T^{2} + 3974 T^{4} - 147920 T^{6} + 4462555 T^{8} - 114064400 T^{10} + 2550078076 T^{12} - 50683682400 T^{14} + 906436851429 T^{16} - 14647584213600 T^{18} + 212985070985596 T^{20} - 2753237325443600 T^{22} + 31129701247121755 T^{24} - 298205817754416080 T^{26} + 2315340770751070214 T^{28} - 13470226124752074320 T^{30} + 48661191875666868481 T^{32} \))
$19$ (\( ( 1 + 10 T + 64 T^{2} + 490 T^{3} + 3215 T^{4} + 17550 T^{5} + 94576 T^{6} + 460240 T^{7} + 2031909 T^{8} + 8744560 T^{9} + 34141936 T^{10} + 120375450 T^{11} + 418982015 T^{12} + 1213288510 T^{13} + 3010936384 T^{14} + 8938717390 T^{15} + 16983563041 T^{16} )^{2} \))(\( ( 1 - 20 T + 194 T^{2} - 1180 T^{3} + 4675 T^{4} - 9700 T^{5} - 16704 T^{6} + 265000 T^{7} - 1452571 T^{8} + 5035000 T^{9} - 6030144 T^{10} - 66532300 T^{11} + 609250675 T^{12} - 2921796820 T^{13} + 9126900914 T^{14} - 17877434780 T^{15} + 16983563041 T^{16} )^{2} \))(\( ( 1 - 10 T + 84 T^{2} - 570 T^{3} + 3295 T^{4} - 17270 T^{5} + 86976 T^{6} - 412920 T^{7} + 1808949 T^{8} - 7845480 T^{9} + 31398336 T^{10} - 118454930 T^{11} + 429407695 T^{12} - 1411376430 T^{13} + 3951854004 T^{14} - 8938717390 T^{15} + 16983563041 T^{16} )^{2} \))
$23$ (\( 1 - 20 T^{2} - 1142 T^{4} + 38020 T^{6} + 43043 T^{8} - 23628420 T^{10} + 598894036 T^{12} + 5084006440 T^{14} - 464001309675 T^{16} + 2689439406760 T^{18} + 167595105928276 T^{20} - 3497854160365380 T^{22} + 3370739739450083 T^{24} + 1575035956342934980 T^{26} - 25026501101367206582 T^{28} - \)\(23\!\cdots\!80\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 - 80 T^{2} + 2318 T^{4} - 3200 T^{6} - 1874717 T^{8} + 61433680 T^{10} - 500626244 T^{12} - 26881238400 T^{14} + 1068565235205 T^{16} - 14220175113600 T^{18} - 140095748747204 T^{20} + 9094389433341520 T^{22} - 146810935393040477 T^{24} - 132564835883676800 T^{26} + 50798099433423104078 T^{28} - \)\(92\!\cdots\!20\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 + 120 T^{2} + 6318 T^{4} + 163820 T^{6} + 993803 T^{8} - 53927760 T^{10} - 610383064 T^{12} + 54470742040 T^{14} + 2185815167925 T^{16} + 28815022539160 T^{18} - 170810207012824 T^{20} - 7983243893378640 T^{22} + 77825692105213643 T^{24} + 6786491067019979180 T^{26} + \)\(13\!\cdots\!78\)\( T^{28} + \)\(13\!\cdots\!80\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 - 192 T^{2} + 16686 T^{4} - 845568 T^{6} + 26196851 T^{8} - 409181376 T^{10} - 4464999396 T^{12} + 516630798336 T^{14} - 19092281863643 T^{16} + 434486501400576 T^{18} - 3158009237802276 T^{20} - 243390624963669696 T^{22} + 13104880743623785811 T^{24} - \)\(35\!\cdots\!68\)\( T^{26} + \)\(59\!\cdots\!26\)\( T^{28} - \)\(57\!\cdots\!52\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( 1 - 32 T^{2} + 1646 T^{4} - 47008 T^{6} + 1481171 T^{8} - 25990176 T^{10} - 202052036 T^{12} + 5111955456 T^{14} - 683802514043 T^{16} + 4299154538496 T^{18} - 142907566074116 T^{20} - 15459562801694496 T^{22} + 740950479731857331 T^{24} - 19776605622975848608 T^{26} + \)\(58\!\cdots\!86\)\( T^{28} - \)\(95\!\cdots\!92\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( 1 + 8 T^{2} + 406 T^{4} + 28072 T^{6} + 29771 T^{8} - 23577976 T^{10} + 27058604 T^{12} - 6773159744 T^{14} - 1045442657003 T^{16} - 5696227344704 T^{18} + 19138036495724 T^{20} - 14024729986778296 T^{22} + 14892835960261931 T^{24} + 11810093453203242472 T^{26} + \)\(14\!\cdots\!46\)\( T^{28} + \)\(23\!\cdots\!48\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( ( 1 - 2 T - 49 T^{2} + 72 T^{3} - 49 T^{4} + 9284 T^{5} + 12189 T^{6} - 289254 T^{7} + 502152 T^{8} - 8966874 T^{9} + 11713629 T^{10} + 276579644 T^{11} - 45252529 T^{12} + 2061298872 T^{13} - 43487680369 T^{14} - 55025228222 T^{15} + 852891037441 T^{16} )^{2} \))(\( ( 1 + 8 T + 46 T^{2} + 132 T^{3} + 1451 T^{4} - 2536 T^{5} - 53176 T^{6} - 483904 T^{7} - 1408043 T^{8} - 15001024 T^{9} - 51102136 T^{10} - 75549976 T^{11} + 1340028971 T^{12} + 3779047932 T^{13} + 40825169326 T^{14} + 220100912888 T^{15} + 852891037441 T^{16} )^{2} \))(\( ( 1 - 22 T + 196 T^{2} - 1098 T^{3} + 6311 T^{4} - 30226 T^{5} + 65144 T^{6} - 236644 T^{7} + 2436517 T^{8} - 7335964 T^{9} + 62603384 T^{10} - 900462766 T^{11} + 5828341031 T^{12} - 31434807798 T^{13} + 173950721476 T^{14} - 605277510442 T^{15} + 852891037441 T^{16} )^{2} \))
$37$ (\( ( 1 - 4 T - 72 T^{2} + 582 T^{3} + 1211 T^{4} - 30884 T^{5} + 87510 T^{6} + 543576 T^{7} - 6157031 T^{8} + 20112312 T^{9} + 119801190 T^{10} - 1564367252 T^{11} + 2269608971 T^{12} + 40358182974 T^{13} - 184732301448 T^{14} - 379727508532 T^{15} + 3512479453921 T^{16} )^{2} \))(\( ( 1 - 26 T + 263 T^{2} - 692 T^{3} - 11549 T^{4} + 133984 T^{5} - 380795 T^{6} - 3856026 T^{7} + 43560144 T^{8} - 142672962 T^{9} - 521308355 T^{10} + 6786691552 T^{11} - 21644685389 T^{12} - 47986018244 T^{13} + 674786045567 T^{14} - 2468228805458 T^{15} + 3512479453921 T^{16} )^{2} \))(\( ( 1 + 16 T + 113 T^{2} + 382 T^{3} + 591 T^{4} + 4486 T^{5} - 17865 T^{6} - 1039164 T^{7} - 9891776 T^{8} - 38449068 T^{9} - 24457185 T^{10} + 227229358 T^{11} + 1107629151 T^{12} + 26489391574 T^{13} + 289927084217 T^{14} + 1518910034128 T^{15} + 3512479453921 T^{16} )^{2} \))
$41$ (\( 1 + 148 T^{2} + 10186 T^{4} + 511812 T^{6} + 22120451 T^{8} + 951728644 T^{10} + 50831464964 T^{12} + 2647072625296 T^{14} + 115842725462677 T^{16} + 4449729083122576 T^{18} + 143637571268137604 T^{20} + 4520810268145579204 T^{22} + \)\(17\!\cdots\!71\)\( T^{24} + \)\(68\!\cdots\!12\)\( T^{26} + \)\(22\!\cdots\!66\)\( T^{28} + \)\(56\!\cdots\!28\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( 1 + 88 T^{2} + 1766 T^{4} - 111688 T^{6} - 6853269 T^{8} + 23392984 T^{10} + 10813494204 T^{12} + 210775851776 T^{14} - 1464522267323 T^{16} + 354314206835456 T^{18} + 30556350195389244 T^{20} + 111119112508045144 T^{22} - 54722840540052846549 T^{24} - \)\(14\!\cdots\!88\)\( T^{26} + \)\(39\!\cdots\!46\)\( T^{28} + \)\(33\!\cdots\!68\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( 1 - 32 T^{2} + 3566 T^{4} - 89088 T^{6} + 5735891 T^{8} - 178432096 T^{10} + 7187309484 T^{12} - 197672681664 T^{14} + 7645459761717 T^{16} - 332287777877184 T^{18} + 20309618834817324 T^{20} - 847571055940119136 T^{22} + 45800660757388081811 T^{24} - \)\(11\!\cdots\!88\)\( T^{26} + \)\(80\!\cdots\!46\)\( T^{28} - \)\(12\!\cdots\!52\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( ( 1 - 10 T + 50 T^{2} - 330 T^{3} + 2078 T^{4} - 14190 T^{5} + 92450 T^{6} - 795070 T^{7} + 3418801 T^{8} )^{4} \))(\( ( 1 - 20 T + 200 T^{2} - 1340 T^{3} + 4516 T^{4} + 580 T^{5} - 17000 T^{6} - 657780 T^{7} + 7777446 T^{8} - 28284540 T^{9} - 31433000 T^{10} + 46114060 T^{11} + 15439305316 T^{12} - 196991313620 T^{13} + 1264272609800 T^{14} - 5436372222140 T^{15} + 11688200277601 T^{16} )^{2} \))(\( ( 1 + 20 T + 200 T^{2} + 1620 T^{3} + 11996 T^{4} + 82100 T^{5} + 555000 T^{6} + 3765140 T^{7} + 25118886 T^{8} + 161901020 T^{9} + 1026195000 T^{10} + 6527524700 T^{11} + 41011936796 T^{12} + 238153677660 T^{13} + 1264272609800 T^{14} + 5436372222140 T^{15} + 11688200277601 T^{16} )^{2} \))
$47$ (\( 1 - 2418 T^{4} + 10664643 T^{8} - 18983595716 T^{12} + 70311124062405 T^{16} - 92633891327046596 T^{20} + \)\(25\!\cdots\!23\)\( T^{24} - \)\(28\!\cdots\!38\)\( T^{28} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 - 80 T^{2} - 898 T^{4} + 279920 T^{6} - 6280317 T^{8} - 360782480 T^{10} + 25011505244 T^{12} + 1426394400 T^{14} - 41525313744635 T^{16} + 3150905229600 T^{18} + 122048166920547164 T^{20} - 3888952038850635920 T^{22} - \)\(14\!\cdots\!37\)\( T^{24} + \)\(14\!\cdots\!80\)\( T^{26} - \)\(10\!\cdots\!18\)\( T^{28} - \)\(20\!\cdots\!20\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 + 222 T^{4} + 113100 T^{6} - 5672277 T^{8} + 25108200 T^{10} + 782541304 T^{12} - 619100250600 T^{14} + 9233449631925 T^{16} - 1367592453575400 T^{18} + 3818551932844024 T^{20} + 270646694323597800 T^{22} - \)\(13\!\cdots\!97\)\( T^{24} + \)\(59\!\cdots\!00\)\( T^{26} + \)\(25\!\cdots\!02\)\( T^{28} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 + 160 T^{2} + 15958 T^{4} + 1479200 T^{6} + 120952283 T^{8} + 8737124640 T^{10} + 582380844716 T^{12} + 34774340762400 T^{14} + 1896903908812405 T^{16} + 97681123201581600 T^{18} + 4595264989995548396 T^{20} + \)\(19\!\cdots\!60\)\( T^{22} + \)\(75\!\cdots\!63\)\( T^{24} + \)\(25\!\cdots\!00\)\( T^{26} + \)\(78\!\cdots\!78\)\( T^{28} + \)\(22\!\cdots\!40\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 - 240 T^{2} + 33963 T^{4} - 3098800 T^{6} + 190267763 T^{8} - 5833470960 T^{10} - 191429745559 T^{12} + 39064778448400 T^{14} - 2711479735984320 T^{16} + 109732962661555600 T^{18} - 1510472770168123879 T^{20} - \)\(12\!\cdots\!40\)\( T^{22} + \)\(11\!\cdots\!43\)\( T^{24} - \)\(54\!\cdots\!00\)\( T^{26} + \)\(16\!\cdots\!83\)\( T^{28} - \)\(33\!\cdots\!60\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 + 80 T^{2} + 9083 T^{4} + 237520 T^{6} + 13372803 T^{8} - 897844720 T^{10} - 19475533479 T^{12} - 4451159434160 T^{14} - 59910604162400 T^{16} - 12503306850555440 T^{18} - 153671326880913399 T^{20} - 19900154611845888880 T^{22} + \)\(83\!\cdots\!83\)\( T^{24} + \)\(41\!\cdots\!80\)\( T^{26} + \)\(44\!\cdots\!03\)\( T^{28} + \)\(11\!\cdots\!20\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 - 260 T^{2} + 30683 T^{4} - 2245240 T^{6} + 115059923 T^{8} - 2957973360 T^{10} - 201554837159 T^{12} + 33046870553100 T^{14} - 2446822440910400 T^{16} + 115036156395341100 T^{18} - 2442312723151817399 T^{20} - \)\(12\!\cdots\!60\)\( T^{22} + \)\(16\!\cdots\!83\)\( T^{24} - \)\(11\!\cdots\!40\)\( T^{26} + \)\(54\!\cdots\!23\)\( T^{28} - \)\(16\!\cdots\!60\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 + 20 T^{2} + 6778 T^{4} + 167940 T^{6} + 29483123 T^{8} - 85841980 T^{10} + 104726571236 T^{12} - 1425328591600 T^{14} + 303412597742005 T^{16} - 4961568827359600 T^{18} + 1269009669958828196 T^{20} - 3620860525200049180 T^{22} + \)\(43\!\cdots\!83\)\( T^{24} + \)\(85\!\cdots\!40\)\( T^{26} + \)\(12\!\cdots\!18\)\( T^{28} + \)\(12\!\cdots\!20\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 - 60 T^{2} + 698 T^{4} + 166340 T^{6} - 5597837 T^{8} + 630362740 T^{10} - 10656119484 T^{12} - 5386959600 T^{14} + 114009289145525 T^{16} - 18752006367600 T^{18} - 129124046646761724 T^{20} + 26589036760602936340 T^{22} - \)\(82\!\cdots\!77\)\( T^{24} + \)\(85\!\cdots\!40\)\( T^{26} + \)\(12\!\cdots\!38\)\( T^{28} - \)\(37\!\cdots\!60\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( ( 1 - 182 T^{2} + 750 T^{3} + 10603 T^{4} - 113200 T^{5} + 287326 T^{6} + 4141900 T^{7} - 58577995 T^{8} + 252655900 T^{9} + 1069140046 T^{10} - 25694249200 T^{11} + 146807452123 T^{12} + 633447225750 T^{13} - 9376708133702 T^{14} + 191707312997281 T^{16} )^{2} \))(\( ( 1 + 20 T + 218 T^{2} + 1620 T^{3} + 10823 T^{4} + 3220 T^{5} - 850044 T^{6} - 11196000 T^{7} - 89705475 T^{8} - 682956000 T^{9} - 3163013724 T^{10} + 730878820 T^{11} + 149853537143 T^{12} + 1368246007620 T^{13} + 11231441610698 T^{14} + 62854856720420 T^{15} + 191707312997281 T^{16} )^{2} \))(\( ( 1 - 30 T + 478 T^{2} - 6450 T^{3} + 82783 T^{4} - 910430 T^{5} + 8719456 T^{6} - 78408700 T^{7} + 650729825 T^{8} - 4782930700 T^{9} + 32445095776 T^{10} - 206650311830 T^{11} + 1146200255503 T^{12} - 5447646141450 T^{13} + 24626738944558 T^{14} - 94282285080630 T^{15} + 191707312997281 T^{16} )^{2} \))
$67$ (\( ( 1 - 12 T + 72 T^{2} - 964 T^{3} + 4023 T^{4} + 41144 T^{5} - 226356 T^{6} + 3623708 T^{7} - 63176047 T^{8} + 242788436 T^{9} - 1016112084 T^{10} + 12374592872 T^{11} + 81067959783 T^{12} - 1301520603148 T^{13} + 6513003516168 T^{14} - 72728539263876 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 + 32 T + 572 T^{2} + 7664 T^{3} + 78423 T^{4} + 617576 T^{5} + 3884304 T^{6} + 20172132 T^{7} + 119941873 T^{8} + 1351532844 T^{9} + 17436640656 T^{10} + 185744010488 T^{11} + 1580311362183 T^{12} + 10347358820048 T^{13} + 51742194600668 T^{14} + 193942771370336 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 + 28 T + 312 T^{2} + 1446 T^{3} - 8557 T^{4} - 190756 T^{5} - 803166 T^{6} + 7412388 T^{7} + 112802353 T^{8} + 496629996 T^{9} - 3605412174 T^{10} - 57372346828 T^{11} - 172433142397 T^{12} + 1952280904722 T^{13} + 28223015236728 T^{14} + 169699924949044 T^{15} + 406067677556641 T^{16} )^{2} \))
$71$ (\( 1 + 240 T^{2} + 28638 T^{4} + 2436480 T^{6} + 132505763 T^{8} + 778560240 T^{10} - 702808292484 T^{12} - 96243271833600 T^{14} - 8226407207003195 T^{16} - 485162333313177600 T^{18} - 17859540132758105604 T^{20} + 99733787793601901040 T^{22} + \)\(85\!\cdots\!43\)\( T^{24} + \)\(79\!\cdots\!80\)\( T^{26} + \)\(46\!\cdots\!58\)\( T^{28} + \)\(19\!\cdots\!40\)\( T^{30} + \)\(41\!\cdots\!21\)\( T^{32} \))(\( 1 + 300 T^{2} + 29338 T^{4} - 689700 T^{6} - 392279837 T^{8} - 32340906500 T^{10} - 382411220684 T^{12} + 141968363327600 T^{14} + 15009370610202005 T^{16} + 715662519534431600 T^{18} - 9717711950842409804 T^{20} - \)\(41\!\cdots\!00\)\( T^{22} - \)\(25\!\cdots\!57\)\( T^{24} - \)\(22\!\cdots\!00\)\( T^{26} + \)\(48\!\cdots\!58\)\( T^{28} + \)\(24\!\cdots\!00\)\( T^{30} + \)\(41\!\cdots\!21\)\( T^{32} \))(\( 1 + 10398 T^{4} + 127680 T^{6} + 63554723 T^{8} + 898867200 T^{10} + 289113089916 T^{12} + 5645744025600 T^{14} + 1031640459128005 T^{16} + 28460195633049600 T^{18} + 7346849613869708796 T^{20} + \)\(11\!\cdots\!00\)\( T^{22} + \)\(41\!\cdots\!03\)\( T^{24} + \)\(41\!\cdots\!80\)\( T^{26} + \)\(17\!\cdots\!18\)\( T^{28} + \)\(41\!\cdots\!21\)\( T^{32} \))
$73$ (\( ( 1 + 38 T + 822 T^{2} + 11634 T^{3} + 108651 T^{4} + 462898 T^{5} - 4203400 T^{6} - 108626388 T^{7} - 1190551871 T^{8} - 7929726324 T^{9} - 22399918600 T^{10} + 180075191266 T^{11} + 3085497282891 T^{12} + 24118114912962 T^{13} + 124396734009558 T^{14} + 419801143725686 T^{15} + 806460091894081 T^{16} )^{2} \))(\( ( 1 + 42 T + 762 T^{2} + 7106 T^{3} + 24251 T^{4} - 107238 T^{5} + 769440 T^{6} + 52193728 T^{7} + 663936289 T^{8} + 3810142144 T^{9} + 4100345760 T^{10} - 41717405046 T^{11} + 688685742491 T^{12} + 14731246739858 T^{13} + 115316680432218 T^{14} + 463990737802074 T^{15} + 806460091894081 T^{16} )^{2} \))(\( ( 1 + 18 T + 182 T^{2} + 2594 T^{3} + 22311 T^{4} + 96938 T^{5} + 775600 T^{6} + 2071352 T^{7} - 37224151 T^{8} + 151208696 T^{9} + 4133172400 T^{10} + 37710529946 T^{11} + 633593154951 T^{12} + 5377547712242 T^{13} + 27542829184598 T^{14} + 198853173343746 T^{15} + 806460091894081 T^{16} )^{2} \))
$79$ (\( ( 1 + 40 T + 754 T^{2} + 8720 T^{3} + 73535 T^{4} + 599640 T^{5} + 5958836 T^{6} + 62841840 T^{7} + 598203469 T^{8} + 4964505360 T^{9} + 37189095476 T^{10} + 295645905960 T^{11} + 2864194206335 T^{12} + 26831931799280 T^{13} + 183287941462834 T^{14} + 768156359446360 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 - 20 T + 374 T^{2} - 4460 T^{3} + 40195 T^{4} - 242420 T^{5} + 51916 T^{6} + 14638920 T^{7} - 204250251 T^{8} + 1156474680 T^{9} + 324007756 T^{10} - 119522514380 T^{11} + 1565598505795 T^{12} - 13723671539540 T^{13} + 90914708364854 T^{14} - 384078179723180 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 + 40 T + 894 T^{2} + 13520 T^{3} + 148475 T^{4} + 1160440 T^{5} + 5467896 T^{6} - 374560 T^{7} - 192027131 T^{8} - 29590240 T^{9} + 34125138936 T^{10} + 572142177160 T^{11} + 5783113276475 T^{12} + 41601802514480 T^{13} + 217320185235774 T^{14} + 768156359446360 T^{15} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( 1 - 580 T^{2} + 154859 T^{4} - 25466920 T^{6} + 2998408675 T^{8} - 294341024800 T^{10} + 28088706305641 T^{12} - 2695917785643540 T^{14} + 238862494458062304 T^{16} - 18572177625298347060 T^{18} + \)\(13\!\cdots\!61\)\( T^{20} - \)\(96\!\cdots\!00\)\( T^{22} + \)\(67\!\cdots\!75\)\( T^{24} - \)\(39\!\cdots\!80\)\( T^{26} + \)\(16\!\cdots\!99\)\( T^{28} - \)\(42\!\cdots\!20\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 + 160 T^{2} - 7746 T^{4} - 3638160 T^{6} - 152423245 T^{8} + 34631145600 T^{10} + 3619940373276 T^{12} - 118847176843360 T^{14} - 34464352785192731 T^{16} - 818738201273907040 T^{18} + \)\(17\!\cdots\!96\)\( T^{20} + \)\(11\!\cdots\!00\)\( T^{22} - \)\(34\!\cdots\!45\)\( T^{24} - \)\(56\!\cdots\!40\)\( T^{26} - \)\(82\!\cdots\!06\)\( T^{28} + \)\(11\!\cdots\!40\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 + 340 T^{2} + 53954 T^{4} + 5481400 T^{6} + 427618795 T^{8} + 29631942460 T^{10} + 1725376515256 T^{12} + 63935663086800 T^{14} + 1996749530688069 T^{16} + 440452783004965200 T^{18} + 81883472506880645176 T^{20} + \)\(96\!\cdots\!40\)\( T^{22} + \)\(96\!\cdots\!95\)\( T^{24} + \)\(85\!\cdots\!00\)\( T^{26} + \)\(57\!\cdots\!94\)\( T^{28} + \)\(25\!\cdots\!60\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( 1 - 240 T^{2} + 32478 T^{4} - 4754580 T^{6} + 555843243 T^{8} - 53178194640 T^{10} + 6042964371256 T^{12} - 599802308936400 T^{14} + 50437204132510005 T^{16} - 4751034089085224400 T^{18} + \)\(37\!\cdots\!96\)\( T^{20} - \)\(26\!\cdots\!40\)\( T^{22} + \)\(21\!\cdots\!83\)\( T^{24} - \)\(14\!\cdots\!80\)\( T^{26} + \)\(80\!\cdots\!38\)\( T^{28} - \)\(46\!\cdots\!40\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( 1 - 600 T^{2} + 154563 T^{4} - 21157800 T^{6} + 1408818603 T^{8} - 669069000 T^{10} - 5363516517119 T^{12} - 36180201070200 T^{14} + 43436587980306480 T^{16} - 286583372677054200 T^{18} - \)\(33\!\cdots\!79\)\( T^{20} - \)\(33\!\cdots\!00\)\( T^{22} + \)\(55\!\cdots\!43\)\( T^{24} - \)\(65\!\cdots\!00\)\( T^{26} + \)\(38\!\cdots\!23\)\( T^{28} - \)\(11\!\cdots\!00\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( 1 - 60 T^{2} - 16237 T^{4} - 66640 T^{6} + 103862803 T^{8} + 6521221240 T^{10} + 1202926888881 T^{12} - 51335664417900 T^{14} - 17826607103823520 T^{16} - 406629797854185900 T^{18} + 75474328767551922321 T^{20} + \)\(32\!\cdots\!40\)\( T^{22} + \)\(40\!\cdots\!43\)\( T^{24} - \)\(20\!\cdots\!40\)\( T^{26} - \)\(40\!\cdots\!77\)\( T^{28} - \)\(11\!\cdots\!60\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( ( 1 + 16 T + 3 T^{2} - 468 T^{3} + 7911 T^{4} + 125516 T^{5} + 616505 T^{6} + 7844856 T^{7} + 144464824 T^{8} + 760951032 T^{9} + 5800695545 T^{10} + 114555064268 T^{11} + 700355141991 T^{12} - 4018875240276 T^{13} + 2498916014787 T^{14} + 1292772551649808 T^{15} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 - 6 T + 18 T^{2} + 2058 T^{3} - 18589 T^{4} + 120954 T^{5} + 752400 T^{6} - 15236496 T^{7} + 198707569 T^{8} - 1477940112 T^{9} + 7079331600 T^{10} + 110391450042 T^{11} - 1645670804509 T^{12} + 17672746248906 T^{13} + 14993496088722 T^{14} - 484789706868678 T^{15} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 + 6 T + 198 T^{2} + 1602 T^{3} + 24431 T^{4} + 29166 T^{5} + 1810440 T^{6} - 6211824 T^{7} + 51901729 T^{8} - 602546928 T^{9} + 17034429960 T^{10} + 26619020718 T^{11} + 2162858864111 T^{12} + 13756919091714 T^{13} + 164928456975942 T^{14} + 484789706868678 T^{15} + 7837433594376961 T^{16} )^{2} \))
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