Properties

Label 170.2.k
Level $170$
Weight $2$
Character orbit 170.k
Rep. character $\chi_{170}(111,\cdot)$
Character field $\Q(\zeta_{8})$
Dimension $24$
Newform subspaces $2$
Sturm bound $54$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.k (of order \(8\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 17 \)
Character field: \(\Q(\zeta_{8})\)
Newform subspaces: \( 2 \)
Sturm bound: \(54\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(170, [\chi])\).

Total New Old
Modular forms 128 24 104
Cusp forms 96 24 72
Eisenstein series 32 0 32

Trace form

\( 24q + 8q^{9} + O(q^{10}) \) \( 24q + 8q^{9} + 8q^{11} - 8q^{12} - 16q^{14} + 16q^{15} - 24q^{16} - 8q^{18} - 16q^{19} - 16q^{22} + 32q^{23} - 8q^{24} - 24q^{27} + 16q^{29} + 16q^{31} + 16q^{34} + 16q^{35} + 8q^{36} - 16q^{37} - 32q^{41} + 32q^{42} - 24q^{43} - 32q^{45} - 32q^{46} - 16q^{49} - 8q^{50} - 48q^{51} + 16q^{52} - 48q^{53} + 40q^{54} - 24q^{59} - 16q^{60} - 48q^{61} + 48q^{62} + 64q^{63} - 16q^{65} + 8q^{66} + 16q^{67} - 32q^{69} + 32q^{71} + 8q^{75} + 16q^{76} + 32q^{77} + 64q^{78} + 32q^{79} + 24q^{82} + 40q^{83} + 32q^{84} - 16q^{86} + 24q^{88} + 16q^{91} + 16q^{92} - 64q^{93} + 32q^{94} + 32q^{95} - 8q^{97} + 16q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(170, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
170.2.k.a \(8\) \(1.357\) \(\Q(\zeta_{16})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{16}^{2}q^{2}+(\zeta_{16}+\zeta_{16}^{2}-\zeta_{16}^{3}+\cdots)q^{3}+\cdots\)
170.2.k.b \(16\) \(1.357\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{9}q^{2}+(-\beta _{6}+\beta _{15})q^{3}-\beta _{3}q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(170, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(170, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(17, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(34, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{4} \))
$3$ (\( 1 - 4 T^{2} - 8 T^{3} + 8 T^{4} + 32 T^{5} + 8 T^{6} - 24 T^{7} - 137 T^{8} - 72 T^{9} + 72 T^{10} + 864 T^{11} + 648 T^{12} - 1944 T^{13} - 2916 T^{14} + 6561 T^{16} \))(\( 1 + 8 T^{3} + 16 T^{5} + 84 T^{6} + 24 T^{7} + 213 T^{8} + 344 T^{9} + 408 T^{10} + 1600 T^{11} + 2312 T^{12} + 4504 T^{13} + 9380 T^{14} + 11040 T^{15} + 26884 T^{16} + 33120 T^{17} + 84420 T^{18} + 121608 T^{19} + 187272 T^{20} + 388800 T^{21} + 297432 T^{22} + 752328 T^{23} + 1397493 T^{24} + 472392 T^{25} + 4960116 T^{26} + 2834352 T^{27} + 12754584 T^{29} + 43046721 T^{32} \))
$5$ (\( 1 + T^{8} \))(\( ( 1 + T^{8} )^{2} \))
$7$ (\( 1 + 4 T^{2} - 32 T^{3} + 8 T^{4} - 160 T^{5} + 572 T^{6} - 576 T^{7} + 4414 T^{8} - 4032 T^{9} + 28028 T^{10} - 54880 T^{11} + 19208 T^{12} - 537824 T^{13} + 470596 T^{14} + 5764801 T^{16} \))(\( 1 + 4 T^{2} + 32 T^{3} + 8 T^{4} + 288 T^{5} + 380 T^{6} + 832 T^{7} + 9984 T^{8} + 3264 T^{9} + 37700 T^{10} + 200992 T^{11} + 77592 T^{12} + 1542176 T^{13} + 2426812 T^{14} + 2781312 T^{15} + 37848190 T^{16} + 19469184 T^{17} + 118913788 T^{18} + 528966368 T^{19} + 186298392 T^{20} + 3378072544 T^{21} + 4435367300 T^{22} + 2688044352 T^{23} + 57555773184 T^{24} + 33574201024 T^{25} + 107340594620 T^{26} + 569470101984 T^{27} + 110730297608 T^{28} + 3100448333024 T^{29} + 2712892291396 T^{30} + 33232930569601 T^{32} \))
$11$ (\( 1 - 16 T + 116 T^{2} - 464 T^{3} + 840 T^{4} + 1136 T^{5} - 10068 T^{6} + 22384 T^{7} - 37474 T^{8} + 246224 T^{9} - 1218228 T^{10} + 1512016 T^{11} + 12298440 T^{12} - 74727664 T^{13} + 205501076 T^{14} - 311794736 T^{15} + 214358881 T^{16} \))(\( 1 + 8 T - 16 T^{2} - 328 T^{3} - 448 T^{4} + 6040 T^{5} + 19712 T^{6} - 66008 T^{7} - 364948 T^{8} + 553512 T^{9} + 5007584 T^{10} - 4614824 T^{11} - 63537600 T^{12} + 36387768 T^{13} + 784865968 T^{14} - 147622328 T^{15} - 9020633882 T^{16} - 1623845608 T^{17} + 94968782128 T^{18} + 48432119208 T^{19} - 930254001600 T^{20} - 743222020024 T^{21} + 8871240518624 T^{22} + 10786382994552 T^{23} - 78229844903188 T^{24} - 155643411187528 T^{25} + 511278513734912 T^{26} + 1723282490490440 T^{27} - 1406015912771008 T^{28} - 11323449583209368 T^{29} - 6075997337331856 T^{30} + 33417985355325208 T^{31} + 45949729863572161 T^{32} \))
$13$ (\( 1 - 52 T^{2} + 1386 T^{4} - 26096 T^{6} + 381683 T^{8} - 4410224 T^{10} + 39585546 T^{12} - 250994068 T^{14} + 815730721 T^{16} \))(\( 1 - 76 T^{2} + 3430 T^{4} - 111192 T^{6} + 2863025 T^{8} - 61251048 T^{10} + 1119763510 T^{12} - 17768536788 T^{14} + 246624737540 T^{16} - 3002882717172 T^{18} + 31981565609110 T^{20} - 295647109745832 T^{22} + 2335457447491025 T^{24} - 15328761425674008 T^{26} + 79912431970109830 T^{28} - 299240605313145964 T^{30} + 665416609183179841 T^{32} \))
$17$ (\( 1 - 48 T^{2} + 1152 T^{4} - 13872 T^{6} + 83521 T^{8} \))(\( 1 - 16 T^{2} + 642 T^{4} - 160 T^{5} - 15280 T^{6} - 2848 T^{7} + 207874 T^{8} - 48416 T^{9} - 4415920 T^{10} - 786080 T^{11} + 53620482 T^{12} - 386201104 T^{14} + 6975757441 T^{16} \))
$19$ (\( 1 + 16 T + 128 T^{2} + 808 T^{3} + 3650 T^{4} + 8088 T^{5} - 11360 T^{6} - 251240 T^{7} - 1571141 T^{8} - 4773560 T^{9} - 4100960 T^{10} + 55475592 T^{11} + 475671650 T^{12} + 2000687992 T^{13} + 6021872768 T^{14} + 14301947824 T^{15} + 16983563041 T^{16} \))(\( 1 + 88 T^{3} + 102 T^{4} + 264 T^{5} + 3872 T^{6} + 34312 T^{7} + 146153 T^{8} + 250536 T^{9} + 2659360 T^{10} + 13384312 T^{11} + 30672606 T^{12} + 263275656 T^{13} + 995458496 T^{14} + 4638677568 T^{15} + 27904305492 T^{16} + 88134873792 T^{17} + 359360517056 T^{18} + 1805807724504 T^{19} + 3997284686526 T^{20} + 33140881558888 T^{21} + 125111934096160 T^{22} + 223947050002104 T^{23} + 2482198689131273 T^{24} + 11072060286193048 T^{25} + 23739488550205472 T^{26} + 30753428349129816 T^{27} + 225758121744748422 T^{28} + 3700662544678621192 T^{29} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( 1 - 24 T + 296 T^{2} - 2376 T^{3} + 13280 T^{4} - 49720 T^{5} + 88072 T^{6} + 294104 T^{7} - 2824990 T^{8} + 6764392 T^{9} + 46590088 T^{10} - 604943240 T^{11} + 3716288480 T^{12} - 15292750968 T^{13} + 43818623144 T^{14} - 81715810728 T^{15} + 78310985281 T^{16} \))(\( 1 - 8 T + 120 T^{2} - 872 T^{3} + 7520 T^{4} - 47768 T^{5} + 313496 T^{6} - 1818424 T^{7} + 10176164 T^{8} - 56108568 T^{9} + 291322632 T^{10} - 1563982584 T^{11} + 7887703072 T^{12} - 40951922824 T^{13} + 203064358952 T^{14} - 1008566309288 T^{15} + 4857811804614 T^{16} - 23197025113624 T^{17} + 107421045885608 T^{18} - 498262044999608 T^{19} + 2207302715371552 T^{20} - 10066328356650312 T^{21} + 43126204813939848 T^{22} - 191039880121129896 T^{23} + 796905429221042084 T^{24} - 3275259227268194312 T^{25} + 12987045559434106904 T^{26} - 45513816516032464936 T^{27} + \)\(16\!\cdots\!20\)\( T^{28} - \)\(43\!\cdots\!76\)\( T^{29} + \)\(13\!\cdots\!80\)\( T^{30} - \)\(21\!\cdots\!56\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 - 8 T + 304 T^{3} - 1408 T^{4} - 824 T^{5} + 41632 T^{6} - 43752 T^{7} - 672337 T^{8} - 1268808 T^{9} + 35012512 T^{10} - 20096536 T^{11} - 995851648 T^{12} + 6235389296 T^{13} - 137999010472 T^{15} + 500246412961 T^{16} \))(\( 1 - 8 T + 32 T^{2} + 416 T^{3} - 3840 T^{4} + 29384 T^{5} - 11840 T^{6} - 651048 T^{7} + 10221809 T^{8} - 36975912 T^{9} + 79144480 T^{10} + 1605711096 T^{11} - 10517929216 T^{12} + 61516520128 T^{13} + 44651055680 T^{14} - 1220613560392 T^{15} + 13350264496192 T^{16} - 35397793251368 T^{17} + 37551537826880 T^{18} + 1500326409401792 T^{19} - 7439131493821696 T^{20} + 32934979541009304 T^{21} + 47076982432418080 T^{22} - 637829908412468808 T^{23} + 5113423286222466449 T^{24} - 9444848373297560712 T^{25} - 4981173642274379840 T^{26} + \)\(35\!\cdots\!36\)\( T^{27} - \)\(13\!\cdots\!40\)\( T^{28} + \)\(42\!\cdots\!24\)\( T^{29} + \)\(95\!\cdots\!92\)\( T^{30} - \)\(69\!\cdots\!92\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 + 16 T + 68 T^{2} - 152 T^{3} - 1144 T^{4} + 5584 T^{5} + 19224 T^{6} - 453096 T^{7} - 4328313 T^{8} - 14045976 T^{9} + 18474264 T^{10} + 166352944 T^{11} - 1056508024 T^{12} - 4351630952 T^{13} + 60350250308 T^{14} + 440201825776 T^{15} + 852891037441 T^{16} \))(\( 1 - 32 T + 548 T^{2} - 6808 T^{3} + 69000 T^{4} - 608640 T^{5} + 4865496 T^{6} - 35869848 T^{7} + 246301929 T^{8} - 1595242696 T^{9} + 9882803936 T^{10} - 59393372880 T^{11} + 349673282200 T^{12} - 2025742485832 T^{13} + 11589796641940 T^{14} - 65585046187504 T^{15} + 367102777561424 T^{16} - 2033136431812624 T^{17} + 11137794572904340 T^{18} - 60348894395421112 T^{19} + 322930619250626200 T^{20} - 1700381840580824880 T^{21} + 8771024871801288416 T^{22} - 43889296708439283256 T^{23} + \)\(21\!\cdots\!89\)\( T^{24} - \)\(94\!\cdots\!08\)\( T^{25} + \)\(39\!\cdots\!96\)\( T^{26} - \)\(15\!\cdots\!40\)\( T^{27} + \)\(54\!\cdots\!00\)\( T^{28} - \)\(16\!\cdots\!28\)\( T^{29} + \)\(41\!\cdots\!08\)\( T^{30} - \)\(75\!\cdots\!32\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( ( 1 + 4 T + 54 T^{2} + 452 T^{3} + 2466 T^{4} + 16724 T^{5} + 73926 T^{6} + 202612 T^{7} + 1874161 T^{8} )^{2} \))(\( 1 + 8 T + 12 T^{2} + 200 T^{3} + 1928 T^{4} + 12312 T^{5} + 148644 T^{6} + 711608 T^{7} + 3174928 T^{8} + 34103656 T^{9} + 148373724 T^{10} + 1112598856 T^{11} + 10158573080 T^{12} + 40152417688 T^{13} + 300897419892 T^{14} + 1928212146840 T^{15} + 6429418423390 T^{16} + 71343849433080 T^{17} + 411928567832148 T^{18} + 2033840413150264 T^{19} + 19038801482185880 T^{20} + 77152007228713192 T^{21} + 380686382068477116 T^{22} + 3237524081178098248 T^{23} + 11151869367678492688 T^{24} + 92481813732095153816 T^{25} + \)\(71\!\cdots\!56\)\( T^{26} + \)\(21\!\cdots\!56\)\( T^{27} + \)\(12\!\cdots\!68\)\( T^{28} + \)\(48\!\cdots\!00\)\( T^{29} + \)\(10\!\cdots\!68\)\( T^{30} + \)\(26\!\cdots\!44\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( 1 + 68 T^{2} + 2312 T^{4} - 9856 T^{5} - 31220 T^{6} - 939904 T^{7} - 3008818 T^{8} - 38536064 T^{9} - 52480820 T^{10} - 679285376 T^{11} + 6533159432 T^{12} + 323007088388 T^{14} + 7984925229121 T^{16} \))(\( 1 + 32 T + 520 T^{2} + 5088 T^{3} + 27680 T^{4} - 17248 T^{5} - 2012456 T^{6} - 22724256 T^{7} - 145472452 T^{8} - 382619616 T^{9} + 3679213384 T^{10} + 59763425248 T^{11} + 446162317792 T^{12} + 1726334249376 T^{13} - 2696546863400 T^{14} - 100460662345632 T^{15} - 856449502946298 T^{16} - 4118887156170912 T^{17} - 4532895277375400 T^{18} + 118980682801243296 T^{19} + 1260748077286239712 T^{20} + 6923963407980762848 T^{21} + 17476647098882361544 T^{22} - 74516805486707049696 T^{23} - \)\(11\!\cdots\!92\)\( T^{24} - \)\(74\!\cdots\!16\)\( T^{25} - \)\(27\!\cdots\!56\)\( T^{26} - \)\(94\!\cdots\!68\)\( T^{27} + \)\(62\!\cdots\!80\)\( T^{28} + \)\(47\!\cdots\!48\)\( T^{29} + \)\(19\!\cdots\!20\)\( T^{30} + \)\(49\!\cdots\!32\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( 1 + 8 T + 32 T^{2} + 200 T^{3} + 4904 T^{4} + 40696 T^{5} + 188640 T^{6} + 1480888 T^{7} + 11243730 T^{8} + 63678184 T^{9} + 348795360 T^{10} + 3235616872 T^{11} + 16765800104 T^{12} + 29401688600 T^{13} + 202283617568 T^{14} + 2174548888856 T^{15} + 11688200277601 T^{16} \))(\( 1 + 16 T + 128 T^{2} + 720 T^{3} + 2756 T^{4} + 21120 T^{5} + 244352 T^{6} + 1810176 T^{7} + 7254104 T^{8} + 17393920 T^{9} + 202697216 T^{10} + 2477751552 T^{11} + 22372983660 T^{12} + 124397766416 T^{13} + 585143289600 T^{14} + 4267259599952 T^{15} + 28438813484718 T^{16} + 183492162797936 T^{17} + 1081929942470400 T^{18} + 9890493214436912 T^{19} + 76488778909791660 T^{20} + 364250397800353536 T^{21} + 1281322691357571584 T^{22} + 4727991176106269440 T^{23} + 84787420386546524504 T^{24} + \)\(90\!\cdots\!68\)\( T^{25} + \)\(52\!\cdots\!48\)\( T^{26} + \)\(19\!\cdots\!40\)\( T^{27} + \)\(11\!\cdots\!56\)\( T^{28} + \)\(12\!\cdots\!60\)\( T^{29} + \)\(94\!\cdots\!72\)\( T^{30} + \)\(50\!\cdots\!12\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 - 292 T^{2} + 40470 T^{4} - 3443528 T^{6} + 195768731 T^{8} - 7606753352 T^{10} + 197480690070 T^{12} - 3147530876068 T^{14} + 23811286661761 T^{16} \))(\( 1 - 188 T^{2} + 22642 T^{4} - 1768080 T^{6} + 95073161 T^{8} - 2786921184 T^{10} - 46621754966 T^{12} + 10885491892716 T^{14} - 693159078792268 T^{16} + 24046051591009644 T^{18} - 227499291894245846 T^{20} - 30040823547287629536 T^{22} + \)\(22\!\cdots\!21\)\( T^{24} - \)\(92\!\cdots\!20\)\( T^{26} + \)\(26\!\cdots\!22\)\( T^{28} - \)\(48\!\cdots\!72\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 + 8 T + 32 T^{2} + 224 T^{3} + 5774 T^{4} + 49528 T^{5} + 236544 T^{6} + 2064840 T^{7} + 16695955 T^{8} + 109436520 T^{9} + 664452096 T^{10} + 7373580056 T^{11} + 45559637294 T^{12} + 93675790432 T^{13} + 709259556128 T^{14} + 9397689118696 T^{15} + 62259690411361 T^{16} \))(\( 1 + 40 T + 800 T^{2} + 11344 T^{3} + 124562 T^{4} + 1010264 T^{5} + 5104128 T^{6} - 5461208 T^{7} - 516663727 T^{8} - 7410897016 T^{9} - 70520559680 T^{10} - 498940491896 T^{11} - 2294252808206 T^{12} + 424322361200 T^{13} + 141643884807200 T^{14} + 1827245915743464 T^{15} + 15562066579376068 T^{16} + 96844033534403592 T^{17} + 397877672423424800 T^{18} + 63171840168372400 T^{19} - 18102758192346087086 T^{20} - \)\(20\!\cdots\!28\)\( T^{21} - \)\(15\!\cdots\!20\)\( T^{22} - \)\(87\!\cdots\!92\)\( T^{23} - \)\(32\!\cdots\!47\)\( T^{24} - \)\(18\!\cdots\!64\)\( T^{25} + \)\(89\!\cdots\!72\)\( T^{26} + \)\(93\!\cdots\!08\)\( T^{27} + \)\(61\!\cdots\!42\)\( T^{28} + \)\(29\!\cdots\!12\)\( T^{29} + \)\(11\!\cdots\!00\)\( T^{30} + \)\(29\!\cdots\!80\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 + 40 T + 800 T^{2} + 11768 T^{3} + 145074 T^{4} + 1520128 T^{5} + 13988832 T^{6} + 118894256 T^{7} + 946277099 T^{8} + 7014761104 T^{9} + 48695124192 T^{10} + 312202368512 T^{11} + 1757914029714 T^{12} + 8413229150632 T^{13} + 33744426912800 T^{14} + 99546059392760 T^{15} + 146830437604321 T^{16} \))(\( 1 - 16 T + 128 T^{2} - 896 T^{3} + 15118 T^{4} - 208456 T^{5} + 1801600 T^{6} - 14575512 T^{7} + 140518089 T^{8} - 1408025728 T^{9} + 12092118176 T^{10} - 107459276400 T^{11} + 935469995214 T^{12} - 7129230334024 T^{13} + 54379192071264 T^{14} - 476400349233432 T^{15} + 4096460971629972 T^{16} - 28107620604772488 T^{17} + 189293967600069984 T^{18} - 1464194196771515096 T^{19} + 11335427636676310254 T^{20} - 76825247851317243600 T^{21} + \)\(51\!\cdots\!16\)\( T^{22} - \)\(35\!\cdots\!32\)\( T^{23} + \)\(20\!\cdots\!69\)\( T^{24} - \)\(12\!\cdots\!68\)\( T^{25} + \)\(92\!\cdots\!00\)\( T^{26} - \)\(62\!\cdots\!04\)\( T^{27} + \)\(26\!\cdots\!58\)\( T^{28} - \)\(94\!\cdots\!84\)\( T^{29} + \)\(79\!\cdots\!08\)\( T^{30} - \)\(58\!\cdots\!84\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 + 24 T + 184 T^{2} - 1312 T^{3} - 37600 T^{4} - 293048 T^{5} - 205024 T^{6} + 17255496 T^{7} + 194698687 T^{8} + 1052585256 T^{9} - 762894304 T^{10} - 66516328088 T^{11} - 520603621600 T^{12} - 1108110346912 T^{13} + 9479748882424 T^{14} + 75425828064504 T^{15} + 191707312997281 T^{16} \))(\( 1 + 24 T + 552 T^{2} + 8160 T^{3} + 113184 T^{4} + 1253720 T^{5} + 13166416 T^{6} + 120528168 T^{7} + 1064498753 T^{8} + 8752960904 T^{9} + 71172146896 T^{10} + 573728005752 T^{11} + 4644413938400 T^{12} + 38440204280720 T^{13} + 313043743583352 T^{14} + 2568662577433176 T^{15} + 20124775252425440 T^{16} + 156688417223423736 T^{17} + 1164835769873652792 T^{18} + 8725196007842106320 T^{19} + 64305816929270194400 T^{20} + \)\(48\!\cdots\!52\)\( T^{21} + \)\(36\!\cdots\!56\)\( T^{22} + \)\(27\!\cdots\!84\)\( T^{23} + \)\(20\!\cdots\!93\)\( T^{24} + \)\(14\!\cdots\!88\)\( T^{25} + \)\(93\!\cdots\!16\)\( T^{26} + \)\(54\!\cdots\!20\)\( T^{27} + \)\(30\!\cdots\!64\)\( T^{28} + \)\(13\!\cdots\!60\)\( T^{29} + \)\(54\!\cdots\!32\)\( T^{30} + \)\(14\!\cdots\!24\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( ( 1 + 212 T^{2} - 224 T^{3} + 19182 T^{4} - 15008 T^{5} + 951668 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 8 T + 276 T^{2} - 2456 T^{3} + 40224 T^{4} - 383416 T^{5} + 4011676 T^{6} - 37708968 T^{7} + 303284510 T^{8} - 2526500856 T^{9} + 18008413564 T^{10} - 115317346408 T^{11} + 810558691104 T^{12} - 3315907262792 T^{13} + 24966513478644 T^{14} - 48485692842584 T^{15} + 406067677556641 T^{16} )^{2} \))
$71$ (\( 1 - 24 T + 372 T^{2} - 5176 T^{3} + 62088 T^{4} - 669672 T^{5} + 6915064 T^{6} - 65111408 T^{7} + 558097111 T^{8} - 4622909968 T^{9} + 34858837624 T^{10} - 239682975192 T^{11} + 1577760449928 T^{12} - 9338691120776 T^{13} + 47653305618612 T^{14} - 218282883801384 T^{15} + 645753531245761 T^{16} \))(\( 1 - 8 T + 324 T^{2} - 1992 T^{3} + 48712 T^{4} - 250440 T^{5} + 4639528 T^{6} - 25517088 T^{7} + 393483513 T^{8} - 2953677392 T^{9} + 37946061664 T^{10} - 310693631608 T^{11} + 3558244901336 T^{12} - 23908134462744 T^{13} + 277605488611268 T^{14} - 1477081706701080 T^{15} + 19716719812522288 T^{16} - 104872801175776680 T^{17} + 1399409268089401988 T^{18} - 8556984313695167784 T^{19} + 90420984352626905816 T^{20} - \)\(56\!\cdots\!08\)\( T^{21} + \)\(48\!\cdots\!44\)\( T^{22} - \)\(26\!\cdots\!72\)\( T^{23} + \)\(25\!\cdots\!93\)\( T^{24} - \)\(11\!\cdots\!28\)\( T^{25} + \)\(15\!\cdots\!28\)\( T^{26} - \)\(57\!\cdots\!40\)\( T^{27} + \)\(79\!\cdots\!92\)\( T^{28} - \)\(23\!\cdots\!12\)\( T^{29} + \)\(26\!\cdots\!44\)\( T^{30} - \)\(46\!\cdots\!08\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))
$73$ (\( 1 + 16 T + 144 T^{2} + 1920 T^{3} + 20608 T^{4} + 212848 T^{5} + 2145152 T^{6} + 18530416 T^{7} + 161770079 T^{8} + 1352720368 T^{9} + 11431515008 T^{10} + 82801490416 T^{11} + 585230950528 T^{12} + 3980297458560 T^{13} + 21792128585616 T^{14} + 176758376305552 T^{15} + 806460091894081 T^{16} \))(\( 1 - 16 T + 276 T^{2} - 2640 T^{3} + 26056 T^{4} - 186192 T^{5} + 1574636 T^{6} - 10818544 T^{7} + 75908205 T^{8} - 182870688 T^{9} + 855139432 T^{10} - 3726715264 T^{11} + 432868265920 T^{12} - 6285826956000 T^{13} + 78502153457000 T^{14} - 718659174458976 T^{15} + 6327370177850484 T^{16} - 52462119735505248 T^{17} + 418337975772353000 T^{18} - 2445293544942252000 T^{19} + 12292697336848246720 T^{20} - 7725747548997895552 T^{21} + \)\(12\!\cdots\!48\)\( T^{22} - \)\(20\!\cdots\!36\)\( T^{23} + \)\(61\!\cdots\!05\)\( T^{24} - \)\(63\!\cdots\!72\)\( T^{25} + \)\(67\!\cdots\!64\)\( T^{26} - \)\(58\!\cdots\!84\)\( T^{27} + \)\(59\!\cdots\!76\)\( T^{28} - \)\(44\!\cdots\!20\)\( T^{29} + \)\(33\!\cdots\!84\)\( T^{30} - \)\(14\!\cdots\!12\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 + 8 T - 40 T^{2} - 1240 T^{3} - 5536 T^{4} + 71320 T^{5} + 838872 T^{6} - 1864712 T^{7} - 47211870 T^{8} - 147312248 T^{9} + 5235400152 T^{10} + 35163541480 T^{11} - 215627648416 T^{12} - 3815549934760 T^{13} - 9723498220840 T^{14} + 153631271889272 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 40 T + 648 T^{2} - 2776 T^{3} - 75808 T^{4} + 1522712 T^{5} - 10404792 T^{6} - 39728344 T^{7} + 1478270884 T^{8} - 13459659000 T^{9} + 34995603032 T^{10} + 626696970424 T^{11} - 10252604611680 T^{12} + 77663922396552 T^{13} - 116715903085032 T^{14} - 5158833680727816 T^{15} + 70986929291924038 T^{16} - 407547860777497464 T^{17} - 728423951153684712 T^{18} + 38291342634473601528 T^{19} - \)\(39\!\cdots\!80\)\( T^{20} + \)\(19\!\cdots\!76\)\( T^{21} + \)\(85\!\cdots\!72\)\( T^{22} - \)\(25\!\cdots\!00\)\( T^{23} + \)\(22\!\cdots\!24\)\( T^{24} - \)\(47\!\cdots\!36\)\( T^{25} - \)\(98\!\cdots\!92\)\( T^{26} + \)\(11\!\cdots\!48\)\( T^{27} - \)\(44\!\cdots\!28\)\( T^{28} - \)\(12\!\cdots\!64\)\( T^{29} + \)\(23\!\cdots\!88\)\( T^{30} - \)\(11\!\cdots\!60\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( 1 - 8 T + 32 T^{2} - 568 T^{3} + 16072 T^{4} - 92744 T^{5} + 388960 T^{6} - 6798072 T^{7} + 117032466 T^{8} - 564239976 T^{9} + 2679545440 T^{10} - 53029813528 T^{11} + 762750135112 T^{12} - 2237375085224 T^{13} + 10462091947808 T^{14} - 217088407917016 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 32 T + 512 T^{2} - 4240 T^{3} + 32036 T^{4} - 651488 T^{5} + 13433984 T^{6} - 155605072 T^{7} + 1085940056 T^{8} - 7946115088 T^{9} + 139887075072 T^{10} - 2116082672992 T^{11} + 19160857300876 T^{12} - 111249085447120 T^{13} + 935146497989248 T^{14} - 14927162026453792 T^{15} + 176840852970889006 T^{16} - 1238954448195664736 T^{17} + 6442224224647929472 T^{18} - 63610780820552403440 T^{19} + \)\(90\!\cdots\!96\)\( T^{20} - \)\(83\!\cdots\!56\)\( T^{21} + \)\(45\!\cdots\!68\)\( T^{22} - \)\(21\!\cdots\!76\)\( T^{23} + \)\(24\!\cdots\!96\)\( T^{24} - \)\(29\!\cdots\!16\)\( T^{25} + \)\(20\!\cdots\!16\)\( T^{26} - \)\(83\!\cdots\!96\)\( T^{27} + \)\(34\!\cdots\!96\)\( T^{28} - \)\(37\!\cdots\!20\)\( T^{29} + \)\(37\!\cdots\!48\)\( T^{30} - \)\(19\!\cdots\!24\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( 1 - 452 T^{2} + 101642 T^{4} - 14938576 T^{6} + 1561986259 T^{8} - 118328460496 T^{10} + 6377246859722 T^{12} - 224635543514372 T^{14} + 3936588805702081 T^{16} \))(\( 1 - 452 T^{2} + 117710 T^{4} - 22520544 T^{6} + 3445383617 T^{8} - 447835564456 T^{10} + 51066923538006 T^{12} - 5226847939440180 T^{14} + 487096621627460484 T^{16} - 41401862528305665780 T^{18} + \)\(32\!\cdots\!46\)\( T^{20} - \)\(22\!\cdots\!16\)\( T^{22} + \)\(13\!\cdots\!77\)\( T^{24} - \)\(70\!\cdots\!44\)\( T^{26} + \)\(29\!\cdots\!10\)\( T^{28} - \)\(88\!\cdots\!32\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( 1 + 32 T + 464 T^{2} + 4544 T^{3} + 40064 T^{4} + 365264 T^{5} + 2561856 T^{6} + 1123488 T^{7} - 134491233 T^{8} + 108978336 T^{9} + 24104503104 T^{10} + 333366590672 T^{11} + 3546837113984 T^{12} + 39020874127808 T^{13} + 386499010287056 T^{14} + 2585545103299616 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 24 T + 620 T^{2} - 11256 T^{3} + 188168 T^{4} - 2644552 T^{5} + 33137396 T^{6} - 363116776 T^{7} + 3495879981 T^{8} - 27290302688 T^{9} + 151068862808 T^{10} + 49701229232 T^{11} - 16205192405056 T^{12} + 307163155178640 T^{13} - 4139600928735048 T^{14} + 47878710235684832 T^{15} - 491134601735993292 T^{16} + 4644234892861428704 T^{17} - 38949505138468066632 T^{18} + \)\(28\!\cdots\!20\)\( T^{19} - \)\(14\!\cdots\!36\)\( T^{20} + \)\(42\!\cdots\!24\)\( T^{21} + \)\(12\!\cdots\!32\)\( T^{22} - \)\(22\!\cdots\!44\)\( T^{23} + \)\(27\!\cdots\!41\)\( T^{24} - \)\(27\!\cdots\!92\)\( T^{25} + \)\(24\!\cdots\!04\)\( T^{26} - \)\(18\!\cdots\!56\)\( T^{27} + \)\(13\!\cdots\!88\)\( T^{28} - \)\(75\!\cdots\!12\)\( T^{29} + \)\(40\!\cdots\!80\)\( T^{30} - \)\(15\!\cdots\!32\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))
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