Properties

Label 229.2
Level 229
Weight 2
Dimension 2072
Nonzero newspaces 8
Newform subspaces 12
Sturm bound 8740
Trace bound 3

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Defining parameters

Level: \( N \) = \( 229 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 12 \)
Sturm bound: \(8740\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(229))\).

Total New Old
Modular forms 2299 2299 0
Cusp forms 2072 2072 0
Eisenstein series 227 227 0

Trace form

\( 2072q - 111q^{2} - 110q^{3} - 107q^{4} - 108q^{5} - 102q^{6} - 106q^{7} - 99q^{8} - 101q^{9} + O(q^{10}) \) \( 2072q - 111q^{2} - 110q^{3} - 107q^{4} - 108q^{5} - 102q^{6} - 106q^{7} - 99q^{8} - 101q^{9} - 96q^{10} - 102q^{11} - 86q^{12} - 100q^{13} - 90q^{14} - 90q^{15} - 83q^{16} - 96q^{17} - 75q^{18} - 94q^{19} - 72q^{20} - 82q^{21} - 78q^{22} - 90q^{23} - 54q^{24} - 83q^{25} - 72q^{26} - 74q^{27} - 58q^{28} - 84q^{29} - 42q^{30} - 82q^{31} - 51q^{32} - 66q^{33} - 60q^{34} - 66q^{35} - 23q^{36} - 76q^{37} - 54q^{38} - 58q^{39} - 24q^{40} - 72q^{41} - 18q^{42} - 70q^{43} - 30q^{44} - 36q^{45} - 42q^{46} - 66q^{47} + 10q^{48} - 57q^{49} - 21q^{50} - 42q^{51} - 16q^{52} - 60q^{53} + 6q^{54} - 42q^{55} + 6q^{56} - 34q^{57} - 24q^{58} - 54q^{59} + 54q^{60} - 52q^{61} - 18q^{62} - 10q^{63} + 13q^{64} - 30q^{65} + 30q^{66} - 46q^{67} + 12q^{68} - 18q^{69} + 30q^{70} - 42q^{71} + 81q^{72} - 40q^{73} + 10q^{75} + 26q^{76} - 18q^{77} + 54q^{78} - 34q^{79} + 72q^{80} + 7q^{81} + 12q^{82} - 30q^{83} + 110q^{84} - 6q^{85} + 18q^{86} + 6q^{87} + 66q^{88} - 24q^{89} + 120q^{90} - 2q^{91} + 54q^{92} + 14q^{93} + 30q^{94} + 6q^{95} + 138q^{96} - 16q^{97} + 57q^{98} + 42q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(229))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
229.2.a \(\chi_{229}(1, \cdot)\) 229.2.a.a 1 1
229.2.a.b 6
229.2.a.c 11
229.2.b \(\chi_{229}(228, \cdot)\) 229.2.b.a 2 1
229.2.b.b 16
229.2.c \(\chi_{229}(94, \cdot)\) 229.2.c.a 36 2
229.2.e \(\chi_{229}(95, \cdot)\) 229.2.e.a 2 2
229.2.e.b 36
229.2.g \(\chi_{229}(16, \cdot)\) 229.2.g.a 306 18
229.2.h \(\chi_{229}(4, \cdot)\) 229.2.h.a 324 18
229.2.i \(\chi_{229}(3, \cdot)\) 229.2.i.a 648 36
229.2.k \(\chi_{229}(5, \cdot)\) 229.2.k.a 684 36

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 + 4 T + 12 T^{2} + 28 T^{3} + 57 T^{4} + 97 T^{5} + 147 T^{6} + 194 T^{7} + 228 T^{8} + 224 T^{9} + 192 T^{10} + 128 T^{11} + 64 T^{12} \))(\( 1 - 5 T + 18 T^{2} - 50 T^{3} + 122 T^{4} - 265 T^{5} + 532 T^{6} - 987 T^{7} + 1721 T^{8} - 2806 T^{9} + 4330 T^{10} - 6287 T^{11} + 8660 T^{12} - 11224 T^{13} + 13768 T^{14} - 15792 T^{15} + 17024 T^{16} - 16960 T^{17} + 15616 T^{18} - 12800 T^{19} + 9216 T^{20} - 5120 T^{21} + 2048 T^{22} \))(\( 1 + T^{2} + 4 T^{4} \))(\( 1 - 8 T^{2} + 37 T^{4} - 118 T^{6} + 311 T^{8} - 757 T^{10} + 1858 T^{12} - 4373 T^{14} + 9365 T^{16} - 17492 T^{18} + 29728 T^{20} - 48448 T^{22} + 79616 T^{24} - 120832 T^{26} + 151552 T^{28} - 131072 T^{30} + 65536 T^{32} \))(\( 1 - T^{2} + 4 T^{4} \))
$3$ (\( 1 - T + 3 T^{2} \))(\( 1 + 6 T + 25 T^{2} + 73 T^{3} + 183 T^{4} + 381 T^{5} + 715 T^{6} + 1143 T^{7} + 1647 T^{8} + 1971 T^{9} + 2025 T^{10} + 1458 T^{11} + 729 T^{12} \))(\( 1 - 3 T + 14 T^{2} - 30 T^{3} + 91 T^{4} - 177 T^{5} + 455 T^{6} - 849 T^{7} + 1912 T^{8} - 3308 T^{9} + 6687 T^{10} - 10738 T^{11} + 20061 T^{12} - 29772 T^{13} + 51624 T^{14} - 68769 T^{15} + 110565 T^{16} - 129033 T^{17} + 199017 T^{18} - 196830 T^{19} + 275562 T^{20} - 177147 T^{21} + 177147 T^{22} \))(\( ( 1 - T + 3 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} - 5 T^{3} + 30 T^{4} - 31 T^{5} + 126 T^{6} - 114 T^{7} + 436 T^{8} - 342 T^{9} + 1134 T^{10} - 837 T^{11} + 2430 T^{12} - 1215 T^{13} + 5103 T^{14} + 6561 T^{16} )^{2} \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 + 3 T + 18 T^{2} + 36 T^{3} + 154 T^{4} + 286 T^{5} + 969 T^{6} + 1430 T^{7} + 3850 T^{8} + 4500 T^{9} + 11250 T^{10} + 9375 T^{11} + 15625 T^{12} \))(\( 1 + 27 T^{2} + 3 T^{3} + 319 T^{4} + 97 T^{5} + 2168 T^{6} + 1410 T^{7} + 9663 T^{8} + 12396 T^{9} + 35526 T^{10} + 73953 T^{11} + 177630 T^{12} + 309900 T^{13} + 1207875 T^{14} + 881250 T^{15} + 6775000 T^{16} + 1515625 T^{17} + 24921875 T^{18} + 1171875 T^{19} + 52734375 T^{20} + 48828125 T^{22} \))(\( ( 1 - 3 T + 5 T^{2} )^{2} \))(\( ( 1 + 6 T + 40 T^{2} + 160 T^{3} + 648 T^{4} + 1993 T^{5} + 6089 T^{6} + 15169 T^{7} + 37231 T^{8} + 75845 T^{9} + 152225 T^{10} + 249125 T^{11} + 405000 T^{12} + 500000 T^{13} + 625000 T^{14} + 468750 T^{15} + 390625 T^{16} )^{2} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))
$7$ (\( 1 - 2 T + 7 T^{2} \))(\( 1 + 5 T + 26 T^{2} + 48 T^{3} + 132 T^{4} - 4 T^{5} + 372 T^{6} - 28 T^{7} + 6468 T^{8} + 16464 T^{9} + 62426 T^{10} + 84035 T^{11} + 117649 T^{12} \))(\( 1 - T + 44 T^{2} - 44 T^{3} + 958 T^{4} - 1042 T^{5} + 13664 T^{6} - 16378 T^{7} + 144436 T^{8} - 183324 T^{9} + 1218159 T^{10} - 1496894 T^{11} + 8527113 T^{12} - 8982876 T^{13} + 49541548 T^{14} - 39323578 T^{15} + 229650848 T^{16} - 122590258 T^{17} + 788954194 T^{18} - 253651244 T^{19} + 1775558708 T^{20} - 282475249 T^{21} + 1977326743 T^{22} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 716 T^{4} - 10574 T^{6} + 124720 T^{8} - 1214049 T^{10} + 10353144 T^{12} - 79384418 T^{14} + 572635904 T^{16} - 3889836482 T^{18} + 24857898744 T^{20} - 142831650801 T^{22} + 718985980720 T^{24} - 2986893282926 T^{26} + 9910361635916 T^{28} - 23059584476866 T^{30} + 33232930569601 T^{32} \))(\( ( 1 - T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))
$11$ (\( 1 + 3 T + 11 T^{2} \))(\( 1 + 22 T + 256 T^{2} + 2025 T^{3} + 11990 T^{4} + 55511 T^{5} + 205343 T^{6} + 610621 T^{7} + 1450790 T^{8} + 2695275 T^{9} + 3748096 T^{10} + 3543122 T^{11} + 1771561 T^{12} \))(\( 1 - 27 T + 409 T^{2} - 4417 T^{3} + 37675 T^{4} - 267294 T^{5} + 1629088 T^{6} - 8705091 T^{7} + 41355517 T^{8} - 176271695 T^{9} + 678133500 T^{10} - 2362262440 T^{11} + 7459468500 T^{12} - 21328875095 T^{13} + 55044193127 T^{14} - 127451237331 T^{15} + 262366251488 T^{16} - 473527625934 T^{17} + 734179167425 T^{18} - 946823177377 T^{19} + 964400605619 T^{20} - 700310464227 T^{21} + 285311670611 T^{22} \))(\( ( 1 - 3 T + 11 T^{2} )^{2} \))(\( ( 1 + 4 T + 50 T^{2} + 185 T^{3} + 1295 T^{4} + 4371 T^{5} + 22737 T^{6} + 69250 T^{7} + 290786 T^{8} + 761750 T^{9} + 2751177 T^{10} + 5817801 T^{11} + 18960095 T^{12} + 29794435 T^{13} + 88578050 T^{14} + 77948684 T^{15} + 214358881 T^{16} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))
$13$ (\( 1 + 6 T + 13 T^{2} \))(\( 1 - T + 44 T^{2} + 56 T^{3} + 656 T^{4} + 3008 T^{5} + 6624 T^{6} + 39104 T^{7} + 110864 T^{8} + 123032 T^{9} + 1256684 T^{10} - 371293 T^{11} + 4826809 T^{12} \))(\( 1 + 7 T + 115 T^{2} + 707 T^{3} + 6330 T^{4} + 33972 T^{5} + 219022 T^{6} + 1022398 T^{7} + 5270221 T^{8} + 21356101 T^{9} + 92339671 T^{10} + 323773790 T^{11} + 1200415723 T^{12} + 3609181069 T^{13} + 11578675537 T^{14} + 29200709278 T^{15} + 81321335446 T^{16} + 163976355348 T^{17} + 397198112610 T^{18} + 576721619747 T^{19} + 1219517427895 T^{20} + 965009442943 T^{21} + 1792160394037 T^{22} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( 1 - 83 T^{2} + 3652 T^{4} - 114842 T^{6} + 2886577 T^{8} - 60801425 T^{10} + 1099853842 T^{12} - 17341481474 T^{14} + 240230833040 T^{16} - 2930710369106 T^{18} + 31412925581362 T^{20} - 293476865402825 T^{22} + 2354669537432017 T^{24} - 15831944920922858 T^{26} + 85084606867300612 T^{28} - 326802240013040987 T^{30} + 665416609183179841 T^{32} \))(\( ( 1 - 13 T^{2} )^{2} \))
$17$ (\( 1 + 7 T + 17 T^{2} \))(\( 1 - 6 T + 81 T^{2} - 329 T^{3} + 2623 T^{4} - 7971 T^{5} + 52173 T^{6} - 135507 T^{7} + 758047 T^{8} - 1616377 T^{9} + 6765201 T^{10} - 8519142 T^{11} + 24137569 T^{12} \))(\( 1 - T + 81 T^{2} - 5 T^{3} + 3142 T^{4} + 2460 T^{5} + 79407 T^{6} + 129763 T^{7} + 1523890 T^{8} + 3699552 T^{9} + 25882886 T^{10} + 73030943 T^{11} + 440009062 T^{12} + 1069170528 T^{13} + 7486871570 T^{14} + 10837935523 T^{15} + 112746584799 T^{16} + 59378419740 T^{17} + 1289284110566 T^{18} - 34878787205 T^{19} + 9605617996257 T^{20} - 2015993900449 T^{21} + 34271896307633 T^{22} \))(\( ( 1 + 3 T + 17 T^{2} )^{2} \))(\( ( 1 - 7 T + 92 T^{2} - 453 T^{3} + 3596 T^{4} - 13372 T^{5} + 85371 T^{6} - 262757 T^{7} + 1557053 T^{8} - 4466869 T^{9} + 24672219 T^{10} - 65696636 T^{11} + 300341516 T^{12} - 643195221 T^{13} + 2220656348 T^{14} - 2872370711 T^{15} + 6975757441 T^{16} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))
$19$ (\( 1 - 3 T + 19 T^{2} \))(\( 1 + 19 T + 242 T^{2} + 2132 T^{3} + 15132 T^{4} + 86016 T^{5} + 412853 T^{6} + 1634304 T^{7} + 5462652 T^{8} + 14623388 T^{9} + 31537682 T^{10} + 47045881 T^{11} + 47045881 T^{12} \))(\( 1 - 8 T + 108 T^{2} - 585 T^{3} + 4544 T^{4} - 16640 T^{5} + 84681 T^{6} - 80981 T^{7} - 213320 T^{8} + 8446280 T^{9} - 44354776 T^{10} + 264036988 T^{11} - 842740744 T^{12} + 3049107080 T^{13} - 1463161880 T^{14} - 10553524901 T^{15} + 209678539419 T^{16} - 782843459840 T^{17} + 4061753182016 T^{18} - 9935384378985 T^{19} + 34850271360132 T^{20} - 49048530062408 T^{21} + 116490258898219 T^{22} \))(\( ( 1 + T + 19 T^{2} )^{2} \))(\( ( 1 + T + 86 T^{2} + 52 T^{3} + 3849 T^{4} + 2105 T^{5} + 116145 T^{6} + 57198 T^{7} + 2545210 T^{8} + 1086762 T^{9} + 41928345 T^{10} + 14438195 T^{11} + 501605529 T^{12} + 128757148 T^{13} + 4045945766 T^{14} + 893871739 T^{15} + 16983563041 T^{16} )^{2} \))(\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \))
$23$ (\( 1 - 4 T + 23 T^{2} \))(\( 1 + 10 T + 121 T^{2} + 805 T^{3} + 6020 T^{4} + 30770 T^{5} + 175232 T^{6} + 707710 T^{7} + 3184580 T^{8} + 9794435 T^{9} + 33860761 T^{10} + 64363430 T^{11} + 148035889 T^{12} \))(\( 1 + 2 T + 105 T^{2} + 155 T^{3} + 5276 T^{4} + 4246 T^{5} + 174344 T^{6} + 58236 T^{7} + 4649848 T^{8} + 1092561 T^{9} + 114455252 T^{10} + 31523320 T^{11} + 2632470796 T^{12} + 577964769 T^{13} + 56574700616 T^{14} + 16296820476 T^{15} + 1122137783992 T^{16} + 628560384694 T^{17} + 17963859058372 T^{18} + 12138202718555 T^{19} + 189121029453615 T^{20} + 82853022427298 T^{21} + 952809757913927 T^{22} \))(\( 1 - 26 T^{2} + 529 T^{4} \))(\( 1 - 111 T^{2} + 7708 T^{4} - 359067 T^{6} + 12935139 T^{8} - 361013314 T^{10} + 8484083118 T^{12} - 175407125231 T^{14} + 3842668659170 T^{16} - 92790369247199 T^{18} + 2374194303824238 T^{20} - 53442926878826146 T^{22} + 1012963479836689059 T^{24} - 14874893101951305483 T^{26} + \)\(16\!\cdots\!68\)\( T^{28} - \)\(12\!\cdots\!99\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 - 3 T + 26 T^{2} - 69 T^{3} + 529 T^{4} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 + 7 T + 85 T^{2} + 230 T^{3} + 1615 T^{4} - 5538 T^{5} + 3872 T^{6} - 160602 T^{7} + 1358215 T^{8} + 5609470 T^{9} + 60118885 T^{10} + 143578043 T^{11} + 594823321 T^{12} \))(\( 1 - 17 T + 342 T^{2} - 4175 T^{3} + 49524 T^{4} - 470482 T^{5} + 4175139 T^{6} - 32368586 T^{7} + 233619080 T^{8} - 1522269325 T^{9} + 9271902326 T^{10} - 51588412974 T^{11} + 268885167454 T^{12} - 1280228502325 T^{13} + 5697735742120 T^{14} - 22893685874666 T^{15} + 85636898124711 T^{16} - 279853665710722 T^{17} + 854282874326916 T^{18} - 2088528774112175 T^{19} + 4961443923747198 T^{20} - 7152022966103417 T^{21} + 12200509765705829 T^{22} \))(\( 1 - 38 T^{2} + 841 T^{4} \))(\( 1 - 191 T^{2} + 18847 T^{4} - 1240684 T^{6} + 59820260 T^{8} - 2223160735 T^{10} + 66767157151 T^{12} - 1767490893446 T^{14} + 48098134782170 T^{16} - 1486459841388086 T^{18} + 47223141676916431 T^{20} - 1322387851509500935 T^{22} + 29924870487394389860 T^{24} - \)\(52\!\cdots\!84\)\( T^{26} + \)\(66\!\cdots\!27\)\( T^{28} - \)\(56\!\cdots\!71\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( 1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4} \))
$31$ (\( 1 - 4 T + 31 T^{2} \))(\( 1 + 3 T + 140 T^{2} + 363 T^{3} + 9096 T^{4} + 20019 T^{5} + 353954 T^{6} + 620589 T^{7} + 8741256 T^{8} + 10814133 T^{9} + 129292940 T^{10} + 85887453 T^{11} + 887503681 T^{12} \))(\( 1 + 3 T + 230 T^{2} + 417 T^{3} + 25240 T^{4} + 26205 T^{5} + 1805514 T^{6} + 1045991 T^{7} + 94279994 T^{8} + 31890065 T^{9} + 3759264739 T^{10} + 929885554 T^{11} + 116537206909 T^{12} + 30646352465 T^{13} + 2808695301254 T^{14} + 965994654311 T^{15} + 51690332938614 T^{16} + 23257033960605 T^{17} + 694418380161640 T^{18} + 355655562612897 T^{19} + 6081113096954330 T^{20} + 2458884860942403 T^{21} + 25408476896404831 T^{22} \))(\( ( 1 - 31 T^{2} )^{2} \))(\( 1 - 240 T^{2} + 31020 T^{4} - 2784587 T^{6} + 192020923 T^{8} - 10703010697 T^{10} + 496384321394 T^{12} - 19480388722295 T^{14} + 652666853734742 T^{16} - 18720653562125495 T^{18} + 458421344878108274 T^{20} - 9498961391369875657 T^{22} + \)\(16\!\cdots\!43\)\( T^{24} - \)\(22\!\cdots\!87\)\( T^{26} + \)\(24\!\cdots\!20\)\( T^{28} - \)\(18\!\cdots\!40\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 - 2 T + 30 T^{2} - 75 T^{3} + 1213 T^{4} - 9844 T^{5} + 49170 T^{6} - 364228 T^{7} + 1660597 T^{8} - 3798975 T^{9} + 56224830 T^{10} - 138687914 T^{11} + 2565726409 T^{12} \))(\( 1 + 14 T + 310 T^{2} + 3653 T^{3} + 48501 T^{4} + 473938 T^{5} + 4779962 T^{6} + 39575868 T^{7} + 327469915 T^{8} + 2326468687 T^{9} + 16324080063 T^{10} + 100102810320 T^{11} + 603990962331 T^{12} + 3184935632503 T^{13} + 16587333604495 T^{14} + 74171548346748 T^{15} + 331461479389634 T^{16} + 1215995242828642 T^{17} + 4604290972827633 T^{18} + 12831087445173413 T^{19} + 40288139336473870 T^{20} + 67320181213849886 T^{21} + 177917621779460413 T^{22} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 17 T + 281 T^{2} + 3142 T^{3} + 33367 T^{4} + 285756 T^{5} + 2303952 T^{6} + 15952696 T^{7} + 104146760 T^{8} + 590249752 T^{9} + 3154110288 T^{10} + 14474398668 T^{11} + 62535130087 T^{12} + 217878712894 T^{13} + 720969120929 T^{14} + 1613841911261 T^{15} + 3512479453921 T^{16} )^{2} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 + 12 T + 221 T^{2} + 1771 T^{3} + 18686 T^{4} + 114550 T^{5} + 926794 T^{6} + 4696550 T^{7} + 31411166 T^{8} + 122059091 T^{9} + 624493181 T^{10} + 1390274412 T^{11} + 4750104241 T^{12} \))(\( 1 - 18 T + 356 T^{2} - 4359 T^{3} + 51715 T^{4} - 479375 T^{5} + 4251298 T^{6} - 31859314 T^{7} + 232844625 T^{8} - 1514648943 T^{9} + 10081363949 T^{10} - 62702924686 T^{11} + 413335921909 T^{12} - 2546124873183 T^{13} + 16047884399625 T^{14} - 90026806987954 T^{15} + 492539235598898 T^{16} - 2277081220529375 T^{17} + 10071717273755915 T^{18} - 34806289073738439 T^{19} + 116547968644250116 T^{20} - 241607867582743218 T^{21} + 550329031716248441 T^{22} \))(\( ( 1 - 12 T + 41 T^{2} )( 1 + 12 T + 41 T^{2} ) \))(\( 1 - 440 T^{2} + 94975 T^{4} - 13407451 T^{6} + 1389803177 T^{8} - 112379988628 T^{10} + 7339258780819 T^{12} - 395046576538985 T^{14} + 17715215154082664 T^{16} - 664073295162033785 T^{18} + 20738991231745878259 T^{20} - \)\(53\!\cdots\!48\)\( T^{22} + \)\(11\!\cdots\!17\)\( T^{24} - \)\(17\!\cdots\!51\)\( T^{26} + \)\(21\!\cdots\!75\)\( T^{28} - \)\(16\!\cdots\!40\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( 1 - 3 T + 44 T^{2} - 123 T^{3} + 1681 T^{4} \))
$43$ (\( 1 - 7 T + 43 T^{2} \))(\( 1 + 9 T + 54 T^{2} + 500 T^{3} + 5830 T^{4} + 30298 T^{5} + 144843 T^{6} + 1302814 T^{7} + 10779670 T^{8} + 39753500 T^{9} + 184615254 T^{10} + 1323075987 T^{11} + 6321363049 T^{12} \))(\( 1 + 2 T + 238 T^{2} + 849 T^{3} + 29022 T^{4} + 132784 T^{5} + 2441971 T^{6} + 12367181 T^{7} + 157924658 T^{8} + 805769938 T^{9} + 8259375292 T^{10} + 39463973852 T^{11} + 355153137556 T^{12} + 1489868615362 T^{13} + 12556115783606 T^{14} + 42280930769981 T^{15} + 358990354561153 T^{16} + 839375871098416 T^{17} + 7888719731547354 T^{18} + 9923282035683249 T^{19} + 119617041640968634 T^{20} + 43222964626568498 T^{21} + 929293739471222707 T^{22} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 - 19 T + 310 T^{2} - 3194 T^{3} + 29757 T^{4} - 206765 T^{5} + 1416853 T^{6} - 8006500 T^{7} + 54339474 T^{8} - 344279500 T^{9} + 2619761197 T^{10} - 16439264855 T^{11} + 101733261357 T^{12} - 469544966942 T^{13} + 1959622545190 T^{14} - 5164553611033 T^{15} + 11688200277601 T^{16} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))
$47$ (\( 1 - 6 T + 47 T^{2} \))(\( 1 + 4 T + 55 T^{2} - 166 T^{3} + 1631 T^{4} + 13731 T^{5} + 250052 T^{6} + 645357 T^{7} + 3602879 T^{8} - 17234618 T^{9} + 268382455 T^{10} + 917380028 T^{11} + 10779215329 T^{12} \))(\( 1 - 14 T + 333 T^{2} - 3474 T^{3} + 51897 T^{4} - 457323 T^{5} + 5383274 T^{6} - 41797065 T^{7} + 414831647 T^{8} - 2859948264 T^{9} + 24684907302 T^{10} - 151609989804 T^{11} + 1160190643194 T^{12} - 6317625715176 T^{13} + 43069066086481 T^{14} - 203956343936265 T^{15} + 1234627013212918 T^{16} - 4929583091904267 T^{17} + 26292220082668311 T^{18} - 82720409862957714 T^{19} + 372670447543221411 T^{20} - 736387851301620686 T^{21} + 2472159215084012303 T^{22} \))(\( 1 - 14 T^{2} + 2209 T^{4} \))(\( 1 - 429 T^{2} + 89248 T^{4} - 12073318 T^{6} + 1201206099 T^{8} - 94346541797 T^{10} + 6146317617691 T^{12} - 345063082939647 T^{14} + 17150099258501604 T^{16} - 762244350213680223 T^{18} + 29992069299012036571 T^{20} - \)\(10\!\cdots\!13\)\( T^{22} + \)\(28\!\cdots\!39\)\( T^{24} - \)\(63\!\cdots\!82\)\( T^{26} + \)\(10\!\cdots\!68\)\( T^{28} - \)\(11\!\cdots\!01\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 - 9 T + 74 T^{2} - 423 T^{3} + 2209 T^{4} \))
$53$ (\( 1 + 10 T + 53 T^{2} \))(\( 1 - 5 T + 270 T^{2} - 1211 T^{3} + 32604 T^{4} - 122945 T^{5} + 2234304 T^{6} - 6516085 T^{7} + 91584636 T^{8} - 180290047 T^{9} + 2130429870 T^{10} - 2090977465 T^{11} + 22164361129 T^{12} \))(\( 1 + 11 T + 354 T^{2} + 3553 T^{3} + 65584 T^{4} + 592461 T^{5} + 8102050 T^{6} + 65518563 T^{7} + 730705114 T^{8} + 5264127809 T^{9} + 50092282289 T^{10} + 319315501126 T^{11} + 2654890961317 T^{12} + 14786935015481 T^{13} + 108785185256978 T^{14} + 516972976498803 T^{15} + 3388240794060650 T^{16} + 13131519558848469 T^{17} + 77042255395069808 T^{18} + 221208680031565633 T^{19} + 1168116311497955082 T^{20} + 1923762174020643539 T^{21} + 9269035929372191597 T^{22} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 14 T + 262 T^{2} + 2641 T^{3} + 30531 T^{4} + 261945 T^{5} + 2383262 T^{6} + 17866475 T^{7} + 140328262 T^{8} + 946923175 T^{9} + 6694582958 T^{10} + 38997585765 T^{11} + 240904275411 T^{12} + 1104454297013 T^{13} + 5807062615798 T^{14} + 16445955957718 T^{15} + 62259690411361 T^{16} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))
$59$ (\( 1 - 4 T + 59 T^{2} \))(\( 1 + 32 T + 715 T^{2} + 11019 T^{3} + 138484 T^{4} + 1386786 T^{5} + 11769428 T^{6} + 81820374 T^{7} + 482062804 T^{8} + 2263071201 T^{9} + 8663913115 T^{10} + 22877577568 T^{11} + 42180533641 T^{12} \))(\( 1 - 52 T + 1679 T^{2} - 39365 T^{3} + 743920 T^{4} - 11786868 T^{5} + 161770508 T^{6} - 1955806202 T^{7} + 21140228280 T^{8} - 205927463199 T^{9} + 1820504676714 T^{10} - 14645035536136 T^{11} + 107409775926126 T^{12} - 716833499395719 T^{13} + 4341758943918120 T^{14} - 23699209795672922 T^{15} + 115653667030773892 T^{16} - 497176382196026388 T^{17} + 1851357612586550480 T^{18} - 5779980176294096165 T^{19} + 14545169979521642581 T^{20} - 26578071171633352852 T^{21} + 30155888444737842659 T^{22} \))(\( 1 - 38 T^{2} + 3481 T^{4} \))(\( 1 - 603 T^{2} + 179192 T^{4} - 34906353 T^{6} + 5003565479 T^{8} - 561658458824 T^{10} + 51272316369020 T^{12} - 3896641819837207 T^{14} + 249793193575778362 T^{16} - 13564210174853317567 T^{18} + \)\(62\!\cdots\!20\)\( T^{20} - \)\(23\!\cdots\!84\)\( T^{22} + \)\(73\!\cdots\!59\)\( T^{24} - \)\(17\!\cdots\!53\)\( T^{26} + \)\(31\!\cdots\!52\)\( T^{28} - \)\(37\!\cdots\!83\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 - 21 T + 206 T^{2} - 1239 T^{3} + 3481 T^{4} \))
$61$ (\( 1 - 5 T + 61 T^{2} \))(\( 1 + 6 T + 78 T^{2} - 288 T^{3} + 2922 T^{4} + 24823 T^{5} + 658309 T^{6} + 1514203 T^{7} + 10872762 T^{8} - 65370528 T^{9} + 1079975598 T^{10} + 5067577806 T^{11} + 51520374361 T^{12} \))(\( 1 + 23 T + 730 T^{2} + 12569 T^{3} + 233280 T^{4} + 3181278 T^{5} + 43747810 T^{6} + 489505637 T^{7} + 5401065376 T^{8} + 50586214408 T^{9} + 463441123067 T^{10} + 3663591433667 T^{11} + 28269908507087 T^{12} + 188231303812168 T^{13} + 1225939220109856 T^{14} + 6777617218505717 T^{15} + 36949238502850810 T^{16} + 163900633506413358 T^{17} + 733139048786978880 T^{18} + 2409569217062824889 T^{19} + 8536726647768922930 T^{20} + 16406886968246299823 T^{21} + 43513917611435838661 T^{22} \))(\( ( 1 - 5 T + 61 T^{2} )^{2} \))(\( ( 1 - 13 T + 375 T^{2} - 4329 T^{3} + 65315 T^{4} - 671683 T^{5} + 7003540 T^{6} - 62874787 T^{7} + 510878041 T^{8} - 3835362007 T^{9} + 26060172340 T^{10} - 152459279023 T^{11} + 904341104915 T^{12} - 3656257387029 T^{13} + 19320140385375 T^{14} - 40855656868273 T^{15} + 191707312997281 T^{16} )^{2} \))(\( ( 1 - 10 T + 61 T^{2} )^{2} \))
$67$ (\( 1 + 10 T + 67 T^{2} \))(\( 1 - 2 T + 123 T^{2} + 578 T^{3} + 8565 T^{4} + 59551 T^{5} + 731956 T^{6} + 3989917 T^{7} + 38448285 T^{8} + 173841014 T^{9} + 2478587883 T^{10} - 2700250214 T^{11} + 90458382169 T^{12} \))(\( 1 - 18 T + 443 T^{2} - 6606 T^{3} + 98499 T^{4} - 1215617 T^{5} + 14570836 T^{6} - 153036301 T^{7} + 1583191611 T^{8} - 14655429456 T^{9} + 133035469476 T^{10} - 1103466095868 T^{11} + 8913376454892 T^{12} - 65788222827984 T^{13} + 476165458499193 T^{14} - 3083853018843421 T^{15} + 19672451513579452 T^{16} - 109962747157133273 T^{17} + 596974032412710177 T^{18} - 2682483077939170446 T^{19} + 12052494737558661521 T^{20} - 32811080481931706082 T^{21} + \)\(12\!\cdots\!83\)\( T^{22} \))(\( 1 + 46 T^{2} + 4489 T^{4} \))(\( 1 - 527 T^{2} + 143352 T^{4} - 26726780 T^{6} + 3818427235 T^{8} - 442828927195 T^{10} + 43091101327723 T^{12} - 3587173226330813 T^{14} + 258101323783461752 T^{16} - 16102820612999019557 T^{18} + \)\(86\!\cdots\!83\)\( T^{20} - \)\(40\!\cdots\!55\)\( T^{22} + \)\(15\!\cdots\!35\)\( T^{24} - \)\(48\!\cdots\!20\)\( T^{26} + \)\(11\!\cdots\!72\)\( T^{28} - \)\(19\!\cdots\!83\)\( T^{30} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( 1 - 3 T + 70 T^{2} - 201 T^{3} + 4489 T^{4} \))
$71$ (\( 1 + 9 T + 71 T^{2} \))(\( 1 + 32 T + 636 T^{2} + 9900 T^{3} + 122326 T^{4} + 1281001 T^{5} + 11678361 T^{6} + 90951071 T^{7} + 616645366 T^{8} + 3543318900 T^{9} + 16161829116 T^{10} + 57735339232 T^{11} + 128100283921 T^{12} \))(\( 1 - 31 T + 971 T^{2} - 18740 T^{3} + 348344 T^{4} - 4998111 T^{5} + 69197006 T^{6} - 800316049 T^{7} + 9017826545 T^{8} - 88139802118 T^{9} + 847167818465 T^{10} - 7184261953662 T^{11} + 60148915111015 T^{12} - 444312742476838 T^{13} + 3227579316547495 T^{14} - 20337376136368369 T^{15} + 124847269226523106 T^{16} - 640259438168673231 T^{17} + 3168230536454554504 T^{18} - 12101421175545561140 T^{19} + 44518894197614009101 T^{20} - \)\(10\!\cdots\!31\)\( T^{21} + \)\(23\!\cdots\!71\)\( T^{22} \))(\( ( 1 + 15 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 368 T^{2} - 1574 T^{3} + 63707 T^{4} - 197279 T^{5} + 7139319 T^{6} - 17214035 T^{7} + 583961990 T^{8} - 1222196485 T^{9} + 35989307079 T^{10} - 70608324169 T^{11} + 1618901961467 T^{12} - 2839856998474 T^{13} + 47140904482928 T^{14} - 54570720950346 T^{15} + 645753531245761 T^{16} )^{2} \))(\( 1 + 9 T + 10 T^{2} + 639 T^{3} + 5041 T^{4} \))
$73$ (\( 1 + 2 T + 73 T^{2} \))(\( 1 - 10 T + 312 T^{2} - 2115 T^{3} + 41777 T^{4} - 211940 T^{5} + 3561314 T^{6} - 15471620 T^{7} + 222629633 T^{8} - 822770955 T^{9} + 8860251192 T^{10} - 20730715930 T^{11} + 151334226289 T^{12} \))(\( 1 + 10 T + 429 T^{2} + 2983 T^{3} + 84095 T^{4} + 430333 T^{5} + 10894315 T^{6} + 41479222 T^{7} + 1084398924 T^{8} + 3149368953 T^{9} + 90417183708 T^{10} + 225688073638 T^{11} + 6600454410684 T^{12} + 16782987150537 T^{13} + 421849616217708 T^{14} + 1177936942848502 T^{15} + 22584694951693795 T^{16} + 65124111601624237 T^{17} + 929030978463462215 T^{18} + 2405670454120043623 T^{19} + 25255910697846934677 T^{20} + 42976258297035576490 T^{21} + \)\(31\!\cdots\!77\)\( T^{22} \))(\( 1 + 34 T^{2} + 5329 T^{4} \))(\( 1 - 540 T^{2} + 153292 T^{4} - 30340255 T^{6} + 4660528956 T^{8} - 585237069848 T^{10} + 61880730121118 T^{12} - 5607687633270269 T^{14} + 439551876124546322 T^{16} - 29883367397697263501 T^{18} + \)\(17\!\cdots\!38\)\( T^{20} - \)\(88\!\cdots\!72\)\( T^{22} + \)\(37\!\cdots\!36\)\( T^{24} - \)\(13\!\cdots\!95\)\( T^{26} + \)\(35\!\cdots\!32\)\( T^{28} - \)\(65\!\cdots\!60\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 - 10 T + 73 T^{2} ) \))
$79$ (\( 1 - 6 T + 79 T^{2} \))(\( 1 + 5 T + 126 T^{2} - 147 T^{3} + 5764 T^{4} - 77387 T^{5} + 176880 T^{6} - 6113573 T^{7} + 35973124 T^{8} - 72476733 T^{9} + 4907710206 T^{10} + 15385281995 T^{11} + 243087455521 T^{12} \))(\( 1 - T + 452 T^{2} - 177 T^{3} + 100814 T^{4} + 38197 T^{5} + 14981916 T^{6} + 18125381 T^{7} + 1691157120 T^{8} + 3150319833 T^{9} + 156470958127 T^{10} + 316775426130 T^{11} + 12361205692033 T^{12} + 19661146077753 T^{13} + 833806415287680 T^{14} + 705985058105861 T^{15} + 46100200497080484 T^{16} + 9285211538535637 T^{17} + 1936022880530633426 T^{18} - 268528259353461297 T^{19} + 54172921384143480188 T^{20} - 9468276082626847201 T^{21} + \)\(74\!\cdots\!79\)\( T^{22} \))(\( 1 + 22 T^{2} + 6241 T^{4} \))(\( 1 - 538 T^{2} + 131982 T^{4} - 20804869 T^{6} + 2625285951 T^{8} - 301355576425 T^{10} + 31081350208218 T^{12} - 2786901221599379 T^{14} + 226521248661222974 T^{16} - 17393050524001724339 T^{18} + \)\(12\!\cdots\!58\)\( T^{20} - \)\(73\!\cdots\!25\)\( T^{22} + \)\(39\!\cdots\!11\)\( T^{24} - \)\(19\!\cdots\!69\)\( T^{26} + \)\(77\!\cdots\!62\)\( T^{28} - \)\(19\!\cdots\!78\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( 1 + 27 T + 322 T^{2} + 2133 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 11 T + 83 T^{2} \))(\( 1 + 14 T + 220 T^{2} + 1979 T^{3} + 12164 T^{4} + 53441 T^{5} + 81605 T^{6} + 4435603 T^{7} + 83797796 T^{8} + 1131566473 T^{9} + 10440830620 T^{10} + 55146569002 T^{11} + 326940373369 T^{12} \))(\( 1 - 37 T + 937 T^{2} - 15039 T^{3} + 188365 T^{4} - 1609640 T^{5} + 10677896 T^{6} - 41094423 T^{7} + 393804009 T^{8} - 6686224079 T^{9} + 121486654562 T^{10} - 1244974710924 T^{11} + 10083392328646 T^{12} - 46061397680231 T^{13} + 225172012894083 T^{14} - 1950272318043783 T^{15} + 42060666325727128 T^{16} - 526256302589677160 T^{17} + 5111482244661089855 T^{18} - 33872222879139037599 T^{19} + \)\(17\!\cdots\!11\)\( T^{20} - \)\(57\!\cdots\!13\)\( T^{21} + \)\(12\!\cdots\!67\)\( T^{22} \))(\( ( 1 + 9 T + 83 T^{2} )^{2} \))(\( ( 1 - 4 T + 426 T^{2} - 1909 T^{3} + 86691 T^{4} - 414059 T^{5} + 11394137 T^{6} - 53249160 T^{7} + 1089259614 T^{8} - 4419680280 T^{9} + 78494209793 T^{10} - 236753553433 T^{11} + 4114209305811 T^{12} - 7519628587487 T^{13} + 139276599055194 T^{14} - 108544203958508 T^{15} + 2252292232139041 T^{16} )^{2} \))(\( 1 + 15 T + 142 T^{2} + 1245 T^{3} + 6889 T^{4} \))
$89$ (\( 1 + 18 T + 89 T^{2} \))(\( 1 + 4 T + 395 T^{2} + 1436 T^{3} + 71541 T^{4} + 228569 T^{5} + 7881488 T^{6} + 20342641 T^{7} + 566676261 T^{8} + 1012335484 T^{9} + 24783185195 T^{10} + 22336237796 T^{11} + 496981290961 T^{12} \))(\( 1 - 4 T + 562 T^{2} - 1396 T^{3} + 156064 T^{4} - 227891 T^{5} + 29094279 T^{6} - 25156606 T^{7} + 4092883616 T^{8} - 2392311257 T^{9} + 454893417150 T^{10} - 218041730972 T^{11} + 40485514126350 T^{12} - 18949497466697 T^{13} + 2885356069887904 T^{14} - 1578381836394046 T^{15} + 162464183561792271 T^{16} - 113257563378393251 T^{17} + 6902919049135837856 T^{18} - 5495477972760105076 T^{19} + \)\(19\!\cdots\!58\)\( T^{20} - \)\(12\!\cdots\!04\)\( T^{21} + \)\(27\!\cdots\!89\)\( T^{22} \))(\( ( 1 - 6 T + 89 T^{2} )( 1 + 6 T + 89 T^{2} ) \))(\( 1 - 764 T^{2} + 301867 T^{4} - 81214762 T^{6} + 16576635524 T^{8} - 2711045835319 T^{10} + 366347122484839 T^{12} - 41631440722564547 T^{14} + 4015766180229082034 T^{16} - \)\(32\!\cdots\!87\)\( T^{18} + \)\(22\!\cdots\!99\)\( T^{20} - \)\(13\!\cdots\!59\)\( T^{22} + \)\(65\!\cdots\!44\)\( T^{24} - \)\(25\!\cdots\!62\)\( T^{26} + \)\(74\!\cdots\!07\)\( T^{28} - \)\(14\!\cdots\!24\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( 1 + 9 T + 116 T^{2} + 801 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + 5 T + 97 T^{2} \))(\( 1 - 22 T + 686 T^{2} - 9964 T^{3} + 174858 T^{4} - 1853451 T^{5} + 22845849 T^{6} - 179784747 T^{7} + 1645238922 T^{8} - 9093873772 T^{9} + 60731086766 T^{10} - 188921485654 T^{11} + 832972004929 T^{12} \))(\( 1 + 17 T + 702 T^{2} + 9571 T^{3} + 232328 T^{4} + 2740222 T^{5} + 50233522 T^{6} + 528331131 T^{7} + 7981871048 T^{8} + 75375511384 T^{9} + 980181063247 T^{10} + 8280895304105 T^{11} + 95077563134959 T^{12} + 709208186612056 T^{13} + 7284838194991304 T^{14} + 46772775157346811 T^{15} + 431372345721495154 T^{16} + 2282528213290554238 T^{17} + 18771703836231037064 T^{18} + 75012076931781893731 T^{19} + \)\(53\!\cdots\!34\)\( T^{20} + \)\(12\!\cdots\!33\)\( T^{21} + \)\(71\!\cdots\!53\)\( T^{22} \))(\( ( 1 + 7 T + 97 T^{2} )^{2} \))(\( ( 1 + 15 T + 681 T^{2} + 8689 T^{3} + 211459 T^{4} + 2267717 T^{5} + 38955080 T^{6} + 346855759 T^{7} + 4644048029 T^{8} + 33645008623 T^{9} + 366528347720 T^{10} + 2069684077541 T^{11} + 18720313230979 T^{12} + 74615399493073 T^{13} + 567253935356649 T^{14} + 1211974267171695 T^{15} + 7837433594376961 T^{16} )^{2} \))(\( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} \))
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