Properties

Label 237.2
Level 237
Weight 2
Dimension 1481
Nonzero newspaces 8
Newform subspaces 15
Sturm bound 8320
Trace bound 1

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Defining parameters

Level: \( N \) = \( 237 = 3 \cdot 79 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 15 \)
Sturm bound: \(8320\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(237))\).

Total New Old
Modular forms 2236 1637 599
Cusp forms 1925 1481 444
Eisenstein series 311 156 155

Trace form

\( 1481q - 3q^{2} - 40q^{3} - 85q^{4} - 6q^{5} - 42q^{6} - 86q^{7} - 15q^{8} - 40q^{9} + O(q^{10}) \) \( 1481q - 3q^{2} - 40q^{3} - 85q^{4} - 6q^{5} - 42q^{6} - 86q^{7} - 15q^{8} - 40q^{9} - 96q^{10} - 12q^{11} - 46q^{12} - 92q^{13} - 24q^{14} - 45q^{15} - 109q^{16} - 18q^{17} - 42q^{18} - 98q^{19} - 42q^{20} - 47q^{21} - 114q^{22} - 24q^{23} - 54q^{24} - 109q^{25} - 42q^{26} - 40q^{27} - 134q^{28} - 30q^{29} - 57q^{30} - 110q^{31} - 63q^{32} - 51q^{33} - 132q^{34} - 48q^{35} - 46q^{36} - 116q^{37} - 60q^{38} - 53q^{39} - 168q^{40} - 42q^{41} - 63q^{42} - 122q^{43} - 84q^{44} - 45q^{45} - 150q^{46} - 48q^{47} - 70q^{48} - 135q^{49} - 93q^{50} - 57q^{51} - 176q^{52} - 54q^{53} - 42q^{54} - 150q^{55} - 120q^{56} - 59q^{57} - 168q^{58} - 60q^{59} - 81q^{60} - 140q^{61} - 96q^{62} - 34q^{63} + 107q^{64} + 72q^{65} + 81q^{66} + 36q^{67} + 186q^{68} + 93q^{69} + 402q^{70} + 84q^{71} + 102q^{72} + 4q^{73} + 198q^{74} + 86q^{75} + 510q^{76} + 138q^{77} + 114q^{78} + 311q^{79} + 594q^{80} - 40q^{81} + 108q^{82} + 150q^{83} + 269q^{84} + 126q^{85} + 180q^{86} + 9q^{87} + 522q^{88} + 66q^{89} + 21q^{90} + 122q^{91} + 144q^{92} + 20q^{93} + 90q^{94} + 36q^{95} + 54q^{96} - 72q^{97} - 171q^{98} - 51q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(237))\)

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
237.2.a \(\chi_{237}(1, \cdot)\) 237.2.a.a 2 1
237.2.a.b 4
237.2.a.c 7
237.2.b \(\chi_{237}(236, \cdot)\) 237.2.b.a 24 1
237.2.e \(\chi_{237}(55, \cdot)\) 237.2.e.a 12 2
237.2.e.b 14
237.2.h \(\chi_{237}(56, \cdot)\) 237.2.h.a 2 2
237.2.h.b 48
237.2.i \(\chi_{237}(10, \cdot)\) 237.2.i.a 72 12
237.2.i.b 96
237.2.l \(\chi_{237}(14, \cdot)\) 237.2.l.a 288 12
237.2.m \(\chi_{237}(4, \cdot)\) 237.2.m.a 144 24
237.2.m.b 168
237.2.n \(\chi_{237}(29, \cdot)\) 237.2.n.a 24 24
237.2.n.b 576

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(237))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(237)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(79))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} \))(\( 1 + 3 T + 7 T^{2} + 13 T^{3} + 21 T^{4} + 26 T^{5} + 28 T^{6} + 24 T^{7} + 16 T^{8} \))(\( 1 - 2 T + 3 T^{2} - 2 T^{3} + 4 T^{4} - 9 T^{5} + 18 T^{6} - 29 T^{7} + 36 T^{8} - 36 T^{9} + 32 T^{10} - 32 T^{11} + 96 T^{12} - 128 T^{13} + 128 T^{14} \))(\( 1 - 4 T + 6 T^{2} - 2 T^{3} - 8 T^{4} + 17 T^{5} - 11 T^{6} - 14 T^{7} + 29 T^{8} - 20 T^{9} + 27 T^{10} - 56 T^{11} + 78 T^{12} - 112 T^{13} + 108 T^{14} - 160 T^{15} + 464 T^{16} - 448 T^{17} - 704 T^{18} + 2176 T^{19} - 2048 T^{20} - 1024 T^{21} + 6144 T^{22} - 8192 T^{23} + 4096 T^{24} \))(\( 1 + 2 T - 2 T^{2} - 10 T^{3} - 8 T^{4} + 17 T^{5} + 31 T^{6} - 14 T^{7} - 57 T^{8} + 18 T^{9} + 141 T^{10} + 20 T^{11} - 220 T^{12} - 72 T^{13} + 220 T^{14} - 144 T^{15} - 880 T^{16} + 160 T^{17} + 2256 T^{18} + 576 T^{19} - 3648 T^{20} - 1792 T^{21} + 7936 T^{22} + 8704 T^{23} - 8192 T^{24} - 20480 T^{25} - 8192 T^{26} + 16384 T^{27} + 16384 T^{28} \))(\( 1 + 2 T^{2} + 4 T^{4} \))
$3$ (\( ( 1 + T )^{2} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{7} \))(\( ( 1 - T + T^{2} )^{6} \))(\( ( 1 + T + T^{2} )^{7} \))(\( 1 + 3 T + 3 T^{2} \))
$5$ (\( ( 1 + 5 T^{2} )^{2} \))(\( 1 + 4 T + 19 T^{2} + 46 T^{3} + 131 T^{4} + 230 T^{5} + 475 T^{6} + 500 T^{7} + 625 T^{8} \))(\( 1 + 2 T + 10 T^{2} + 28 T^{3} + 91 T^{4} + 212 T^{5} + 574 T^{6} + 1252 T^{7} + 2870 T^{8} + 5300 T^{9} + 11375 T^{10} + 17500 T^{11} + 31250 T^{12} + 31250 T^{13} + 78125 T^{14} \))(\( 1 - T - 13 T^{2} + 26 T^{3} + 64 T^{4} - 240 T^{5} - 120 T^{6} + 1376 T^{7} - 720 T^{8} - 6648 T^{9} + 13614 T^{10} + 16111 T^{11} - 97799 T^{12} + 80555 T^{13} + 340350 T^{14} - 831000 T^{15} - 450000 T^{16} + 4300000 T^{17} - 1875000 T^{18} - 18750000 T^{19} + 25000000 T^{20} + 50781250 T^{21} - 126953125 T^{22} - 48828125 T^{23} + 244140625 T^{24} \))(\( 1 + T - 12 T^{2} - 9 T^{3} + 64 T^{4} + 31 T^{5} - 29 T^{6} + 90 T^{7} - 1416 T^{8} - 460 T^{9} + 7752 T^{10} - 2835 T^{11} - 11687 T^{12} + 12002 T^{13} - 23641 T^{14} + 60010 T^{15} - 292175 T^{16} - 354375 T^{17} + 4845000 T^{18} - 1437500 T^{19} - 22125000 T^{20} + 7031250 T^{21} - 11328125 T^{22} + 60546875 T^{23} + 625000000 T^{24} - 439453125 T^{25} - 2929687500 T^{26} + 1220703125 T^{27} + 6103515625 T^{28} \))(\( 1 + 5 T^{2} + 25 T^{4} \))
$7$ (\( ( 1 - T + 7 T^{2} )^{2} \))(\( 1 + 2 T + 8 T^{2} + 2 T^{3} - 2 T^{4} + 14 T^{5} + 392 T^{6} + 686 T^{7} + 2401 T^{8} \))(\( 1 - 4 T + 26 T^{2} - 70 T^{3} + 236 T^{4} - 460 T^{5} + 1035 T^{6} - 2196 T^{7} + 7245 T^{8} - 22540 T^{9} + 80948 T^{10} - 168070 T^{11} + 436982 T^{12} - 470596 T^{13} + 823543 T^{14} \))(\( 1 - 19 T^{2} + 16 T^{3} + 178 T^{4} - 237 T^{5} - 649 T^{6} + 1171 T^{7} - 2416 T^{8} + 740 T^{9} + 44686 T^{10} - 23572 T^{11} - 369171 T^{12} - 165004 T^{13} + 2189614 T^{14} + 253820 T^{15} - 5800816 T^{16} + 19680997 T^{17} - 76354201 T^{18} - 195179691 T^{19} + 1026134578 T^{20} + 645657712 T^{21} - 5367029731 T^{22} + 13841287201 T^{24} \))(\( 1 + 4 T - 10 T^{2} - 24 T^{3} + 184 T^{4} + 179 T^{5} - 1126 T^{6} + 1246 T^{7} + 4593 T^{8} - 17490 T^{9} + 33193 T^{10} + 128603 T^{11} - 633569 T^{12} - 472788 T^{13} + 5014545 T^{14} - 3309516 T^{15} - 31044881 T^{16} + 44110829 T^{17} + 79696393 T^{18} - 293954430 T^{19} + 540361857 T^{20} + 1026134578 T^{21} - 6491165926 T^{22} + 7223295653 T^{23} + 51975445816 T^{24} - 47455841832 T^{25} - 138412872010 T^{26} + 387556041628 T^{27} + 678223072849 T^{28} \))(\( ( 1 + T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))
$11$ (\( 1 - 6 T + 29 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 + 8 T + 55 T^{2} + 222 T^{3} + 879 T^{4} + 2442 T^{5} + 6655 T^{6} + 10648 T^{7} + 14641 T^{8} \))(\( 1 - 2 T + 35 T^{2} - 92 T^{3} + 647 T^{4} - 1922 T^{5} + 8882 T^{6} - 25228 T^{7} + 97702 T^{8} - 232562 T^{9} + 861157 T^{10} - 1346972 T^{11} + 5636785 T^{12} - 3543122 T^{13} + 19487171 T^{14} \))(\( 1 + T - 38 T^{2} + 41 T^{3} + 736 T^{4} - 1700 T^{5} - 8708 T^{6} + 16105 T^{7} + 94451 T^{8} + 11489 T^{9} - 1733866 T^{10} - 561566 T^{11} + 25696509 T^{12} - 6177226 T^{13} - 209797786 T^{14} + 15291859 T^{15} + 1382857091 T^{16} + 2593726355 T^{17} - 15426753188 T^{18} - 33128190700 T^{19} + 157768136416 T^{20} + 96675855331 T^{21} - 985622134838 T^{22} + 285311670611 T^{23} + 3138428376721 T^{24} \))(\( 1 - T - 55 T^{2} + 36 T^{3} + 1639 T^{4} - 638 T^{5} - 33588 T^{6} + 5212 T^{7} + 528275 T^{8} + 4310 T^{9} - 6798723 T^{10} - 525944 T^{11} + 77364436 T^{12} + 3220293 T^{13} - 847514395 T^{14} + 35423223 T^{15} + 9361096756 T^{16} - 700031464 T^{17} - 99540103443 T^{18} + 694129810 T^{19} + 935871387275 T^{20} + 101567135252 T^{21} - 7199886095028 T^{22} - 1504370626858 T^{23} + 42511438921039 T^{24} + 10271220141996 T^{25} - 172613560719655 T^{26} - 34522712143931 T^{27} + 379749833583241 T^{28} \))(\( 1 + 11 T^{2} + 121 T^{4} \))
$13$ (\( 1 + 2 T + 19 T^{2} + 26 T^{3} + 169 T^{4} \))(\( 1 + 6 T + 31 T^{2} + 160 T^{3} + 609 T^{4} + 2080 T^{5} + 5239 T^{6} + 13182 T^{7} + 28561 T^{8} \))(\( 1 - 6 T + 75 T^{2} - 274 T^{3} + 1981 T^{4} - 4506 T^{5} + 28948 T^{6} - 50850 T^{7} + 376324 T^{8} - 761514 T^{9} + 4352257 T^{10} - 7825714 T^{11} + 27846975 T^{12} - 28960854 T^{13} + 62748517 T^{14} \))(\( 1 - 5 T - 28 T^{2} + 177 T^{3} + 239 T^{4} - 1957 T^{5} - 5580 T^{6} + 12142 T^{7} + 189395 T^{8} - 268153 T^{9} - 2742489 T^{10} + 2536361 T^{11} + 29352114 T^{12} + 32972693 T^{13} - 463480641 T^{14} - 589132141 T^{15} + 5409310595 T^{16} + 4508239606 T^{17} - 26933594220 T^{18} - 122798847769 T^{19} + 194959642319 T^{20} + 1876996389021 T^{21} - 3860037771772 T^{22} - 8960801970185 T^{23} + 23298085122481 T^{24} \))(\( 1 - 45 T^{2} - 38 T^{3} + 896 T^{4} + 1245 T^{5} - 10632 T^{6} - 8234 T^{7} + 112683 T^{8} - 204263 T^{9} - 1615052 T^{10} + 3961661 T^{11} + 26278989 T^{12} - 21354923 T^{13} - 371772352 T^{14} - 277613999 T^{15} + 4441149141 T^{16} + 8703769217 T^{17} - 46127500172 T^{18} - 75841422059 T^{19} + 543899318547 T^{20} - 516671288978 T^{21} - 8672849025672 T^{22} + 13202601719385 T^{23} + 123521208696704 T^{24} - 68102094973406 T^{25} - 1048413830511645 T^{26} + 3937376385699289 T^{28} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))
$17$ (\( 1 - 2 T + 33 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 + 8 T + 68 T^{2} + 352 T^{3} + 1782 T^{4} + 5984 T^{5} + 19652 T^{6} + 39304 T^{7} + 83521 T^{8} \))(\( 1 + 8 T + 58 T^{2} + 274 T^{3} + 1828 T^{4} + 8632 T^{5} + 43073 T^{6} + 159548 T^{7} + 732241 T^{8} + 2494648 T^{9} + 8980964 T^{10} + 22884754 T^{11} + 82351706 T^{12} + 193100552 T^{13} + 410338673 T^{14} \))(\( ( 1 + 3 T + 73 T^{2} + 224 T^{3} + 2565 T^{4} + 7184 T^{5} + 54464 T^{6} + 122128 T^{7} + 741285 T^{8} + 1100512 T^{9} + 6097033 T^{10} + 4259571 T^{11} + 24137569 T^{12} )^{2} \))(\( ( 1 - T + 52 T^{2} - 89 T^{3} + 1606 T^{4} - 3716 T^{5} + 34949 T^{6} - 78550 T^{7} + 594133 T^{8} - 1073924 T^{9} + 7890278 T^{10} - 7433369 T^{11} + 73832564 T^{12} - 24137569 T^{13} + 410338673 T^{14} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))
$19$ (\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( 1 + 4 T + 49 T^{2} + 108 T^{3} + 1093 T^{4} + 2052 T^{5} + 17689 T^{6} + 27436 T^{7} + 130321 T^{8} \))(\( 1 - 4 T + 62 T^{2} - 312 T^{3} + 2089 T^{4} - 11492 T^{5} + 53852 T^{6} - 266384 T^{7} + 1023188 T^{8} - 4148612 T^{9} + 14328451 T^{10} - 40660152 T^{11} + 153518138 T^{12} - 188183524 T^{13} + 893871739 T^{14} \))(\( 1 - 14 T + 71 T^{2} - 66 T^{3} - 918 T^{4} + 4799 T^{5} + 167 T^{6} - 131653 T^{7} + 682010 T^{8} - 1266660 T^{9} - 766434 T^{10} + 15221078 T^{11} - 90042003 T^{12} + 289200482 T^{13} - 276682674 T^{14} - 8688020940 T^{15} + 88880225210 T^{16} - 325985861647 T^{17} + 7856662127 T^{18} + 4289690475461 T^{19} - 15590910871638 T^{20} - 21297388053414 T^{21} + 435305704303871 T^{22} - 1630863624575066 T^{23} + 2213314919066161 T^{24} \))(\( 1 + 10 T - 28 T^{2} - 496 T^{3} + 738 T^{4} + 14763 T^{5} - 31248 T^{6} - 382432 T^{7} + 878783 T^{8} + 7323190 T^{9} - 23661041 T^{10} - 106538395 T^{11} + 557112717 T^{12} + 843379564 T^{13} - 10753514507 T^{14} + 16024211716 T^{15} + 201117690837 T^{16} - 730746851305 T^{17} - 3083530524161 T^{18} + 18132943435810 T^{19} + 41343120442823 T^{20} - 341845156889248 T^{21} - 530702377905168 T^{22} + 4763838482311377 T^{23} + 4524726898257138 T^{24} - 57779168413516624 T^{25} - 61972817733852508 T^{26} + 420529834622570590 T^{27} + 799006685782884121 T^{28} \))(\( ( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))
$23$ (\( 1 - 6 T + 37 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 18 T + 177 T^{2} + 1210 T^{3} + 6471 T^{4} + 27830 T^{5} + 93633 T^{6} + 219006 T^{7} + 279841 T^{8} \))(\( 1 - 8 T + 137 T^{2} - 772 T^{3} + 7751 T^{4} - 34138 T^{5} + 261448 T^{6} - 950172 T^{7} + 6013304 T^{8} - 18059002 T^{9} + 94306417 T^{10} - 216037252 T^{11} + 881778991 T^{12} - 1184287112 T^{13} + 3404825447 T^{14} \))(\( 1 - 12 T - 31 T^{2} + 596 T^{3} + 3378 T^{4} - 31279 T^{5} - 156093 T^{6} + 867089 T^{7} + 7081314 T^{8} - 19727492 T^{9} - 218157812 T^{10} + 146487534 T^{11} + 5892821905 T^{12} + 3369213282 T^{13} - 115405482548 T^{14} - 240024395164 T^{15} + 1981641991074 T^{16} + 5580882215527 T^{17} - 23107366021677 T^{18} - 106499535156713 T^{19} + 264534508279218 T^{20} + 1073486986231948 T^{21} - 1284221847623119 T^{22} - 11433717094967124 T^{23} + 21914624432020321 T^{24} \))(\( 1 + 2 T - 64 T^{2} - 340 T^{3} + 1704 T^{4} + 14785 T^{5} + 23820 T^{6} - 225324 T^{7} - 2122525 T^{8} - 8279236 T^{9} + 39564187 T^{10} + 398296169 T^{11} + 718363107 T^{12} - 5230348054 T^{13} - 36010780655 T^{14} - 120298005242 T^{15} + 380014083603 T^{16} + 4846069488223 T^{17} + 11071681654267 T^{18} - 53288002673948 T^{19} - 314209875299725 T^{20} - 767188889019828 T^{21} + 1865367669393420 T^{22} + 26630042099730455 T^{23} + 70590775108057896 T^{24} - 323955317690735180 T^{25} - 1402535963649300544 T^{26} + 1008072723872934766 T^{27} + 11592836324538749809 T^{28} \))(\( 1 + 23 T^{2} + 529 T^{4} \))
$29$ (\( 1 - 6 T + 65 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 + 2 T + 48 T^{2} + 214 T^{3} + 1150 T^{4} + 6206 T^{5} + 40368 T^{6} + 48778 T^{7} + 707281 T^{8} \))(\( 1 + 10 T + 114 T^{2} + 840 T^{3} + 7444 T^{4} + 47574 T^{5} + 311525 T^{6} + 1600752 T^{7} + 9034225 T^{8} + 40009734 T^{9} + 181551716 T^{10} + 594116040 T^{11} + 2338270986 T^{12} + 5948233210 T^{13} + 17249876309 T^{14} \))(\( 1 + 7 T - 121 T^{2} - 702 T^{3} + 10162 T^{4} + 44152 T^{5} - 604328 T^{6} - 1748492 T^{7} + 28862380 T^{8} + 46985952 T^{9} - 1106765400 T^{10} - 536389445 T^{11} + 35382392505 T^{12} - 15555293905 T^{13} - 930789701400 T^{14} + 1145940383328 T^{15} + 20413812988780 T^{16} - 35863579937308 T^{17} - 359468387933288 T^{18} + 761616538794968 T^{19} + 5083504048509682 T^{20} - 10184016475060038 T^{21} - 50905575229324321 T^{22} + 85403568359940803 T^{23} + 353814783205469041 T^{24} \))(\( 1 + 5 T - 2 T^{2} + 189 T^{3} + 122 T^{4} - 9583 T^{5} + 20785 T^{6} + 69388 T^{7} - 1113054 T^{8} + 2725022 T^{9} + 1654250 T^{10} - 164770649 T^{11} + 595514475 T^{12} + 5602523324 T^{13} + 19236111863 T^{14} + 162473176396 T^{15} + 500827673475 T^{16} - 4018591358461 T^{17} + 1170019594250 T^{18} + 55893332270278 T^{19} - 662070476732334 T^{20} + 1196934417328892 T^{21} + 10397621693394385 T^{22} - 139021979886752627 T^{23} + 51326282462624522 T^{24} + 2305896345718401681 T^{25} - 707629566410938082 T^{26} + 51303143564793010945 T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \))(\( 1 - 29 T^{2} + 841 T^{4} \))
$31$ (\( 1 + 4 T + 58 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 + 57 T^{2} + 136 T^{3} + 1985 T^{4} + 4216 T^{5} + 54777 T^{6} + 923521 T^{8} \))(\( 1 - 4 T + 138 T^{2} - 348 T^{3} + 9049 T^{4} - 16880 T^{5} + 397000 T^{6} - 613472 T^{7} + 12307000 T^{8} - 16221680 T^{9} + 269578759 T^{10} - 321385308 T^{11} + 3950822838 T^{12} - 3550014724 T^{13} + 27512614111 T^{14} \))(\( 1 - T - 108 T^{2} + 507 T^{3} + 6028 T^{4} - 45190 T^{5} - 147455 T^{6} + 2346139 T^{7} - 1139161 T^{8} - 71107129 T^{9} + 283822908 T^{10} + 969133759 T^{11} - 12090324542 T^{12} + 30043146529 T^{13} + 272753814588 T^{14} - 2118352480039 T^{15} - 1052039105881 T^{16} + 67167967697989 T^{17} - 130866855281855 T^{18} - 1243295031676090 T^{19} + 5141227173694348 T^{20} + 13404888435460197 T^{21} - 88519854993926508 T^{22} - 25408476896404831 T^{23} + 787662783788549761 T^{24} \))(\( 1 - 8 T - 59 T^{2} + 132 T^{3} + 6629 T^{4} - 3597 T^{5} - 194568 T^{6} - 1688577 T^{7} + 7575934 T^{8} + 72019553 T^{9} + 208115715 T^{10} - 3464123395 T^{11} - 11782284919 T^{12} + 22266485812 T^{13} + 723964091030 T^{14} + 690261060172 T^{15} - 11322775807159 T^{16} - 103199700060445 T^{17} + 192199233232515 T^{18} + 2061858657789503 T^{19} + 6723669312013054 T^{20} - 46457167397710047 T^{21} - 165945303372820488 T^{22} - 95103320911933587 T^{23} + 5433315914395729829 T^{24} + 3353918950325437692 T^{25} - 46472104243524435899 T^{26} - \)\(19\!\cdots\!28\)\( T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))
$37$ (\( 1 + 2 T^{2} + 1369 T^{4} \))(\( 1 - 4 T + 100 T^{2} - 308 T^{3} + 5030 T^{4} - 11396 T^{5} + 136900 T^{6} - 202612 T^{7} + 1874161 T^{8} \))(\( 1 - 10 T + 163 T^{2} - 1332 T^{3} + 13421 T^{4} - 96422 T^{5} + 717047 T^{6} - 4367384 T^{7} + 26530739 T^{8} - 132001718 T^{9} + 679813913 T^{10} - 2496382452 T^{11} + 11303064991 T^{12} - 25657264090 T^{13} + 94931877133 T^{14} \))(\( 1 + 16 T + 46 T^{2} + 220 T^{3} + 8544 T^{4} + 42999 T^{5} + 92670 T^{6} + 2890155 T^{7} + 15972996 T^{8} + 26498210 T^{9} + 820205093 T^{10} + 4748434031 T^{11} + 5487583594 T^{12} + 175692059147 T^{13} + 1122860772317 T^{14} + 1342213831130 T^{15} + 29935966156356 T^{16} + 200414784043335 T^{17} + 237765866322030 T^{18} + 4081975784841867 T^{19} + 30010624454301024 T^{20} + 28591582754916940 T^{21} + 221194881131221054 T^{22} + 2846681948471366608 T^{23} + 6582952005840035281 T^{24} \))(\( 1 + 7 T - 78 T^{2} - 815 T^{3} + 1049 T^{4} + 32329 T^{5} + 90045 T^{6} - 44727 T^{7} - 2150270 T^{8} - 24457001 T^{9} - 62059470 T^{10} + 132639729 T^{11} - 1004059939 T^{12} + 9875771030 T^{13} + 209686537612 T^{14} + 365403528110 T^{15} - 1374558056491 T^{16} + 6718600193037 T^{17} - 116309438354670 T^{18} - 1695945225692957 T^{19} - 5517004525480430 T^{20} - 4246018068527691 T^{21} + 316281212428316445 T^{22} + 4201533085835044333 T^{23} + 5044205006666323601 T^{24} - \)\(14\!\cdots\!95\)\( T^{25} - \)\(51\!\cdots\!18\)\( T^{26} + \)\(17\!\cdots\!79\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \))(\( ( 1 + 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))
$41$ (\( 1 + 74 T^{2} + 1681 T^{4} \))(\( 1 - 2 T + 88 T^{2} - 254 T^{3} + 4222 T^{4} - 10414 T^{5} + 147928 T^{6} - 137842 T^{7} + 2825761 T^{8} \))(\( 1 + 20 T + 303 T^{2} + 3464 T^{3} + 34933 T^{4} + 302412 T^{5} + 2301099 T^{6} + 15564336 T^{7} + 94345059 T^{8} + 508354572 T^{9} + 2407617293 T^{10} + 9788436104 T^{11} + 35104428903 T^{12} + 95002084820 T^{13} + 194754273881 T^{14} \))(\( ( 1 + 16 T + 291 T^{2} + 3183 T^{3} + 32230 T^{4} + 256934 T^{5} + 1802584 T^{6} + 10534294 T^{7} + 54178630 T^{8} + 219375543 T^{9} + 822296451 T^{10} + 1853699216 T^{11} + 4750104241 T^{12} )^{2} \))(\( ( 1 - 4 T + 174 T^{2} - 349 T^{3} + 13543 T^{4} - 8247 T^{5} + 695076 T^{6} - 47130 T^{7} + 28498116 T^{8} - 13863207 T^{9} + 933397103 T^{10} - 986190589 T^{11} + 20158978974 T^{12} - 19000416964 T^{13} + 194754273881 T^{14} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))
$43$ (\( 1 - 14 T + 127 T^{2} - 602 T^{3} + 1849 T^{4} \))(\( 1 + 16 T + 240 T^{2} + 2064 T^{3} + 16638 T^{4} + 88752 T^{5} + 443760 T^{6} + 1272112 T^{7} + 3418801 T^{8} \))(\( 1 - 22 T + 362 T^{2} - 4096 T^{3} + 41340 T^{4} - 352314 T^{5} + 2798799 T^{6} - 19169920 T^{7} + 120348357 T^{8} - 651428586 T^{9} + 3286819380 T^{10} - 14003408896 T^{11} + 53217056366 T^{12} - 139069987078 T^{13} + 271818611107 T^{14} \))(\( 1 + 13 T - 106 T^{2} - 1629 T^{3} + 11602 T^{4} + 145926 T^{5} - 856847 T^{6} - 8075305 T^{7} + 57945799 T^{8} + 316956221 T^{9} - 3111148334 T^{10} - 5203258249 T^{11} + 147733034850 T^{12} - 223740104707 T^{13} - 5752513269566 T^{14} + 25200238263047 T^{15} + 198105155566999 T^{16} - 1187138014800115 T^{17} - 5416440964446503 T^{18} + 39665402644400082 T^{19} + 135606499620726802 T^{20} - 818723364845117247 T^{21} - 2290817125208130394 T^{22} + 12080818613125895191 T^{23} + 39959630797262576401 T^{24} \))(\( 1 - 14 T + 17 T^{2} - 186 T^{3} + 9021 T^{4} - 21535 T^{5} - 132494 T^{6} - 3241595 T^{7} + 14766188 T^{8} + 128563547 T^{9} + 553244987 T^{10} - 8838926105 T^{11} - 34450122597 T^{12} + 65601097412 T^{13} + 2938191531330 T^{14} + 2820847188716 T^{15} - 63698276681853 T^{16} - 702756497830235 T^{17} + 1891434514800587 T^{18} + 18899926871027321 T^{19} + 93342435197787212 T^{20} - 881125850671395665 T^{21} - 1548616407580466894 T^{22} - 10823331898059914005 T^{23} + \)\(19\!\cdots\!29\)\( T^{24} - \)\(17\!\cdots\!02\)\( T^{25} + \)\(67\!\cdots\!17\)\( T^{26} - \)\(24\!\cdots\!02\)\( T^{27} + \)\(73\!\cdots\!49\)\( T^{28} \))(\( ( 1 - 13 T + 43 T^{2} )( 1 - 8 T + 43 T^{2} ) \))
$47$ (\( 1 - 4 T + 66 T^{2} - 188 T^{3} + 2209 T^{4} \))(\( 1 + 6 T + 120 T^{2} + 422 T^{3} + 6750 T^{4} + 19834 T^{5} + 265080 T^{6} + 622938 T^{7} + 4879681 T^{8} \))(\( 1 + 10 T + 225 T^{2} + 1668 T^{3} + 20573 T^{4} + 117238 T^{5} + 1143773 T^{6} + 5726584 T^{7} + 53757331 T^{8} + 258978742 T^{9} + 2135950579 T^{10} + 8139307908 T^{11} + 51602626575 T^{12} + 107792153290 T^{13} + 506623120463 T^{14} \))(\( 1 + 10 T - 131 T^{2} - 1422 T^{3} + 10690 T^{4} + 107089 T^{5} - 733659 T^{6} - 5576839 T^{7} + 44582566 T^{8} + 192138342 T^{9} - 2584964368 T^{10} - 3138022686 T^{11} + 133217586745 T^{12} - 147487066242 T^{13} - 5710186288912 T^{14} + 19948379081466 T^{15} + 217548700241446 T^{16} - 1279020179492873 T^{17} - 7908268339058811 T^{18} + 54253763347262207 T^{19} + 254542654414225090 T^{20} - 1591403532752134674 T^{21} - 6890486322893736419 T^{22} + 24721592150840123030 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 - 10 T - 50 T^{2} + 480 T^{3} + 2542 T^{4} - 13197 T^{5} + 194616 T^{6} - 1162104 T^{7} - 19083907 T^{8} + 63886850 T^{9} + 618375631 T^{10} - 1887746987 T^{11} + 35724811093 T^{12} - 22647754332 T^{13} - 2877234181747 T^{14} - 1064444453604 T^{15} + 78916107704437 T^{16} - 195991555431301 T^{17} + 3017475817453711 T^{18} + 14652130060457950 T^{19} - 205709542871610403 T^{20} - 588748754782534152 T^{21} + 4634057364965278776 T^{22} - 14769164853537216099 T^{23} + \)\(13\!\cdots\!58\)\( T^{24} + \)\(11\!\cdots\!40\)\( T^{25} - \)\(58\!\cdots\!50\)\( T^{26} - \)\(54\!\cdots\!70\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))
$53$ (\( 1 - 8 T + 114 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 + 6 T + 68 T^{2} + 122 T^{3} + 2758 T^{4} + 6466 T^{5} + 191012 T^{6} + 893262 T^{7} + 7890481 T^{8} \))(\( 1 + 135 T^{2} - 96 T^{3} + 11281 T^{4} - 31072 T^{5} + 642143 T^{6} - 2090496 T^{7} + 34033579 T^{8} - 87281248 T^{9} + 1679481437 T^{10} - 757486176 T^{11} + 56456391555 T^{12} + 1174711139837 T^{14} \))(\( 1 + 5 T - 106 T^{2} - 1679 T^{3} - 1458 T^{4} + 109574 T^{5} + 1124632 T^{6} + 1901025 T^{7} - 48168315 T^{8} - 671040283 T^{9} - 2466020596 T^{10} + 20490147830 T^{11} + 331154507893 T^{12} + 1085977834990 T^{13} - 6927051854164 T^{14} - 99902464212191 T^{15} - 380071174309515 T^{16} + 795000087080325 T^{17} + 24926749785229528 T^{18} + 128717798436499438 T^{19} - 90774628619764338 T^{20} - 5540303070635781307 T^{21} - 18538071858744383194 T^{22} + 46345179646860957985 T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 + 21 T + 123 T^{2} - 162 T^{3} - 4813 T^{4} - 38222 T^{5} - 308606 T^{6} - 2907558 T^{7} - 21668541 T^{8} + 5038310 T^{9} + 1098307833 T^{10} + 5070572306 T^{11} + 13175448060 T^{12} + 92376155435 T^{13} + 539578269815 T^{14} + 4895936238055 T^{15} + 37009833600540 T^{16} + 754891593200362 T^{17} + 8666177088437673 T^{18} + 2106998534336830 T^{19} - 480269367862542789 T^{20} - 3415540772322188046 T^{21} - 19213714019088472766 T^{22} - \)\(12\!\cdots\!26\)\( T^{23} - \)\(84\!\cdots\!37\)\( T^{24} - \)\(15\!\cdots\!14\)\( T^{25} + \)\(60\!\cdots\!43\)\( T^{26} + \)\(54\!\cdots\!33\)\( T^{27} + \)\(13\!\cdots\!69\)\( T^{28} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))
$59$ (\( 1 - 12 T + 122 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 + 2 T + 156 T^{2} + 466 T^{3} + 11878 T^{4} + 27494 T^{5} + 543036 T^{6} + 410758 T^{7} + 12117361 T^{8} \))(\( 1 + 2 T + 49 T^{2} + 716 T^{3} + 10537 T^{4} + 20366 T^{5} + 595329 T^{6} + 4889512 T^{7} + 35124411 T^{8} + 70894046 T^{9} + 2164078523 T^{10} + 8676030476 T^{11} + 35031290651 T^{12} + 84361067282 T^{13} + 2488651484819 T^{14} \))(\( 1 - 14 T - 136 T^{2} + 2286 T^{3} + 16231 T^{4} - 248906 T^{5} - 1328567 T^{6} + 18105901 T^{7} + 84767428 T^{8} - 866977326 T^{9} - 4910082816 T^{10} + 21158482657 T^{11} + 259370370471 T^{12} + 1248350476763 T^{13} - 17091998282496 T^{14} - 178058936236554 T^{15} + 1027157526117508 T^{16} + 12944348580188399 T^{17} - 56039665037822447 T^{18} - 619440286480358014 T^{19} + 2383204832755734151 T^{20} + 19803608441445190554 T^{21} - 69511878448887230536 T^{22} - \)\(42\!\cdots\!26\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 + 16 T - 231 T^{2} - 3610 T^{3} + 51560 T^{4} + 598566 T^{5} - 7616776 T^{6} - 61700087 T^{7} + 949402948 T^{8} + 4879635385 T^{9} - 91212656209 T^{10} - 244848917268 T^{11} + 7324669806301 T^{12} + 6216850173568 T^{13} - 471887351007689 T^{14} + 366794160240512 T^{15} + 25497175595733781 T^{16} - 50286825779584572 T^{17} - 1105256683053344449 T^{18} + 3488569906996720115 T^{19} + 40046322986978573668 T^{20} - \)\(15\!\cdots\!53\)\( T^{21} - \)\(11\!\cdots\!96\)\( T^{22} + \)\(51\!\cdots\!74\)\( T^{23} + \)\(26\!\cdots\!60\)\( T^{24} - \)\(10\!\cdots\!90\)\( T^{25} - \)\(41\!\cdots\!11\)\( T^{26} + \)\(16\!\cdots\!64\)\( T^{27} + \)\(61\!\cdots\!61\)\( T^{28} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))
$61$ (\( 1 + 12 T + 86 T^{2} + 732 T^{3} + 3721 T^{4} \))(\( 1 - 18 T + 316 T^{2} - 3214 T^{3} + 31126 T^{4} - 196054 T^{5} + 1175836 T^{6} - 4085658 T^{7} + 13845841 T^{8} \))(\( 1 - 4 T + 235 T^{2} - 936 T^{3} + 30301 T^{4} - 110204 T^{5} + 2627055 T^{6} - 8298736 T^{7} + 160250355 T^{8} - 410069084 T^{9} + 6877751281 T^{10} - 12959707176 T^{11} + 198480130735 T^{12} - 206081497444 T^{13} + 3142742836021 T^{14} \))(\( ( 1 + 9 T + 267 T^{2} + 2030 T^{3} + 35431 T^{4} + 218142 T^{5} + 2730976 T^{6} + 13306662 T^{7} + 131838751 T^{8} + 460771430 T^{9} + 3696839547 T^{10} + 7601366709 T^{11} + 51520374361 T^{12} )^{2} \))(\( ( 1 - 25 T + 610 T^{2} - 9099 T^{3} + 128374 T^{4} - 1364822 T^{5} + 13732467 T^{6} - 110189848 T^{7} + 837680487 T^{8} - 5078502662 T^{9} + 29138458894 T^{10} - 125983307259 T^{11} + 515203743610 T^{12} - 1288009359025 T^{13} + 3142742836021 T^{14} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))
$67$ (\( 1 - 16 T + 190 T^{2} - 1072 T^{3} + 4489 T^{4} \))(\( 1 + 40 T + 793 T^{2} + 10200 T^{3} + 95961 T^{4} + 683400 T^{5} + 3559777 T^{6} + 12030520 T^{7} + 20151121 T^{8} \))(\( 1 - 20 T + 378 T^{2} - 5132 T^{3} + 63293 T^{4} - 659568 T^{5} + 6462696 T^{6} - 54343904 T^{7} + 433000632 T^{8} - 2960800752 T^{9} + 19036192559 T^{10} - 103415552972 T^{11} + 510347290446 T^{12} - 1809167643380 T^{13} + 6060711605323 T^{14} \))(\( ( 1 + 24 T + 502 T^{2} + 6422 T^{3} + 76690 T^{4} + 692715 T^{5} + 6304637 T^{6} + 46411905 T^{7} + 344261410 T^{8} + 1931499986 T^{9} + 10115862742 T^{10} + 32403002568 T^{11} + 90458382169 T^{12} )^{2} \))(\( ( 1 - 23 T + 537 T^{2} - 7844 T^{3} + 110198 T^{4} - 1192269 T^{5} + 12286050 T^{6} - 103096913 T^{7} + 823165350 T^{8} - 5352095541 T^{9} + 33143481074 T^{10} - 158065393124 T^{11} + 725017182459 T^{12} - 2080542789887 T^{13} + 6060711605323 T^{14} )^{2} \))(\( ( 1 + 11 T + 67 T^{2} )^{2} \))
$71$ (\( 1 + 134 T^{2} + 5041 T^{4} \))(\( 1 + 20 T + 380 T^{2} + 4348 T^{3} + 43830 T^{4} + 308708 T^{5} + 1915580 T^{6} + 7158220 T^{7} + 25411681 T^{8} \))(\( 1 + 20 T + 537 T^{2} + 7712 T^{3} + 119181 T^{4} + 1293004 T^{5} + 14341893 T^{6} + 120175744 T^{7} + 1018274403 T^{8} + 6518033164 T^{9} + 42656190891 T^{10} + 195974883872 T^{11} + 968871161487 T^{12} + 2562005678420 T^{13} + 9095120158391 T^{14} \))(\( ( 1 - 22 T + 403 T^{2} - 4999 T^{3} + 59990 T^{4} - 588642 T^{5} + 5492554 T^{6} - 41793582 T^{7} + 302409590 T^{8} - 1789197089 T^{9} + 10240907443 T^{10} - 39693045722 T^{11} + 128100283921 T^{12} )^{2} \))(\( ( 1 + 14 T + 414 T^{2} + 4853 T^{3} + 79407 T^{4} + 768763 T^{5} + 8941020 T^{6} + 70112530 T^{7} + 634812420 T^{8} + 3875334283 T^{9} + 28420638777 T^{10} + 123322887893 T^{11} + 746950951314 T^{12} + 1793403974894 T^{13} + 9095120158391 T^{14} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))
$73$ (\( 1 + 6 T + 83 T^{2} + 438 T^{3} + 5329 T^{4} \))(\( 1 - 14 T + 219 T^{2} - 2664 T^{3} + 21353 T^{4} - 194472 T^{5} + 1167051 T^{6} - 5446238 T^{7} + 28398241 T^{8} \))(\( 1 - 2 T + 291 T^{2} - 614 T^{3} + 42469 T^{4} - 99734 T^{5} + 4203292 T^{6} - 9581654 T^{7} + 306840316 T^{8} - 531482486 T^{9} + 16521162973 T^{10} - 17436519974 T^{11} + 603263833563 T^{12} - 302668452578 T^{13} + 11047398519097 T^{14} \))(\( 1 + 14 T - 91 T^{2} - 262 T^{3} + 27170 T^{4} + 3379 T^{5} - 1150171 T^{6} + 25964141 T^{7} + 83424478 T^{8} - 1465280076 T^{9} + 15566934196 T^{10} + 111166638340 T^{11} - 905949147397 T^{12} + 8115164598820 T^{13} + 82956192330484 T^{14} - 570018859325292 T^{15} + 2369108431543198 T^{16} + 53825523143746613 T^{17} - 174060238385045419 T^{18} + 37329159596028763 T^{19} + 21911520696762180770 T^{20} - 15424355717566193206 T^{21} - \)\(39\!\cdots\!59\)\( T^{22} + \)\(43\!\cdots\!78\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 + 32 T + 376 T^{2} + 3368 T^{3} + 52246 T^{4} + 582865 T^{5} + 4571978 T^{6} + 62595162 T^{7} + 709398547 T^{8} + 4934852864 T^{9} + 54322492269 T^{10} + 617270673805 T^{11} + 4165385816541 T^{12} + 39420062826658 T^{13} + 442782923082577 T^{14} + 2877664586346034 T^{15} + 22197341016346989 T^{16} + 240128785711599685 T^{17} + 1542663227175698829 T^{18} + 10230303287993092352 T^{19} + \)\(10\!\cdots\!83\)\( T^{20} + \)\(69\!\cdots\!14\)\( T^{21} + \)\(36\!\cdots\!18\)\( T^{22} + \)\(34\!\cdots\!45\)\( T^{23} + \)\(22\!\cdots\!54\)\( T^{24} + \)\(10\!\cdots\!36\)\( T^{25} + \)\(86\!\cdots\!96\)\( T^{26} + \)\(53\!\cdots\!56\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))
$79$ (\( ( 1 - T )^{2} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{7} \))(\( 1 - 8 T + 71 T^{2} - 1365 T^{3} + 6961 T^{4} - 49507 T^{5} + 723073 T^{6} - 3911053 T^{7} + 43443601 T^{8} - 672998235 T^{9} + 2765455751 T^{10} - 24616451192 T^{11} + 243087455521 T^{12} \))(\( 1 + 11 T + 34 T^{2} + 292 T^{3} + 5759 T^{4} - 20843 T^{5} - 401781 T^{6} - 2757039 T^{7} - 31740699 T^{8} - 130081163 T^{9} + 2839411601 T^{10} + 11373423652 T^{11} + 104619917566 T^{12} + 2673962010731 T^{13} + 19203908986159 T^{14} \))(\( 1 + 17 T + 79 T^{2} \))
$83$ (\( 1 - 14 T + 165 T^{2} - 1162 T^{3} + 6889 T^{4} \))(\( 1 + 8 T + 76 T^{2} + 968 T^{3} + 11126 T^{4} + 80344 T^{5} + 523564 T^{6} + 4574296 T^{7} + 47458321 T^{8} \))(\( 1 - 10 T + 412 T^{2} - 4496 T^{3} + 82086 T^{4} - 866006 T^{5} + 10219863 T^{6} - 93230464 T^{7} + 848248629 T^{8} - 5965915334 T^{9} + 46935707682 T^{10} - 213372611216 T^{11} + 1622884744916 T^{12} - 3269403733690 T^{13} + 27136050989627 T^{14} \))(\( 1 + 12 T - 237 T^{2} - 3488 T^{3} + 27516 T^{4} + 473555 T^{5} - 2657889 T^{6} - 40574429 T^{7} + 296013562 T^{8} + 2478410100 T^{9} - 31489365696 T^{10} - 76889102878 T^{11} + 2822798830709 T^{12} - 6381795538874 T^{13} - 216930240279744 T^{14} + 1417122675848700 T^{15} + 14048306645749402 T^{16} - 159824324897517847 T^{17} - 868971222033358041 T^{18} + 12850412626392813985 T^{19} + 61974073059537852156 T^{20} - \)\(65\!\cdots\!64\)\( T^{21} - \)\(36\!\cdots\!13\)\( T^{22} + \)\(15\!\cdots\!04\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 + 10 T - 234 T^{2} - 1608 T^{3} + 38156 T^{4} + 96991 T^{5} - 3628708 T^{6} + 20129832 T^{7} + 258971539 T^{8} - 3910880496 T^{9} + 787318419 T^{10} + 384137697115 T^{11} - 1967087753977 T^{12} - 12188118628546 T^{13} + 269296406932941 T^{14} - 1011613846169318 T^{15} - 13551267537147553 T^{16} + 219644941420294505 T^{17} + 37364810258114499 T^{18} - 15405117223659998928 T^{19} + 84668251652604544891 T^{20} + \)\(54\!\cdots\!64\)\( T^{21} - \)\(81\!\cdots\!28\)\( T^{22} + \)\(18\!\cdots\!73\)\( T^{23} + \)\(59\!\cdots\!44\)\( T^{24} - \)\(20\!\cdots\!36\)\( T^{25} - \)\(25\!\cdots\!74\)\( T^{26} + \)\(88\!\cdots\!30\)\( T^{27} + \)\(73\!\cdots\!29\)\( T^{28} \))(\( 1 + 83 T^{2} + 6889 T^{4} \))
$89$ (\( 1 + 50 T^{2} + 7921 T^{4} \))(\( 1 - 2 T + 241 T^{2} - 534 T^{3} + 27075 T^{4} - 47526 T^{5} + 1908961 T^{6} - 1409938 T^{7} + 62742241 T^{8} \))(\( 1 + 20 T + 644 T^{2} + 9066 T^{3} + 168861 T^{4} + 1831726 T^{5} + 24665898 T^{6} + 210567376 T^{7} + 2195264922 T^{8} + 14509101646 T^{9} + 119041770309 T^{10} + 568821156906 T^{11} + 3596134285156 T^{12} + 9939625819220 T^{13} + 44231334895529 T^{14} \))(\( ( 1 - 5 T + 345 T^{2} - 424 T^{3} + 47751 T^{4} + 81028 T^{5} + 4451204 T^{6} + 7211492 T^{7} + 378235671 T^{8} - 298906856 T^{9} + 21646073145 T^{10} - 27920297245 T^{11} + 496981290961 T^{12} )^{2} \))(\( ( 1 - 13 T + 422 T^{2} - 4713 T^{3} + 91818 T^{4} - 855866 T^{5} + 12327267 T^{6} - 95013968 T^{7} + 1097126763 T^{8} - 6779314586 T^{9} + 64728843642 T^{10} - 295704181833 T^{11} + 2356473087478 T^{12} - 6460756782493 T^{13} + 44231334895529 T^{14} )^{2} \))(\( ( 1 - 89 T^{2} )^{2} \))
$97$ (\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 - 22 T + 463 T^{2} - 5796 T^{3} + 68533 T^{4} - 562212 T^{5} + 4356367 T^{6} - 20078806 T^{7} + 88529281 T^{8} \))(\( 1 - 14 T + 543 T^{2} - 6190 T^{3} + 136345 T^{4} - 1283466 T^{5} + 20459672 T^{6} - 157691574 T^{7} + 1984588184 T^{8} - 12076131594 T^{9} + 124438400185 T^{10} - 547996249390 T^{11} + 4662925759551 T^{12} - 11661608069006 T^{13} + 80798284478113 T^{14} \))(\( ( 1 + 19 T + 578 T^{2} + 7807 T^{3} + 136575 T^{4} + 1393086 T^{5} + 17451260 T^{6} + 135129342 T^{7} + 1285034175 T^{8} + 7125238111 T^{9} + 51169924418 T^{10} + 163159464883 T^{11} + 832972004929 T^{12} )^{2} \))(\( ( 1 - 2 T + 480 T^{2} - 136 T^{3} + 106498 T^{4} + 82866 T^{5} + 14780621 T^{6} + 15161712 T^{7} + 1433720237 T^{8} + 779686194 T^{9} + 97197849154 T^{10} - 12039982216 T^{11} + 4121923323360 T^{12} - 1665944009858 T^{13} + 80798284478113 T^{14} )^{2} \))(\( ( 1 - 19 T + 97 T^{2} )^{2} \))
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