Properties

Label 525.2.i
Level 525
Weight 2
Character orbit i
Rep. character \(\chi_{525}(151,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 50
Newform subspaces 11
Sturm bound 160
Trace bound 2

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Defining parameters

Level: \( N \) = \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 525.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 11 \)
Sturm bound: \(160\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(525, [\chi])\).

Total New Old
Modular forms 184 50 134
Cusp forms 136 50 86
Eisenstein series 48 0 48

Trace form

\( 50q - 2q^{2} - q^{3} - 26q^{4} - 4q^{6} + 3q^{7} + 24q^{8} - 25q^{9} + O(q^{10}) \) \( 50q - 2q^{2} - q^{3} - 26q^{4} - 4q^{6} + 3q^{7} + 24q^{8} - 25q^{9} - 10q^{11} - 2q^{12} + 2q^{13} - 10q^{14} - 36q^{16} + 8q^{17} - 2q^{18} + 9q^{19} + 2q^{21} - 16q^{22} - 8q^{23} + 12q^{24} + 22q^{26} + 2q^{27} - 32q^{28} + 8q^{29} - q^{31} - 16q^{32} - 10q^{33} + 32q^{34} + 52q^{36} - 7q^{37} - 30q^{38} + q^{39} + 20q^{41} + 14q^{42} + 34q^{43} - 64q^{44} - 32q^{46} + 2q^{47} + 24q^{48} - 31q^{49} - 8q^{51} + 2q^{52} + 28q^{53} + 2q^{54} + 12q^{56} - 14q^{57} + 28q^{58} - 28q^{59} + 28q^{61} - 60q^{62} - 3q^{63} + 144q^{64} - 4q^{66} - 19q^{67} - 12q^{68} - 16q^{69} + 4q^{71} - 12q^{72} + 7q^{73} - 22q^{74} - 116q^{76} - 36q^{77} - 12q^{78} + 21q^{79} - 25q^{81} + 8q^{82} - 20q^{83} + 10q^{84} - 46q^{86} - 20q^{87} - 8q^{88} + 16q^{89} + 61q^{91} + 88q^{92} + 15q^{93} - 4q^{94} + 16q^{96} + 44q^{97} - 4q^{98} + 20q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(525, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
525.2.i.a \(2\) \(4.192\) \(\Q(\sqrt{-3}) \) None \(-2\) \(1\) \(0\) \(1\) \(q-2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
525.2.i.b \(2\) \(4.192\) \(\Q(\sqrt{-3}) \) None \(-1\) \(-1\) \(0\) \(-1\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
525.2.i.c \(2\) \(4.192\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(0\) \(-5\) \(q+(1-\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}+(-2-\zeta_{6})q^{7}+\cdots\)
525.2.i.d \(2\) \(4.192\) \(\Q(\sqrt{-3}) \) None \(1\) \(1\) \(0\) \(1\) \(q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
525.2.i.e \(2\) \(4.192\) \(\Q(\sqrt{-3}) \) None \(2\) \(1\) \(0\) \(5\) \(q+2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
525.2.i.f \(4\) \(4.192\) \(\Q(\zeta_{12})\) None \(-2\) \(-2\) \(0\) \(0\) \(q+(\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(-1+\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
525.2.i.g \(4\) \(4.192\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(-2\) \(0\) \(2\) \(q+\beta _{1}q^{2}+\beta _{2}q^{3}+\beta _{3}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
525.2.i.h \(8\) \(4.192\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-2\) \(4\) \(0\) \(2\) \(q+(-1-\beta _{1}-\beta _{3}+\beta _{4})q^{2}+\beta _{4}q^{3}+\cdots\)
525.2.i.i \(8\) \(4.192\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-1\) \(4\) \(0\) \(1\) \(q-\beta _{1}q^{2}-\beta _{5}q^{3}+(\beta _{2}-\beta _{3}+2\beta _{5}+\cdots)q^{4}+\cdots\)
525.2.i.j \(8\) \(4.192\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(1\) \(-4\) \(0\) \(-1\) \(q+\beta _{1}q^{2}+\beta _{5}q^{3}+(\beta _{2}-\beta _{3}+2\beta _{5}+\cdots)q^{4}+\cdots\)
525.2.i.k \(8\) \(4.192\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(2\) \(-4\) \(0\) \(-2\) \(q+(1+\beta _{1}+\beta _{3}-\beta _{4})q^{2}-\beta _{4}q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(525, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(525, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 - 2 T^{2} + 4 T^{4} \))(\( 1 - T - T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} \))(\( ( 1 + 2 T + 2 T^{2} )^{2}( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} ) \))(\( ( 1 - 2 T^{2} )^{2}( 1 + 2 T^{2} + 4 T^{4} ) \))(\( 1 + 2 T - 4 T^{3} - 6 T^{4} - 4 T^{5} + 4 T^{6} + 12 T^{7} + 12 T^{8} + 24 T^{9} + 16 T^{10} - 32 T^{11} - 96 T^{12} - 128 T^{13} + 256 T^{15} + 256 T^{16} \))(\( 1 + T - T^{3} + 3 T^{5} + 7 T^{6} + 8 T^{7} - 6 T^{8} + 16 T^{9} + 28 T^{10} + 24 T^{11} - 32 T^{13} + 128 T^{15} + 256 T^{16} \))(\( 1 - T + T^{3} - 3 T^{5} + 7 T^{6} - 8 T^{7} - 6 T^{8} - 16 T^{9} + 28 T^{10} - 24 T^{11} + 32 T^{13} - 128 T^{15} + 256 T^{16} \))(\( 1 - 2 T + 4 T^{3} - 6 T^{4} + 4 T^{5} + 4 T^{6} - 12 T^{7} + 12 T^{8} - 24 T^{9} + 16 T^{10} + 32 T^{11} - 96 T^{12} + 128 T^{13} - 256 T^{15} + 256 T^{16} \))
$3$ (\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{4} \))(\( ( 1 + T + T^{2} )^{4} \))(\( ( 1 + T + T^{2} )^{4} \))
$5$ 1
$7$ (\( 1 - T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 - 2 T - 8 T^{2} + 14 T^{3} + 41 T^{4} + 98 T^{5} - 392 T^{6} - 686 T^{7} + 2401 T^{8} \))(\( 1 - T + 11 T^{2} - 12 T^{3} + 116 T^{4} - 84 T^{5} + 539 T^{6} - 343 T^{7} + 2401 T^{8} \))(\( 1 + T + 11 T^{2} + 12 T^{3} + 116 T^{4} + 84 T^{5} + 539 T^{6} + 343 T^{7} + 2401 T^{8} \))(\( 1 + 2 T - 8 T^{2} - 14 T^{3} + 41 T^{4} - 98 T^{5} - 392 T^{6} + 686 T^{7} + 2401 T^{8} \))
$11$ (\( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 44 T^{5} - 1936 T^{6} - 2662 T^{7} + 14641 T^{8} \))(\( 1 + 4 T - 8 T^{2} + 8 T^{3} + 279 T^{4} + 88 T^{5} - 968 T^{6} + 5324 T^{7} + 14641 T^{8} \))(\( 1 - 26 T^{2} - 28 T^{3} + 316 T^{4} + 518 T^{5} - 2872 T^{6} - 2968 T^{7} + 31007 T^{8} - 32648 T^{9} - 347512 T^{10} + 689458 T^{11} + 4626556 T^{12} - 4509428 T^{13} - 46060586 T^{14} + 214358881 T^{16} \))(\( 1 + 8 T + 28 T^{2} + 120 T^{3} + 402 T^{4} + 396 T^{5} + 384 T^{6} - 2384 T^{7} - 29421 T^{8} - 26224 T^{9} + 46464 T^{10} + 527076 T^{11} + 5885682 T^{12} + 19326120 T^{13} + 49603708 T^{14} + 155897368 T^{15} + 214358881 T^{16} \))(\( 1 + 8 T + 28 T^{2} + 120 T^{3} + 402 T^{4} + 396 T^{5} + 384 T^{6} - 2384 T^{7} - 29421 T^{8} - 26224 T^{9} + 46464 T^{10} + 527076 T^{11} + 5885682 T^{12} + 19326120 T^{13} + 49603708 T^{14} + 155897368 T^{15} + 214358881 T^{16} \))(\( 1 - 26 T^{2} - 28 T^{3} + 316 T^{4} + 518 T^{5} - 2872 T^{6} - 2968 T^{7} + 31007 T^{8} - 32648 T^{9} - 347512 T^{10} + 689458 T^{11} + 4626556 T^{12} - 4509428 T^{13} - 46060586 T^{14} + 214358881 T^{16} \))
$13$ (\( ( 1 - 3 T + 13 T^{2} )^{2} \))(\( ( 1 + 3 T + 13 T^{2} )^{2} \))(\( ( 1 - T + 13 T^{2} )^{2} \))(\( ( 1 - 3 T + 13 T^{2} )^{2} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 + 8 T + 39 T^{2} + 104 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - 6 T + 33 T^{2} - 78 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - 2 T + 24 T^{2} - 42 T^{3} + 413 T^{4} - 546 T^{5} + 4056 T^{6} - 4394 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 + 7 T + 33 T^{2} + 94 T^{3} + 326 T^{4} + 1222 T^{5} + 5577 T^{6} + 15379 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 - 7 T + 33 T^{2} - 94 T^{3} + 326 T^{4} - 1222 T^{5} + 5577 T^{6} - 15379 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 + 2 T + 24 T^{2} + 42 T^{3} + 413 T^{4} + 546 T^{5} + 4056 T^{6} + 4394 T^{7} + 28561 T^{8} )^{2} \))
$17$ (\( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 + 2 T - 13 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 10 T + 44 T^{2} - 220 T^{3} + 1147 T^{4} - 3740 T^{5} + 12716 T^{6} - 49130 T^{7} + 83521 T^{8} \))(\( 1 + 4 T - 4 T^{2} - 56 T^{3} - 161 T^{4} - 952 T^{5} - 1156 T^{6} + 19652 T^{7} + 83521 T^{8} \))(\( 1 + 2 T - 36 T^{2} - 80 T^{3} + 524 T^{4} + 896 T^{5} - 9840 T^{6} - 2574 T^{7} + 223639 T^{8} - 43758 T^{9} - 2843760 T^{10} + 4402048 T^{11} + 43765004 T^{12} - 113588560 T^{13} - 868952484 T^{14} + 820677346 T^{15} + 6975757441 T^{16} \))(\( 1 - 6 T - 12 T^{2} + 108 T^{3} + 22 T^{4} + 258 T^{5} - 5760 T^{6} - 6186 T^{7} + 134727 T^{8} - 105162 T^{9} - 1664640 T^{10} + 1267554 T^{11} + 1837462 T^{12} + 153344556 T^{13} - 289650828 T^{14} - 2462032038 T^{15} + 6975757441 T^{16} \))(\( 1 + 6 T - 12 T^{2} - 108 T^{3} + 22 T^{4} - 258 T^{5} - 5760 T^{6} + 6186 T^{7} + 134727 T^{8} + 105162 T^{9} - 1664640 T^{10} - 1267554 T^{11} + 1837462 T^{12} - 153344556 T^{13} - 289650828 T^{14} + 2462032038 T^{15} + 6975757441 T^{16} \))(\( 1 - 2 T - 36 T^{2} + 80 T^{3} + 524 T^{4} - 896 T^{5} - 9840 T^{6} + 2574 T^{7} + 223639 T^{8} + 43758 T^{9} - 2843760 T^{10} - 4402048 T^{11} + 43765004 T^{12} + 113588560 T^{13} - 868952484 T^{14} - 820677346 T^{15} + 6975757441 T^{16} \))
$19$ (\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 + 2 T - 23 T^{2} - 22 T^{3} + 292 T^{4} - 418 T^{5} - 8303 T^{6} + 13718 T^{7} + 130321 T^{8} \))(\( 1 - 2 T - 3 T^{2} + 62 T^{3} - 388 T^{4} + 1178 T^{5} - 1083 T^{6} - 13718 T^{7} + 130321 T^{8} \))(\( 1 - 12 T + 30 T^{2} + 80 T^{3} + 689 T^{4} - 9436 T^{5} + 28846 T^{6} - 59944 T^{7} + 291332 T^{8} - 1138936 T^{9} + 10413406 T^{10} - 64721524 T^{11} + 89791169 T^{12} + 198087920 T^{13} + 1411376430 T^{14} - 10726460868 T^{15} + 16983563041 T^{16} \))(\( 1 + 3 T - 30 T^{2} + 125 T^{3} + 969 T^{4} - 3616 T^{5} + 6406 T^{6} + 66486 T^{7} - 238148 T^{8} + 1263234 T^{9} + 2312566 T^{10} - 24802144 T^{11} + 126281049 T^{12} + 309512375 T^{13} - 1411376430 T^{14} + 2681615217 T^{15} + 16983563041 T^{16} \))(\( 1 + 3 T - 30 T^{2} + 125 T^{3} + 969 T^{4} - 3616 T^{5} + 6406 T^{6} + 66486 T^{7} - 238148 T^{8} + 1263234 T^{9} + 2312566 T^{10} - 24802144 T^{11} + 126281049 T^{12} + 309512375 T^{13} - 1411376430 T^{14} + 2681615217 T^{15} + 16983563041 T^{16} \))(\( 1 - 12 T + 30 T^{2} + 80 T^{3} + 689 T^{4} - 9436 T^{5} + 28846 T^{6} - 59944 T^{7} + 291332 T^{8} - 1138936 T^{9} + 10413406 T^{10} - 64721524 T^{11} + 89791169 T^{12} + 198087920 T^{13} + 1411376430 T^{14} - 10726460868 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 + 2 T - 19 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 - 2 T - 19 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 + 6 T - 16 T^{2} + 36 T^{3} + 1347 T^{4} + 828 T^{5} - 8464 T^{6} + 73002 T^{7} + 279841 T^{8} \))(\( 1 + 4 T - 16 T^{2} - 56 T^{3} + 127 T^{4} - 1288 T^{5} - 8464 T^{6} + 48668 T^{7} + 279841 T^{8} \))(\( 1 + 10 T + 40 T^{2} + 392 T^{3} + 2364 T^{4} + 4712 T^{5} + 38616 T^{6} + 172414 T^{7} + 38807 T^{8} + 3965522 T^{9} + 20427864 T^{10} + 57330904 T^{11} + 661544124 T^{12} + 2523046456 T^{13} + 5921435560 T^{14} + 34048254470 T^{15} + 78310985281 T^{16} \))(\( 1 + 2 T - 20 T^{2} - 84 T^{3} - 562 T^{4} - 922 T^{5} + 2568 T^{6} + 43258 T^{7} + 424639 T^{8} + 994934 T^{9} + 1358472 T^{10} - 11217974 T^{11} - 157270642 T^{12} - 540652812 T^{13} - 2960717780 T^{14} + 6809650894 T^{15} + 78310985281 T^{16} \))(\( 1 - 2 T - 20 T^{2} + 84 T^{3} - 562 T^{4} + 922 T^{5} + 2568 T^{6} - 43258 T^{7} + 424639 T^{8} - 994934 T^{9} + 1358472 T^{10} + 11217974 T^{11} - 157270642 T^{12} + 540652812 T^{13} - 2960717780 T^{14} - 6809650894 T^{15} + 78310985281 T^{16} \))(\( 1 - 10 T + 40 T^{2} - 392 T^{3} + 2364 T^{4} - 4712 T^{5} + 38616 T^{6} - 172414 T^{7} + 38807 T^{8} - 3965522 T^{9} + 20427864 T^{10} - 57330904 T^{11} + 661544124 T^{12} - 2523046456 T^{13} + 5921435560 T^{14} - 34048254470 T^{15} + 78310985281 T^{16} \))
$29$ (\( ( 1 + 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 8 T + 29 T^{2} )^{2} \))(\( ( 1 - 4 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 32 T^{2} - 58 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 8 T + 56 T^{2} - 232 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 6 T + 78 T^{2} + 332 T^{3} + 2820 T^{4} + 9628 T^{5} + 65598 T^{6} + 146334 T^{7} + 707281 T^{8} )^{2} \))(\( ( 1 - 4 T + 29 T^{2} )^{8} \))(\( ( 1 - 4 T + 29 T^{2} )^{8} \))(\( ( 1 + 6 T + 78 T^{2} + 332 T^{3} + 2820 T^{4} + 9628 T^{5} + 65598 T^{6} + 146334 T^{7} + 707281 T^{8} )^{2} \))
$31$ (\( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \))(\( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} \))(\( 1 + 9 T + 50 T^{2} + 279 T^{3} + 961 T^{4} \))(\( 1 + 6 T - 23 T^{2} - 18 T^{3} + 1404 T^{4} - 558 T^{5} - 22103 T^{6} + 178746 T^{7} + 923521 T^{8} \))(\( 1 - 6 T - 27 T^{2} - 6 T^{3} + 1892 T^{4} - 186 T^{5} - 25947 T^{6} - 178746 T^{7} + 923521 T^{8} \))(\( 1 - 8 T - 54 T^{2} + 352 T^{3} + 3285 T^{4} - 9272 T^{5} - 157694 T^{6} + 60416 T^{7} + 6248764 T^{8} + 1872896 T^{9} - 151543934 T^{10} - 276222152 T^{11} + 3033766485 T^{12} + 10077461152 T^{13} - 47925198774 T^{14} - 220100912888 T^{15} + 852891037441 T^{16} \))(\( 1 + 9 T + 13 T^{2} + 90 T^{3} + 749 T^{4} - 1629 T^{5} + 22230 T^{6} + 154161 T^{7} - 324118 T^{8} + 4778991 T^{9} + 21363030 T^{10} - 48529539 T^{11} + 691717229 T^{12} + 2576623590 T^{13} + 11537547853 T^{14} + 247613526999 T^{15} + 852891037441 T^{16} \))(\( 1 + 9 T + 13 T^{2} + 90 T^{3} + 749 T^{4} - 1629 T^{5} + 22230 T^{6} + 154161 T^{7} - 324118 T^{8} + 4778991 T^{9} + 21363030 T^{10} - 48529539 T^{11} + 691717229 T^{12} + 2576623590 T^{13} + 11537547853 T^{14} + 247613526999 T^{15} + 852891037441 T^{16} \))(\( 1 - 8 T - 54 T^{2} + 352 T^{3} + 3285 T^{4} - 9272 T^{5} - 157694 T^{6} + 60416 T^{7} + 6248764 T^{8} + 1872896 T^{9} - 151543934 T^{10} - 276222152 T^{11} + 3033766485 T^{12} + 10077461152 T^{13} - 47925198774 T^{14} - 220100912888 T^{15} + 852891037441 T^{16} \))
$37$ (\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))(\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))(\( 1 - 3 T - 28 T^{2} - 111 T^{3} + 1369 T^{4} \))(\( 1 - 4 T - 35 T^{2} + 92 T^{3} + 640 T^{4} + 3404 T^{5} - 47915 T^{6} - 202612 T^{7} + 1874161 T^{8} \))(\( 1 + 14 T + 75 T^{2} + 658 T^{3} + 6020 T^{4} + 24346 T^{5} + 102675 T^{6} + 709142 T^{7} + 1874161 T^{8} \))(\( 1 + 24 T + 224 T^{2} + 1676 T^{3} + 18423 T^{4} + 154738 T^{5} + 897620 T^{6} + 6581218 T^{7} + 49111756 T^{8} + 243505066 T^{9} + 1228841780 T^{10} + 7837943914 T^{11} + 34527668103 T^{12} + 116220471932 T^{13} + 574722715616 T^{14} + 2278365051192 T^{15} + 3512479453921 T^{16} \))(\( 1 + 8 T - 22 T^{2} - 800 T^{3} - 2955 T^{4} + 21128 T^{5} + 190466 T^{6} - 114864 T^{7} - 6645332 T^{8} - 4249968 T^{9} + 260747954 T^{10} + 1070196584 T^{11} - 5538145755 T^{12} - 55475165600 T^{13} - 56445980998 T^{14} + 759455017064 T^{15} + 3512479453921 T^{16} \))(\( 1 - 8 T - 22 T^{2} + 800 T^{3} - 2955 T^{4} - 21128 T^{5} + 190466 T^{6} + 114864 T^{7} - 6645332 T^{8} + 4249968 T^{9} + 260747954 T^{10} - 1070196584 T^{11} - 5538145755 T^{12} + 55475165600 T^{13} - 56445980998 T^{14} - 759455017064 T^{15} + 3512479453921 T^{16} \))(\( 1 - 24 T + 224 T^{2} - 1676 T^{3} + 18423 T^{4} - 154738 T^{5} + 897620 T^{6} - 6581218 T^{7} + 49111756 T^{8} - 243505066 T^{9} + 1228841780 T^{10} - 7837943914 T^{11} + 34527668103 T^{12} - 116220471932 T^{13} + 574722715616 T^{14} - 2278365051192 T^{15} + 3512479453921 T^{16} \))
$41$ (\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 12 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 80 T^{2} - 82 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 4 T + 114 T^{2} - 346 T^{3} + 5976 T^{4} - 14186 T^{5} + 191634 T^{6} - 275684 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 - 4 T + 104 T^{2} - 256 T^{3} + 4966 T^{4} - 10496 T^{5} + 174824 T^{6} - 275684 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 - 4 T + 104 T^{2} - 256 T^{3} + 4966 T^{4} - 10496 T^{5} + 174824 T^{6} - 275684 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 - 4 T + 114 T^{2} - 346 T^{3} + 5976 T^{4} - 14186 T^{5} + 191634 T^{6} - 275684 T^{7} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 - T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 5 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 63 T^{2} - 172 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 18 T + 165 T^{2} - 774 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 8 T + 140 T^{2} - 878 T^{3} + 8293 T^{4} - 37754 T^{5} + 258860 T^{6} - 636056 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 - 5 T + 120 T^{2} - 677 T^{3} + 6782 T^{4} - 29111 T^{5} + 221880 T^{6} - 397535 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 + 5 T + 120 T^{2} + 677 T^{3} + 6782 T^{4} + 29111 T^{5} + 221880 T^{6} + 397535 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 + 8 T + 140 T^{2} + 878 T^{3} + 8293 T^{4} + 37754 T^{5} + 258860 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} \))(\( 1 - 10 T + 53 T^{2} - 470 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 + 10 T + 53 T^{2} + 470 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( ( 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} )^{2} \))(\( 1 + 4 T + 46 T^{2} - 496 T^{3} - 2061 T^{4} - 23312 T^{5} + 101614 T^{6} + 415292 T^{7} + 4879681 T^{8} \))(\( 1 - 10 T - 104 T^{2} + 652 T^{3} + 13518 T^{4} - 45086 T^{5} - 905784 T^{6} + 563558 T^{7} + 53075567 T^{8} + 26487226 T^{9} - 2000876856 T^{10} - 4680963778 T^{11} + 65963527758 T^{12} + 149532944564 T^{13} - 1121038394216 T^{14} - 5066231204630 T^{15} + 23811286661761 T^{16} \))(\( 1 + 6 T - 88 T^{2} + 52 T^{3} + 6918 T^{4} - 17418 T^{5} - 221392 T^{6} + 441630 T^{7} + 3733263 T^{8} + 20756610 T^{9} - 489054928 T^{10} - 1808389014 T^{11} + 33757633158 T^{12} + 11925940364 T^{13} - 948570948952 T^{14} + 3039738722778 T^{15} + 23811286661761 T^{16} \))(\( 1 - 6 T - 88 T^{2} - 52 T^{3} + 6918 T^{4} + 17418 T^{5} - 221392 T^{6} - 441630 T^{7} + 3733263 T^{8} - 20756610 T^{9} - 489054928 T^{10} + 1808389014 T^{11} + 33757633158 T^{12} - 11925940364 T^{13} - 948570948952 T^{14} - 3039738722778 T^{15} + 23811286661761 T^{16} \))(\( 1 + 10 T - 104 T^{2} - 652 T^{3} + 13518 T^{4} + 45086 T^{5} - 905784 T^{6} - 563558 T^{7} + 53075567 T^{8} - 26487226 T^{9} - 2000876856 T^{10} + 4680963778 T^{11} + 65963527758 T^{12} - 149532944564 T^{13} - 1121038394216 T^{14} + 5066231204630 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} \))(\( 1 - 14 T + 143 T^{2} - 742 T^{3} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 + 14 T + 143 T^{2} + 742 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 - 4 T + 14 T^{2} + 416 T^{3} - 3653 T^{4} + 22048 T^{5} + 39326 T^{6} - 595508 T^{7} + 7890481 T^{8} \))(\( 1 - 8 T - 50 T^{2} - 64 T^{3} + 6795 T^{4} - 3392 T^{5} - 140450 T^{6} - 1191016 T^{7} + 7890481 T^{8} \))(\( 1 + 20 T + 68 T^{2} - 136 T^{3} + 12794 T^{4} + 117580 T^{5} - 98000 T^{6} + 2920764 T^{7} + 73537971 T^{8} + 154800492 T^{9} - 275282000 T^{10} + 17504957660 T^{11} + 100950813914 T^{12} - 56874587048 T^{13} + 1507176556772 T^{14} + 23494222796740 T^{15} + 62259690411361 T^{16} \))(\( 1 - 6 T - 100 T^{2} + 508 T^{3} + 4614 T^{4} - 6198 T^{5} - 355984 T^{6} - 202434 T^{7} + 25854567 T^{8} - 10729002 T^{9} - 999959056 T^{10} - 922739646 T^{11} + 36406679334 T^{12} + 212443310444 T^{13} - 2216436112900 T^{14} - 7048266839022 T^{15} + 62259690411361 T^{16} \))(\( 1 + 6 T - 100 T^{2} - 508 T^{3} + 4614 T^{4} + 6198 T^{5} - 355984 T^{6} + 202434 T^{7} + 25854567 T^{8} + 10729002 T^{9} - 999959056 T^{10} + 922739646 T^{11} + 36406679334 T^{12} - 212443310444 T^{13} - 2216436112900 T^{14} + 7048266839022 T^{15} + 62259690411361 T^{16} \))(\( 1 - 20 T + 68 T^{2} + 136 T^{3} + 12794 T^{4} - 117580 T^{5} - 98000 T^{6} - 2920764 T^{7} + 73537971 T^{8} - 154800492 T^{9} - 275282000 T^{10} - 17504957660 T^{11} + 100950813914 T^{12} + 56874587048 T^{13} + 1507176556772 T^{14} - 23494222796740 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 - 8 T + 5 T^{2} - 472 T^{3} + 3481 T^{4} \))(\( 1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4} \))(\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 + 10 T - 16 T^{2} - 20 T^{3} + 4075 T^{4} - 1180 T^{5} - 55696 T^{6} + 2053790 T^{7} + 12117361 T^{8} \))(\( 1 - 116 T^{2} + 9975 T^{4} - 403796 T^{6} + 12117361 T^{8} \))(\( 1 + 2 T - 226 T^{2} - 236 T^{3} + 32016 T^{4} + 19820 T^{5} - 2961940 T^{6} - 407278 T^{7} + 205790979 T^{8} - 24029402 T^{9} - 10310513140 T^{10} + 4070611780 T^{11} + 387949429776 T^{12} - 168722134564 T^{13} - 9532800602866 T^{14} + 4977302969638 T^{15} + 146830437604321 T^{16} \))(\( 1 + 10 T - 96 T^{2} - 1492 T^{3} + 4998 T^{4} + 116970 T^{5} + 158320 T^{6} - 3153670 T^{7} - 19452177 T^{8} - 186066530 T^{9} + 551111920 T^{10} + 24023181630 T^{11} + 60562570278 T^{12} - 1066667054108 T^{13} - 4049331229536 T^{14} + 24886514848190 T^{15} + 146830437604321 T^{16} \))(\( 1 + 10 T - 96 T^{2} - 1492 T^{3} + 4998 T^{4} + 116970 T^{5} + 158320 T^{6} - 3153670 T^{7} - 19452177 T^{8} - 186066530 T^{9} + 551111920 T^{10} + 24023181630 T^{11} + 60562570278 T^{12} - 1066667054108 T^{13} - 4049331229536 T^{14} + 24886514848190 T^{15} + 146830437604321 T^{16} \))(\( 1 + 2 T - 226 T^{2} - 236 T^{3} + 32016 T^{4} + 19820 T^{5} - 2961940 T^{6} - 407278 T^{7} + 205790979 T^{8} - 24029402 T^{9} - 10310513140 T^{10} + 4070611780 T^{11} + 387949429776 T^{12} - 168722134564 T^{13} - 9532800602866 T^{14} + 4977302969638 T^{15} + 146830437604321 T^{16} \))
$61$ (\( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \))(\( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \))(\( ( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} )^{2} \))(\( 1 + 8 T - 2 T^{2} - 448 T^{3} - 3269 T^{4} - 27328 T^{5} - 7442 T^{6} + 1815848 T^{7} + 13845841 T^{8} \))(\( 1 - 8 T - 80 T^{2} + 376 T^{3} + 3998 T^{4} + 13396 T^{5} - 247632 T^{6} - 798752 T^{7} + 15355491 T^{8} - 48723872 T^{9} - 921438672 T^{10} + 3040637476 T^{11} + 55355672318 T^{12} + 317568209176 T^{13} - 4121629948880 T^{14} - 25141942688168 T^{15} + 191707312997281 T^{16} \))(\( ( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} )^{4} \))(\( 1 - 8 T - 80 T^{2} + 376 T^{3} + 3998 T^{4} + 13396 T^{5} - 247632 T^{6} - 798752 T^{7} + 15355491 T^{8} - 48723872 T^{9} - 921438672 T^{10} + 3040637476 T^{11} + 55355672318 T^{12} + 317568209176 T^{13} - 4121629948880 T^{14} - 25141942688168 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 + 12 T + 49 T^{2} - 468 T^{3} - 5112 T^{4} - 31356 T^{5} + 219961 T^{6} + 3609156 T^{7} + 20151121 T^{8} \))(\( 1 + 2 T + 31 T^{2} - 322 T^{3} - 4028 T^{4} - 21574 T^{5} + 139159 T^{6} + 601526 T^{7} + 20151121 T^{8} \))(\( 1 + 6 T - 160 T^{2} - 148 T^{3} + 18063 T^{4} - 22466 T^{5} - 1278868 T^{6} + 541096 T^{7} + 68029204 T^{8} + 36253432 T^{9} - 5740838452 T^{10} - 6756941558 T^{11} + 363989698623 T^{12} - 199818515836 T^{13} - 14473341147040 T^{14} + 36364269631938 T^{15} + 406067677556641 T^{16} \))(\( 1 + T - 250 T^{2} - 133 T^{3} + 38093 T^{4} + 11624 T^{5} - 3926238 T^{6} - 284754 T^{7} + 306049684 T^{8} - 19078518 T^{9} - 17624882382 T^{10} + 3496069112 T^{11} + 767616652253 T^{12} - 179566639231 T^{13} - 22614595542250 T^{14} + 6060711605323 T^{15} + 406067677556641 T^{16} \))(\( 1 - T - 250 T^{2} + 133 T^{3} + 38093 T^{4} - 11624 T^{5} - 3926238 T^{6} + 284754 T^{7} + 306049684 T^{8} + 19078518 T^{9} - 17624882382 T^{10} - 3496069112 T^{11} + 767616652253 T^{12} + 179566639231 T^{13} - 22614595542250 T^{14} - 6060711605323 T^{15} + 406067677556641 T^{16} \))(\( 1 - 6 T - 160 T^{2} + 148 T^{3} + 18063 T^{4} + 22466 T^{5} - 1278868 T^{6} - 541096 T^{7} + 68029204 T^{8} - 36253432 T^{9} - 5740838452 T^{10} + 6756941558 T^{11} + 363989698623 T^{12} + 199818515836 T^{13} - 14473341147040 T^{14} - 36364269631938 T^{15} + 406067677556641 T^{16} \))
$71$ (\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 2 T + 116 T^{2} - 142 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 4 T + 144 T^{2} + 284 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 14 T + 194 T^{2} + 1648 T^{3} + 14264 T^{4} + 117008 T^{5} + 977954 T^{6} + 5010754 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 - 22 T + 252 T^{2} - 1806 T^{3} + 12966 T^{4} - 128226 T^{5} + 1270332 T^{6} - 7874042 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 - 22 T + 252 T^{2} - 1806 T^{3} + 12966 T^{4} - 128226 T^{5} + 1270332 T^{6} - 7874042 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 + 14 T + 194 T^{2} + 1648 T^{3} + 14264 T^{4} + 117008 T^{5} + 977954 T^{6} + 5010754 T^{7} + 25411681 T^{8} )^{2} \))
$73$ (\( 1 - T - 72 T^{2} - 73 T^{3} + 5329 T^{4} \))(\( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \))(\( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 + 3 T - 64 T^{2} + 219 T^{3} + 5329 T^{4} \))(\( 1 - 8 T - 23 T^{2} + 472 T^{3} - 2432 T^{4} + 34456 T^{5} - 122567 T^{6} - 3112136 T^{7} + 28398241 T^{8} \))(\( 1 + 10 T - 21 T^{2} - 250 T^{3} + 2012 T^{4} - 18250 T^{5} - 111909 T^{6} + 3890170 T^{7} + 28398241 T^{8} \))(\( 1 + 12 T - 84 T^{2} - 932 T^{3} + 8735 T^{4} + 42250 T^{5} - 655952 T^{6} - 2845850 T^{7} + 12256412 T^{8} - 207747050 T^{9} - 3495568208 T^{10} + 16435968250 T^{11} + 248058635135 T^{12} - 1932102724676 T^{13} - 12712075008276 T^{14} + 132568782229164 T^{15} + 806460091894081 T^{16} \))(\( 1 + 4 T - 154 T^{2} + 776 T^{3} + 16497 T^{4} - 112424 T^{5} - 312970 T^{6} + 5766588 T^{7} - 11491964 T^{8} + 420960924 T^{9} - 1667817130 T^{10} - 43734847208 T^{11} + 468485781777 T^{12} + 1608703556168 T^{13} - 23305470848506 T^{14} + 44189594076388 T^{15} + 806460091894081 T^{16} \))(\( 1 - 4 T - 154 T^{2} - 776 T^{3} + 16497 T^{4} + 112424 T^{5} - 312970 T^{6} - 5766588 T^{7} - 11491964 T^{8} - 420960924 T^{9} - 1667817130 T^{10} + 43734847208 T^{11} + 468485781777 T^{12} - 1608703556168 T^{13} - 23305470848506 T^{14} - 44189594076388 T^{15} + 806460091894081 T^{16} \))(\( 1 - 12 T - 84 T^{2} + 932 T^{3} + 8735 T^{4} - 42250 T^{5} - 655952 T^{6} + 2845850 T^{7} + 12256412 T^{8} + 207747050 T^{9} - 3495568208 T^{10} - 16435968250 T^{11} + 248058635135 T^{12} + 1932102724676 T^{13} - 12712075008276 T^{14} - 132568782229164 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 + 6 T - 23 T^{2} - 594 T^{3} - 5604 T^{4} - 46926 T^{5} - 143543 T^{6} + 2958234 T^{7} + 38950081 T^{8} \))(\( 1 + 2 T - 123 T^{2} - 62 T^{3} + 9572 T^{4} - 4898 T^{5} - 767643 T^{6} + 986078 T^{7} + 38950081 T^{8} \))(\( 1 - 8 T - 98 T^{2} + 1128 T^{3} - 3647 T^{4} + 31028 T^{5} - 546546 T^{6} - 5486636 T^{7} + 152161556 T^{8} - 433444244 T^{9} - 3410993586 T^{10} + 15298014092 T^{11} - 142050945407 T^{12} + 3470919618072 T^{13} - 23822570641058 T^{14} - 153631271889272 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 8 T - 182 T^{2} - 1560 T^{3} + 19865 T^{4} + 148676 T^{5} - 1461174 T^{6} - 5521756 T^{7} + 104973092 T^{8} - 436218724 T^{9} - 9119186934 T^{10} + 73303066364 T^{11} + 773743359065 T^{12} - 4800207982440 T^{13} - 44241916904822 T^{14} + 153631271889272 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 8 T - 182 T^{2} - 1560 T^{3} + 19865 T^{4} + 148676 T^{5} - 1461174 T^{6} - 5521756 T^{7} + 104973092 T^{8} - 436218724 T^{9} - 9119186934 T^{10} + 73303066364 T^{11} + 773743359065 T^{12} - 4800207982440 T^{13} - 44241916904822 T^{14} + 153631271889272 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 8 T - 98 T^{2} + 1128 T^{3} - 3647 T^{4} + 31028 T^{5} - 546546 T^{6} - 5486636 T^{7} + 152161556 T^{8} - 433444244 T^{9} - 3410993586 T^{10} + 15298014092 T^{11} - 142050945407 T^{12} + 3470919618072 T^{13} - 23822570641058 T^{14} - 153631271889272 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( ( 1 + 2 T + 83 T^{2} )^{2} \))(\( ( 1 - 14 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 14 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 28 T^{2} + 498 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 8 T + 180 T^{2} + 664 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 6 T + 276 T^{2} + 1256 T^{3} + 32400 T^{4} + 104248 T^{5} + 1901364 T^{6} + 3430722 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 - 2 T + 220 T^{2} - 894 T^{3} + 22430 T^{4} - 74202 T^{5} + 1515580 T^{6} - 1143574 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 + 2 T + 220 T^{2} + 894 T^{3} + 22430 T^{4} + 74202 T^{5} + 1515580 T^{6} + 1143574 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 - 6 T + 276 T^{2} - 1256 T^{3} + 32400 T^{4} - 104248 T^{5} + 1901364 T^{6} - 3430722 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( 1 - 12 T + 55 T^{2} - 1068 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 16 T + 167 T^{2} + 1424 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 4 T^{2} - 828 T^{3} - 9525 T^{4} - 73692 T^{5} - 31684 T^{6} + 4229814 T^{7} + 62742241 T^{8} \))(\( 1 - 16 T + 32 T^{2} - 736 T^{3} + 19471 T^{4} - 65504 T^{5} + 253472 T^{6} - 11279504 T^{7} + 62742241 T^{8} \))(\( 1 + 8 T - 98 T^{2} - 1004 T^{3} + 4584 T^{4} + 89290 T^{5} + 1051584 T^{6} - 2927032 T^{7} - 142835833 T^{8} - 260505848 T^{9} + 8329596864 T^{10} + 62946682010 T^{11} + 287610432744 T^{12} - 5606395686796 T^{13} - 48704166514178 T^{14} + 353850679164232 T^{15} + 3936588805702081 T^{16} \))(\( 1 - 10 T - 260 T^{2} + 1572 T^{3} + 55478 T^{4} - 195874 T^{5} - 7285344 T^{6} + 5340058 T^{7} + 785732359 T^{8} + 475265162 T^{9} - 57707209824 T^{10} - 138085097906 T^{11} + 3480814046198 T^{12} + 8778141453828 T^{13} - 129215135649860 T^{14} - 442313348955290 T^{15} + 3936588805702081 T^{16} \))(\( 1 - 10 T - 260 T^{2} + 1572 T^{3} + 55478 T^{4} - 195874 T^{5} - 7285344 T^{6} + 5340058 T^{7} + 785732359 T^{8} + 475265162 T^{9} - 57707209824 T^{10} - 138085097906 T^{11} + 3480814046198 T^{12} + 8778141453828 T^{13} - 129215135649860 T^{14} - 442313348955290 T^{15} + 3936588805702081 T^{16} \))(\( 1 + 8 T - 98 T^{2} - 1004 T^{3} + 4584 T^{4} + 89290 T^{5} + 1051584 T^{6} - 2927032 T^{7} - 142835833 T^{8} - 260505848 T^{9} + 8329596864 T^{10} + 62946682010 T^{11} + 287610432744 T^{12} - 5606395686796 T^{13} - 48704166514178 T^{14} + 353850679164232 T^{15} + 3936588805702081 T^{16} \))
$97$ (\( ( 1 - 6 T + 97 T^{2} )^{2} \))(\( ( 1 - 9 T + 97 T^{2} )^{2} \))(\( ( 1 - 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 9 T + 97 T^{2} )^{2} \))(\( ( 1 - 6 T + 97 T^{2} )^{2} \))(\( ( 1 + 16 T + 210 T^{2} + 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 16 T + 250 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 2 T + 380 T^{2} + 566 T^{3} + 54898 T^{4} + 54902 T^{5} + 3575420 T^{6} + 1825346 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 + 12 T + 330 T^{2} + 3296 T^{3} + 45123 T^{4} + 319712 T^{5} + 3104970 T^{6} + 10952076 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 - 12 T + 330 T^{2} - 3296 T^{3} + 45123 T^{4} - 319712 T^{5} + 3104970 T^{6} - 10952076 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 - 2 T + 380 T^{2} - 566 T^{3} + 54898 T^{4} - 54902 T^{5} + 3575420 T^{6} - 1825346 T^{7} + 88529281 T^{8} )^{2} \))
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