Properties

Label 315.2.i
Level 315
Weight 2
Character orbit i
Rep. character \(\chi_{315}(106,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 48
Newform subspaces 6
Sturm bound 96
Trace bound 3

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Defining parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 6 \)
Sturm bound: \(96\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(315, [\chi])\).

Total New Old
Modular forms 104 48 56
Cusp forms 88 48 40
Eisenstein series 16 0 16

Trace form

\( 48q + 4q^{2} + 4q^{3} - 24q^{4} + 16q^{6} + 8q^{9} + O(q^{10}) \) \( 48q + 4q^{2} + 4q^{3} - 24q^{4} + 16q^{6} + 8q^{9} - 6q^{11} - 48q^{12} + 2q^{15} - 24q^{16} + 24q^{17} + 4q^{18} + 24q^{19} + 2q^{21} - 12q^{22} + 24q^{23} - 20q^{24} - 24q^{25} - 16q^{26} - 8q^{27} - 8q^{29} - 32q^{30} + 4q^{32} - 32q^{33} - 16q^{35} + 64q^{36} - 4q^{38} + 30q^{39} + 12q^{40} - 36q^{41} - 12q^{43} - 24q^{44} + 8q^{45} - 24q^{46} + 24q^{47} + 28q^{48} - 24q^{49} + 4q^{50} - 2q^{51} - 36q^{52} - 32q^{53} + 44q^{54} - 48q^{57} - 36q^{58} - 12q^{59} + 8q^{60} + 72q^{62} + 8q^{63} + 96q^{64} + 14q^{65} + 44q^{66} - 12q^{67} - 80q^{68} + 36q^{69} - 16q^{71} + 32q^{72} + 72q^{73} - 44q^{74} + 4q^{75} - 24q^{76} + 16q^{77} - 72q^{78} - 6q^{79} + 32q^{80} - 8q^{81} + 96q^{82} - 20q^{83} + 12q^{84} - 6q^{85} + 84q^{86} - 12q^{87} - 36q^{88} + 40q^{89} - 32q^{90} + 12q^{91} + 104q^{92} + 24q^{93} - 36q^{94} + 24q^{95} - 148q^{96} - 36q^{97} - 8q^{98} + 40q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(315, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
315.2.i.a \(2\) \(2.515\) \(\Q(\sqrt{-3}) \) None \(-2\) \(-3\) \(1\) \(-1\) \(q+(-2+2\zeta_{6})q^{2}+(-1-\zeta_{6})q^{3}-2\zeta_{6}q^{4}+\cdots\)
315.2.i.b \(2\) \(2.515\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(1\) \(-1\) \(q+(2-\zeta_{6})q^{3}+2\zeta_{6}q^{4}+\zeta_{6}q^{5}+(-1+\cdots)q^{7}+\cdots\)
315.2.i.c \(8\) \(2.515\) 8.0.142635249.1 None \(1\) \(-1\) \(-4\) \(-4\) \(q+\beta _{7}q^{2}+(-\beta _{1}+\beta _{2}+\beta _{4}+\beta _{6})q^{3}+\cdots\)
315.2.i.d \(8\) \(2.515\) \(\Q(\zeta_{15})\) None \(1\) \(3\) \(4\) \(4\) \(q+(1+\zeta_{15}-\zeta_{15}^{2}+\zeta_{15}^{3})q^{2}+(1+\cdots)q^{3}+\cdots\)
315.2.i.e \(12\) \(2.515\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(3\) \(1\) \(6\) \(-6\) \(q+(1-\beta _{2}-\beta _{4})q^{2}+\beta _{9}q^{3}+(-\beta _{1}+\cdots)q^{4}+\cdots\)
315.2.i.f \(16\) \(2.515\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(1\) \(1\) \(-8\) \(8\) \(q+(\beta _{2}+\beta _{13})q^{2}+\beta _{5}q^{3}+(-2-2\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(315, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(315, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 - 2 T^{2} + 4 T^{4} \))(\( 1 - T - 3 T^{2} + 4 T^{3} + 3 T^{4} - 6 T^{5} + 4 T^{6} + 4 T^{7} - 15 T^{8} + 8 T^{9} + 16 T^{10} - 48 T^{11} + 48 T^{12} + 128 T^{13} - 192 T^{14} - 128 T^{15} + 256 T^{16} \))(\( ( 1 - 5 T + 13 T^{2} - 25 T^{3} + 39 T^{4} - 50 T^{5} + 52 T^{6} - 40 T^{7} + 16 T^{8} )( 1 + 4 T + 4 T^{2} - 7 T^{3} - 21 T^{4} - 14 T^{5} + 16 T^{6} + 32 T^{7} + 16 T^{8} ) \))(\( 1 - 3 T + 4 T^{2} - 9 T^{3} + 17 T^{4} - 15 T^{5} + 18 T^{6} - 24 T^{7} - 14 T^{8} + 72 T^{9} - 127 T^{10} + 258 T^{11} - 443 T^{12} + 516 T^{13} - 508 T^{14} + 576 T^{15} - 224 T^{16} - 768 T^{17} + 1152 T^{18} - 1920 T^{19} + 4352 T^{20} - 4608 T^{21} + 4096 T^{22} - 6144 T^{23} + 4096 T^{24} \))(\( 1 - T - 2 T^{2} + 3 T^{3} + T^{4} - 5 T^{5} - 2 T^{6} + 6 T^{7} - 2 T^{9} + 13 T^{10} - 34 T^{11} + 77 T^{12} + 46 T^{13} - 264 T^{14} - 24 T^{15} + 500 T^{16} - 48 T^{17} - 1056 T^{18} + 368 T^{19} + 1232 T^{20} - 1088 T^{21} + 832 T^{22} - 256 T^{23} + 3072 T^{25} - 2048 T^{26} - 10240 T^{27} + 4096 T^{28} + 24576 T^{29} - 32768 T^{30} - 32768 T^{31} + 65536 T^{32} \))
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 15 T^{4} + 9 T^{5} - 18 T^{6} + 27 T^{7} + 81 T^{8} \))(\( 1 - 3 T + 6 T^{2} - 9 T^{3} + 9 T^{4} - 27 T^{5} + 54 T^{6} - 81 T^{7} + 81 T^{8} \))(\( 1 - T + 2 T^{2} - T^{3} - 4 T^{4} + 20 T^{5} - 38 T^{6} + 60 T^{7} - 36 T^{8} - 27 T^{9} + 162 T^{10} - 243 T^{11} + 729 T^{12} \))(\( 1 - T - T^{2} - 2 T^{3} + T^{4} + 6 T^{5} + 6 T^{6} - 45 T^{7} + 90 T^{8} - 135 T^{9} + 54 T^{10} + 162 T^{11} + 81 T^{12} - 486 T^{13} - 729 T^{14} - 2187 T^{15} + 6561 T^{16} \))
$5$ (\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{6} \))(\( ( 1 + T + T^{2} )^{8} \))
$7$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( ( 1 + T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{4} \))(\( ( 1 + T + T^{2} )^{6} \))(\( ( 1 - T + T^{2} )^{8} \))
$11$ (\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - T - 9 T^{2} + 80 T^{3} - 100 T^{4} - 635 T^{5} + 2400 T^{6} + 384 T^{7} - 22329 T^{8} + 4224 T^{9} + 290400 T^{10} - 845185 T^{11} - 1464100 T^{12} + 12884080 T^{13} - 15944049 T^{14} - 19487171 T^{15} + 214358881 T^{16} \))(\( 1 + 7 T - 9 T^{2} - 72 T^{3} + 656 T^{4} + 2001 T^{5} - 7276 T^{6} + 844 T^{7} + 151647 T^{8} + 9284 T^{9} - 880396 T^{10} + 2663331 T^{11} + 9604496 T^{12} - 11595672 T^{13} - 15944049 T^{14} + 136410197 T^{15} + 214358881 T^{16} \))(\( 1 - 7 T + 2 T^{2} - 101 T^{3} + 1217 T^{4} - 1299 T^{5} + 3761 T^{6} - 100272 T^{7} + 184115 T^{8} + 8971 T^{9} + 4867308 T^{10} - 13424374 T^{11} - 3841008 T^{12} - 147668114 T^{13} + 588944268 T^{14} + 11940401 T^{15} + 2695627715 T^{16} - 16148905872 T^{17} + 6662840921 T^{18} - 25313835129 T^{19} + 260874758177 T^{20} - 238152716791 T^{21} + 51874849202 T^{22} - 1997181694277 T^{23} + 3138428376721 T^{24} \))(\( 1 + 4 T - 8 T^{2} - 252 T^{3} - 884 T^{4} + 761 T^{5} + 29542 T^{6} + 109554 T^{7} + 21618 T^{8} - 2194876 T^{9} - 9004202 T^{10} - 8688626 T^{11} + 112135271 T^{12} + 523246502 T^{13} + 801407592 T^{14} - 4004853117 T^{15} - 22200406036 T^{16} - 44053384287 T^{17} + 96970318632 T^{18} + 696441094162 T^{19} + 1641772502711 T^{20} - 1399311905926 T^{21} - 15951493099322 T^{22} - 42771923935796 T^{23} + 4634010289458 T^{24} + 258322601339814 T^{25} + 766243397562742 T^{26} + 217122181334971 T^{27} - 2774370685021364 T^{28} - 8699723460270612 T^{29} - 3037998668665928 T^{30} + 16708992677662604 T^{31} + 45949729863572161 T^{32} \))
$13$ (\( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} \))(\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \))(\( 1 - 12 T + 46 T^{2} - 114 T^{3} + 995 T^{4} - 5253 T^{5} + 13023 T^{6} - 59967 T^{7} + 319775 T^{8} - 779571 T^{9} + 2200887 T^{10} - 11540841 T^{11} + 28418195 T^{12} - 42327402 T^{13} + 222033214 T^{14} - 752982204 T^{15} + 815730721 T^{16} \))(\( 1 - 8 T - 2 T^{2} + 50 T^{3} + 725 T^{4} - 1727 T^{5} - 11281 T^{6} + 7925 T^{7} + 170479 T^{8} + 103025 T^{9} - 1906489 T^{10} - 3794219 T^{11} + 20706725 T^{12} + 18564650 T^{13} - 9653618 T^{14} - 501988136 T^{15} + 815730721 T^{16} \))(\( 1 + 10 T + 13 T^{2} - 140 T^{3} - 193 T^{4} + 1289 T^{5} - 4213 T^{6} - 33844 T^{7} + 25985 T^{8} + 133822 T^{9} - 1152335 T^{10} + 1550359 T^{11} + 32289884 T^{12} + 20154667 T^{13} - 194744615 T^{14} + 294006934 T^{15} + 742157585 T^{16} - 12566040292 T^{17} - 20335346317 T^{18} + 80882838413 T^{19} - 157436029153 T^{20} - 1484629912220 T^{21} + 1792160394037 T^{22} + 17921603940370 T^{23} + 23298085122481 T^{24} \))(\( 1 + 5 T - 24 T^{2} - 129 T^{3} + 240 T^{4} + 1174 T^{5} - 1462 T^{6} - 4299 T^{7} - 36 T^{8} + 124186 T^{9} + 631224 T^{10} - 2988076 T^{11} - 15052803 T^{12} + 29465898 T^{13} + 184401460 T^{14} - 98215947 T^{15} - 1918362832 T^{16} - 1276807311 T^{17} + 31163846740 T^{18} + 64736577906 T^{19} - 429923106483 T^{20} - 1109451702268 T^{21} + 3046797684216 T^{22} + 7792487332162 T^{23} - 29366305956 T^{24} - 45588742804527 T^{25} - 201549115083238 T^{26} + 2103996302599438 T^{27} + 5591540429395440 T^{28} - 39070888750400637 T^{29} - 94497033256782936 T^{30} + 255929465070453785 T^{31} + 665416609183179841 T^{32} \))
$17$ (\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 3 T + 17 T^{2} )^{2} \))(\( ( 1 - 5 T + 14 T^{2} + 61 T^{3} - 359 T^{4} + 1037 T^{5} + 4046 T^{6} - 24565 T^{7} + 83521 T^{8} )^{2} \))(\( ( 1 - T + 64 T^{2} - 47 T^{3} + 1599 T^{4} - 799 T^{5} + 18496 T^{6} - 4913 T^{7} + 83521 T^{8} )^{2} \))(\( ( 1 - 7 T + 51 T^{2} - 194 T^{3} + 1000 T^{4} - 2743 T^{5} + 13847 T^{6} - 46631 T^{7} + 289000 T^{8} - 953122 T^{9} + 4259571 T^{10} - 9938999 T^{11} + 24137569 T^{12} )^{2} \))(\( ( 1 - 4 T + 48 T^{2} - 60 T^{3} + 722 T^{4} + 693 T^{5} + 14099 T^{6} - 6718 T^{7} + 352974 T^{8} - 114206 T^{9} + 4074611 T^{10} + 3404709 T^{11} + 60302162 T^{12} - 85191420 T^{13} + 1158603312 T^{14} - 1641354692 T^{15} + 6975757441 T^{16} )^{2} \))
$19$ (\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 9 T + 75 T^{2} + 394 T^{3} + 1947 T^{4} + 7486 T^{5} + 27075 T^{6} + 61731 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 - 3 T + 45 T^{2} - 168 T^{3} + 989 T^{4} - 3192 T^{5} + 16245 T^{6} - 20577 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 - 15 T + 143 T^{2} - 992 T^{3} + 5604 T^{4} - 27269 T^{5} + 123824 T^{6} - 518111 T^{7} + 2023044 T^{8} - 6804128 T^{9} + 18635903 T^{10} - 37141485 T^{11} + 47045881 T^{12} )^{2} \))(\( ( 1 - 3 T + 51 T^{2} - 156 T^{3} + 1837 T^{4} - 5706 T^{5} + 46629 T^{6} - 140085 T^{7} + 980820 T^{8} - 2661615 T^{9} + 16833069 T^{10} - 39137454 T^{11} + 239399677 T^{12} - 386271444 T^{13} + 2399339931 T^{14} - 2681615217 T^{15} + 16983563041 T^{16} )^{2} \))
$23$ (\( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( ( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} )^{4} \))(\( 1 - 8 T + 18 T^{2} - 240 T^{3} + 1445 T^{4} - 1872 T^{5} + 33674 T^{6} - 166760 T^{7} + 89244 T^{8} - 3835480 T^{9} + 17813546 T^{10} - 22776624 T^{11} + 404370245 T^{12} - 1544722320 T^{13} + 2664646002 T^{14} - 27238603576 T^{15} + 78310985281 T^{16} \))(\( 1 - 14 T + 34 T^{2} + 44 T^{3} + 2382 T^{4} - 11350 T^{5} - 42548 T^{6} - 846 T^{7} + 1532858 T^{8} + 1867708 T^{9} - 19984074 T^{10} - 41596486 T^{11} + 327972422 T^{12} - 956719178 T^{13} - 10571575146 T^{14} + 22724403236 T^{15} + 428956515578 T^{16} - 5445146178 T^{17} - 6298631005172 T^{18} - 38644768823450 T^{19} + 186536766939342 T^{20} + 79250717104372 T^{21} + 1408501381264066 T^{22} - 13339336610794978 T^{23} + 21914624432020321 T^{24} \))(\( 1 - 8 T - 56 T^{2} + 636 T^{3} + 781 T^{4} - 20716 T^{5} + 6724 T^{6} + 401256 T^{7} - 165759 T^{8} - 5155168 T^{9} - 11298644 T^{10} + 13905856 T^{11} + 907093574 T^{12} + 431304176 T^{13} - 30428900436 T^{14} - 606821352 T^{15} + 718124779886 T^{16} - 13956891096 T^{17} - 16096888330644 T^{18} + 5247677909392 T^{19} + 253841972841734 T^{20} + 89502858924608 T^{21} - 1672604809034516 T^{22} - 17552447189960096 T^{23} - 12980750609193279 T^{24} + 722723312327997528 T^{25} + 278551861400575876 T^{26} - 19738406944944911732 T^{27} + 17115321681407870701 T^{28} + \)\(32\!\cdots\!88\)\( T^{29} - \)\(64\!\cdots\!04\)\( T^{30} - \)\(21\!\cdots\!56\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \))(\( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} \))(\( 1 - 15 T + 39 T^{2} + 24 T^{3} + 4348 T^{4} - 28215 T^{5} - 20142 T^{6} - 397710 T^{7} + 7357161 T^{8} - 11533590 T^{9} - 16939422 T^{10} - 688135635 T^{11} + 3075257788 T^{12} + 492267576 T^{13} + 23198109519 T^{14} - 258748144635 T^{15} + 500246412961 T^{16} \))(\( 1 + T - 61 T^{2} + 156 T^{3} + 1796 T^{4} - 8467 T^{5} - 22034 T^{6} + 147078 T^{7} + 433657 T^{8} + 4265262 T^{9} - 18530594 T^{10} - 206501663 T^{11} + 1270276676 T^{12} + 3199739244 T^{13} - 36284222581 T^{14} + 17249876309 T^{15} + 500246412961 T^{16} \))(\( 1 + 13 T - 59 T^{2} - 922 T^{3} + 7941 T^{4} + 68816 T^{5} - 482663 T^{6} - 2453358 T^{7} + 29192036 T^{8} + 77970550 T^{9} - 1143760647 T^{10} - 657761371 T^{11} + 39749720555 T^{12} - 19075079759 T^{13} - 961902704127 T^{14} + 1901623743950 T^{15} + 20646972414116 T^{16} - 50321191488342 T^{17} - 287099208583823 T^{18} + 1187067488080144 T^{19} + 3972456765323301 T^{20} - 13375588589751218 T^{21} - 24821726764711859 T^{22} + 158606626954175777 T^{23} + 353814783205469041 T^{24} \))(\( 1 + 19 T + 128 T^{2} + 803 T^{3} + 9164 T^{4} + 53572 T^{5} + 135245 T^{6} + 1457003 T^{7} + 11306929 T^{8} + 8567345 T^{9} + 77536775 T^{10} + 2101449919 T^{11} + 3528685908 T^{12} - 3024827910 T^{13} + 382687393437 T^{14} + 1623090782861 T^{15} - 344149034518 T^{16} + 47069632702969 T^{17} + 321840097880517 T^{18} - 73772527896990 T^{19} + 2495772497696148 T^{20} + 43103152404646931 T^{21} + 46120682005129775 T^{22} + 147785641546529605 T^{23} + 5656250673854706769 T^{24} + 21136955208279060607 T^{25} + 56898549767685684245 T^{26} + \)\(65\!\cdots\!88\)\( T^{27} + \)\(32\!\cdots\!24\)\( T^{28} + \)\(82\!\cdots\!67\)\( T^{29} + \)\(38\!\cdots\!68\)\( T^{30} + \)\(16\!\cdots\!31\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 - 16 T + 64 T^{2} - 106 T^{3} + 3435 T^{4} - 20277 T^{5} - 29847 T^{6} - 161799 T^{7} + 4834609 T^{8} - 5015769 T^{9} - 28682967 T^{10} - 604072107 T^{11} + 3172294635 T^{12} - 3034690006 T^{13} + 56800235584 T^{14} - 440201825776 T^{15} + 852891037441 T^{16} \))(\( 1 - 94 T^{2} - 90 T^{3} + 4885 T^{4} + 5625 T^{5} - 188701 T^{6} - 84465 T^{7} + 6132709 T^{8} - 2618415 T^{9} - 181341661 T^{10} + 167574375 T^{11} + 4511400085 T^{12} - 2576623590 T^{13} - 83425346014 T^{14} + 852891037441 T^{16} \))(\( 1 + 10 T - 32 T^{2} + 202 T^{3} + 8390 T^{4} + 1445 T^{5} - 77485 T^{6} + 2882135 T^{7} + 6359417 T^{8} - 43083374 T^{9} + 605642146 T^{10} + 2584009420 T^{11} - 10106229799 T^{12} + 80104292020 T^{13} + 582022102306 T^{14} - 1283496794834 T^{15} + 5873055147257 T^{16} + 82513078117385 T^{17} - 68768222722285 T^{18} + 39755727390395 T^{19} + 7155755804129990 T^{20} + 5340803676455542 T^{21} - 26228105183385632 T^{22} + 254084768964048310 T^{23} + 787662783788549761 T^{24} \))(\( 1 - 140 T^{2} + 126 T^{3} + 9063 T^{4} - 13239 T^{5} - 402475 T^{6} + 353781 T^{7} + 16648256 T^{8} + 3630141 T^{9} - 737222427 T^{10} - 31707297 T^{11} + 29744064802 T^{12} - 15980009589 T^{13} - 962711725271 T^{14} + 358072466157 T^{15} + 28652584678461 T^{16} + 11100246450867 T^{17} - 925165967985431 T^{18} - 476060465665899 T^{19} + 27469268470007842 T^{20} - 907752993614847 T^{21} - 654287617678253787 T^{22} + 99874668501519651 T^{23} + 14199148331423352896 T^{24} + 9353835967624347051 T^{25} - \)\(32\!\cdots\!75\)\( T^{26} - \)\(33\!\cdots\!09\)\( T^{27} + \)\(71\!\cdots\!43\)\( T^{28} + \)\(30\!\cdots\!66\)\( T^{29} - \)\(10\!\cdots\!40\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( ( 1 + 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{2} \))(\( ( 1 + 15 T + 213 T^{2} + 1714 T^{3} + 12885 T^{4} + 63418 T^{5} + 291597 T^{6} + 759795 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 + 21 T + 289 T^{2} + 2592 T^{3} + 18369 T^{4} + 95904 T^{5} + 395641 T^{6} + 1063713 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 - 21 T + 257 T^{2} - 2192 T^{3} + 15138 T^{4} - 92879 T^{5} + 560324 T^{6} - 3436523 T^{7} + 20723922 T^{8} - 111031376 T^{9} + 481659377 T^{10} - 1456223097 T^{11} + 2565726409 T^{12} )^{2} \))(\( ( 1 - 21 T + 401 T^{2} - 5016 T^{3} + 57403 T^{4} - 523584 T^{5} + 4405763 T^{6} - 31274655 T^{7} + 205649488 T^{8} - 1157162235 T^{9} + 6031489547 T^{10} - 26521100352 T^{11} + 107582463883 T^{12} - 347829288312 T^{13} + 1028856290009 T^{14} - 1993569419793 T^{15} + 3512479453921 T^{16} )^{2} \))
$41$ (\( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} \))(\( 1 - 41 T^{2} + 1681 T^{4} \))(\( 1 - 2 T - 44 T^{2} - 426 T^{3} + 1985 T^{4} + 17281 T^{5} + 170979 T^{6} - 841573 T^{7} - 6842441 T^{8} - 34504493 T^{9} + 287415699 T^{10} + 1191023801 T^{11} + 5609135585 T^{12} - 49354741626 T^{13} - 209004586604 T^{14} - 389508547762 T^{15} + 7984925229121 T^{16} \))(\( 1 + 20 T + 126 T^{2} + 390 T^{3} + 5225 T^{4} + 43635 T^{5} + 118739 T^{6} + 1639535 T^{7} + 19690659 T^{8} + 67220935 T^{9} + 199600259 T^{10} + 3007367835 T^{11} + 14764601225 T^{12} + 45183918390 T^{13} + 598513134366 T^{14} + 3895085477620 T^{15} + 7984925229121 T^{16} \))(\( 1 + 4 T - 158 T^{2} + 14 T^{3} + 16080 T^{4} - 33799 T^{5} - 940223 T^{6} + 3880863 T^{7} + 37316861 T^{8} - 184680932 T^{9} - 912886272 T^{10} + 3736722920 T^{11} + 26275273283 T^{12} + 153205639720 T^{13} - 1534561823232 T^{14} - 12728394514372 T^{15} + 105448530456221 T^{16} + 449622043781463 T^{17} - 4466157259785743 T^{18} - 6582499702903919 T^{19} + 128397597684265680 T^{20} + 4583347081515454 T^{21} - 2120780171004079358 T^{22} + 2201316126864993764 T^{23} + 22563490300366186081 T^{24} \))(\( 1 + 20 T + 118 T^{2} - 150 T^{3} - 4985 T^{4} - 42197 T^{5} - 220145 T^{6} + 1270731 T^{7} + 23848134 T^{8} + 110309977 T^{9} + 134825125 T^{10} - 2724509833 T^{11} - 47046106654 T^{12} - 288572614127 T^{13} - 46076715579 T^{14} + 8512166036907 T^{15} + 65404558839995 T^{16} + 348998807513187 T^{17} - 77454958888299 T^{18} - 19888713138246967 T^{19} - 132941053384713694 T^{20} - 315651358838524433 T^{21} + 640433398055855125 T^{22} + 21483339472464810737 T^{23} + \)\(19\!\cdots\!14\)\( T^{24} + \)\(41\!\cdots\!91\)\( T^{25} - \)\(29\!\cdots\!45\)\( T^{26} - \)\(23\!\cdots\!77\)\( T^{27} - \)\(11\!\cdots\!85\)\( T^{28} - \)\(13\!\cdots\!50\)\( T^{29} + \)\(44\!\cdots\!98\)\( T^{30} + \)\(31\!\cdots\!20\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( 1 + 4 T - 27 T^{2} + 172 T^{3} + 1849 T^{4} \))(\( 1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 - 7 T + 31 T^{2} - 88 T^{3} - 1116 T^{4} + 17139 T^{5} + 37680 T^{6} - 784944 T^{7} + 5466535 T^{8} - 33752592 T^{9} + 69670320 T^{10} + 1362670473 T^{11} - 3815381916 T^{12} - 12936742984 T^{13} + 195962254519 T^{14} - 1902730277749 T^{15} + 11688200277601 T^{16} \))(\( 1 - 7 T - 77 T^{2} + 160 T^{3} + 5570 T^{4} + 5177 T^{5} - 258466 T^{6} - 11150 T^{7} + 7688479 T^{8} - 479450 T^{9} - 477903634 T^{10} + 411607739 T^{11} + 19042721570 T^{12} + 23521350880 T^{13} - 486744954773 T^{14} - 1902730277749 T^{15} + 11688200277601 T^{16} \))(\( 1 + 13 T - 59 T^{2} - 1298 T^{3} + 1643 T^{4} + 58718 T^{5} - 206473 T^{6} - 2100910 T^{7} + 20656952 T^{8} + 88473724 T^{9} - 1065054743 T^{10} - 2017009199 T^{11} + 42128186297 T^{12} - 86731395557 T^{13} - 1969286219807 T^{14} + 7034280374068 T^{15} + 70622008154552 T^{16} - 308851507983130 T^{17} - 1305190792816177 T^{18} + 15960645206980826 T^{19} + 19203713056098443 T^{20} - 652365210294022214 T^{21} - 1275077456483770691 T^{22} + 12080818613125895191 T^{23} + 39959630797262576401 T^{24} \))(\( 1 + 13 T - 93 T^{2} - 1248 T^{3} + 9810 T^{4} + 69431 T^{5} - 827740 T^{6} - 2844144 T^{7} + 45303354 T^{8} + 67606073 T^{9} - 1695183309 T^{10} + 999801391 T^{11} + 52470440688 T^{12} - 81486657567 T^{13} - 1021133493305 T^{14} + 1151633415525 T^{15} + 13207491736631 T^{16} + 49520236867575 T^{17} - 1888075829120945 T^{18} - 6478759683179469 T^{19} + 179385995094575088 T^{20} + 146979245800144213 T^{21} - 10715869130794149141 T^{22} + 18376588865258452811 T^{23} + \)\(52\!\cdots\!54\)\( T^{24} - \)\(14\!\cdots\!92\)\( T^{25} - \)\(17\!\cdots\!60\)\( T^{26} + \)\(64\!\cdots\!17\)\( T^{27} + \)\(39\!\cdots\!10\)\( T^{28} - \)\(21\!\cdots\!64\)\( T^{29} - \)\(68\!\cdots\!57\)\( T^{30} + \)\(41\!\cdots\!91\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 + 7 T + 2 T^{2} + 329 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 10 T - 86 T^{2} + 618 T^{3} + 9423 T^{4} - 33027 T^{5} - 603099 T^{6} + 677569 T^{7} + 30716757 T^{8} + 31845743 T^{9} - 1332245691 T^{10} - 3428962221 T^{11} + 45981234063 T^{12} + 141735214326 T^{13} - 927012518294 T^{14} - 5066231204630 T^{15} + 23811286661761 T^{16} \))(\( 1 - 2 T - 58 T^{2} - 714 T^{3} + 2873 T^{4} + 38399 T^{5} + 291889 T^{6} - 1740327 T^{7} - 14193947 T^{8} - 81795369 T^{9} + 644782801 T^{10} + 3986699377 T^{11} + 14019323513 T^{12} - 163752334998 T^{13} - 625194489082 T^{14} - 1013246240926 T^{15} + 23811286661761 T^{16} \))(\( 1 - 8 T - 163 T^{2} + 1298 T^{3} + 15011 T^{4} - 102465 T^{5} - 1195399 T^{6} + 5422512 T^{7} + 87138701 T^{8} - 218512510 T^{9} - 5181134511 T^{10} + 4257232207 T^{11} + 259375228416 T^{12} + 200089913729 T^{13} - 11445126134799 T^{14} - 22686624325730 T^{15} + 425209063634381 T^{16} + 1243626052597584 T^{17} - 12885463225071271 T^{18} - 51911138038241295 T^{19} + 357431224079694371 T^{20} + 1452631354087391566 T^{21} - 8573658554440297987 T^{22} - 19777273720672098424 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 - 11 T - 184 T^{2} + 2519 T^{3} + 18578 T^{4} - 328940 T^{5} - 1113262 T^{6} + 30708371 T^{7} + 18431464 T^{8} - 2157799474 T^{9} + 3826585394 T^{10} + 115855042036 T^{11} - 510755934975 T^{12} - 4424444319954 T^{13} + 38393050326972 T^{14} + 80181278073827 T^{15} - 2076366795569416 T^{16} + 3768520069469869 T^{17} + 84810248172281148 T^{18} - 459359082630584142 T^{19} - 2492326031534742975 T^{20} + 26570775426731714252 T^{21} + 41247587936732304626 T^{22} - \)\(10\!\cdots\!62\)\( T^{23} + \)\(43\!\cdots\!04\)\( T^{24} + \)\(34\!\cdots\!57\)\( T^{25} - \)\(58\!\cdots\!38\)\( T^{26} - \)\(81\!\cdots\!20\)\( T^{27} + \)\(21\!\cdots\!98\)\( T^{28} + \)\(13\!\cdots\!13\)\( T^{29} - \)\(47\!\cdots\!96\)\( T^{30} - \)\(13\!\cdots\!73\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( ( 1 + 4 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 112 T^{2} - 119 T^{3} + 7405 T^{4} - 6307 T^{5} + 314608 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 + 8 T + 46 T^{2} - 401 T^{3} - 3531 T^{4} - 21253 T^{5} + 129214 T^{6} + 1191016 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 - 10 T + 194 T^{2} - 1431 T^{3} + 17912 T^{4} - 104359 T^{5} + 1085054 T^{6} - 5531027 T^{7} + 50314808 T^{8} - 213042987 T^{9} + 1530753314 T^{10} - 4181954930 T^{11} + 22164361129 T^{12} )^{2} \))(\( ( 1 + 8 T + 306 T^{2} + 2229 T^{3} + 44063 T^{4} + 295215 T^{5} + 3976442 T^{6} + 23818424 T^{7} + 249123624 T^{8} + 1262376472 T^{9} + 11169825578 T^{10} + 43950723555 T^{11} + 347678264303 T^{12} + 932157753897 T^{13} + 6782294505474 T^{14} + 9397689118696 T^{15} + 62259690411361 T^{16} )^{2} \))
$59$ (\( 1 + 14 T + 137 T^{2} + 826 T^{3} + 3481 T^{4} \))(\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 - 13 T - 39 T^{2} + 934 T^{3} + 1620 T^{4} - 17751 T^{5} - 381692 T^{6} - 173558 T^{7} + 35373867 T^{8} - 10239922 T^{9} - 1328669852 T^{10} - 3645682629 T^{11} + 19630124820 T^{12} + 667739295266 T^{13} - 1645040811999 T^{14} - 32352469302647 T^{15} + 146830437604321 T^{16} \))(\( 1 + 19 T + 59 T^{2} - 876 T^{3} - 244 T^{4} + 106457 T^{5} + 689896 T^{6} + 657732 T^{7} - 7735313 T^{8} + 38806188 T^{9} + 2401527976 T^{10} + 21864032203 T^{11} - 2956636084 T^{12} - 626273685924 T^{13} + 2488651484819 T^{14} + 47284378211561 T^{15} + 146830437604321 T^{16} \))(\( 1 - 21 T + 31 T^{2} + 1548 T^{3} + 1085 T^{4} - 95916 T^{5} - 961065 T^{6} + 11023368 T^{7} + 57284362 T^{8} - 412976394 T^{9} - 5014563901 T^{10} - 1966652877 T^{11} + 492853263145 T^{12} - 116032519743 T^{13} - 17455696939381 T^{14} - 84816678823326 T^{15} + 694135294008682 T^{16} + 7880873640019032 T^{17} - 40538234563687665 T^{18} - 238701495817899204 T^{19} + 159311024800688285 T^{20} + 13410317527277845572 T^{21} + 15844619352319883431 T^{22} - \)\(63\!\cdots\!39\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 + 7 T - 253 T^{2} - 1432 T^{3} + 34994 T^{4} + 154249 T^{5} - 3340312 T^{6} - 13153948 T^{7} + 235615798 T^{8} + 1028459735 T^{9} - 13624035805 T^{10} - 71911153079 T^{11} + 749305929396 T^{12} + 3788827420407 T^{13} - 43280647002525 T^{14} - 92100685540933 T^{15} + 2571044091919703 T^{16} - 5433940446915047 T^{17} - 150659932215789525 T^{18} + 778145586775769253 T^{19} + 9079610445931843956 T^{20} - 51411030705285766621 T^{21} - \)\(57\!\cdots\!05\)\( T^{22} + \)\(25\!\cdots\!65\)\( T^{23} + \)\(34\!\cdots\!58\)\( T^{24} - \)\(11\!\cdots\!72\)\( T^{25} - \)\(17\!\cdots\!12\)\( T^{26} + \)\(46\!\cdots\!91\)\( T^{27} + \)\(62\!\cdots\!14\)\( T^{28} - \)\(15\!\cdots\!28\)\( T^{29} - \)\(15\!\cdots\!33\)\( T^{30} + \)\(25\!\cdots\!93\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} \))(\( ( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 - 32 T + 429 T^{2} - 4324 T^{3} + 51554 T^{4} - 537888 T^{5} + 4331413 T^{6} - 37090382 T^{7} + 322144443 T^{8} - 2262513302 T^{9} + 16117187773 T^{10} - 122090356128 T^{11} + 713808486914 T^{12} - 3652034405524 T^{13} + 22102240600869 T^{14} - 100567770752672 T^{15} + 191707312997281 T^{16} \))(\( 1 - 12 T - 49 T^{2} + 852 T^{3} + 1876 T^{4} - 9576 T^{5} - 372571 T^{6} + 33096 T^{7} + 27681967 T^{8} + 2018856 T^{9} - 1386336691 T^{10} - 2173570056 T^{11} + 25974797716 T^{12} + 719596048452 T^{13} - 2524498343689 T^{14} - 37712914032252 T^{15} + 191707312997281 T^{16} \))(\( 1 + 2 T - 229 T^{2} + 106 T^{3} + 27669 T^{4} - 60786 T^{5} - 2252600 T^{6} + 7627838 T^{7} + 144871004 T^{8} - 483509158 T^{9} - 8189045917 T^{10} + 12920273306 T^{11} + 479371943129 T^{12} + 788136671666 T^{13} - 30471439857157 T^{14} - 109747392191998 T^{15} + 2005860886894364 T^{16} + 6442443759427238 T^{17} - 116054795285588600 T^{18} - 191034766030372506 T^{19} + 5304349643321767989 T^{20} + 1239579485840418946 T^{21} - \)\(16\!\cdots\!29\)\( T^{22} + 87027835222871677322 T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( 1 + 24 T + 19 T^{2} - 3276 T^{3} - 8256 T^{4} + 259110 T^{5} - 422515 T^{6} - 22871148 T^{7} + 156657818 T^{8} + 2168029494 T^{9} - 15387409002 T^{10} - 138383636634 T^{11} + 1199692712956 T^{12} + 4463711557584 T^{13} - 105028925164004 T^{14} - 58126091327262 T^{15} + 7596603299571471 T^{16} - 3545691570962982 T^{17} - 390812630535258884 T^{18} + 1013177713051973904 T^{19} + 16610754552447415996 T^{20} - \)\(11\!\cdots\!34\)\( T^{21} - \)\(79\!\cdots\!22\)\( T^{22} + \)\(68\!\cdots\!74\)\( T^{23} + \)\(30\!\cdots\!58\)\( T^{24} - \)\(26\!\cdots\!68\)\( T^{25} - \)\(30\!\cdots\!15\)\( T^{26} + \)\(11\!\cdots\!10\)\( T^{27} - \)\(21\!\cdots\!76\)\( T^{28} - \)\(53\!\cdots\!56\)\( T^{29} + \)\(18\!\cdots\!79\)\( T^{30} + \)\(14\!\cdots\!24\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( 1 - 2 T - 63 T^{2} - 134 T^{3} + 4489 T^{4} \))(\( 1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4} \))(\( 1 - 209 T^{2} + 72 T^{3} + 24028 T^{4} - 9936 T^{5} - 2229779 T^{6} + 300708 T^{7} + 169918447 T^{8} + 20147436 T^{9} - 10009477931 T^{10} - 2988381168 T^{11} + 484191135388 T^{12} + 97209007704 T^{13} - 18905801873321 T^{14} + 406067677556641 T^{16} \))(\( ( 1 - 11 T - 42 T^{2} - 319 T^{3} + 13973 T^{4} - 21373 T^{5} - 188538 T^{6} - 3308393 T^{7} + 20151121 T^{8} )^{2} \))(\( 1 + 6 T - 135 T^{2} - 1082 T^{3} + 1827 T^{4} + 35682 T^{5} + 102358 T^{6} + 2130660 T^{7} + 45193590 T^{8} + 41844472 T^{9} - 3116106003 T^{10} - 9659182248 T^{11} + 64516763713 T^{12} - 647165210616 T^{13} - 13988199847467 T^{14} + 12585268932136 T^{15} + 910701500514390 T^{16} + 2876657560480620 T^{17} + 9259139082054502 T^{18} + 216258311501135286 T^{19} + 741885646895983107 T^{20} - 29437470216791132654 T^{21} - \)\(24\!\cdots\!15\)\( T^{22} + \)\(73\!\cdots\!98\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 + 16 T - 303 T^{2} - 5196 T^{3} + 65028 T^{4} + 1020572 T^{5} - 10522045 T^{6} - 137230818 T^{7} + 1450070274 T^{8} + 14061798986 T^{9} - 167697342162 T^{10} - 1081437670724 T^{11} + 16651518835248 T^{12} + 59637358262580 T^{13} - 1404138817994528 T^{14} - 1533447489642366 T^{15} + 101851891881704651 T^{16} - 102740981806038522 T^{17} - 6303179153977436192 T^{18} + 17936710783128348540 T^{19} + \)\(33\!\cdots\!08\)\( T^{20} - \)\(14\!\cdots\!68\)\( T^{21} - \)\(15\!\cdots\!78\)\( T^{22} + \)\(85\!\cdots\!78\)\( T^{23} + \)\(58\!\cdots\!34\)\( T^{24} - \)\(37\!\cdots\!46\)\( T^{25} - \)\(19\!\cdots\!05\)\( T^{26} + \)\(12\!\cdots\!76\)\( T^{27} + \)\(53\!\cdots\!08\)\( T^{28} - \)\(28\!\cdots\!52\)\( T^{29} - \)\(11\!\cdots\!87\)\( T^{30} + \)\(39\!\cdots\!88\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( ( 1 + 9 T + 71 T^{2} )^{2} \))(\( ( 1 - 9 T + 71 T^{2} )^{2} \))(\( ( 1 - 13 T + 268 T^{2} - 2163 T^{3} + 26345 T^{4} - 153573 T^{5} + 1350988 T^{6} - 4652843 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 - 13 T + 248 T^{2} - 1951 T^{3} + 23005 T^{4} - 138521 T^{5} + 1250168 T^{6} - 4652843 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 + 29 T + 653 T^{2} + 10236 T^{3} + 134114 T^{4} + 1436171 T^{5} + 13182683 T^{6} + 101968141 T^{7} + 676068674 T^{8} + 3663576996 T^{9} + 16593827693 T^{10} + 52322651179 T^{11} + 128100283921 T^{12} )^{2} \))(\( ( 1 + 5 T + 278 T^{2} + 1065 T^{3} + 37904 T^{4} + 109648 T^{5} + 3610954 T^{6} + 7649988 T^{7} + 277532887 T^{8} + 543149148 T^{9} + 18202819114 T^{10} + 39244225328 T^{11} + 963204356624 T^{12} + 1921504258815 T^{13} + 35611878930038 T^{14} + 45475600791955 T^{15} + 645753531245761 T^{16} )^{2} \))
$73$ (\( ( 1 - 13 T + 73 T^{2} )^{2} \))(\( ( 1 - 11 T + 73 T^{2} )^{2} \))(\( ( 1 + 2 T + 68 T^{2} + 507 T^{3} + 5887 T^{4} + 37011 T^{5} + 362372 T^{6} + 778034 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 + 4 T + 78 T^{2} + 545 T^{3} + 761 T^{4} + 39785 T^{5} + 415662 T^{6} + 1556068 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 - 8 T + 205 T^{2} - 1029 T^{3} + 18060 T^{4} - 73748 T^{5} + 1316041 T^{6} - 5383604 T^{7} + 96241740 T^{8} - 400298493 T^{9} + 5821639405 T^{10} - 16584572744 T^{11} + 151334226289 T^{12} )^{2} \))(\( ( 1 - 10 T + 496 T^{2} - 3839 T^{3} + 108764 T^{4} - 681709 T^{5} + 14326022 T^{6} - 74456369 T^{7} + 1261625443 T^{8} - 5435314937 T^{9} + 76343371238 T^{10} - 265196390053 T^{11} + 3088706284124 T^{12} - 7958521845527 T^{13} + 75061776239344 T^{14} - 110473985190970 T^{15} + 806460091894081 T^{16} )^{2} \))
$79$ (\( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 177 T^{2} - 8 T^{3} + 11052 T^{4} + 1024 T^{5} - 1379699 T^{6} - 30756 T^{7} + 176681095 T^{8} - 2429724 T^{9} - 8610701459 T^{10} + 504871936 T^{11} + 430476295212 T^{12} - 24616451192 T^{13} - 43026479627217 T^{14} + 1517108809906561 T^{16} \))(\( 1 - 24 T + 279 T^{2} - 2304 T^{3} + 10516 T^{4} + 43728 T^{5} - 930339 T^{6} + 9097068 T^{7} - 88845273 T^{8} + 718668372 T^{9} - 5806245699 T^{10} + 21559609392 T^{11} + 409599051796 T^{12} - 7089537943296 T^{13} + 67821400090359 T^{14} - 460893815667816 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 22 T + 12 T^{2} + 2648 T^{3} - 8141 T^{4} - 130972 T^{5} - 734750 T^{6} + 15896118 T^{7} + 81246412 T^{8} - 907461286 T^{9} - 9153348513 T^{10} - 1468497462 T^{11} + 1287301310934 T^{12} - 116011299498 T^{13} - 57126048069633 T^{14} - 447413804988154 T^{15} + 3164554328359372 T^{16} + 48913251611159082 T^{17} - 178608507944054750 T^{18} - 2515174367735216548 T^{19} - 12350782821449313101 T^{20} + \)\(31\!\cdots\!12\)\( T^{21} + \)\(11\!\cdots\!12\)\( T^{22} - \)\(16\!\cdots\!38\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( 1 + 27 T - 135 T^{2} - 7902 T^{3} + 39995 T^{4} + 1914831 T^{5} - 6362838 T^{6} - 289531215 T^{7} + 1468553911 T^{8} + 37012370742 T^{9} - 224794814526 T^{10} - 3282480268776 T^{11} + 32122316535770 T^{12} + 227052617209830 T^{13} - 3341971702380213 T^{14} - 6208633901935233 T^{15} + 304269296557139782 T^{16} - 490482078252883407 T^{17} - 20857245394554909333 T^{18} + \)\(11\!\cdots\!70\)\( T^{19} + \)\(12\!\cdots\!70\)\( T^{20} - \)\(10\!\cdots\!24\)\( T^{21} - \)\(54\!\cdots\!46\)\( T^{22} + \)\(71\!\cdots\!78\)\( T^{23} + \)\(22\!\cdots\!71\)\( T^{24} - \)\(34\!\cdots\!85\)\( T^{25} - \)\(60\!\cdots\!38\)\( T^{26} + \)\(14\!\cdots\!49\)\( T^{27} + \)\(23\!\cdots\!95\)\( T^{28} - \)\(36\!\cdots\!78\)\( T^{29} - \)\(49\!\cdots\!35\)\( T^{30} + \)\(78\!\cdots\!73\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( 1 + 13 T + 86 T^{2} + 1079 T^{3} + 6889 T^{4} \))(\( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \))(\( 1 + 17 T - 38 T^{2} - 1593 T^{3} + 7218 T^{4} + 142914 T^{5} - 479067 T^{6} - 6921758 T^{7} - 296163 T^{8} - 574505914 T^{9} - 3300292563 T^{10} + 81716367318 T^{11} + 342554160978 T^{12} - 6274891744299 T^{13} - 12423734188022 T^{14} + 461312866823659 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 19 T + 8 T^{2} + 1653 T^{3} - 472 T^{4} - 87362 T^{5} - 374159 T^{6} - 2991816 T^{7} + 141658213 T^{8} - 248320728 T^{9} - 2577581351 T^{10} - 49952455894 T^{11} - 22400327512 T^{12} + 6511234182879 T^{13} + 2615522986952 T^{14} - 515584968802913 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 5 T - 361 T^{2} + 1484 T^{3} + 71060 T^{4} - 224895 T^{5} - 10522018 T^{6} + 24042225 T^{7} + 1282356419 T^{8} - 1800322606 T^{9} - 131803281744 T^{10} + 61097984545 T^{11} + 11701835672526 T^{12} + 5071132717235 T^{13} - 907992807934416 T^{14} - 1029401061916922 T^{15} + 60858482569312499 T^{16} + 94703301423150675 T^{17} - 3440072493515338642 T^{18} - 6102762187312164165 T^{19} + \)\(16\!\cdots\!60\)\( T^{20} + \)\(27\!\cdots\!52\)\( T^{21} - \)\(56\!\cdots\!89\)\( T^{22} - \)\(64\!\cdots\!35\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 + 5 T - 412 T^{2} - 1919 T^{3} + 85319 T^{4} + 351005 T^{5} - 12811975 T^{6} - 43639307 T^{7} + 1623479161 T^{8} + 4403947840 T^{9} - 179310486826 T^{10} - 369180098536 T^{11} + 17626003433352 T^{12} + 22874863773276 T^{13} - 1609250765486967 T^{14} - 692777265967880 T^{15} + 138259235603031206 T^{16} - 57500513075334040 T^{17} - 11086128523439715663 T^{18} + 13079549732330164212 T^{19} + \)\(83\!\cdots\!92\)\( T^{20} - \)\(14\!\cdots\!48\)\( T^{21} - \)\(58\!\cdots\!94\)\( T^{22} + \)\(11\!\cdots\!80\)\( T^{23} + \)\(36\!\cdots\!01\)\( T^{24} - \)\(81\!\cdots\!21\)\( T^{25} - \)\(19\!\cdots\!75\)\( T^{26} + \)\(45\!\cdots\!35\)\( T^{27} + \)\(91\!\cdots\!59\)\( T^{28} - \)\(17\!\cdots\!97\)\( T^{29} - \)\(30\!\cdots\!48\)\( T^{30} + \)\(30\!\cdots\!35\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 12 T + 89 T^{2} )^{2} \))(\( ( 1 + 17 T + 309 T^{2} + 2514 T^{3} + 30601 T^{4} + 223746 T^{5} + 2447589 T^{6} + 11984473 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 15 T + 241 T^{2} - 1380 T^{3} + 17331 T^{4} - 122820 T^{5} + 1908961 T^{6} - 10574535 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - T + 315 T^{2} - 80 T^{3} + 52930 T^{4} - 6403 T^{5} + 5804756 T^{6} - 569867 T^{7} + 419258530 T^{8} - 56397520 T^{9} + 19763805915 T^{10} - 5584059449 T^{11} + 496981290961 T^{12} )^{2} \))(\( ( 1 - 27 T + 659 T^{2} - 11088 T^{3} + 177613 T^{4} - 2313648 T^{5} + 28600757 T^{6} - 303824349 T^{7} + 3074549044 T^{8} - 27040367061 T^{9} + 226546596197 T^{10} - 1631050116912 T^{11} + 11143837650733 T^{12} - 61916051170512 T^{13} + 327510670743299 T^{14} - 1194246042179283 T^{15} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( 1 - 9 T - 16 T^{2} - 873 T^{3} + 9409 T^{4} \))(\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 - 4 T - 167 T^{2} + 888 T^{3} + 5042 T^{4} - 25144 T^{5} - 827367 T^{6} - 2199130 T^{7} + 197329403 T^{8} - 213315610 T^{9} - 7784696103 T^{10} - 22948249912 T^{11} + 446364634802 T^{12} + 7625558148216 T^{13} - 139106324823143 T^{14} - 323193137912452 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 12 T - 33 T^{2} + 1956 T^{3} - 21452 T^{4} + 120264 T^{5} + 108549 T^{6} - 18621312 T^{7} + 276817383 T^{8} - 1806267264 T^{9} + 1021337541 T^{10} + 109761705672 T^{11} - 1899130136012 T^{12} + 16796837542692 T^{13} - 27488076162657 T^{14} - 969579413737356 T^{15} + 7837433594376961 T^{16} \))(\( 1 + 32 T + 96 T^{2} - 4240 T^{3} + 41557 T^{4} + 1467386 T^{5} - 2429426 T^{6} - 116703246 T^{7} + 1775844382 T^{8} + 18457608542 T^{9} - 163047635337 T^{10} - 64121694798 T^{11} + 28748647747254 T^{12} - 6219804395406 T^{13} - 1534115200885833 T^{14} + 16845760960852766 T^{15} + 157214226306349342 T^{16} - 1002170482498374222 T^{17} - 2023643846046640754 T^{18} + \)\(11\!\cdots\!18\)\( T^{19} + \)\(32\!\cdots\!77\)\( T^{20} - \)\(32\!\cdots\!80\)\( T^{21} + \)\(70\!\cdots\!04\)\( T^{22} + \)\(22\!\cdots\!96\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 + 27 T + 15 T^{2} - 9096 T^{3} - 108991 T^{4} + 887943 T^{5} + 28936428 T^{6} + 99887319 T^{7} - 3490694579 T^{8} - 39893502474 T^{9} + 151428852180 T^{10} + 5743284567084 T^{11} + 21023505312788 T^{12} - 484348205982834 T^{13} - 5169233095099629 T^{14} + 18335501604402927 T^{15} + 619275518773189270 T^{16} + 1778543655627083919 T^{17} - 48637314191792409261 T^{18} - \)\(44\!\cdots\!82\)\( T^{19} + \)\(18\!\cdots\!28\)\( T^{20} + \)\(49\!\cdots\!88\)\( T^{21} + \)\(12\!\cdots\!20\)\( T^{22} - \)\(32\!\cdots\!62\)\( T^{23} - \)\(27\!\cdots\!19\)\( T^{24} + \)\(75\!\cdots\!23\)\( T^{25} + \)\(21\!\cdots\!72\)\( T^{26} + \)\(63\!\cdots\!79\)\( T^{27} - \)\(75\!\cdots\!31\)\( T^{28} - \)\(61\!\cdots\!92\)\( T^{29} + \)\(97\!\cdots\!35\)\( T^{30} + \)\(17\!\cdots\!11\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))
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