Properties

Label 1925.2.b
Level $1925$
Weight $2$
Character orbit 1925.b
Rep. character $\chi_{1925}(1849,\cdot)$
Character field $\Q$
Dimension $88$
Newform subspaces $18$
Sturm bound $480$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1925.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(480\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\), \(3\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1925, [\chi])\).

Total New Old
Modular forms 252 88 164
Cusp forms 228 88 140
Eisenstein series 24 0 24

Trace form

\( 88q - 100q^{4} + 16q^{6} - 76q^{9} + O(q^{10}) \) \( 88q - 100q^{4} + 16q^{6} - 76q^{9} - 8q^{11} + 8q^{14} + 124q^{16} - 32q^{19} + 8q^{24} - 24q^{26} + 40q^{29} + 20q^{31} - 40q^{34} + 68q^{36} - 16q^{39} - 16q^{41} + 20q^{44} + 80q^{46} - 88q^{49} + 24q^{51} + 40q^{54} - 24q^{56} - 44q^{59} - 196q^{64} - 36q^{69} - 44q^{71} + 208q^{76} - 24q^{79} + 8q^{81} - 40q^{86} + 108q^{89} - 16q^{91} - 16q^{94} - 80q^{96} + 44q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1925, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1925.2.b.a \(2\) \(15.371\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+2iq^{2}-2q^{4}-iq^{7}+3q^{9}+q^{11}+\cdots\)
1925.2.b.b \(2\) \(15.371\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}+q^{4}+iq^{7}+3iq^{8}+3q^{9}+\cdots\)
1925.2.b.c \(2\) \(15.371\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}-2iq^{3}+q^{4}+2q^{6}-iq^{7}+\cdots\)
1925.2.b.d \(2\) \(15.371\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}-2iq^{3}+q^{4}+2q^{6}-iq^{7}+\cdots\)
1925.2.b.e \(2\) \(15.371\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+3iq^{3}+2q^{4}-iq^{7}-6q^{9}-q^{11}+\cdots\)
1925.2.b.f \(2\) \(15.371\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+2iq^{3}+2q^{4}+iq^{7}-q^{9}-q^{11}+\cdots\)
1925.2.b.g \(2\) \(15.371\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-iq^{3}+2q^{4}+iq^{7}+2q^{9}-q^{11}+\cdots\)
1925.2.b.h \(4\) \(15.371\) \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{2}+(\beta _{1}+\beta _{2})q^{3}-3q^{4}+(-5+\cdots)q^{6}+\cdots\)
1925.2.b.i \(4\) \(15.371\) \(\Q(i, \sqrt{13})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}-\beta _{2}q^{3}+(-2+\beta _{3})q^{4}+(-1+\cdots)q^{6}+\cdots\)
1925.2.b.j \(4\) \(15.371\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{8}+\zeta_{8}^{2})q^{2}-\zeta_{8}^{2}q^{3}+(-1-2\zeta_{8}^{3})q^{4}+\cdots\)
1925.2.b.k \(4\) \(15.371\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}^{2}q^{2}+(\zeta_{12}-\zeta_{12}^{2})q^{3}-q^{4}+\cdots\)
1925.2.b.l \(4\) \(15.371\) \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+(2\beta _{1}+\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1925.2.b.m \(6\) \(15.371\) 6.0.350464.1 None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{3}-\beta _{5})q^{2}+(-\beta _{3}-\beta _{4})q^{3}+(-2+\cdots)q^{4}+\cdots\)
1925.2.b.n \(6\) \(15.371\) 6.0.350464.1 None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{3}-\beta _{5})q^{2}+(-\beta _{3}+\beta _{4})q^{3}+(-2+\cdots)q^{4}+\cdots\)
1925.2.b.o \(6\) \(15.371\) 6.0.350464.1 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{4}q^{2}-\beta _{5}q^{3}+(-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1925.2.b.p \(8\) \(15.371\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{4}-\beta _{6})q^{2}-\beta _{7}q^{3}+(-2+\beta _{2}+\cdots)q^{4}+\cdots\)
1925.2.b.q \(14\) \(15.371\) \(\mathbb{Q}[x]/(x^{14} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+\beta _{6}q^{3}+(-2-\beta _{7}+\beta _{8}+\cdots)q^{4}+\cdots\)
1925.2.b.r \(14\) \(15.371\) \(\mathbb{Q}[x]/(x^{14} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+\beta _{9}q^{3}+(-2+\beta _{2})q^{4}-\beta _{10}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1925, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1925, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(55, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(275, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(385, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T + 2 T^{2} )( 1 + 2 T + 2 T^{2} ) \))(\( 1 - 3 T^{2} + 4 T^{4} \))(\( 1 - 3 T^{2} + 4 T^{4} \))(\( 1 - 3 T^{2} + 4 T^{4} \))(\( ( 1 - 2 T^{2} )^{2} \))(\( ( 1 - 2 T^{2} )^{2} \))(\( ( 1 - 2 T^{2} )^{2} \))(\( ( 1 + T^{2} + 4 T^{4} )^{2} \))(\( 1 - T^{2} + 5 T^{4} - 4 T^{6} + 16 T^{8} \))(\( 1 - 2 T^{2} + T^{4} - 8 T^{6} + 16 T^{8} \))(\( ( 1 - T^{2} + 4 T^{4} )^{2} \))(\( 1 - 5 T^{2} + 13 T^{4} - 20 T^{6} + 16 T^{8} \))(\( 1 - T^{2} + 3 T^{4} + 5 T^{6} + 12 T^{8} - 16 T^{10} + 64 T^{12} \))(\( 1 - T^{2} + 3 T^{4} + 5 T^{6} + 12 T^{8} - 16 T^{10} + 64 T^{12} \))(\( 1 - 5 T^{2} + 15 T^{4} - 35 T^{6} + 60 T^{8} - 80 T^{10} + 64 T^{12} \))(\( 1 + 2 T^{4} + 4 T^{6} - 15 T^{8} + 16 T^{10} + 32 T^{12} + 256 T^{16} \))(\( 1 - T^{2} - T^{4} - 2 T^{6} + 23 T^{8} + 7 T^{10} + 13 T^{12} - 76 T^{14} + 52 T^{16} + 112 T^{18} + 1472 T^{20} - 512 T^{22} - 1024 T^{24} - 4096 T^{26} + 16384 T^{28} \))(\( 1 - T^{2} + 3 T^{4} - 14 T^{6} + 15 T^{8} - 49 T^{10} + 129 T^{12} - 200 T^{14} + 516 T^{16} - 784 T^{18} + 960 T^{20} - 3584 T^{22} + 3072 T^{24} - 4096 T^{26} + 16384 T^{28} \))
$3$ (\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 - 5 T^{2} + 9 T^{4} \))(\( ( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} )( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} ) \))(\( ( 1 - 5 T^{2} + 9 T^{4} )^{2} \))(\( ( 1 - 4 T^{2} + 9 T^{4} )^{2} \))(\( 1 - 4 T^{2} + 10 T^{4} - 36 T^{6} + 81 T^{8} \))(\( ( 1 - T^{2} + 9 T^{4} )^{2} \))(\( 1 - 6 T^{2} + 11 T^{4} - 8 T^{6} + 99 T^{8} - 486 T^{10} + 729 T^{12} \))(\( 1 - 10 T^{2} + 51 T^{4} - 176 T^{6} + 459 T^{8} - 810 T^{10} + 729 T^{12} \))(\( 1 - 10 T^{2} + 55 T^{4} - 200 T^{6} + 495 T^{8} - 810 T^{10} + 729 T^{12} \))(\( 1 - 4 T^{2} + 28 T^{4} - 88 T^{6} + 334 T^{8} - 792 T^{10} + 2268 T^{12} - 2916 T^{14} + 6561 T^{16} \))(\( 1 - 8 T^{2} + 38 T^{4} - 130 T^{6} + 427 T^{8} - 1548 T^{10} + 6047 T^{12} - 20438 T^{14} + 54423 T^{16} - 125388 T^{18} + 311283 T^{20} - 852930 T^{22} + 2243862 T^{24} - 4251528 T^{26} + 4782969 T^{28} \))(\( 1 - 12 T^{2} + 86 T^{4} - 486 T^{6} + 2251 T^{8} - 8948 T^{10} + 31567 T^{12} - 99770 T^{14} + 284103 T^{16} - 724788 T^{18} + 1640979 T^{20} - 3188646 T^{22} + 5078214 T^{24} - 6377292 T^{26} + 4782969 T^{28} \))
$5$ 1
$7$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{3} \))(\( ( 1 + T^{2} )^{3} \))(\( ( 1 + T^{2} )^{3} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{7} \))(\( ( 1 + T^{2} )^{7} \))
$11$ (\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{6} \))(\( ( 1 + T )^{6} \))(\( ( 1 - T )^{6} \))(\( ( 1 + T )^{8} \))(\( ( 1 + T )^{14} \))(\( ( 1 - T )^{14} \))
$13$ (\( 1 - 17 T^{2} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( 1 - 25 T^{2} + 169 T^{4} \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( 1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8} \))(\( ( 1 - 13 T^{2} + 169 T^{4} )^{2} \))(\( 1 - 40 T^{2} + 706 T^{4} - 6760 T^{6} + 28561 T^{8} \))(\( 1 - 44 T^{2} + 810 T^{4} - 7436 T^{6} + 28561 T^{8} \))(\( 1 - 34 T^{2} + 547 T^{4} - 5746 T^{6} + 28561 T^{8} \))(\( 1 - 50 T^{2} + 1243 T^{4} - 19712 T^{6} + 210067 T^{8} - 1428050 T^{10} + 4826809 T^{12} \))(\( 1 - 30 T^{2} + 515 T^{4} - 7640 T^{6} + 87035 T^{8} - 856830 T^{10} + 4826809 T^{12} \))(\( 1 - 70 T^{2} + 2135 T^{4} - 36240 T^{6} + 360815 T^{8} - 1999270 T^{10} + 4826809 T^{12} \))(\( 1 - 24 T^{2} + 676 T^{4} - 11332 T^{6} + 170174 T^{8} - 1915108 T^{10} + 19307236 T^{12} - 115843416 T^{14} + 815730721 T^{16} \))(\( 1 - 75 T^{2} + 2874 T^{4} - 71883 T^{6} + 1292748 T^{8} - 17812313 T^{10} + 207668689 T^{12} - 2486972978 T^{14} + 35096008441 T^{16} - 508737471593 T^{18} + 6239847681132 T^{20} - 58637171417643 T^{22} + 396205305574026 T^{24} - 1747356384186075 T^{26} + 3937376385699289 T^{28} \))(\( 1 - 67 T^{2} + 2154 T^{4} - 45555 T^{6} + 732220 T^{8} - 9544865 T^{10} + 107828273 T^{12} - 1269413986 T^{14} + 18222978137 T^{16} - 272610889265 T^{18} + 3534286085980 T^{20} - 37160612995155 T^{22} + 296947191442746 T^{24} - 1560971703206227 T^{26} + 3937376385699289 T^{28} \))
$17$ (\( 1 - 25 T^{2} + 289 T^{4} \))(\( 1 + 2 T^{2} + 289 T^{4} \))(\( 1 - 18 T^{2} + 289 T^{4} \))(\( 1 - 18 T^{2} + 289 T^{4} \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( 1 - 25 T^{2} + 289 T^{4} \))(\( 1 + 2 T^{2} + 289 T^{4} \))(\( 1 - 56 T^{2} + 1342 T^{4} - 16184 T^{6} + 83521 T^{8} \))(\( 1 - 61 T^{2} + 1505 T^{4} - 17629 T^{6} + 83521 T^{8} \))(\( 1 - 56 T^{2} + 1330 T^{4} - 16184 T^{6} + 83521 T^{8} \))(\( 1 + 4 T^{2} - 390 T^{4} + 1156 T^{6} + 83521 T^{8} \))(\( 1 - 5 T^{2} + 553 T^{4} - 1445 T^{6} + 83521 T^{8} \))(\( 1 - 30 T^{2} + 875 T^{4} - 14840 T^{6} + 252875 T^{8} - 2505630 T^{10} + 24137569 T^{12} \))(\( 1 - 42 T^{2} + 1155 T^{4} - 24816 T^{6} + 333795 T^{8} - 3507882 T^{10} + 24137569 T^{12} \))(\( 1 - 82 T^{2} + 3015 T^{4} - 64936 T^{6} + 871335 T^{8} - 6848722 T^{10} + 24137569 T^{12} \))(\( 1 - 108 T^{2} + 5428 T^{4} - 166640 T^{6} + 3422558 T^{8} - 48158960 T^{10} + 453351988 T^{12} - 2606857452 T^{14} + 6975757441 T^{16} \))(\( 1 - 78 T^{2} + 2929 T^{4} - 70475 T^{6} + 1331396 T^{8} - 25389734 T^{10} + 525828026 T^{12} - 9885378585 T^{14} + 151964299514 T^{16} - 2120575973414 T^{18} + 32136662816324 T^{20} - 491616505654475 T^{22} + 5904846134415121 T^{24} - 45444534503921358 T^{26} + 168377826559400929 T^{28} \))(\( 1 - 86 T^{2} + 4753 T^{4} - 189995 T^{6} + 6008924 T^{8} - 156145310 T^{10} + 3395650762 T^{12} - 62667505313 T^{14} + 981343070218 T^{16} - 13041412436510 T^{18} + 145040817665756 T^{20} - 1325359035002795 T^{22} + 9582019008834097 T^{24} - 50105512401759446 T^{26} + 168377826559400929 T^{28} \))
$19$ (\( ( 1 - 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 - 6 T + 19 T^{2} )^{2} \))(\( ( 1 - T + 19 T^{2} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 9 T + 55 T^{2} - 171 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 19 T^{2} )^{4} \))(\( ( 1 - 4 T + 19 T^{2} )^{4} \))(\( ( 1 - 9 T + 47 T^{2} - 171 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 2 T - 3 T^{2} - 124 T^{3} - 57 T^{4} + 722 T^{5} + 6859 T^{6} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} + 220 T^{3} + 1007 T^{4} + 2166 T^{5} + 6859 T^{6} )^{2} \))(\( ( 1 + 10 T + 69 T^{2} + 340 T^{3} + 1311 T^{4} + 3610 T^{5} + 6859 T^{6} )^{2} \))(\( ( 1 + 6 T + 48 T^{2} + 222 T^{3} + 1134 T^{4} + 4218 T^{5} + 17328 T^{6} + 41154 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 + 18 T + 185 T^{2} + 1459 T^{3} + 10121 T^{4} + 60906 T^{5} + 315433 T^{6} + 1443042 T^{7} + 5993227 T^{8} + 21987066 T^{9} + 69419939 T^{10} + 190138339 T^{11} + 458078315 T^{12} + 846825858 T^{13} + 893871739 T^{14} )^{2} \))(\( ( 1 + 20 T + 279 T^{2} + 2725 T^{3} + 21795 T^{4} + 141440 T^{5} + 786093 T^{6} + 3682030 T^{7} + 14935767 T^{8} + 51059840 T^{9} + 149491905 T^{10} + 355124725 T^{11} + 690831621 T^{12} + 940917620 T^{13} + 893871739 T^{14} )^{2} \))
$23$ (\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 + 18 T^{2} + 529 T^{4} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 - 21 T^{2} + 529 T^{4} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 - 37 T^{2} + 529 T^{4} \))(\( 1 - 44 T^{2} + 1222 T^{4} - 23276 T^{6} + 279841 T^{8} \))(\( 1 - 29 T^{2} + 1005 T^{4} - 15341 T^{6} + 279841 T^{8} \))(\( 1 - 4 T^{2} - 90 T^{4} - 2116 T^{6} + 279841 T^{8} \))(\( ( 1 - 10 T^{2} + 529 T^{4} )^{2} \))(\( 1 - 85 T^{2} + 2853 T^{4} - 44965 T^{6} + 279841 T^{8} \))(\( 1 - 46 T^{2} + 2031 T^{4} - 47188 T^{6} + 1074399 T^{8} - 12872686 T^{10} + 148035889 T^{12} \))(\( 1 - 94 T^{2} + 4271 T^{4} - 120948 T^{6} + 2259359 T^{8} - 26305054 T^{10} + 148035889 T^{12} \))(\( 1 - 126 T^{2} + 6863 T^{4} - 206708 T^{6} + 3630527 T^{8} - 35259966 T^{10} + 148035889 T^{12} \))(\( 1 - 44 T^{2} + 2276 T^{4} - 59268 T^{6} + 1781910 T^{8} - 31352772 T^{10} + 636918116 T^{12} - 6513579116 T^{14} + 78310985281 T^{16} \))(\( 1 - 123 T^{2} + 6888 T^{4} - 224652 T^{6} + 4621074 T^{8} - 64290365 T^{10} + 798183093 T^{12} - 14390041784 T^{14} + 422238856197 T^{16} - 17991080031965 T^{18} + 684084797724786 T^{20} - 17592719465347212 T^{22} + 285345809239614312 T^{24} - 2695498805138499483 T^{26} + 11592836324538749809 T^{28} \))(\( 1 - 131 T^{2} + 8848 T^{4} - 429692 T^{6} + 16961690 T^{8} - 563988533 T^{10} + 16018669573 T^{12} - 394335260888 T^{14} + 8473876204117 T^{16} - 157827115063253 T^{18} + 2510938858092410 T^{20} - 33649603887363452 T^{22} + 366541771218366352 T^{24} - 2870815800594662051 T^{26} + 11592836324538749809 T^{28} \))
$29$ (\( ( 1 - 4 T + 29 T^{2} )^{2} \))(\( ( 1 - 10 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 10 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 8 T + 54 T^{2} + 232 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 3 T + 29 T^{2} )^{4} \))(\( ( 1 + 4 T + 54 T^{2} + 116 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 46 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 8 T + 69 T^{2} - 232 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 2 T + 51 T^{2} + 12 T^{3} + 1479 T^{4} + 1682 T^{5} + 24389 T^{6} )^{2} \))(\( ( 1 - 10 T + 99 T^{2} - 540 T^{3} + 2871 T^{4} - 8410 T^{5} + 24389 T^{6} )^{2} \))(\( ( 1 + 2 T + 35 T^{2} + 76 T^{3} + 1015 T^{4} + 1682 T^{5} + 24389 T^{6} )^{2} \))(\( ( 1 - 4 T + 36 T^{2} - 76 T^{3} + 710 T^{4} - 2204 T^{5} + 30276 T^{6} - 97556 T^{7} + 707281 T^{8} )^{2} \))(\( ( 1 - 2 T + 160 T^{2} - 304 T^{3} + 11826 T^{4} - 20366 T^{5} + 526289 T^{6} - 767200 T^{7} + 15262381 T^{8} - 17127806 T^{9} + 288424314 T^{10} - 215013424 T^{11} + 3281783840 T^{12} - 1189646642 T^{13} + 17249876309 T^{14} )^{2} \))(\( ( 1 + 4 T + 74 T^{2} - 8 T^{3} + 2316 T^{4} - 5492 T^{5} + 93861 T^{6} - 110928 T^{7} + 2721969 T^{8} - 4618772 T^{9} + 56484924 T^{10} - 5658248 T^{11} + 1517825026 T^{12} + 2379293284 T^{13} + 17249876309 T^{14} )^{2} \))
$31$ (\( ( 1 - 2 T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 6 T + 31 T^{2} )^{2} \))(\( ( 1 - 10 T + 31 T^{2} )^{2} \))(\( ( 1 - T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 5 T + 31 T^{2} )^{2} \))(\( ( 1 + 10 T + 82 T^{2} + 310 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 10 T + 74 T^{2} + 310 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 12 T + 80 T^{2} + 372 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 10 T + 84 T^{2} - 310 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 6 T + 26 T^{2} + 186 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 4 T + 33 T^{2} + 298 T^{3} + 1023 T^{4} + 3844 T^{5} + 29791 T^{6} )^{2} \))(\( ( 1 + 10 T + 113 T^{2} + 594 T^{3} + 3503 T^{4} + 9610 T^{5} + 29791 T^{6} )^{2} \))(\( ( 1 - 8 T + 99 T^{2} - 494 T^{3} + 3069 T^{4} - 7688 T^{5} + 29791 T^{6} )^{2} \))(\( ( 1 - 10 T + 94 T^{2} - 628 T^{3} + 4210 T^{4} - 19468 T^{5} + 90334 T^{6} - 297910 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 - 24 T + 377 T^{2} - 4310 T^{3} + 40198 T^{4} - 315114 T^{5} + 2137726 T^{6} - 12692896 T^{7} + 66269506 T^{8} - 302824554 T^{9} + 1197538618 T^{10} - 3980375510 T^{11} + 10793189927 T^{12} - 21300088344 T^{13} + 27512614111 T^{14} )^{2} \))(\( ( 1 - 6 T + 121 T^{2} - 562 T^{3} + 8078 T^{4} - 33066 T^{5} + 353226 T^{6} - 1195140 T^{7} + 10950006 T^{8} - 31776426 T^{9} + 240651698 T^{10} - 519018802 T^{11} + 3464127271 T^{12} - 5325022086 T^{13} + 27512614111 T^{14} )^{2} \))
$37$ (\( 1 - 65 T^{2} + 1369 T^{4} \))(\( 1 - 38 T^{2} + 1369 T^{4} \))(\( 1 - 38 T^{2} + 1369 T^{4} \))(\( 1 - 38 T^{2} + 1369 T^{4} \))(\( 1 - 49 T^{2} + 1369 T^{4} \))(\( 1 - 49 T^{2} + 1369 T^{4} \))(\( 1 + 47 T^{2} + 1369 T^{4} \))(\( 1 - 76 T^{2} + 2902 T^{4} - 104044 T^{6} + 1874161 T^{8} \))(\( 1 - 72 T^{2} + 2734 T^{4} - 98568 T^{6} + 1874161 T^{8} \))(\( 1 - 100 T^{2} + 4726 T^{4} - 136900 T^{6} + 1874161 T^{8} \))(\( 1 - 20 T^{2} - 234 T^{4} - 27380 T^{6} + 1874161 T^{8} \))(\( 1 - 40 T^{2} + 2158 T^{4} - 54760 T^{6} + 1874161 T^{8} \))(\( 1 - 6 T^{2} + 1319 T^{4} + 7612 T^{6} + 1805711 T^{8} - 11244966 T^{10} + 2565726409 T^{12} \))(\( 1 - 70 T^{2} + 3943 T^{4} - 191428 T^{6} + 5397967 T^{8} - 131191270 T^{10} + 2565726409 T^{12} \))(\( 1 - 118 T^{2} + 7943 T^{4} - 356004 T^{6} + 10873967 T^{8} - 221150998 T^{10} + 2565726409 T^{12} \))(\( 1 - 236 T^{2} + 25892 T^{4} - 1733860 T^{6} + 77573302 T^{8} - 2373654340 T^{10} + 48525776612 T^{12} - 605511432524 T^{14} + 3512479453921 T^{16} \))(\( 1 - 213 T^{2} + 21413 T^{4} - 1394124 T^{6} + 69620392 T^{8} - 3040633556 T^{10} + 125038461586 T^{12} - 4821513483110 T^{14} + 171177653911234 T^{16} - 5698636825946516 T^{18} + 178626878359332328 T^{20} - 4896831906218160204 T^{22} + \)\(10\!\cdots\!37\)\( T^{24} - \)\(14\!\cdots\!53\)\( T^{26} + \)\(90\!\cdots\!89\)\( T^{28} \))(\( 1 - 389 T^{2} + 72165 T^{4} - 8526348 T^{6} + 722005416 T^{8} - 46643679252 T^{10} + 2381748954450 T^{12} - 97925153644422 T^{14} + 3260614318642050 T^{16} - 87417764550607572 T^{18} + 1852468363272231144 T^{20} - 29948622166980410508 T^{22} + \)\(34\!\cdots\!85\)\( T^{24} - \)\(25\!\cdots\!09\)\( T^{26} + \)\(90\!\cdots\!89\)\( T^{28} \))
$41$ (\( ( 1 - 5 T + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 4 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 3 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 18 T + 158 T^{2} + 738 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 3 T + 55 T^{2} + 123 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 64 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 18 T + 160 T^{2} - 738 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 15 T + 127 T^{2} + 615 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 10 T + 135 T^{2} + 822 T^{3} + 5535 T^{4} + 16810 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 87 T^{2} - 54 T^{3} + 3567 T^{4} + 68921 T^{6} )^{2} \))(\( ( 1 - 10 T + 97 T^{2} - 486 T^{3} + 3977 T^{4} - 16810 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 10 T + 150 T^{2} + 928 T^{3} + 8510 T^{4} + 38048 T^{5} + 252150 T^{6} + 689210 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + 10 T + 255 T^{2} + 1921 T^{3} + 27881 T^{4} + 166322 T^{5} + 1770983 T^{6} + 8556586 T^{7} + 72610303 T^{8} + 279587282 T^{9} + 1921586401 T^{10} + 5428286881 T^{11} + 29543331255 T^{12} + 47501042410 T^{13} + 194754273881 T^{14} )^{2} \))(\( ( 1 - 24 T + 389 T^{2} - 4601 T^{3} + 45383 T^{4} - 378812 T^{5} + 2829723 T^{6} - 18901258 T^{7} + 116018643 T^{8} - 636782972 T^{9} + 3127841743 T^{10} - 13001326361 T^{11} + 45068062189 T^{12} - 114002501784 T^{13} + 194754273881 T^{14} )^{2} \))
$43$ (\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 + 58 T^{2} + 1849 T^{4} \))(\( 1 - 22 T^{2} + 1849 T^{4} \))(\( 1 - 22 T^{2} + 1849 T^{4} \))(\( 1 - 22 T^{2} + 1849 T^{4} \))(\( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 1849 T^{4} )^{2} \))(\( 1 - 116 T^{2} + 6294 T^{4} - 214484 T^{6} + 3418801 T^{8} \))(\( 1 - 10 T^{2} - 2757 T^{4} - 18490 T^{6} + 3418801 T^{8} \))(\( 1 - 166 T^{2} + 13927 T^{4} - 742468 T^{6} + 25751023 T^{8} - 567520966 T^{10} + 6321363049 T^{12} \))(\( 1 - 70 T^{2} + 2791 T^{4} - 68356 T^{6} + 5160559 T^{8} - 239316070 T^{10} + 6321363049 T^{12} \))(\( 1 - 166 T^{2} + 14471 T^{4} - 767748 T^{6} + 26756879 T^{8} - 567520966 T^{10} + 6321363049 T^{12} \))(\( 1 - 220 T^{2} + 23780 T^{4} - 1668212 T^{6} + 83747254 T^{8} - 3084523988 T^{10} + 81299087780 T^{12} - 1390699870780 T^{14} + 11688200277601 T^{16} \))(\( 1 - 332 T^{2} + 50938 T^{4} - 4716582 T^{6} + 287361931 T^{8} - 11808167820 T^{10} + 342715711523 T^{12} - 10574277880730 T^{14} + 633681350606027 T^{16} - 40369775951183820 T^{18} + 1816519092312687619 T^{20} - 55128355041727879782 T^{22} + \)\(11\!\cdots\!62\)\( T^{24} - \)\(13\!\cdots\!32\)\( T^{26} + \)\(73\!\cdots\!49\)\( T^{28} \))(\( 1 - 396 T^{2} + 76666 T^{4} - 9614518 T^{6} + 876626507 T^{8} - 61962221996 T^{10} + 3534161695395 T^{12} - 166534468471882 T^{14} + 6534664974785355 T^{16} - 211836506522146796 T^{18} + 5541474409123739843 T^{20} - \)\(11\!\cdots\!18\)\( T^{22} + \)\(16\!\cdots\!34\)\( T^{24} - \)\(15\!\cdots\!96\)\( T^{26} + \)\(73\!\cdots\!49\)\( T^{28} \))
$47$ (\( 1 - 30 T^{2} + 2209 T^{4} \))(\( 1 - 78 T^{2} + 2209 T^{4} \))(\( 1 - 58 T^{2} + 2209 T^{4} \))(\( 1 + 6 T^{2} + 2209 T^{4} \))(\( 1 - 30 T^{2} + 2209 T^{4} \))(\( 1 + 50 T^{2} + 2209 T^{4} \))(\( ( 1 - 47 T^{2} )^{2} \))(\( 1 - 128 T^{2} + 8014 T^{4} - 282752 T^{6} + 4879681 T^{8} \))(\( 1 - 125 T^{2} + 8061 T^{4} - 276125 T^{6} + 4879681 T^{8} \))(\( ( 1 - 92 T^{2} + 2209 T^{4} )^{2} \))(\( 1 - 164 T^{2} + 11034 T^{4} - 362276 T^{6} + 4879681 T^{8} \))(\( 1 - 25 T^{2} - 387 T^{4} - 55225 T^{6} + 4879681 T^{8} \))(\( 1 - 50 T^{2} + 6459 T^{4} - 207008 T^{6} + 14267931 T^{8} - 243984050 T^{10} + 10779215329 T^{12} \))(\( 1 - 102 T^{2} + 5075 T^{4} - 209096 T^{6} + 11210675 T^{8} - 497727462 T^{10} + 10779215329 T^{12} \))(\( 1 - 150 T^{2} + 9095 T^{4} - 399200 T^{6} + 20090855 T^{8} - 731952150 T^{10} + 10779215329 T^{12} \))(\( 1 + 24 T^{2} + 6092 T^{4} + 110956 T^{6} + 18607278 T^{8} + 245101804 T^{10} + 29727016652 T^{12} + 258701167896 T^{14} + 23811286661761 T^{16} \))(\( 1 - 507 T^{2} + 123382 T^{4} - 19151794 T^{6} + 2124233871 T^{8} - 178577667484 T^{10} + 11761265538551 T^{12} - 617459414203988 T^{14} + 25980635574659159 T^{16} - 871402051045992604 T^{18} + 22897574304664208559 T^{20} - \)\(45\!\cdots\!34\)\( T^{22} + \)\(64\!\cdots\!18\)\( T^{24} - \)\(58\!\cdots\!87\)\( T^{26} + \)\(25\!\cdots\!69\)\( T^{28} \))(\( 1 - 279 T^{2} + 36014 T^{4} - 2901450 T^{6} + 175458439 T^{8} - 9612863124 T^{10} + 523448793295 T^{12} - 26366424181532 T^{14} + 1156298384388655 T^{16} - 46907705541783444 T^{18} + 1891304295271211431 T^{20} - 69087257684766453450 T^{22} + \)\(18\!\cdots\!86\)\( T^{24} - \)\(32\!\cdots\!39\)\( T^{26} + \)\(25\!\cdots\!69\)\( T^{28} \))
$53$ (\( 1 - 25 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 6 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 25 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 140 T^{2} + 9238 T^{4} - 393260 T^{6} + 7890481 T^{8} \))(\( 1 - 154 T^{2} + 10715 T^{4} - 432586 T^{6} + 7890481 T^{8} \))(\( ( 1 - 98 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 58 T^{2} + 2809 T^{4} )^{2} \))(\( 1 - 130 T^{2} + 9123 T^{4} - 365170 T^{6} + 7890481 T^{8} \))(\( 1 - 118 T^{2} + 7655 T^{4} - 441636 T^{6} + 21502895 T^{8} - 931076758 T^{10} + 22164361129 T^{12} \))(\( 1 - 214 T^{2} + 20391 T^{4} - 1256932 T^{6} + 57278319 T^{8} - 1688562934 T^{10} + 22164361129 T^{12} \))(\( 1 - 150 T^{2} + 13575 T^{4} - 822180 T^{6} + 38132175 T^{8} - 1183572150 T^{10} + 22164361129 T^{12} \))(\( 1 + 52 T^{2} + 8900 T^{4} + 437628 T^{6} + 34349046 T^{8} + 1229297052 T^{10} + 70225280900 T^{12} + 1152546778708 T^{14} + 62259690411361 T^{16} \))(\( 1 - 399 T^{2} + 76924 T^{4} - 9472269 T^{6} + 838079105 T^{8} - 57700076298 T^{10} + 3359096356493 T^{12} - 180955601681513 T^{14} + 9435701665388837 T^{16} - 455281355727919338 T^{18} + 18575487937889109545 T^{20} - \)\(58\!\cdots\!09\)\( T^{22} + \)\(13\!\cdots\!76\)\( T^{24} - \)\(19\!\cdots\!59\)\( T^{26} + \)\(13\!\cdots\!69\)\( T^{28} \))(\( 1 - 407 T^{2} + 80524 T^{4} - 10550805 T^{6} + 1047471025 T^{8} - 84402133530 T^{10} + 5701705132253 T^{12} - 327253549079281 T^{14} + 16016089716498677 T^{16} - 665973430977927930 T^{18} + 23216526070263787225 T^{20} - \)\(65\!\cdots\!05\)\( T^{22} + \)\(14\!\cdots\!76\)\( T^{24} - \)\(19\!\cdots\!87\)\( T^{26} + \)\(13\!\cdots\!69\)\( T^{28} \))
$59$ (\( ( 1 - 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - 14 T + 59 T^{2} )^{2} \))(\( ( 1 + 2 T + 59 T^{2} )^{2} \))(\( ( 1 + 3 T + 59 T^{2} )^{2} \))(\( ( 1 - 12 T + 59 T^{2} )^{2} \))(\( ( 1 - 9 T + 59 T^{2} )^{2} \))(\( ( 1 + 2 T + 114 T^{2} + 118 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + T + 115 T^{2} + 59 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 4 T + 24 T^{2} + 236 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 6 T + 100 T^{2} + 354 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + T + 57 T^{2} + 59 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 8 T + 161 T^{2} - 814 T^{3} + 9499 T^{4} - 27848 T^{5} + 205379 T^{6} )^{2} \))(\( ( 1 + 14 T + 177 T^{2} + 1578 T^{3} + 10443 T^{4} + 48734 T^{5} + 205379 T^{6} )^{2} \))(\( ( 1 + 8 T + 195 T^{2} + 954 T^{3} + 11505 T^{4} + 27848 T^{5} + 205379 T^{6} )^{2} \))(\( ( 1 + 26 T + 346 T^{2} + 2920 T^{3} + 22018 T^{4} + 172280 T^{5} + 1204426 T^{6} + 5339854 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 + T + 134 T^{2} - 678 T^{3} + 8727 T^{4} - 69216 T^{5} + 851597 T^{6} - 3045606 T^{7} + 50244223 T^{8} - 240940896 T^{9} + 1792342533 T^{10} - 8215570758 T^{11} + 95799856066 T^{12} + 42180533641 T^{13} + 2488651484819 T^{14} )^{2} \))(\( ( 1 + 9 T + 252 T^{2} + 2286 T^{3} + 34021 T^{4} + 267344 T^{5} + 2975391 T^{6} + 19435778 T^{7} + 175548069 T^{8} + 930624464 T^{9} + 6987198959 T^{10} + 27700287246 T^{11} + 180160923348 T^{12} + 379624802769 T^{13} + 2488651484819 T^{14} )^{2} \))
$61$ (\( ( 1 + 9 T + 61 T^{2} )^{2} \))(\( ( 1 + 6 T + 61 T^{2} )^{2} \))(\( ( 1 - 12 T + 61 T^{2} )^{2} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{2} \))(\( ( 1 + 7 T + 61 T^{2} )^{2} \))(\( ( 1 + 10 T + 61 T^{2} )^{2} \))(\( ( 1 + 10 T + 142 T^{2} + 610 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + T + 41 T^{2} + 61 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 8 T + 40 T^{2} + 488 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 2 T + 120 T^{2} + 122 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 7 T + 133 T^{2} - 427 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 8 T + 191 T^{2} + 978 T^{3} + 11651 T^{4} + 29768 T^{5} + 226981 T^{6} )^{2} \))(\( ( 1 - 10 T + 123 T^{2} - 1282 T^{3} + 7503 T^{4} - 37210 T^{5} + 226981 T^{6} )^{2} \))(\( ( 1 + 65 T^{2} + 358 T^{3} + 3965 T^{4} + 226981 T^{6} )^{2} \))(\( ( 1 + 12 T + 274 T^{2} + 2158 T^{3} + 25982 T^{4} + 131638 T^{5} + 1019554 T^{6} + 2723772 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 - 28 T + 679 T^{2} - 10771 T^{3} + 150205 T^{4} - 1658112 T^{5} + 16369827 T^{6} - 134783758 T^{7} + 998559447 T^{8} - 6169834752 T^{9} + 34093681105 T^{10} - 149133553411 T^{11} + 573480888379 T^{12} - 1442570482108 T^{13} + 3142742836021 T^{14} )^{2} \))(\( ( 1 - 18 T + 373 T^{2} - 4081 T^{3} + 50635 T^{4} - 416554 T^{5} + 4100887 T^{6} - 28795402 T^{7} + 250154107 T^{8} - 1549997434 T^{9} + 11493182935 T^{10} - 56504877121 T^{11} + 315034420273 T^{12} - 927366738498 T^{13} + 3142742836021 T^{14} )^{2} \))
$67$ (\( 1 + 35 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 + 10 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( 1 - 125 T^{2} + 4489 T^{4} \))(\( 1 - 85 T^{2} + 4489 T^{4} \))(\( 1 - 109 T^{2} + 4489 T^{4} \))(\( 1 - 28 T^{2} + 1174 T^{4} - 125692 T^{6} + 20151121 T^{8} \))(\( 1 - 105 T^{2} + 11653 T^{4} - 471345 T^{6} + 20151121 T^{8} \))(\( 1 - 116 T^{2} + 11190 T^{4} - 520724 T^{6} + 20151121 T^{8} \))(\( 1 - 164 T^{2} + 14934 T^{4} - 736196 T^{6} + 20151121 T^{8} \))(\( 1 - 185 T^{2} + 16173 T^{4} - 830465 T^{6} + 20151121 T^{8} \))(\( 1 - 190 T^{2} + 17351 T^{4} - 1214964 T^{6} + 77888639 T^{8} - 3828712990 T^{10} + 90458382169 T^{12} \))(\( 1 - 174 T^{2} + 19463 T^{4} - 1617812 T^{6} + 87369407 T^{8} - 3506295054 T^{10} + 90458382169 T^{12} \))(\( 1 - 382 T^{2} + 62023 T^{4} - 5482996 T^{6} + 278421247 T^{8} - 7697728222 T^{10} + 90458382169 T^{12} \))(\( 1 - 52 T^{2} + 11204 T^{4} - 335580 T^{6} + 60079542 T^{8} - 1506418620 T^{10} + 225773159684 T^{12} - 4703835872788 T^{14} + 406067677556641 T^{16} \))(\( 1 - 526 T^{2} + 134815 T^{4} - 22676851 T^{6} + 2856501869 T^{8} - 290267710230 T^{10} + 24735759835075 T^{12} - 1793877555190130 T^{14} + 111038825899651675 T^{16} - 5849219751237667830 T^{18} + \)\(25\!\cdots\!61\)\( T^{20} - \)\(92\!\cdots\!91\)\( T^{22} + \)\(24\!\cdots\!35\)\( T^{24} - \)\(43\!\cdots\!86\)\( T^{26} + \)\(36\!\cdots\!29\)\( T^{28} \))(\( 1 - 342 T^{2} + 64543 T^{4} - 8318467 T^{6} + 818361485 T^{8} - 65460107534 T^{10} + 4619022147363 T^{12} - 310261882714834 T^{14} + 20734790419512507 T^{16} - 1319094547590645614 T^{18} + 74027655962520360965 T^{20} - \)\(33\!\cdots\!47\)\( T^{22} + \)\(11\!\cdots\!07\)\( T^{24} - \)\(27\!\cdots\!62\)\( T^{26} + \)\(36\!\cdots\!29\)\( T^{28} \))
$71$ (\( ( 1 - 3 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 - T + 71 T^{2} )^{2} \))(\( ( 1 + 9 T + 71 T^{2} )^{2} \))(\( ( 1 - 9 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 158 T^{2} + 852 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 9 T + 133 T^{2} + 639 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 24 T + 278 T^{2} - 1704 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 12 T + 166 T^{2} - 852 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 9 T + 161 T^{2} + 639 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 20 T + 325 T^{2} + 3000 T^{3} + 23075 T^{4} + 100820 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 + 24 T + 293 T^{2} + 2608 T^{3} + 20803 T^{4} + 120984 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 + 4 T + 133 T^{2} + 504 T^{3} + 9443 T^{4} + 20164 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{8} \))(\( ( 1 + 10 T + 325 T^{2} + 2911 T^{3} + 52753 T^{4} + 413890 T^{5} + 5452989 T^{6} + 36222970 T^{7} + 387162219 T^{8} + 2086419490 T^{9} + 18880878983 T^{10} + 73973403391 T^{11} + 586374539075 T^{12} + 1281002839210 T^{13} + 9095120158391 T^{14} )^{2} \))(\( ( 1 - 18 T + 189 T^{2} - 369 T^{3} - 6119 T^{4} + 78846 T^{5} + 55309 T^{6} - 2373798 T^{7} + 3926939 T^{8} + 397462686 T^{9} - 2190057409 T^{10} - 9376910289 T^{11} + 340999347339 T^{12} - 2305805110578 T^{13} + 9095120158391 T^{14} )^{2} \))
$73$ (\( 1 - 137 T^{2} + 5329 T^{4} \))(\( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \))(\( 1 - 82 T^{2} + 5329 T^{4} \))(\( 1 - 82 T^{2} + 5329 T^{4} \))(\( 1 - 46 T^{2} + 5329 T^{4} \))(\( 1 - 121 T^{2} + 5329 T^{4} \))(\( 1 - 142 T^{2} + 5329 T^{4} \))(\( 1 - 264 T^{2} + 27902 T^{4} - 1406856 T^{6} + 28398241 T^{8} \))(\( 1 - 248 T^{2} + 25566 T^{4} - 1321592 T^{6} + 28398241 T^{8} \))(\( 1 - 184 T^{2} + 18322 T^{4} - 980536 T^{6} + 28398241 T^{8} \))(\( 1 - 188 T^{2} + 16794 T^{4} - 1001852 T^{6} + 28398241 T^{8} \))(\( 1 - 184 T^{2} + 18142 T^{4} - 980536 T^{6} + 28398241 T^{8} \))(\( 1 - 286 T^{2} + 39467 T^{4} - 3489912 T^{6} + 210319643 T^{8} - 8121896926 T^{10} + 151334226289 T^{12} \))(\( 1 - 106 T^{2} + 9475 T^{4} - 995536 T^{6} + 50492275 T^{8} - 3010213546 T^{10} + 151334226289 T^{12} \))(\( 1 - 130 T^{2} + 19895 T^{4} - 1371720 T^{6} + 106020455 T^{8} - 3691771330 T^{10} + 151334226289 T^{12} \))(\( 1 - 348 T^{2} + 49012 T^{4} - 3832288 T^{6} + 251118878 T^{8} - 20422262752 T^{10} + 1391854587892 T^{12} - 52664310748572 T^{14} + 806460091894081 T^{16} \))(\( 1 - 805 T^{2} + 312401 T^{4} - 77421408 T^{6} + 13684077884 T^{8} - 1824365628996 T^{10} + 189176272640170 T^{12} - 15498766818951438 T^{14} + 1008120356899465930 T^{16} - 51808774804344996036 T^{18} + \)\(20\!\cdots\!76\)\( T^{20} - \)\(62\!\cdots\!48\)\( T^{22} + \)\(13\!\cdots\!49\)\( T^{24} - \)\(18\!\cdots\!05\)\( T^{26} + \)\(12\!\cdots\!09\)\( T^{28} \))(\( 1 - 213 T^{2} + 28889 T^{4} - 2855816 T^{6} + 238013852 T^{8} - 17642224284 T^{10} + 1295071237346 T^{12} - 92053907050734 T^{14} + 6901434623816834 T^{16} - 501008136993084444 T^{18} + 36019642138484555228 T^{20} - \)\(23\!\cdots\!96\)\( T^{22} + \)\(12\!\cdots\!61\)\( T^{24} - \)\(48\!\cdots\!73\)\( T^{26} + \)\(12\!\cdots\!09\)\( T^{28} \))
$79$ (\( ( 1 + 10 T + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 6 T + 79 T^{2} )^{2} \))(\( ( 1 - 4 T + 79 T^{2} )^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{2} \))(\( ( 1 + 78 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 17 T + 201 T^{2} - 1343 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 4 T + 90 T^{2} - 316 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 20 T + 210 T^{2} - 1580 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 25 T + 303 T^{2} - 1975 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 8 T + 125 T^{2} - 1020 T^{3} + 9875 T^{4} - 49928 T^{5} + 493039 T^{6} )^{2} \))(\( ( 1 + 8 T + 125 T^{2} + 1020 T^{3} + 9875 T^{4} + 49928 T^{5} + 493039 T^{6} )^{2} \))(\( ( 1 + 4 T + 233 T^{2} + 612 T^{3} + 18407 T^{4} + 24964 T^{5} + 493039 T^{6} )^{2} \))(\( ( 1 + 20 T + 368 T^{2} + 4168 T^{3} + 45118 T^{4} + 329272 T^{5} + 2296688 T^{6} + 9860780 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 + 19 T + 278 T^{2} + 2808 T^{3} + 26403 T^{4} + 181292 T^{5} + 1490709 T^{6} + 11552842 T^{7} + 117766011 T^{8} + 1131443372 T^{9} + 13017708717 T^{10} + 109371827448 T^{11} + 855421678922 T^{12} + 4618661654899 T^{13} + 19203908986159 T^{14} )^{2} \))(\( ( 1 + 25 T + 478 T^{2} + 6646 T^{3} + 87747 T^{4} + 960800 T^{5} + 9906269 T^{6} + 88850016 T^{7} + 782595251 T^{8} + 5996352800 T^{9} + 43262693133 T^{10} + 258862238326 T^{11} + 1470832958722 T^{12} + 6077186388025 T^{13} + 19203908986159 T^{14} )^{2} \))
$83$ (\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 + 90 T^{2} + 6889 T^{4} \))(\( ( 1 - 83 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 + 36 T^{2} + 11222 T^{4} + 248004 T^{6} + 47458321 T^{8} \))(\( 1 - 109 T^{2} + 6917 T^{4} - 750901 T^{6} + 47458321 T^{8} \))(\( ( 1 - 22 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( 1 - 269 T^{2} + 30957 T^{4} - 1853141 T^{6} + 47458321 T^{8} \))(\( 1 - 102 T^{2} + 3527 T^{4} - 85556 T^{6} + 24297503 T^{8} - 4840748742 T^{10} + 326940373369 T^{12} \))(\( 1 - 166 T^{2} + 28487 T^{4} - 2384052 T^{6} + 196246943 T^{8} - 7878081286 T^{10} + 326940373369 T^{12} \))(\( 1 - 246 T^{2} + 34311 T^{4} - 3388308 T^{6} + 236368479 T^{8} - 11674746966 T^{10} + 326940373369 T^{12} \))(\( 1 - 508 T^{2} + 123444 T^{4} - 18468260 T^{6} + 1851922742 T^{8} - 127227843140 T^{10} + 5858444977524 T^{12} - 166085709671452 T^{14} + 2252292232139041 T^{16} \))(\( 1 - 647 T^{2} + 206352 T^{4} - 43124860 T^{6} + 6663917658 T^{8} - 818445689153 T^{10} + 84035330651789 T^{12} - 7457467650895768 T^{14} + 578919392860174421 T^{16} - 38842058236889292113 T^{18} + \)\(21\!\cdots\!02\)\( T^{20} - \)\(97\!\cdots\!60\)\( T^{22} + \)\(32\!\cdots\!48\)\( T^{24} - \)\(69\!\cdots\!67\)\( T^{26} + \)\(73\!\cdots\!29\)\( T^{28} \))(\( 1 - 847 T^{2} + 334320 T^{4} - 81958540 T^{6} + 14105059386 T^{8} - 1841387987065 T^{10} + 194050814302669 T^{12} - 17326976482574456 T^{14} + 1336816059731086741 T^{16} - 87389182175674617865 T^{18} + \)\(46\!\cdots\!34\)\( T^{20} - \)\(18\!\cdots\!40\)\( T^{22} + \)\(51\!\cdots\!80\)\( T^{24} - \)\(90\!\cdots\!67\)\( T^{26} + \)\(73\!\cdots\!29\)\( T^{28} \))
$89$ (\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 14 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 15 T + 89 T^{2} )^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 - 3 T + 89 T^{2} )^{2} \))(\( ( 1 + 2 T + 89 T^{2} )^{4} \))(\( ( 1 - 14 T + 214 T^{2} - 1246 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 4 T + 174 T^{2} - 356 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 70 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 10 T + 198 T^{2} - 890 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 16 T + 331 T^{2} - 2880 T^{3} + 29459 T^{4} - 126736 T^{5} + 704969 T^{6} )^{2} \))(\( ( 1 + 20 T + 315 T^{2} + 3240 T^{3} + 28035 T^{4} + 158420 T^{5} + 704969 T^{6} )^{2} \))(\( ( 1 + 12 T + 171 T^{2} + 2056 T^{3} + 15219 T^{4} + 95052 T^{5} + 704969 T^{6} )^{2} \))(\( ( 1 - 10 T + 372 T^{2} - 2638 T^{3} + 50342 T^{4} - 234782 T^{5} + 2946612 T^{6} - 7049690 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 2 T + 355 T^{2} - 508 T^{3} + 65625 T^{4} - 68894 T^{5} + 8090387 T^{6} - 6332296 T^{7} + 720044443 T^{8} - 545709374 T^{9} + 46263590625 T^{10} - 31873058428 T^{11} + 1982341104395 T^{12} - 993962581922 T^{13} + 44231334895529 T^{14} )^{2} \))(\( ( 1 - 16 T + 495 T^{2} - 5560 T^{3} + 102741 T^{4} - 892816 T^{5} + 12867243 T^{6} - 93437520 T^{7} + 1145184627 T^{8} - 7071995536 T^{9} + 72429220029 T^{10} - 348846859960 T^{11} + 2764109427255 T^{12} - 7951700655376 T^{13} + 44231334895529 T^{14} )^{2} \))
$97$ (\( 1 - 178 T^{2} + 9409 T^{4} \))(\( 1 - 94 T^{2} + 9409 T^{4} \))(\( 1 - 190 T^{2} + 9409 T^{4} \))(\( 1 - 94 T^{2} + 9409 T^{4} \))(\( 1 - 169 T^{2} + 9409 T^{4} \))(\( 1 - 94 T^{2} + 9409 T^{4} \))(\( 1 - 193 T^{2} + 9409 T^{4} \))(\( 1 + 4 T^{2} + 7302 T^{4} + 37636 T^{6} + 88529281 T^{8} \))(\( 1 - 290 T^{2} + 37971 T^{4} - 2728610 T^{6} + 88529281 T^{8} \))(\( 1 - 124 T^{2} + 20614 T^{4} - 1166716 T^{6} + 88529281 T^{8} \))(\( ( 1 - 190 T^{2} + 9409 T^{4} )^{2} \))(\( 1 - 106 T^{2} + 11947 T^{4} - 997354 T^{6} + 88529281 T^{8} \))(\( 1 - 454 T^{2} + 95567 T^{4} - 11805588 T^{6} + 899189903 T^{8} - 40192293574 T^{10} + 832972004929 T^{12} \))(\( 1 - 198 T^{2} + 29007 T^{4} - 3701140 T^{6} + 272926863 T^{8} - 17528797638 T^{10} + 832972004929 T^{12} \))(\( 1 - 102 T^{2} + 8655 T^{4} - 435796 T^{6} + 81434895 T^{8} - 9029986662 T^{10} + 832972004929 T^{12} \))(\( 1 - 388 T^{2} + 81220 T^{4} - 12091516 T^{6} + 1353693814 T^{8} - 113769074044 T^{10} + 7190348202820 T^{12} - 323193137912452 T^{14} + 7837433594376961 T^{16} \))(\( 1 - 544 T^{2} + 145588 T^{4} - 26811014 T^{6} + 4003836014 T^{8} - 526202800616 T^{10} + 61846383096861 T^{12} - 6413113738177764 T^{14} + 581912618558365149 T^{16} - 46584355598720837096 T^{18} + \)\(33\!\cdots\!06\)\( T^{20} - \)\(21\!\cdots\!54\)\( T^{22} + \)\(10\!\cdots\!12\)\( T^{24} - \)\(37\!\cdots\!04\)\( T^{26} + \)\(65\!\cdots\!69\)\( T^{28} \))(\( 1 - 836 T^{2} + 340396 T^{4} - 89911234 T^{6} + 17365917414 T^{8} - 2628719777716 T^{10} + 326610398494637 T^{12} - 34289460876308108 T^{14} + 3073077239436039533 T^{16} - \)\(23\!\cdots\!96\)\( T^{18} + \)\(14\!\cdots\!06\)\( T^{20} - \)\(70\!\cdots\!74\)\( T^{22} + \)\(25\!\cdots\!04\)\( T^{24} - \)\(58\!\cdots\!76\)\( T^{26} + \)\(65\!\cdots\!69\)\( T^{28} \))
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