Properties

Label 670.2
Level 670
Weight 2
Dimension 4115
Nonzero newspaces 12
Newform subspaces 39
Sturm bound 53856
Trace bound 4

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 670 = 2 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 12 \)
Newform subspaces: \( 39 \)
Sturm bound: \(53856\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(670))\).

Total New Old
Modular forms 13992 4115 9877
Cusp forms 12937 4115 8822
Eisenstein series 1055 0 1055

Trace form

\( 4115q + q^{2} + 4q^{3} + q^{4} + q^{5} + 4q^{6} + 8q^{7} + q^{8} + 13q^{9} + O(q^{10}) \) \( 4115q + q^{2} + 4q^{3} + q^{4} + q^{5} + 4q^{6} + 8q^{7} + q^{8} + 13q^{9} + q^{10} + 12q^{11} + 4q^{12} + 14q^{13} + 8q^{14} + 4q^{15} + q^{16} + 18q^{17} + 13q^{18} + 20q^{19} + q^{20} + 32q^{21} + 12q^{22} + 24q^{23} + 4q^{24} + q^{25} + 14q^{26} + 40q^{27} + 8q^{28} + 30q^{29} + 4q^{30} + 32q^{31} + q^{32} + 48q^{33} + 18q^{34} + 8q^{35} + 13q^{36} + 38q^{37} + 20q^{38} + 56q^{39} + q^{40} + 42q^{41} + 32q^{42} + 44q^{43} + 12q^{44} + 13q^{45} + 24q^{46} + 48q^{47} + 4q^{48} + 57q^{49} + q^{50} + 72q^{51} - 30q^{52} - 78q^{53} - 158q^{54} - 186q^{55} + 8q^{56} - 228q^{57} - 234q^{58} - 204q^{59} - 128q^{60} - 466q^{61} - 100q^{62} - 468q^{63} + q^{64} - 250q^{65} - 480q^{66} - 197q^{67} - 114q^{68} - 168q^{69} - 256q^{70} - 456q^{71} + 13q^{72} - 498q^{73} - 94q^{74} - 260q^{75} - 244q^{76} - 168q^{77} - 208q^{78} - 228q^{79} + q^{80} - 275q^{81} - 156q^{82} - 48q^{83} - 12q^{84} + 18q^{85} + 44q^{86} + 120q^{87} + 12q^{88} + 90q^{89} + 13q^{90} + 112q^{91} + 24q^{92} + 128q^{93} + 48q^{94} + 20q^{95} + 4q^{96} + 98q^{97} + 57q^{98} + 156q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(670))\)

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

Label \(\chi\) Newforms Dimension \(\chi\) degree
670.2.a \(\chi_{670}(1, \cdot)\) 670.2.a.a 1 1
670.2.a.b 1
670.2.a.c 1
670.2.a.d 1
670.2.a.e 2
670.2.a.f 2
670.2.a.g 3
670.2.a.h 3
670.2.a.i 3
670.2.a.j 4
670.2.c \(\chi_{670}(269, \cdot)\) 670.2.c.a 10 1
670.2.c.b 24
670.2.e \(\chi_{670}(171, \cdot)\) 670.2.e.a 2 2
670.2.e.b 2
670.2.e.c 2
670.2.e.d 2
670.2.e.e 4
670.2.e.f 4
670.2.e.g 6
670.2.e.h 6
670.2.e.i 8
670.2.e.j 12
670.2.f \(\chi_{670}(133, \cdot)\) 670.2.f.a 68 2
670.2.h \(\chi_{670}(29, \cdot)\) 670.2.h.a 4 2
670.2.h.b 12
670.2.h.c 52
670.2.k \(\chi_{670}(81, \cdot)\) 670.2.k.a 40 10
670.2.k.b 50
670.2.k.c 50
670.2.k.d 60
670.2.m \(\chi_{670}(97, \cdot)\) 670.2.m.a 136 4
670.2.o \(\chi_{670}(9, \cdot)\) 670.2.o.a 340 10
670.2.q \(\chi_{670}(21, \cdot)\) 670.2.q.a 100 20
670.2.q.b 120
670.2.q.c 120
670.2.q.d 140
670.2.s \(\chi_{670}(3, \cdot)\) 670.2.s.a 680 20
670.2.v \(\chi_{670}(19, \cdot)\) 670.2.v.a 680 20
670.2.w \(\chi_{670}(7, \cdot)\) 670.2.w.a 1360 40

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(670))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(670)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(67))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(134))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(335))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T^{2} )^{5} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3} \))(\( ( 1 + T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{6} \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 - T^{2} + T^{4} )^{3} \))
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 4 T^{2} + 9 T^{4} \))(\( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4} \))(\( 1 - 4 T + 11 T^{2} - 22 T^{3} + 33 T^{4} - 36 T^{5} + 27 T^{6} \))(\( 1 + 3 T^{2} + 2 T^{3} + 9 T^{4} + 27 T^{6} \))(\( 1 - 2 T + 5 T^{2} - 6 T^{3} + 15 T^{4} - 18 T^{5} + 27 T^{6} \))(\( 1 + 2 T + 4 T^{2} - 2 T^{4} + 36 T^{6} + 54 T^{7} + 81 T^{8} \))(\( 1 - 18 T^{2} + 165 T^{4} - 1004 T^{6} + 4474 T^{8} - 15248 T^{10} + 40266 T^{12} - 81324 T^{14} + 120285 T^{16} - 118098 T^{18} + 59049 T^{20} \))(\( ( 1 + 3 T + 3 T^{2} )^{2} \))(\( ( 1 + 3 T + 3 T^{2} )^{2} \))(\( ( 1 - T + 3 T^{2} )^{2} \))(\( ( 1 - T + 3 T^{2} )^{2} \))(\( ( 1 - T + 3 T^{2} )^{4} \))(\( ( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} )^{2} \))(\( ( 1 + T + 4 T^{2} + 3 T^{3} + 12 T^{4} + 9 T^{5} + 27 T^{6} )^{2} \))(\( ( 1 - T + 2 T^{2} - 7 T^{3} + 6 T^{4} - 9 T^{5} + 27 T^{6} )^{2} \))(\( ( 1 - 2 T + 5 T^{2} - 6 T^{3} + 9 T^{4} )^{4} \))(\( ( 1 + 2 T + 5 T^{2} + 8 T^{3} + 20 T^{4} + 30 T^{5} + 67 T^{6} + 90 T^{7} + 180 T^{8} + 216 T^{9} + 405 T^{10} + 486 T^{11} + 729 T^{12} )^{2} \))(\( ( 1 - 5 T^{2} + 9 T^{4} )^{2} \))(\( ( 1 - 7 T^{2} + 10 T^{4} + 13 T^{6} + 90 T^{8} - 567 T^{10} + 729 T^{12} )^{2} \))
$5$ (\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{4} \))(\( 1 + 2 T + T^{2} + 8 T^{3} + 2 T^{4} - 52 T^{5} + 10 T^{6} + 200 T^{7} + 125 T^{8} + 1250 T^{9} + 3125 T^{10} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{6} \))(\( ( 1 - T )^{6} \))(\( ( 1 - T )^{8} \))(\( ( 1 + T )^{12} \))(\( ( 1 + 2 T + 5 T^{2} )^{2} \))(\( ( 1 - 4 T + 5 T^{2} )^{6} \))
$7$ (\( 1 + T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - 2 T + 13 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 + 6 T + 21 T^{2} + 42 T^{3} + 49 T^{4} \))(\( 1 - T + 18 T^{2} - 13 T^{3} + 126 T^{4} - 49 T^{5} + 343 T^{6} \))(\( 1 - 3 T + 18 T^{2} - 35 T^{3} + 126 T^{4} - 147 T^{5} + 343 T^{6} \))(\( 1 - T + 16 T^{2} - 15 T^{3} + 112 T^{4} - 49 T^{5} + 343 T^{6} \))(\( 1 + 5 T + 19 T^{2} + 30 T^{3} + 80 T^{4} + 210 T^{5} + 931 T^{6} + 1715 T^{7} + 2401 T^{8} \))(\( 1 - 45 T^{2} + 1011 T^{4} - 14862 T^{6} + 157941 T^{8} - 1265687 T^{10} + 7739109 T^{12} - 35683662 T^{14} + 118943139 T^{16} - 259416045 T^{18} + 282475249 T^{20} \))(\( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} \))(\( ( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))(\( ( 1 - 4 T + 7 T^{2} )( 1 - T + 7 T^{2} ) \))(\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))(\( ( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} )^{2} \))(\( 1 + 4 T + 4 T^{2} - 8 T^{3} - 17 T^{4} - 56 T^{5} + 196 T^{6} + 1372 T^{7} + 2401 T^{8} \))(\( 1 + 2 T - 7 T^{2} - 30 T^{3} - 8 T^{4} + 76 T^{5} + 319 T^{6} + 532 T^{7} - 392 T^{8} - 10290 T^{9} - 16807 T^{10} + 33614 T^{11} + 117649 T^{12} \))(\( 1 - 2 T - 11 T^{2} + 10 T^{3} + 80 T^{4} + 22 T^{5} - 685 T^{6} + 154 T^{7} + 3920 T^{8} + 3430 T^{9} - 26411 T^{10} - 33614 T^{11} + 117649 T^{12} \))(\( ( 1 - 12 T^{2} + 95 T^{4} - 588 T^{6} + 2401 T^{8} )^{2} \))(\( 1 - 3 T + 5 T^{2} - 10 T^{3} - 68 T^{4} + 300 T^{5} + 123 T^{6} - 1387 T^{7} + 8070 T^{8} - 30067 T^{9} + 12793 T^{10} + 117978 T^{11} - 261036 T^{12} + 825846 T^{13} + 626857 T^{14} - 10312981 T^{15} + 19376070 T^{16} - 23311309 T^{17} + 14470827 T^{18} + 247062900 T^{19} - 392006468 T^{20} - 403536070 T^{21} + 1412376245 T^{22} - 5931980229 T^{23} + 13841287201 T^{24} \))(\( 1 + 5 T^{2} - 24 T^{4} + 245 T^{6} + 2401 T^{8} \))(\( 1 + 14 T^{2} + 141 T^{4} + 802 T^{6} + 554 T^{8} - 42726 T^{10} - 383267 T^{12} - 2093574 T^{14} + 1330154 T^{16} + 94354498 T^{18} + 812836941 T^{20} + 3954653486 T^{22} + 13841287201 T^{24} \))
$11$ (\( 1 - 3 T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( ( 1 + 5 T + 11 T^{2} )^{2} \))(\( ( 1 - T + 11 T^{2} )^{3} \))(\( ( 1 - 3 T + 11 T^{2} )^{3} \))(\( ( 1 - T + 11 T^{2} )^{3} \))(\( 1 - 9 T + 47 T^{2} - 212 T^{3} + 800 T^{4} - 2332 T^{5} + 5687 T^{6} - 11979 T^{7} + 14641 T^{8} \))(\( ( 1 + 5 T + 41 T^{2} + 158 T^{3} + 777 T^{4} + 2275 T^{5} + 8547 T^{6} + 19118 T^{7} + 54571 T^{8} + 73205 T^{9} + 161051 T^{10} )^{2} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \))(\( ( 1 - 11 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 11 T^{2} + 121 T^{4} )^{2} \))(\( 1 - 8 T + 19 T^{2} - 32 T^{3} + 218 T^{4} - 428 T^{5} - 461 T^{6} - 4708 T^{7} + 26378 T^{8} - 42592 T^{9} + 278179 T^{10} - 1288408 T^{11} + 1771561 T^{12} \))(\( ( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} )^{3} \))(\( ( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{4} \))(\( 1 - 30 T^{2} + 8 T^{3} + 349 T^{4} - 204 T^{5} - 2618 T^{6} + 1912 T^{7} + 28154 T^{8} - 7772 T^{9} - 263958 T^{10} + 8212 T^{11} + 1870181 T^{12} + 90332 T^{13} - 31938918 T^{14} - 10344532 T^{15} + 412202714 T^{16} + 307929512 T^{17} - 4637946698 T^{18} - 3975382884 T^{19} + 74811249469 T^{20} + 18863581528 T^{21} - 778122738030 T^{22} + 3138428376721 T^{24} \))(\( ( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 4 T - T^{2} - 44 T^{3} - 122 T^{4} - 60 T^{5} + 683 T^{6} - 660 T^{7} - 14762 T^{8} - 58564 T^{9} - 14641 T^{10} + 644204 T^{11} + 1771561 T^{12} )^{2} \))
$13$ (\( 1 + 4 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 + 8 T^{2} + 169 T^{4} \))(\( 1 + 4 T + 12 T^{2} + 52 T^{3} + 169 T^{4} \))(\( 1 - 4 T + 23 T^{2} - 114 T^{3} + 299 T^{4} - 676 T^{5} + 2197 T^{6} \))(\( 1 + 27 T^{2} - 2 T^{3} + 351 T^{4} + 2197 T^{6} \))(\( 1 - 2 T + 33 T^{2} - 54 T^{3} + 429 T^{4} - 338 T^{5} + 2197 T^{6} \))(\( 1 + 12 T + 98 T^{2} + 526 T^{3} + 2214 T^{4} + 6838 T^{5} + 16562 T^{6} + 26364 T^{7} + 28561 T^{8} \))(\( 1 - 94 T^{2} + 4253 T^{4} - 122540 T^{6} + 2494726 T^{8} - 37505152 T^{10} + 421608694 T^{12} - 3499864940 T^{14} + 20528418677 T^{16} - 76678687774 T^{18} + 137858491849 T^{20} \))(\( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} )^{2} \))(\( 1 + 2 T - 25 T^{2} - 42 T^{3} + 364 T^{4} + 280 T^{5} - 4715 T^{6} + 3640 T^{7} + 61516 T^{8} - 92274 T^{9} - 714025 T^{10} + 742586 T^{11} + 4826809 T^{12} \))(\( 1 - 2 T - 19 T^{2} + 26 T^{3} + 158 T^{4} + 30 T^{5} - 1863 T^{6} + 390 T^{7} + 26702 T^{8} + 57122 T^{9} - 542659 T^{10} - 742586 T^{11} + 4826809 T^{12} \))(\( 1 + 6 T^{2} + 32 T^{3} - 222 T^{4} - 112 T^{5} - 224 T^{6} - 6624 T^{7} + 28591 T^{8} - 86112 T^{9} - 37856 T^{10} - 246064 T^{11} - 6340542 T^{12} + 11881376 T^{13} + 28960854 T^{14} + 815730721 T^{16} \))(\( 1 - 22 T^{2} - 100 T^{3} + 89 T^{4} + 1706 T^{5} + 9418 T^{6} + 964 T^{7} - 112718 T^{8} - 587414 T^{9} - 800366 T^{10} + 4260054 T^{11} + 36575217 T^{12} + 55380702 T^{13} - 135261854 T^{14} - 1290548558 T^{15} - 3219338798 T^{16} + 357926452 T^{17} + 45458887162 T^{18} + 107048970002 T^{19} + 72600034169 T^{20} - 1060449937300 T^{21} - 3032886820678 T^{22} + 23298085122481 T^{24} \))(\( ( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} )( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} ) \))(\( 1 + 66 T^{2} + 2413 T^{4} + 63062 T^{6} + 1296074 T^{8} + 21859066 T^{10} + 309202565 T^{12} + 3694182154 T^{14} + 37017169514 T^{16} + 304388229158 T^{18} + 1968358229773 T^{20} + 9098660462034 T^{22} + 23298085122481 T^{24} \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 4 T + 36 T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 + 16 T^{2} + 289 T^{4} \))(\( 1 - 8 T + 57 T^{2} - 270 T^{3} + 969 T^{4} - 2312 T^{5} + 4913 T^{6} \))(\( 1 + 9 T^{2} - 98 T^{3} + 153 T^{4} + 4913 T^{6} \))(\( 1 + 2 T + 47 T^{2} + 62 T^{3} + 799 T^{4} + 578 T^{5} + 4913 T^{6} \))(\( 1 + 24 T^{2} - 98 T^{3} + 290 T^{4} - 1666 T^{5} + 6936 T^{6} + 83521 T^{8} \))(\( 1 - 82 T^{2} + 3789 T^{4} - 123128 T^{6} + 3017618 T^{8} - 57758216 T^{10} + 872091602 T^{12} - 10283773688 T^{14} + 91457248941 T^{16} - 572012110162 T^{18} + 2015993900449 T^{20} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - T + 15 T^{2} + 48 T^{3} - 110 T^{4} + 816 T^{5} + 4335 T^{6} - 4913 T^{7} + 83521 T^{8} \))(\( 1 + 4 T - 16 T^{2} - 8 T^{3} + 463 T^{4} - 136 T^{5} - 4624 T^{6} + 19652 T^{7} + 83521 T^{8} \))(\( 1 - 3 T - 27 T^{2} + 114 T^{3} + 351 T^{4} - 1191 T^{5} - 2486 T^{6} - 20247 T^{7} + 101439 T^{8} + 560082 T^{9} - 2255067 T^{10} - 4259571 T^{11} + 24137569 T^{12} \))(\( 1 + 8 T + 7 T^{2} + 456 T^{4} - 196 T^{5} - 11919 T^{6} - 3332 T^{7} + 131784 T^{8} + 584647 T^{10} + 11358856 T^{11} + 24137569 T^{12} \))(\( 1 - 2 T - 33 T^{2} + 34 T^{3} + 455 T^{4} + 308 T^{5} - 7356 T^{6} - 6804 T^{7} + 160206 T^{8} - 115668 T^{9} - 2125884 T^{10} + 1513204 T^{11} + 38002055 T^{12} + 48275138 T^{13} - 796539777 T^{14} - 820677346 T^{15} + 6975757441 T^{16} \))(\( 1 - 13 T + 40 T^{2} + 75 T^{3} + 2 T^{4} - 1883 T^{5} - 12464 T^{6} + 42003 T^{7} + 336334 T^{8} - 678947 T^{9} - 6215028 T^{10} + 19094437 T^{11} - 8287190 T^{12} + 324605429 T^{13} - 1796143092 T^{14} - 3335666611 T^{15} + 28090952014 T^{16} + 59638253571 T^{17} - 300850660016 T^{18} - 772667721259 T^{19} + 13951514882 T^{20} + 8894090737275 T^{21} + 80639756017960 T^{22} - 445534651999229 T^{23} + 582622237229761 T^{24} \))(\( 1 + 18 T^{2} + 35 T^{4} + 5202 T^{6} + 83521 T^{8} \))(\( 1 + 30 T^{2} + 537 T^{4} + 6234 T^{6} - 42978 T^{8} - 2943726 T^{10} - 59555419 T^{12} - 850736814 T^{14} - 3589565538 T^{16} + 150473605146 T^{18} + 3745981745817 T^{20} + 60479817013470 T^{22} + 582622237229761 T^{24} \))
$19$ (\( 1 - 2 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 4 T + 34 T^{2} + 76 T^{3} + 361 T^{4} \))(\( 1 + 4 T + 34 T^{2} + 76 T^{3} + 361 T^{4} \))(\( 1 + 8 T + 17 T^{2} + 323 T^{4} + 2888 T^{5} + 6859 T^{6} \))(\( 1 + 33 T^{2} - 16 T^{3} + 627 T^{4} + 6859 T^{6} \))(\( 1 + 25 T^{2} + 32 T^{3} + 475 T^{4} + 6859 T^{6} \))(\( 1 - 4 T - 4 T^{2} - 68 T^{3} + 790 T^{4} - 1292 T^{5} - 1444 T^{6} - 27436 T^{7} + 130321 T^{8} \))(\( ( 1 - 16 T + 187 T^{2} - 1440 T^{3} + 9054 T^{4} - 43192 T^{5} + 172026 T^{6} - 519840 T^{7} + 1282633 T^{8} - 2085136 T^{9} + 2476099 T^{10} )^{2} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - 7 T + 19 T^{2} )( 1 - T + 19 T^{2} ) \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 + 3 T + 17 T^{2} - 138 T^{3} - 468 T^{4} - 2622 T^{5} + 6137 T^{6} + 20577 T^{7} + 130321 T^{8} \))(\( 1 - 4 T - 2 T^{2} + 80 T^{3} - 341 T^{4} + 1520 T^{5} - 722 T^{6} - 27436 T^{7} + 130321 T^{8} \))(\( ( 1 - 8 T + 19 T^{2} )^{3}( 1 + 7 T + 19 T^{2} )^{3} \))(\( 1 + 10 T + 71 T^{2} + 218 T^{3} - 70 T^{4} - 6366 T^{5} - 37065 T^{6} - 120954 T^{7} - 25270 T^{8} + 1495262 T^{9} + 9252791 T^{10} + 24760990 T^{11} + 47045881 T^{12} \))(\( 1 - 6 T - 25 T^{2} + 174 T^{3} + 369 T^{4} - 816 T^{5} - 19490 T^{6} - 15060 T^{7} + 634970 T^{8} - 286140 T^{9} - 7035890 T^{10} - 5596944 T^{11} + 48088449 T^{12} + 430841226 T^{13} - 1176147025 T^{14} - 5363230434 T^{15} + 16983563041 T^{16} \))(\( 1 + 9 T - 44 T^{2} - 447 T^{3} + 2388 T^{4} + 16025 T^{5} - 95700 T^{6} - 368991 T^{7} + 3307180 T^{8} + 6564321 T^{9} - 84607156 T^{10} - 46389887 T^{11} + 1807770198 T^{12} - 881407853 T^{13} - 30543183316 T^{14} + 45024677739 T^{15} + 430995004780 T^{16} - 913658246109 T^{17} - 4502290811700 T^{18} + 14324294617475 T^{19} + 40556748541908 T^{20} - 144241400907213 T^{21} - 269766915343244 T^{22} + 1048412330083971 T^{23} + 2213314919066161 T^{24} \))(\( ( 1 - 6 T + 17 T^{2} - 114 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 6 T - 17 T^{2} - 122 T^{3} + 482 T^{4} + 1742 T^{5} - 4249 T^{6} + 33098 T^{7} + 174002 T^{8} - 836798 T^{9} - 2215457 T^{10} + 14856594 T^{11} + 47045881 T^{12} )^{2} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( 1 + 14 T^{2} + 529 T^{4} \))(\( 1 - 10 T + 89 T^{2} - 452 T^{3} + 2047 T^{4} - 5290 T^{5} + 12167 T^{6} \))(\( 1 - 6 T + 57 T^{2} - 252 T^{3} + 1311 T^{4} - 3174 T^{5} + 12167 T^{6} \))(\( ( 1 - 2 T + 23 T^{2} )^{3} \))(\( 1 - 6 T + 48 T^{2} - 230 T^{3} + 1118 T^{4} - 5290 T^{5} + 25392 T^{6} - 73002 T^{7} + 279841 T^{8} \))(\( 1 - 62 T^{2} + 1437 T^{4} - 23000 T^{6} + 937394 T^{8} - 31808148 T^{10} + 495881426 T^{12} - 6436343000 T^{14} + 212727572493 T^{16} - 4855281087422 T^{18} + 41426511213649 T^{20} \))(\( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 + 7 T + 26 T^{2} + 161 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( ( 1 - 23 T^{2} + 529 T^{4} )^{2} \))(\( 1 - 40 T^{2} + 1071 T^{4} - 21160 T^{6} + 279841 T^{8} \))(\( 1 - 4 T - 47 T^{2} + 80 T^{3} + 1856 T^{4} - 982 T^{5} - 47861 T^{6} - 22586 T^{7} + 981824 T^{8} + 973360 T^{9} - 13152527 T^{10} - 25745372 T^{11} + 148035889 T^{12} \))(\( 1 - 8 T - 19 T^{2} + 64 T^{3} + 2068 T^{4} - 3900 T^{5} - 35473 T^{6} - 89700 T^{7} + 1093972 T^{8} + 778688 T^{9} - 5316979 T^{10} - 51490744 T^{11} + 148035889 T^{12} \))(\( 1 + 4 T + 36 T^{2} - 40 T^{3} - 258 T^{4} - 5964 T^{5} - 4160 T^{6} - 26812 T^{7} + 433347 T^{8} - 616676 T^{9} - 2200640 T^{10} - 72563988 T^{11} - 72198978 T^{12} - 257453720 T^{13} + 5329292004 T^{14} + 13619301788 T^{15} + 78310985281 T^{16} \))(\( 1 + 3 T - 61 T^{2} + 152 T^{3} + 3112 T^{4} - 12886 T^{5} - 45691 T^{6} + 688011 T^{7} - 564098 T^{8} - 15294841 T^{9} + 80978327 T^{10} + 183557680 T^{11} - 2300581128 T^{12} + 4221826640 T^{13} + 42837534983 T^{14} - 186092330447 T^{15} - 157857748418 T^{16} + 4428274783773 T^{17} - 6763907804299 T^{18} - 43874580710042 T^{19} + 243703786194472 T^{20} + 273775204542376 T^{21} - 2527017184032589 T^{22} + 2858429273741781 T^{23} + 21914624432020321 T^{24} \))(\( 1 + 21 T^{2} - 88 T^{4} + 11109 T^{6} + 279841 T^{8} \))(\( 1 + 74 T^{2} + 2245 T^{4} + 60710 T^{6} + 2129954 T^{8} + 55926134 T^{10} + 1196453357 T^{12} + 29584924886 T^{14} + 596048457314 T^{16} + 8987258821190 T^{18} + 175808161955845 T^{20} + 3065561829810026 T^{22} + 21914624432020321 T^{24} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 10 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 4 T + 60 T^{2} + 116 T^{3} + 841 T^{4} \))(\( 1 + 8 T^{2} + 841 T^{4} \))(\( 1 + 2 T + 67 T^{2} + 66 T^{3} + 1943 T^{4} + 1682 T^{5} + 24389 T^{6} \))(\( 1 - 6 T + 87 T^{2} - 330 T^{3} + 2523 T^{4} - 5046 T^{5} + 24389 T^{6} \))(\( 1 - 4 T + 73 T^{2} - 178 T^{3} + 2117 T^{4} - 3364 T^{5} + 24389 T^{6} \))(\( 1 - 18 T + 206 T^{2} - 1600 T^{3} + 9854 T^{4} - 46400 T^{5} + 173246 T^{6} - 439002 T^{7} + 707281 T^{8} \))(\( ( 1 - 6 T + 101 T^{2} - 478 T^{3} + 4940 T^{4} - 18130 T^{5} + 143260 T^{6} - 401998 T^{7} + 2463289 T^{8} - 4243686 T^{9} + 20511149 T^{10} )^{2} \))(\( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \))(\( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} \))(\( ( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} )^{2} \))(\( 1 + 6 T - 27 T^{2} - 354 T^{3} - 54 T^{4} + 6180 T^{5} + 37357 T^{6} + 179220 T^{7} - 45414 T^{8} - 8633706 T^{9} - 19096587 T^{10} + 123066894 T^{11} + 594823321 T^{12} \))(\( 1 - 2 T - 15 T^{2} + 178 T^{3} - 406 T^{4} - 1530 T^{5} + 46609 T^{6} - 44370 T^{7} - 341446 T^{8} + 4341242 T^{9} - 10609215 T^{10} - 41022298 T^{11} + 594823321 T^{12} \))(\( 1 - 8 T + 6 T^{2} + 176 T^{3} - 630 T^{4} - 4896 T^{5} + 42544 T^{6} + 32264 T^{7} - 1327857 T^{8} + 935656 T^{9} + 35779504 T^{10} - 119408544 T^{11} - 445587030 T^{12} + 3609962224 T^{13} + 3568939926 T^{14} - 137999010472 T^{15} + 500246412961 T^{16} \))(\( 1 + 5 T - 85 T^{2} - 500 T^{3} + 2870 T^{4} + 17958 T^{5} - 106967 T^{6} - 359533 T^{7} + 5761950 T^{8} + 9857485 T^{9} - 199566941 T^{10} - 170362208 T^{11} + 5343888062 T^{12} - 4940504032 T^{13} - 167835797381 T^{14} + 240414201665 T^{15} + 4075317757950 T^{16} - 7374434933417 T^{17} - 63626466177407 T^{18} + 309773278757022 T^{19} + 1435707205198070 T^{20} - 7253572987934500 T^{21} - 35760114830517085 T^{22} + 61002548828529145 T^{23} + 353814783205469041 T^{24} \))(\( ( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 4 T - 55 T^{2} + 116 T^{3} + 2182 T^{4} - 1092 T^{5} - 70867 T^{6} - 31668 T^{7} + 1835062 T^{8} + 2829124 T^{9} - 38900455 T^{10} - 82044596 T^{11} + 594823321 T^{12} )^{2} \))
$31$ (\( 1 - 2 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 48 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 - 8 T + 76 T^{2} - 248 T^{3} + 961 T^{4} \))(\( 1 + 89 T^{2} + 2 T^{3} + 2759 T^{4} + 29791 T^{6} \))(\( 1 + 45 T^{2} + 114 T^{3} + 1395 T^{4} + 29791 T^{6} \))(\( 1 + 2 T + 35 T^{2} + 142 T^{3} + 1085 T^{4} + 1922 T^{5} + 29791 T^{6} \))(\( 1 - 2 T + 82 T^{2} - 240 T^{3} + 3146 T^{4} - 7440 T^{5} + 78802 T^{6} - 59582 T^{7} + 923521 T^{8} \))(\( ( 1 + 22 T + 303 T^{2} + 2872 T^{3} + 21804 T^{4} + 132454 T^{5} + 675924 T^{6} + 2759992 T^{7} + 9026673 T^{8} + 20317462 T^{9} + 28629151 T^{10} )^{2} \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} \))(\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 - T - 13 T^{2} + 48 T^{3} - 796 T^{4} + 1488 T^{5} - 12493 T^{6} - 29791 T^{7} + 923521 T^{8} \))(\( 1 - 6 T - 29 T^{2} - 18 T^{3} + 2148 T^{4} - 558 T^{5} - 27869 T^{6} - 178746 T^{7} + 923521 T^{8} \))(\( 1 + 16 T + 99 T^{2} + 432 T^{3} + 3142 T^{4} + 22912 T^{5} + 131047 T^{6} + 710272 T^{7} + 3019462 T^{8} + 12869712 T^{9} + 91428579 T^{10} + 458066416 T^{11} + 887503681 T^{12} \))(\( 1 - 15 T + 91 T^{2} - 392 T^{3} + 1667 T^{4} + 1799 T^{5} - 58798 T^{6} + 55769 T^{7} + 1601987 T^{8} - 11678072 T^{9} + 84040411 T^{10} - 429437265 T^{11} + 887503681 T^{12} \))(\( 1 - 6 T - 7 T^{2} - 582 T^{3} + 3477 T^{4} + 4116 T^{5} + 167230 T^{6} - 964848 T^{7} - 938686 T^{8} - 29910288 T^{9} + 160708030 T^{10} + 122619756 T^{11} + 3211082517 T^{12} - 16662165882 T^{13} - 6212525767 T^{14} - 165075684666 T^{15} + 852891037441 T^{16} \))(\( 1 - 14 T + 5 T^{2} + 30 T^{3} + 7646 T^{4} - 26950 T^{5} - 117941 T^{6} - 1352166 T^{7} + 9773992 T^{8} + 17741470 T^{9} + 157961425 T^{10} - 1351375662 T^{11} - 1247034236 T^{12} - 41892645522 T^{13} + 151800929425 T^{14} + 528536132770 T^{15} + 9026486865832 T^{16} - 38711364591066 T^{17} - 104673071640821 T^{18} - 741464950291450 T^{19} + 6521204872273886 T^{20} + 793188664820130 T^{21} + 4098141434904005 T^{22} - 355718676549667634 T^{23} + 787662783788549761 T^{24} \))(\( ( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 9 T + 19 T^{2} + 76 T^{3} - 247 T^{4} - 11005 T^{5} - 79342 T^{6} - 341155 T^{7} - 237367 T^{8} + 2264116 T^{9} + 17546899 T^{10} + 257662359 T^{11} + 887503681 T^{12} )^{2} \))
$37$ (\( 1 + 7 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( 1 - T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 + 2 T + 25 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 + 14 T + 121 T^{2} + 518 T^{3} + 1369 T^{4} \))(\( 1 + T + 46 T^{2} - 77 T^{3} + 1702 T^{4} + 1369 T^{5} + 50653 T^{6} \))(\( 1 - 3 T + 66 T^{2} - 289 T^{3} + 2442 T^{4} - 4107 T^{5} + 50653 T^{6} \))(\( 1 - 3 T + 48 T^{2} + 21 T^{3} + 1776 T^{4} - 4107 T^{5} + 50653 T^{6} \))(\( 1 - 9 T + 107 T^{2} - 670 T^{3} + 5406 T^{4} - 24790 T^{5} + 146483 T^{6} - 455877 T^{7} + 1874161 T^{8} \))(\( 1 - 313 T^{2} + 45847 T^{4} - 4132514 T^{6} + 253562625 T^{8} - 11058842659 T^{10} + 347127233625 T^{12} - 7744996570754 T^{14} + 117630858673423 T^{16} - 1099406069077273 T^{18} + 4808584372417849 T^{20} \))(\( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( ( 1 - T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))(\( 1 - 3 T - 19 T^{2} + 138 T^{3} - 918 T^{4} + 5106 T^{5} - 26011 T^{6} - 151959 T^{7} + 1874161 T^{8} \))(\( 1 + 2 T - 47 T^{2} - 46 T^{3} + 1084 T^{4} - 1702 T^{5} - 64343 T^{6} + 101306 T^{7} + 1874161 T^{8} \))(\( 1 + 10 T + 105 T^{2} + 726 T^{3} + 4090 T^{4} + 19330 T^{5} + 114025 T^{6} + 715210 T^{7} + 5599210 T^{8} + 36774078 T^{9} + 196786905 T^{10} + 693439570 T^{11} + 2565726409 T^{12} \))(\( 1 - T - 45 T^{2} - 284 T^{3} + 533 T^{4} + 7509 T^{5} + 35806 T^{6} + 277833 T^{7} + 729677 T^{8} - 14385452 T^{9} - 84337245 T^{10} - 69343957 T^{11} + 2565726409 T^{12} \))(\( 1 - 2 T - 101 T^{2} + 210 T^{3} + 5537 T^{4} - 9548 T^{5} - 216090 T^{6} + 178624 T^{7} + 7490762 T^{8} + 6609088 T^{9} - 295827210 T^{10} - 483634844 T^{11} + 10377229457 T^{12} + 14562230970 T^{13} - 259138367309 T^{14} - 189863754266 T^{15} + 3512479453921 T^{16} \))(\( 1 - 2 T - 167 T^{2} + 214 T^{3} + 15106 T^{4} - 11398 T^{5} - 1004433 T^{6} + 492074 T^{7} + 54283648 T^{8} - 15920658 T^{9} - 2515349755 T^{10} + 229313142 T^{11} + 100683594736 T^{12} + 8484586254 T^{13} - 3443513814595 T^{14} - 806429089674 T^{15} + 101736296019328 T^{16} + 34122358296818 T^{17} - 2577100274171097 T^{18} - 1082033535561934 T^{19} + 53059514630930626 T^{20} + 27811812316146478 T^{21} - 803033590193780783 T^{22} - 355835243558920826 T^{23} + 6582952005840035281 T^{24} \))(\( ( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} )( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} ) \))(\( 1 + 171 T^{2} + 15667 T^{4} + 1047668 T^{6} + 56787617 T^{8} + 2608311361 T^{10} + 103527750422 T^{12} + 3570778253209 T^{14} + 106429137064337 T^{16} + 2688029455464212 T^{18} + 55030015604580307 T^{20} + 822267927683452179 T^{22} + 6582952005840035281 T^{24} \))
$41$ (\( 1 + 12 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 12 T + 100 T^{2} + 492 T^{3} + 1681 T^{4} \))(\( 1 - 16 T^{2} + 1681 T^{4} \))(\( 1 - 12 T + 113 T^{2} - 646 T^{3} + 4633 T^{4} - 20172 T^{5} + 68921 T^{6} \))(\( 1 - 12 T + 129 T^{2} - 782 T^{3} + 5289 T^{4} - 20172 T^{5} + 68921 T^{6} \))(\( 1 + 10 T + 131 T^{2} + 738 T^{3} + 5371 T^{4} + 16810 T^{5} + 68921 T^{6} \))(\( 1 - 12 T + 124 T^{2} - 702 T^{3} + 4810 T^{4} - 28782 T^{5} + 208444 T^{6} - 827052 T^{7} + 2825761 T^{8} \))(\( ( 1 + 14 T + 191 T^{2} + 1564 T^{3} + 13928 T^{4} + 86898 T^{5} + 571048 T^{6} + 2629084 T^{7} + 13163911 T^{8} + 39560654 T^{9} + 115856201 T^{10} )^{2} \))(\( 1 + 10 T + 59 T^{2} + 410 T^{3} + 1681 T^{4} \))(\( 1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4} \))(\( 1 - 7 T + 8 T^{2} - 287 T^{3} + 1681 T^{4} \))(\( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \))(\( 1 + 5 T - 15 T^{2} - 210 T^{3} - 1106 T^{4} - 8610 T^{5} - 25215 T^{6} + 344605 T^{7} + 2825761 T^{8} \))(\( 1 - 14 T + 89 T^{2} - 350 T^{3} + 1732 T^{4} - 14350 T^{5} + 149609 T^{6} - 964894 T^{7} + 2825761 T^{8} \))(\( 1 + 10 T - 47 T^{2} - 194 T^{3} + 7262 T^{4} + 18574 T^{5} - 224003 T^{6} + 761534 T^{7} + 12207422 T^{8} - 13370674 T^{9} - 132810767 T^{10} + 1158562010 T^{11} + 4750104241 T^{12} \))(\( 1 - 3 T - 89 T^{2} + 112 T^{3} + 4977 T^{4} - 829 T^{5} - 229842 T^{6} - 33989 T^{7} + 8366337 T^{8} + 7719152 T^{9} - 251492729 T^{10} - 347568603 T^{11} + 4750104241 T^{12} \))(\( 1 + 10 T - 9 T^{2} - 442 T^{3} - 963 T^{4} + 11220 T^{5} + 74098 T^{6} - 354064 T^{7} - 6196662 T^{8} - 14516624 T^{9} + 124558738 T^{10} + 773293620 T^{11} - 2721207843 T^{12} - 51208440842 T^{13} - 42750938169 T^{14} + 1947542738810 T^{15} + 7984925229121 T^{16} \))(\( 1 - 15 T + 10 T^{2} + 1143 T^{3} - 7544 T^{4} + 3513 T^{5} + 117690 T^{6} - 1404453 T^{7} + 17153184 T^{8} - 62509119 T^{9} - 393210722 T^{10} + 3101818011 T^{11} - 9679576502 T^{12} + 127174538451 T^{13} - 660987223682 T^{14} - 4308190990599 T^{15} + 48470798373024 T^{16} - 162714589063053 T^{17} + 559039768123290 T^{18} + 684171764143953 T^{19} - 60238275928488824 T^{20} + 374197551012297423 T^{21} + 134226593101524010 T^{22} - 8254935475743726615 T^{23} + 22563490300366186081 T^{24} \))(\( ( 1 + 9 T + 40 T^{2} + 369 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 19 T + 127 T^{2} - 960 T^{3} + 12045 T^{4} - 75229 T^{5} + 324054 T^{6} - 3084389 T^{7} + 20247645 T^{8} - 66164160 T^{9} + 358871647 T^{10} - 2201267819 T^{11} + 4750104241 T^{12} )^{2} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 2 T + 43 T^{2} \))(\( 1 - 12 T + 104 T^{2} - 516 T^{3} + 1849 T^{4} \))(\( 1 + 16 T + 132 T^{2} + 688 T^{3} + 1849 T^{4} \))(\( 1 + 6 T + 3 T^{2} - 290 T^{3} + 129 T^{4} + 11094 T^{5} + 79507 T^{6} \))(\( 1 - 6 T + 75 T^{2} - 394 T^{3} + 3225 T^{4} - 11094 T^{5} + 79507 T^{6} \))(\( 1 - 4 T + 109 T^{2} - 278 T^{3} + 4687 T^{4} - 7396 T^{5} + 79507 T^{6} \))(\( 1 + 16 T + 236 T^{2} + 2034 T^{3} + 16350 T^{4} + 87462 T^{5} + 436364 T^{6} + 1272112 T^{7} + 3418801 T^{8} \))(\( 1 - 354 T^{2} + 58877 T^{4} - 6058604 T^{6} + 427212514 T^{8} - 21571299984 T^{10} + 789915938386 T^{12} - 20713161413804 T^{14} + 372182892235973 T^{16} - 4137622898270754 T^{18} + 21611482313284249 T^{20} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 + 12 T + 43 T^{2} )^{2} \))(\( ( 1 - 7 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 9 T + 58 T^{2} + 387 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{4} \))(\( ( 1 - T + 124 T^{2} - 83 T^{3} + 5332 T^{4} - 1849 T^{5} + 79507 T^{6} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{6} \))(\( ( 1 - 6 T + 107 T^{2} - 822 T^{3} + 5640 T^{4} - 35346 T^{5} + 197843 T^{6} - 477042 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 - 2 T + 124 T^{2} - 178 T^{3} + 9494 T^{4} - 11492 T^{5} + 473530 T^{6} - 494156 T^{7} + 17554406 T^{8} - 14152246 T^{9} + 423931324 T^{10} - 294016886 T^{11} + 6321363049 T^{12} )^{2} \))(\( ( 1 - 37 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 82 T^{2} + 7063 T^{4} - 303196 T^{6} + 13059487 T^{8} - 280341682 T^{10} + 6321363049 T^{12} )^{2} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - 2 T + 47 T^{2} \))(\( 1 + 8 T + 108 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 + 4 T + 80 T^{2} + 188 T^{3} + 2209 T^{4} \))(\( 1 - 8 T + 125 T^{2} - 622 T^{3} + 5875 T^{4} - 17672 T^{5} + 103823 T^{6} \))(\( 1 + 33 T^{2} + 54 T^{3} + 1551 T^{4} + 103823 T^{6} \))(\( 1 + 2 T + 47 T^{2} + 446 T^{3} + 2209 T^{4} + 4418 T^{5} + 103823 T^{6} \))(\( 1 - 10 T + 78 T^{2} - 612 T^{3} + 5978 T^{4} - 28764 T^{5} + 172302 T^{6} - 1038230 T^{7} + 4879681 T^{8} \))(\( 1 - 262 T^{2} + 33957 T^{4} - 2943524 T^{6} + 192523542 T^{8} - 10045496272 T^{10} + 425284504278 T^{12} - 14363458135844 T^{14} + 366029814926853 T^{16} - 6238557105381382 T^{18} + 52599132235830049 T^{20} \))(\( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 7 T + 2 T^{2} - 329 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( ( 1 - 47 T^{2} + 2209 T^{4} )^{2} \))(\( 1 - 40 T^{2} - 609 T^{4} - 88360 T^{6} + 4879681 T^{8} \))(\( ( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} )^{3} \))(\( 1 - 28 T + 389 T^{2} - 4300 T^{3} + 42820 T^{4} - 357312 T^{5} + 2567015 T^{6} - 16793664 T^{7} + 94589380 T^{8} - 446438900 T^{9} + 1898195909 T^{10} - 6421660196 T^{11} + 10779215329 T^{12} \))(\( 1 - 106 T^{2} - 160 T^{3} + 5466 T^{4} + 12240 T^{5} - 136912 T^{6} - 335520 T^{7} + 3292095 T^{8} - 15769440 T^{9} - 302438608 T^{10} + 1270793520 T^{11} + 26672336346 T^{12} - 36695201120 T^{13} - 1142596824874 T^{14} + 23811286661761 T^{16} \))(\( 1 + 11 T + 21 T^{2} + 238 T^{3} + 2416 T^{4} - 1266 T^{5} + 27593 T^{6} + 237409 T^{7} - 3540598 T^{8} - 23835901 T^{9} - 225483611 T^{10} - 2925072706 T^{11} - 23217991888 T^{12} - 137478417182 T^{13} - 498093296699 T^{14} - 2474714749523 T^{15} - 17276988789238 T^{16} + 54448568766863 T^{17} + 297430888573097 T^{18} - 641384870506158 T^{19} + 57528068574814576 T^{20} + 266353052598458546 T^{21} + 1104581776952431029 T^{22} + 27193751365924135333 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8} \))(\( 1 + 198 T^{2} + 19869 T^{4} + 1555434 T^{6} + 106114506 T^{8} + 6025836978 T^{10} + 295951998413 T^{12} + 13311073884402 T^{14} + 517804938752586 T^{16} + 16766358016047786 T^{18} + 473106454682529309 T^{20} + 10414628182694349702 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 + 4 T + 108 T^{2} + 212 T^{3} + 2809 T^{4} \))(\( 1 + 16 T + 168 T^{2} + 848 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 87 T^{2} - 354 T^{3} + 4611 T^{4} - 33708 T^{5} + 148877 T^{6} \))(\( 1 + 147 T^{2} - 2 T^{3} + 7791 T^{4} + 148877 T^{6} \))(\( 1 + 2 T + 101 T^{2} + 230 T^{3} + 5353 T^{4} + 5618 T^{5} + 148877 T^{6} \))(\( 1 - 8 T + 102 T^{2} - 810 T^{3} + 8870 T^{4} - 42930 T^{5} + 286518 T^{6} - 1191016 T^{7} + 7890481 T^{8} \))(\( 1 - 170 T^{2} + 12485 T^{4} - 655208 T^{6} + 35704230 T^{8} - 1969785216 T^{10} + 100293182070 T^{12} - 5169906275048 T^{14} + 276722048695565 T^{16} - 10584147369931370 T^{18} + 174887470365513049 T^{20} \))(\( ( 1 + 2 T + 53 T^{2} )^{2} \))(\( ( 1 + 9 T + 53 T^{2} )^{2} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( ( 1 + 3 T + 53 T^{2} )^{2} \))(\( ( 1 - 5 T + 64 T^{2} - 265 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 7 T + 53 T^{2} )^{4} \))(\( ( 1 - 12 T + 189 T^{2} - 1290 T^{3} + 10017 T^{4} - 33708 T^{5} + 148877 T^{6} )^{2} \))(\( ( 1 - 3 T + 6 T^{2} - 291 T^{3} + 318 T^{4} - 8427 T^{5} + 148877 T^{6} )^{2} \))(\( ( 1 + 6 T + 155 T^{2} + 846 T^{3} + 10954 T^{4} + 44838 T^{5} + 435395 T^{6} + 893262 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 + 26 T + 463 T^{2} + 6128 T^{3} + 66707 T^{4} + 609960 T^{5} + 4803618 T^{6} + 32327880 T^{7} + 187379963 T^{8} + 912318256 T^{9} + 3653292703 T^{10} + 10873082818 T^{11} + 22164361129 T^{12} )^{2} \))(\( ( 1 - 102 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 283 T^{2} + 35030 T^{4} - 2420679 T^{6} + 98399270 T^{8} - 2233006123 T^{10} + 22164361129 T^{12} )^{2} \))
$59$ (\( 1 + 12 T + 59 T^{2} \))(\( 1 - 14 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 12 T + 136 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 + 20 T^{2} + 3481 T^{4} \))(\( 1 + 4 T + 23 T^{2} + 482 T^{3} + 1357 T^{4} + 13924 T^{5} + 205379 T^{6} \))(\( 1 + 12 T + 183 T^{2} + 1214 T^{3} + 10797 T^{4} + 41772 T^{5} + 205379 T^{6} \))(\( 1 - 6 T + 145 T^{2} - 622 T^{3} + 8555 T^{4} - 20886 T^{5} + 205379 T^{6} \))(\( 1 - 18 T + 200 T^{2} - 1280 T^{3} + 8822 T^{4} - 75520 T^{5} + 696200 T^{6} - 3696822 T^{7} + 12117361 T^{8} \))(\( ( 1 - 14 T + 257 T^{2} - 2624 T^{3} + 26124 T^{4} - 210486 T^{5} + 1541316 T^{6} - 9134144 T^{7} + 52782403 T^{8} - 169643054 T^{9} + 714924299 T^{10} )^{2} \))(\( ( 1 + 9 T + 59 T^{2} )^{2} \))(\( ( 1 + 10 T + 59 T^{2} )^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 6 T + 59 T^{2} )^{2} \))(\( ( 1 - 13 T + 112 T^{2} - 767 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 4 T + 98 T^{2} + 236 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + T + 96 T^{2} + 361 T^{3} + 5664 T^{4} + 3481 T^{5} + 205379 T^{6} )^{2} \))(\( ( 1 - 2 T + 149 T^{2} - 244 T^{3} + 8791 T^{4} - 6962 T^{5} + 205379 T^{6} )^{2} \))(\( ( 1 + 2 T + 209 T^{2} + 326 T^{3} + 17888 T^{4} + 19234 T^{5} + 727529 T^{6} + 410758 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 - 27 T + 477 T^{2} - 6482 T^{3} + 73823 T^{4} - 703863 T^{5} + 5830086 T^{6} - 41527917 T^{7} + 256977863 T^{8} - 1331266678 T^{9} + 5779981197 T^{10} - 19302956073 T^{11} + 42180533641 T^{12} )^{2} \))(\( ( 1 - 14 T + 59 T^{2} )^{4} \))(\( ( 1 + 18 T + 269 T^{2} + 2228 T^{3} + 15871 T^{4} + 62658 T^{5} + 205379 T^{6} )^{4} \))
$61$ (\( 1 - 11 T + 61 T^{2} \))(\( 1 + 15 T + 61 T^{2} \))(\( 1 + 3 T + 61 T^{2} \))(\( 1 + 13 T + 61 T^{2} \))(\( 1 - 2 T - 39 T^{2} - 122 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 145 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 - 3 T + 86 T^{2} - 17 T^{3} + 5246 T^{4} - 11163 T^{5} + 226981 T^{6} \))(\( 1 + 3 T + 138 T^{2} + 433 T^{3} + 8418 T^{4} + 11163 T^{5} + 226981 T^{6} \))(\( 1 + 19 T + 296 T^{2} + 2519 T^{3} + 18056 T^{4} + 70699 T^{5} + 226981 T^{6} \))(\( 1 + 11 T + 139 T^{2} + 918 T^{3} + 9230 T^{4} + 55998 T^{5} + 517219 T^{6} + 2496791 T^{7} + 13845841 T^{8} \))(\( ( 1 + 17 T + 343 T^{2} + 3856 T^{3} + 43731 T^{4} + 344465 T^{5} + 2667591 T^{6} + 14348176 T^{7} + 77854483 T^{8} + 235379297 T^{9} + 844596301 T^{10} )^{2} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( ( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2} \))(\( 1 + 4 T - 104 T^{2} - 8 T^{3} + 9703 T^{4} - 488 T^{5} - 386984 T^{6} + 907924 T^{7} + 13845841 T^{8} \))(\( 1 - 14 T - 15 T^{2} + 414 T^{3} + 8074 T^{4} - 47030 T^{5} - 155279 T^{6} - 2868830 T^{7} + 30043354 T^{8} + 93970134 T^{9} - 207687615 T^{10} - 11824348214 T^{11} + 51520374361 T^{12} \))(\( 1 + 12 T + 67 T^{2} + 548 T^{3} - 1024 T^{4} - 53600 T^{5} - 408187 T^{6} - 3269600 T^{7} - 3810304 T^{8} + 124385588 T^{9} + 927671347 T^{10} + 10135155612 T^{11} + 51520374361 T^{12} \))(\( 1 - 106 T^{2} + 288 T^{3} + 5634 T^{4} - 24048 T^{5} + 215776 T^{6} + 1055520 T^{7} - 25537489 T^{8} + 64386720 T^{9} + 802902496 T^{10} - 5458439088 T^{11} + 78007468194 T^{12} + 243243734688 T^{13} - 5461159682266 T^{14} + 191707312997281 T^{16} \))(\( 1 + 6 T - 96 T^{2} - 1652 T^{3} - 2003 T^{4} + 93280 T^{5} + 1366160 T^{6} + 4921878 T^{7} - 64022702 T^{8} - 1092093406 T^{9} - 3096375760 T^{10} + 35452199832 T^{11} + 535442747805 T^{12} + 2162584189752 T^{13} - 11521614202960 T^{14} - 247884453387286 T^{15} - 886448152282382 T^{16} + 4156999952773278 T^{17} + 70385074637023760 T^{18} + 293155051744038880 T^{19} - 383989747933553843 T^{20} - 19318729345362000932 T^{21} - 68480919519636729696 T^{22} + \)\(26\!\cdots\!66\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( ( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 2 T - 17 T^{2} + 178 T^{3} - 1060 T^{4} - 4628 T^{5} + 424277 T^{6} - 282308 T^{7} - 3944260 T^{8} + 40402618 T^{9} - 235379297 T^{10} - 1689192602 T^{11} + 51520374361 T^{12} )^{2} \))
$67$ (\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T^{2} )^{5} \))(\( 1 + 16 T + 67 T^{2} \))(\( 1 - 5 T + 67 T^{2} \))(\( 1 - 11 T + 67 T^{2} \))(\( 1 - 5 T + 67 T^{2} \))(\( 1 + 15 T + 142 T^{2} + 1005 T^{3} + 4489 T^{4} \))(\( 1 - 14 T + 129 T^{2} - 938 T^{3} + 4489 T^{4} \))(\( 1 + 18 T + 225 T^{2} + 1960 T^{3} + 15075 T^{4} + 80802 T^{5} + 300763 T^{6} \))(\( 1 + 15 T - 30 T^{2} - 1447 T^{3} - 2010 T^{4} + 67335 T^{5} + 300763 T^{6} \))(\( 1 + 30 T + 455 T^{2} + 4650 T^{3} + 39636 T^{4} + 311550 T^{5} + 2042495 T^{6} + 9022890 T^{7} + 20151121 T^{8} \))(\( 1 + 5 T - 175 T^{2} - 600 T^{3} + 18649 T^{4} + 24883 T^{5} - 1350818 T^{6} + 1667161 T^{7} + 83715361 T^{8} - 180457800 T^{9} - 3526446175 T^{10} + 6750625535 T^{11} + 90458382169 T^{12} \))(\( 1 - 13 T^{2} + 4489 T^{4} \))(\( 1 - 123 T^{2} + 8382 T^{4} - 459619 T^{6} + 37626798 T^{8} - 2478587883 T^{10} + 90458382169 T^{12} \))
$71$ (\( 1 + 9 T + 71 T^{2} \))(\( 1 - 5 T + 71 T^{2} \))(\( 1 + 9 T + 71 T^{2} \))(\( 1 + 15 T + 71 T^{2} \))(\( 1 + 18 T + 173 T^{2} + 1278 T^{3} + 5041 T^{4} \))(\( 1 - 10 T + 149 T^{2} - 710 T^{3} + 5041 T^{4} \))(\( 1 - 13 T + 176 T^{2} - 1859 T^{3} + 12496 T^{4} - 65533 T^{5} + 357911 T^{6} \))(\( 1 - 9 T + 228 T^{2} - 1271 T^{3} + 16188 T^{4} - 45369 T^{5} + 357911 T^{6} \))(\( 1 + T + 118 T^{2} - 211 T^{3} + 8378 T^{4} + 5041 T^{5} + 357911 T^{6} \))(\( 1 - 27 T + 405 T^{2} - 4664 T^{3} + 44204 T^{4} - 331144 T^{5} + 2041605 T^{6} - 9663597 T^{7} + 25411681 T^{8} \))(\( ( 1 - 21 T + 419 T^{2} - 5154 T^{3} + 59679 T^{4} - 516319 T^{5} + 4237209 T^{6} - 25981314 T^{7} + 149964709 T^{8} - 533645301 T^{9} + 1804229351 T^{10} )^{2} \))(\( 1 + 6 T - 35 T^{2} + 426 T^{3} + 5041 T^{4} \))(\( 1 - T - 70 T^{2} - 71 T^{3} + 5041 T^{4} \))(\( 1 - 71 T^{2} + 5041 T^{4} \))(\( 1 - 15 T + 154 T^{2} - 1065 T^{3} + 5041 T^{4} \))(\( 1 + 5 T - 75 T^{2} - 210 T^{3} + 3184 T^{4} - 14910 T^{5} - 378075 T^{6} + 1789555 T^{7} + 25411681 T^{8} \))(\( 1 - 14 T + 59 T^{2} + 70 T^{3} - 188 T^{4} + 4970 T^{5} + 297419 T^{6} - 5010754 T^{7} + 25411681 T^{8} \))(\( 1 - 22 T + 115 T^{2} - 1162 T^{3} + 33446 T^{4} - 241528 T^{5} + 705031 T^{6} - 17148488 T^{7} + 168601286 T^{8} - 415892582 T^{9} + 2922343315 T^{10} - 39693045722 T^{11} + 128100283921 T^{12} \))(\( 1 + 13 T + 11 T^{2} - 1376 T^{3} - 8549 T^{4} + 44835 T^{5} + 1032674 T^{6} + 3183285 T^{7} - 43095509 T^{8} - 492485536 T^{9} + 279528491 T^{10} + 23454981563 T^{11} + 128100283921 T^{12} \))(\( 1 - 2 T - 249 T^{2} + 314 T^{3} + 37203 T^{4} - 28896 T^{5} - 3913832 T^{6} + 808832 T^{7} + 318119334 T^{8} + 57427072 T^{9} - 19729627112 T^{10} - 10342196256 T^{11} + 945390768243 T^{12} + 566528016214 T^{13} - 31896970696329 T^{14} - 18190240316782 T^{15} + 645753531245761 T^{16} \))(\( 1 + 6 T - 305 T^{2} - 910 T^{3} + 57340 T^{4} + 65544 T^{5} - 7089129 T^{6} + 1095488 T^{7} + 656935218 T^{8} - 448771246 T^{9} - 50069217633 T^{10} + 18113601166 T^{11} + 3541402908076 T^{12} + 1286065682786 T^{13} - 252398926087953 T^{14} - 160620165427106 T^{15} + 16693828197481458 T^{16} + 1976511603268288 T^{17} - 908119437652594809 T^{18} + 596130555661579704 T^{19} + 37027507481631935740 T^{20} - 41722135653788618210 T^{21} - \)\(99\!\cdots\!05\)\( T^{22} + \)\(13\!\cdots\!26\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( ( 1 - 8 T - 7 T^{2} - 568 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 3 T - 173 T^{2} + 116 T^{3} + 19209 T^{4} + 2167 T^{5} - 1571790 T^{6} + 153857 T^{7} + 96832569 T^{8} + 41517676 T^{9} - 4396220813 T^{10} - 5412688053 T^{11} + 128100283921 T^{12} )^{2} \))
$73$ (\( 1 + 10 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 8 T + 73 T^{2} \))(\( 1 - 8 T + 112 T^{2} - 584 T^{3} + 5329 T^{4} \))(\( 1 - 4 T + 100 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 - 16 T + 245 T^{2} - 1998 T^{3} + 17885 T^{4} - 85264 T^{5} + 389017 T^{6} \))(\( 1 + 24 T + 405 T^{2} + 3966 T^{3} + 29565 T^{4} + 127896 T^{5} + 389017 T^{6} \))(\( 1 + 14 T + 143 T^{2} + 1178 T^{3} + 10439 T^{4} + 74606 T^{5} + 389017 T^{6} \))(\( 1 - 4 T + 208 T^{2} - 458 T^{3} + 19258 T^{4} - 33434 T^{5} + 1108432 T^{6} - 1556068 T^{7} + 28398241 T^{8} \))(\( 1 - 562 T^{2} + 150869 T^{4} - 25463572 T^{6} + 2988190594 T^{8} - 254492780640 T^{10} + 15924067675426 T^{12} - 723120654376852 T^{14} + 22831643385995141 T^{16} - 453230571644473522 T^{18} + 4297625829703557649 T^{20} \))(\( 1 + 13 T + 96 T^{2} + 949 T^{3} + 5329 T^{4} \))(\( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 + 17 T + 119 T^{2} + 408 T^{3} + 2474 T^{4} + 29784 T^{5} + 634151 T^{6} + 6613289 T^{7} + 28398241 T^{8} \))(\( 1 - 12 T - 32 T^{2} - 360 T^{3} + 15135 T^{4} - 26280 T^{5} - 170528 T^{6} - 4668204 T^{7} + 28398241 T^{8} \))(\( 1 + 3 T - 195 T^{2} - 178 T^{3} + 25539 T^{4} + 7215 T^{5} - 2151870 T^{6} + 526695 T^{7} + 136097331 T^{8} - 69245026 T^{9} - 5537656995 T^{10} + 6219214779 T^{11} + 151334226289 T^{12} \))(\( 1 - 8 T - 121 T^{2} + 320 T^{3} + 13520 T^{4} + 6948 T^{5} - 1251447 T^{6} + 507204 T^{7} + 72048080 T^{8} + 124485440 T^{9} - 3436187161 T^{10} - 16584572744 T^{11} + 151334226289 T^{12} \))(\( 1 + 6 T - 141 T^{2} - 982 T^{3} + 8907 T^{4} + 60020 T^{5} - 400016 T^{6} - 1768644 T^{7} + 21775510 T^{8} - 129111012 T^{9} - 2131685264 T^{10} + 23348800340 T^{11} + 252943132587 T^{12} - 2035756304326 T^{13} - 21338125906749 T^{14} + 66284391114582 T^{15} + 806460091894081 T^{16} \))(\( 1 + 15 T - 106 T^{2} - 1163 T^{3} + 21544 T^{4} + 60457 T^{5} - 2079088 T^{6} + 511371 T^{7} + 104840020 T^{8} - 300419013 T^{9} - 1687664590 T^{10} + 28478406421 T^{11} + 187295331994 T^{12} + 2078923668733 T^{13} - 8993564600110 T^{14} - 116868103180221 T^{15} + 2977272154404820 T^{16} + 1060108693584003 T^{17} - 314637173866744432 T^{18} + 667892572269047329 T^{19} + 17374376219766081064 T^{20} - 68467655341715582819 T^{21} - \)\(45\!\cdots\!94\)\( T^{22} + \)\(47\!\cdots\!55\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 + 130 T^{2} + 11571 T^{4} + 692770 T^{6} + 28398241 T^{8} \))(\( 1 + 386 T^{2} + 83553 T^{4} + 12827782 T^{6} + 1534429814 T^{8} + 148563481686 T^{10} + 11894691241285 T^{12} + 791694793904694 T^{14} + 43575107655557174 T^{16} + 1941282463973960998 T^{18} + 67382160058026149793 T^{20} + \)\(16\!\cdots\!14\)\( T^{22} + \)\(22\!\cdots\!21\)\( T^{24} \))
$79$ (\( 1 - 2 T + 79 T^{2} \))(\( 1 - 14 T + 79 T^{2} \))(\( 1 - 6 T + 79 T^{2} \))(\( 1 + 14 T + 79 T^{2} \))(\( ( 1 + 10 T + 79 T^{2} )^{2} \))(\( 1 - 4 T + 130 T^{2} - 316 T^{3} + 6241 T^{4} \))(\( 1 + 12 T + 141 T^{2} + 1816 T^{3} + 11139 T^{4} + 74892 T^{5} + 493039 T^{6} \))(\( 1 + 189 T^{2} + 16 T^{3} + 14931 T^{4} + 493039 T^{6} \))(\( 1 - 8 T + 189 T^{2} - 976 T^{3} + 14931 T^{4} - 49928 T^{5} + 493039 T^{6} \))(\( 1 + 20 T + 380 T^{2} + 4388 T^{3} + 47814 T^{4} + 346652 T^{5} + 2371580 T^{6} + 9860780 T^{7} + 38950081 T^{8} \))(\( ( 1 - 16 T + 375 T^{2} - 3532 T^{3} + 49326 T^{4} - 346096 T^{5} + 3896754 T^{6} - 22043212 T^{7} + 184889625 T^{8} - 623201296 T^{9} + 3077056399 T^{10} )^{2} \))(\( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 6 T - 43 T^{2} - 474 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( ( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} )^{2} \))(\( 1 - 12 T - 45 T^{2} + 1460 T^{3} - 1578 T^{4} - 68844 T^{5} + 739911 T^{6} - 5438676 T^{7} - 9848298 T^{8} + 719836940 T^{9} - 1752753645 T^{10} - 36924676788 T^{11} + 243087455521 T^{12} \))(\( 1 - 81 T^{2} - 256 T^{3} + 162 T^{4} + 10368 T^{5} + 484143 T^{6} + 819072 T^{7} + 1011042 T^{8} - 126217984 T^{9} - 3154956561 T^{10} + 243087455521 T^{12} \))(\( 1 + 4 T - 128 T^{2} - 1128 T^{3} + 6482 T^{4} + 103228 T^{5} + 692256 T^{6} - 4576940 T^{7} - 92296909 T^{8} - 361578260 T^{9} + 4320369696 T^{10} + 50895429892 T^{11} + 252474425042 T^{12} - 3470919618072 T^{13} - 31115194306688 T^{14} + 76815635944636 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 2 T - 242 T^{2} + 112 T^{3} + 26953 T^{4} - 57548 T^{5} - 1783510 T^{6} + 957694 T^{7} + 110363922 T^{8} + 507417094 T^{9} - 13686678458 T^{10} - 27884490580 T^{11} + 1457355588761 T^{12} - 2202874755820 T^{13} - 85418560256378 T^{14} + 250176416608666 T^{15} + 4298683701377682 T^{16} + 2946878450983906 T^{17} - 433548907796258710 T^{18} - 1105146554335478132 T^{19} + 40890633753411538633 T^{20} + 13423378750053251728 T^{21} - \)\(22\!\cdots\!42\)\( T^{22} + \)\(14\!\cdots\!58\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( ( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 22 T + 119 T^{2} + 926 T^{3} + 26246 T^{4} + 165042 T^{5} + 57363 T^{6} + 13038318 T^{7} + 163801286 T^{8} + 456554114 T^{9} + 4635059639 T^{10} + 67695240778 T^{11} + 243087455521 T^{12} )^{2} \))
$83$ (\( 1 + 3 T + 83 T^{2} \))(\( 1 + 7 T + 83 T^{2} \))(\( 1 - 5 T + 83 T^{2} \))(\( 1 - 9 T + 83 T^{2} \))(\( 1 - 18 T + 239 T^{2} - 1494 T^{3} + 6889 T^{4} \))(\( 1 - 2 T + 95 T^{2} - 166 T^{3} + 6889 T^{4} \))(\( 1 - 31 T + 536 T^{2} - 5831 T^{3} + 44488 T^{4} - 213559 T^{5} + 571787 T^{6} \))(\( 1 + 9 T + 192 T^{2} + 1465 T^{3} + 15936 T^{4} + 62001 T^{5} + 571787 T^{6} \))(\( 1 - 5 T + 240 T^{2} - 821 T^{3} + 19920 T^{4} - 34445 T^{5} + 571787 T^{6} \))(\( 1 - 9 T + 223 T^{2} - 2340 T^{3} + 23724 T^{4} - 194220 T^{5} + 1536247 T^{6} - 5146083 T^{7} + 47458321 T^{8} \))(\( 1 + 43 T^{2} + 14155 T^{4} + 472430 T^{6} + 146527317 T^{8} + 6307431145 T^{10} + 1009426686813 T^{12} + 22420734590030 T^{14} + 4627840985038195 T^{16} + 96848565981978763 T^{18} + 15516041187205853449 T^{20} \))(\( 1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4} \))(\( 1 + 5 T - 58 T^{2} + 415 T^{3} + 6889 T^{4} \))(\( 1 - 13 T + 86 T^{2} - 1079 T^{3} + 6889 T^{4} \))(\( 1 - 15 T + 142 T^{2} - 1245 T^{3} + 6889 T^{4} \))(\( 1 - 7 T - 81 T^{2} + 252 T^{3} + 5944 T^{4} + 20916 T^{5} - 558009 T^{6} - 4002509 T^{7} + 47458321 T^{8} \))(\( 1 + 10 T - 37 T^{2} - 290 T^{3} + 3580 T^{4} - 24070 T^{5} - 254893 T^{6} + 5717870 T^{7} + 47458321 T^{8} \))(\( 1 - 10 T - 101 T^{2} + 1382 T^{3} + 6758 T^{4} - 76126 T^{5} - 22193 T^{6} - 6318458 T^{7} + 46555862 T^{8} + 790209634 T^{9} - 4793290421 T^{10} - 39390406430 T^{11} + 326940373369 T^{12} \))(\( 1 + 13 T + 47 T^{2} - 452 T^{3} - 8489 T^{4} - 49401 T^{5} - 217678 T^{6} - 4100283 T^{7} - 58480721 T^{8} - 258447724 T^{9} + 2230541087 T^{10} + 51207528359 T^{11} + 326940373369 T^{12} \))(\( 1 + 14 T - 59 T^{2} - 18 T^{3} + 16865 T^{4} - 38860 T^{5} - 1229022 T^{6} - 2000336 T^{7} - 7379294 T^{8} - 166027888 T^{9} - 8466732558 T^{10} - 22219642820 T^{11} + 800384583665 T^{12} - 70902731574 T^{13} - 19289482028771 T^{14} + 379904713854778 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 15 T - 144 T^{2} - 2385 T^{3} + 15192 T^{4} + 158043 T^{5} - 2280884 T^{6} - 15564285 T^{7} + 184361076 T^{8} + 834889563 T^{9} - 18985402836 T^{10} + 2183972751 T^{11} + 2269595958342 T^{12} + 181269738333 T^{13} - 130790440137204 T^{14} + 477378998559081 T^{15} + 8749467124713396 T^{16} - 61308351194235255 T^{17} - 745713066571378196 T^{18} + 4288662906553619961 T^{19} + 34216823590656310872 T^{20} - \)\(44\!\cdots\!55\)\( T^{21} - \)\(22\!\cdots\!56\)\( T^{22} + \)\(19\!\cdots\!05\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 + 157 T^{2} + 17760 T^{4} + 1081573 T^{6} + 47458321 T^{8} \))(\( 1 + 351 T^{2} + 66199 T^{4} + 8468096 T^{6} + 831340373 T^{8} + 69882063121 T^{10} + 5713663080470 T^{12} + 481417532840569 T^{14} + 39454018282093733 T^{16} + 2768562467964535424 T^{18} + \)\(14\!\cdots\!59\)\( T^{20} + \)\(54\!\cdots\!99\)\( T^{22} + \)\(10\!\cdots\!61\)\( T^{24} \))
$89$ (\( 1 + 3 T + 89 T^{2} \))(\( 1 + 15 T + 89 T^{2} \))(\( 1 - 13 T + 89 T^{2} \))(\( 1 + 15 T + 89 T^{2} \))(\( 1 + 6 T + 115 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 14 T + 219 T^{2} + 1246 T^{3} + 7921 T^{4} \))(\( 1 - 11 T + 270 T^{2} - 1791 T^{3} + 24030 T^{4} - 87131 T^{5} + 704969 T^{6} \))(\( 1 + 21 T + 366 T^{2} + 3649 T^{3} + 32574 T^{4} + 166341 T^{5} + 704969 T^{6} \))(\( 1 + T + 182 T^{2} + 325 T^{3} + 16198 T^{4} + 7921 T^{5} + 704969 T^{6} \))(\( 1 - 17 T + 437 T^{2} - 4606 T^{3} + 61882 T^{4} - 409934 T^{5} + 3461477 T^{6} - 11984473 T^{7} + 62742241 T^{8} \))(\( ( 1 - 11 T + 303 T^{2} - 3546 T^{3} + 46941 T^{4} - 445557 T^{5} + 4177749 T^{6} - 28087866 T^{7} + 213605607 T^{8} - 690164651 T^{9} + 5584059449 T^{10} )^{2} \))(\( ( 1 - 15 T + 89 T^{2} )^{2} \))(\( ( 1 + 15 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 9 T + 89 T^{2} )^{2} \))(\( ( 1 + 4 T - 11 T^{2} + 356 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 14 T + 203 T^{2} + 1246 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 3 T + 198 T^{2} + 255 T^{3} + 17622 T^{4} + 23763 T^{5} + 704969 T^{6} )^{2} \))(\( ( 1 + 25 T + 414 T^{2} + 4341 T^{3} + 36846 T^{4} + 198025 T^{5} + 704969 T^{6} )^{2} \))(\( ( 1 + 24 T + 458 T^{2} + 5904 T^{3} + 65995 T^{4} + 525456 T^{5} + 3627818 T^{6} + 16919256 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 7 T + 474 T^{2} + 2757 T^{3} + 98380 T^{4} + 461827 T^{5} + 11406836 T^{6} + 41102603 T^{7} + 779267980 T^{8} + 1943599533 T^{9} + 29739822234 T^{10} + 39088416143 T^{11} + 496981290961 T^{12} )^{2} \))(\( ( 1 + 2 T + 89 T^{2} )^{4} \))(\( ( 1 + 5 T + 89 T^{2} )^{12} \))
$97$ (\( 1 - 5 T + 97 T^{2} \))(\( 1 - 13 T + 97 T^{2} \))(\( 1 - 19 T + 97 T^{2} \))(\( 1 - 3 T + 97 T^{2} \))(\( 1 + 22 T + 307 T^{2} + 2134 T^{3} + 9409 T^{4} \))(\( 1 + 18 T + 267 T^{2} + 1746 T^{3} + 9409 T^{4} \))(\( 1 - 9 T + 86 T^{2} + 283 T^{3} + 8342 T^{4} - 84681 T^{5} + 912673 T^{6} \))(\( 1 + 21 T + 414 T^{2} + 4233 T^{3} + 40158 T^{4} + 197589 T^{5} + 912673 T^{6} \))(\( 1 + 9 T + 286 T^{2} + 1645 T^{3} + 27742 T^{4} + 84681 T^{5} + 912673 T^{6} \))(\( 1 + 5 T + 277 T^{2} + 1126 T^{3} + 35698 T^{4} + 109222 T^{5} + 2606293 T^{6} + 4563365 T^{7} + 88529281 T^{8} \))(\( 1 - 677 T^{2} + 227007 T^{4} - 48829758 T^{6} + 7431063213 T^{8} - 834105913299 T^{10} + 69918873771117 T^{12} - 4322863367143998 T^{14} + 189090475922917503 T^{16} - 5305942543393202597 T^{18} + 73742412689492826049 T^{20} \))(\( 1 + 15 T + 128 T^{2} + 1455 T^{3} + 9409 T^{4} \))(\( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} \))(\( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} \))(\( 1 - 10 T + 3 T^{2} - 970 T^{3} + 9409 T^{4} \))(\( 1 + 5 T - 127 T^{2} - 210 T^{3} + 11270 T^{4} - 20370 T^{5} - 1194943 T^{6} + 4563365 T^{7} + 88529281 T^{8} \))(\( 1 + 12 T + 10 T^{2} - 720 T^{3} - 5421 T^{4} - 69840 T^{5} + 94090 T^{6} + 10952076 T^{7} + 88529281 T^{8} \))(\( 1 + 17 T - 31 T^{2} - 1074 T^{3} + 15991 T^{4} + 146833 T^{5} - 218606 T^{6} + 14242801 T^{7} + 150459319 T^{8} - 980210802 T^{9} - 2744407711 T^{10} + 145984784369 T^{11} + 832972004929 T^{12} \))(\( 1 + 6 T - 155 T^{2} - 1006 T^{3} + 11498 T^{4} + 39646 T^{5} - 894259 T^{6} + 3845662 T^{7} + 108184682 T^{8} - 918149038 T^{9} - 13722038555 T^{10} + 51524041542 T^{11} + 832972004929 T^{12} \))(\( 1 + 14 T - 147 T^{2} - 3290 T^{3} + 13285 T^{4} + 444080 T^{5} + 962262 T^{6} - 18194372 T^{7} - 148423758 T^{8} - 1764854084 T^{9} + 9053923158 T^{10} + 405299825840 T^{11} + 1176111498085 T^{12} - 28252349445530 T^{13} - 122446884724563 T^{14} + 1131175982693582 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 21 T - 142 T^{2} + 4297 T^{3} + 29014 T^{4} - 522941 T^{5} - 7276852 T^{6} + 62270277 T^{7} + 1168472998 T^{8} - 5690613441 T^{9} - 141398916082 T^{10} + 210890785117 T^{11} + 14999641117474 T^{12} + 20456406156349 T^{13} - 1330422401415538 T^{14} - 5193669241037793 T^{15} + 103444074380854438 T^{16} + 534736056496641189 T^{17} - 6061414000011603508 T^{18} - 42252735683268890333 T^{19} + \)\(22\!\cdots\!54\)\( T^{20} + \)\(32\!\cdots\!49\)\( T^{21} - \)\(10\!\cdots\!58\)\( T^{22} - \)\(15\!\cdots\!13\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 + 190 T^{2} + 26691 T^{4} + 1787710 T^{6} + 88529281 T^{8} \))(\( 1 + 198 T^{2} + 10197 T^{4} - 1658894 T^{6} - 164346534 T^{8} + 16808727870 T^{10} + 3976889862093 T^{12} + 158153320528830 T^{14} - 14549480489862054 T^{16} - 1381812261144688526 T^{18} + 79918310361861871317 T^{20} + \)\(14\!\cdots\!02\)\( T^{22} + \)\(69\!\cdots\!41\)\( T^{24} \))
show more
show less