Properties

Label 100.6
Level 100
Weight 6
Dimension 786
Nonzero newspaces 6
Newform subspaces 18
Sturm bound 3600
Trace bound 3

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Defining parameters

Level: \( N \) = \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 18 \)
Sturm bound: \(3600\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(100))\).

Total New Old
Modular forms 1570 826 744
Cusp forms 1430 786 644
Eisenstein series 140 40 100

Trace form

\( 786q - 6q^{2} - 8q^{3} - 10q^{4} - 35q^{5} + 350q^{6} - 172q^{7} - 498q^{8} - 681q^{9} + O(q^{10}) \) \( 786q - 6q^{2} - 8q^{3} - 10q^{4} - 35q^{5} + 350q^{6} - 172q^{7} - 498q^{8} - 681q^{9} - 576q^{10} - 260q^{11} + 2550q^{12} + 2242q^{13} - 10q^{14} + 2438q^{15} - 3970q^{16} - 6096q^{17} + 6482q^{18} - 296q^{19} + 1354q^{20} - 9704q^{21} + 4870q^{22} + 9464q^{23} + 103q^{25} - 13700q^{26} - 14654q^{27} - 11850q^{28} - 26526q^{29} - 8610q^{30} + 996q^{31} - 35226q^{32} + 27120q^{33} - 10q^{34} - 7768q^{35} + 63990q^{36} + 39071q^{37} + 113040q^{38} - 27576q^{39} - 46516q^{40} - 15874q^{41} - 122110q^{42} - 22920q^{43} - 51740q^{44} - 5665q^{45} - 1550q^{46} + 38036q^{47} + 64600q^{48} + 66588q^{49} + 97394q^{50} + 89852q^{51} + 47328q^{52} + 91655q^{53} - 66180q^{54} - 106920q^{55} + 10250q^{56} - 285508q^{57} + 75504q^{58} - 161182q^{59} - 90070q^{60} - 167298q^{61} + 267340q^{62} + 295378q^{63} + 345620q^{64} + 343015q^{65} - 309810q^{66} + 76008q^{67} - 627986q^{68} + 129922q^{69} - 364250q^{70} - 78952q^{71} - 622096q^{72} - 553138q^{73} - 498238q^{75} + 586980q^{76} + 143900q^{77} + 1174820q^{78} + 449592q^{79} + 467814q^{80} + 1218359q^{81} + 151438q^{82} + 517734q^{83} - 78490q^{84} - 74853q^{85} - 632150q^{86} - 236738q^{87} - 233090q^{88} - 1440329q^{89} + 135974q^{90} - 661156q^{91} - 887710q^{92} - 1017338q^{93} - 385770q^{94} + 136036q^{95} + 1013450q^{96} + 1736426q^{97} + 770582q^{98} + 1027400q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(100))\)

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
100.6.a \(\chi_{100}(1, \cdot)\) 100.6.a.a 1 1
100.6.a.b 1
100.6.a.c 2
100.6.a.d 2
100.6.a.e 2
100.6.c \(\chi_{100}(49, \cdot)\) 100.6.c.a 2 1
100.6.c.b 2
100.6.c.c 4
100.6.e \(\chi_{100}(7, \cdot)\) 100.6.e.a 2 2
100.6.e.b 2
100.6.e.c 2
100.6.e.d 16
100.6.e.e 24
100.6.e.f 40
100.6.g \(\chi_{100}(21, \cdot)\) 100.6.g.a 52 4
100.6.i \(\chi_{100}(9, \cdot)\) 100.6.i.a 48 4
100.6.l \(\chi_{100}(3, \cdot)\) 100.6.l.a 8 8
100.6.l.b 576

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(100))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(100)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 8 T + 32 T^{2} \))(\( 1 + 8 T + 32 T^{2} \))(\( 1 - 8 T + 32 T^{2} \))(\( 1 + 516 T^{4} - 644864 T^{8} + 541065216 T^{12} + 1099511627776 T^{16} \))(\( 1 + 8 T + 32 T^{2} - 1024 T^{4} + 32768 T^{6} + 262144 T^{7} + 1048576 T^{8} \))
$3$ (\( 1 + 22 T + 243 T^{2} \))(\( 1 - 12 T + 243 T^{2} \))(\( 1 + 20 T + 177 T^{2} + 4860 T^{3} + 59049 T^{4} \))(\( 1 + 362 T^{2} + 59049 T^{4} \))(\( 1 - 20 T + 177 T^{2} - 4860 T^{3} + 59049 T^{4} \))(\( 1 - 2 T^{2} + 59049 T^{4} \))(\( 1 - 342 T^{2} + 59049 T^{4} \))(\( 1 + 46 T^{2} - 44973 T^{4} + 2716254 T^{6} + 3486784401 T^{8} \))(\( 1 + 38 T + 722 T^{2} + 9234 T^{3} + 59049 T^{4} \))(\( 1 + 59049 T^{4} \))(\( 1 - 38 T + 722 T^{2} - 9234 T^{3} + 59049 T^{4} \))(\( ( 1 - 42884 T^{4} - 724593114 T^{8} - 149527262252484 T^{12} + 12157665459056928801 T^{16} )^{2} \))(\( 1 - 59049 T^{4} + 3486784401 T^{8} - 205891132094649 T^{12} + 12157665459056928801 T^{16} \))
$5$ (\( 1 + 76 T + 2651 T^{2} - 36024 T^{3} - 11022199 T^{4} - 112575000 T^{5} + 25888671875 T^{6} + 2319335937500 T^{7} + 95367431640625 T^{8} \))
$7$ (\( 1 + 218 T + 16807 T^{2} \))(\( 1 - 88 T + 16807 T^{2} \))(\( 1 - 40 T + 19290 T^{2} - 672280 T^{3} + 282475249 T^{4} \))(\( 1 + 18610 T^{2} + 282475249 T^{4} \))(\( 1 + 40 T + 19290 T^{2} + 672280 T^{3} + 282475249 T^{4} \))(\( 1 + 13910 T^{2} + 282475249 T^{4} \))(\( 1 - 25870 T^{2} + 282475249 T^{4} \))(\( 1 - 36980 T^{2} + 883272198 T^{4} - 10445934708020 T^{6} + 79792266297612001 T^{8} \))(\( 1 + 366 T + 66978 T^{2} + 6151362 T^{3} + 282475249 T^{4} \))(\( 1 + 282475249 T^{4} \))(\( 1 - 366 T + 66978 T^{2} - 6151362 T^{3} + 282475249 T^{4} \))(\( ( 1 - 136496804 T^{4} + 12656061493785606 T^{8} - \)\(10\!\cdots\!04\)\( T^{12} + \)\(63\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 + 282475249 T^{4} )^{4} \))
$11$ (\( 1 + 480 T + 161051 T^{2} \))(\( 1 - 540 T + 161051 T^{2} \))(\( 1 + 60 T + 230977 T^{2} + 9663060 T^{3} + 25937424601 T^{4} \))(\( ( 1 + 100 T + 161051 T^{2} )^{2} \))(\( 1 + 60 T + 230977 T^{2} + 9663060 T^{3} + 25937424601 T^{4} \))(\( ( 1 + 480 T + 161051 T^{2} )^{2} \))(\( ( 1 - 540 T + 161051 T^{2} )^{2} \))(\( ( 1 + 60 T + 230977 T^{2} + 9663060 T^{3} + 25937424601 T^{4} )^{2} \))(\( ( 1 - 161051 T^{2} )^{2} \))(\( ( 1 - 161051 T^{2} )^{2} \))(\( ( 1 - 161051 T^{2} )^{2} \))(\( ( 1 - 306604 T^{2} + 65336293526 T^{4} - 7952518132365004 T^{6} + \)\(67\!\cdots\!01\)\( T^{8} )^{4} \))(\( ( 1 + 161051 T^{2} + 25937424601 T^{4} + 4177248169415651 T^{6} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \))
$13$ (\( 1 - 622 T + 371293 T^{2} \))(\( 1 - 418 T + 371293 T^{2} \))(\( 1 + 920 T + 718602 T^{2} + 341589560 T^{3} + 137858491849 T^{4} \))(\( 1 + 202442 T^{2} + 137858491849 T^{4} \))(\( 1 - 920 T + 718602 T^{2} - 341589560 T^{3} + 137858491849 T^{4} \))(\( 1 - 355702 T^{2} + 137858491849 T^{4} \))(\( 1 - 567862 T^{2} + 137858491849 T^{4} \))(\( 1 - 590804 T^{2} + 163581027702 T^{4} - 81447348418356596 T^{6} + \)\(19\!\cdots\!01\)\( T^{8} \))(\( 1 + 137858491849 T^{4} \))(\( ( 1 + 244 T + 371293 T^{2} )( 1 + 1194 T + 371293 T^{2} ) \))(\( 1 + 137858491849 T^{4} \))(\( ( 1 - 358018338524 T^{4} + \)\(63\!\cdots\!26\)\( T^{8} - \)\(68\!\cdots\!24\)\( T^{12} + \)\(36\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 + 244 T - 311757 T^{2} - 166664200 T^{3} + 75087127001 T^{4} - 61881250810600 T^{5} - 42978349843368693 T^{6} + 12489357895438144708 T^{7} + \)\(19\!\cdots\!01\)\( T^{8} )( 1 + 1194 T + 1054343 T^{2} + 815561700 T^{3} + 582310494301 T^{4} + 302812350278100 T^{5} + 145350135871550207 T^{6} + 61115956258824363858 T^{7} + \)\(19\!\cdots\!01\)\( T^{8} ) \))
$17$ (\( 1 + 186 T + 1419857 T^{2} \))(\( 1 + 594 T + 1419857 T^{2} \))(\( 1 + 2910 T + 4897843 T^{2} + 4131783870 T^{3} + 2015993900449 T^{4} \))(\( 1 + 1879458 T^{2} + 2015993900449 T^{4} \))(\( 1 - 2910 T + 4897843 T^{2} - 4131783870 T^{3} + 2015993900449 T^{4} \))(\( 1 - 2805118 T^{2} + 2015993900449 T^{4} \))(\( 1 - 2486878 T^{2} + 2015993900449 T^{4} \))(\( 1 - 1327586 T^{2} + 3973871730147 T^{4} - 2676405278321486114 T^{6} + \)\(40\!\cdots\!01\)\( T^{8} \))(\( 1 + 2015993900449 T^{4} \))(\( ( 1 - 2242 T + 1419857 T^{2} )( 1 + 808 T + 1419857 T^{2} ) \))(\( 1 + 2015993900449 T^{4} \))(\( ( 1 + 564889729156 T^{4} - \)\(30\!\cdots\!34\)\( T^{8} + \)\(22\!\cdots\!56\)\( T^{12} + \)\(16\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 2242 T + 3606707 T^{2} - 4902917700 T^{3} + 5871333302501 T^{4} - 6961442016768900 T^{5} + 7271099312706711443 T^{6} - \)\(64\!\cdots\!06\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} )( 1 + 808 T - 766993 T^{2} - 1766974800 T^{3} - 338695258399 T^{4} - 2508851538603600 T^{5} - 1546253209687079857 T^{6} + \)\(23\!\cdots\!44\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} ) \))
$19$ (\( 1 + 1204 T + 2476099 T^{2} \))(\( 1 - 836 T + 2476099 T^{2} \))(\( 1 - 2092 T + 5218089 T^{2} - 5179999108 T^{3} + 6131066257801 T^{4} \))(\( ( 1 + 2244 T + 2476099 T^{2} )^{2} \))(\( 1 - 2092 T + 5218089 T^{2} - 5179999108 T^{3} + 6131066257801 T^{4} \))(\( ( 1 - 1204 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 836 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 2092 T + 5218089 T^{2} + 5179999108 T^{3} + 6131066257801 T^{4} )^{2} \))(\( ( 1 + 2476099 T^{2} )^{2} \))(\( ( 1 + 2476099 T^{2} )^{2} \))(\( ( 1 + 2476099 T^{2} )^{2} \))(\( ( 1 - 2521524 T^{2} + 7745160464566 T^{4} - 15459630714635408724 T^{6} + \)\(37\!\cdots\!01\)\( T^{8} )^{4} \))(\( ( 1 - 2476099 T^{2} + 6131066257801 T^{4} - 15181127029874798299 T^{6} + \)\(37\!\cdots\!01\)\( T^{8} )^{2} \))
$23$ (\( 1 - 3186 T + 6436343 T^{2} \))(\( 1 - 4104 T + 6436343 T^{2} \))(\( 1 + 120 T + 5087290 T^{2} + 772361160 T^{3} + 41426511213649 T^{4} \))(\( 1 + 1185810 T^{2} + 41426511213649 T^{4} \))(\( 1 - 120 T + 5087290 T^{2} - 772361160 T^{3} + 41426511213649 T^{4} \))(\( 1 - 2722090 T^{2} + 41426511213649 T^{4} \))(\( 1 + 3970130 T^{2} + 41426511213649 T^{4} \))(\( 1 - 10160180 T^{2} + 108548175292998 T^{4} - \)\(42\!\cdots\!20\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} \))(\( 1 + 4838 T + 11703122 T^{2} + 31139027434 T^{3} + 41426511213649 T^{4} \))(\( 1 + 41426511213649 T^{4} \))(\( 1 - 4838 T + 11703122 T^{2} - 31139027434 T^{3} + 41426511213649 T^{4} \))(\( ( 1 + 135199478790876 T^{4} + \)\(79\!\cdots\!66\)\( T^{8} + \)\(23\!\cdots\!76\)\( T^{12} + \)\(29\!\cdots\!01\)\( T^{16} )^{2} \))(\( 1 - 41426511213649 T^{4} + \)\(17\!\cdots\!01\)\( T^{8} - \)\(71\!\cdots\!49\)\( T^{12} + \)\(29\!\cdots\!01\)\( T^{16} \))
$29$ (\( 1 - 5526 T + 20511149 T^{2} \))(\( 1 + 594 T + 20511149 T^{2} \))(\( 1 - 3552 T + 20618074 T^{2} - 72855601248 T^{3} + 420707233300201 T^{4} \))(\( ( 1 + 7854 T + 20511149 T^{2} )^{2} \))(\( 1 - 3552 T + 20618074 T^{2} - 72855601248 T^{3} + 420707233300201 T^{4} \))(\( ( 1 + 5526 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 594 T + 20511149 T^{2} )^{2} \))(\( ( 1 + 3552 T + 20618074 T^{2} + 72855601248 T^{3} + 420707233300201 T^{4} )^{2} \))(\( 1 - 38179702 T^{2} + 420707233300201 T^{4} \))(\( ( 1 - 2950 T + 20511149 T^{2} )( 1 + 2950 T + 20511149 T^{2} ) \))(\( 1 - 38179702 T^{2} + 420707233300201 T^{4} \))(\( ( 1 - 30854764 T^{2} + 962616580074806 T^{4} - \)\(12\!\cdots\!64\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} )^{4} \))(\( ( 1 - 2950 T - 11808649 T^{2} + 95343404100 T^{3} - 39054082967299 T^{4} + 1955602767662310900 T^{5} - \)\(49\!\cdots\!49\)\( T^{6} - \)\(25\!\cdots\!50\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} )( 1 + 2950 T - 11808649 T^{2} - 95343404100 T^{3} - 39054082967299 T^{4} - 1955602767662310900 T^{5} - \)\(49\!\cdots\!49\)\( T^{6} + \)\(25\!\cdots\!50\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} ) \))
$31$ (\( 1 - 9356 T + 28629151 T^{2} \))(\( 1 - 4256 T + 28629151 T^{2} \))(\( 1 + 8888 T + 67804938 T^{2} + 254455894088 T^{3} + 819628286980801 T^{4} \))(\( ( 1 + 2144 T + 28629151 T^{2} )^{2} \))(\( 1 + 8888 T + 67804938 T^{2} + 254455894088 T^{3} + 819628286980801 T^{4} \))(\( ( 1 - 9356 T + 28629151 T^{2} )^{2} \))(\( ( 1 - 4256 T + 28629151 T^{2} )^{2} \))(\( ( 1 + 8888 T + 67804938 T^{2} + 254455894088 T^{3} + 819628286980801 T^{4} )^{2} \))(\( ( 1 - 28629151 T^{2} )^{2} \))(\( ( 1 - 28629151 T^{2} )^{2} \))(\( ( 1 - 28629151 T^{2} )^{2} \))(\( ( 1 - 11513724 T^{2} + 684086233912966 T^{4} - \)\(94\!\cdots\!24\)\( T^{6} + \)\(67\!\cdots\!01\)\( T^{8} )^{4} \))(\( ( 1 + 28629151 T^{2} + 819628286980801 T^{4} + \)\(23\!\cdots\!51\)\( T^{6} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \))
$37$ (\( 1 + 5618 T + 69343957 T^{2} \))(\( 1 - 298 T + 69343957 T^{2} \))(\( 1 + 12140 T + 90486990 T^{2} + 841835637980 T^{3} + 4808584372417849 T^{4} \))(\( 1 + 30515770 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 12140 T + 90486990 T^{2} - 841835637980 T^{3} + 4808584372417849 T^{4} \))(\( 1 - 107125990 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 138599110 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 33594380 T^{2} - 2634705186058602 T^{4} - \)\(16\!\cdots\!20\)\( T^{6} + \)\(23\!\cdots\!01\)\( T^{8} \))(\( 1 + 4808584372417849 T^{4} \))(\( ( 1 - 12242 T + 69343957 T^{2} )( 1 - 11292 T + 69343957 T^{2} ) \))(\( 1 + 4808584372417849 T^{4} \))(\( ( 1 - 12873900293860124 T^{4} + \)\(85\!\cdots\!46\)\( T^{8} - \)\(29\!\cdots\!24\)\( T^{12} + \)\(53\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 + 11292 T + 69343957 T^{2} )^{4}( 1 - 12242 T + 80522607 T^{2} - 136849033300 T^{3} - 3908450331677299 T^{4} - 9489653480646768100 T^{5} + \)\(38\!\cdots\!43\)\( T^{6} - \)\(40\!\cdots\!06\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} ) \))
$41$ (\( 1 + 14394 T + 115856201 T^{2} \))(\( 1 - 17226 T + 115856201 T^{2} \))(\( 1 + 12438 T + 217381963 T^{2} + 1441019428038 T^{3} + 13422659310152401 T^{4} \))(\( ( 1 + 7414 T + 115856201 T^{2} )^{2} \))(\( 1 + 12438 T + 217381963 T^{2} + 1441019428038 T^{3} + 13422659310152401 T^{4} \))(\( ( 1 + 14394 T + 115856201 T^{2} )^{2} \))(\( ( 1 - 17226 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 12438 T + 217381963 T^{2} + 1441019428038 T^{3} + 13422659310152401 T^{4} )^{2} \))(\( ( 1 + 4548 T + 115856201 T^{2} )^{2} \))(\( ( 1 - 4952 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 4548 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 3776 T + 126948226 T^{2} + 437473014976 T^{3} + 13422659310152401 T^{4} )^{8} \))(\( ( 1 + 4952 T - 91333897 T^{2} - 1026005365296 T^{3} + 5500819759999505 T^{4} - \)\(11\!\cdots\!96\)\( T^{5} - \)\(12\!\cdots\!97\)\( T^{6} + \)\(77\!\cdots\!52\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )^{2} \))
$43$ (\( 1 - 370 T + 147008443 T^{2} \))(\( 1 - 12100 T + 147008443 T^{2} \))(\( 1 + 1160 T + 270794886 T^{2} + 170529793880 T^{3} + 21611482313284249 T^{4} \))(\( 1 - 21442214 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 1160 T + 270794886 T^{2} - 170529793880 T^{3} + 21611482313284249 T^{4} \))(\( 1 - 293879986 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 147606886 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 540244172 T^{2} + 116157205788519894 T^{4} - \)\(11\!\cdots\!28\)\( T^{6} + \)\(46\!\cdots\!01\)\( T^{8} \))(\( 1 - 11862 T + 70353522 T^{2} - 1743814150866 T^{3} + 21611482313284249 T^{4} \))(\( 1 + 21611482313284249 T^{4} \))(\( 1 + 11862 T + 70353522 T^{2} + 1743814150866 T^{3} + 21611482313284249 T^{4} \))(\( ( 1 - 42096508454785604 T^{4} + \)\(93\!\cdots\!06\)\( T^{8} - \)\(19\!\cdots\!04\)\( T^{12} + \)\(21\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 + 21611482313284249 T^{4} )^{4} \))
$47$ (\( 1 + 16146 T + 229345007 T^{2} \))(\( 1 - 1296 T + 229345007 T^{2} \))(\( 1 - 1200 T + 451923598 T^{2} - 275214008400 T^{3} + 52599132235830049 T^{4} \))(\( 1 + 369731298 T^{2} + 52599132235830049 T^{4} \))(\( 1 + 1200 T + 451923598 T^{2} + 275214008400 T^{3} + 52599132235830049 T^{4} \))(\( 1 - 197996698 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 457010398 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 902407196 T^{2} + 308772689280765702 T^{4} - \)\(47\!\cdots\!04\)\( T^{6} + \)\(27\!\cdots\!01\)\( T^{8} \))(\( 1 - 33334 T + 555577778 T^{2} - 7644986463338 T^{3} + 52599132235830049 T^{4} \))(\( 1 + 52599132235830049 T^{4} \))(\( 1 + 33334 T + 555577778 T^{2} + 7644986463338 T^{3} + 52599132235830049 T^{4} \))(\( ( 1 - 119422202718033764 T^{4} + \)\(89\!\cdots\!46\)\( T^{8} - \)\(33\!\cdots\!64\)\( T^{12} + \)\(76\!\cdots\!01\)\( T^{16} )^{2} \))(\( 1 - 52599132235830049 T^{4} + \)\(27\!\cdots\!01\)\( T^{8} - \)\(14\!\cdots\!49\)\( T^{12} + \)\(76\!\cdots\!01\)\( T^{16} \))
$53$ (\( 1 - 4374 T + 418195493 T^{2} \))(\( 1 + 19494 T + 418195493 T^{2} \))(\( 1 + 26340 T + 941756110 T^{2} + 11015269285620 T^{3} + 174887470365513049 T^{4} \))(\( 1 + 248174170 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 26340 T + 941756110 T^{2} - 11015269285620 T^{3} + 174887470365513049 T^{4} \))(\( 1 - 817259110 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 456374950 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 1189716620 T^{2} + 656395125486896598 T^{4} - \)\(20\!\cdots\!80\)\( T^{6} + \)\(30\!\cdots\!01\)\( T^{8} \))(\( 1 + 174887470365513049 T^{4} \))(\( ( 1 + 7294 T + 418195493 T^{2} )( 1 + 40244 T + 418195493 T^{2} ) \))(\( 1 + 174887470365513049 T^{4} \))(\( ( 1 - 321387478479003804 T^{4} + \)\(86\!\cdots\!06\)\( T^{8} - \)\(98\!\cdots\!04\)\( T^{12} + \)\(93\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 40244 T + 418195493 T^{2} )^{4}( 1 + 7294 T - 364993057 T^{2} - 5712577283700 T^{3} + 110970912706384301 T^{4} - \)\(23\!\cdots\!00\)\( T^{5} - \)\(63\!\cdots\!93\)\( T^{6} + \)\(53\!\cdots\!58\)\( T^{7} + \)\(30\!\cdots\!01\)\( T^{8} ) \))
$59$ (\( 1 + 11748 T + 714924299 T^{2} \))(\( 1 + 7668 T + 714924299 T^{2} \))(\( 1 + 36696 T + 1234961302 T^{2} + 26234862076104 T^{3} + 511116753300641401 T^{4} \))(\( ( 1 - 25972 T + 714924299 T^{2} )^{2} \))(\( 1 + 36696 T + 1234961302 T^{2} + 26234862076104 T^{3} + 511116753300641401 T^{4} \))(\( ( 1 - 11748 T + 714924299 T^{2} )^{2} \))(\( ( 1 - 7668 T + 714924299 T^{2} )^{2} \))(\( ( 1 - 36696 T + 1234961302 T^{2} - 26234862076104 T^{3} + 511116753300641401 T^{4} )^{2} \))(\( ( 1 + 714924299 T^{2} )^{2} \))(\( ( 1 + 714924299 T^{2} )^{2} \))(\( ( 1 + 714924299 T^{2} )^{2} \))(\( ( 1 + 2094416236 T^{2} + 2095075619918645846 T^{4} + \)\(10\!\cdots\!36\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} )^{4} \))(\( ( 1 - 714924299 T^{2} + 511116753300641401 T^{4} - \)\(36\!\cdots\!99\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} )^{2} \))
$61$ (\( 1 - 13202 T + 844596301 T^{2} \))(\( 1 + 34738 T + 844596301 T^{2} \))(\( 1 - 19204 T + 627029406 T^{2} - 16219627364404 T^{3} + 713342911662882601 T^{4} \))(\( ( 1 + 3058 T + 844596301 T^{2} )^{2} \))(\( 1 - 19204 T + 627029406 T^{2} - 16219627364404 T^{3} + 713342911662882601 T^{4} \))(\( ( 1 - 13202 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 34738 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 19204 T + 627029406 T^{2} - 16219627364404 T^{3} + 713342911662882601 T^{4} )^{2} \))(\( ( 1 + 25448 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 54948 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 25448 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 16384 T + 1092947546 T^{2} - 13837865795584 T^{3} + 713342911662882601 T^{4} )^{8} \))(\( ( 1 - 54948 T + 2174686403 T^{2} - 73085790924696 T^{3} + 2179185947921400505 T^{4} - \)\(61\!\cdots\!96\)\( T^{5} + \)\(15\!\cdots\!03\)\( T^{6} - \)\(33\!\cdots\!48\)\( T^{7} + \)\(50\!\cdots\!01\)\( T^{8} )^{2} \))
$67$ (\( 1 - 11542 T + 1350125107 T^{2} \))(\( 1 + 21812 T + 1350125107 T^{2} \))(\( 1 - 90460 T + 4578999825 T^{2} - 122132317179220 T^{3} + 1822837804551761449 T^{4} \))(\( 1 - 755362070 T^{2} + 1822837804551761449 T^{4} \))(\( 1 + 90460 T + 4578999825 T^{2} + 122132317179220 T^{3} + 1822837804551761449 T^{4} \))(\( 1 - 2567032450 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 2224486870 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 974988050 T^{2} + 2516736182389071123 T^{4} - \)\(17\!\cdots\!50\)\( T^{6} + \)\(33\!\cdots\!01\)\( T^{8} \))(\( 1 - 100434 T + 5043494178 T^{2} - 135598464996438 T^{3} + 1822837804551761449 T^{4} \))(\( 1 + 1822837804551761449 T^{4} \))(\( 1 + 100434 T + 5043494178 T^{2} + 135598464996438 T^{3} + 1822837804551761449 T^{4} \))(\( ( 1 - 6389567977600689284 T^{4} + \)\(16\!\cdots\!86\)\( T^{8} - \)\(21\!\cdots\!84\)\( T^{12} + \)\(11\!\cdots\!01\)\( T^{16} )^{2} \))(\( 1 - 1822837804551761449 T^{4} + \)\(33\!\cdots\!01\)\( T^{8} - \)\(60\!\cdots\!49\)\( T^{12} + \)\(11\!\cdots\!01\)\( T^{16} \))
$71$ (\( 1 + 29532 T + 1804229351 T^{2} \))(\( 1 + 46872 T + 1804229351 T^{2} \))(\( 1 - 2736 T + 2006886526 T^{2} - 4936371504336 T^{3} + 3255243551009881201 T^{4} \))(\( ( 1 - 37608 T + 1804229351 T^{2} )^{2} \))(\( 1 - 2736 T + 2006886526 T^{2} - 4936371504336 T^{3} + 3255243551009881201 T^{4} \))(\( ( 1 + 29532 T + 1804229351 T^{2} )^{2} \))(\( ( 1 + 46872 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 2736 T + 2006886526 T^{2} - 4936371504336 T^{3} + 3255243551009881201 T^{4} )^{2} \))(\( ( 1 - 1804229351 T^{2} )^{2} \))(\( ( 1 - 1804229351 T^{2} )^{2} \))(\( ( 1 - 1804229351 T^{2} )^{2} \))(\( ( 1 - 4148035484 T^{2} + 9418407296761234086 T^{4} - \)\(13\!\cdots\!84\)\( T^{6} + \)\(10\!\cdots\!01\)\( T^{8} )^{4} \))(\( ( 1 + 1804229351 T^{2} + 3255243551009881201 T^{4} + \)\(58\!\cdots\!51\)\( T^{6} + \)\(10\!\cdots\!01\)\( T^{8} )^{2} \))
$73$ (\( 1 + 33698 T + 2073071593 T^{2} \))(\( 1 + 67562 T + 2073071593 T^{2} \))(\( 1 + 12770 T + 4184025507 T^{2} + 26473124242610 T^{3} + 4297625829703557649 T^{4} \))(\( 1 + 3569749522 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 12770 T + 4184025507 T^{2} - 26473124242610 T^{3} + 4297625829703557649 T^{4} \))(\( 1 - 3010587982 T^{2} + 4297625829703557649 T^{4} \))(\( 1 + 418480658 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 8204978114 T^{2} + 25425197509477462947 T^{4} - \)\(35\!\cdots\!86\)\( T^{6} + \)\(18\!\cdots\!01\)\( T^{8} \))(\( 1 + 4297625829703557649 T^{4} \))(\( ( 1 - 88806 T + 2073071593 T^{2} )( 1 + 20144 T + 2073071593 T^{2} ) \))(\( 1 + 4297625829703557649 T^{4} \))(\( ( 1 + 1929098357836241476 T^{4} - \)\(10\!\cdots\!74\)\( T^{8} + \)\(35\!\cdots\!76\)\( T^{12} + \)\(34\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 88806 T + 5813434043 T^{2} - 332166627734700 T^{3} + 17446724570285327701 T^{4} - \)\(68\!\cdots\!00\)\( T^{5} + \)\(24\!\cdots\!07\)\( T^{6} - \)\(79\!\cdots\!42\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )( 1 + 20144 T - 1667290857 T^{2} - 75345861192800 T^{3} + 1938646285047562001 T^{4} - \)\(15\!\cdots\!00\)\( T^{5} - \)\(71\!\cdots\!93\)\( T^{6} + \)\(17\!\cdots\!08\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} ) \))
$79$ (\( 1 - 31208 T + 3077056399 T^{2} \))(\( 1 + 76912 T + 3077056399 T^{2} \))(\( 1 + 16184 T + 6157384362 T^{2} + 49799080761416 T^{3} + 9468276082626847201 T^{4} \))(\( ( 1 - 79728 T + 3077056399 T^{2} )^{2} \))(\( 1 + 16184 T + 6157384362 T^{2} + 49799080761416 T^{3} + 9468276082626847201 T^{4} \))(\( ( 1 + 31208 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 76912 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 16184 T + 6157384362 T^{2} - 49799080761416 T^{3} + 9468276082626847201 T^{4} )^{2} \))(\( ( 1 + 3077056399 T^{2} )^{2} \))(\( ( 1 + 3077056399 T^{2} )^{2} \))(\( ( 1 + 3077056399 T^{2} )^{2} \))(\( ( 1 + 7125302076 T^{2} + 27017788725885808966 T^{4} + \)\(67\!\cdots\!76\)\( T^{6} + \)\(89\!\cdots\!01\)\( T^{8} )^{4} \))(\( ( 1 - 3077056399 T^{2} + 9468276082626847201 T^{4} - \)\(29\!\cdots\!99\)\( T^{6} + \)\(89\!\cdots\!01\)\( T^{8} )^{2} \))
$83$ (\( 1 - 38466 T + 3939040643 T^{2} \))(\( 1 + 67716 T + 3939040643 T^{2} \))(\( 1 - 30300 T + 4057027705 T^{2} - 119352931482900 T^{3} + 15516041187205853449 T^{4} \))(\( 1 + 7612675530 T^{2} + 15516041187205853449 T^{4} \))(\( 1 + 30300 T + 4057027705 T^{2} + 119352931482900 T^{3} + 15516041187205853449 T^{4} \))(\( 1 - 6398448130 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 3292624630 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 7195965410 T^{2} + 40258768525685533923 T^{4} - \)\(11\!\cdots\!90\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} \))(\( 1 - 163262 T + 13327240322 T^{2} - 643095653457466 T^{3} + 15516041187205853449 T^{4} \))(\( 1 + 15516041187205853449 T^{4} \))(\( 1 + 163262 T + 13327240322 T^{2} + 643095653457466 T^{3} + 15516041187205853449 T^{4} \))(\( ( 1 - 2758744208432235524 T^{4} + \)\(22\!\cdots\!66\)\( T^{8} - \)\(66\!\cdots\!24\)\( T^{12} + \)\(57\!\cdots\!01\)\( T^{16} )^{2} \))(\( 1 - 15516041187205853449 T^{4} + \)\(24\!\cdots\!01\)\( T^{8} - \)\(37\!\cdots\!49\)\( T^{12} + \)\(57\!\cdots\!01\)\( T^{16} \))
$89$ (\( 1 - 119514 T + 5584059449 T^{2} \))(\( 1 - 29754 T + 5584059449 T^{2} \))(\( 1 + 47322 T - 1006825781 T^{2} + 264248861245578 T^{3} + 31181719929966183601 T^{4} \))(\( ( 1 + 826 T + 5584059449 T^{2} )^{2} \))(\( 1 + 47322 T - 1006825781 T^{2} + 264248861245578 T^{3} + 31181719929966183601 T^{4} \))(\( ( 1 + 119514 T + 5584059449 T^{2} )^{2} \))(\( ( 1 + 29754 T + 5584059449 T^{2} )^{2} \))(\( ( 1 - 47322 T - 1006825781 T^{2} - 264248861245578 T^{3} + 31181719929966183601 T^{4} )^{2} \))(\( 1 + 11118190898 T^{2} + 31181719929966183601 T^{4} \))(\( ( 1 - 51050 T + 5584059449 T^{2} )( 1 + 51050 T + 5584059449 T^{2} ) \))(\( 1 + 11118190898 T^{2} + 31181719929966183601 T^{4} \))(\( ( 1 - 19759605404 T^{2} + \)\(15\!\cdots\!06\)\( T^{4} - \)\(61\!\cdots\!04\)\( T^{6} + \)\(97\!\cdots\!01\)\( T^{8} )^{4} \))(\( ( 1 + 51050 T + 5584059449 T^{2} )^{4}( 1 + 51050 T - 2977956949 T^{2} - 437090937117900 T^{3} - 5684403700090133899 T^{4} - \)\(24\!\cdots\!00\)\( T^{5} - \)\(92\!\cdots\!49\)\( T^{6} + \)\(88\!\cdots\!50\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} ) \))
$97$ (\( 1 + 94658 T + 8587340257 T^{2} \))(\( 1 - 122398 T + 8587340257 T^{2} \))(\( 1 - 2980 T + 14115250950 T^{2} - 25590273965860 T^{3} + 73742412689492826049 T^{4} \))(\( 1 + 15761405890 T^{2} + 73742412689492826049 T^{4} \))(\( 1 + 2980 T + 14115250950 T^{2} + 25590273965860 T^{3} + 73742412689492826049 T^{4} \))(\( 1 - 8214543550 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 2193410110 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 28221621500 T^{2} + \)\(34\!\cdots\!98\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{6} + \)\(54\!\cdots\!01\)\( T^{8} \))(\( 1 + 73742412689492826049 T^{4} \))(\( ( 1 - 92142 T + 8587340257 T^{2} )( 1 + 160808 T + 8587340257 T^{2} ) \))(\( 1 + 73742412689492826049 T^{4} \))(\( ( 1 - \)\(16\!\cdots\!64\)\( T^{4} + \)\(16\!\cdots\!06\)\( T^{8} - \)\(87\!\cdots\!64\)\( T^{12} + \)\(29\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 92142 T - 97192093 T^{2} + 800210179793700 T^{3} - 72898344813670117499 T^{4} + \)\(68\!\cdots\!00\)\( T^{5} - \)\(71\!\cdots\!57\)\( T^{6} - \)\(58\!\cdots\!06\)\( T^{7} + \)\(54\!\cdots\!01\)\( T^{8} )( 1 + 160808 T + 17271872607 T^{2} + 1396542278138800 T^{3} + 76255723711077510401 T^{4} + \)\(11\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} + \)\(10\!\cdots\!44\)\( T^{7} + \)\(54\!\cdots\!01\)\( T^{8} ) \))
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