# Properties

 Label 81.4 Level 81 Weight 4 Dimension 548 Nonzero newspaces 4 Newform subspaces 14 Sturm bound 1944 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$81 = 3^{4}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$4$$ Newform subspaces: $$14$$ Sturm bound: $$1944$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(81))$$.

Total New Old
Modular forms 783 604 179
Cusp forms 675 548 127
Eisenstein series 108 56 52

## Trace form

 $$548q - 12q^{2} - 18q^{3} - 28q^{4} - 24q^{5} - 18q^{6} - 2q^{7} + 57q^{8} - 18q^{9} + O(q^{10})$$ $$548q - 12q^{2} - 18q^{3} - 28q^{4} - 24q^{5} - 18q^{6} - 2q^{7} + 57q^{8} - 18q^{9} - 9q^{10} - 75q^{11} - 18q^{12} - 74q^{13} - 69q^{14} - 18q^{15} + 152q^{16} + 189q^{17} + 396q^{18} + 421q^{19} + 1155q^{20} + 90q^{21} - 198q^{22} - 798q^{23} - 1098q^{24} - 883q^{25} - 3327q^{26} - 720q^{27} - 1577q^{28} - 1038q^{29} - 774q^{30} + 34q^{31} + 846q^{32} + 414q^{33} + 864q^{34} + 2481q^{35} + 2286q^{36} + 1501q^{37} + 2460q^{38} - 18q^{39} + 1989q^{40} + 3567q^{41} + 3537q^{42} + 1429q^{43} + 4857q^{44} + 1008q^{45} - 225q^{46} - 1596q^{47} - 2295q^{48} - 1821q^{49} - 9174q^{50} - 2979q^{51} - 4097q^{52} - 5607q^{53} - 6696q^{54} - 4275q^{55} - 11775q^{56} - 2232q^{57} - 3519q^{58} - 3705q^{59} - 1305q^{60} - 56q^{61} + 3981q^{62} + 1980q^{63} + 5705q^{64} + 5478q^{65} - 1962q^{66} + 6433q^{67} + 8640q^{68} - 396q^{69} + 6039q^{70} + 2367q^{71} + 1710q^{72} - 389q^{73} + 1911q^{74} + 4482q^{75} - 4646q^{76} + 5052q^{77} + 10881q^{78} - 3944q^{79} + 10254q^{80} + 5742q^{81} - 630q^{82} + 1842q^{83} + 17361q^{84} - 3429q^{85} + 3558q^{86} + 5454q^{87} + 2412q^{88} + 12249q^{89} + 11079q^{90} + 9707q^{91} + 19173q^{92} + 3780q^{93} + 13329q^{94} + 6027q^{95} - 17253q^{96} + 5155q^{97} - 14805q^{98} - 13230q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(81))$$

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
81.4.a $$\chi_{81}(1, \cdot)$$ 81.4.a.a 2 1
81.4.a.b 2
81.4.a.c 2
81.4.a.d 2
81.4.a.e 2
81.4.c $$\chi_{81}(28, \cdot)$$ 81.4.c.a 2 2
81.4.c.b 2
81.4.c.c 2
81.4.c.d 4
81.4.c.e 4
81.4.c.f 4
81.4.c.g 4
81.4.e $$\chi_{81}(10, \cdot)$$ 81.4.e.a 48 6
81.4.g $$\chi_{81}(4, \cdot)$$ 81.4.g.a 468 18

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(81))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(81)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 3 T + 10 T^{2} + 24 T^{3} + 64 T^{4}$$)($$1 + 3 T + 4 T^{2} + 24 T^{3} + 64 T^{4}$$)($$1 + 13 T^{2} + 64 T^{4}$$)($$1 - 3 T + 10 T^{2} - 24 T^{3} + 64 T^{4}$$)($$1 - 3 T + 4 T^{2} - 24 T^{3} + 64 T^{4}$$)($$1 + 3 T + T^{2} + 24 T^{3} + 64 T^{4}$$)($$1 - 8 T^{2} + 64 T^{4}$$)($$1 - 3 T + T^{2} - 24 T^{3} + 64 T^{4}$$)($$1 + 3 T + 5 T^{2} - 36 T^{3} - 120 T^{4} - 288 T^{5} + 320 T^{6} + 1536 T^{7} + 4096 T^{8}$$)($$1 + 2 T^{2} - 60 T^{4} + 128 T^{6} + 4096 T^{8}$$)($$1 - 13 T^{2} + 105 T^{4} - 832 T^{6} + 4096 T^{8}$$)($$1 - 3 T + 5 T^{2} + 36 T^{3} - 120 T^{4} + 288 T^{5} + 320 T^{6} - 1536 T^{7} + 4096 T^{8}$$)
$3$ 1
$5$ ($$1 + 15 T + 298 T^{2} + 1875 T^{3} + 15625 T^{4}$$)($$1 - 12 T + 229 T^{2} - 1500 T^{3} + 15625 T^{4}$$)($$1 + 103 T^{2} + 15625 T^{4}$$)($$1 - 15 T + 298 T^{2} - 1875 T^{3} + 15625 T^{4}$$)($$1 + 12 T + 229 T^{2} + 1500 T^{3} + 15625 T^{4}$$)($$1 + 15 T + 100 T^{2} + 1875 T^{3} + 15625 T^{4}$$)($$1 - 125 T^{2} + 15625 T^{4}$$)($$1 - 15 T + 100 T^{2} - 1875 T^{3} + 15625 T^{4}$$)($$1 - 12 T - 85 T^{2} + 252 T^{3} + 18816 T^{4} + 31500 T^{5} - 1328125 T^{6} - 23437500 T^{7} + 244140625 T^{8}$$)($$1 + 38 T^{2} - 14181 T^{4} + 593750 T^{6} + 244140625 T^{8}$$)($$1 - 103 T^{2} - 5016 T^{4} - 1609375 T^{6} + 244140625 T^{8}$$)($$1 + 12 T - 85 T^{2} - 252 T^{3} + 18816 T^{4} - 31500 T^{5} - 1328125 T^{6} + 23437500 T^{7} + 244140625 T^{8}$$)
$7$ ($$1 - 7 T + 624 T^{2} - 2401 T^{3} + 117649 T^{4}$$)($$1 - 10 T + 198 T^{2} - 3430 T^{3} + 117649 T^{4}$$)($$( 1 + 22 T + 343 T^{2} )^{2}$$)($$1 - 7 T + 624 T^{2} - 2401 T^{3} + 117649 T^{4}$$)($$1 - 10 T + 198 T^{2} - 3430 T^{3} + 117649 T^{4}$$)($$1 - 25 T + 282 T^{2} - 8575 T^{3} + 117649 T^{4}$$)($$( 1 - 17 T + 343 T^{2} )( 1 + 37 T + 343 T^{2} )$$)($$1 - 25 T + 282 T^{2} - 8575 T^{3} + 117649 T^{4}$$)($$1 + 10 T - 98 T^{2} - 4880 T^{3} - 112745 T^{4} - 1673840 T^{5} - 11529602 T^{6} + 403536070 T^{7} + 13841287201 T^{8}$$)($$( 1 + 11 T - 222 T^{2} + 3773 T^{3} + 117649 T^{4} )^{2}$$)($$( 1 - 22 T + 141 T^{2} - 7546 T^{3} + 117649 T^{4} )^{2}$$)($$1 + 10 T - 98 T^{2} - 4880 T^{3} - 112745 T^{4} - 1673840 T^{5} - 11529602 T^{6} + 403536070 T^{7} + 13841287201 T^{8}$$)
$11$ ($$1 + 66 T + 3223 T^{2} + 87846 T^{3} + 1771561 T^{4}$$)($$1 - 42 T + 3046 T^{2} - 55902 T^{3} + 1771561 T^{4}$$)($$1 - 806 T^{2} + 1771561 T^{4}$$)($$1 - 66 T + 3223 T^{2} - 87846 T^{3} + 1771561 T^{4}$$)($$1 + 42 T + 3046 T^{2} + 55902 T^{3} + 1771561 T^{4}$$)($$1 - 15 T - 1106 T^{2} - 19965 T^{3} + 1771561 T^{4}$$)($$1 - 1331 T^{2} + 1771561 T^{4}$$)($$1 + 15 T - 1106 T^{2} + 19965 T^{3} + 1771561 T^{4}$$)($$1 - 42 T - 1282 T^{2} - 16128 T^{3} + 5158671 T^{4} - 21466368 T^{5} - 2271141202 T^{6} - 99033803022 T^{7} + 3138428376721 T^{8}$$)($$1 - 2374 T^{2} + 3864315 T^{4} - 4205685814 T^{6} + 3138428376721 T^{8}$$)($$1 + 806 T^{2} - 1121925 T^{4} + 1427878166 T^{6} + 3138428376721 T^{8}$$)($$1 + 42 T - 1282 T^{2} + 16128 T^{3} + 5158671 T^{4} + 21466368 T^{5} - 2271141202 T^{6} + 99033803022 T^{7} + 3138428376721 T^{8}$$)
$13$ ($$1 + 11 T + 2568 T^{2} + 24167 T^{3} + 4826809 T^{4}$$)($$1 - 64 T + 4905 T^{2} - 140608 T^{3} + 4826809 T^{4}$$)($$( 1 + 49 T + 2197 T^{2} )^{2}$$)($$1 + 11 T + 2568 T^{2} + 24167 T^{3} + 4826809 T^{4}$$)($$1 - 64 T + 4905 T^{2} - 140608 T^{3} + 4826809 T^{4}$$)($$1 + 20 T - 1797 T^{2} + 43940 T^{3} + 4826809 T^{4}$$)($$( 1 - 89 T + 2197 T^{2} )( 1 + 19 T + 2197 T^{2} )$$)($$1 + 20 T - 1797 T^{2} + 43940 T^{3} + 4826809 T^{4}$$)($$1 + 64 T - 809 T^{2} + 32704 T^{3} + 10233304 T^{4} + 71850688 T^{5} - 3904888481 T^{6} + 678687959872 T^{7} + 23298085122481 T^{8}$$)($$( 1 + 29 T - 1356 T^{2} + 63713 T^{3} + 4826809 T^{4} )^{2}$$)($$( 1 - 49 T + 204 T^{2} - 107653 T^{3} + 4826809 T^{4} )^{2}$$)($$1 + 64 T - 809 T^{2} + 32704 T^{3} + 10233304 T^{4} + 71850688 T^{5} - 3904888481 T^{6} + 678687959872 T^{7} + 23298085122481 T^{8}$$)
$17$ ($$1 + 99 T + 11608 T^{2} + 486387 T^{3} + 24137569 T^{4}$$)($$1 + 18 T + 1699 T^{2} + 88434 T^{3} + 24137569 T^{4}$$)($$1 + 8503 T^{2} + 24137569 T^{4}$$)($$1 - 99 T + 11608 T^{2} - 486387 T^{3} + 24137569 T^{4}$$)($$1 - 18 T + 1699 T^{2} - 88434 T^{3} + 24137569 T^{4}$$)($$( 1 - 72 T + 4913 T^{2} )^{2}$$)($$( 1 + 4913 T^{2} )^{2}$$)($$( 1 + 72 T + 4913 T^{2} )^{2}$$)($$( 1 - 18 T + 1699 T^{2} - 88434 T^{3} + 24137569 T^{4} )^{2}$$)($$( 1 + 7234 T^{2} + 24137569 T^{4} )^{2}$$)($$( 1 + 8503 T^{2} + 24137569 T^{4} )^{2}$$)($$( 1 + 18 T + 1699 T^{2} + 88434 T^{3} + 24137569 T^{4} )^{2}$$)
$19$ ($$1 + 77 T + 9186 T^{2} + 528143 T^{3} + 47045881 T^{4}$$)($$1 - 130 T + 13326 T^{2} - 891670 T^{3} + 47045881 T^{4}$$)($$( 1 + 70 T + 6859 T^{2} )^{2}$$)($$1 + 77 T + 9186 T^{2} + 528143 T^{3} + 47045881 T^{4}$$)($$1 - 130 T + 13326 T^{2} - 891670 T^{3} + 47045881 T^{4}$$)($$( 1 - 2 T + 6859 T^{2} )^{2}$$)($$( 1 - 56 T + 6859 T^{2} )^{2}$$)($$( 1 - 2 T + 6859 T^{2} )^{2}$$)($$( 1 - 130 T + 13326 T^{2} - 891670 T^{3} + 47045881 T^{4} )^{2}$$)($$( 1 - 29 T + 6859 T^{2} )^{4}$$)($$( 1 + 70 T + 6859 T^{2} )^{4}$$)($$( 1 - 130 T + 13326 T^{2} - 891670 T^{3} + 47045881 T^{4} )^{2}$$)
$23$ ($$1 + 33 T + 21628 T^{2} + 401511 T^{3} + 148035889 T^{4}$$)($$1 + 114 T + 20686 T^{2} + 1387038 T^{3} + 148035889 T^{4}$$)($$1 + 23746 T^{2} + 148035889 T^{4}$$)($$1 - 33 T + 21628 T^{2} - 401511 T^{3} + 148035889 T^{4}$$)($$1 - 114 T + 20686 T^{2} - 1387038 T^{3} + 148035889 T^{4}$$)($$1 + 114 T + 829 T^{2} + 1387038 T^{3} + 148035889 T^{4}$$)($$1 - 12167 T^{2} + 148035889 T^{4}$$)($$1 - 114 T + 829 T^{2} - 1387038 T^{3} + 148035889 T^{4}$$)($$1 + 114 T - 7690 T^{2} - 415872 T^{3} + 121752375 T^{4} - 5059914624 T^{5} - 1138395986410 T^{6} + 205331403406782 T^{7} + 21914624432020321 T^{8}$$)($$1 - 17134 T^{2} + 145538067 T^{4} - 2536446922126 T^{6} + 21914624432020321 T^{8}$$)($$1 - 23746 T^{2} + 415836627 T^{4} - 3515260220194 T^{6} + 21914624432020321 T^{8}$$)($$1 - 114 T - 7690 T^{2} + 415872 T^{3} + 121752375 T^{4} + 5059914624 T^{5} - 1138395986410 T^{6} - 205331403406782 T^{7} + 21914624432020321 T^{8}$$)
$29$ ($$1 - 51 T + 49420 T^{2} - 1243839 T^{3} + 594823321 T^{4}$$)($$1 - 240 T + 53545 T^{2} - 5853360 T^{3} + 594823321 T^{4}$$)($$1 - 28985 T^{2} + 594823321 T^{4}$$)($$1 + 51 T + 49420 T^{2} + 1243839 T^{3} + 594823321 T^{4}$$)($$1 + 240 T + 53545 T^{2} + 5853360 T^{3} + 594823321 T^{4}$$)($$1 + 30 T - 23489 T^{2} + 731670 T^{3} + 594823321 T^{4}$$)($$1 - 24389 T^{2} + 594823321 T^{4}$$)($$1 - 30 T - 23489 T^{2} - 731670 T^{3} + 594823321 T^{4}$$)($$1 - 240 T + 4055 T^{2} - 1144080 T^{3} + 867437304 T^{4} - 27902967120 T^{5} + 2412008566655 T^{6} - 3481715034208560 T^{7} + 353814783205469041 T^{8}$$)($$1 + 24950 T^{2} + 27679179 T^{4} + 14840841858950 T^{6} + 353814783205469041 T^{8}$$)($$1 + 28985 T^{2} + 245306904 T^{4} + 17240953959185 T^{6} + 353814783205469041 T^{8}$$)($$1 + 240 T + 4055 T^{2} + 1144080 T^{3} + 867437304 T^{4} + 27902967120 T^{5} + 2412008566655 T^{6} + 3481715034208560 T^{7} + 353814783205469041 T^{8}$$)
$31$ ($$1 - 43 T + 59970 T^{2} - 1281013 T^{3} + 887503681 T^{4}$$)($$1 + 8 T + 26766 T^{2} + 238328 T^{3} + 887503681 T^{4}$$)($$( 1 + 112 T + 29791 T^{2} )^{2}$$)($$1 - 43 T + 59970 T^{2} - 1281013 T^{3} + 887503681 T^{4}$$)($$1 + 8 T + 26766 T^{2} + 238328 T^{3} + 887503681 T^{4}$$)($$1 + 101 T - 19590 T^{2} + 3008891 T^{3} + 887503681 T^{4}$$)($$( 1 + 19 T + 29791 T^{2} )( 1 + 289 T + 29791 T^{2} )$$)($$1 + 101 T - 19590 T^{2} + 3008891 T^{3} + 887503681 T^{4}$$)($$1 - 8 T - 26702 T^{2} + 262528 T^{3} - 172991549 T^{4} + 7820971648 T^{5} - 23698123290062 T^{6} - 211516977285368 T^{7} + 787662783788549761 T^{8}$$)($$( 1 - 268 T + 42033 T^{2} - 7983988 T^{3} + 887503681 T^{4} )^{2}$$)($$( 1 - 112 T - 17247 T^{2} - 3336592 T^{3} + 887503681 T^{4} )^{2}$$)($$1 - 8 T - 26702 T^{2} + 262528 T^{3} - 172991549 T^{4} + 7820971648 T^{5} - 23698123290062 T^{6} - 211516977285368 T^{7} + 787662783788549761 T^{8}$$)
$37$ ($$1 + 50 T + 77874 T^{2} + 2532650 T^{3} + 2565726409 T^{4}$$)($$1 - 184 T + 105153 T^{2} - 9320152 T^{3} + 2565726409 T^{4}$$)($$( 1 - 281 T + 50653 T^{2} )^{2}$$)($$1 + 50 T + 77874 T^{2} + 2532650 T^{3} + 2565726409 T^{4}$$)($$1 - 184 T + 105153 T^{2} - 9320152 T^{3} + 2565726409 T^{4}$$)($$( 1 + 430 T + 50653 T^{2} )^{2}$$)($$( 1 - 110 T + 50653 T^{2} )^{2}$$)($$( 1 + 430 T + 50653 T^{2} )^{2}$$)($$( 1 - 184 T + 105153 T^{2} - 9320152 T^{3} + 2565726409 T^{4} )^{2}$$)($$( 1 - 83 T + 50653 T^{2} )^{4}$$)($$( 1 - 281 T + 50653 T^{2} )^{4}$$)($$( 1 - 184 T + 105153 T^{2} - 9320152 T^{3} + 2565726409 T^{4} )^{2}$$)
$41$ ($$1 + 132 T + 62965 T^{2} + 9097572 T^{3} + 4750104241 T^{4}$$)($$1 + 672 T + 239566 T^{2} + 46314912 T^{3} + 4750104241 T^{4}$$)($$1 + 135490 T^{2} + 4750104241 T^{4}$$)($$1 - 132 T + 62965 T^{2} - 9097572 T^{3} + 4750104241 T^{4}$$)($$1 - 672 T + 239566 T^{2} - 46314912 T^{3} + 4750104241 T^{4}$$)($$1 - 30 T - 68021 T^{2} - 2067630 T^{3} + 4750104241 T^{4}$$)($$1 - 68921 T^{2} + 4750104241 T^{4}$$)($$1 + 30 T - 68021 T^{2} + 2067630 T^{3} + 4750104241 T^{4}$$)($$1 + 672 T + 212018 T^{2} + 68358528 T^{3} + 21518143251 T^{4} + 4711338108288 T^{5} + 1007107600968338 T^{6} + 220000659912741792 T^{7} + 22563490300366186081 T^{8}$$)($$1 - 64114 T^{2} - 639499245 T^{4} - 304548183307474 T^{6} + 22563490300366186081 T^{8}$$)($$1 - 135490 T^{2} + 13607435859 T^{4} - 643591623613090 T^{6} + 22563490300366186081 T^{8}$$)($$1 - 672 T + 212018 T^{2} - 68358528 T^{3} + 21518143251 T^{4} - 4711338108288 T^{5} + 1007107600968338 T^{6} - 220000659912741792 T^{7} + 22563490300366186081 T^{8}$$)
$43$ ($$1 - 88 T + 160653 T^{2} - 6996616 T^{3} + 6321363049 T^{4}$$)($$1 - 370 T + 192726 T^{2} - 29417590 T^{3} + 6321363049 T^{4}$$)($$( 1 - 50 T + 79507 T^{2} )^{2}$$)($$1 - 88 T + 160653 T^{2} - 6996616 T^{3} + 6321363049 T^{4}$$)($$1 - 370 T + 192726 T^{2} - 29417590 T^{3} + 6321363049 T^{4}$$)($$1 + 110 T - 67407 T^{2} + 8745770 T^{3} + 6321363049 T^{4}$$)($$( 1 - 449 T + 79507 T^{2} )( 1 - 71 T + 79507 T^{2} )$$)($$1 + 110 T - 67407 T^{2} + 8745770 T^{3} + 6321363049 T^{4}$$)($$1 + 370 T - 55826 T^{2} + 12473440 T^{3} + 19937439727 T^{4} + 991725794080 T^{5} - 352896413573474 T^{6} + 185959266416631910 T^{7} + 39959630797262576401 T^{8}$$)($$( 1 - 232 T - 25683 T^{2} - 18445624 T^{3} + 6321363049 T^{4} )^{2}$$)($$( 1 + 50 T - 77007 T^{2} + 3975350 T^{3} + 6321363049 T^{4} )^{2}$$)($$1 + 370 T - 55826 T^{2} + 12473440 T^{3} + 19937439727 T^{4} + 991725794080 T^{5} - 352896413573474 T^{6} + 185959266416631910 T^{7} + 39959630797262576401 T^{8}$$)
$47$ ($$1 + 399 T + 179128 T^{2} + 41425377 T^{3} + 10779215329 T^{4}$$)($$1 - 276 T + 226462 T^{2} - 28655148 T^{3} + 10779215329 T^{4}$$)($$1 + 148846 T^{2} + 10779215329 T^{4}$$)($$1 - 399 T + 179128 T^{2} - 41425377 T^{3} + 10779215329 T^{4}$$)($$1 + 276 T + 226462 T^{2} + 28655148 T^{3} + 10779215329 T^{4}$$)($$1 - 330 T + 5077 T^{2} - 34261590 T^{3} + 10779215329 T^{4}$$)($$1 - 103823 T^{2} + 10779215329 T^{4}$$)($$1 + 330 T + 5077 T^{2} + 34261590 T^{3} + 10779215329 T^{4}$$)($$1 - 276 T - 150286 T^{2} - 5193216 T^{3} + 32597001267 T^{4} - 539175264768 T^{5} - 1619965154934094 T^{6} - 308880010576363692 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$)($$1 - 55294 T^{2} - 7721788893 T^{4} - 596025932401726 T^{6} +$$$$11\!\cdots\!41$$$$T^{8}$$)($$1 - 148846 T^{2} + 11375916387 T^{4} - 1604443084860334 T^{6} +$$$$11\!\cdots\!41$$$$T^{8}$$)($$1 + 276 T - 150286 T^{2} + 5193216 T^{3} + 32597001267 T^{4} + 539175264768 T^{5} - 1619965154934094 T^{6} + 308880010576363692 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$)
$53$ ($$1 + 54 T + 81970 T^{2} + 8039358 T^{3} + 22164361129 T^{4}$$)($$( 1 + 162 T + 148877 T^{2} )^{2}$$)($$1 + 157786 T^{2} + 22164361129 T^{4}$$)($$1 - 54 T + 81970 T^{2} - 8039358 T^{3} + 22164361129 T^{4}$$)($$( 1 - 162 T + 148877 T^{2} )^{2}$$)($$( 1 - 621 T + 148877 T^{2} )^{2}$$)($$( 1 + 148877 T^{2} )^{2}$$)($$( 1 + 621 T + 148877 T^{2} )^{2}$$)($$( 1 - 162 T + 148877 T^{2} )^{4}$$)($$( 1 + 204442 T^{2} + 22164361129 T^{4} )^{2}$$)($$( 1 + 157786 T^{2} + 22164361129 T^{4} )^{2}$$)($$( 1 + 162 T + 148877 T^{2} )^{4}$$)
$59$ ($$1 + 798 T + 417367 T^{2} + 163892442 T^{3} + 42180533641 T^{4}$$)($$1 + 204 T + 300550 T^{2} + 41897316 T^{3} + 42180533641 T^{4}$$)($$1 + 401350 T^{2} + 42180533641 T^{4}$$)($$1 - 798 T + 417367 T^{2} - 163892442 T^{3} + 42180533641 T^{4}$$)($$1 - 204 T + 300550 T^{2} - 41897316 T^{3} + 42180533641 T^{4}$$)($$1 - 660 T + 230221 T^{2} - 135550140 T^{3} + 42180533641 T^{4}$$)($$1 - 205379 T^{2} + 42180533641 T^{4}$$)($$1 + 660 T + 230221 T^{2} + 135550140 T^{3} + 42180533641 T^{4}$$)($$1 + 204 T - 258934 T^{2} - 22482432 T^{3} + 39602716395 T^{4} - 4617419401728 T^{5} - 10921974297798694 T^{6} + 1767251147005607556 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$)($$1 - 327526 T^{2} + 65092747035 T^{4} - 13815221461302166 T^{6} +$$$$17\!\cdots\!81$$$$T^{8}$$)($$1 - 401350 T^{2} + 118901288859 T^{4} - 16929157176815350 T^{6} +$$$$17\!\cdots\!81$$$$T^{8}$$)($$1 - 204 T - 258934 T^{2} + 22482432 T^{3} + 39602716395 T^{4} + 4617419401728 T^{5} - 10921974297798694 T^{6} - 1767251147005607556 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$)
$61$ ($$1 - 439 T + 411186 T^{2} - 99644659 T^{3} + 51520374361 T^{4}$$)($$1 - 208 T + 193401 T^{2} - 47212048 T^{3} + 51520374361 T^{4}$$)($$( 1 + 679 T + 226981 T^{2} )^{2}$$)($$1 - 439 T + 411186 T^{2} - 99644659 T^{3} + 51520374361 T^{4}$$)($$1 - 208 T + 193401 T^{2} - 47212048 T^{3} + 51520374361 T^{4}$$)($$1 - 376 T - 85605 T^{2} - 85344856 T^{3} + 51520374361 T^{4}$$)($$( 1 - 719 T + 226981 T^{2} )( 1 + 901 T + 226981 T^{2} )$$)($$1 - 376 T - 85605 T^{2} - 85344856 T^{3} + 51520374361 T^{4}$$)($$1 + 208 T - 150137 T^{2} - 54196688 T^{3} - 23936533544 T^{4} - 12301618438928 T^{5} - 7735114445437457 T^{6} + 2432382387309501328 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$)($$( 1 + 767 T + 361308 T^{2} + 174094427 T^{3} + 51520374361 T^{4} )^{2}$$)($$( 1 - 679 T + 234060 T^{2} - 154120099 T^{3} + 51520374361 T^{4} )^{2}$$)($$1 + 208 T - 150137 T^{2} - 54196688 T^{3} - 23936533544 T^{4} - 12301618438928 T^{5} - 7735114445437457 T^{6} + 2432382387309501328 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$)
$67$ ($$1 - 988 T + 809625 T^{2} - 297153844 T^{3} + 90458382169 T^{4}$$)($$1 - 802 T + 675630 T^{2} - 241211926 T^{3} + 90458382169 T^{4}$$)($$( 1 + 274 T + 300763 T^{2} )^{2}$$)($$1 - 988 T + 809625 T^{2} - 297153844 T^{3} + 90458382169 T^{4}$$)($$1 - 802 T + 675630 T^{2} - 241211926 T^{3} + 90458382169 T^{4}$$)($$1 - 250 T - 238263 T^{2} - 75190750 T^{3} + 90458382169 T^{4}$$)($$( 1 - 1007 T + 300763 T^{2} )( 1 + 127 T + 300763 T^{2} )$$)($$1 - 250 T - 238263 T^{2} - 75190750 T^{3} + 90458382169 T^{4}$$)($$1 + 802 T - 32426 T^{2} + 59431408 T^{3} + 172565550079 T^{4} + 17874768564304 T^{5} - 2933203500211994 T^{6} + 21819640585828547494 T^{7} +$$$$81\!\cdots\!61$$$$T^{8}$$)($$( 1 - 511 T - 39642 T^{2} - 153689893 T^{3} + 90458382169 T^{4} )^{2}$$)($$( 1 - 274 T - 225687 T^{2} - 82409062 T^{3} + 90458382169 T^{4} )^{2}$$)($$1 + 802 T - 32426 T^{2} + 59431408 T^{3} + 172565550079 T^{4} + 17874768564304 T^{5} - 2933203500211994 T^{6} + 21819640585828547494 T^{7} +$$$$81\!\cdots\!61$$$$T^{8}$$)
$71$ ($$1 + 1368 T + 1012606 T^{2} + 489622248 T^{3} + 128100283921 T^{4}$$)($$1 + 126 T + 534598 T^{2} + 45096786 T^{3} + 128100283921 T^{4}$$)($$1 + 516130 T^{2} + 128100283921 T^{4}$$)($$1 - 1368 T + 1012606 T^{2} - 489622248 T^{3} + 128100283921 T^{4}$$)($$1 - 126 T + 534598 T^{2} - 45096786 T^{3} + 128100283921 T^{4}$$)($$( 1 + 360 T + 357911 T^{2} )^{2}$$)($$( 1 + 357911 T^{2} )^{2}$$)($$( 1 - 360 T + 357911 T^{2} )^{2}$$)($$( 1 - 126 T + 534598 T^{2} - 45096786 T^{3} + 128100283921 T^{4} )^{2}$$)($$( 1 + 207790 T^{2} + 128100283921 T^{4} )^{2}$$)($$( 1 + 516130 T^{2} + 128100283921 T^{4} )^{2}$$)($$( 1 + 126 T + 534598 T^{2} + 45096786 T^{3} + 128100283921 T^{4} )^{2}$$)
$73$ ($$1 + 455 T + 342636 T^{2} + 177002735 T^{3} + 151334226289 T^{4}$$)($$1 + 374 T + 646791 T^{2} + 145492358 T^{3} + 151334226289 T^{4}$$)($$( 1 + 511 T + 389017 T^{2} )^{2}$$)($$1 + 455 T + 342636 T^{2} + 177002735 T^{3} + 151334226289 T^{4}$$)($$1 + 374 T + 646791 T^{2} + 145492358 T^{3} + 151334226289 T^{4}$$)($$( 1 - 785 T + 389017 T^{2} )^{2}$$)($$( 1 - 1190 T + 389017 T^{2} )^{2}$$)($$( 1 - 785 T + 389017 T^{2} )^{2}$$)($$( 1 + 374 T + 646791 T^{2} + 145492358 T^{3} + 151334226289 T^{4} )^{2}$$)($$( 1 - 137 T + 389017 T^{2} )^{4}$$)($$( 1 + 511 T + 389017 T^{2} )^{4}$$)($$( 1 + 374 T + 646791 T^{2} + 145492358 T^{3} + 151334226289 T^{4} )^{2}$$)
$79$ ($$1 + 803 T + 359562 T^{2} + 395910317 T^{3} + 243087455521 T^{4}$$)($$1 - 1378 T + 1460286 T^{2} - 679407742 T^{3} + 243087455521 T^{4}$$)($$( 1 + 526 T + 493039 T^{2} )^{2}$$)($$1 + 803 T + 359562 T^{2} + 395910317 T^{3} + 243087455521 T^{4}$$)($$1 - 1378 T + 1460286 T^{2} - 679407742 T^{3} + 243087455521 T^{4}$$)($$1 + 488 T - 254895 T^{2} + 240603032 T^{3} + 243087455521 T^{4}$$)($$( 1 - 503 T + 493039 T^{2} )( 1 + 1387 T + 493039 T^{2} )$$)($$1 + 488 T - 254895 T^{2} + 240603032 T^{3} + 243087455521 T^{4}$$)($$1 + 1378 T + 438598 T^{2} + 653458624 T^{3} + 953123877799 T^{4} + 322180586518336 T^{5} + 106617671816599558 T^{6} +$$$$16\!\cdots\!82$$$$T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$)($$( 1 - 475 T - 267414 T^{2} - 234193525 T^{3} + 243087455521 T^{4} )^{2}$$)($$( 1 - 526 T - 216363 T^{2} - 259338514 T^{3} + 243087455521 T^{4} )^{2}$$)($$1 + 1378 T + 438598 T^{2} + 653458624 T^{3} + 953123877799 T^{4} + 322180586518336 T^{5} + 106617671816599558 T^{6} +$$$$16\!\cdots\!82$$$$T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$)
$83$ ($$1 + 813 T + 1214362 T^{2} + 464862831 T^{3} + 326940373369 T^{4}$$)($$1 + 1164 T + 1339798 T^{2} + 665560068 T^{3} + 326940373369 T^{4}$$)($$1 + 993046 T^{2} + 326940373369 T^{4}$$)($$1 - 813 T + 1214362 T^{2} - 464862831 T^{3} + 326940373369 T^{4}$$)($$1 - 1164 T + 1339798 T^{2} - 665560068 T^{3} + 326940373369 T^{4}$$)($$1 + 489 T - 332666 T^{2} + 279603843 T^{3} + 326940373369 T^{4}$$)($$1 - 571787 T^{2} + 326940373369 T^{4}$$)($$1 - 489 T - 332666 T^{2} - 279603843 T^{3} + 326940373369 T^{4}$$)($$1 + 1164 T + 15098 T^{2} + 228404736 T^{3} + 693406388283 T^{4} + 130598858783232 T^{5} + 4936145757125162 T^{6} +$$$$21\!\cdots\!92$$$$T^{7} +$$$$10\!\cdots\!61$$$$T^{8}$$)($$1 - 810646 T^{2} + 330206563947 T^{4} - 265032905910086374 T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$)($$1 - 993046 T^{2} + 659199984747 T^{4} - 324666830012591974 T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$)($$1 - 1164 T + 15098 T^{2} - 228404736 T^{3} + 693406388283 T^{4} - 130598858783232 T^{5} + 4936145757125162 T^{6} -$$$$21\!\cdots\!92$$$$T^{7} +$$$$10\!\cdots\!61$$$$T^{8}$$)
$89$ ($$1 - 396 T + 1406374 T^{2} - 279167724 T^{3} + 496981290961 T^{4}$$)($$1 + 954 T + 816667 T^{2} + 672540426 T^{3} + 496981290961 T^{4}$$)($$1 + 1408615 T^{2} + 496981290961 T^{4}$$)($$1 + 396 T + 1406374 T^{2} + 279167724 T^{3} + 496981290961 T^{4}$$)($$1 - 954 T + 816667 T^{2} - 672540426 T^{3} + 496981290961 T^{4}$$)($$( 1 + 450 T + 704969 T^{2} )^{2}$$)($$( 1 + 704969 T^{2} )^{2}$$)($$( 1 - 450 T + 704969 T^{2} )^{2}$$)($$( 1 - 954 T + 816667 T^{2} - 672540426 T^{3} + 496981290961 T^{4} )^{2}$$)($$( 1 + 1345138 T^{2} + 496981290961 T^{4} )^{2}$$)($$( 1 + 1408615 T^{2} + 496981290961 T^{4} )^{2}$$)($$( 1 + 954 T + 816667 T^{2} + 672540426 T^{3} + 496981290961 T^{4} )^{2}$$)
$97$ ($$1 - 736 T + 1874937 T^{2} - 671727328 T^{3} + 832972004929 T^{4}$$)($$1 + 872 T + 1964142 T^{2} + 795850856 T^{3} + 832972004929 T^{4}$$)($$( 1 - 1778 T + 912673 T^{2} )^{2}$$)($$1 - 736 T + 1874937 T^{2} - 671727328 T^{3} + 832972004929 T^{4}$$)($$1 + 872 T + 1964142 T^{2} + 795850856 T^{3} + 832972004929 T^{4}$$)($$1 - 1105 T + 308352 T^{2} - 1008503665 T^{3} + 832972004929 T^{4}$$)($$( 1 - 1853 T + 912673 T^{2} )( 1 + 523 T + 912673 T^{2} )$$)($$1 - 1105 T + 308352 T^{2} - 1008503665 T^{3} + 832972004929 T^{4}$$)($$1 - 872 T - 1203758 T^{2} - 121030112 T^{3} + 2330899844803 T^{4} - 110460915409376 T^{5} - 1002696714709323182 T^{6} -$$$$66\!\cdots\!24$$$$T^{7} +$$$$69\!\cdots\!41$$$$T^{8}$$)($$( 1 + 821 T - 238632 T^{2} + 749304533 T^{3} + 832972004929 T^{4} )^{2}$$)($$( 1 + 1778 T + 2248611 T^{2} + 1622732594 T^{3} + 832972004929 T^{4} )^{2}$$)($$1 - 872 T - 1203758 T^{2} - 121030112 T^{3} + 2330899844803 T^{4} - 110460915409376 T^{5} - 1002696714709323182 T^{6} -$$$$66\!\cdots\!24$$$$T^{7} +$$$$69\!\cdots\!41$$$$T^{8}$$)