# Properties

 Label 175.2 Level 175 Weight 2 Dimension 959 Nonzero newspaces 12 Newform subspaces 32 Sturm bound 4800 Trace bound 2

## Defining parameters

 Level: $$N$$ = $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$32$$ Sturm bound: $$4800$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(175))$$.

Total New Old
Modular forms 1368 1163 205
Cusp forms 1033 959 74
Eisenstein series 335 204 131

## Trace form

 $$959q - 25q^{2} - 28q^{3} - 37q^{4} - 38q^{5} - 60q^{6} - 41q^{7} - 97q^{8} - 55q^{9} + O(q^{10})$$ $$959q - 25q^{2} - 28q^{3} - 37q^{4} - 38q^{5} - 60q^{6} - 41q^{7} - 97q^{8} - 55q^{9} - 58q^{10} - 60q^{11} - 100q^{12} - 58q^{13} - 71q^{14} - 116q^{15} - 85q^{16} - 50q^{17} - 83q^{18} - 36q^{19} - 28q^{20} - 82q^{21} - 80q^{22} - 48q^{23} + 40q^{24} - 6q^{25} - 102q^{26} - 4q^{27} + 25q^{28} - 54q^{29} + 4q^{30} - 32q^{31} + 9q^{32} - 8q^{34} - 32q^{35} - 53q^{36} - 48q^{37} - 40q^{38} + 16q^{39} - 6q^{40} - 62q^{41} + 38q^{42} - 56q^{43} - 32q^{44} + 38q^{45} - 72q^{46} - 56q^{47} + 56q^{48} - 105q^{49} - 46q^{50} - 160q^{51} + 6q^{52} - 88q^{53} + 28q^{54} - 44q^{55} - 79q^{56} - 36q^{57} + 46q^{58} + 92q^{60} + 14q^{61} + 132q^{62} + 61q^{63} + 75q^{64} + 14q^{65} + 48q^{66} + 52q^{67} + 158q^{68} + 52q^{69} + 102q^{70} - 124q^{71} + 147q^{72} + 34q^{73} + 126q^{74} + 28q^{75} + 28q^{76} + 18q^{77} - 8q^{78} + 16q^{79} + 98q^{80} - 91q^{81} + 86q^{82} + 48q^{83} + 90q^{84} + 22q^{85} - 36q^{86} + 4q^{87} + 180q^{88} + 84q^{89} + 170q^{90} - 56q^{91} + 12q^{92} + 28q^{93} + 64q^{94} + 28q^{95} - 4q^{96} + 114q^{97} + 125q^{98} - 68q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(175))$$

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
175.2.a $$\chi_{175}(1, \cdot)$$ 175.2.a.a 1 1
175.2.a.b 1
175.2.a.c 1
175.2.a.d 2
175.2.a.e 2
175.2.a.f 2
175.2.b $$\chi_{175}(99, \cdot)$$ 175.2.b.a 2 1
175.2.b.b 4
175.2.b.c 4
175.2.e $$\chi_{175}(51, \cdot)$$ 175.2.e.a 2 2
175.2.e.b 2
175.2.e.c 4
175.2.e.d 6
175.2.e.e 6
175.2.f $$\chi_{175}(118, \cdot)$$ 175.2.f.a 4 2
175.2.f.b 4
175.2.f.c 4
175.2.f.d 8
175.2.h $$\chi_{175}(36, \cdot)$$ 175.2.h.a 4 4
175.2.h.b 28
175.2.h.c 32
175.2.k $$\chi_{175}(74, \cdot)$$ 175.2.k.a 8 2
175.2.k.b 12
175.2.n $$\chi_{175}(29, \cdot)$$ 175.2.n.a 56 4
175.2.o $$\chi_{175}(68, \cdot)$$ 175.2.o.a 4 4
175.2.o.b 4
175.2.o.c 8
175.2.o.d 24
175.2.q $$\chi_{175}(11, \cdot)$$ 175.2.q.a 144 8
175.2.s $$\chi_{175}(13, \cdot)$$ 175.2.s.a 144 8
175.2.t $$\chi_{175}(4, \cdot)$$ 175.2.t.a 144 8
175.2.x $$\chi_{175}(3, \cdot)$$ 175.2.x.a 288 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(175))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(175)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + 2 T + 2 T^{2}$$)($$1 + 2 T^{2}$$)($$1 - 2 T + 2 T^{2}$$)($$1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4}$$)($$1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4}$$)($$1 - T - 2 T^{3} + 4 T^{4}$$)($$( 1 - 2 T^{2} )^{2}$$)($$1 + T^{2} + 4 T^{4} + 4 T^{6} + 16 T^{8}$$)($$1 - 5 T^{2} + 13 T^{4} - 20 T^{6} + 16 T^{8}$$)($$1 + T - T^{2} + 2 T^{3} + 4 T^{4}$$)($$1 - T - T^{2} - 2 T^{3} + 4 T^{4}$$)($$1 - 2 T + T^{2} + 2 T^{3} - 3 T^{4} + 4 T^{5} + 4 T^{6} - 16 T^{7} + 16 T^{8}$$)($$1 + T - T^{2} - T^{4} - 2 T^{5} + 5 T^{6} - 4 T^{7} - 4 T^{8} - 16 T^{10} + 32 T^{11} + 64 T^{12}$$)($$1 - T - T^{2} - T^{4} + 2 T^{5} + 5 T^{6} + 4 T^{7} - 4 T^{8} - 16 T^{10} - 32 T^{11} + 64 T^{12}$$)($$( 1 - 2 T + 2 T^{2} )^{2}( 1 + 2 T + 2 T^{2} )^{2}$$)($$1 + T^{4} + 16 T^{8}$$)($$( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2}$$)($$1 - T^{4} - 15 T^{8} - 16 T^{12} + 256 T^{16}$$)($$1 + 5 T + 13 T^{2} + 25 T^{3} + 39 T^{4} + 50 T^{5} + 52 T^{6} + 40 T^{7} + 16 T^{8}$$)($$1 + 2 T^{2} + 3 T^{4} - 14 T^{6} - 31 T^{8} - 56 T^{10} + 48 T^{12} + 128 T^{14} + 256 T^{16}$$)($$1 + 3 T^{2} - T^{4} - 16 T^{6} - 9 T^{8} + 38 T^{10} + 113 T^{12} + 152 T^{14} - 144 T^{16} - 1024 T^{18} - 256 T^{20} + 3072 T^{22} + 4096 T^{24}$$)($$1 + 2 T + 5 T^{2} + 8 T^{3} + 13 T^{4} + 16 T^{5} + 20 T^{6} + 16 T^{7} + 16 T^{8}$$)($$1 - 4 T + 5 T^{2} + 2 T^{3} - 11 T^{4} + 4 T^{5} + 20 T^{6} - 32 T^{7} + 16 T^{8}$$)($$1 - T^{4} - 15 T^{8} - 16 T^{12} + 256 T^{16}$$)
$3$ ($$1 + T + 3 T^{2}$$)($$1 + T + 3 T^{2}$$)($$1 - T + 3 T^{2}$$)($$1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4}$$)($$1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}$$)($$1 - T + 2 T^{2} - 3 T^{3} + 9 T^{4}$$)($$1 - 5 T^{2} + 9 T^{4}$$)($$1 - 3 T^{2} + 16 T^{4} - 27 T^{6} + 81 T^{8}$$)($$( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} )( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} )$$)($$1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}$$)($$1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4}$$)($$1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 6 T^{5} - 9 T^{6} - 54 T^{7} + 81 T^{8}$$)($$1 - 3 T + 2 T^{2} + T^{3} - 5 T^{4} + 22 T^{5} - 53 T^{6} + 66 T^{7} - 45 T^{8} + 27 T^{9} + 162 T^{10} - 729 T^{11} + 729 T^{12}$$)($$1 + 3 T + 2 T^{2} - T^{3} - 5 T^{4} - 22 T^{5} - 53 T^{6} - 66 T^{7} - 45 T^{8} - 27 T^{9} + 162 T^{10} + 729 T^{11} + 729 T^{12}$$)($$1 - 17 T^{4} + 81 T^{8}$$)($$( 1 + 9 T^{4} )^{2}$$)($$1 - 17 T^{4} + 81 T^{8}$$)($$( 1 + 9 T^{4} )^{4}$$)($$1 + 4 T + 13 T^{2} + 30 T^{3} + 61 T^{4} + 90 T^{5} + 117 T^{6} + 108 T^{7} + 81 T^{8}$$)($$1 + 6 T^{2} + 17 T^{4} + 6 T^{6} - 44 T^{8} + 54 T^{10} + 1377 T^{12} + 4374 T^{14} + 6561 T^{16}$$)($$1 + 5 T^{2} - 5 T^{6} + 75 T^{8} - 220 T^{10} - 1967 T^{12} - 1980 T^{14} + 6075 T^{16} - 3645 T^{18} + 295245 T^{22} + 531441 T^{24}$$)($$1 - 4 T + 5 T^{2} + 4 T^{3} - 20 T^{4} + 12 T^{5} + 45 T^{6} - 108 T^{7} + 81 T^{8}$$)($$1 - 2 T + 5 T^{2} - 10 T^{3} + 16 T^{4} - 30 T^{5} + 45 T^{6} - 54 T^{7} + 81 T^{8}$$)($$( 1 + 9 T^{4} )^{2}( 1 - 9 T^{4} + 81 T^{8} )$$)
$5$ ()()()()()()()()()()()()()()()()()()($$1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4}$$)()()()()()
$7$ ($$1 - T$$)($$1 + T$$)($$1 + T$$)($$( 1 + T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{2}$$)($$1 + T^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$1 + T + 7 T^{2}$$)($$1 - T + 7 T^{2}$$)($$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$$)($$1 - 2 T - 4 T^{2} + 13 T^{3} - 28 T^{4} - 98 T^{5} + 343 T^{6}$$)($$1 + 2 T - 4 T^{2} - 13 T^{3} - 28 T^{4} + 98 T^{5} + 343 T^{6}$$)($$1 + 49 T^{4}$$)($$1 + 49 T^{4}$$)($$1 + 4 T + 8 T^{2} + 28 T^{3} + 49 T^{4}$$)($$( 1 + 49 T^{4} )^{2}$$)($$( 1 + T )^{4}$$)($$1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8}$$)($$1 + 12 T^{2} + 12 T^{4} - 349 T^{6} + 588 T^{8} + 28812 T^{10} + 117649 T^{12}$$)($$( 1 - 5 T + 7 T^{2} )^{2}$$)($$1 + 11 T^{2} + 49 T^{4}$$)($$1 + 23 T^{4} + 2401 T^{8}$$)
$11$ ($$1 + 3 T + 11 T^{2}$$)($$1 + 3 T + 11 T^{2}$$)($$1 + 3 T + 11 T^{2}$$)($$1 - 4 T + 21 T^{2} - 44 T^{3} + 121 T^{4}$$)($$1 - 4 T + 21 T^{2} - 44 T^{3} + 121 T^{4}$$)($$1 - T + 18 T^{2} - 11 T^{3} + 121 T^{4}$$)($$( 1 + 3 T + 11 T^{2} )^{2}$$)($$( 1 - T + 18 T^{2} - 11 T^{3} + 121 T^{4} )^{2}$$)($$( 1 - 4 T + 21 T^{2} - 44 T^{3} + 121 T^{4} )^{2}$$)($$1 - 11 T^{2} + 121 T^{4}$$)($$1 - 11 T^{2} + 121 T^{4}$$)($$1 + 4 T - 2 T^{2} - 16 T^{3} + 27 T^{4} - 176 T^{5} - 242 T^{6} + 5324 T^{7} + 14641 T^{8}$$)($$1 - 2 T - 10 T^{2} - 30 T^{3} + 44 T^{4} + 388 T^{5} + 23 T^{6} + 4268 T^{7} + 5324 T^{8} - 39930 T^{9} - 146410 T^{10} - 322102 T^{11} + 1771561 T^{12}$$)($$1 - 2 T - 10 T^{2} - 30 T^{3} + 44 T^{4} + 388 T^{5} + 23 T^{6} + 4268 T^{7} + 5324 T^{8} - 39930 T^{9} - 146410 T^{10} - 322102 T^{11} + 1771561 T^{12}$$)($$( 1 + 3 T + 11 T^{2} )^{4}$$)($$( 1 - 4 T + 11 T^{2} )^{4}$$)($$( 1 + T + 11 T^{2} )^{4}$$)($$( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} )^{4}$$)($$1 + 2 T - 7 T^{2} - 36 T^{3} + 5 T^{4} - 396 T^{5} - 847 T^{6} + 2662 T^{7} + 14641 T^{8}$$)($$( 1 + 4 T - 2 T^{2} - 16 T^{3} + 27 T^{4} - 176 T^{5} - 242 T^{6} + 5324 T^{7} + 14641 T^{8} )^{2}$$)($$( 1 - 2 T - 10 T^{2} - 30 T^{3} + 44 T^{4} + 388 T^{5} + 23 T^{6} + 4268 T^{7} + 5324 T^{8} - 39930 T^{9} - 146410 T^{10} - 322102 T^{11} + 1771561 T^{12} )^{2}$$)($$1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 44 T^{5} - 1936 T^{6} - 2662 T^{7} + 14641 T^{8}$$)($$1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 44 T^{5} - 1936 T^{6} - 2662 T^{7} + 14641 T^{8}$$)($$( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} )^{4}$$)
$13$ ($$1 + T + 13 T^{2}$$)($$1 + 5 T + 13 T^{2}$$)($$1 - T + 13 T^{2}$$)($$1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4}$$)($$1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4}$$)($$1 + 5 T + 28 T^{2} + 65 T^{3} + 169 T^{4}$$)($$1 - T^{2} + 169 T^{4}$$)($$1 - 31 T^{2} + 472 T^{4} - 5239 T^{6} + 28561 T^{8}$$)($$1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8}$$)($$( 1 - 2 T + 13 T^{2} )^{2}$$)($$( 1 + 2 T + 13 T^{2} )^{2}$$)($$( 1 - 4 T + 22 T^{2} - 52 T^{3} + 169 T^{4} )^{2}$$)($$( 1 + 4 T + 24 T^{2} + 111 T^{3} + 312 T^{4} + 676 T^{5} + 2197 T^{6} )^{2}$$)($$( 1 - 4 T + 24 T^{2} - 111 T^{3} + 312 T^{4} - 676 T^{5} + 2197 T^{6} )^{2}$$)($$1 + 23 T^{4} + 28561 T^{8}$$)($$( 1 + 169 T^{4} )^{2}$$)($$1 + 103 T^{4} + 28561 T^{8}$$)($$( 1 + 169 T^{4} )^{4}$$)($$1 - 3 T^{2} + 40 T^{3} + 129 T^{4} + 520 T^{5} - 507 T^{6} + 28561 T^{8}$$)($$( 1 - 28 T^{2} + 406 T^{4} - 4732 T^{6} + 28561 T^{8} )^{2}$$)($$( 1 - 32 T^{2} + 312 T^{4} - 1641 T^{6} + 52728 T^{8} - 913952 T^{10} + 4826809 T^{12} )^{2}$$)($$( 1 - 4 T + 8 T^{2} - 52 T^{3} + 169 T^{4} )^{2}$$)($$( 1 + 4 T + 8 T^{2} + 52 T^{3} + 169 T^{4} )^{2}$$)($$( 1 - 142 T^{4} + 28561 T^{8} )^{2}$$)
$17$ ($$1 + 7 T + 17 T^{2}$$)($$1 + 3 T + 17 T^{2}$$)($$1 - 7 T + 17 T^{2}$$)($$1 - 4 T + 18 T^{2} - 68 T^{3} + 289 T^{4}$$)($$1 + 4 T + 18 T^{2} + 68 T^{3} + 289 T^{4}$$)($$1 - 5 T + 36 T^{2} - 85 T^{3} + 289 T^{4}$$)($$1 - 25 T^{2} + 289 T^{4}$$)($$1 - 47 T^{2} + 1024 T^{4} - 13583 T^{6} + 83521 T^{8}$$)($$1 - 20 T^{2} + 358 T^{4} - 5780 T^{6} + 83521 T^{8}$$)($$1 + 2 T - 13 T^{2} + 34 T^{3} + 289 T^{4}$$)($$1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4}$$)($$1 - 4 T - 14 T^{2} + 16 T^{3} + 339 T^{4} + 272 T^{5} - 4046 T^{6} - 19652 T^{7} + 83521 T^{8}$$)($$1 - 10 T + 20 T^{2} - 78 T^{3} + 1430 T^{4} - 4570 T^{5} + 2447 T^{6} - 77690 T^{7} + 413270 T^{8} - 383214 T^{9} + 1670420 T^{10} - 14198570 T^{11} + 24137569 T^{12}$$)($$1 + 10 T + 20 T^{2} + 78 T^{3} + 1430 T^{4} + 4570 T^{5} + 2447 T^{6} + 77690 T^{7} + 413270 T^{8} + 383214 T^{9} + 1670420 T^{10} + 14198570 T^{11} + 24137569 T^{12}$$)($$1 + 263 T^{4} + 83521 T^{8}$$)($$( 1 + 289 T^{4} )^{2}$$)($$1 + 263 T^{4} + 83521 T^{8}$$)($$( 1 + 289 T^{4} )^{4}$$)($$1 + 12 T + 77 T^{2} + 420 T^{3} + 1981 T^{4} + 7140 T^{5} + 22253 T^{6} + 58956 T^{7} + 83521 T^{8}$$)($$1 + 44 T^{2} + 1002 T^{4} + 15664 T^{6} + 229331 T^{8} + 4526896 T^{10} + 83688042 T^{12} + 1062053036 T^{14} + 6975757441 T^{16}$$)($$1 + 60 T^{2} + 1700 T^{4} + 35390 T^{6} + 702600 T^{8} + 13798760 T^{10} + 249192563 T^{12} + 3987841640 T^{14} + 58681854600 T^{16} + 854228566910 T^{18} + 11858787649700 T^{20} + 120959634026940 T^{22} + 582622237229761 T^{24}$$)($$1 + 8 T + 20 T^{2} - 52 T^{3} - 545 T^{4} - 884 T^{5} + 5780 T^{6} + 39304 T^{7} + 83521 T^{8}$$)($$1 + 4 T + 20 T^{2} + 100 T^{3} + 271 T^{4} + 1700 T^{5} + 5780 T^{6} + 19652 T^{7} + 83521 T^{8}$$)($$1 + 94 T^{4} - 74685 T^{8} + 7850974 T^{12} + 6975757441 T^{16}$$)
$19$ ($$1 + 19 T^{2}$$)($$1 - 2 T + 19 T^{2}$$)($$1 + 19 T^{2}$$)($$1 + 18 T^{2} + 361 T^{4}$$)($$1 + 18 T^{2} + 361 T^{4}$$)($$1 + 6 T + 30 T^{2} + 114 T^{3} + 361 T^{4}$$)($$( 1 + 2 T + 19 T^{2} )^{2}$$)($$( 1 - 6 T + 30 T^{2} - 114 T^{3} + 361 T^{4} )^{2}$$)($$( 1 + 18 T^{2} + 361 T^{4} )^{2}$$)($$1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4}$$)($$1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4}$$)($$1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8}$$)($$1 - 4 T - 24 T^{2} + 158 T^{3} + 204 T^{4} - 1724 T^{5} + 2823 T^{6} - 32756 T^{7} + 73644 T^{8} + 1083722 T^{9} - 3127704 T^{10} - 9904396 T^{11} + 47045881 T^{12}$$)($$1 - 4 T - 24 T^{2} + 158 T^{3} + 204 T^{4} - 1724 T^{5} + 2823 T^{6} - 32756 T^{7} + 73644 T^{8} + 1083722 T^{9} - 3127704 T^{10} - 9904396 T^{11} + 47045881 T^{12}$$)($$( 1 + 19 T^{2} )^{4}$$)($$( 1 + 19 T^{2} )^{4}$$)($$( 1 + 28 T^{2} + 361 T^{4} )^{2}$$)($$( 1 + 19 T^{2} )^{8}$$)($$1 + 2 T - 15 T^{2} - 68 T^{3} + 149 T^{4} - 1292 T^{5} - 5415 T^{6} + 13718 T^{7} + 130321 T^{8}$$)($$( 1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 + 4 T - 24 T^{2} - 158 T^{3} + 204 T^{4} + 1724 T^{5} + 2823 T^{6} + 32756 T^{7} + 73644 T^{8} - 1083722 T^{9} - 3127704 T^{10} + 9904396 T^{11} + 47045881 T^{12} )^{2}$$)($$1 + 2 T - 32 T^{2} - 4 T^{3} + 859 T^{4} - 76 T^{5} - 11552 T^{6} + 13718 T^{7} + 130321 T^{8}$$)($$1 - 2 T - 32 T^{2} + 4 T^{3} + 859 T^{4} + 76 T^{5} - 11552 T^{6} - 13718 T^{7} + 130321 T^{8}$$)($$( 1 - 37 T^{2} + 361 T^{4} )^{2}( 1 + 11 T^{2} + 361 T^{4} )^{2}$$)
$23$ ($$1 + 6 T + 23 T^{2}$$)($$1 - 6 T + 23 T^{2}$$)($$1 - 6 T + 23 T^{2}$$)($$1 + 8 T + 57 T^{2} + 184 T^{3} + 529 T^{4}$$)($$1 - 8 T + 57 T^{2} - 184 T^{3} + 529 T^{4}$$)($$1 - 2 T + 30 T^{2} - 46 T^{3} + 529 T^{4}$$)($$1 - 10 T^{2} + 529 T^{4}$$)($$1 - 56 T^{2} + 1774 T^{4} - 29624 T^{6} + 279841 T^{8}$$)($$1 - 50 T^{2} + 1363 T^{4} - 26450 T^{6} + 279841 T^{8}$$)($$1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4}$$)($$1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4}$$)($$1 + 2 T - 41 T^{2} - 2 T^{3} + 1404 T^{4} - 46 T^{5} - 21689 T^{6} + 24334 T^{7} + 279841 T^{8}$$)($$1 - T - 58 T^{2} + 51 T^{3} + 2069 T^{4} - 1060 T^{5} - 53233 T^{6} - 24380 T^{7} + 1094501 T^{8} + 620517 T^{9} - 16230778 T^{10} - 6436343 T^{11} + 148035889 T^{12}$$)($$1 + T - 58 T^{2} - 51 T^{3} + 2069 T^{4} + 1060 T^{5} - 53233 T^{6} + 24380 T^{7} + 1094501 T^{8} - 620517 T^{9} - 16230778 T^{10} + 6436343 T^{11} + 148035889 T^{12}$$)($$( 1 + 529 T^{4} )^{2}$$)($$1 - 734 T^{4} + 279841 T^{8}$$)($$( 1 + 4 T + 8 T^{2} + 92 T^{3} + 529 T^{4} )^{2}$$)($$1 + 734 T^{4} + 258915 T^{8} + 205403294 T^{12} + 78310985281 T^{16}$$)($$1 + 16 T + 113 T^{2} + 570 T^{3} + 2741 T^{4} + 13110 T^{5} + 59777 T^{6} + 194672 T^{7} + 279841 T^{8}$$)($$1 + 86 T^{2} + 4497 T^{4} + 158326 T^{6} + 4211876 T^{8} + 83754454 T^{10} + 1258444977 T^{12} + 12731086454 T^{14} + 78310985281 T^{16}$$)($$1 + 117 T^{2} + 7604 T^{4} + 351191 T^{6} + 12704151 T^{8} + 377097632 T^{10} + 9413788433 T^{12} + 199484647328 T^{14} + 3555142319991 T^{16} + 51988871893799 T^{18} + 595476732076724 T^{20} + 4846901811996933 T^{22} + 21914624432020321 T^{24}$$)($$1 + 14 T + 53 T^{2} - 226 T^{3} - 2552 T^{4} - 5198 T^{5} + 28037 T^{6} + 170338 T^{7} + 279841 T^{8}$$)($$1 - 4 T + 53 T^{2} - 244 T^{3} + 1588 T^{4} - 5612 T^{5} + 28037 T^{6} - 48668 T^{7} + 279841 T^{8}$$)($$1 + 1049 T^{4} + 820560 T^{8} + 293553209 T^{12} + 78310985281 T^{16}$$)
$29$ ($$1 + 5 T + 29 T^{2}$$)($$1 - 3 T + 29 T^{2}$$)($$1 + 5 T + 29 T^{2}$$)($$( 1 - 5 T + 29 T^{2} )^{2}$$)($$( 1 - 5 T + 29 T^{2} )^{2}$$)($$1 - T + 20 T^{2} - 29 T^{3} + 841 T^{4}$$)($$( 1 + 3 T + 29 T^{2} )^{2}$$)($$( 1 + T + 20 T^{2} + 29 T^{3} + 841 T^{4} )^{2}$$)($$( 1 + 5 T + 29 T^{2} )^{4}$$)($$( 1 - 7 T + 29 T^{2} )^{2}$$)($$( 1 - 7 T + 29 T^{2} )^{2}$$)($$( 1 + T + 29 T^{2} )^{4}$$)($$( 1 + 8 T + 82 T^{2} + 389 T^{3} + 2378 T^{4} + 6728 T^{5} + 24389 T^{6} )^{2}$$)($$( 1 + 8 T + 82 T^{2} + 389 T^{3} + 2378 T^{4} + 6728 T^{5} + 24389 T^{6} )^{2}$$)($$( 1 + 23 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 54 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 49 T^{2} + 841 T^{4} )^{2}$$)($$( 1 + 54 T^{2} + 2075 T^{4} + 45414 T^{6} + 707281 T^{8} )^{2}$$)($$1 - 2 T - 25 T^{2} - 32 T^{3} + 929 T^{4} - 928 T^{5} - 21025 T^{6} - 48778 T^{7} + 707281 T^{8}$$)($$( 1 - T + 29 T^{2} )^{8}$$)($$( 1 - 8 T + 82 T^{2} - 389 T^{3} + 2378 T^{4} - 6728 T^{5} + 24389 T^{6} )^{4}$$)($$( 1 - 49 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 49 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 57 T^{2} + 841 T^{4} )^{4}$$)
$31$ ($$1 - 2 T + 31 T^{2}$$)($$1 + 4 T + 31 T^{2}$$)($$1 - 2 T + 31 T^{2}$$)($$1 + 6 T + 26 T^{2} + 186 T^{3} + 961 T^{4}$$)($$1 + 6 T + 26 T^{2} + 186 T^{3} + 961 T^{4}$$)($$( 1 + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 + 31 T^{2} )^{4}$$)($$( 1 + 6 T + 26 T^{2} + 186 T^{3} + 961 T^{4} )^{2}$$)($$1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4}$$)($$1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4}$$)($$( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} )^{2}$$)($$1 + 5 T - 50 T^{2} - 119 T^{3} + 2065 T^{4} + 80 T^{5} - 77809 T^{6} + 2480 T^{7} + 1984465 T^{8} - 3545129 T^{9} - 46176050 T^{10} + 143145755 T^{11} + 887503681 T^{12}$$)($$1 + 5 T - 50 T^{2} - 119 T^{3} + 2065 T^{4} + 80 T^{5} - 77809 T^{6} + 2480 T^{7} + 1984465 T^{8} - 3545129 T^{9} - 46176050 T^{10} + 143145755 T^{11} + 887503681 T^{12}$$)($$( 1 - 31 T^{2} )^{4}$$)($$( 1 - 31 T^{2} )^{4}$$)($$( 1 - 52 T^{2} + 961 T^{4} )^{2}$$)($$( 1 - 31 T^{2} )^{8}$$)($$1 + 16 T + 105 T^{2} + 554 T^{3} + 3269 T^{4} + 17174 T^{5} + 100905 T^{6} + 476656 T^{7} + 923521 T^{8}$$)($$( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} )^{4}$$)($$( 1 + 5 T - 50 T^{2} - 119 T^{3} + 2065 T^{4} + 80 T^{5} - 77809 T^{6} + 2480 T^{7} + 1984465 T^{8} - 3545129 T^{9} - 46176050 T^{10} + 143145755 T^{11} + 887503681 T^{12} )^{2}$$)($$1 + 12 T + 106 T^{2} + 696 T^{3} + 3891 T^{4} + 21576 T^{5} + 101866 T^{6} + 357492 T^{7} + 923521 T^{8}$$)($$1 + 12 T + 106 T^{2} + 696 T^{3} + 3891 T^{4} + 21576 T^{5} + 101866 T^{6} + 357492 T^{7} + 923521 T^{8}$$)($$( 1 - 11 T + 31 T^{2} )^{4}( 1 - 7 T + 31 T^{2} )^{4}$$)
$37$ ($$1 + 2 T + 37 T^{2}$$)($$1 + 2 T + 37 T^{2}$$)($$1 - 2 T + 37 T^{2}$$)($$( 1 + 3 T + 37 T^{2} )^{2}$$)($$( 1 - 3 T + 37 T^{2} )^{2}$$)($$( 1 + 6 T + 37 T^{2} )^{2}$$)($$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$)($$( 1 - 38 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 65 T^{2} + 1369 T^{4} )^{2}$$)($$1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4}$$)($$1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4}$$)($$( 1 - 37 T^{2} + 1369 T^{4} )^{2}$$)($$1 + 10 T - 20 T^{2} - 118 T^{3} + 3370 T^{4} + 3970 T^{5} - 134113 T^{6} + 146890 T^{7} + 4613530 T^{8} - 5977054 T^{9} - 37483220 T^{10} + 693439570 T^{11} + 2565726409 T^{12}$$)($$1 - 10 T - 20 T^{2} + 118 T^{3} + 3370 T^{4} - 3970 T^{5} - 134113 T^{6} - 146890 T^{7} + 4613530 T^{8} + 5977054 T^{9} - 37483220 T^{10} - 693439570 T^{11} + 2565726409 T^{12}$$)($$( 1 + 1369 T^{4} )^{2}$$)($$1 - 1294 T^{4} + 1874161 T^{8}$$)($$( 1 - 12 T + 72 T^{2} - 444 T^{3} + 1369 T^{4} )^{2}$$)($$1 + 1294 T^{4} - 199725 T^{8} + 2425164334 T^{12} + 3512479453921 T^{16}$$)($$1 + 3 T - 33 T^{2} - 35 T^{3} + 1296 T^{4} - 1295 T^{5} - 45177 T^{6} + 151959 T^{7} + 1874161 T^{8}$$)($$( 1 + 37 T^{2} + 1369 T^{4} )^{4}$$)($$1 + 140 T^{2} + 9500 T^{4} + 496350 T^{6} + 23947600 T^{8} + 1024983040 T^{10} + 39266587843 T^{12} + 1403201781760 T^{14} + 44881657963600 T^{16} + 1273498303107150 T^{18} + 33368554812249500 T^{20} + 673201812138498860 T^{22} + 6582952005840035281 T^{24}$$)($$1 - 12 T + 72 T^{2} - 288 T^{3} + 983 T^{4} - 10656 T^{5} + 98568 T^{6} - 607836 T^{7} + 1874161 T^{8}$$)($$1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 10656 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8}$$)($$1 - 626 T^{4} - 1482285 T^{8} - 1173224786 T^{12} + 3512479453921 T^{16}$$)
$41$ ($$1 - 2 T + 41 T^{2}$$)($$1 + 12 T + 41 T^{2}$$)($$1 - 2 T + 41 T^{2}$$)($$1 - 14 T + 126 T^{2} - 574 T^{3} + 1681 T^{4}$$)($$1 - 14 T + 126 T^{2} - 574 T^{3} + 1681 T^{4}$$)($$1 - 2 T + 66 T^{2} - 82 T^{3} + 1681 T^{4}$$)($$( 1 + 12 T + 41 T^{2} )^{2}$$)($$( 1 - 2 T + 66 T^{2} - 82 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 - 14 T + 126 T^{2} - 574 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 - 5 T + 41 T^{2} )^{2}$$)($$( 1 - 5 T + 41 T^{2} )^{2}$$)($$( 1 + 10 T + 99 T^{2} + 410 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + T + 77 T^{2} + 187 T^{3} + 3157 T^{4} + 1681 T^{5} + 68921 T^{6} )^{2}$$)($$( 1 + T + 77 T^{2} + 187 T^{3} + 3157 T^{4} + 1681 T^{5} + 68921 T^{6} )^{2}$$)($$( 1 - 41 T^{2} )^{4}$$)($$( 1 - 41 T^{2} )^{4}$$)($$( 1 + 8 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 41 T^{2} )^{8}$$)($$1 + 8 T - 7 T^{2} + 36 T^{3} + 1925 T^{4} + 1476 T^{5} - 11767 T^{6} + 551368 T^{7} + 2825761 T^{8}$$)($$( 1 + 10 T + 99 T^{2} + 410 T^{3} + 1681 T^{4} )^{4}$$)($$( 1 + T + 77 T^{2} + 187 T^{3} + 3157 T^{4} + 1681 T^{5} + 68921 T^{6} )^{4}$$)($$1 - 122 T^{2} + 6651 T^{4} - 205082 T^{6} + 2825761 T^{8}$$)($$1 - 122 T^{2} + 6651 T^{4} - 205082 T^{6} + 2825761 T^{8}$$)($$( 1 - 7 T^{2} + 1681 T^{4} )^{4}$$)
$43$ ($$1 - 4 T + 43 T^{2}$$)($$1 - 10 T + 43 T^{2}$$)($$1 + 4 T + 43 T^{2}$$)($$1 + 8 T + 97 T^{2} + 344 T^{3} + 1849 T^{4}$$)($$1 - 8 T + 97 T^{2} - 344 T^{3} + 1849 T^{4}$$)($$1 + 10 T + 94 T^{2} + 430 T^{3} + 1849 T^{4}$$)($$1 + 14 T^{2} + 1849 T^{4}$$)($$1 - 88 T^{2} + 3934 T^{4} - 162712 T^{6} + 3418801 T^{8}$$)($$1 - 130 T^{2} + 7603 T^{4} - 240370 T^{6} + 3418801 T^{8}$$)($$( 1 - 7 T + 43 T^{2} )^{2}$$)($$( 1 + 7 T + 43 T^{2} )^{2}$$)($$( 1 + 10 T + 109 T^{2} + 430 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 + 9 T + 109 T^{2} + 721 T^{3} + 4687 T^{4} + 16641 T^{5} + 79507 T^{6} )^{2}$$)($$( 1 - 9 T + 109 T^{2} - 721 T^{3} + 4687 T^{4} - 16641 T^{5} + 79507 T^{6} )^{2}$$)($$( 1 + 1849 T^{4} )^{2}$$)($$1 - 334 T^{4} + 3418801 T^{8}$$)($$( 1 - 6 T + 18 T^{2} - 258 T^{3} + 1849 T^{4} )^{2}$$)($$1 + 334 T^{4} - 3307245 T^{8} + 1141879534 T^{12} + 11688200277601 T^{16}$$)($$( 1 + 10 T + 106 T^{2} + 430 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 - 118 T^{2} + 6979 T^{4} - 218182 T^{6} + 3418801 T^{8} )^{2}$$)($$( 1 - 137 T^{2} + 8277 T^{4} - 361401 T^{6} + 15304173 T^{8} - 468375737 T^{10} + 6321363049 T^{12} )^{2}$$)($$1 - 6 T + 18 T^{2} - 60 T^{3} - 889 T^{4} - 2580 T^{5} + 33282 T^{6} - 477042 T^{7} + 3418801 T^{8}$$)($$1 - 6 T + 18 T^{2} - 60 T^{3} - 889 T^{4} - 2580 T^{5} + 33282 T^{6} - 477042 T^{7} + 3418801 T^{8}$$)($$( 1 + 23 T^{4} + 3418801 T^{8} )^{2}$$)
$47$ ($$1 - 3 T + 47 T^{2}$$)($$1 + 9 T + 47 T^{2}$$)($$1 + 3 T + 47 T^{2}$$)($$( 1 - 2 T + 47 T^{2} )^{2}$$)($$( 1 + 2 T + 47 T^{2} )^{2}$$)($$1 - 5 T + 62 T^{2} - 235 T^{3} + 2209 T^{4}$$)($$1 - 13 T^{2} + 2209 T^{4}$$)($$1 - 99 T^{2} + 5912 T^{4} - 218691 T^{6} + 4879681 T^{8}$$)($$( 1 - 90 T^{2} + 2209 T^{4} )^{2}$$)($$1 - 47 T^{2} + 2209 T^{4}$$)($$1 - 47 T^{2} + 2209 T^{4}$$)($$( 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} )^{2}$$)($$1 - 28 T + 390 T^{2} - 4298 T^{3} + 42442 T^{4} - 351442 T^{5} + 2516387 T^{6} - 16517774 T^{7} + 93754378 T^{8} - 446231254 T^{9} + 1903075590 T^{10} - 6421660196 T^{11} + 10779215329 T^{12}$$)($$1 + 28 T + 390 T^{2} + 4298 T^{3} + 42442 T^{4} + 351442 T^{5} + 2516387 T^{6} + 16517774 T^{7} + 93754378 T^{8} + 446231254 T^{9} + 1903075590 T^{10} + 6421660196 T^{11} + 10779215329 T^{12}$$)($$1 - 3457 T^{4} + 4879681 T^{8}$$)($$( 1 + 2209 T^{4} )^{2}$$)($$1 - 2017 T^{4} + 4879681 T^{8}$$)($$( 1 + 2209 T^{4} )^{4}$$)($$1 + 16 T + 49 T^{2} - 628 T^{3} - 7311 T^{4} - 29516 T^{5} + 108241 T^{6} + 1661168 T^{7} + 4879681 T^{8}$$)($$( 1 + 90 T^{2} + 5891 T^{4} + 198810 T^{6} + 4879681 T^{8} )^{2}$$)($$1 + 4 T^{2} - 3704 T^{4} + 16022 T^{6} + 5623204 T^{8} - 48345236 T^{10} - 9088343797 T^{12} - 106794626324 T^{14} + 27439441717924 T^{16} + 172704588001238 T^{18} - 88197005795162744 T^{20} + 210396528943320196 T^{22} +$$$$11\!\cdots\!41$$$$T^{24}$$)($$1 + 6 T + 90 T^{2} + 672 T^{3} + 5159 T^{4} + 31584 T^{5} + 198810 T^{6} + 622938 T^{7} + 4879681 T^{8}$$)($$1 + 18 T + 90 T^{2} - 528 T^{3} - 8377 T^{4} - 24816 T^{5} + 198810 T^{6} + 1868814 T^{7} + 4879681 T^{8}$$)($$1 - 2306 T^{4} + 437955 T^{8} - 11252544386 T^{12} + 23811286661761 T^{16}$$)
$53$ ($$1 + 6 T + 53 T^{2}$$)($$1 + 12 T + 53 T^{2}$$)($$1 - 6 T + 53 T^{2}$$)($$1 + 8 T + 102 T^{2} + 424 T^{3} + 2809 T^{4}$$)($$1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4}$$)($$1 - 2 T + 90 T^{2} - 106 T^{3} + 2809 T^{4}$$)($$1 + 38 T^{2} + 2809 T^{4}$$)($$1 - 176 T^{2} + 13294 T^{4} - 494384 T^{6} + 7890481 T^{8}$$)($$1 - 140 T^{2} + 9238 T^{4} - 393260 T^{6} + 7890481 T^{8}$$)($$1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4}$$)($$1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4}$$)($$1 + 8 T - 50 T^{2} + 64 T^{3} + 6795 T^{4} + 3392 T^{5} - 140450 T^{6} + 1191016 T^{7} + 7890481 T^{8}$$)($$1 - 10 T - 40 T^{2} + 474 T^{3} + 2810 T^{4} - 10870 T^{5} - 143977 T^{6} - 576110 T^{7} + 7893290 T^{8} + 70567698 T^{9} - 315619240 T^{10} - 4181954930 T^{11} + 22164361129 T^{12}$$)($$1 + 10 T - 40 T^{2} - 474 T^{3} + 2810 T^{4} + 10870 T^{5} - 143977 T^{6} + 576110 T^{7} + 7893290 T^{8} - 70567698 T^{9} - 315619240 T^{10} + 4181954930 T^{11} + 22164361129 T^{12}$$)($$( 1 + 2809 T^{4} )^{2}$$)($$1 - 5582 T^{4} + 7890481 T^{8}$$)($$( 1 + 2 T + 2 T^{2} + 106 T^{3} + 2809 T^{4} )^{2}$$)($$( 1 - 5582 T^{4} + 7890481 T^{8} )^{2}$$)($$1 - 17 T + 71 T^{2} + 849 T^{3} - 12256 T^{4} + 44997 T^{5} + 199439 T^{6} - 2530909 T^{7} + 7890481 T^{8}$$)($$1 + 164 T^{2} + 15066 T^{4} + 1018768 T^{6} + 56507555 T^{8} + 2861719312 T^{10} + 118877986746 T^{12} + 3634955225156 T^{14} + 62259690411361 T^{16}$$)($$1 + 180 T^{2} + 16700 T^{4} + 954830 T^{6} + 33983400 T^{8} + 232503080 T^{10} - 28393979917 T^{12} + 653101151720 T^{14} + 268145372015400 T^{16} + 21163196936803070 T^{18} + 1039736829869728700 T^{20} + 31479744665792348820 T^{22} +$$$$49\!\cdots\!41$$$$T^{24}$$)($$( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} )( 1 + 14 T + 143 T^{2} + 742 T^{3} + 2809 T^{4} )$$)($$( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} )( 1 + 14 T + 143 T^{2} + 742 T^{3} + 2809 T^{4} )$$)($$1 - 4786 T^{4} + 15015315 T^{8} - 37763842066 T^{12} + 62259690411361 T^{16}$$)
$59$ ($$1 - 10 T + 59 T^{2}$$)($$1 + 59 T^{2}$$)($$1 - 10 T + 59 T^{2}$$)($$1 - 10 T + 98 T^{2} - 590 T^{3} + 3481 T^{4}$$)($$1 - 10 T + 98 T^{2} - 590 T^{3} + 3481 T^{4}$$)($$( 1 + 4 T + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 - 4 T + 59 T^{2} )^{4}$$)($$( 1 + 10 T + 98 T^{2} + 590 T^{3} + 3481 T^{4} )^{2}$$)($$1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4}$$)($$1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4}$$)($$1 - 8 T + 2 T^{2} + 448 T^{3} - 3413 T^{4} + 26432 T^{5} + 6962 T^{6} - 1643032 T^{7} + 12117361 T^{8}$$)($$1 - 5 T - 90 T^{2} + 815 T^{3} + 2965 T^{4} - 29480 T^{5} + 26483 T^{6} - 1739320 T^{7} + 10321165 T^{8} + 167383885 T^{9} - 1090562490 T^{10} - 3574621495 T^{11} + 42180533641 T^{12}$$)($$1 - 5 T - 90 T^{2} + 815 T^{3} + 2965 T^{4} - 29480 T^{5} + 26483 T^{6} - 1739320 T^{7} + 10321165 T^{8} + 167383885 T^{9} - 1090562490 T^{10} - 3574621495 T^{11} + 42180533641 T^{12}$$)($$( 1 + 59 T^{2} )^{4}$$)($$( 1 + 59 T^{2} )^{4}$$)($$( 1 + 28 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 59 T^{2} )^{8}$$)($$1 - 19 T^{2} - 390 T^{3} + 2701 T^{4} - 23010 T^{5} - 66139 T^{6} + 12117361 T^{8}$$)($$( 1 + 8 T + 2 T^{2} - 448 T^{3} - 3413 T^{4} - 26432 T^{5} + 6962 T^{6} + 1643032 T^{7} + 12117361 T^{8} )^{2}$$)($$( 1 + 5 T - 90 T^{2} - 815 T^{3} + 2965 T^{4} + 29480 T^{5} + 26483 T^{6} + 1739320 T^{7} + 10321165 T^{8} - 167383885 T^{9} - 1090562490 T^{10} + 3574621495 T^{11} + 42180533641 T^{12} )^{2}$$)($$1 + 6 T - 64 T^{2} - 108 T^{3} + 4395 T^{4} - 6372 T^{5} - 222784 T^{6} + 1232274 T^{7} + 12117361 T^{8}$$)($$1 - 6 T - 64 T^{2} + 108 T^{3} + 4395 T^{4} + 6372 T^{5} - 222784 T^{6} - 1232274 T^{7} + 12117361 T^{8}$$)($$( 1 - 106 T^{2} + 7755 T^{4} - 368986 T^{6} + 12117361 T^{8} )^{2}$$)
$61$ ($$1 + 8 T + 61 T^{2}$$)($$1 - 8 T + 61 T^{2}$$)($$1 + 8 T + 61 T^{2}$$)($$1 + 6 T + 86 T^{2} + 366 T^{3} + 3721 T^{4}$$)($$1 + 6 T + 86 T^{2} + 366 T^{3} + 3721 T^{4}$$)($$1 - 6 T - 22 T^{2} - 366 T^{3} + 3721 T^{4}$$)($$( 1 - 8 T + 61 T^{2} )^{2}$$)($$( 1 - 6 T - 22 T^{2} - 366 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 6 T + 86 T^{2} + 366 T^{3} + 3721 T^{4} )^{2}$$)($$1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4}$$)($$1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4}$$)($$1 - 6 T - 23 T^{2} + 378 T^{3} - 2436 T^{4} + 23058 T^{5} - 85583 T^{6} - 1361886 T^{7} + 13845841 T^{8}$$)($$1 + 5 T - 140 T^{2} - 269 T^{3} + 14425 T^{4} + 8210 T^{5} - 1000579 T^{6} + 500810 T^{7} + 53675425 T^{8} - 61057889 T^{9} - 1938417740 T^{10} + 4222981505 T^{11} + 51520374361 T^{12}$$)($$1 + 5 T - 140 T^{2} - 269 T^{3} + 14425 T^{4} + 8210 T^{5} - 1000579 T^{6} + 500810 T^{7} + 53675425 T^{8} - 61057889 T^{9} - 1938417740 T^{10} + 4222981505 T^{11} + 51520374361 T^{12}$$)($$( 1 - 61 T^{2} )^{4}$$)($$( 1 - 61 T^{2} )^{4}$$)($$( 1 - 82 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 - 61 T^{2} )^{8}$$)($$1 + 8 T - 27 T^{2} - 584 T^{3} - 2575 T^{4} - 35624 T^{5} - 100467 T^{6} + 1815848 T^{7} + 13845841 T^{8}$$)($$( 1 - 6 T - 23 T^{2} + 378 T^{3} - 2436 T^{4} + 23058 T^{5} - 85583 T^{6} - 1361886 T^{7} + 13845841 T^{8} )^{2}$$)($$( 1 + 5 T - 140 T^{2} - 269 T^{3} + 14425 T^{4} + 8210 T^{5} - 1000579 T^{6} + 500810 T^{7} + 53675425 T^{8} - 61057889 T^{9} - 1938417740 T^{10} + 4222981505 T^{11} + 51520374361 T^{12} )^{2}$$)($$1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 79788 T^{5} + 584197 T^{6} + 2723772 T^{7} + 13845841 T^{8}$$)($$1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 79788 T^{5} + 584197 T^{6} + 2723772 T^{7} + 13845841 T^{8}$$)($$( 1 - 3 T + 64 T^{2} - 183 T^{3} + 3721 T^{4} )^{4}$$)
$67$ ($$1 + 2 T + 67 T^{2}$$)($$1 - 4 T + 67 T^{2}$$)($$1 - 2 T + 67 T^{2}$$)($$1 - 4 T + 133 T^{2} - 268 T^{3} + 4489 T^{4}$$)($$1 + 4 T + 133 T^{2} + 268 T^{3} + 4489 T^{4}$$)($$1 + 4 T + 70 T^{2} + 268 T^{3} + 4489 T^{4}$$)($$1 - 118 T^{2} + 4489 T^{4}$$)($$1 - 124 T^{2} + 11734 T^{4} - 556636 T^{6} + 20151121 T^{8}$$)($$1 - 250 T^{2} + 24523 T^{4} - 1122250 T^{6} + 20151121 T^{8}$$)($$( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} )$$)($$( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)($$1 - 22 T + 231 T^{2} - 2618 T^{3} + 27092 T^{4} - 175406 T^{5} + 1036959 T^{6} - 6616786 T^{7} + 20151121 T^{8}$$)($$1 + 12 T - 85 T^{2} - 388 T^{3} + 18282 T^{4} + 44708 T^{5} - 1099813 T^{6} + 2995436 T^{7} + 82067898 T^{8} - 116696044 T^{9} - 1712845285 T^{10} + 16201501284 T^{11} + 90458382169 T^{12}$$)($$1 - 12 T - 85 T^{2} + 388 T^{3} + 18282 T^{4} - 44708 T^{5} - 1099813 T^{6} - 2995436 T^{7} + 82067898 T^{8} + 116696044 T^{9} - 1712845285 T^{10} - 16201501284 T^{11} + 90458382169 T^{12}$$)($$( 1 + 4489 T^{4} )^{2}$$)($$1 + 4946 T^{4} + 20151121 T^{8}$$)($$( 1 - 2 T + 2 T^{2} - 134 T^{3} + 4489 T^{4} )^{2}$$)($$1 - 4946 T^{4} + 4311795 T^{8} - 99667444466 T^{12} + 406067677556641 T^{16}$$)($$1 - 6 T + 69 T^{2} - 382 T^{3} + 1329 T^{4} - 25594 T^{5} + 309741 T^{6} - 1804578 T^{7} + 20151121 T^{8}$$)($$1 + 22 T^{2} - 7647 T^{4} - 18634 T^{6} + 43789364 T^{8} - 83648026 T^{10} - 154095622287 T^{12} + 1990084407718 T^{14} + 406067677556641 T^{16}$$)($$1 + 314 T^{2} + 53101 T^{4} + 6531102 T^{6} + 648138474 T^{8} + 54464437634 T^{10} + 3935187653893 T^{12} + 244490860539026 T^{14} + 13060716814329354 T^{16} + 590792920700720238 T^{18} + 21562599745935193741 T^{20} +$$$$57\!\cdots\!86$$$$T^{22} +$$$$81\!\cdots\!61$$$$T^{24}$$)($$1 + 8 T + 137 T^{2} + 1224 T^{3} + 11492 T^{4} + 82008 T^{5} + 614993 T^{6} + 2406104 T^{7} + 20151121 T^{8}$$)($$1 - 22 T + 137 T^{2} + 834 T^{3} - 16648 T^{4} + 55878 T^{5} + 614993 T^{6} - 6616786 T^{7} + 20151121 T^{8}$$)($$1 + 6169 T^{4} + 17905440 T^{8} + 124312265449 T^{12} + 406067677556641 T^{16}$$)
$71$ ($$1 + 8 T + 71 T^{2}$$)($$1 + 71 T^{2}$$)($$1 + 8 T + 71 T^{2}$$)($$1 - 4 T + 101 T^{2} - 284 T^{3} + 5041 T^{4}$$)($$1 - 4 T + 101 T^{2} - 284 T^{3} + 5041 T^{4}$$)($$( 1 - 8 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 - 8 T + 71 T^{2} )^{4}$$)($$( 1 - 4 T + 101 T^{2} - 284 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 + 2 T + 71 T^{2} )^{2}$$)($$( 1 + 2 T + 71 T^{2} )^{2}$$)($$( 1 + 8 T + 86 T^{2} + 568 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 - 7 T + 124 T^{2} - 571 T^{3} + 8804 T^{4} - 35287 T^{5} + 357911 T^{6} )^{2}$$)($$( 1 - 7 T + 124 T^{2} - 571 T^{3} + 8804 T^{4} - 35287 T^{5} + 357911 T^{6} )^{2}$$)($$( 1 - 12 T + 71 T^{2} )^{4}$$)($$( 1 + 16 T + 71 T^{2} )^{4}$$)($$( 1 + 6 T + 71 T^{2} )^{4}$$)($$( 1 - 16 T + 185 T^{2} - 1136 T^{3} + 5041 T^{4} )^{4}$$)($$1 + 20 T + 169 T^{2} + 1600 T^{3} + 17121 T^{4} + 113600 T^{5} + 851929 T^{6} + 7158220 T^{7} + 25411681 T^{8}$$)($$( 1 + 8 T + 86 T^{2} + 568 T^{3} + 5041 T^{4} )^{4}$$)($$( 1 - 7 T + 124 T^{2} - 571 T^{3} + 8804 T^{4} - 35287 T^{5} + 357911 T^{6} )^{4}$$)($$( 1 - 6 T + 148 T^{2} - 426 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 - 6 T + 148 T^{2} - 426 T^{3} + 5041 T^{4} )^{2}$$)($$( 1 - 2 T + 71 T^{2} )^{8}$$)
$73$ ($$1 + 6 T + 73 T^{2}$$)($$1 + 2 T + 73 T^{2}$$)($$1 - 6 T + 73 T^{2}$$)($$1 - 22 T + 262 T^{2} - 1606 T^{3} + 5329 T^{4}$$)($$1 + 22 T + 262 T^{2} + 1606 T^{3} + 5329 T^{4}$$)($$1 - 8 T + 94 T^{2} - 584 T^{3} + 5329 T^{4}$$)($$1 - 142 T^{2} + 5329 T^{4}$$)($$1 - 124 T^{2} + 10150 T^{4} - 660796 T^{6} + 28398241 T^{8}$$)($$1 - 40 T^{2} + 8638 T^{4} - 213160 T^{6} + 28398241 T^{8}$$)($$1 - 6 T - 37 T^{2} - 438 T^{3} + 5329 T^{4}$$)($$1 + 6 T - 37 T^{2} + 438 T^{3} + 5329 T^{4}$$)($$1 - 4 T - 126 T^{2} + 16 T^{3} + 13667 T^{4} + 1168 T^{5} - 671454 T^{6} - 1556068 T^{7} + 28398241 T^{8}$$)($$1 + 12 T - 43 T^{2} - 364 T^{3} + 5670 T^{4} - 19828 T^{5} - 841663 T^{6} - 1447444 T^{7} + 30215430 T^{8} - 141602188 T^{9} - 1221124363 T^{10} + 24876859116 T^{11} + 151334226289 T^{12}$$)($$1 - 12 T - 43 T^{2} + 364 T^{3} + 5670 T^{4} + 19828 T^{5} - 841663 T^{6} + 1447444 T^{7} + 30215430 T^{8} + 141602188 T^{9} - 1221124363 T^{10} - 24876859116 T^{11} + 151334226289 T^{12}$$)($$1 - 9502 T^{4} + 28398241 T^{8}$$)($$( 1 + 5329 T^{4} )^{2}$$)($$( 1 + 5329 T^{4} )^{2}$$)($$( 1 + 5329 T^{4} )^{4}$$)($$1 - 12 T + 21 T^{2} - 476 T^{3} + 9429 T^{4} - 34748 T^{5} + 111909 T^{6} - 4668204 T^{7} + 28398241 T^{8}$$)($$1 + 268 T^{2} + 43338 T^{4} + 4777904 T^{6} + 400303859 T^{8} + 25461450416 T^{10} + 1230722968458 T^{12} + 40557572645452 T^{14} + 806460091894081 T^{16}$$)($$1 + 230 T^{2} + 21925 T^{4} + 1827570 T^{6} + 185266650 T^{8} + 12633670430 T^{10} + 701193878653 T^{12} + 67324829721470 T^{14} + 5261246975962650 T^{16} + 276573891938987730 T^{18} + 17681637514777725925 T^{20} +$$$$98\!\cdots\!70$$$$T^{22} +$$$$22\!\cdots\!21$$$$T^{24}$$)($$1 + 144 T^{2} - 600 T^{3} + 10991 T^{4} - 43800 T^{5} + 767376 T^{6} + 28398241 T^{8}$$)($$1 - 24 T + 144 T^{2} + 1752 T^{3} - 31057 T^{4} + 127896 T^{5} + 767376 T^{6} - 9336408 T^{7} + 28398241 T^{8}$$)($$1 + 9214 T^{4} + 56499555 T^{8} + 261661392574 T^{12} + 806460091894081 T^{16}$$)
$79$ ($$1 + 5 T + 79 T^{2}$$)($$1 + T + 79 T^{2}$$)($$1 + 5 T + 79 T^{2}$$)($$1 + 33 T^{2} + 6241 T^{4}$$)($$1 + 33 T^{2} + 6241 T^{4}$$)($$1 + 9 T + 174 T^{2} + 711 T^{3} + 6241 T^{4}$$)($$( 1 - T + 79 T^{2} )^{2}$$)($$( 1 - 9 T + 174 T^{2} - 711 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 + 33 T^{2} + 6241 T^{4} )^{2}$$)($$1 - 2 T - 75 T^{2} - 158 T^{3} + 6241 T^{4}$$)($$1 - 2 T - 75 T^{2} - 158 T^{3} + 6241 T^{4}$$)($$1 + 24 T + 282 T^{2} + 3264 T^{3} + 34691 T^{4} + 257856 T^{5} + 1759962 T^{6} + 11832936 T^{7} + 38950081 T^{8}$$)($$1 - 7 T - 96 T^{2} + 1499 T^{3} + 771 T^{4} - 65582 T^{5} + 599343 T^{6} - 5180978 T^{7} + 4811811 T^{8} + 739065461 T^{9} - 3739207776 T^{10} - 21539394793 T^{11} + 243087455521 T^{12}$$)($$1 - 7 T - 96 T^{2} + 1499 T^{3} + 771 T^{4} - 65582 T^{5} + 599343 T^{6} - 5180978 T^{7} + 4811811 T^{8} + 739065461 T^{9} - 3739207776 T^{10} - 21539394793 T^{11} + 243087455521 T^{12}$$)($$( 1 - 157 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 94 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 + 11 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 + 94 T^{2} + 2595 T^{4} + 586654 T^{6} + 38950081 T^{8} )^{2}$$)($$1 - 6 T + 57 T^{2} - 118 T^{3} + 105 T^{4} - 9322 T^{5} + 355737 T^{6} - 2958234 T^{7} + 38950081 T^{8}$$)($$( 1 - 24 T + 282 T^{2} - 3264 T^{3} + 34691 T^{4} - 257856 T^{5} + 1759962 T^{6} - 11832936 T^{7} + 38950081 T^{8} )^{2}$$)($$( 1 + 7 T - 96 T^{2} - 1499 T^{3} + 771 T^{4} + 65582 T^{5} + 599343 T^{6} + 5180978 T^{7} + 4811811 T^{8} - 739065461 T^{9} - 3739207776 T^{10} + 21539394793 T^{11} + 243087455521 T^{12} )^{2}$$)($$1 + 6 T + 148 T^{2} + 816 T^{3} + 13203 T^{4} + 64464 T^{5} + 923668 T^{6} + 2958234 T^{7} + 38950081 T^{8}$$)($$1 - 6 T + 148 T^{2} - 816 T^{3} + 13203 T^{4} - 64464 T^{5} + 923668 T^{6} - 2958234 T^{7} + 38950081 T^{8}$$)($$( 1 + 154 T^{2} + 17475 T^{4} + 961114 T^{6} + 38950081 T^{8} )^{2}$$)
$83$ ($$1 - 4 T + 83 T^{2}$$)($$1 + 12 T + 83 T^{2}$$)($$1 + 4 T + 83 T^{2}$$)($$1 - 2 T + 122 T^{2} - 166 T^{3} + 6889 T^{4}$$)($$1 + 2 T + 122 T^{2} + 166 T^{3} + 6889 T^{4}$$)($$( 1 + 4 T + 83 T^{2} )^{2}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$( 1 - 150 T^{2} + 6889 T^{4} )^{2}$$)($$1 - 240 T^{2} + 27998 T^{4} - 1653360 T^{6} + 47458321 T^{8}$$)($$( 1 + 11 T + 83 T^{2} )^{2}$$)($$( 1 - 11 T + 83 T^{2} )^{2}$$)($$( 1 + 2 T + 5 T^{2} + 166 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 + 26 T + 448 T^{2} + 4715 T^{3} + 37184 T^{4} + 179114 T^{5} + 571787 T^{6} )^{2}$$)($$( 1 - 26 T + 448 T^{2} - 4715 T^{3} + 37184 T^{4} - 179114 T^{5} + 571787 T^{6} )^{2}$$)($$1 - 6382 T^{4} + 47458321 T^{8}$$)($$( 1 + 6889 T^{4} )^{2}$$)($$1 + 7538 T^{4} + 47458321 T^{8}$$)($$( 1 + 6889 T^{4} )^{4}$$)($$1 - 12 T - 19 T^{2} + 204 T^{3} + 4489 T^{4} + 16932 T^{5} - 130891 T^{6} - 6861444 T^{7} + 47458321 T^{8}$$)($$( 1 - 6 T^{2} + 13139 T^{4} - 41334 T^{6} + 47458321 T^{8} )^{2}$$)($$( 1 - 220 T^{2} + 29892 T^{4} - 2915285 T^{6} + 205925988 T^{8} - 10440830620 T^{10} + 326940373369 T^{12} )^{2}$$)($$1 + 2 T + 2 T^{2} + 140 T^{3} + 9631 T^{4} + 11620 T^{5} + 13778 T^{6} + 1143574 T^{7} + 47458321 T^{8}$$)($$1 - 2 T + 2 T^{2} - 140 T^{3} + 9631 T^{4} - 11620 T^{5} + 13778 T^{6} - 1143574 T^{7} + 47458321 T^{8}$$)($$( 1 + 5543 T^{4} + 47458321 T^{8} )^{2}$$)
$89$ ($$1 + 89 T^{2}$$)($$1 + 12 T + 89 T^{2}$$)($$1 + 89 T^{2}$$)($$1 - 30 T + 398 T^{2} - 2670 T^{3} + 7921 T^{4}$$)($$1 - 30 T + 398 T^{2} - 2670 T^{3} + 7921 T^{4}$$)($$1 - 6 T + 170 T^{2} - 534 T^{3} + 7921 T^{4}$$)($$( 1 - 12 T + 89 T^{2} )^{2}$$)($$( 1 + 6 T + 170 T^{2} + 534 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 30 T + 398 T^{2} + 2670 T^{3} + 7921 T^{4} )^{2}$$)($$1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4}$$)($$1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4}$$)($$1 - 6 T - 119 T^{2} + 138 T^{3} + 12900 T^{4} + 12282 T^{5} - 942599 T^{6} - 4229814 T^{7} + 62742241 T^{8}$$)($$1 - 6 T - 94 T^{2} - 198 T^{3} + 3584 T^{4} + 53484 T^{5} - 294637 T^{6} + 4760076 T^{7} + 28388864 T^{8} - 139583862 T^{9} - 5897770654 T^{10} - 33504356694 T^{11} + 496981290961 T^{12}$$)($$1 - 6 T - 94 T^{2} - 198 T^{3} + 3584 T^{4} + 53484 T^{5} - 294637 T^{6} + 4760076 T^{7} + 28388864 T^{8} - 139583862 T^{9} - 5897770654 T^{10} - 33504356694 T^{11} + 496981290961 T^{12}$$)($$( 1 + 89 T^{2} )^{4}$$)($$( 1 + 89 T^{2} )^{4}$$)($$( 1 + 138 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 89 T^{2} )^{8}$$)($$1 + 3 T - 35 T^{2} + 693 T^{3} + 9604 T^{4} + 61677 T^{5} - 277235 T^{6} + 2114907 T^{7} + 62742241 T^{8}$$)($$( 1 + 6 T - 119 T^{2} - 138 T^{3} + 12900 T^{4} - 12282 T^{5} - 942599 T^{6} + 4229814 T^{7} + 62742241 T^{8} )^{2}$$)($$( 1 + 6 T - 94 T^{2} + 198 T^{3} + 3584 T^{4} - 53484 T^{5} - 294637 T^{6} - 4760076 T^{7} + 28388864 T^{8} + 139583862 T^{9} - 5897770654 T^{10} + 33504356694 T^{11} + 496981290961 T^{12} )^{2}$$)($$( 1 - 16 T + 89 T^{2} )^{2}( 1 + 16 T + 167 T^{2} + 1424 T^{3} + 7921 T^{4} )$$)($$( 1 + 16 T + 89 T^{2} )^{2}( 1 - 16 T + 167 T^{2} - 1424 T^{3} + 7921 T^{4} )$$)($$( 1 - 31 T^{2} - 6960 T^{4} - 245551 T^{6} + 62742241 T^{8} )^{2}$$)
$97$ ($$1 + 7 T + 97 T^{2}$$)($$1 - T + 97 T^{2}$$)($$1 - 7 T + 97 T^{2}$$)($$1 + 6 T + 198 T^{2} + 582 T^{3} + 9409 T^{4}$$)($$1 - 6 T + 198 T^{2} - 582 T^{3} + 9409 T^{4}$$)($$1 - 9 T + 108 T^{2} - 873 T^{3} + 9409 T^{4}$$)($$1 - 193 T^{2} + 9409 T^{4}$$)($$1 - 135 T^{2} + 14768 T^{4} - 1270215 T^{6} + 88529281 T^{8}$$)($$1 - 360 T^{2} + 51038 T^{4} - 3387240 T^{6} + 88529281 T^{8}$$)($$( 1 + 16 T + 97 T^{2} )^{2}$$)($$( 1 - 16 T + 97 T^{2} )^{2}$$)($$( 1 + 12 T + 198 T^{2} + 1164 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 - 7 T + 181 T^{2} - 1617 T^{3} + 17557 T^{4} - 65863 T^{5} + 912673 T^{6} )^{2}$$)($$( 1 + 7 T + 181 T^{2} + 1617 T^{3} + 17557 T^{4} + 65863 T^{5} + 912673 T^{6} )^{2}$$)($$1 + 3383 T^{4} + 88529281 T^{8}$$)($$( 1 + 9409 T^{4} )^{2}$$)($$1 + 16903 T^{4} + 88529281 T^{8}$$)($$( 1 + 9409 T^{4} )^{4}$$)($$1 + 36 T + 549 T^{2} + 5092 T^{3} + 43869 T^{4} + 493924 T^{5} + 5165541 T^{6} + 32856228 T^{7} + 88529281 T^{8}$$)($$( 1 - 252 T^{2} + 30086 T^{4} - 2371068 T^{6} + 88529281 T^{8} )^{2}$$)($$( 1 - 313 T^{2} + 45237 T^{4} - 4644209 T^{6} + 425634933 T^{8} - 27709664953 T^{10} + 832972004929 T^{12} )^{2}$$)($$1 + 4 T + 8 T^{2} + 12 T^{3} - 8818 T^{4} + 1164 T^{5} + 75272 T^{6} + 3650692 T^{7} + 88529281 T^{8}$$)($$1 - 4 T + 8 T^{2} - 12 T^{3} - 8818 T^{4} - 1164 T^{5} + 75272 T^{6} - 3650692 T^{7} + 88529281 T^{8}$$)($$( 1 + 2498 T^{4} + 88529281 T^{8} )^{2}$$)