LMFDB - Space of modular forms of level 336 and weight 2

Defining parameters

Level:	$ N $	=	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	=	$ 2 $
Nonzero newspaces:			$ 16 $
Newform subspaces:			$ 56 $
Sturm bound:			$12288$
Trace bound:			$8$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_1(336))$.

	Total	New	Old
Modular forms	3408	1288	2120
Cusp forms	2737	1196	1541
Eisenstein series	671	92	579

Trace form

$ 1196q - 7q^{3} - 8q^{4} + 4q^{5} + 4q^{6} - 10q^{7} + 24q^{8} + 7q^{9} + O(q^{10}) $

\( 1196q - 7q^{3} - 8q^{4} + 4q^{5} + 4q^{6} - 10q^{7} + 24q^{8} + 7q^{9} - 8q^{10} + 24q^{11} - 12q^{12} - 4q^{13} - 12q^{14} + 2q^{15} - 56q^{16} - 4q^{17} - 28q^{18} + 26q^{19} - 32q^{20} - 7q^{21} - 80q^{22} + 20q^{23} - 68q^{24} + 24q^{25} - 40q^{26} - 28q^{27} - 24q^{28} + 44q^{29} - 52q^{30} - 2q^{31} + 13q^{33} - 8q^{34} + 12q^{35} - 56q^{36} + 62q^{37} + 16q^{38} - 48q^{39} + 24q^{40} + 12q^{41} - 40q^{42} - 48q^{43} - 16q^{44} - 35q^{45} - 48q^{46} - 36q^{47} - 4q^{48} - 84q^{49} - 96q^{50} - 107q^{51} - 176q^{52} - 124q^{53} - 84q^{54} - 160q^{55} - 168q^{56} - 86q^{57} - 224q^{58} - 104q^{59} - 196q^{60} - 234q^{61} - 192q^{62} - 47q^{63} - 344q^{64} - 72q^{65} - 236q^{66} - 86q^{67} - 184q^{68} - 140q^{69} - 192q^{70} + 4q^{71} - 180q^{72} - 66q^{73} - 104q^{74} - 40q^{75} - 152q^{76} - 16q^{77} - 120q^{78} + 14q^{79} - 16q^{80} - 89q^{81} - 72q^{82} + 72q^{83} - 4q^{84} - 84q^{85} + 32q^{86} + 36q^{87} + 40q^{88} + 12q^{89} + 36q^{90} - 164q^{91} + 56q^{92} - 43q^{93} + 200q^{94} - 68q^{95} + 220q^{96} + 20q^{97} + 208q^{98} + 14q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_1(336))$

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
336.2.a	$\chi_{336}(1, \cdot)$	336.2.a.a	1	1
		336.2.a.b	1
		336.2.a.c	1
		336.2.a.d	1
		336.2.a.e	1
		336.2.a.f	1
336.2.b	$\chi_{336}(223, \cdot)$	336.2.b.a	2	1
		336.2.b.b	2
		336.2.b.c	2
		336.2.b.d	2
336.2.c	$\chi_{336}(169, \cdot)$	None	0	1
336.2.h	$\chi_{336}(239, \cdot)$	336.2.h.a	4	1
336.2.h	$\chi_{336}(239, \cdot)$	336.2.h.b	8	1
336.2.i	$\chi_{336}(41, \cdot)$	None	0	1
336.2.j	$\chi_{336}(71, \cdot)$	None	0	1
336.2.k	$\chi_{336}(209, \cdot)$	336.2.k.a	2	1
		336.2.k.b	4
		336.2.k.c	8
336.2.p	$\chi_{336}(55, \cdot)$	None	0	1
336.2.q	$\chi_{336}(193, \cdot)$	336.2.q.a	2	2
		336.2.q.b	2
		336.2.q.c	2
		336.2.q.d	2
		336.2.q.e	2
		336.2.q.f	2
		336.2.q.g	4
336.2.s	$\chi_{336}(155, \cdot)$	336.2.s.a	4	2
		336.2.s.b	4
		336.2.s.c	40
		336.2.s.d	48
336.2.u	$\chi_{336}(139, \cdot)$	336.2.u.a	64	2
336.2.w	$\chi_{336}(85, \cdot)$	336.2.w.a	20	2
336.2.w	$\chi_{336}(85, \cdot)$	336.2.w.b	28	2
336.2.y	$\chi_{336}(125, \cdot)$	336.2.y.a	120	2
336.2.bb	$\chi_{336}(103, \cdot)$	None	0	2
336.2.bc	$\chi_{336}(17, \cdot)$	336.2.bc.a	2	2
		336.2.bc.b	2
		336.2.bc.c	2
		336.2.bc.d	2
		336.2.bc.e	4
		336.2.bc.f	16
336.2.bd	$\chi_{336}(23, \cdot)$	None	0	2
336.2.bi	$\chi_{336}(89, \cdot)$	None	0	2
336.2.bj	$\chi_{336}(95, \cdot)$	336.2.bj.a	2	2
		336.2.bj.b	2
		336.2.bj.c	2
		336.2.bj.d	2
		336.2.bj.e	8
		336.2.bj.f	8
		336.2.bj.g	8
336.2.bk	$\chi_{336}(25, \cdot)$	None	0	2
336.2.bl	$\chi_{336}(31, \cdot)$	336.2.bl.a	2	2
		336.2.bl.b	2
		336.2.bl.c	2
		336.2.bl.d	2
		336.2.bl.e	2
		336.2.bl.f	2
		336.2.bl.g	2
		336.2.bl.h	2
336.2.bo	$\chi_{336}(5, \cdot)$	336.2.bo.a	240	4
336.2.bq	$\chi_{336}(37, \cdot)$	336.2.bq.a	8	4
336.2.bq	$\chi_{336}(37, \cdot)$	336.2.bq.b	120	4
336.2.bs	$\chi_{336}(19, \cdot)$	336.2.bs.a	128	4
336.2.bu	$\chi_{336}(11, \cdot)$	336.2.bu.a	240	4

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_1(336))$ into lower level spaces

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	($ ( 1 + 2 T + 2 T^{2} )^{2} $)($ ( 1 - 2 T + 2 T^{2} )^{2} $)($ 1 - 2 T^{2} + 2 T^{4} + 4 T^{6} - 8 T^{7} - 4 T^{8} - 16 T^{12} - 64 T^{13} + 64 T^{14} + 128 T^{16} - 512 T^{18} + 1024 T^{20} $)($ ( 1 - 2 T^{2} + 4 T^{4} )^{2} $)
$3$	($ 1 + T $)($ 1 + T $)($ 1 + T $)($ 1 - T $)($ 1 - T $)($ 1 - T $)($ ( 1 + T )^{2} $)($ ( 1 + T )^{2} $)($ ( 1 - T )^{2} $)($ ( 1 - T )^{2} $)($ 1 - 2 T^{2} + 9 T^{4} $)($ 1 - 4 T^{2} + 10 T^{4} - 36 T^{6} + 81 T^{8} $)($ 1 + 3 T^{2} $)($ 1 + 9 T^{4} $)($ 1 - 2 T^{2} + 2 T^{4} - 18 T^{6} + 81 T^{8} $)($ 1 + T + T^{2} $)($ 1 + T + T^{2} $)($ 1 + T + T^{2} $)($ 1 - T + T^{2} $)($ 1 - T + T^{2} $)($ 1 - T + T^{2} $)($ ( 1 + T + T^{2} )^{2} $)($ ( 1 + 2 T + 3 T^{2} )^{2} $)($ 1 - 2 T^{2} + 9 T^{4} $)($ ( 1 + T^{4} )^{5} $)($ 1 + 3 T + 3 T^{2} $)($ 1 + 3 T^{2} $)($ 1 - 3 T + 3 T^{2} $)($ 1 - 3 T + 3 T^{2} $)($ 1 + 3 T^{2} + 9 T^{4} $)($ 1 - T^{2} - 7 T^{4} + 24 T^{5} + 22 T^{6} - 48 T^{7} + 10 T^{8} - 144 T^{9} + 198 T^{10} + 648 T^{11} - 567 T^{12} - 729 T^{14} + 6561 T^{16} $)($ 1 + 3 T + 3 T^{2} $)($ 1 + 3 T + 3 T^{2} $)($ 1 - 3 T + 3 T^{2} $)($ 1 - 3 T + 3 T^{2} $)($ 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} + 18 T^{5} + 45 T^{6} + 81 T^{7} + 81 T^{8} $)($ 1 + 5 T^{2} + 16 T^{4} + 45 T^{6} + 81 T^{8} $)($ 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} - 18 T^{5} + 45 T^{6} - 81 T^{7} + 81 T^{8} $)($ 1 + T + T^{2} $)($ 1 + T + T^{2} $)($ 1 + T + T^{2} $)($ 1 + T + T^{2} $)($ 1 - T + T^{2} $)($ 1 - T + T^{2} $)($ 1 - T + T^{2} $)($ 1 - T + T^{2} $)($ 1 - T^{4} + T^{8} $)
$5$	($ 1 + 2 T + 5 T^{2} $)($ 1 + 5 T^{2} $)($ 1 - 2 T + 5 T^{2} $)($ 1 + 2 T + 5 T^{2} $)($ 1 - 2 T + 5 T^{2} $)($ 1 - 4 T + 5 T^{2} $)($ ( 1 - 5 T^{2} )^{2} $)($ 1 - 4 T^{2} + 25 T^{4} $)($ 1 - 4 T^{2} + 25 T^{4} $)($ ( 1 - 5 T^{2} )^{2} $)($ ( 1 - 8 T^{2} + 25 T^{4} )^{2} $)($ ( 1 - 4 T^{2} + 42 T^{4} - 100 T^{6} + 625 T^{8} )^{2} $)($ ( 1 + 5 T^{2} )^{2} $)($ ( 1 + 4 T^{2} + 25 T^{4} )^{2} $)($ ( 1 + 10 T^{2} + 58 T^{4} + 250 T^{6} + 625 T^{8} )^{2} $)($ 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} $)($ 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} $)($ 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} $)($ 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} $)($ 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} $)($ 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} $)($ ( 1 - T + 5 T^{2} )^{2}( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} ) $)($ 1 + 4 T + 8 T^{2} + 12 T^{3} + 14 T^{4} + 60 T^{5} + 200 T^{6} + 500 T^{7} + 625 T^{8} $)($ 1 - 4 T + 8 T^{2} - 12 T^{3} + 14 T^{4} - 60 T^{5} + 200 T^{6} - 500 T^{7} + 625 T^{8} $)($ 1 - 24 T^{3} + 10 T^{4} + 24 T^{5} + 288 T^{6} - 320 T^{7} + 485 T^{8} - 2720 T^{9} + 5088 T^{10} - 20208 T^{11} + 28392 T^{12} - 21904 T^{13} + 280352 T^{14} - 374560 T^{15} + 85402 T^{16} - 2818304 T^{17} + 5181408 T^{18} - 7099744 T^{19} + 30439356 T^{20} - 35498720 T^{21} + 129535200 T^{22} - 352288000 T^{23} + 53376250 T^{24} - 1170500000 T^{25} + 4380500000 T^{26} - 1711250000 T^{27} + 11090625000 T^{28} - 39468750000 T^{29} + 49687500000 T^{30} - 132812500000 T^{31} + 118408203125 T^{32} - 390625000000 T^{33} + 1757812500000 T^{34} + 732421875000 T^{35} + 1525878906250 T^{36} - 18310546875000 T^{37} + 95367431640625 T^{40} $)($ 1 - 5 T^{2} + 25 T^{4} $)($ 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} $)($ 1 - 5 T^{2} + 25 T^{4} $)($ 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} $)($ 1 - 7 T^{2} + 24 T^{4} - 175 T^{6} + 625 T^{8} $)($ ( 1 - 13 T^{2} + 93 T^{4} - 434 T^{6} + 1886 T^{8} - 10850 T^{10} + 58125 T^{12} - 203125 T^{14} + 390625 T^{16} )( 1 + 2 T^{2} - 39 T^{4} - 38 T^{6} + 836 T^{8} - 950 T^{10} - 24375 T^{12} + 31250 T^{14} + 390625 T^{16} ) $)($ 1 + 5 T^{2} + 25 T^{4} $)($ 1 + 5 T^{2} + 25 T^{4} $)($ 1 + 5 T^{2} + 25 T^{4} $)($ 1 + 5 T^{2} + 25 T^{4} $)($ 1 + 4 T^{2} + 19 T^{4} - 212 T^{6} - 1016 T^{8} - 5300 T^{10} + 11875 T^{12} + 62500 T^{14} + 390625 T^{16} $)($ ( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} )^{2}( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} )^{2} $)($ 1 + 4 T^{2} + 19 T^{4} - 212 T^{6} - 1016 T^{8} - 5300 T^{10} + 11875 T^{12} + 62500 T^{14} + 390625 T^{16} $)($ 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} $)($ 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} $)($ 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} $)($ 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} $)($ 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} $)($ 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} $)($ 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} $)($ 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} $)($ 1 - 8 T + 32 T^{2} - 64 T^{3} + 17 T^{4} + 272 T^{5} - 672 T^{6} + 376 T^{7} + 816 T^{8} + 1880 T^{9} - 16800 T^{10} + 34000 T^{11} + 10625 T^{12} - 200000 T^{13} + 500000 T^{14} - 625000 T^{15} + 390625 T^{16} $)
$7$	($ 1 - T $)($ 1 + T $)($ 1 - T $)($ 1 - T $)($ 1 + T $)($ 1 - T $)($ 1 + 4 T + 7 T^{2} $)($ 1 - 2 T + 7 T^{2} $)($ 1 + 2 T + 7 T^{2} $)($ 1 - 4 T + 7 T^{2} $)($ ( 1 + T^{2} )^{2} $)($ ( 1 + T^{2} )^{4} $)($ 1 + 4 T + 7 T^{2} $)($ ( 1 - 2 T + 7 T^{2} )^{2} $)($ ( 1 - 2 T - 2 T^{2} - 14 T^{3} + 49 T^{4} )^{2} $)($ 1 + 5 T + 7 T^{2} $)($ 1 + T + 7 T^{2} $)($ 1 + T + 7 T^{2} $)($ 1 + 5 T + 7 T^{2} $)($ 1 + T + 7 T^{2} $)($ 1 - 5 T + 7 T^{2} $)($ ( 1 - 3 T + 7 T^{2} )^{2} $)($ ( 1 + T )^{4} $)($ ( 1 + T )^{4} $)($ ( 1 + T^{2} )^{10} $)($ 1 - 5 T + 7 T^{2} $)($ 1 + 4 T + 7 T^{2} $)($ 1 + T + 7 T^{2} $)($ 1 + 4 T + 7 T^{2} $)($ ( 1 - 5 T + 7 T^{2} )^{2} $)($ ( 1 + 2 T + 9 T^{2} + 10 T^{3} + 44 T^{4} + 70 T^{5} + 441 T^{6} + 686 T^{7} + 2401 T^{8} )^{2} $)($ 1 + 5 T + 7 T^{2} $)($ 1 + T + 7 T^{2} $)($ 1 - T + 7 T^{2} $)($ 1 - 5 T + 7 T^{2} $)($ ( 1 + 3 T + 2 T^{2} + 21 T^{3} + 49 T^{4} )^{2} $)($ ( 1 + 2 T^{2} + 49 T^{4} )^{2} $)($ ( 1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4} )^{2} $)($ 1 - T + 7 T^{2} $)($ 1 + 5 T + 7 T^{2} $)($ 1 - T + 7 T^{2} $)($ 1 - T + 7 T^{2} $)($ 1 + T + 7 T^{2} $)($ 1 - 5 T + 7 T^{2} $)($ 1 + T + 7 T^{2} $)($ 1 + T + 7 T^{2} $)($ 1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8} $)
$11$	($ 1 + 4 T + 11 T^{2} $)($ 1 - 6 T + 11 T^{2} $)($ 1 + 11 T^{2} $)($ 1 - 4 T + 11 T^{2} $)($ 1 + 11 T^{2} $)($ 1 + 2 T + 11 T^{2} $)($ 1 - 10 T^{2} + 121 T^{4} $)($ 1 - 16 T^{2} + 121 T^{4} $)($ 1 - 16 T^{2} + 121 T^{4} $)($ 1 - 10 T^{2} + 121 T^{4} $)($ ( 1 + 20 T^{2} + 121 T^{4} )^{2} $)($ ( 1 + 4 T^{2} + 54 T^{4} + 484 T^{6} + 14641 T^{8} )^{2} $)($ ( 1 - 11 T^{2} )^{2} $)($ ( 1 - 11 T^{2} )^{4} $)($ ( 1 - 8 T^{2} + 190 T^{4} - 968 T^{6} + 14641 T^{8} )^{2} $)($ 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} $)($ 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} $)($ 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} $)($ 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} $)($ 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} $)($ 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} $)($ 1 + T - 7 T^{2} - 14 T^{3} - 68 T^{4} - 154 T^{5} - 847 T^{6} + 1331 T^{7} + 14641 T^{8} $)($ 1 + 4 T + 8 T^{2} - 12 T^{3} - 178 T^{4} - 132 T^{5} + 968 T^{6} + 5324 T^{7} + 14641 T^{8} $)($ 1 - 4 T + 8 T^{2} + 12 T^{3} - 178 T^{4} + 132 T^{5} + 968 T^{6} - 5324 T^{7} + 14641 T^{8} $)($ 1 - 12 T + 72 T^{2} - 404 T^{3} + 2214 T^{4} - 9404 T^{5} + 35048 T^{6} - 135156 T^{7} + 456709 T^{8} - 1318720 T^{9} + 4165952 T^{10} - 13629648 T^{11} + 41111256 T^{12} - 141076784 T^{13} + 543949056 T^{14} - 1968344544 T^{15} + 6732047034 T^{16} - 23311377560 T^{17} + 80768058832 T^{18} - 261514302488 T^{19} + 834968255588 T^{20} - 2876657327368 T^{21} + 9772935118672 T^{22} - 31027443532360 T^{23} + 98563900624794 T^{24} - 317003857155744 T^{25} + 963638933596416 T^{26} - 2749187413938064 T^{27} + 8812562832664536 T^{28} - 32137997030742768 T^{29} + 108054065891385152 T^{30} - 376246206268137920 T^{31} + 1433348485503871189 T^{32} - 4665951682525138236 T^{33} + 13309472167425430568 T^{34} - 39282841785184782004 T^{35} + $$10\!\cdots\!54$$ T^{36} - $$20\!\cdots\!84$$ T^{37} + $$40\!\cdots\!32$$ T^{38} - $$73\!\cdots\!92$$ T^{39} + $$67\!\cdots\!01$$ T^{40} $)($ 1 + 11 T^{2} + 121 T^{4} $)($ 1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4} $)($ 1 + 11 T^{2} + 121 T^{4} $)($ 1 - 9 T + 38 T^{2} - 99 T^{3} + 121 T^{4} $)($ 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} $)($ 1 + 49 T^{2} + 1300 T^{4} + 22265 T^{6} + 252641 T^{8} + 1395328 T^{10} - 12534274 T^{12} - 440430082 T^{14} - 6222779240 T^{16} - 53292039922 T^{18} - 183514305634 T^{20} + 2471908667008 T^{22} + 54155842054721 T^{24} + 577496758741265 T^{26} + 4079956889737300 T^{28} + 18607741845578809 T^{30} + 45949729863572161 T^{32} $)($ 1 - 11 T^{2} + 121 T^{4} $)($ 1 - 11 T^{2} + 121 T^{4} $)($ 1 - 11 T^{2} + 121 T^{4} $)($ 1 - 11 T^{2} + 121 T^{4} $)($ 1 - 4 T^{2} - 173 T^{4} + 212 T^{6} + 19144 T^{8} + 25652 T^{10} - 2532893 T^{12} - 7086244 T^{14} + 214358881 T^{16} $)($ ( 1 - 11 T^{2} )^{4}( 1 + 11 T^{2} + 121 T^{4} )^{2} $)($ 1 - 4 T^{2} - 173 T^{4} + 212 T^{6} + 19144 T^{8} + 25652 T^{10} - 2532893 T^{12} - 7086244 T^{14} + 214358881 T^{16} $)($ 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} $)($ 1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4} $)($ 1 - 3 T + 14 T^{2} - 33 T^{3} + 121 T^{4} $)($ 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} $)($ 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} $)($ 1 - 9 T + 38 T^{2} - 99 T^{3} + 121 T^{4} $)($ 1 + 3 T + 14 T^{2} + 33 T^{3} + 121 T^{4} $)($ 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} $)($ 1 - 4 T + 8 T^{2} + 64 T^{3} - 415 T^{4} + 1384 T^{5} - 168 T^{6} - 14524 T^{7} + 77232 T^{8} - 159764 T^{9} - 20328 T^{10} + 1842104 T^{11} - 6076015 T^{12} + 10307264 T^{13} + 14172488 T^{14} - 77948684 T^{15} + 214358881 T^{16} $)
$13$	($ 1 + 2 T + 13 T^{2} $)($ 1 - 2 T + 13 T^{2} $)($ 1 + 2 T + 13 T^{2} $)($ 1 - 6 T + 13 T^{2} $)($ 1 - 6 T + 13 T^{2} $)($ 1 + 6 T + 13 T^{2} $)($ ( 1 - 13 T^{2} )^{2} $)($ 1 - 2 T^{2} + 169 T^{4} $)($ 1 - 2 T^{2} + 169 T^{4} $)($ ( 1 - 13 T^{2} )^{2} $)($ ( 1 - 2 T + 13 T^{2} )^{4} $)($ ( 1 + 2 T + 26 T^{3} + 169 T^{4} )^{4} $)($ ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) $)($ ( 1 - 20 T^{2} + 169 T^{4} )^{2} $)($ ( 1 - 38 T^{2} + 682 T^{4} - 6422 T^{6} + 28561 T^{8} )^{2} $)($ ( 1 + 3 T + 13 T^{2} )^{2} $)($ ( 1 + 13 T^{2} )^{2} $)($ ( 1 + 3 T + 13 T^{2} )^{2} $)($ ( 1 + 4 T + 13 T^{2} )^{2} $)($ ( 1 - 4 T + 13 T^{2} )^{2} $)($ ( 1 - T + 13 T^{2} )^{2} $)($ ( 1 - 5 T + 18 T^{2} - 65 T^{3} + 169 T^{4} )^{2} $)($ 1 + 12 T + 72 T^{2} + 324 T^{3} + 1262 T^{4} + 4212 T^{5} + 12168 T^{6} + 26364 T^{7} + 28561 T^{8} $)($ 1 + 12 T + 72 T^{2} + 324 T^{3} + 1262 T^{4} + 4212 T^{5} + 12168 T^{6} + 26364 T^{7} + 28561 T^{8} $)($ 1 - 80 T^{3} - 118 T^{4} + 1616 T^{5} + 3200 T^{6} + 22560 T^{7} - 156451 T^{8} - 297120 T^{9} - 121472 T^{10} + 6082400 T^{11} + 43092280 T^{12} - 92564192 T^{13} - 225951616 T^{14} - 2287891296 T^{15} + 3788315570 T^{16} + 30959844064 T^{17} + 31244658048 T^{18} + 7310121920 T^{19} - 2235843760900 T^{20} + 95031584960 T^{21} + 5280347210112 T^{22} + 68018777408608 T^{23} + 108198080994770 T^{24} - 849478022965728 T^{25} - 1090625293673344 T^{26} - 5808265775303264 T^{27} + 35151696633933880 T^{28} + 64500806986335200 T^{29} - 16745946721881728 T^{30} - 532486696276273440 T^{31} - 3645008715497274931 T^{32} + 6832862404721227680 T^{33} + 12599604434237724800 T^{34} + 82716403110770663312 T^{35} - 78519159883615221238 T^{36} - $$69\!\cdots\!40$$ T^{37} + $$19\!\cdots\!01$$ T^{40} $)($ ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) $)($ ( 1 - 13 T^{2} )^{2} $)($ ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) $)($ ( 1 - 13 T^{2} )^{2} $)($ ( 1 - 14 T^{2} + 169 T^{4} )^{2} $)($ ( 1 - 49 T^{2} + 1278 T^{4} - 23495 T^{6} + 341186 T^{8} - 3970655 T^{10} + 36500958 T^{12} - 236513641 T^{14} + 815730721 T^{16} )^{2} $)($ ( 1 + 5 T + 13 T^{2} )^{2} $)($ ( 1 - 7 T + 13 T^{2} )^{2} $)($ ( 1 - 7 T + 13 T^{2} )^{2} $)($ ( 1 + 5 T + 13 T^{2} )^{2} $)($ ( 1 - 2 T + 13 T^{2} )^{8} $)($ ( 1 + 4 T + 13 T^{2} )^{8} $)($ ( 1 - 2 T + 13 T^{2} )^{8} $)($ ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) $)($ ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) $)($ ( 1 - 13 T^{2} )^{2} $)($ ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) $)($ ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) $)($ ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) $)($ ( 1 - 13 T^{2} )^{2} $)($ ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) $)($ ( 1 + 4 T + 8 T^{2} + 44 T^{3} + 238 T^{4} + 572 T^{5} + 1352 T^{6} + 8788 T^{7} + 28561 T^{8} )^{2} $)
$17$	($ 1 + 6 T + 17 T^{2} $)($ 1 + 17 T^{2} $)($ 1 - 6 T + 17 T^{2} $)($ 1 - 2 T + 17 T^{2} $)($ 1 + 2 T + 17 T^{2} $)($ 1 + 4 T + 17 T^{2} $)($ 1 + 14 T^{2} + 289 T^{4} $)($ 1 - 28 T^{2} + 289 T^{4} $)($ 1 - 28 T^{2} + 289 T^{4} $)($ 1 + 14 T^{2} + 289 T^{4} $)($ ( 1 + 16 T^{2} + 289 T^{4} )^{2} $)($ ( 1 - 28 T^{2} + 582 T^{4} - 8092 T^{6} + 83521 T^{8} )^{2} $)($ ( 1 + 17 T^{2} )^{2} $)($ ( 1 + 10 T^{2} + 289 T^{4} )^{2} $)($ ( 1 + 48 T^{2} + 1086 T^{4} + 13872 T^{6} + 83521 T^{8} )^{2} $)($ 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} $)($ 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} $)($ 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} $)($ 1 - 17 T^{2} + 289 T^{4} $)($ 1 - 17 T^{2} + 289 T^{4} $)($ 1 - 17 T^{2} + 289 T^{4} $)($ ( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} )^{2} $)($ 1 - 20 T^{2} + 166 T^{4} - 5780 T^{6} + 83521 T^{8} $)($ 1 - 20 T^{2} + 166 T^{4} - 5780 T^{6} + 83521 T^{8} $)($ ( 1 + 94 T^{2} + 88 T^{3} + 4293 T^{4} + 6432 T^{5} + 134312 T^{6} + 219984 T^{7} + 3234206 T^{8} + 4965576 T^{9} + 61650188 T^{10} + 84414792 T^{11} + 934685534 T^{12} + 1080781392 T^{13} + 11217872552 T^{14} + 9132520224 T^{15} + 103622583717 T^{16} + 36109803224 T^{17} + 655721199454 T^{18} + 2015993900449 T^{20} )^{2} $)($ 1 - 17 T^{2} + 289 T^{4} $)($ 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} $)($ 1 - 17 T^{2} + 289 T^{4} $)($ 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} $)($ 1 - 22 T^{2} + 195 T^{4} - 6358 T^{6} + 83521 T^{8} $)($ 1 - 42 T^{2} + 1059 T^{4} - 6894 T^{6} - 204407 T^{8} + 7559412 T^{10} - 49293810 T^{12} - 1407253056 T^{14} + 49740922386 T^{16} - 406696133184 T^{18} - 4117068305010 T^{20} + 182465828749428 T^{22} - 1425893651242487 T^{24} - 13898261949695406 T^{26} + 616996949226316899 T^{28} - 7071868715494839018 T^{30} + 48661191875666868481 T^{32} $)($ 1 + 17 T^{2} + 289 T^{4} $)($ 1 + 17 T^{2} + 289 T^{4} $)($ 1 + 17 T^{2} + 289 T^{4} $)($ 1 + 17 T^{2} + 289 T^{4} $)($ 1 + 25 T^{2} - 95 T^{4} + 3550 T^{6} + 254254 T^{8} + 1025950 T^{10} - 7934495 T^{12} + 603439225 T^{14} + 6975757441 T^{16} $)($ ( 1 + 23 T^{2} + 240 T^{4} + 6647 T^{6} + 83521 T^{8} )^{2} $)($ 1 + 25 T^{2} - 95 T^{4} + 3550 T^{6} + 254254 T^{8} + 1025950 T^{10} - 7934495 T^{12} + 603439225 T^{14} + 6975757441 T^{16} $)($ 1 + 12 T + 65 T^{2} + 204 T^{3} + 289 T^{4} $)($ 1 - 6 T + 29 T^{2} - 102 T^{3} + 289 T^{4} $)($ 1 - 6 T + 29 T^{2} - 102 T^{3} + 289 T^{4} $)($ 1 + 17 T^{2} + 289 T^{4} $)($ 1 + 12 T + 65 T^{2} + 204 T^{3} + 289 T^{4} $)($ 1 - 6 T + 29 T^{2} - 102 T^{3} + 289 T^{4} $)($ 1 - 6 T + 29 T^{2} - 102 T^{3} + 289 T^{4} $)($ 1 + 17 T^{2} + 289 T^{4} $)($ ( 1 - 8 T + 22 T^{2} - 64 T^{3} + 387 T^{4} - 1088 T^{5} + 6358 T^{6} - 39304 T^{7} + 83521 T^{8} )^{2} $)
$19$	($ 1 + 4 T + 19 T^{2} $)($ 1 - 4 T + 19 T^{2} $)($ 1 - 4 T + 19 T^{2} $)($ 1 - 4 T + 19 T^{2} $)($ 1 + 4 T + 19 T^{2} $)($ 1 - 4 T + 19 T^{2} $)($ ( 1 + 4 T + 19 T^{2} )^{2} $)($ ( 1 - 2 T + 19 T^{2} )^{2} $)($ ( 1 + 2 T + 19 T^{2} )^{2} $)($ ( 1 - 4 T + 19 T^{2} )^{2} $)($ ( 1 - 22 T^{2} + 361 T^{4} )^{2} $)($ ( 1 - 68 T^{2} + 1866 T^{4} - 24548 T^{6} + 130321 T^{8} )^{2} $)($ ( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) $)($ ( 1 - 32 T^{2} + 361 T^{4} )^{2} $)($ ( 1 - 2 T^{2} + 706 T^{4} - 722 T^{6} + 130321 T^{8} )^{2} $)($ 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} $)($ ( 1 - 7 T + 19 T^{2} )( 1 - T + 19 T^{2} ) $)($ ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) $)($ 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} $)($ 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} $)($ ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) $)($ 1 + 5 T - 5 T^{2} - 40 T^{3} + 64 T^{4} - 760 T^{5} - 1805 T^{6} + 34295 T^{7} + 130321 T^{8} $)($ 1 + 4 T + 8 T^{2} + 68 T^{3} + 574 T^{4} + 1292 T^{5} + 2888 T^{6} + 27436 T^{7} + 130321 T^{8} $)($ 1 + 4 T + 8 T^{2} + 68 T^{3} + 574 T^{4} + 1292 T^{5} + 2888 T^{6} + 27436 T^{7} + 130321 T^{8} $)($ 1 - 8 T + 32 T^{2} - 120 T^{3} + 170 T^{4} + 664 T^{5} - 3552 T^{6} + 30376 T^{7} - 137251 T^{8} + 141472 T^{9} + 439424 T^{10} - 1977376 T^{11} + 92609976 T^{12} - 734762848 T^{13} + 3144939136 T^{14} - 11850068256 T^{15} + 18020441650 T^{16} + 30523407824 T^{17} - 82150980672 T^{18} + 1163681013680 T^{19} - 8113217288772 T^{20} + 22109939259920 T^{21} - 29656504022592 T^{22} + 209360054264816 T^{23} + 2348441976269650 T^{24} - 29341942158613344 T^{25} + 147956432344498816 T^{26} - 656783744694352672 T^{27} + 1572847365621497016 T^{28} - 638074909083447904 T^{29} + 2694137659267946624 T^{30} + 16480109906848838368 T^{31} - $$30\!\cdots\!11$$ T^{32} + $$12\!\cdots\!84$$ T^{33} - $$28\!\cdots\!92$$ T^{34} + $$10\!\cdots\!36$$ T^{35} + $$49\!\cdots\!70$$ T^{36} - $$65\!\cdots\!80$$ T^{37} + $$33\!\cdots\!12$$ T^{38} - $$15\!\cdots\!32$$ T^{39} + $$37\!\cdots\!01$$ T^{40} $)($ ( 1 - 8 T + 19 T^{2} )( 1 - 7 T + 19 T^{2} ) $)($ 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} $)($ ( 1 - 8 T + 19 T^{2} )( 1 - T + 19 T^{2} ) $)($ 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} $)($ ( 1 - T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} $)($ ( 1 - 3 T + 62 T^{2} - 177 T^{3} + 2049 T^{4} - 6000 T^{5} + 55390 T^{6} - 152826 T^{7} + 1217108 T^{8} - 2903694 T^{9} + 19995790 T^{10} - 41154000 T^{11} + 267027729 T^{12} - 438269523 T^{13} + 2916844622 T^{14} - 2681615217 T^{15} + 16983563041 T^{16} )^{2} $)($ ( 1 + 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) $)($ ( 1 - 8 T + 19 T^{2} )( 1 - T + 19 T^{2} ) $)($ ( 1 + T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) $)($ ( 1 - 8 T + 19 T^{2} )( 1 - 7 T + 19 T^{2} ) $)($ ( 1 - 3 T + 37 T^{2} - 102 T^{3} + 852 T^{4} - 1938 T^{5} + 13357 T^{6} - 20577 T^{7} + 130321 T^{8} )^{2} $)($ ( 1 - 37 T^{2} + 361 T^{4} )^{2}( 1 + 26 T^{2} + 361 T^{4} )^{2} $)($ ( 1 + 3 T + 37 T^{2} + 102 T^{3} + 852 T^{4} + 1938 T^{5} + 13357 T^{6} + 20577 T^{7} + 130321 T^{8} )^{2} $)($ ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) $)($ 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} $)($ 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} $)($ 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} $)($ ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) $)($ 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} $)($ 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} $)($ 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} $)($ ( 1 - 4 T + 8 T^{2} + 120 T^{3} - 601 T^{4} + 2280 T^{5} + 2888 T^{6} - 27436 T^{7} + 130321 T^{8} )^{2} $)
$23$	($ 1 + 23 T^{2} $)($ 1 - 6 T + 23 T^{2} $)($ 1 - 4 T + 23 T^{2} $)($ 1 + 8 T + 23 T^{2} $)($ 1 - 4 T + 23 T^{2} $)($ 1 + 2 T + 23 T^{2} $)($ 1 - 34 T^{2} + 529 T^{4} $)($ 1 + 8 T^{2} + 529 T^{4} $)($ 1 + 8 T^{2} + 529 T^{4} $)($ 1 - 34 T^{2} + 529 T^{4} $)($ ( 1 - 4 T^{2} + 529 T^{4} )^{2} $)($ ( 1 + 28 T^{2} + 1062 T^{4} + 14812 T^{6} + 279841 T^{8} )^{2} $)($ ( 1 - 23 T^{2} )^{2} $)($ ( 1 - 10 T^{2} + 529 T^{4} )^{2} $)($ ( 1 - 42 T^{2} + 529 T^{4} )^{4} $)($ 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 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} $)($ 1 - 23 T^{2} + 529 T^{4} $)($ 1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} $)($ 1 - 23 T^{2} + 529 T^{4} $)($ ( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} )^{2} $)($ 1 - 20 T^{2} + 646 T^{4} - 10580 T^{6} + 279841 T^{8} $)($ 1 - 20 T^{2} + 646 T^{4} - 10580 T^{6} + 279841 T^{8} $)($ 1 - 228 T^{2} + 24654 T^{4} - 1671172 T^{6} + 79227605 T^{8} - 2786970288 T^{10} + 76106467832 T^{12} - 1708777858416 T^{14} + 34629617977562 T^{16} - 710121266972024 T^{18} + 15689370725592020 T^{20} - 375654150228200696 T^{22} + 9690786924458927642 T^{24} - $$25\!\cdots\!24$$ T^{26} + $$59\!\cdots\!92$$ T^{28} - $$11\!\cdots\!12$$ T^{30} + $$17\!\cdots\!05$$ T^{32} - $$19\!\cdots\!48$$ T^{34} + $$15\!\cdots\!94$$ T^{36} - $$73\!\cdots\!32$$ T^{38} + $$17\!\cdots\!01$$ T^{40} $)($ 1 + 23 T^{2} + 529 T^{4} $)($ 1 - 9 T + 50 T^{2} - 207 T^{3} + 529 T^{4} $)($ 1 + 23 T^{2} + 529 T^{4} $)($ 1 + 9 T + 50 T^{2} + 207 T^{3} + 529 T^{4} $)($ 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} $)($ 1 + 102 T^{2} + 6323 T^{4} + 248898 T^{6} + 6664873 T^{8} + 89132724 T^{10} - 1162854034 T^{12} - 108042764448 T^{14} - 3248548802990 T^{16} - 57154622392992 T^{18} - 325414235728594 T^{20} + 13194842036331636 T^{22} + 521932771402734313 T^{24} + 10310975788054808802 T^{26} + $$13\!\cdots\!83$$ T^{28} + $$11\!\cdots\!18$$ T^{30} + $$61\!\cdots\!61$$ T^{32} $)($ 1 - 23 T^{2} + 529 T^{4} $)($ 1 - 23 T^{2} + 529 T^{4} $)($ 1 - 23 T^{2} + 529 T^{4} $)($ 1 - 23 T^{2} + 529 T^{4} $)($ 1 - 79 T^{2} + 3637 T^{4} - 122134 T^{6} + 3199486 T^{8} - 64608886 T^{10} + 1017781717 T^{12} - 11694835231 T^{14} + 78310985281 T^{16} $)($ ( 1 - 35 T^{2} + 696 T^{4} - 18515 T^{6} + 279841 T^{8} )^{2} $)($ 1 - 79 T^{2} + 3637 T^{4} - 122134 T^{6} + 3199486 T^{8} - 64608886 T^{10} + 1017781717 T^{12} - 11694835231 T^{14} + 78310985281 T^{16} $)($ 1 + 23 T^{2} + 529 T^{4} $)($ 1 + 12 T + 71 T^{2} + 276 T^{3} + 529 T^{4} $)($ 1 + 23 T^{2} + 529 T^{4} $)($ 1 - 12 T + 71 T^{2} - 276 T^{3} + 529 T^{4} $)($ 1 + 23 T^{2} + 529 T^{4} $)($ 1 - 12 T + 71 T^{2} - 276 T^{3} + 529 T^{4} $)($ 1 + 23 T^{2} + 529 T^{4} $)($ 1 + 12 T + 71 T^{2} + 276 T^{3} + 529 T^{4} $)($ 1 + 24 T^{2} + 526 T^{4} - 24192 T^{6} - 582045 T^{8} - 12797568 T^{10} + 147196366 T^{12} + 3552861336 T^{14} + 78310985281 T^{16} $)
$29$	($ 1 + 2 T + 29 T^{2} $)($ 1 - 6 T + 29 T^{2} $)($ 1 - 6 T + 29 T^{2} $)($ 1 + 2 T + 29 T^{2} $)($ 1 + 10 T + 29 T^{2} $)($ 1 + 2 T + 29 T^{2} $)($ ( 1 + 6 T + 29 T^{2} )^{2} $)($ ( 1 - 6 T + 29 T^{2} )^{2} $)($ ( 1 - 6 T + 29 T^{2} )^{2} $)($ ( 1 + 6 T + 29 T^{2} )^{2} $)($ ( 1 - 50 T^{2} + 841 T^{4} )^{2} $)($ ( 1 - 76 T^{2} + 2934 T^{4} - 63916 T^{6} + 707281 T^{8} )^{2} $)($ ( 1 - 29 T^{2} )^{2} $)($ ( 1 - 22 T^{2} + 841 T^{4} )^{2} $)($ ( 1 - 32 T^{2} + 238 T^{4} - 26912 T^{6} + 707281 T^{8} )^{2} $)($ ( 1 + 4 T + 29 T^{2} )^{2} $)($ ( 1 + 5 T + 29 T^{2} )^{2} $)($ ( 1 - 4 T + 29 T^{2} )^{2} $)($ ( 1 - 9 T + 29 T^{2} )^{2} $)($ ( 1 + 3 T + 29 T^{2} )^{2} $)($ ( 1 - 4 T + 29 T^{2} )^{2} $)($ ( 1 - 3 T + 46 T^{2} - 87 T^{3} + 841 T^{4} )^{2} $)($ ( 1 - 4 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2} $)($ ( 1 - 10 T + 29 T^{2} )^{2}( 1 + 4 T + 29 T^{2} )^{2} $)($ 1 - 12 T + 72 T^{2} - 660 T^{3} + 5062 T^{4} - 19812 T^{5} + 91080 T^{6} - 552988 T^{7} + 898845 T^{8} + 3230928 T^{9} - 3305184 T^{10} + 180759600 T^{11} - 2401479032 T^{12} + 11590609840 T^{13} - 61731406432 T^{14} + 542524821712 T^{15} - 2180038368078 T^{16} + 2872412612888 T^{17} - 25762237441424 T^{18} + 2546561760168 T^{19} + 1284480158865124 T^{20} + 73850291044872 T^{21} - 21666041688237584 T^{22} + 70055271215725432 T^{23} - 1541899717012575918 T^{24} + 11127807454333267088 T^{25} - 36719280183883000672 T^{26} + $$19\!\cdots\!60$$ T^{27} - $$12\!\cdots\!52$$ T^{28} + $$26\!\cdots\!00$$ T^{29} - $$13\!\cdots\!84$$ T^{30} + $$39\!\cdots\!12$$ T^{31} + $$31\!\cdots\!45$$ T^{32} - $$56\!\cdots\!32$$ T^{33} + $$27\!\cdots\!80$$ T^{34} - $$17\!\cdots\!88$$ T^{35} + $$12\!\cdots\!02$$ T^{36} - $$47\!\cdots\!40$$ T^{37} + $$15\!\cdots\!92$$ T^{38} - $$73\!\cdots\!28$$ T^{39} + $$17\!\cdots\!01$$ T^{40} $)($ ( 1 - 29 T^{2} )^{2} $)($ ( 1 - 29 T^{2} )^{2} $)($ ( 1 - 29 T^{2} )^{2} $)($ ( 1 - 29 T^{2} )^{2} $)($ ( 1 - 49 T^{2} + 841 T^{4} )^{2} $)($ ( 1 - 103 T^{2} + 6606 T^{4} - 293969 T^{6} + 9810626 T^{8} - 247227929 T^{10} + 4672298286 T^{12} - 61266802063 T^{14} + 500246412961 T^{16} )^{2} $)($ ( 1 - 29 T^{2} )^{2} $)($ ( 1 - 29 T^{2} )^{2} $)($ ( 1 - 29 T^{2} )^{2} $)($ ( 1 - 29 T^{2} )^{2} $)($ ( 1 - 31 T^{2} + 1908 T^{4} - 26071 T^{6} + 707281 T^{8} )^{2} $)($ ( 1 - 14 T^{2} + 841 T^{4} )^{4} $)($ ( 1 - 31 T^{2} + 1908 T^{4} - 26071 T^{6} + 707281 T^{8} )^{2} $)($ ( 1 + 29 T^{2} )^{2} $)($ ( 1 + 9 T + 29 T^{2} )^{2} $)($ ( 1 - 9 T + 29 T^{2} )^{2} $)($ ( 1 + 29 T^{2} )^{2} $)($ ( 1 + 29 T^{2} )^{2} $)($ ( 1 + 9 T + 29 T^{2} )^{2} $)($ ( 1 - 9 T + 29 T^{2} )^{2} $)($ ( 1 + 29 T^{2} )^{2} $)($ ( 1 - 16 T + 128 T^{2} - 832 T^{3} + 4879 T^{4} - 24128 T^{5} + 107648 T^{6} - 390224 T^{7} + 707281 T^{8} )^{2} $)
$31$	($ 1 + 31 T^{2} $)($ 1 + 8 T + 31 T^{2} $)($ 1 - 8 T + 31 T^{2} $)($ 1 + 31 T^{2} $)($ 1 - 8 T + 31 T^{2} $)($ 1 + 31 T^{2} $)($ ( 1 - 4 T + 31 T^{2} )^{2} $)($ ( 1 + 8 T + 31 T^{2} )^{2} $)($ ( 1 - 8 T + 31 T^{2} )^{2} $)($ ( 1 + 4 T + 31 T^{2} )^{2} $)($ ( 1 - 26 T^{2} + 961 T^{4} )^{2} $)($ ( 1 - 28 T^{2} + 390 T^{4} - 26908 T^{6} + 923521 T^{8} )^{2} $)($ ( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) $)($ ( 1 - 31 T^{2} )^{4} $)($ ( 1 - 32 T^{2} + 2110 T^{4} - 30752 T^{6} + 923521 T^{8} )^{2} $)($ ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) $)($ 1 - 3 T - 22 T^{2} - 93 T^{3} + 961 T^{4} $)($ 1 - 3 T - 22 T^{2} - 93 T^{3} + 961 T^{4} $)($ 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} $)($ 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} $)($ 1 - 9 T + 50 T^{2} - 279 T^{3} + 961 T^{4} $)($ ( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} )^{2} $)($ 1 + 28 T^{2} + 966 T^{4} + 26908 T^{6} + 923521 T^{8} $)($ 1 + 28 T^{2} + 966 T^{4} + 26908 T^{6} + 923521 T^{8} $)($ ( 1 + 134 T^{2} - 64 T^{3} + 7637 T^{4} + 64 T^{5} + 250776 T^{6} + 647232 T^{7} + 5787898 T^{8} + 46731200 T^{9} + 141742500 T^{10} + 1448667200 T^{11} + 5562169978 T^{12} + 19281688512 T^{13} + 231596902296 T^{14} + 1832265664 T^{15} + 6777865611797 T^{16} - 1760807303104 T^{17} + 114287399017094 T^{18} + 819628286980801 T^{20} )^{2} $)($ ( 1 - 7 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) $)($ ( 1 - 7 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) $)($ ( 1 + 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) $)($ ( 1 - 7 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) $)($ ( 1 - 7 T + 31 T^{2} )^{2}( 1 + 4 T + 31 T^{2} )^{2} $)($ ( 1 - 24 T + 350 T^{2} - 3792 T^{3} + 33057 T^{4} - 243840 T^{5} + 1597150 T^{6} - 9593544 T^{7} + 54548420 T^{8} - 297399864 T^{9} + 1534861150 T^{10} - 7264237440 T^{11} + 30528833697 T^{12} - 108561740592 T^{13} + 310626288350 T^{14} - 660302738664 T^{15} + 852891037441 T^{16} )^{2} $)($ ( 1 - 7 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) $)($ ( 1 - 11 T + 31 T^{2} )( 1 - 4 T + 31 T^{2} ) $)($ ( 1 + 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) $)($ ( 1 - 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) $)($ ( 1 - 4 T + 31 T^{2} )^{4}( 1 + 7 T + 31 T^{2} )^{4} $)($ ( 1 + 53 T^{2} + 1848 T^{4} + 50933 T^{6} + 923521 T^{8} )^{2} $)($ ( 1 - 7 T + 31 T^{2} )^{4}( 1 + 4 T + 31 T^{2} )^{4} $)($ 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} $)($ 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} $)($ 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} $)($ 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} $)($ 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} $)($ 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} $)($ 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} $)($ 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} $)($ ( 1 - 2 T - 57 T^{2} + 2 T^{3} + 2636 T^{4} + 62 T^{5} - 54777 T^{6} - 59582 T^{7} + 923521 T^{8} )^{2} $)
$37$	($ 1 - 6 T + 37 T^{2} $)($ 1 - 2 T + 37 T^{2} $)($ 1 + 10 T + 37 T^{2} $)($ 1 + 10 T + 37 T^{2} $)($ 1 - 6 T + 37 T^{2} $)($ 1 - 2 T + 37 T^{2} $)($ ( 1 + 2 T + 37 T^{2} )^{2} $)($ ( 1 - 4 T + 37 T^{2} )^{2} $)($ ( 1 - 4 T + 37 T^{2} )^{2} $)($ ( 1 + 2 T + 37 T^{2} )^{2} $)($ ( 1 + 4 T + 37 T^{2} )^{4} $)($ ( 1 - 16 T + 126 T^{2} - 592 T^{3} + 1369 T^{4} )^{4} $)($ ( 1 - 10 T + 37 T^{2} )^{2} $)($ ( 1 + 2 T + 37 T^{2} )^{4} $)($ ( 1 + 2 T + 37 T^{2} )^{8} $)($ 1 - 9 T + 44 T^{2} - 333 T^{3} + 1369 T^{4} $)($ 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} $)($ ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{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 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} $)($ 1 + 3 T - 53 T^{2} - 36 T^{3} + 2142 T^{4} - 1332 T^{5} - 72557 T^{6} + 151959 T^{7} + 1874161 T^{8} $)($ 1 + 4 T + 8 T^{2} + 92 T^{3} + 862 T^{4} + 3404 T^{5} + 10952 T^{6} + 202612 T^{7} + 1874161 T^{8} $)($ 1 + 4 T + 8 T^{2} + 92 T^{3} + 862 T^{4} + 3404 T^{5} + 10952 T^{6} + 202612 T^{7} + 1874161 T^{8} $)($ 1 - 12 T + 72 T^{2} - 692 T^{3} + 4646 T^{4} - 25508 T^{5} + 211016 T^{6} - 1455420 T^{7} + 12729021 T^{8} - 102763568 T^{9} + 668772640 T^{10} - 4907491152 T^{11} + 30685175816 T^{12} - 181058288208 T^{13} + 1246622676896 T^{14} - 7968453274800 T^{15} + 52782260071730 T^{16} - 346974533424104 T^{17} + 2138919388551792 T^{18} - 13427579452955416 T^{19} + 82729716000045604 T^{20} - 496820439759350392 T^{21} + 2928180642927403248 T^{22} - 17575301041531139912 T^{23} + 98922453318293568530 T^{24} - $$55\!\cdots\!00$$ T^{25} + $$31\!\cdots\!64$$ T^{26} - $$17\!\cdots\!64$$ T^{27} + $$10\!\cdots\!36$$ T^{28} - $$63\!\cdots\!04$$ T^{29} + $$32\!\cdots\!60$$ T^{30} - $$18\!\cdots\!84$$ T^{31} + $$83\!\cdots\!01$$ T^{32} - $$35\!\cdots\!40$$ T^{33} + $$19\!\cdots\!24$$ T^{34} - $$85\!\cdots\!44$$ T^{35} + $$57\!\cdots\!86$$ T^{36} - $$31\!\cdots\!64$$ T^{37} + $$12\!\cdots\!88$$ T^{38} - $$74\!\cdots\!76$$ T^{39} + $$23\!\cdots\!01$$ T^{40} $)($ ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) $)($ 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} $)($ ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) $)($ 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} $)($ ( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} )^{2} $)($ ( 1 + T - 60 T^{2} + 689 T^{3} + 2741 T^{4} - 33360 T^{5} + 198298 T^{6} + 1145242 T^{7} - 8840520 T^{8} + 42373954 T^{9} + 271469962 T^{10} - 1689784080 T^{11} + 5137075301 T^{12} + 47777986373 T^{13} - 153943584540 T^{14} + 94931877133 T^{15} + 3512479453921 T^{16} )^{2} $)($ ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) $)($ ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) $)($ ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) $)($ ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) $)($ ( 1 - 5 T - 41 T^{2} + 40 T^{3} + 2062 T^{4} + 1480 T^{5} - 56129 T^{6} - 253265 T^{7} + 1874161 T^{8} )^{2} $)($ ( 1 - 11 T + 37 T^{2} )^{4}( 1 + 10 T + 37 T^{2} )^{4} $)($ ( 1 - 5 T - 41 T^{2} + 40 T^{3} + 2062 T^{4} + 1480 T^{5} - 56129 T^{6} - 253265 T^{7} + 1874161 T^{8} )^{2} $)($ ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) $)($ 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} $)($ ( 1 - T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) $)($ ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) $)($ ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) $)($ 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} $)($ ( 1 - T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) $)($ ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) $)($ ( 1 + 4 T + 8 T^{2} - 264 T^{3} - 1897 T^{4} - 9768 T^{5} + 10952 T^{6} + 202612 T^{7} + 1874161 T^{8} )^{2} $)
$41$	($ 1 - 2 T + 41 T^{2} $)($ 1 - 12 T + 41 T^{2} $)($ 1 + 10 T + 41 T^{2} $)($ 1 + 6 T + 41 T^{2} $)($ 1 + 2 T + 41 T^{2} $)($ 1 + 41 T^{2} $)($ 1 - 34 T^{2} + 1681 T^{4} $)($ 1 - 28 T^{2} + 1681 T^{4} $)($ 1 - 28 T^{2} + 1681 T^{4} $)($ 1 - 34 T^{2} + 1681 T^{4} $)($ ( 1 - 80 T^{2} + 1681 T^{4} )^{2} $)($ ( 1 - 76 T^{2} + 3078 T^{4} - 127756 T^{6} + 2825761 T^{8} )^{2} $)($ ( 1 + 41 T^{2} )^{2} $)($ ( 1 + 58 T^{2} + 1681 T^{4} )^{2} $)($ ( 1 + 48 T^{2} + 606 T^{4} + 80688 T^{6} + 2825761 T^{8} )^{2} $)($ ( 1 + 2 T + 41 T^{2} )^{2} $)($ ( 1 + 41 T^{2} )^{2} $)($ ( 1 - 6 T + 41 T^{2} )^{2} $)($ ( 1 + 41 T^{2} )^{2} $)($ ( 1 - 8 T + 41 T^{2} )^{2} $)($ ( 1 + 10 T + 41 T^{2} )^{2} $)($ ( 1 - 6 T + 34 T^{2} - 246 T^{3} + 1681 T^{4} )^{2} $)($ ( 1 + 12 T + 86 T^{2} + 492 T^{3} + 1681 T^{4} )^{2} $)($ ( 1 - 12 T + 86 T^{2} - 492 T^{3} + 1681 T^{4} )^{2} $)($ 1 - 444 T^{2} + 98574 T^{4} - 14512124 T^{6} + 1588488149 T^{8} - 137675000112 T^{10} + 9846636632440 T^{12} - 599501624736304 T^{14} + 31880388630283994 T^{16} - 1512585137889846120 T^{18} + 64990434418739739988 T^{20} - $$25\!\cdots\!20$$ T^{22} + $$90\!\cdots\!34$$ T^{24} - $$28\!\cdots\!64$$ T^{26} + $$78\!\cdots\!40$$ T^{28} - $$18\!\cdots\!12$$ T^{30} + $$35\!\cdots\!69$$ T^{32} - $$55\!\cdots\!64$$ T^{34} + $$62\!\cdots\!34$$ T^{36} - $$47\!\cdots\!24$$ T^{38} + $$18\!\cdots\!01$$ T^{40} $)($ ( 1 + 41 T^{2} )^{2} $)($ ( 1 - 6 T + 41 T^{2} )^{2} $)($ ( 1 + 41 T^{2} )^{2} $)($ ( 1 + 6 T + 41 T^{2} )^{2} $)($ ( 1 + 34 T^{2} + 1681 T^{4} )^{2} $)($ ( 1 + 240 T^{2} + 27996 T^{4} + 2035728 T^{6} + 100303238 T^{8} + 3422058768 T^{10} + 79110004956 T^{12} + 1140025017840 T^{14} + 7984925229121 T^{16} )^{2} $)($ ( 1 - 41 T^{2} )^{2} $)($ ( 1 - 41 T^{2} )^{2} $)($ ( 1 - 41 T^{2} )^{2} $)($ ( 1 - 41 T^{2} )^{2} $)($ ( 1 - 64 T^{2} + 2334 T^{4} - 107584 T^{6} + 2825761 T^{8} )^{2} $)($ ( 1 - 38 T^{2} + 1681 T^{4} )^{4} $)($ ( 1 - 64 T^{2} + 2334 T^{4} - 107584 T^{6} + 2825761 T^{8} )^{2} $)($ 1 + 26 T^{2} + 1681 T^{4} $)($ 1 - 70 T^{2} + 1681 T^{4} $)($ 1 + 26 T^{2} + 1681 T^{4} $)($ 1 - 70 T^{2} + 1681 T^{4} $)($ 1 + 26 T^{2} + 1681 T^{4} $)($ 1 - 70 T^{2} + 1681 T^{4} $)($ 1 + 26 T^{2} + 1681 T^{4} $)($ 1 - 70 T^{2} + 1681 T^{4} $)($ ( 1 - 152 T^{2} + 9106 T^{4} - 255512 T^{6} + 2825761 T^{8} )^{2} $)
$43$	($ 1 - 4 T + 43 T^{2} $)($ 1 - 4 T + 43 T^{2} $)($ 1 + 12 T + 43 T^{2} $)($ 1 - 4 T + 43 T^{2} $)($ 1 - 4 T + 43 T^{2} $)($ 1 - 4 T + 43 T^{2} $)($ 1 - 74 T^{2} + 1849 T^{4} $)($ 1 - 62 T^{2} + 1849 T^{4} $)($ 1 - 62 T^{2} + 1849 T^{4} $)($ 1 - 74 T^{2} + 1849 T^{4} $)($ ( 1 - 70 T^{2} + 1849 T^{4} )^{2} $)($ ( 1 - 82 T^{2} + 1849 T^{4} )^{4} $)($ ( 1 - 8 T + 43 T^{2} )^{2} $)($ ( 1 + 4 T + 43 T^{2} )^{4} $)($ ( 1 - 4 T + 43 T^{2} )^{8} $)($ ( 1 - T + 43 T^{2} )^{2} $)($ ( 1 + 2 T + 43 T^{2} )^{2} $)($ ( 1 + 11 T + 43 T^{2} )^{2} $)($ ( 1 - 10 T + 43 T^{2} )^{2} $)($ ( 1 + 6 T + 43 T^{2} )^{2} $)($ ( 1 + 5 T + 43 T^{2} )^{2} $)($ ( 1 - 7 T + 84 T^{2} - 301 T^{3} + 1849 T^{4} )^{2} $)($ 1 + 4 T + 8 T^{2} + 116 T^{3} + 1486 T^{4} + 4988 T^{5} + 14792 T^{6} + 318028 T^{7} + 3418801 T^{8} $)($ 1 + 4 T + 8 T^{2} + 116 T^{3} + 1486 T^{4} + 4988 T^{5} + 14792 T^{6} + 318028 T^{7} + 3418801 T^{8} $)($ 1 - 4 T + 8 T^{2} + 300 T^{3} + 7078 T^{4} - 44284 T^{5} + 165512 T^{6} + 1626836 T^{7} + 19994909 T^{8} - 198352400 T^{9} + 930543520 T^{10} + 1629993648 T^{11} + 32879446344 T^{12} - 523812434096 T^{13} + 2532563214112 T^{14} - 8844397053040 T^{15} + 52614175288210 T^{16} - 1034528298704760 T^{17} + 4610401791613808 T^{18} - 37116758465801176 T^{19} + 100269344691342500 T^{20} - 1596020614029450568 T^{21} + 8524632912693930992 T^{22} - 82252241445119353320 T^{23} + $$17\!\cdots\!10$$ T^{24} - $$13\!\cdots\!20$$ T^{25} + $$16\!\cdots\!88$$ T^{26} - $$14\!\cdots\!72$$ T^{27} + $$38\!\cdots\!44$$ T^{28} + $$81\!\cdots\!64$$ T^{29} + $$20\!\cdots\!80$$ T^{30} - $$18\!\cdots\!00$$ T^{31} + $$79\!\cdots\!09$$ T^{32} + $$27\!\cdots\!48$$ T^{33} + $$12\!\cdots\!88$$ T^{34} - $$14\!\cdots\!88$$ T^{35} + $$96\!\cdots\!78$$ T^{36} + $$17\!\cdots\!00$$ T^{37} + $$20\!\cdots\!92$$ T^{38} - $$43\!\cdots\!28$$ T^{39} + $$46\!\cdots\!01$$ T^{40} $)($ ( 1 + 13 T + 43 T^{2} )^{2} $)($ ( 1 + 4 T + 43 T^{2} )^{2} $)($ ( 1 - 5 T + 43 T^{2} )^{2} $)($ ( 1 + 4 T + 43 T^{2} )^{2} $)($ ( 1 - 8 T + 43 T^{2} )^{4} $)($ ( 1 + 5 T + 76 T^{2} + 341 T^{3} + 2710 T^{4} + 14663 T^{5} + 140524 T^{6} + 397535 T^{7} + 3418801 T^{8} )^{4} $)($ ( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) $)($ ( 1 - 5 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} ) $)($ ( 1 - 5 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} ) $)($ ( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) $)($ ( 1 - 80 T^{2} + 3246 T^{4} - 147920 T^{6} + 3418801 T^{8} )^{2} $)($ ( 1 - 82 T^{2} + 1849 T^{4} )^{4} $)($ ( 1 - 80 T^{2} + 3246 T^{4} - 147920 T^{6} + 3418801 T^{8} )^{2} $)($ ( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) $)($ 1 - 74 T^{2} + 1849 T^{4} $)($ 1 - 74 T^{2} + 1849 T^{4} $)($ 1 - 11 T^{2} + 1849 T^{4} $)($ ( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) $)($ 1 - 74 T^{2} + 1849 T^{4} $)($ 1 - 74 T^{2} + 1849 T^{4} $)($ 1 - 11 T^{2} + 1849 T^{4} $)($ ( 1 - 24 T + 288 T^{2} - 2664 T^{3} + 20018 T^{4} - 114552 T^{5} + 532512 T^{6} - 1908168 T^{7} + 3418801 T^{8} )^{2} $)
$47$	($ 1 + 47 T^{2} $)($ 1 + 12 T + 47 T^{2} $)($ 1 - 8 T + 47 T^{2} $)($ 1 + 47 T^{2} $)($ 1 + 8 T + 47 T^{2} $)($ 1 + 12 T + 47 T^{2} $)($ ( 1 + 47 T^{2} )^{2} $)($ ( 1 - 12 T + 47 T^{2} )^{2} $)($ ( 1 + 12 T + 47 T^{2} )^{2} $)($ ( 1 + 47 T^{2} )^{2} $)($ ( 1 + 22 T^{2} + 2209 T^{4} )^{2} $)($ ( 1 + 47 T^{2} )^{8} $)($ ( 1 + 47 T^{2} )^{2} $)($ ( 1 + 70 T^{2} + 2209 T^{4} )^{2} $)($ ( 1 + 4 T^{2} + 4150 T^{4} + 8836 T^{6} + 4879681 T^{8} )^{2} $)($ 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} $)($ 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} $)($ 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} $)($ 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} $)($ 1 - 10 T + 53 T^{2} - 470 T^{3} + 2209 T^{4} $)($ 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} $)($ ( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} )^{2} $)($ ( 1 + 62 T^{2} + 2209 T^{4} )^{2} $)($ ( 1 + 62 T^{2} + 2209 T^{4} )^{2} $)($ ( 1 + 254 T^{2} - 128 T^{3} + 33997 T^{4} - 26752 T^{5} + 3110696 T^{6} - 2769024 T^{7} + 212743186 T^{8} - 186077312 T^{9} + 11293892148 T^{10} - 8745633664 T^{11} + 469949697874 T^{12} - 287488378752 T^{13} + 15179204167976 T^{14} - 6135437627264 T^{15} + 366460983540013 T^{16} - 64847759419264 T^{17} + 6048066812087294 T^{18} + 52599132235830049 T^{20} )^{2} $)($ 1 - 47 T^{2} + 2209 T^{4} $)($ 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} $)($ 1 - 47 T^{2} + 2209 T^{4} $)($ 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} $)($ 1 - 46 T^{2} - 93 T^{4} - 101614 T^{6} + 4879681 T^{8} $)($ 1 - 158 T^{2} + 13171 T^{4} - 396034 T^{6} - 16747399 T^{8} + 2338864468 T^{10} - 73391270338 T^{12} - 1850417430616 T^{14} + 239524986298546 T^{16} - 4087572104230744 T^{18} - 358125987434202178 T^{20} + 25211123725919029972 T^{22} - $$39\!\cdots\!39$$ T^{24} - $$20\!\cdots\!66$$ T^{26} + $$15\!\cdots\!11$$ T^{28} - $$40\!\cdots\!02$$ T^{30} + $$56\!\cdots\!21$$ T^{32} $)($ 1 - 47 T^{2} + 2209 T^{4} $)($ 1 - 47 T^{2} + 2209 T^{4} $)($ 1 - 47 T^{2} + 2209 T^{4} $)($ 1 - 47 T^{2} + 2209 T^{4} $)($ 1 - 71 T^{2} + 517 T^{4} - 7526 T^{6} + 4451326 T^{8} - 16624934 T^{10} + 2522795077 T^{12} - 765324288359 T^{14} + 23811286661761 T^{16} $)($ ( 1 + 5 T^{2} - 2184 T^{4} + 11045 T^{6} + 4879681 T^{8} )^{2} $)($ 1 - 71 T^{2} + 517 T^{4} - 7526 T^{6} + 4451326 T^{8} - 16624934 T^{10} + 2522795077 T^{12} - 765324288359 T^{14} + 23811286661761 T^{16} $)($ 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} $)($ 1 - 47 T^{2} + 2209 T^{4} $)($ 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} $)($ 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} $)($ 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} $)($ 1 - 47 T^{2} + 2209 T^{4} $)($ 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} $)($ 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} $)($ ( 1 - 4 T - 64 T^{2} + 56 T^{3} + 3439 T^{4} + 2632 T^{5} - 141376 T^{6} - 415292 T^{7} + 4879681 T^{8} )^{2} $)
$53$	($ 1 - 6 T + 53 T^{2} $)($ 1 + 6 T + 53 T^{2} $)($ 1 - 6 T + 53 T^{2} $)($ 1 - 6 T + 53 T^{2} $)($ 1 + 10 T + 53 T^{2} $)($ 1 + 6 T + 53 T^{2} $)($ ( 1 + 6 T + 53 T^{2} )^{2} $)($ ( 1 + 6 T + 53 T^{2} )^{2} $)($ ( 1 + 6 T + 53 T^{2} )^{2} $)($ ( 1 + 6 T + 53 T^{2} )^{2} $)($ ( 1 + 22 T^{2} + 2809 T^{4} )^{2} $)($ ( 1 - 124 T^{2} + 7734 T^{4} - 348316 T^{6} + 7890481 T^{8} )^{2} $)($ ( 1 - 53 T^{2} )^{2} $)($ ( 1 - 70 T^{2} + 2809 T^{4} )^{2} $)($ ( 1 - 128 T^{2} + 8014 T^{4} - 359552 T^{6} + 7890481 T^{8} )^{2} $)($ 1 + 8 T + 11 T^{2} + 424 T^{3} + 2809 T^{4} $)($ 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} $)($ 1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4} $)($ 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} $)($ 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} $)($ 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} $)($ 1 + 11 T - T^{2} + 176 T^{3} + 5662 T^{4} + 9328 T^{5} - 2809 T^{6} + 1637647 T^{7} + 7890481 T^{8} $)($ 1 + 12 T + 72 T^{2} + 660 T^{3} + 6046 T^{4} + 34980 T^{5} + 202248 T^{6} + 1786524 T^{7} + 7890481 T^{8} $)($ 1 - 12 T + 72 T^{2} - 660 T^{3} + 6046 T^{4} - 34980 T^{5} + 202248 T^{6} - 1786524 T^{7} + 7890481 T^{8} $)($ 1 + 36 T + 648 T^{2} + 8284 T^{3} + 81222 T^{4} + 595308 T^{5} + 3111560 T^{6} + 8277492 T^{7} - 42534915 T^{8} - 754029808 T^{9} - 6543023328 T^{10} - 50633847440 T^{11} - 420252241016 T^{12} - 3590853938064 T^{13} - 27232240956512 T^{14} - 163759336672624 T^{15} - 644108178966222 T^{16} - 225724213191624 T^{17} + 21359012626695920 T^{18} + 241077022660459848 T^{19} + 1941342482920284900 T^{20} + 12777082201004371944 T^{21} + 59997466468388839280 T^{22} - 33605143687329406248 T^{23} - $$50\!\cdots\!82$$ T^{24} - $$68\!\cdots\!32$$ T^{25} - $$60\!\cdots\!48$$ T^{26} - $$42\!\cdots\!68$$ T^{27} - $$26\!\cdots\!76$$ T^{28} - $$16\!\cdots\!20$$ T^{29} - $$11\!\cdots\!72$$ T^{30} - $$69\!\cdots\!76$$ T^{31} - $$20\!\cdots\!15$$ T^{32} + $$21\!\cdots\!16$$ T^{33} + $$42\!\cdots\!40$$ T^{34} + $$43\!\cdots\!56$$ T^{35} + $$31\!\cdots\!62$$ T^{36} + $$17\!\cdots\!92$$ T^{37} + $$70\!\cdots\!72$$ T^{38} + $$20\!\cdots\!12$$ T^{39} + $$30\!\cdots\!01$$ T^{40} $)($ 1 + 53 T^{2} + 2809 T^{4} $)($ 1 + 9 T + 80 T^{2} + 477 T^{3} + 2809 T^{4} $)($ 1 + 53 T^{2} + 2809 T^{4} $)($ 1 - 9 T + 80 T^{2} - 477 T^{3} + 2809 T^{4} $)($ 1 + 25 T^{2} - 2184 T^{4} + 70225 T^{6} + 7890481 T^{8} $)($ 1 + 265 T^{2} + 32968 T^{4} + 3168029 T^{6} + 288393293 T^{8} + 22350648928 T^{10} + 1460664537746 T^{12} + 90342579430370 T^{14} + 5160533915433520 T^{16} + 253772305619909330 T^{18} + 11525345782458595826 T^{20} + $$49\!\cdots\!12$$ T^{22} + $$17\!\cdots\!73$$ T^{24} + $$55\!\cdots\!21$$ T^{26} + $$16\!\cdots\!88$$ T^{28} + $$36\!\cdots\!85$$ T^{30} + $$38\!\cdots\!21$$ T^{32} $)($ 1 + 53 T^{2} + 2809 T^{4} $)($ 1 + 53 T^{2} + 2809 T^{4} $)($ 1 + 53 T^{2} + 2809 T^{4} $)($ 1 + 53 T^{2} + 2809 T^{4} $)($ 1 - 20 T^{2} - 4805 T^{4} + 8260 T^{6} + 18077944 T^{8} + 23202340 T^{10} - 37913761205 T^{12} - 443287222580 T^{14} + 62259690411361 T^{16} $)($ ( 1 + 95 T^{2} + 6216 T^{4} + 266855 T^{6} + 7890481 T^{8} )^{2} $)($ 1 - 20 T^{2} - 4805 T^{4} + 8260 T^{6} + 18077944 T^{8} + 23202340 T^{10} - 37913761205 T^{12} - 443287222580 T^{14} + 62259690411361 T^{16} $)($ 1 - 53 T^{2} + 2809 T^{4} $)($ 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} $)($ 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} $)($ 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} $)($ 1 - 53 T^{2} + 2809 T^{4} $)($ 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} $)($ 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} $)($ 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} $)($ 1 + 16 T + 128 T^{2} - 384 T^{3} - 14095 T^{4} - 132640 T^{5} - 244352 T^{6} + 5040144 T^{7} + 65449648 T^{8} + 267127632 T^{9} - 686384768 T^{10} - 19747045280 T^{11} - 111216329695 T^{12} - 160587069312 T^{13} + 2837038224512 T^{14} + 18795378237392 T^{15} + 62259690411361 T^{16} $)
$59$	($ 1 + 12 T + 59 T^{2} $)($ 1 + 59 T^{2} $)($ 1 + 4 T + 59 T^{2} $)($ 1 + 4 T + 59 T^{2} $)($ 1 + 12 T + 59 T^{2} $)($ 1 - 8 T + 59 T^{2} $)($ ( 1 - 12 T + 59 T^{2} )^{2} $)($ ( 1 + 12 T + 59 T^{2} )^{2} $)($ ( 1 - 12 T + 59 T^{2} )^{2} $)($ ( 1 + 12 T + 59 T^{2} )^{2} $)($ ( 1 - 10 T^{2} + 3481 T^{4} )^{2} $)($ ( 1 + 76 T^{2} + 8298 T^{4} + 264556 T^{6} + 12117361 T^{8} )^{2} $)($ ( 1 + 59 T^{2} )^{2} $)($ ( 1 - 32 T^{2} + 3481 T^{4} )^{2} $)($ ( 1 + 110 T^{2} + 8610 T^{4} + 382910 T^{6} + 12117361 T^{8} )^{2} $)($ 1 - 59 T^{2} + 3481 T^{4} $)($ 1 + 11 T + 62 T^{2} + 649 T^{3} + 3481 T^{4} $)($ 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} $)($ 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} $)($ 1 + 5 T - 34 T^{2} + 295 T^{3} + 3481 T^{4} $)($ 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} $)($ 1 - 5 T - 85 T^{2} + 40 T^{3} + 7144 T^{4} + 2360 T^{5} - 295885 T^{6} - 1026895 T^{7} + 12117361 T^{8} $)($ ( 1 + 6 T + 59 T^{2} )^{2}( 1 - 82 T^{2} + 3481 T^{4} ) $)($ ( 1 - 6 T + 59 T^{2} )^{2}( 1 - 82 T^{2} + 3481 T^{4} ) $)($ 1 + 64 T^{3} + 17498 T^{4} + 16320 T^{5} + 2048 T^{6} + 964352 T^{7} + 139305821 T^{8} + 265011968 T^{9} + 159053824 T^{10} + 7117236224 T^{11} + 675196772280 T^{12} + 1865453307392 T^{13} + 2177063692288 T^{14} + 46993822767872 T^{15} + 2378495821379602 T^{16} + 7365185499789056 T^{17} + 13796808199690240 T^{18} + 308926250030895744 T^{19} + 7830691115365441372 T^{20} + 18226648751822848896 T^{21} + 48026689343121725440 T^{22} + $$15\!\cdots\!24$$ T^{23} + $$28\!\cdots\!22$$ T^{24} + $$33\!\cdots\!28$$ T^{25} + $$91\!\cdots\!08$$ T^{26} + $$46\!\cdots\!48$$ T^{27} + $$99\!\cdots\!80$$ T^{28} + $$61\!\cdots\!36$$ T^{29} + $$81\!\cdots\!24$$ T^{30} + $$79\!\cdots\!12$$ T^{31} + $$24\!\cdots\!01$$ T^{32} + $$10\!\cdots\!08$$ T^{33} + $$12\!\cdots\!28$$ T^{34} + $$59\!\cdots\!80$$ T^{35} + $$37\!\cdots\!18$$ T^{36} + $$81\!\cdots\!16$$ T^{37} + $$26\!\cdots\!01$$ T^{40} $)($ 1 - 59 T^{2} + 3481 T^{4} $)($ 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} $)($ 1 - 59 T^{2} + 3481 T^{4} $)($ 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} $)($ 1 - 115 T^{2} + 9744 T^{4} - 400315 T^{6} + 12117361 T^{8} $)($ 1 - 187 T^{2} + 11904 T^{4} - 267479 T^{6} + 18518765 T^{8} - 3131679552 T^{10} + 215589458578 T^{12} - 8824777678534 T^{14} + 393339242196864 T^{16} - 30719051098976854 T^{18} + 2612375297384172658 T^{20} - $$13\!\cdots\!32$$ T^{22} + $$27\!\cdots\!65$$ T^{24} - $$13\!\cdots\!79$$ T^{26} + $$21\!\cdots\!24$$ T^{28} - $$11\!\cdots\!07$$ T^{30} + $$21\!\cdots\!41$$ T^{32} $)($ 1 - 59 T^{2} + 3481 T^{4} $)($ 1 - 59 T^{2} + 3481 T^{4} $)($ 1 - 59 T^{2} + 3481 T^{4} $)($ 1 - 59 T^{2} + 3481 T^{4} $)($ 1 - 76 T^{2} - 2117 T^{4} - 70756 T^{6} + 30075832 T^{8} - 246301636 T^{10} - 25652453237 T^{12} - 3205720556716 T^{14} + 146830437604321 T^{16} $)($ ( 1 - 107 T^{2} + 7968 T^{4} - 372467 T^{6} + 12117361 T^{8} )^{2} $)($ 1 - 76 T^{2} - 2117 T^{4} - 70756 T^{6} + 30075832 T^{8} - 246301636 T^{10} - 25652453237 T^{12} - 3205720556716 T^{14} + 146830437604321 T^{16} $)($ 1 - 59 T^{2} + 3481 T^{4} $)($ 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} $)($ 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} $)($ 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} $)($ 1 - 59 T^{2} + 3481 T^{4} $)($ 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} $)($ 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} $)($ 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} $)($ 1 + 20 T + 200 T^{2} + 640 T^{3} - 7087 T^{4} - 104360 T^{5} - 465000 T^{6} + 1041740 T^{7} + 28323408 T^{8} + 61462660 T^{9} - 1618665000 T^{10} - 21433352440 T^{11} - 85875737407 T^{12} + 457551551360 T^{13} + 8436106728200 T^{14} + 49773029696380 T^{15} + 146830437604321 T^{16} $)
$61$	($ 1 + 2 T + 61 T^{2} $)($ 1 + 10 T + 61 T^{2} $)($ 1 + 10 T + 61 T^{2} $)($ 1 - 6 T + 61 T^{2} $)($ 1 + 2 T + 61 T^{2} $)($ 1 - 6 T + 61 T^{2} $)($ 1 + 70 T^{2} + 3721 T^{4} $)($ ( 1 - 61 T^{2} )^{2} $)($ ( 1 - 61 T^{2} )^{2} $)($ 1 + 70 T^{2} + 3721 T^{4} $)($ ( 1 + 6 T + 61 T^{2} )^{4} $)($ ( 1 + 6 T + 128 T^{2} + 366 T^{3} + 3721 T^{4} )^{4} $)($ ( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) $)($ ( 1 + 28 T^{2} + 3721 T^{4} )^{2} $)($ ( 1 - 118 T^{2} + 9546 T^{4} - 439078 T^{6} + 13845841 T^{8} )^{2} $)($ 1 + 10 T + 39 T^{2} + 610 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 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} $)($ 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} $)($ 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} $)($ ( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} )^{2} $)($ 1 - 4 T + 8 T^{2} - 236 T^{3} + 6958 T^{4} - 14396 T^{5} + 29768 T^{6} - 907924 T^{7} + 13845841 T^{8} $)($ 1 - 4 T + 8 T^{2} - 236 T^{3} + 6958 T^{4} - 14396 T^{5} + 29768 T^{6} - 907924 T^{7} + 13845841 T^{8} $)($ 1 - 8 T + 32 T^{2} - 24 T^{3} - 22 T^{4} + 28776 T^{5} - 229216 T^{6} + 1701688 T^{7} - 7327587 T^{8} + 193356480 T^{9} - 944223232 T^{10} + 7441249856 T^{11} + 25030907320 T^{12} - 252322194176 T^{13} + 6350522522880 T^{14} - 30700611174144 T^{15} + 72405460654514 T^{16} + 1378721064691056 T^{17} + 10443615781913408 T^{18} + 4859962566983248 T^{19} + 374225284282585404 T^{20} + 296457716585978128 T^{21} + 38860694324499791168 T^{22} + $$31\!\cdots\!36$$ T^{23} + $$10\!\cdots\!74$$ T^{24} - $$25\!\cdots\!44$$ T^{25} + $$32\!\cdots\!80$$ T^{26} - $$79\!\cdots\!96$$ T^{27} + $$47\!\cdots\!20$$ T^{28} + $$87\!\cdots\!96$$ T^{29} - $$67\!\cdots\!32$$ T^{30} + $$84\!\cdots\!80$$ T^{31} - $$19\!\cdots\!27$$ T^{32} + $$27\!\cdots\!28$$ T^{33} - $$22\!\cdots\!56$$ T^{34} + $$17\!\cdots\!76$$ T^{35} - $$80\!\cdots\!42$$ T^{36} - $$53\!\cdots\!04$$ T^{37} + $$43\!\cdots\!92$$ T^{38} - $$66\!\cdots\!28$$ T^{39} + $$50\!\cdots\!01$$ T^{40} $)($ ( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) $)($ 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} $)($ ( 1 - 13 T + 61 T^{2} )( 1 + T + 61 T^{2} ) $)($ 1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4} $)($ ( 1 + 61 T^{2} + 3721 T^{4} )^{2} $)($ ( 1 - 18 T + 331 T^{2} - 4014 T^{3} + 46777 T^{4} - 446148 T^{5} + 4203790 T^{6} - 34615152 T^{7} + 285617722 T^{8} - 2111524272 T^{9} + 15642302590 T^{10} - 101267119188 T^{11} + 647666904457 T^{12} - 3390209552214 T^{13} + 17053243913491 T^{14} - 56569371048378 T^{15} + 191707312997281 T^{16} )^{2} $)($ ( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) $)($ ( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) $)($ ( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) $)($ ( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) $)($ ( 1 - 15 T + 61 T^{2} )^{4}( 1 + 15 T + 164 T^{2} + 915 T^{3} + 3721 T^{4} )^{2} $)($ ( 1 + 3 T - 52 T^{2} + 183 T^{3} + 3721 T^{4} )^{4} $)($ ( 1 - 15 T + 61 T^{2} )^{4}( 1 + 15 T + 164 T^{2} + 915 T^{3} + 3721 T^{4} )^{2} $)($ 1 + 61 T^{2} + 3721 T^{4} $)($ ( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) $)($ 1 + 61 T^{2} + 3721 T^{4} $)($ 1 + 24 T + 253 T^{2} + 1464 T^{3} + 3721 T^{4} $)($ 1 + 61 T^{2} + 3721 T^{4} $)($ ( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) $)($ 1 + 61 T^{2} + 3721 T^{4} $)($ 1 + 24 T + 253 T^{2} + 1464 T^{3} + 3721 T^{4} $)($ 1 + 4 T + 8 T^{2} - 344 T^{3} - 4910 T^{4} - 20804 T^{5} + 15232 T^{6} + 1637244 T^{7} + 8848707 T^{8} + 99871884 T^{9} + 56678272 T^{10} - 4722112724 T^{11} - 67983079310 T^{12} - 290541127544 T^{13} + 412162994888 T^{14} + 12570971344084 T^{15} + 191707312997281 T^{16} $)
$67$	($ 1 + 4 T + 67 T^{2} $)($ 1 + 8 T + 67 T^{2} $)($ 1 + 12 T + 67 T^{2} $)($ 1 + 4 T + 67 T^{2} $)($ 1 + 12 T + 67 T^{2} $)($ 1 - 8 T + 67 T^{2} $)($ ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) $)($ ( 1 - 67 T^{2} )^{2} $)($ ( 1 - 67 T^{2} )^{2} $)($ ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) $)($ ( 1 + 10 T^{2} + 4489 T^{4} )^{2} $)($ ( 1 - 16 T + 67 T^{2} )^{4}( 1 + 16 T + 67 T^{2} )^{4} $)($ ( 1 - 16 T + 67 T^{2} )^{2} $)($ ( 1 + 8 T + 67 T^{2} )^{4} $)($ ( 1 + 4 T + 70 T^{2} + 268 T^{3} + 4489 T^{4} )^{4} $)($ 1 + 15 T + 158 T^{2} + 1005 T^{3} + 4489 T^{4} $)($ 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} $)($ 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} $)($ 1 + 10 T + 33 T^{2} + 670 T^{3} + 4489 T^{4} $)($ 1 - 6 T - 31 T^{2} - 402 T^{3} + 4489 T^{4} $)($ ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) $)($ 1 - 7 T - 83 T^{2} + 14 T^{3} + 9652 T^{4} + 938 T^{5} - 372587 T^{6} - 2105341 T^{7} + 20151121 T^{8} $)($ ( 1 + 14 T + 98 T^{2} + 938 T^{3} + 4489 T^{4} )^{2} $)($ ( 1 + 14 T + 98 T^{2} + 938 T^{3} + 4489 T^{4} )^{2} $)($ 1 + 12 T + 72 T^{2} + 444 T^{3} + 4102 T^{4} + 54484 T^{5} + 457032 T^{6} + 4150628 T^{7} - 7647939 T^{8} - 292442192 T^{9} - 1506268000 T^{10} - 8570912144 T^{11} - 68934819128 T^{12} - 1058425781488 T^{13} - 9188787191264 T^{14} - 93290480178608 T^{15} + 105617726671378 T^{16} + 7399255991086056 T^{17} + 46962879527876336 T^{18} + 365051212101087880 T^{19} + 2665710348810285028 T^{20} + 24458431210772887960 T^{21} + $$21\!\cdots\!04$$ T^{22} + $$22\!\cdots\!28$$ T^{23} + $$21\!\cdots\!38$$ T^{24} - $$12\!\cdots\!56$$ T^{25} - $$83\!\cdots\!16$$ T^{26} - $$64\!\cdots\!24$$ T^{27} - $$27\!\cdots\!48$$ T^{28} - $$23\!\cdots\!68$$ T^{29} - $$27\!\cdots\!00$$ T^{30} - $$35\!\cdots\!36$$ T^{31} - $$62\!\cdots\!79$$ T^{32} + $$22\!\cdots\!36$$ T^{33} + $$16\!\cdots\!28$$ T^{34} + $$13\!\cdots\!12$$ T^{35} + $$67\!\cdots\!62$$ T^{36} + $$49\!\cdots\!88$$ T^{37} + $$53\!\cdots\!48$$ T^{38} + $$59\!\cdots\!36$$ T^{39} + $$33\!\cdots\!01$$ T^{40} $)($ ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) $)($ ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) $)($ ( 1 - 16 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) $)($ ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) $)($ ( 1 - 2 T - 63 T^{2} - 134 T^{3} + 4489 T^{4} )^{2} $)($ ( 1 + 7 T - 200 T^{2} - 865 T^{3} + 28259 T^{4} + 67468 T^{5} - 2804524 T^{6} - 1482940 T^{7} + 222922288 T^{8} - 99356980 T^{9} - 12589508236 T^{10} + 20291878084 T^{11} + 569450528339 T^{12} - 1167858217555 T^{13} - 18091676433800 T^{14} + 42424981237261 T^{15} + 406067677556641 T^{16} )^{2} $)($ ( 1 - 16 T + 67 T^{2} )( 1 - 11 T + 67 T^{2} ) $)($ ( 1 + 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) $)($ ( 1 - 16 T + 67 T^{2} )( 1 - 5 T + 67 T^{2} ) $)($ ( 1 + 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) $)($ ( 1 - 33 T + 545 T^{2} - 6006 T^{3} + 52956 T^{4} - 402402 T^{5} + 2446505 T^{6} - 9925179 T^{7} + 20151121 T^{8} )^{2} $)($ ( 1 + 53 T^{2} - 1680 T^{4} + 237917 T^{6} + 20151121 T^{8} )^{2} $)($ ( 1 + 33 T + 545 T^{2} + 6006 T^{3} + 52956 T^{4} + 402402 T^{5} + 2446505 T^{6} + 9925179 T^{7} + 20151121 T^{8} )^{2} $)($ 1 - 3 T + 70 T^{2} - 201 T^{3} + 4489 T^{4} $)($ 1 + 67 T^{2} + 4489 T^{4} $)($ 1 + 24 T + 259 T^{2} + 1608 T^{3} + 4489 T^{4} $)($ 1 - 15 T + 142 T^{2} - 1005 T^{3} + 4489 T^{4} $)($ 1 + 3 T + 70 T^{2} + 201 T^{3} + 4489 T^{4} $)($ 1 + 67 T^{2} + 4489 T^{4} $)($ 1 - 24 T + 259 T^{2} - 1608 T^{3} + 4489 T^{4} $)($ 1 + 15 T + 142 T^{2} + 1005 T^{3} + 4489 T^{4} $)($ 1 + 32 T + 512 T^{2} + 4160 T^{3} + 8974 T^{4} - 146656 T^{5} - 634880 T^{6} + 15805728 T^{7} + 232306035 T^{8} + 1058983776 T^{9} - 2849976320 T^{10} - 44108698528 T^{11} + 180836159854 T^{12} + 5616520445120 T^{13} + 46314691670528 T^{14} + 193942771370336 T^{15} + 406067677556641 T^{16} $)
$71$	($ 1 + 71 T^{2} $)($ 1 + 6 T + 71 T^{2} $)($ 1 + 4 T + 71 T^{2} $)($ 1 + 8 T + 71 T^{2} $)($ 1 - 12 T + 71 T^{2} $)($ 1 + 14 T + 71 T^{2} $)($ 1 - 34 T^{2} + 5041 T^{4} $)($ 1 + 8 T^{2} + 5041 T^{4} $)($ 1 + 8 T^{2} + 5041 T^{4} $)($ 1 - 34 T^{2} + 5041 T^{4} $)($ ( 1 + 92 T^{2} + 5041 T^{4} )^{2} $)($ ( 1 + 196 T^{2} + 17958 T^{4} + 988036 T^{6} + 25411681 T^{8} )^{2} $)($ ( 1 - 71 T^{2} )^{2} $)($ ( 1 - 71 T^{2} )^{4} $)($ ( 1 - 88 T^{2} + 6510 T^{4} - 443608 T^{6} + 25411681 T^{8} )^{2} $)($ ( 1 - 6 T + 71 T^{2} )^{2} $)($ ( 1 + 2 T + 71 T^{2} )^{2} $)($ ( 1 - 10 T + 71 T^{2} )^{2} $)($ ( 1 - 6 T + 71 T^{2} )^{2} $)($ ( 1 + 10 T + 71 T^{2} )^{2} $)($ ( 1 - 6 T + 71 T^{2} )^{2} $)($ ( 1 + 2 T + 71 T^{2} )^{4} $)($ ( 1 - 106 T^{2} + 5041 T^{4} )^{2} $)($ ( 1 - 106 T^{2} + 5041 T^{4} )^{2} $)($ 1 - 588 T^{2} + 187310 T^{4} - 41853292 T^{6} + 7286062485 T^{8} - 1044227363664 T^{10} + 127464606182264 T^{12} - 13544991508155920 T^{14} + 1271322412181875034 T^{16} - $$10\!\cdots\!40$$ T^{18} + $$79\!\cdots\!36$$ T^{20} - $$53\!\cdots\!40$$ T^{22} + $$32\!\cdots\!54$$ T^{24} - $$17\!\cdots\!20$$ T^{26} + $$82\!\cdots\!04$$ T^{28} - $$33\!\cdots\!64$$ T^{30} + $$11\!\cdots\!85$$ T^{32} - $$34\!\cdots\!52$$ T^{34} + $$78\!\cdots\!10$$ T^{36} - $$12\!\cdots\!68$$ T^{38} + $$10\!\cdots\!01$$ T^{40} $)($ ( 1 - 71 T^{2} )^{2} $)($ 1 - 34 T^{2} + 5041 T^{4} $)($ ( 1 - 71 T^{2} )^{2} $)($ 1 - 34 T^{2} + 5041 T^{4} $)($ ( 1 + 2 T^{2} + 5041 T^{4} )^{2} $)($ ( 1 - 224 T^{2} + 21244 T^{4} - 988448 T^{6} + 37865158 T^{8} - 4982766368 T^{10} + 539845751164 T^{12} - 28694463598304 T^{14} + 645753531245761 T^{16} )^{2} $)($ ( 1 + 71 T^{2} )^{2} $)($ ( 1 + 71 T^{2} )^{2} $)($ ( 1 + 71 T^{2} )^{2} $)($ ( 1 + 71 T^{2} )^{2} $)($ ( 1 + 160 T^{2} + 14430 T^{4} + 806560 T^{6} + 25411681 T^{8} )^{2} $)($ ( 1 - 34 T^{2} + 5041 T^{4} )^{4} $)($ ( 1 + 160 T^{2} + 14430 T^{4} + 806560 T^{6} + 25411681 T^{8} )^{2} $)($ 1 - 130 T^{2} + 5041 T^{4} $)($ 1 - 94 T^{2} + 5041 T^{4} $)($ 1 + 50 T^{2} + 5041 T^{4} $)($ 1 - 130 T^{2} + 5041 T^{4} $)($ 1 - 130 T^{2} + 5041 T^{4} $)($ 1 - 94 T^{2} + 5041 T^{4} $)($ 1 + 50 T^{2} + 5041 T^{4} $)($ 1 - 130 T^{2} + 5041 T^{4} $)($ ( 1 - 16 T^{2} - 3966 T^{4} - 80656 T^{6} + 25411681 T^{8} )^{2} $)
$73$	($ 1 + 6 T + 73 T^{2} $)($ 1 + 10 T + 73 T^{2} $)($ 1 - 2 T + 73 T^{2} $)($ 1 - 10 T + 73 T^{2} $)($ 1 + 14 T + 73 T^{2} $)($ 1 + 2 T + 73 T^{2} $)($ ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) $)($ 1 + 70 T^{2} + 5329 T^{4} $)($ 1 + 70 T^{2} + 5329 T^{4} $)($ ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) $)($ ( 1 + 2 T + 73 T^{2} )^{4} $)($ ( 1 + 16 T + 198 T^{2} + 1168 T^{3} + 5329 T^{4} )^{4} $)($ ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) $)($ ( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2} $)($ ( 1 - 132 T^{2} + 10662 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{2} $)($ 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} $)($ ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) $)($ 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} $)($ 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} $)($ 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} $)($ 1 - 3 T - 64 T^{2} - 219 T^{3} + 5329 T^{4} $)($ 1 - T - 131 T^{2} + 14 T^{3} + 12022 T^{4} + 1022 T^{5} - 698099 T^{6} - 389017 T^{7} + 28398241 T^{8} $)($ 1 - 148 T^{2} + 14086 T^{4} - 788692 T^{6} + 28398241 T^{8} $)($ 1 - 148 T^{2} + 14086 T^{4} - 788692 T^{6} + 28398241 T^{8} $)($ 1 - 588 T^{2} + 185438 T^{4} - 41247404 T^{6} + 7184467213 T^{8} - 1034846312688 T^{10} + 127434000595944 T^{12} - 13711360420506544 T^{14} + 1308031080629694098 T^{16} - $$11\!\cdots\!16$$ T^{18} + $$85\!\cdots\!04$$ T^{20} - $$59\!\cdots\!64$$ T^{22} + $$37\!\cdots\!18$$ T^{24} - $$20\!\cdots\!16$$ T^{26} + $$10\!\cdots\!64$$ T^{28} - $$44\!\cdots\!12$$ T^{30} + $$16\!\cdots\!73$$ T^{32} - $$50\!\cdots\!36$$ T^{34} + $$12\!\cdots\!18$$ T^{36} - $$20\!\cdots\!72$$ T^{38} + $$18\!\cdots\!01$$ T^{40} $)($ ( 1 - 7 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) $)($ 1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4} $)($ ( 1 + 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) $)($ 1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4} $)($ ( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} )^{2} $)($ ( 1 - 15 T + 312 T^{2} - 3555 T^{3} + 48909 T^{4} - 547104 T^{5} + 5639826 T^{6} - 55083990 T^{7} + 457851344 T^{8} - 4021131270 T^{9} + 30054632754 T^{10} - 212832756768 T^{11} + 1388929569069 T^{12} - 7369769513115 T^{13} + 47216278602168 T^{14} - 165710977786455 T^{15} + 806460091894081 T^{16} )^{2} $)($ ( 1 - 10 T + 73 T^{2} )( 1 - 7 T + 73 T^{2} ) $)($ ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) $)($ ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) $)($ ( 1 - 10 T + 73 T^{2} )( 1 - 7 T + 73 T^{2} ) $)($ ( 1 - 7 T - 95 T^{2} + 14 T^{3} + 11830 T^{4} + 1022 T^{5} - 506255 T^{6} - 2723119 T^{7} + 28398241 T^{8} )^{2} $)($ ( 1 - 10 T + 73 T^{2} )^{4}( 1 + 17 T + 73 T^{2} )^{4} $)($ ( 1 - 7 T - 95 T^{2} + 14 T^{3} + 11830 T^{4} + 1022 T^{5} - 506255 T^{6} - 2723119 T^{7} + 28398241 T^{8} )^{2} $)($ 1 - 15 T + 148 T^{2} - 1095 T^{3} + 5329 T^{4} $)($ 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} $)($ 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} $)($ 1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4} $)($ 1 - 15 T + 148 T^{2} - 1095 T^{3} + 5329 T^{4} $)($ 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} $)($ 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} $)($ 1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4} $)($ ( 1 + 130 T^{2} + 11571 T^{4} + 692770 T^{6} + 28398241 T^{8} )^{2} $)
$79$	($ 1 - 16 T + 79 T^{2} $)($ 1 - 4 T + 79 T^{2} $)($ 1 + 8 T + 79 T^{2} $)($ 1 + 79 T^{2} $)($ 1 - 8 T + 79 T^{2} $)($ 1 + 12 T + 79 T^{2} $)($ 1 - 50 T^{2} + 6241 T^{4} $)($ 1 - 62 T^{2} + 6241 T^{4} $)($ 1 - 62 T^{2} + 6241 T^{4} $)($ 1 - 50 T^{2} + 6241 T^{4} $)($ ( 1 - 14 T^{2} + 6241 T^{4} )^{2} $)($ ( 1 - 14 T^{2} + 6241 T^{4} )^{4} $)($ ( 1 - 4 T + 79 T^{2} )^{2} $)($ ( 1 - 10 T + 79 T^{2} )^{4} $)($ ( 1 + 14 T + 190 T^{2} + 1106 T^{3} + 6241 T^{4} )^{4} $)($ 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} $)($ 1 - 3 T - 70 T^{2} - 237 T^{3} + 6241 T^{4} $)($ 1 + 3 T - 70 T^{2} + 237 T^{3} + 6241 T^{4} $)($ 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} $)($ 1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4} $)($ 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} $)($ 1 - 8 T - 53 T^{2} + 328 T^{3} + 2392 T^{4} + 25912 T^{5} - 330773 T^{6} - 3944312 T^{7} + 38950081 T^{8} $)($ ( 1 - 154 T^{2} + 6241 T^{4} )^{2} $)($ ( 1 - 154 T^{2} + 6241 T^{4} )^{2} $)($ ( 1 - 12 T + 410 T^{2} - 3636 T^{3} + 71661 T^{4} - 457104 T^{5} + 6962296 T^{6} - 26461552 T^{7} + 422059634 T^{8} - 425195336 T^{9} + 24696310492 T^{10} - 33590431544 T^{11} + 2634074175794 T^{12} - 13046577136528 T^{13} + 271181993145976 T^{14} - 1406534788208496 T^{15} + 17419890150090381 T^{16} - 69825413073674124 T^{17} + 622014612061690010 T^{18} - 1438219151791419828 T^{19} + 9468276082626847201 T^{20} )^{2} $)($ ( 1 - 13 T + 79 T^{2} )( 1 - 4 T + 79 T^{2} ) $)($ 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} $)($ ( 1 - 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) $)($ 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} $)($ ( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} )^{2} $)($ ( 1 + 14 T - 152 T^{2} - 1448 T^{3} + 34763 T^{4} + 170492 T^{5} - 3816772 T^{6} - 3401582 T^{7} + 382142944 T^{8} - 268724978 T^{9} - 23820474052 T^{10} + 84059205188 T^{11} + 1354021665803 T^{12} - 4455577665752 T^{13} - 36949293239192 T^{14} + 268854725806226 T^{15} + 1517108809906561 T^{16} )^{2} $)($ ( 1 - 13 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) $)($ ( 1 - 17 T + 79 T^{2} )( 1 - 4 T + 79 T^{2} ) $)($ ( 1 + 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) $)($ ( 1 - 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) $)($ ( 1 - 4 T + 79 T^{2} )^{4}( 1 + 13 T + 79 T^{2} )^{4} $)($ ( 1 + 77 T^{2} - 312 T^{4} + 480557 T^{6} + 38950081 T^{8} )^{2} $)($ ( 1 - 13 T + 79 T^{2} )^{4}( 1 + 4 T + 79 T^{2} )^{4} $)($ 1 + 27 T + 322 T^{2} + 2133 T^{3} + 6241 T^{4} $)($ 1 - 3 T + 82 T^{2} - 237 T^{3} + 6241 T^{4} $)($ ( 1 - 13 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) $)($ ( 1 - 17 T + 79 T^{2} )( 1 - 4 T + 79 T^{2} ) $)($ 1 - 27 T + 322 T^{2} - 2133 T^{3} + 6241 T^{4} $)($ 1 + 3 T + 82 T^{2} + 237 T^{3} + 6241 T^{4} $)($ ( 1 - 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) $)($ ( 1 + 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) $)($ ( 1 + 14 T + 7 T^{2} + 434 T^{3} + 13996 T^{4} + 34286 T^{5} + 43687 T^{6} + 6902546 T^{7} + 38950081 T^{8} )^{2} $)
$83$	($ 1 - 12 T + 83 T^{2} $)($ 1 - 12 T + 83 T^{2} $)($ 1 + 4 T + 83 T^{2} $)($ 1 - 4 T + 83 T^{2} $)($ 1 + 12 T + 83 T^{2} $)($ 1 - 4 T + 83 T^{2} $)($ ( 1 - 12 T + 83 T^{2} )^{2} $)($ ( 1 + 83 T^{2} )^{2} $)($ ( 1 + 83 T^{2} )^{2} $)($ ( 1 + 12 T + 83 T^{2} )^{2} $)($ ( 1 - 34 T^{2} + 6889 T^{4} )^{2} $)($ ( 1 + 172 T^{2} + 21066 T^{4} + 1184908 T^{6} + 47458321 T^{8} )^{2} $)($ ( 1 + 83 T^{2} )^{2} $)($ ( 1 + 160 T^{2} + 6889 T^{4} )^{2} $)($ ( 1 + 318 T^{2} + 39042 T^{4} + 2190702 T^{6} + 47458321 T^{8} )^{2} $)($ ( 1 + 6 T + 83 T^{2} )^{2} $)($ ( 1 - 7 T + 83 T^{2} )^{2} $)($ ( 1 + 2 T + 83 T^{2} )^{2} $)($ ( 1 - 9 T + 83 T^{2} )^{2} $)($ ( 1 + 7 T + 83 T^{2} )^{2} $)($ ( 1 + 6 T + 83 T^{2} )^{2} $)($ ( 1 + 7 T + 164 T^{2} + 581 T^{3} + 6889 T^{4} )^{2} $)($ 1 + 4 T + 8 T^{2} - 444 T^{3} - 12994 T^{4} - 36852 T^{5} + 55112 T^{6} + 2287148 T^{7} + 47458321 T^{8} $)($ 1 - 4 T + 8 T^{2} + 444 T^{3} - 12994 T^{4} + 36852 T^{5} + 55112 T^{6} - 2287148 T^{7} + 47458321 T^{8} $)($ 1 - 40 T + 800 T^{2} - 11352 T^{3} + 130714 T^{4} - 1275720 T^{5} + 10891552 T^{6} - 81158136 T^{7} + 533032573 T^{8} - 3591219744 T^{9} + 28582322304 T^{10} - 250649734880 T^{11} + 2277897381816 T^{12} - 18560960241440 T^{13} + 98547001707136 T^{14} + 326327140527904 T^{15} - 18693002705618542 T^{16} + 320247589539147216 T^{17} - 4017303386407089728 T^{18} + 43457553116792444976 T^{19} - $$41\!\cdots\!72$$ T^{20} + $$36\!\cdots\!08$$ T^{21} - $$27\!\cdots\!92$$ T^{22} + $$18\!\cdots\!92$$ T^{23} - $$88\!\cdots\!82$$ T^{24} + $$12\!\cdots\!72$$ T^{25} + $$32\!\cdots\!84$$ T^{26} - $$50\!\cdots\!80$$ T^{27} + $$51\!\cdots\!56$$ T^{28} - $$46\!\cdots\!40$$ T^{29} + $$44\!\cdots\!96$$ T^{30} - $$46\!\cdots\!48$$ T^{31} + $$56\!\cdots\!53$$ T^{32} - $$72\!\cdots\!68$$ T^{33} + $$80\!\cdots\!08$$ T^{34} - $$77\!\cdots\!40$$ T^{35} + $$66\!\cdots\!34$$ T^{36} - $$47\!\cdots\!96$$ T^{37} + $$27\!\cdots\!00$$ T^{38} - $$11\!\cdots\!80$$ T^{39} + $$24\!\cdots\!01$$ T^{40} $)($ ( 1 + 83 T^{2} )^{2} $)($ ( 1 - 12 T + 83 T^{2} )^{2} $)($ ( 1 + 83 T^{2} )^{2} $)($ ( 1 + 12 T + 83 T^{2} )^{2} $)($ ( 1 + 91 T^{2} + 6889 T^{4} )^{2} $)($ ( 1 + 141 T^{2} + 22278 T^{4} + 1798779 T^{6} + 189218258 T^{8} + 12391788531 T^{10} + 1057276475238 T^{12} + 46098592645029 T^{14} + 2252292232139041 T^{16} )^{2} $)($ ( 1 + 83 T^{2} )^{2} $)($ ( 1 + 83 T^{2} )^{2} $)($ ( 1 + 83 T^{2} )^{2} $)($ ( 1 + 83 T^{2} )^{2} $)($ ( 1 + 301 T^{2} + 36300 T^{4} + 2073589 T^{6} + 47458321 T^{8} )^{2} $)($ ( 1 - 10 T^{2} + 6889 T^{4} )^{4} $)($ ( 1 + 301 T^{2} + 36300 T^{4} + 2073589 T^{6} + 47458321 T^{8} )^{2} $)($ ( 1 + 6 T + 83 T^{2} )^{2} $)($ ( 1 - 15 T + 83 T^{2} )^{2} $)($ ( 1 - 3 T + 83 T^{2} )^{2} $)($ ( 1 + 18 T + 83 T^{2} )^{2} $)($ ( 1 - 6 T + 83 T^{2} )^{2} $)($ ( 1 + 15 T + 83 T^{2} )^{2} $)($ ( 1 + 3 T + 83 T^{2} )^{2} $)($ ( 1 - 18 T + 83 T^{2} )^{2} $)($ ( 1 + 4 T + 8 T^{2} - 144 T^{3} - 11569 T^{4} - 11952 T^{5} + 55112 T^{6} + 2287148 T^{7} + 47458321 T^{8} )^{2} $)
$89$	($ 1 + 14 T + 89 T^{2} $)($ 1 - 12 T + 89 T^{2} $)($ 1 - 6 T + 89 T^{2} $)($ 1 + 6 T + 89 T^{2} $)($ 1 + 2 T + 89 T^{2} $)($ 1 + 89 T^{2} $)($ 1 - 130 T^{2} + 7921 T^{4} $)($ 1 - 28 T^{2} + 7921 T^{4} $)($ 1 - 28 T^{2} + 7921 T^{4} $)($ 1 - 130 T^{2} + 7921 T^{4} $)($ ( 1 - 80 T^{2} + 7921 T^{4} )^{2} $)($ ( 1 + 20 T^{2} + 15750 T^{4} + 158420 T^{6} + 62742241 T^{8} )^{2} $)($ ( 1 + 89 T^{2} )^{2} $)($ ( 1 + 89 T^{2} )^{4} $)($ ( 1 - 24 T^{2} + 12654 T^{4} - 190104 T^{6} + 62742241 T^{8} )^{2} $)($ 1 - 8 T - 25 T^{2} - 712 T^{3} + 7921 T^{4} $)($ 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} $)($ 1 - 89 T^{2} + 7921 T^{4} $)($ 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} $)($ 1 - 18 T + 235 T^{2} - 1602 T^{3} + 7921 T^{4} $)($ 1 + 16 T + 167 T^{2} + 1424 T^{3} + 7921 T^{4} $)($ 1 - 6 T - 94 T^{2} + 288 T^{3} + 5775 T^{4} + 25632 T^{5} - 744574 T^{6} - 4229814 T^{7} + 62742241 T^{8} $)($ ( 1 + 20 T + 246 T^{2} + 1780 T^{3} + 7921 T^{4} )^{2} $)($ ( 1 - 20 T + 246 T^{2} - 1780 T^{3} + 7921 T^{4} )^{2} $)($ 1 - 700 T^{2} + 256046 T^{4} - 64249660 T^{6} + 12305003221 T^{8} - 1904804480368 T^{10} + 247648665045240 T^{12} - 27922948227608560 T^{14} + 2822449388027483546 T^{16} - $$26\!\cdots\!68$$ T^{18} + $$23\!\cdots\!96$$ T^{20} - $$21\!\cdots\!28$$ T^{22} + $$17\!\cdots\!86$$ T^{24} - $$13\!\cdots\!60$$ T^{26} + $$97\!\cdots\!40$$ T^{28} - $$59\!\cdots\!68$$ T^{30} + $$30\!\cdots\!41$$ T^{32} - $$12\!\cdots\!60$$ T^{34} + $$39\!\cdots\!06$$ T^{36} - $$85\!\cdots\!00$$ T^{38} + $$97\!\cdots\!01$$ T^{40} $)($ 1 - 89 T^{2} + 7921 T^{4} $)($ 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} $)($ 1 - 89 T^{2} + 7921 T^{4} $)($ 1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4} $)($ 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} $)($ 1 - 378 T^{2} + 69795 T^{4} - 8481534 T^{6} + 768803689 T^{8} - 53847736620 T^{10} + 2382276614382 T^{12} + 24847322779392 T^{14} - 11761219221673230 T^{16} + 196815643735564032 T^{18} + $$14\!\cdots\!62$$ T^{20} - $$26\!\cdots\!20$$ T^{22} + $$30\!\cdots\!09$$ T^{24} - $$26\!\cdots\!34$$ T^{26} + $$17\!\cdots\!95$$ T^{28} - $$73\!\cdots\!98$$ T^{30} + $$15\!\cdots\!61$$ T^{32} $)($ 1 + 89 T^{2} + 7921 T^{4} $)($ 1 + 89 T^{2} + 7921 T^{4} $)($ 1 + 89 T^{2} + 7921 T^{4} $)($ 1 + 89 T^{2} + 7921 T^{4} $)($ 1 + 109 T^{2} - 1787 T^{4} - 236966 T^{6} + 29962582 T^{8} - 1877007686 T^{10} - 112120384667 T^{12} + 54170960714749 T^{14} + 3936588805702081 T^{16} $)($ ( 1 + 167 T^{2} + 19968 T^{4} + 1322807 T^{6} + 62742241 T^{8} )^{2} $)($ 1 + 109 T^{2} - 1787 T^{4} - 236966 T^{6} + 29962582 T^{8} - 1877007686 T^{10} - 112120384667 T^{12} + 54170960714749 T^{14} + 3936588805702081 T^{16} $)($ 1 - 12 T + 137 T^{2} - 1068 T^{3} + 7921 T^{4} $)($ 1 + 18 T + 197 T^{2} + 1602 T^{3} + 7921 T^{4} $)($ 1 + 6 T + 101 T^{2} + 534 T^{3} + 7921 T^{4} $)($ 1 - 12 T + 137 T^{2} - 1068 T^{3} + 7921 T^{4} $)($ 1 - 12 T + 137 T^{2} - 1068 T^{3} + 7921 T^{4} $)($ 1 + 18 T + 197 T^{2} + 1602 T^{3} + 7921 T^{4} $)($ 1 + 6 T + 101 T^{2} + 534 T^{3} + 7921 T^{4} $)($ 1 - 12 T + 137 T^{2} - 1068 T^{3} + 7921 T^{4} $)($ ( 1 - 64 T^{2} - 3825 T^{4} - 506944 T^{6} + 62742241 T^{8} )^{2} $)
$97$	($ 1 - 18 T + 97 T^{2} $)($ 1 + 10 T + 97 T^{2} $)($ 1 - 10 T + 97 T^{2} $)($ 1 + 14 T + 97 T^{2} $)($ 1 - 10 T + 97 T^{2} $)($ 1 + 2 T + 97 T^{2} $)($ ( 1 - 97 T^{2} )^{2} $)($ 1 - 170 T^{2} + 9409 T^{4} $)($ 1 - 170 T^{2} + 9409 T^{4} $)($ ( 1 - 97 T^{2} )^{2} $)($ ( 1 - 14 T + 97 T^{2} )^{4} $)($ ( 1 - 4 T + 150 T^{2} - 388 T^{3} + 9409 T^{4} )^{4} $)($ ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) $)($ ( 1 - 170 T^{2} + 9409 T^{4} )^{2} $)($ ( 1 - 204 T^{2} + 28950 T^{4} - 1919436 T^{6} + 88529281 T^{8} )^{2} $)($ ( 1 + 14 T + 97 T^{2} )^{2} $)($ ( 1 - 7 T + 97 T^{2} )^{2} $)($ ( 1 - 10 T + 97 T^{2} )^{2} $)($ ( 1 + T + 97 T^{2} )^{2} $)($ ( 1 + 17 T + 97 T^{2} )^{2} $)($ ( 1 + 6 T + 97 T^{2} )^{2} $)($ ( 1 - 25 T + 336 T^{2} - 2425 T^{3} + 9409 T^{4} )^{2} $)($ ( 1 + 2 T + 97 T^{2} )^{4} $)($ ( 1 + 2 T + 97 T^{2} )^{4} $)($ ( 1 + 36 T + 1006 T^{2} + 17860 T^{3} + 282157 T^{4} + 3451280 T^{5} + 41738920 T^{6} + 425104976 T^{7} + 4638925202 T^{8} + 43768470968 T^{9} + 461725986516 T^{10} + 4245541683896 T^{11} + 43647647225618 T^{12} + 387981833760848 T^{13} + 3695116577316520 T^{14} + 29637315682178960 T^{15} + 235028881994751853 T^{16} + 1443057360779098180 T^{17} + 7884458195943222766 T^{18} + 27368318111564347812 T^{19} + 73742412689492826049 T^{20} )^{2} $)($ ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) $)($ 1 - 146 T^{2} + 9409 T^{4} $)($ ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) $)($ 1 - 146 T^{2} + 9409 T^{4} $)($ ( 1 - 19 T + 97 T^{2} )^{2}( 1 + 19 T + 97 T^{2} )^{2} $)($ ( 1 - 429 T^{2} + 87258 T^{4} - 11450019 T^{6} + 1191212138 T^{8} - 107733228771 T^{10} + 7724888001498 T^{12} - 357344990114541 T^{14} + 7837433594376961 T^{16} )^{2} $)($ ( 1 + 14 T + 97 T^{2} )^{2} $)($ ( 1 + 14 T + 97 T^{2} )^{2} $)($ ( 1 + 14 T + 97 T^{2} )^{2} $)($ ( 1 + 14 T + 97 T^{2} )^{2} $)($ ( 1 - T + 180 T^{2} - 97 T^{3} + 9409 T^{4} )^{4} $)($ ( 1 - 8 T + 97 T^{2} )^{8} $)($ ( 1 - T + 180 T^{2} - 97 T^{3} + 9409 T^{4} )^{4} $)($ 1 - 146 T^{2} + 9409 T^{4} $)($ 1 - 119 T^{2} + 9409 T^{4} $)($ ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) $)($ 1 - 146 T^{2} + 9409 T^{4} $)($ 1 - 146 T^{2} + 9409 T^{4} $)($ 1 - 119 T^{2} + 9409 T^{4} $)($ ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) $)($ 1 - 146 T^{2} + 9409 T^{4} $)($ ( 1 + 6 T + 75 T^{2} + 582 T^{3} + 9409 T^{4} )^{4} $)
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Overview	Features
Universe	Knowledge

Degree 1	Degree 2
Degree 3	Degree 4
$\zeta$ zeros

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	=	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	=	\( 2 \)
Nonzero newspaces:			\( 16 \)
Newform subspaces:			\( 56 \)
Sturm bound:			\(12288\)
Trace bound:			\(8\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Downloads

Learn more about

Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(336))\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(336))\) into lower level spaces

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	336.2

Level	336
Weight	2
Dimension	1196
Nonzero newspaces	16
Newform subspaces	56
Sturm bound	12288
Trace bound	8

Label	\(\chi\)	Newforms	Dimension	\(\chi\) degree
336.2.a	\(\chi_{336}(1, \cdot)\)	336.2.a.a	1	1
		336.2.a.b	1
		336.2.a.c	1
		336.2.a.d	1
		336.2.a.e	1
		336.2.a.f	1
336.2.b	\(\chi_{336}(223, \cdot)\)	336.2.b.a	2	1
		336.2.b.b	2
		336.2.b.c	2
		336.2.b.d	2
336.2.c	\(\chi_{336}(169, \cdot)\)	None	0	1
336.2.h	\(\chi_{336}(239, \cdot)\)	336.2.h.a	4	1
336.2.h	\(\chi_{336}(239, \cdot)\)	336.2.h.b	8	1
336.2.i	\(\chi_{336}(41, \cdot)\)	None	0	1
336.2.j	\(\chi_{336}(71, \cdot)\)	None	0	1
336.2.k	\(\chi_{336}(209, \cdot)\)	336.2.k.a	2	1
		336.2.k.b	4
		336.2.k.c	8
336.2.p	\(\chi_{336}(55, \cdot)\)	None	0	1
336.2.q	\(\chi_{336}(193, \cdot)\)	336.2.q.a	2	2
		336.2.q.b	2
		336.2.q.c	2
		336.2.q.d	2
		336.2.q.e	2
		336.2.q.f	2
		336.2.q.g	4
336.2.s	\(\chi_{336}(155, \cdot)\)	336.2.s.a	4	2
		336.2.s.b	4
		336.2.s.c	40
		336.2.s.d	48
336.2.u	\(\chi_{336}(139, \cdot)\)	336.2.u.a	64	2
336.2.w	\(\chi_{336}(85, \cdot)\)	336.2.w.a	20	2
336.2.w	\(\chi_{336}(85, \cdot)\)	336.2.w.b	28	2
336.2.y	\(\chi_{336}(125, \cdot)\)	336.2.y.a	120	2
336.2.bb	\(\chi_{336}(103, \cdot)\)	None	0	2
336.2.bc	\(\chi_{336}(17, \cdot)\)	336.2.bc.a	2	2
		336.2.bc.b	2
		336.2.bc.c	2
		336.2.bc.d	2
		336.2.bc.e	4
		336.2.bc.f	16
336.2.bd	\(\chi_{336}(23, \cdot)\)	None	0	2
336.2.bi	\(\chi_{336}(89, \cdot)\)	None	0	2
336.2.bj	\(\chi_{336}(95, \cdot)\)	336.2.bj.a	2	2
		336.2.bj.b	2
		336.2.bj.c	2
		336.2.bj.d	2
		336.2.bj.e	8
		336.2.bj.f	8
		336.2.bj.g	8
336.2.bk	\(\chi_{336}(25, \cdot)\)	None	0	2
336.2.bl	\(\chi_{336}(31, \cdot)\)	336.2.bl.a	2	2
		336.2.bl.b	2
		336.2.bl.c	2
		336.2.bl.d	2
		336.2.bl.e	2
		336.2.bl.f	2
		336.2.bl.g	2
		336.2.bl.h	2
336.2.bo	\(\chi_{336}(5, \cdot)\)	336.2.bo.a	240	4
336.2.bq	\(\chi_{336}(37, \cdot)\)	336.2.bq.a	8	4
336.2.bq	\(\chi_{336}(37, \cdot)\)	336.2.bq.b	120	4
336.2.bs	\(\chi_{336}(19, \cdot)\)	336.2.bs.a	128	4
336.2.bu	\(\chi_{336}(11, \cdot)\)	336.2.bu.a	240	4