# Properties

 Label 1575.2.bk Level 1575 Weight 2 Character orbit bk Rep. character $$\chi_{1575}(26,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 100 Newform subspaces 9 Sturm bound 480 Trace bound 11

# Related objects

## Defining parameters

 Level: $$N$$ = $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1575.bk (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$21$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$9$$ Sturm bound: $$480$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$2$$, $$11$$, $$37$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1575, [\chi])$$.

Total New Old
Modular forms 528 100 428
Cusp forms 432 100 332
Eisenstein series 96 0 96

## Trace form

 $$100q + 48q^{4} + 6q^{7} + O(q^{10})$$ $$100q + 48q^{4} + 6q^{7} - 40q^{16} + 18q^{19} - 56q^{22} + 40q^{28} - 18q^{31} + 18q^{37} + 12q^{43} - 8q^{46} + 18q^{49} - 32q^{58} - 24q^{61} + 64q^{64} + 10q^{67} + 78q^{73} + 30q^{79} + 180q^{82} - 16q^{88} + 82q^{91} - 132q^{94} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1575, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1575.2.bk.a $$4$$ $$12.576$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+(-2+2\beta _{2})q^{4}+(-1-2\beta _{2})q^{7}+\cdots$$
1575.2.bk.b $$4$$ $$12.576$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+(-2+2\beta _{2})q^{4}+(1+2\beta _{2})q^{7}+\beta _{1}q^{11}+\cdots$$
1575.2.bk.c $$4$$ $$12.576$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+\beta _{1}q^{2}+(-1+3\beta _{2})q^{7}-2\beta _{3}q^{8}+\cdots$$
1575.2.bk.d $$8$$ $$12.576$$ 8.0.796594176.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-2\beta _{2}q^{4}+(\beta _{6}+\beta _{7})q^{7}+2\beta _{3}q^{11}+\cdots$$
1575.2.bk.e $$12$$ $$12.576$$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+\beta _{3}q^{2}+(-\beta _{2}+\beta _{4}+\beta _{5})q^{4}+(-\beta _{1}+\cdots)q^{7}+\cdots$$
1575.2.bk.f $$12$$ $$12.576$$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+(\beta _{1}-\beta _{3})q^{2}+(1-\beta _{4}-\beta _{5})q^{4}+(\beta _{1}+\cdots)q^{7}+\cdots$$
1575.2.bk.g $$16$$ $$12.576$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+(-\beta _{8}-\beta _{9}+\beta _{15})q^{2}+(2-2\beta _{3}+\cdots)q^{4}+\cdots$$
1575.2.bk.h $$16$$ $$12.576$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+(-\beta _{8}-\beta _{9}+\beta _{15})q^{2}+(2-2\beta _{3}+\cdots)q^{4}+\cdots$$
1575.2.bk.i $$24$$ $$12.576$$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1575, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1575, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(315, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(525, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + 2 T^{2} + 4 T^{4} )^{2}$$)($$( 1 + 2 T^{2} + 4 T^{4} )^{2}$$)($$( 1 + 2 T^{2} )^{2}( 1 - 2 T^{2} + 4 T^{4} )$$)($$( 1 + 2 T^{2} + 4 T^{4} )^{4}$$)($$1 + 2 T^{2} + 2 T^{4} - 12 T^{5} - 24 T^{7} - 8 T^{8} - 24 T^{9} + 52 T^{10} + 24 T^{11} + 148 T^{12} + 48 T^{13} + 208 T^{14} - 192 T^{15} - 128 T^{16} - 768 T^{17} - 1536 T^{19} + 512 T^{20} + 2048 T^{22} + 4096 T^{24}$$)($$1 + 2 T^{2} + 2 T^{4} + 12 T^{5} + 24 T^{7} - 8 T^{8} + 24 T^{9} + 52 T^{10} - 24 T^{11} + 148 T^{12} - 48 T^{13} + 208 T^{14} + 192 T^{15} - 128 T^{16} + 768 T^{17} + 1536 T^{19} + 512 T^{20} + 2048 T^{22} + 4096 T^{24}$$)($$1 + 2 T^{2} - 6 T^{4} - 20 T^{6} + 11 T^{8} + 84 T^{10} + 70 T^{12} - 172 T^{14} - 543 T^{16} - 688 T^{18} + 1120 T^{20} + 5376 T^{22} + 2816 T^{24} - 20480 T^{26} - 24576 T^{28} + 32768 T^{30} + 65536 T^{32}$$)($$1 + 2 T^{2} - 6 T^{4} - 20 T^{6} + 11 T^{8} + 84 T^{10} + 70 T^{12} - 172 T^{14} - 543 T^{16} - 688 T^{18} + 1120 T^{20} + 5376 T^{22} + 2816 T^{24} - 20480 T^{26} - 24576 T^{28} + 32768 T^{30} + 65536 T^{32}$$)
$3$ 1
$5$ 1
$7$ ($$( 1 + 4 T + 7 T^{2} )^{2}$$)($$( 1 - 4 T + 7 T^{2} )^{2}$$)($$( 1 - T + 7 T^{2} )^{2}$$)($$( 1 - 7 T^{2} + 49 T^{4} )^{2}$$)($$1 - 2 T - T^{2} - 6 T^{3} + 130 T^{4} - 130 T^{5} - 131 T^{6} - 910 T^{7} + 6370 T^{8} - 2058 T^{9} - 2401 T^{10} - 33614 T^{11} + 117649 T^{12}$$)($$1 - 2 T - T^{2} - 6 T^{3} + 130 T^{4} - 130 T^{5} - 131 T^{6} - 910 T^{7} + 6370 T^{8} - 2058 T^{9} - 2401 T^{10} - 33614 T^{11} + 117649 T^{12}$$)($$( 1 + 4 T + 4 T^{2} - 26 T^{3} - 143 T^{4} - 182 T^{5} + 196 T^{6} + 1372 T^{7} + 2401 T^{8} )^{2}$$)($$( 1 - 4 T + 4 T^{2} + 26 T^{3} - 143 T^{4} + 182 T^{5} + 196 T^{6} - 1372 T^{7} + 2401 T^{8} )^{2}$$)
$11$ ($$1 + 4 T^{2} - 105 T^{4} + 484 T^{6} + 14641 T^{8}$$)($$1 + 4 T^{2} - 105 T^{4} + 484 T^{6} + 14641 T^{8}$$)($$1 + 20 T^{2} + 279 T^{4} + 2420 T^{6} + 14641 T^{8}$$)($$( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2}( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{2}$$)($$1 + 12 T + 104 T^{2} + 672 T^{3} + 3641 T^{4} + 16332 T^{5} + 62340 T^{6} + 195852 T^{7} + 477010 T^{8} + 649368 T^{9} - 1262450 T^{10} - 12891636 T^{11} - 53694803 T^{12} - 141807996 T^{13} - 152756450 T^{14} + 864308808 T^{15} + 6983903410 T^{16} + 31542160452 T^{17} + 110439112740 T^{18} + 318264476772 T^{19} + 780480685721 T^{20} + 1584540848352 T^{21} + 2697492158504 T^{22} + 3423740047332 T^{23} + 3138428376721 T^{24}$$)($$1 - 12 T + 104 T^{2} - 672 T^{3} + 3641 T^{4} - 16332 T^{5} + 62340 T^{6} - 195852 T^{7} + 477010 T^{8} - 649368 T^{9} - 1262450 T^{10} + 12891636 T^{11} - 53694803 T^{12} + 141807996 T^{13} - 152756450 T^{14} - 864308808 T^{15} + 6983903410 T^{16} - 31542160452 T^{17} + 110439112740 T^{18} - 318264476772 T^{19} + 780480685721 T^{20} - 1584540848352 T^{21} + 2697492158504 T^{22} - 3423740047332 T^{23} + 3138428376721 T^{24}$$)($$1 + 38 T^{2} + 624 T^{4} + 5140 T^{6} + 15002 T^{8} - 146958 T^{10} - 223400 T^{12} + 62692658 T^{14} + 1151697111 T^{16} + 7585811618 T^{18} - 3270799400 T^{20} - 260345061438 T^{22} + 3215811932762 T^{24} + 133318362449140 T^{26} + 1958379307073904 T^{28} + 14430493676163158 T^{30} + 45949729863572161 T^{32}$$)($$1 + 38 T^{2} + 624 T^{4} + 5140 T^{6} + 15002 T^{8} - 146958 T^{10} - 223400 T^{12} + 62692658 T^{14} + 1151697111 T^{16} + 7585811618 T^{18} - 3270799400 T^{20} - 260345061438 T^{22} + 3215811932762 T^{24} + 133318362449140 T^{26} + 1958379307073904 T^{28} + 14430493676163158 T^{30} + 45949729863572161 T^{32}$$)
$13$ ($$( 1 - 14 T^{2} + 169 T^{4} )^{2}$$)($$( 1 - 14 T^{2} + 169 T^{4} )^{2}$$)($$( 1 - 5 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2}$$)($$( 1 - 19 T^{2} + 169 T^{4} )^{4}$$)($$1 - 54 T^{2} + 1725 T^{4} - 42910 T^{6} + 855078 T^{8} - 14180646 T^{10} + 200139573 T^{12} - 2396529174 T^{14} + 24421882758 T^{16} - 207118374190 T^{18} + 1407135493725 T^{20} - 7444358559846 T^{22} + 23298085122481 T^{24}$$)($$1 - 54 T^{2} + 1725 T^{4} - 42910 T^{6} + 855078 T^{8} - 14180646 T^{10} + 200139573 T^{12} - 2396529174 T^{14} + 24421882758 T^{16} - 207118374190 T^{18} + 1407135493725 T^{20} - 7444358559846 T^{22} + 23298085122481 T^{24}$$)($$( 1 - 68 T^{2} + 2146 T^{4} - 43136 T^{6} + 637867 T^{8} - 7289984 T^{10} + 61291906 T^{12} - 328223012 T^{14} + 815730721 T^{16} )^{2}$$)($$( 1 - 68 T^{2} + 2146 T^{4} - 43136 T^{6} + 637867 T^{8} - 7289984 T^{10} + 61291906 T^{12} - 328223012 T^{14} + 815730721 T^{16} )^{2}$$)
$17$ ($$1 + 20 T^{2} + 111 T^{4} + 5780 T^{6} + 83521 T^{8}$$)($$1 + 20 T^{2} + 111 T^{4} + 5780 T^{6} + 83521 T^{8}$$)($$1 - 10 T^{2} - 189 T^{4} - 2890 T^{6} + 83521 T^{8}$$)($$( 1 - 20 T^{2} + 111 T^{4} - 5780 T^{6} + 83521 T^{8} )^{2}$$)($$1 - 42 T^{2} + 120 T^{3} + 831 T^{4} - 5256 T^{5} - 3958 T^{6} + 127332 T^{7} - 227496 T^{8} - 2018952 T^{9} + 9082140 T^{10} + 14965308 T^{11} - 192427863 T^{12} + 254410236 T^{13} + 2624738460 T^{14} - 9919111176 T^{15} - 19000693416 T^{16} + 180793231524 T^{17} - 95536498102 T^{18} - 2156740065288 T^{19} + 5796854433471 T^{20} + 14230545179640 T^{21} - 84671743818858 T^{22} + 582622237229761 T^{24}$$)($$1 - 42 T^{2} - 120 T^{3} + 831 T^{4} + 5256 T^{5} - 3958 T^{6} - 127332 T^{7} - 227496 T^{8} + 2018952 T^{9} + 9082140 T^{10} - 14965308 T^{11} - 192427863 T^{12} - 254410236 T^{13} + 2624738460 T^{14} + 9919111176 T^{15} - 19000693416 T^{16} - 180793231524 T^{17} - 95536498102 T^{18} + 2156740065288 T^{19} + 5796854433471 T^{20} - 14230545179640 T^{21} - 84671743818858 T^{22} + 582622237229761 T^{24}$$)($$1 - 82 T^{2} + 3288 T^{4} - 91868 T^{6} + 2140106 T^{8} - 45216606 T^{10} + 875760904 T^{12} - 15517704646 T^{14} + 263733650775 T^{16} - 4484616642694 T^{18} + 73144426462984 T^{20} - 1091418947270814 T^{22} + 14928860354028746 T^{24} - 185205327646448732 T^{26} + 1915661916011454168 T^{28} - 13806981777870876178 T^{30} + 48661191875666868481 T^{32}$$)($$1 - 82 T^{2} + 3288 T^{4} - 91868 T^{6} + 2140106 T^{8} - 45216606 T^{10} + 875760904 T^{12} - 15517704646 T^{14} + 263733650775 T^{16} - 4484616642694 T^{18} + 73144426462984 T^{20} - 1091418947270814 T^{22} + 14928860354028746 T^{24} - 185205327646448732 T^{26} + 1915661916011454168 T^{28} - 13806981777870876178 T^{30} + 48661191875666868481 T^{32}$$)
$19$ ($$( 1 - 12 T + 67 T^{2} - 228 T^{3} + 361 T^{4} )^{2}$$)($$( 1 - 12 T + 67 T^{2} - 228 T^{3} + 361 T^{4} )^{2}$$)($$( 1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4} )^{2}$$)($$( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} )^{4}$$)($$1 - 6 T + 75 T^{2} - 378 T^{3} + 2703 T^{4} - 15096 T^{5} + 85196 T^{6} - 494928 T^{7} + 2381289 T^{8} - 12436578 T^{9} + 56967225 T^{10} - 267312150 T^{11} + 1188448446 T^{12} - 5078930850 T^{13} + 20565168225 T^{14} - 85302488502 T^{15} + 310331963769 T^{16} - 1225490725872 T^{17} + 4008120877676 T^{18} - 13493887771944 T^{19} + 45906570899823 T^{20} - 121975949760462 T^{21} + 459829969335075 T^{22} - 698941553389314 T^{23} + 2213314919066161 T^{24}$$)($$1 - 6 T + 75 T^{2} - 378 T^{3} + 2703 T^{4} - 15096 T^{5} + 85196 T^{6} - 494928 T^{7} + 2381289 T^{8} - 12436578 T^{9} + 56967225 T^{10} - 267312150 T^{11} + 1188448446 T^{12} - 5078930850 T^{13} + 20565168225 T^{14} - 85302488502 T^{15} + 310331963769 T^{16} - 1225490725872 T^{17} + 4008120877676 T^{18} - 13493887771944 T^{19} + 45906570899823 T^{20} - 121975949760462 T^{21} + 459829969335075 T^{22} - 698941553389314 T^{23} + 2213314919066161 T^{24}$$)($$( 1 + 18 T + 196 T^{2} + 1584 T^{3} + 10591 T^{4} + 63342 T^{5} + 346888 T^{6} + 1747512 T^{7} + 8010724 T^{8} + 33202728 T^{9} + 125226568 T^{10} + 434462778 T^{11} + 1380229711 T^{12} + 3922140816 T^{13} + 9220992676 T^{14} + 16089691302 T^{15} + 16983563041 T^{16} )^{2}$$)($$( 1 + 18 T + 196 T^{2} + 1584 T^{3} + 10591 T^{4} + 63342 T^{5} + 346888 T^{6} + 1747512 T^{7} + 8010724 T^{8} + 33202728 T^{9} + 125226568 T^{10} + 434462778 T^{11} + 1380229711 T^{12} + 3922140816 T^{13} + 9220992676 T^{14} + 16089691302 T^{15} + 16983563041 T^{16} )^{2}$$)
$23$ ($$1 - 26 T^{2} + 147 T^{4} - 13754 T^{6} + 279841 T^{8}$$)($$1 - 26 T^{2} + 147 T^{4} - 13754 T^{6} + 279841 T^{8}$$)($$1 + 14 T^{2} - 333 T^{4} + 7406 T^{6} + 279841 T^{8}$$)($$( 1 + 4 T^{2} - 513 T^{4} + 2116 T^{6} + 279841 T^{8} )^{2}$$)($$1 - 12 T + 110 T^{2} - 744 T^{3} + 3947 T^{4} - 16104 T^{5} + 39450 T^{6} + 32328 T^{7} - 1232708 T^{8} + 9040500 T^{9} - 50848868 T^{10} + 256937364 T^{11} - 1217123567 T^{12} + 5909559372 T^{13} - 26899051172 T^{14} + 109995763500 T^{15} - 344962239428 T^{16} + 208074096504 T^{17} + 5840015821050 T^{18} - 54831308998488 T^{19} + 309093458904107 T^{20} - 1340057580128472 T^{21} + 4556916233501390 T^{22} - 11433717094967124 T^{23} + 21914624432020321 T^{24}$$)($$1 + 12 T + 110 T^{2} + 744 T^{3} + 3947 T^{4} + 16104 T^{5} + 39450 T^{6} - 32328 T^{7} - 1232708 T^{8} - 9040500 T^{9} - 50848868 T^{10} - 256937364 T^{11} - 1217123567 T^{12} - 5909559372 T^{13} - 26899051172 T^{14} - 109995763500 T^{15} - 344962239428 T^{16} - 208074096504 T^{17} + 5840015821050 T^{18} + 54831308998488 T^{19} + 309093458904107 T^{20} + 1340057580128472 T^{21} + 4556916233501390 T^{22} + 11433717094967124 T^{23} + 21914624432020321 T^{24}$$)($$( 1 + 44 T^{2} + 1407 T^{4} + 23276 T^{6} + 279841 T^{8} )^{4}$$)($$( 1 + 44 T^{2} + 1407 T^{4} + 23276 T^{6} + 279841 T^{8} )^{4}$$)
$29$ ($$( 1 + 14 T^{2} + 841 T^{4} )^{2}$$)($$( 1 + 14 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 50 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 56 T^{2} + 841 T^{4} )^{4}$$)($$1 - 184 T^{2} + 16934 T^{4} - 1029132 T^{6} + 46624399 T^{8} - 1703051756 T^{10} + 52962115192 T^{12} - 1432266526796 T^{14} + 32976551549119 T^{16} - 612151713987372 T^{18} + 8471172757081574 T^{20} - 77410130927236984 T^{22} + 353814783205469041 T^{24}$$)($$1 - 184 T^{2} + 16934 T^{4} - 1029132 T^{6} + 46624399 T^{8} - 1703051756 T^{10} + 52962115192 T^{12} - 1432266526796 T^{14} + 32976551549119 T^{16} - 612151713987372 T^{18} + 8471172757081574 T^{20} - 77410130927236984 T^{22} + 353814783205469041 T^{24}$$)($$( 1 - 128 T^{2} + 9244 T^{4} - 439136 T^{6} + 14973286 T^{8} - 369313376 T^{10} + 6538105564 T^{12} - 76137385088 T^{14} + 500246412961 T^{16} )^{2}$$)($$( 1 - 128 T^{2} + 9244 T^{4} - 439136 T^{6} + 14973286 T^{8} - 369313376 T^{10} + 6538105564 T^{12} - 76137385088 T^{14} + 500246412961 T^{16} )^{2}$$)
$31$ ($$( 1 - 9 T + 58 T^{2} - 279 T^{3} + 961 T^{4} )^{2}$$)($$( 1 - 9 T + 58 T^{2} - 279 T^{3} + 961 T^{4} )^{2}$$)($$( 1 - 7 T + 31 T^{2} )^{2}( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 - 9 T + 58 T^{2} - 279 T^{3} + 961 T^{4} )^{4}$$)($$1 - 6 T + 75 T^{2} - 378 T^{3} + 2451 T^{4} - 14268 T^{5} + 15464 T^{6} - 69696 T^{7} - 2002827 T^{8} + 11519550 T^{9} - 72958179 T^{10} + 685842102 T^{11} - 2463303858 T^{12} + 21261105162 T^{13} - 70112810019 T^{14} + 343178914050 T^{15} - 1849652793867 T^{16} - 1995337308096 T^{17} + 13724356922984 T^{18} - 392549978135748 T^{19} + 2090435932767891 T^{20} - 9994177176733638 T^{21} + 61472121523560075 T^{22} - 152450861378428986 T^{23} + 787662783788549761 T^{24}$$)($$1 - 6 T + 75 T^{2} - 378 T^{3} + 2451 T^{4} - 14268 T^{5} + 15464 T^{6} - 69696 T^{7} - 2002827 T^{8} + 11519550 T^{9} - 72958179 T^{10} + 685842102 T^{11} - 2463303858 T^{12} + 21261105162 T^{13} - 70112810019 T^{14} + 343178914050 T^{15} - 1849652793867 T^{16} - 1995337308096 T^{17} + 13724356922984 T^{18} - 392549978135748 T^{19} + 2090435932767891 T^{20} - 9994177176733638 T^{21} + 61472121523560075 T^{22} - 152450861378428986 T^{23} + 787662783788549761 T^{24}$$)($$( 1 + 12 T + 124 T^{2} + 912 T^{3} + 6826 T^{4} + 43740 T^{5} + 264016 T^{6} + 1524612 T^{7} + 8482819 T^{8} + 47262972 T^{9} + 253719376 T^{10} + 1303058340 T^{11} + 6303954346 T^{12} + 26109785712 T^{13} + 110050456444 T^{14} + 330151369332 T^{15} + 852891037441 T^{16} )^{2}$$)($$( 1 + 12 T + 124 T^{2} + 912 T^{3} + 6826 T^{4} + 43740 T^{5} + 264016 T^{6} + 1524612 T^{7} + 8482819 T^{8} + 47262972 T^{9} + 253719376 T^{10} + 1303058340 T^{11} + 6303954346 T^{12} + 26109785712 T^{13} + 110050456444 T^{14} + 330151369332 T^{15} + 852891037441 T^{16} )^{2}$$)
$37$ ($$( 1 - 11 T + 37 T^{2} )^{2}( 1 + 10 T + 37 T^{2} )^{2}$$)($$( 1 - 10 T + 37 T^{2} )^{2}( 1 + 11 T + 37 T^{2} )^{2}$$)($$( 1 - 10 T + 37 T^{2} )^{2}( 1 + 11 T + 37 T^{2} )^{2}$$)($$( 1 - 53 T^{2} + 1440 T^{4} - 72557 T^{6} + 1874161 T^{8} )^{2}$$)($$1 - 10 T - T^{2} - 206 T^{3} + 1893 T^{4} + 19356 T^{5} - 25070 T^{6} + 11384 T^{7} - 4236169 T^{8} - 22256290 T^{9} - 6662617 T^{10} + 355484390 T^{11} + 9259385786 T^{12} + 13152922430 T^{13} - 9121122673 T^{14} - 1127347857370 T^{15} - 7939262729209 T^{16} + 789411606488 T^{17} - 64322761073630 T^{18} + 1837501413786348 T^{19} + 6649123606272453 T^{20} - 26772118397785862 T^{21} - 4808584372417849 T^{22} - 1779176217794604130 T^{23} + 6582952005840035281 T^{24}$$)($$1 - 10 T - T^{2} - 206 T^{3} + 1893 T^{4} + 19356 T^{5} - 25070 T^{6} + 11384 T^{7} - 4236169 T^{8} - 22256290 T^{9} - 6662617 T^{10} + 355484390 T^{11} + 9259385786 T^{12} + 13152922430 T^{13} - 9121122673 T^{14} - 1127347857370 T^{15} - 7939262729209 T^{16} + 789411606488 T^{17} - 64322761073630 T^{18} + 1837501413786348 T^{19} + 6649123606272453 T^{20} - 26772118397785862 T^{21} - 4808584372417849 T^{22} - 1779176217794604130 T^{23} + 6582952005840035281 T^{24}$$)($$( 1 - 16 T + 54 T^{2} - 56 T^{3} + 4049 T^{4} - 16428 T^{5} - 157370 T^{6} + 483332 T^{7} + 3649380 T^{8} + 17883284 T^{9} - 215439530 T^{10} - 832127484 T^{11} + 7588477889 T^{12} - 3883261592 T^{13} + 138549226086 T^{14} - 1518910034128 T^{15} + 3512479453921 T^{16} )^{2}$$)($$( 1 + 16 T + 54 T^{2} + 56 T^{3} + 4049 T^{4} + 16428 T^{5} - 157370 T^{6} - 483332 T^{7} + 3649380 T^{8} - 17883284 T^{9} - 215439530 T^{10} + 832127484 T^{11} + 7588477889 T^{12} + 3883261592 T^{13} + 138549226086 T^{14} + 1518910034128 T^{15} + 3512479453921 T^{16} )^{2}$$)
$41$ ($$( 1 + 28 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 28 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 28 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 58 T^{2} + 1681 T^{4} )^{4}$$)($$( 1 + 12 T + 192 T^{2} + 1704 T^{3} + 18087 T^{4} + 122340 T^{5} + 939022 T^{6} + 5015940 T^{7} + 30404247 T^{8} + 117441384 T^{9} + 542546112 T^{10} + 1390274412 T^{11} + 4750104241 T^{12} )^{2}$$)($$( 1 - 12 T + 192 T^{2} - 1704 T^{3} + 18087 T^{4} - 122340 T^{5} + 939022 T^{6} - 5015940 T^{7} + 30404247 T^{8} - 117441384 T^{9} + 542546112 T^{10} - 1390274412 T^{11} + 4750104241 T^{12} )^{2}$$)($$( 1 + 166 T^{2} + 15208 T^{4} + 966850 T^{6} + 45385570 T^{8} + 1625274850 T^{10} + 42974173288 T^{12} + 788517304006 T^{14} + 7984925229121 T^{16} )^{2}$$)($$( 1 + 166 T^{2} + 15208 T^{4} + 966850 T^{6} + 45385570 T^{8} + 1625274850 T^{10} + 42974173288 T^{12} + 788517304006 T^{14} + 7984925229121 T^{16} )^{2}$$)
$43$ ($$( 1 + T + 43 T^{2} )^{4}$$)($$( 1 - T + 43 T^{2} )^{4}$$)($$( 1 - T + 43 T^{2} )^{4}$$)($$( 1 + 65 T^{2} + 1849 T^{4} )^{4}$$)($$( 1 - 2 T + 65 T^{2} - 318 T^{3} + 5080 T^{4} - 14554 T^{5} + 241357 T^{6} - 625822 T^{7} + 9392920 T^{8} - 25283226 T^{9} + 222222065 T^{10} - 294016886 T^{11} + 6321363049 T^{12} )^{2}$$)($$( 1 - 2 T + 65 T^{2} - 318 T^{3} + 5080 T^{4} - 14554 T^{5} + 241357 T^{6} - 625822 T^{7} + 9392920 T^{8} - 25283226 T^{9} + 222222065 T^{10} - 294016886 T^{11} + 6321363049 T^{12} )^{2}$$)($$( 1 - 14 T + 196 T^{2} - 1574 T^{3} + 12670 T^{4} - 67682 T^{5} + 362404 T^{6} - 1113098 T^{7} + 3418801 T^{8} )^{4}$$)($$( 1 + 14 T + 196 T^{2} + 1574 T^{3} + 12670 T^{4} + 67682 T^{5} + 362404 T^{6} + 1113098 T^{7} + 3418801 T^{8} )^{4}$$)
$47$ ($$1 - 40 T^{2} - 609 T^{4} - 88360 T^{6} + 4879681 T^{8}$$)($$1 - 40 T^{2} - 609 T^{4} - 88360 T^{6} + 4879681 T^{8}$$)($$1 + 56 T^{2} + 927 T^{4} + 123704 T^{6} + 4879681 T^{8}$$)($$( 1 - 80 T^{2} + 4191 T^{4} - 176720 T^{6} + 4879681 T^{8} )^{2}$$)($$1 - 162 T^{2} + 192 T^{3} + 11733 T^{4} - 23328 T^{5} - 672838 T^{6} + 760896 T^{7} + 42600858 T^{8} + 8087904 T^{9} - 2486828586 T^{10} - 573202080 T^{11} + 123583933965 T^{12} - 26940497760 T^{13} - 5493404346474 T^{14} + 839710456992 T^{15} + 207878597366298 T^{16} + 174507698446272 T^{17} - 7252665683533702 T^{18} - 11818504154160864 T^{19} + 279377826402441813 T^{20} + 214873050835731264 T^{21} - 8521059422204467938 T^{22} +$$$$11\!\cdots\!41$$$$T^{24}$$)($$1 - 162 T^{2} - 192 T^{3} + 11733 T^{4} + 23328 T^{5} - 672838 T^{6} - 760896 T^{7} + 42600858 T^{8} - 8087904 T^{9} - 2486828586 T^{10} + 573202080 T^{11} + 123583933965 T^{12} + 26940497760 T^{13} - 5493404346474 T^{14} - 839710456992 T^{15} + 207878597366298 T^{16} - 174507698446272 T^{17} - 7252665683533702 T^{18} + 11818504154160864 T^{19} + 279377826402441813 T^{20} - 214873050835731264 T^{21} - 8521059422204467938 T^{22} +$$$$11\!\cdots\!41$$$$T^{24}$$)($$1 - 22 T^{2} - 2964 T^{4} + 321316 T^{6} + 96242 T^{8} - 796763922 T^{10} + 37839327784 T^{12} + 943600073510 T^{14} - 103395703967577 T^{16} + 2084412562383590 T^{18} + 184643848840356904 T^{20} - 8588489881616560338 T^{22} + 2291645850901202162 T^{24} +$$$$16\!\cdots\!84$$$$T^{26} -$$$$34\!\cdots\!24$$$$T^{28} -$$$$56\!\cdots\!18$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$)($$1 - 22 T^{2} - 2964 T^{4} + 321316 T^{6} + 96242 T^{8} - 796763922 T^{10} + 37839327784 T^{12} + 943600073510 T^{14} - 103395703967577 T^{16} + 2084412562383590 T^{18} + 184643848840356904 T^{20} - 8588489881616560338 T^{22} + 2291645850901202162 T^{24} +$$$$16\!\cdots\!84$$$$T^{26} -$$$$34\!\cdots\!24$$$$T^{28} -$$$$56\!\cdots\!18$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$)
$53$ ($$1 + 88 T^{2} + 4935 T^{4} + 247192 T^{6} + 7890481 T^{8}$$)($$1 + 88 T^{2} + 4935 T^{4} + 247192 T^{6} + 7890481 T^{8}$$)($$1 + 98 T^{2} + 6795 T^{4} + 275282 T^{6} + 7890481 T^{8}$$)($$( 1 - 62 T^{2} + 1035 T^{4} - 174158 T^{6} + 7890481 T^{8} )^{2}$$)($$1 - 12 T + 290 T^{2} - 2904 T^{3} + 41501 T^{4} - 314064 T^{5} + 3427926 T^{6} - 18869580 T^{7} + 172940890 T^{8} - 534860628 T^{9} + 5470600330 T^{10} + 1694225472 T^{11} + 171503876389 T^{12} + 89793950016 T^{13} + 15366916326970 T^{14} - 79628445714756 T^{15} + 1364586806668090 T^{16} - 7891173310802940 T^{17} + 75977789787488454 T^{18} - 368934479421767568 T^{19} + 2583839411761892861 T^{20} - 9582513470593394232 T^{21} + 50717366405998784210 T^{22} -$$$$11\!\cdots\!64$$$$T^{23} +$$$$49\!\cdots\!41$$$$T^{24}$$)($$1 + 12 T + 290 T^{2} + 2904 T^{3} + 41501 T^{4} + 314064 T^{5} + 3427926 T^{6} + 18869580 T^{7} + 172940890 T^{8} + 534860628 T^{9} + 5470600330 T^{10} - 1694225472 T^{11} + 171503876389 T^{12} - 89793950016 T^{13} + 15366916326970 T^{14} + 79628445714756 T^{15} + 1364586806668090 T^{16} + 7891173310802940 T^{17} + 75977789787488454 T^{18} + 368934479421767568 T^{19} + 2583839411761892861 T^{20} + 9582513470593394232 T^{21} + 50717366405998784210 T^{22} +$$$$11\!\cdots\!64$$$$T^{23} +$$$$49\!\cdots\!41$$$$T^{24}$$)($$1 + 326 T^{2} + 56544 T^{4} + 6845812 T^{6} + 648146234 T^{8} + 51557322450 T^{10} + 3593647250584 T^{12} + 223852743083186 T^{14} + 12523118248684503 T^{16} + 628802355320669474 T^{18} + 28355605351435290904 T^{20} +$$$$11\!\cdots\!50$$$$T^{22} +$$$$40\!\cdots\!74$$$$T^{24} +$$$$11\!\cdots\!88$$$$T^{26} +$$$$27\!\cdots\!04$$$$T^{28} +$$$$44\!\cdots\!94$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$)($$1 + 326 T^{2} + 56544 T^{4} + 6845812 T^{6} + 648146234 T^{8} + 51557322450 T^{10} + 3593647250584 T^{12} + 223852743083186 T^{14} + 12523118248684503 T^{16} + 628802355320669474 T^{18} + 28355605351435290904 T^{20} +$$$$11\!\cdots\!50$$$$T^{22} +$$$$40\!\cdots\!74$$$$T^{24} +$$$$11\!\cdots\!88$$$$T^{26} +$$$$27\!\cdots\!04$$$$T^{28} +$$$$44\!\cdots\!94$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$)
$59$ ($$1 - 64 T^{2} + 615 T^{4} - 222784 T^{6} + 12117361 T^{8}$$)($$1 - 64 T^{2} + 615 T^{4} - 222784 T^{6} + 12117361 T^{8}$$)($$1 - 94 T^{2} + 5355 T^{4} - 327214 T^{6} + 12117361 T^{8}$$)($$( 1 - 64 T^{2} + 615 T^{4} - 222784 T^{6} + 12117361 T^{8} )^{2}$$)($$1 - 24 T + 84 T^{2} + 1128 T^{3} + 9885 T^{4} - 197484 T^{5} - 900376 T^{6} + 12268224 T^{7} + 92002038 T^{8} - 411544404 T^{9} - 9205294494 T^{10} + 19136016684 T^{11} + 471190351413 T^{12} + 1129024984356 T^{13} - 32043630133614 T^{14} - 84522578149116 T^{15} + 1114821907181718 T^{16} + 8770851443174976 T^{17} - 37978340157549016 T^{18} - 491468849827995396 T^{19} + 1451418875718713085 T^{20} + 9771859283442771192 T^{21} + 42933807277253877684 T^{22} -$$$$72\!\cdots\!16$$$$T^{23} +$$$$17\!\cdots\!81$$$$T^{24}$$)($$1 + 24 T + 84 T^{2} - 1128 T^{3} + 9885 T^{4} + 197484 T^{5} - 900376 T^{6} - 12268224 T^{7} + 92002038 T^{8} + 411544404 T^{9} - 9205294494 T^{10} - 19136016684 T^{11} + 471190351413 T^{12} - 1129024984356 T^{13} - 32043630133614 T^{14} + 84522578149116 T^{15} + 1114821907181718 T^{16} - 8770851443174976 T^{17} - 37978340157549016 T^{18} + 491468849827995396 T^{19} + 1451418875718713085 T^{20} - 9771859283442771192 T^{21} + 42933807277253877684 T^{22} +$$$$72\!\cdots\!16$$$$T^{23} +$$$$17\!\cdots\!81$$$$T^{24}$$)($$1 - 22 T^{2} + 3828 T^{4} + 95908 T^{6} + 52130 T^{8} + 477386214 T^{10} - 45373908104 T^{12} - 418149177034 T^{14} - 123462803610873 T^{16} - 1455577285255354 T^{18} - 549812024476993544 T^{20} + 20136405259376625174 T^{22} + 7654270712313253730 T^{24} +$$$$49\!\cdots\!08$$$$T^{26} +$$$$68\!\cdots\!68$$$$T^{28} -$$$$13\!\cdots\!42$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$)($$1 - 22 T^{2} + 3828 T^{4} + 95908 T^{6} + 52130 T^{8} + 477386214 T^{10} - 45373908104 T^{12} - 418149177034 T^{14} - 123462803610873 T^{16} - 1455577285255354 T^{18} - 549812024476993544 T^{20} + 20136405259376625174 T^{22} + 7654270712313253730 T^{24} +$$$$49\!\cdots\!08$$$$T^{26} +$$$$68\!\cdots\!68$$$$T^{28} -$$$$13\!\cdots\!42$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$)
$61$ ($$( 1 + 3 T + 64 T^{2} + 183 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 3 T + 64 T^{2} + 183 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 6 T + 73 T^{2} + 366 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 18 T + 169 T^{2} + 1098 T^{3} + 3721 T^{4} )^{4}$$)($$1 + 138 T^{2} + 5889 T^{4} - 41004 T^{5} - 102418 T^{6} - 5059224 T^{7} - 8898402 T^{8} - 158765292 T^{9} + 1660743870 T^{10} + 9579987756 T^{11} + 189206615181 T^{12} + 584379253116 T^{13} + 6179627940270 T^{14} - 36036704743452 T^{15} - 123205859246082 T^{16} - 4273001876330424 T^{17} - 5276613701304898 T^{18} - 128865027248205084 T^{19} + 1128964366240987809 T^{20} + 98441321809477798938 T^{22} +$$$$26\!\cdots\!21$$$$T^{24}$$)($$1 + 138 T^{2} + 5889 T^{4} - 41004 T^{5} - 102418 T^{6} - 5059224 T^{7} - 8898402 T^{8} - 158765292 T^{9} + 1660743870 T^{10} + 9579987756 T^{11} + 189206615181 T^{12} + 584379253116 T^{13} + 6179627940270 T^{14} - 36036704743452 T^{15} - 123205859246082 T^{16} - 4273001876330424 T^{17} - 5276613701304898 T^{18} - 128865027248205084 T^{19} + 1128964366240987809 T^{20} + 98441321809477798938 T^{22} +$$$$26\!\cdots\!21$$$$T^{24}$$)($$( 1 - 6 T + 76 T^{2} - 384 T^{3} + 199 T^{4} - 1944 T^{5} - 83024 T^{6} - 574314 T^{7} + 970696 T^{8} - 35033154 T^{9} - 308932304 T^{10} - 441251064 T^{11} + 2755322359 T^{12} - 324324979584 T^{13} + 3915548451436 T^{14} - 18856457016126 T^{15} + 191707312997281 T^{16} )^{2}$$)($$( 1 - 6 T + 76 T^{2} - 384 T^{3} + 199 T^{4} - 1944 T^{5} - 83024 T^{6} - 574314 T^{7} + 970696 T^{8} - 35033154 T^{9} - 308932304 T^{10} - 441251064 T^{11} + 2755322359 T^{12} - 324324979584 T^{13} + 3915548451436 T^{14} - 18856457016126 T^{15} + 191707312997281 T^{16} )^{2}$$)
$67$ ($$( 1 - 2 T - 63 T^{2} - 134 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 - 16 T + 67 T^{2} )^{2}( 1 + 5 T + 67 T^{2} )^{2}$$)($$( 1 + 55 T^{2} - 1464 T^{4} + 246895 T^{6} + 20151121 T^{8} )^{2}$$)($$1 + 6 T - 219 T^{2} - 1306 T^{3} + 22971 T^{4} + 120540 T^{5} - 1928636 T^{6} - 7225512 T^{7} + 163638939 T^{8} + 309627182 T^{9} - 13816766313 T^{10} - 6348931902 T^{11} + 1027244050774 T^{12} - 425378437434 T^{13} - 62023463979057 T^{14} + 93124400139866 T^{15} + 3297508060100619 T^{16} - 9755345162129784 T^{17} - 174461292352891484 T^{18} + 730558176905634420 T^{19} + 9327780621153600411 T^{20} - 35531733921561200782 T^{21} -$$$$39\!\cdots\!31$$$$T^{22} +$$$$73\!\cdots\!98$$$$T^{23} +$$$$81\!\cdots\!61$$$$T^{24}$$)($$1 + 6 T - 219 T^{2} - 1306 T^{3} + 22971 T^{4} + 120540 T^{5} - 1928636 T^{6} - 7225512 T^{7} + 163638939 T^{8} + 309627182 T^{9} - 13816766313 T^{10} - 6348931902 T^{11} + 1027244050774 T^{12} - 425378437434 T^{13} - 62023463979057 T^{14} + 93124400139866 T^{15} + 3297508060100619 T^{16} - 9755345162129784 T^{17} - 174461292352891484 T^{18} + 730558176905634420 T^{19} + 9327780621153600411 T^{20} - 35531733921561200782 T^{21} -$$$$39\!\cdots\!31$$$$T^{22} +$$$$73\!\cdots\!98$$$$T^{23} +$$$$81\!\cdots\!61$$$$T^{24}$$)($$( 1 - 10 T - 126 T^{2} + 1216 T^{3} + 11501 T^{4} - 65334 T^{5} - 1130858 T^{6} + 905912 T^{7} + 100956192 T^{8} + 60696104 T^{9} - 5076421562 T^{10} - 19650049842 T^{11} + 231758042621 T^{12} + 1641752130112 T^{13} - 11397756153294 T^{14} - 60607116053230 T^{15} + 406067677556641 T^{16} )^{2}$$)($$( 1 + 10 T - 126 T^{2} - 1216 T^{3} + 11501 T^{4} + 65334 T^{5} - 1130858 T^{6} - 905912 T^{7} + 100956192 T^{8} - 60696104 T^{9} - 5076421562 T^{10} + 19650049842 T^{11} + 231758042621 T^{12} - 1641752130112 T^{13} - 11397756153294 T^{14} + 60607116053230 T^{15} + 406067677556641 T^{16} )^{2}$$)
$71$ ($$( 1 - 124 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 124 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 92 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 - 14 T^{2} + 5041 T^{4} )^{4}$$)($$1 - 460 T^{2} + 108410 T^{4} - 17445312 T^{6} + 2127087511 T^{8} - 205744741412 T^{10} + 16152616390624 T^{12} - 1037159241457892 T^{14} + 54052869288615991 T^{16} - 2234749420290428352 T^{18} + 70006140322352950010 T^{20} -$$$$14\!\cdots\!60$$$$T^{22} +$$$$16\!\cdots\!41$$$$T^{24}$$)($$1 - 460 T^{2} + 108410 T^{4} - 17445312 T^{6} + 2127087511 T^{8} - 205744741412 T^{10} + 16152616390624 T^{12} - 1037159241457892 T^{14} + 54052869288615991 T^{16} - 2234749420290428352 T^{18} + 70006140322352950010 T^{20} -$$$$14\!\cdots\!60$$$$T^{22} +$$$$16\!\cdots\!41$$$$T^{24}$$)($$( 1 - 224 T^{2} + 32620 T^{4} - 3468128 T^{6} + 273760102 T^{8} - 17482833248 T^{10} + 828929034220 T^{12} - 28694463598304 T^{14} + 645753531245761 T^{16} )^{2}$$)($$( 1 - 224 T^{2} + 32620 T^{4} - 3468128 T^{6} + 273760102 T^{8} - 17482833248 T^{10} + 828929034220 T^{12} - 28694463598304 T^{14} + 645753531245761 T^{16} )^{2}$$)
$73$ ($$( 1 - 15 T + 148 T^{2} - 1095 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 + 15 T + 148 T^{2} + 1095 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 - 7 T + 73 T^{2} )^{2}( 1 + 10 T + 73 T^{2} )^{2}$$)($$( 1 - 29 T^{2} - 4488 T^{4} - 154541 T^{6} + 28398241 T^{8} )^{2}$$)($$1 - 42 T + 1119 T^{2} - 22302 T^{3} + 373665 T^{4} - 5463480 T^{5} + 72046694 T^{6} - 868793688 T^{7} + 9715685451 T^{8} - 101446836642 T^{9} + 996453337299 T^{10} - 9237792089286 T^{11} + 81107402998458 T^{12} - 674358822517878 T^{13} + 5310099834466371 T^{14} - 39464544049960914 T^{15} + 275908376917691691 T^{16} - 1801071514770504984 T^{17} + 10903130693170338566 T^{18} - 60357240861116077560 T^{19} +$$$$30\!\cdots\!65$$$$T^{20} -$$$$13\!\cdots\!26$$$$T^{21} +$$$$48\!\cdots\!31$$$$T^{22} -$$$$13\!\cdots\!34$$$$T^{23} +$$$$22\!\cdots\!21$$$$T^{24}$$)($$1 - 42 T + 1119 T^{2} - 22302 T^{3} + 373665 T^{4} - 5463480 T^{5} + 72046694 T^{6} - 868793688 T^{7} + 9715685451 T^{8} - 101446836642 T^{9} + 996453337299 T^{10} - 9237792089286 T^{11} + 81107402998458 T^{12} - 674358822517878 T^{13} + 5310099834466371 T^{14} - 39464544049960914 T^{15} + 275908376917691691 T^{16} - 1801071514770504984 T^{17} + 10903130693170338566 T^{18} - 60357240861116077560 T^{19} +$$$$30\!\cdots\!65$$$$T^{20} -$$$$13\!\cdots\!26$$$$T^{21} +$$$$48\!\cdots\!31$$$$T^{22} -$$$$13\!\cdots\!34$$$$T^{23} +$$$$22\!\cdots\!21$$$$T^{24}$$)($$( 1 + 30 T + 592 T^{2} + 8760 T^{3} + 110131 T^{4} + 1245672 T^{5} + 12867868 T^{6} + 123737394 T^{7} + 1095916576 T^{8} + 9032829762 T^{9} + 68572868572 T^{10} + 484587584424 T^{11} + 3127526679571 T^{12} + 18160107154680 T^{13} + 89589861963088 T^{14} + 331421955572910 T^{15} + 806460091894081 T^{16} )^{2}$$)($$( 1 - 30 T + 592 T^{2} - 8760 T^{3} + 110131 T^{4} - 1245672 T^{5} + 12867868 T^{6} - 123737394 T^{7} + 1095916576 T^{8} - 9032829762 T^{9} + 68572868572 T^{10} - 484587584424 T^{11} + 3127526679571 T^{12} - 18160107154680 T^{13} + 89589861963088 T^{14} - 331421955572910 T^{15} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$( 1 - 17 T + 79 T^{2} )^{2}( 1 + 4 T + 79 T^{2} )^{2}$$)($$( 1 - 17 T + 79 T^{2} )^{2}( 1 + 4 T + 79 T^{2} )^{2}$$)($$( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 + 7 T - 30 T^{2} + 553 T^{3} + 6241 T^{4} )^{4}$$)($$1 - 18 T - 177 T^{2} + 3346 T^{3} + 40923 T^{4} - 454344 T^{5} - 6959816 T^{6} + 45880152 T^{7} + 889910193 T^{8} - 3054243566 T^{9} - 93786004551 T^{10} + 82100387598 T^{11} + 8378036930038 T^{12} + 6485930620242 T^{13} - 585318454402791 T^{14} - 1505861193537074 T^{15} + 34662074100075633 T^{16} + 141175815298692648 T^{17} - 1691843962334344136 T^{18} - 8725180824407424696 T^{19} + 62084643827806195803 T^{20} +$$$$40\!\cdots\!74$$$$T^{21} -$$$$16\!\cdots\!77$$$$T^{22} -$$$$13\!\cdots\!22$$$$T^{23} +$$$$59\!\cdots\!41$$$$T^{24}$$)($$1 - 18 T - 177 T^{2} + 3346 T^{3} + 40923 T^{4} - 454344 T^{5} - 6959816 T^{6} + 45880152 T^{7} + 889910193 T^{8} - 3054243566 T^{9} - 93786004551 T^{10} + 82100387598 T^{11} + 8378036930038 T^{12} + 6485930620242 T^{13} - 585318454402791 T^{14} - 1505861193537074 T^{15} + 34662074100075633 T^{16} + 141175815298692648 T^{17} - 1691843962334344136 T^{18} - 8725180824407424696 T^{19} + 62084643827806195803 T^{20} +$$$$40\!\cdots\!74$$$$T^{21} -$$$$16\!\cdots\!77$$$$T^{22} -$$$$13\!\cdots\!22$$$$T^{23} +$$$$59\!\cdots\!41$$$$T^{24}$$)($$( 1 + 20 T + 48 T^{2} - 932 T^{3} + 3095 T^{4} + 74406 T^{5} - 479348 T^{6} - 4107202 T^{7} + 15334692 T^{8} - 324468958 T^{9} - 2991610868 T^{10} + 36685059834 T^{11} + 120550500695 T^{12} - 2867816563868 T^{13} + 11668197865008 T^{14} + 384078179723180 T^{15} + 1517108809906561 T^{16} )^{2}$$)($$( 1 + 20 T + 48 T^{2} - 932 T^{3} + 3095 T^{4} + 74406 T^{5} - 479348 T^{6} - 4107202 T^{7} + 15334692 T^{8} - 324468958 T^{9} - 2991610868 T^{10} + 36685059834 T^{11} + 120550500695 T^{12} - 2867816563868 T^{13} + 11668197865008 T^{14} + 384078179723180 T^{15} + 1517108809906561 T^{16} )^{2}$$)
$83$ ($$( 1 + 112 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 + 112 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 + 112 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 - 58 T^{2} + 6889 T^{4} )^{4}$$)($$( 1 + 12 T + 300 T^{2} + 3204 T^{3} + 50769 T^{4} + 448284 T^{5} + 5149954 T^{6} + 37207572 T^{7} + 349747641 T^{8} + 1832005548 T^{9} + 14237496300 T^{10} + 47268487716 T^{11} + 326940373369 T^{12} )^{2}$$)($$( 1 - 12 T + 300 T^{2} - 3204 T^{3} + 50769 T^{4} - 448284 T^{5} + 5149954 T^{6} - 37207572 T^{7} + 349747641 T^{8} - 1832005548 T^{9} + 14237496300 T^{10} - 47268487716 T^{11} + 326940373369 T^{12} )^{2}$$)($$( 1 - 26 T^{2} + 11944 T^{4} - 144062 T^{6} + 103766434 T^{8} - 992443118 T^{10} + 566842186024 T^{12} - 8500449707594 T^{14} + 2252292232139041 T^{16} )^{2}$$)($$( 1 - 26 T^{2} + 11944 T^{4} - 144062 T^{6} + 103766434 T^{8} - 992443118 T^{10} + 566842186024 T^{12} - 8500449707594 T^{14} + 2252292232139041 T^{16} )^{2}$$)
$89$ ($$1 - 124 T^{2} + 7455 T^{4} - 982204 T^{6} + 62742241 T^{8}$$)($$1 - 124 T^{2} + 7455 T^{4} - 982204 T^{6} + 62742241 T^{8}$$)($$1 - 154 T^{2} + 15795 T^{4} - 1219834 T^{6} + 62742241 T^{8}$$)($$( 1 + 116 T^{2} + 5535 T^{4} + 918836 T^{6} + 62742241 T^{8} )^{2}$$)($$1 + 12 T - 282 T^{2} - 648 T^{3} + 81261 T^{4} - 90852 T^{5} - 10133890 T^{6} + 68061132 T^{7} + 906987750 T^{8} - 7493819688 T^{9} - 22404081252 T^{10} + 403242866028 T^{11} + 351385075713 T^{12} + 35888615076492 T^{13} - 177462727597092 T^{14} - 5282910571629672 T^{15} + 56906443994547750 T^{16} + 380057407254236268 T^{17} - 5036353734656768290 T^{18} - 4018505237928600708 T^{19} +$$$$31\!\cdots\!41$$$$T^{20} -$$$$22\!\cdots\!32$$$$T^{21} -$$$$87\!\cdots\!82$$$$T^{22} +$$$$33\!\cdots\!68$$$$T^{23} +$$$$24\!\cdots\!21$$$$T^{24}$$)($$1 - 12 T - 282 T^{2} + 648 T^{3} + 81261 T^{4} + 90852 T^{5} - 10133890 T^{6} - 68061132 T^{7} + 906987750 T^{8} + 7493819688 T^{9} - 22404081252 T^{10} - 403242866028 T^{11} + 351385075713 T^{12} - 35888615076492 T^{13} - 177462727597092 T^{14} + 5282910571629672 T^{15} + 56906443994547750 T^{16} - 380057407254236268 T^{17} - 5036353734656768290 T^{18} + 4018505237928600708 T^{19} +$$$$31\!\cdots\!41$$$$T^{20} +$$$$22\!\cdots\!32$$$$T^{21} -$$$$87\!\cdots\!82$$$$T^{22} -$$$$33\!\cdots\!68$$$$T^{23} +$$$$24\!\cdots\!21$$$$T^{24}$$)($$1 - 274 T^{2} + 39456 T^{4} - 2343500 T^{6} - 104444614 T^{8} + 34279054002 T^{10} - 2550582735800 T^{12} + 19462808419946 T^{14} + 9741779321027607 T^{16} + 154164905494392266 T^{18} -$$$$16\!\cdots\!00$$$$T^{20} +$$$$17\!\cdots\!22$$$$T^{22} -$$$$41\!\cdots\!34$$$$T^{24} -$$$$73\!\cdots\!00$$$$T^{26} +$$$$97\!\cdots\!76$$$$T^{28} -$$$$53\!\cdots\!34$$$$T^{30} +$$$$15\!\cdots\!61$$$$T^{32}$$)($$1 - 274 T^{2} + 39456 T^{4} - 2343500 T^{6} - 104444614 T^{8} + 34279054002 T^{10} - 2550582735800 T^{12} + 19462808419946 T^{14} + 9741779321027607 T^{16} + 154164905494392266 T^{18} -$$$$16\!\cdots\!00$$$$T^{20} +$$$$17\!\cdots\!22$$$$T^{22} -$$$$41\!\cdots\!34$$$$T^{24} -$$$$73\!\cdots\!00$$$$T^{26} +$$$$97\!\cdots\!76$$$$T^{28} -$$$$53\!\cdots\!34$$$$T^{30} +$$$$15\!\cdots\!61$$$$T^{32}$$)
$97$ ($$( 1 - 191 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 - 191 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 - 86 T^{2} + 9409 T^{4} )^{2}$$)($$( 1 - 166 T^{2} + 9409 T^{4} )^{4}$$)($$1 - 432 T^{2} + 85686 T^{4} - 12118408 T^{6} + 1564041327 T^{8} - 184182428136 T^{10} + 19001698536324 T^{12} - 1732972466331624 T^{14} + 138463454133595887 T^{16} - 10094294608307633032 T^{18} +$$$$67\!\cdots\!46$$$$T^{20} -$$$$31\!\cdots\!68$$$$T^{22} +$$$$69\!\cdots\!41$$$$T^{24}$$)($$1 - 432 T^{2} + 85686 T^{4} - 12118408 T^{6} + 1564041327 T^{8} - 184182428136 T^{10} + 19001698536324 T^{12} - 1732972466331624 T^{14} + 138463454133595887 T^{16} - 10094294608307633032 T^{18} +$$$$67\!\cdots\!46$$$$T^{20} -$$$$31\!\cdots\!68$$$$T^{22} +$$$$69\!\cdots\!41$$$$T^{24}$$)($$( 1 - 380 T^{2} + 72514 T^{4} - 9832688 T^{6} + 1060287259 T^{8} - 92515761392 T^{10} + 6419612282434 T^{12} - 316529361873020 T^{14} + 7837433594376961 T^{16} )^{2}$$)($$( 1 - 380 T^{2} + 72514 T^{4} - 9832688 T^{6} + 1060287259 T^{8} - 92515761392 T^{10} + 6419612282434 T^{12} - 316529361873020 T^{14} + 7837433594376961 T^{16} )^{2}$$)