# Properties

 Label 162.4 Level 162 Weight 4 Dimension 576 Nonzero newspaces 4 Newform subspaces 22 Sturm bound 5832 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$162\( 162 = 2 \cdot 3^{4}$$ \) Weight: $$k$$ = $$4$$ Nonzero newspaces: $$4$$ Newform subspaces: $$22$$ Sturm bound: $$5832$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(162))$$.

Total New Old
Modular forms 2295 576 1719
Cusp forms 2079 576 1503
Eisenstein series 216 0 216

## Trace form

 $$576q + 24q^{5} - 36q^{7} - 24q^{8} + O(q^{10})$$ $$576q + 24q^{5} - 36q^{7} - 24q^{8} - 36q^{10} + 51q^{11} + 90q^{13} + 132q^{14} - 204q^{17} - 414q^{18} - 810q^{19} - 480q^{20} - 216q^{21} + 126q^{22} + 1416q^{23} + 1242q^{25} + 2340q^{26} + 1404q^{27} + 576q^{28} + 1842q^{29} + 756q^{30} + 162q^{31} - 864q^{33} - 234q^{34} - 3732q^{35} - 1152q^{36} - 2070q^{37} - 1626q^{38} - 144q^{40} - 3873q^{41} - 2079q^{43} - 264q^{44} - 1026q^{45} - 504q^{46} + 432q^{47} + 1530q^{49} + 792q^{50} + 2961q^{51} + 360q^{52} + 6450q^{53} + 4320q^{55} + 528q^{56} + 2214q^{57} + 72q^{58} + 1911q^{59} - 972q^{61} - 1488q^{62} - 1998q^{63} + 576q^{64} - 2130q^{65} + 9360q^{66} - 7749q^{67} + 336q^{68} + 4122q^{69} - 1368q^{70} + 1200q^{71} - 576q^{72} + 1692q^{73} - 2388q^{74} - 9000q^{75} + 1980q^{76} - 9276q^{77} - 10656q^{78} + 8190q^{79} - 1920q^{80} - 11520q^{81} + 1044q^{82} - 5958q^{83} - 2448q^{84} + 6480q^{85} - 3582q^{86} - 10944q^{87} + 504q^{88} - 7224q^{89} + 1440q^{90} - 6876q^{91} + 1488q^{92} + 9090q^{93} - 6156q^{94} + 2172q^{95} + 2880q^{96} - 9441q^{97} + 10914q^{98} + 20754q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(162))$$

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
162.4.a $$\chi_{162}(1, \cdot)$$ 162.4.a.a 1 1
162.4.a.b 1
162.4.a.c 1
162.4.a.d 1
162.4.a.e 2
162.4.a.f 2
162.4.a.g 2
162.4.a.h 2
162.4.c $$\chi_{162}(55, \cdot)$$ 162.4.c.a 2 2
162.4.c.b 2
162.4.c.c 2
162.4.c.d 2
162.4.c.e 2
162.4.c.f 2
162.4.c.g 2
162.4.c.h 2
162.4.c.i 4
162.4.c.j 4
162.4.e $$\chi_{162}(19, \cdot)$$ 162.4.e.a 24 6
162.4.e.b 30
162.4.g $$\chi_{162}(7, \cdot)$$ 162.4.g.a 234 18
162.4.g.b 252

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(162))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(162)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 2 T$$)($$1 + 2 T$$)($$1 - 2 T$$)($$1 - 2 T$$)($$( 1 + 2 T )^{2}$$)($$( 1 + 2 T )^{2}$$)($$( 1 - 2 T )^{2}$$)($$( 1 - 2 T )^{2}$$)($$1 + 2 T + 4 T^{2}$$)($$1 + 2 T + 4 T^{2}$$)($$1 + 2 T + 4 T^{2}$$)($$1 + 2 T + 4 T^{2}$$)($$1 - 2 T + 4 T^{2}$$)($$1 - 2 T + 4 T^{2}$$)($$1 - 2 T + 4 T^{2}$$)($$1 - 2 T + 4 T^{2}$$)($$( 1 + 2 T + 4 T^{2} )^{2}$$)($$( 1 - 2 T + 4 T^{2} )^{2}$$)
$3$ 1
$5$ ($$1 - 9 T + 125 T^{2}$$)($$1 - 21 T + 125 T^{2}$$)($$1 + 21 T + 125 T^{2}$$)($$1 + 9 T + 125 T^{2}$$)($$1 + 12 T + 259 T^{2} + 1500 T^{3} + 15625 T^{4}$$)($$1 + 9 T + 34 T^{2} + 1125 T^{3} + 15625 T^{4}$$)($$1 - 9 T + 34 T^{2} - 1125 T^{3} + 15625 T^{4}$$)($$1 - 12 T + 259 T^{2} - 1500 T^{3} + 15625 T^{4}$$)($$1 + 12 T + 19 T^{2} + 1500 T^{3} + 15625 T^{4}$$)($$1 + 3 T - 116 T^{2} + 375 T^{3} + 15625 T^{4}$$)($$1 - 6 T - 89 T^{2} - 750 T^{3} + 15625 T^{4}$$)($$1 - 21 T + 316 T^{2} - 2625 T^{3} + 15625 T^{4}$$)($$1 + 21 T + 316 T^{2} + 2625 T^{3} + 15625 T^{4}$$)($$1 + 6 T - 89 T^{2} + 750 T^{3} + 15625 T^{4}$$)($$1 - 3 T - 116 T^{2} - 375 T^{3} + 15625 T^{4}$$)($$1 - 12 T + 19 T^{2} - 1500 T^{3} + 15625 T^{4}$$)($$1 + 12 T - 115 T^{2} + 108 T^{3} + 33456 T^{4} + 13500 T^{5} - 1796875 T^{6} + 23437500 T^{7} + 244140625 T^{8}$$)($$1 - 12 T - 115 T^{2} - 108 T^{3} + 33456 T^{4} - 13500 T^{5} - 1796875 T^{6} - 23437500 T^{7} + 244140625 T^{8}$$)
$7$ ($$1 + 31 T + 343 T^{2}$$)($$1 - 8 T + 343 T^{2}$$)($$1 - 8 T + 343 T^{2}$$)($$1 + 31 T + 343 T^{2}$$)($$1 - 16 T + 642 T^{2} - 5488 T^{3} + 117649 T^{4}$$)($$1 - 19 T + 540 T^{2} - 6517 T^{3} + 117649 T^{4}$$)($$1 - 19 T + 540 T^{2} - 6517 T^{3} + 117649 T^{4}$$)($$1 - 16 T + 642 T^{2} - 5488 T^{3} + 117649 T^{4}$$)($$( 1 - 35 T + 343 T^{2} )( 1 + 28 T + 343 T^{2} )$$)($$1 + 29 T + 498 T^{2} + 9947 T^{3} + 117649 T^{4}$$)($$1 - 16 T - 87 T^{2} - 5488 T^{3} + 117649 T^{4}$$)($$1 + 8 T - 279 T^{2} + 2744 T^{3} + 117649 T^{4}$$)($$1 + 8 T - 279 T^{2} + 2744 T^{3} + 117649 T^{4}$$)($$1 - 16 T - 87 T^{2} - 5488 T^{3} + 117649 T^{4}$$)($$1 + 29 T + 498 T^{2} + 9947 T^{3} + 117649 T^{4}$$)($$( 1 - 35 T + 343 T^{2} )( 1 + 28 T + 343 T^{2} )$$)($$1 + 16 T - 386 T^{2} - 704 T^{3} + 206707 T^{4} - 241472 T^{5} - 45412514 T^{6} + 645657712 T^{7} + 13841287201 T^{8}$$)($$1 + 16 T - 386 T^{2} - 704 T^{3} + 206707 T^{4} - 241472 T^{5} - 45412514 T^{6} + 645657712 T^{7} + 13841287201 T^{8}$$)
$11$ ($$1 - 15 T + 1331 T^{2}$$)($$1 - 36 T + 1331 T^{2}$$)($$1 + 36 T + 1331 T^{2}$$)($$1 + 15 T + 1331 T^{2}$$)($$1 + 36 T + 1258 T^{2} + 47916 T^{3} + 1771561 T^{4}$$)($$1 + 24 T + 1861 T^{2} + 31944 T^{3} + 1771561 T^{4}$$)($$1 - 24 T + 1861 T^{2} - 31944 T^{3} + 1771561 T^{4}$$)($$1 - 36 T + 1258 T^{2} - 47916 T^{3} + 1771561 T^{4}$$)($$1 + 60 T + 2269 T^{2} + 79860 T^{3} + 1771561 T^{4}$$)($$1 - 57 T + 1918 T^{2} - 75867 T^{3} + 1771561 T^{4}$$)($$1 - 12 T - 1187 T^{2} - 15972 T^{3} + 1771561 T^{4}$$)($$1 - 36 T - 35 T^{2} - 47916 T^{3} + 1771561 T^{4}$$)($$1 + 36 T - 35 T^{2} + 47916 T^{3} + 1771561 T^{4}$$)($$1 + 12 T - 1187 T^{2} + 15972 T^{3} + 1771561 T^{4}$$)($$1 + 57 T + 1918 T^{2} + 75867 T^{3} + 1771561 T^{4}$$)($$1 - 60 T + 2269 T^{2} - 79860 T^{3} + 1771561 T^{4}$$)($$1 + 36 T + 38 T^{2} - 50544 T^{3} - 1913973 T^{4} - 67274064 T^{5} + 67319318 T^{6} + 84886116876 T^{7} + 3138428376721 T^{8}$$)($$1 - 36 T + 38 T^{2} + 50544 T^{3} - 1913973 T^{4} + 67274064 T^{5} + 67319318 T^{6} - 84886116876 T^{7} + 3138428376721 T^{8}$$)
$13$ ($$1 + 37 T + 2197 T^{2}$$)($$1 + 49 T + 2197 T^{2}$$)($$1 + 49 T + 2197 T^{2}$$)($$1 + 37 T + 2197 T^{2}$$)($$1 - 10 T - 873 T^{2} - 21970 T^{3} + 4826809 T^{4}$$)($$1 - 61 T + 5088 T^{2} - 134017 T^{3} + 4826809 T^{4}$$)($$1 - 61 T + 5088 T^{2} - 134017 T^{3} + 4826809 T^{4}$$)($$1 - 10 T - 873 T^{2} - 21970 T^{3} + 4826809 T^{4}$$)($$1 - 79 T + 4044 T^{2} - 173563 T^{3} + 4826809 T^{4}$$)($$1 + 20 T - 1797 T^{2} + 43940 T^{3} + 4826809 T^{4}$$)($$1 + 38 T - 753 T^{2} + 83486 T^{3} + 4826809 T^{4}$$)($$1 - 49 T + 204 T^{2} - 107653 T^{3} + 4826809 T^{4}$$)($$1 - 49 T + 204 T^{2} - 107653 T^{3} + 4826809 T^{4}$$)($$1 + 38 T - 753 T^{2} + 83486 T^{3} + 4826809 T^{4}$$)($$1 + 20 T - 1797 T^{2} + 43940 T^{3} + 4826809 T^{4}$$)($$1 - 79 T + 4044 T^{2} - 173563 T^{3} + 4826809 T^{4}$$)($$1 + 10 T + 973 T^{2} - 52670 T^{3} - 4284380 T^{4} - 115715990 T^{5} + 4696485157 T^{6} + 106044993730 T^{7} + 23298085122481 T^{8}$$)($$1 + 10 T + 973 T^{2} - 52670 T^{3} - 4284380 T^{4} - 115715990 T^{5} + 4696485157 T^{6} + 106044993730 T^{7} + 23298085122481 T^{8}$$)
$17$ ($$1 - 42 T + 4913 T^{2}$$)($$1 - 21 T + 4913 T^{2}$$)($$1 + 21 T + 4913 T^{2}$$)($$1 + 42 T + 4913 T^{2}$$)($$1 + 120 T + 10159 T^{2} + 589560 T^{3} + 24137569 T^{4}$$)($$1 - 3 T + 9592 T^{2} - 14739 T^{3} + 24137569 T^{4}$$)($$1 + 3 T + 9592 T^{2} + 14739 T^{3} + 24137569 T^{4}$$)($$1 - 120 T + 10159 T^{2} - 589560 T^{3} + 24137569 T^{4}$$)($$( 1 + 108 T + 4913 T^{2} )^{2}$$)($$( 1 + 72 T + 4913 T^{2} )^{2}$$)($$( 1 - 126 T + 4913 T^{2} )^{2}$$)($$( 1 + 21 T + 4913 T^{2} )^{2}$$)($$( 1 - 21 T + 4913 T^{2} )^{2}$$)($$( 1 + 126 T + 4913 T^{2} )^{2}$$)($$( 1 - 72 T + 4913 T^{2} )^{2}$$)($$( 1 - 108 T + 4913 T^{2} )^{2}$$)($$( 1 - 120 T + 10159 T^{2} - 589560 T^{3} + 24137569 T^{4} )^{2}$$)($$( 1 + 120 T + 10159 T^{2} + 589560 T^{3} + 24137569 T^{4} )^{2}$$)
$19$ ($$1 + 28 T + 6859 T^{2}$$)($$1 + 112 T + 6859 T^{2}$$)($$1 + 112 T + 6859 T^{2}$$)($$1 + 28 T + 6859 T^{2}$$)($$1 + 8 T + 666 T^{2} + 54872 T^{3} + 47045881 T^{4}$$)($$1 - 133 T + 12234 T^{2} - 912247 T^{3} + 47045881 T^{4}$$)($$1 - 133 T + 12234 T^{2} - 912247 T^{3} + 47045881 T^{4}$$)($$1 + 8 T + 666 T^{2} + 54872 T^{3} + 47045881 T^{4}$$)($$( 1 - 11 T + 6859 T^{2} )^{2}$$)($$( 1 + 106 T + 6859 T^{2} )^{2}$$)($$( 1 - 20 T + 6859 T^{2} )^{2}$$)($$( 1 + 112 T + 6859 T^{2} )^{2}$$)($$( 1 + 112 T + 6859 T^{2} )^{2}$$)($$( 1 - 20 T + 6859 T^{2} )^{2}$$)($$( 1 + 106 T + 6859 T^{2} )^{2}$$)($$( 1 - 11 T + 6859 T^{2} )^{2}$$)($$( 1 + 8 T + 666 T^{2} + 54872 T^{3} + 47045881 T^{4} )^{2}$$)($$( 1 + 8 T + 666 T^{2} + 54872 T^{3} + 47045881 T^{4} )^{2}$$)
$23$ ($$1 + 195 T + 12167 T^{2}$$)($$1 - 180 T + 12167 T^{2}$$)($$1 + 180 T + 12167 T^{2}$$)($$1 - 195 T + 12167 T^{2}$$)($$1 + 180 T + 32002 T^{2} + 2190060 T^{3} + 148035889 T^{4}$$)($$1 - 69 T + 25288 T^{2} - 839523 T^{3} + 148035889 T^{4}$$)($$1 + 69 T + 25288 T^{2} + 839523 T^{3} + 148035889 T^{4}$$)($$1 - 180 T + 32002 T^{2} - 2190060 T^{3} + 148035889 T^{4}$$)($$1 - 132 T + 5257 T^{2} - 1606044 T^{3} + 148035889 T^{4}$$)($$1 + 174 T + 18109 T^{2} + 2117058 T^{3} + 148035889 T^{4}$$)($$1 - 168 T + 16057 T^{2} - 2044056 T^{3} + 148035889 T^{4}$$)($$1 - 180 T + 20233 T^{2} - 2190060 T^{3} + 148035889 T^{4}$$)($$1 + 180 T + 20233 T^{2} + 2190060 T^{3} + 148035889 T^{4}$$)($$1 + 168 T + 16057 T^{2} + 2044056 T^{3} + 148035889 T^{4}$$)($$1 - 174 T + 18109 T^{2} - 2117058 T^{3} + 148035889 T^{4}$$)($$1 + 132 T + 5257 T^{2} + 1606044 T^{3} + 148035889 T^{4}$$)($$1 + 180 T + 398 T^{2} + 1380240 T^{3} + 481881315 T^{4} + 16793380080 T^{5} + 58918283822 T^{6} + 324207479063340 T^{7} + 21914624432020321 T^{8}$$)($$1 - 180 T + 398 T^{2} - 1380240 T^{3} + 481881315 T^{4} - 16793380080 T^{5} + 58918283822 T^{6} - 324207479063340 T^{7} + 21914624432020321 T^{8}$$)
$29$ ($$1 + 111 T + 24389 T^{2}$$)($$1 + 135 T + 24389 T^{2}$$)($$1 - 135 T + 24389 T^{2}$$)($$1 - 111 T + 24389 T^{2}$$)($$1 + 324 T + 73699 T^{2} + 7902036 T^{3} + 594823321 T^{4}$$)($$1 - 237 T + 51244 T^{2} - 5780193 T^{3} + 594823321 T^{4}$$)($$1 + 237 T + 51244 T^{2} + 5780193 T^{3} + 594823321 T^{4}$$)($$1 - 324 T + 73699 T^{2} - 7902036 T^{3} + 594823321 T^{4}$$)($$1 + 96 T - 15173 T^{2} + 2341344 T^{3} + 594823321 T^{4}$$)($$1 - 210 T + 19711 T^{2} - 5121690 T^{3} + 594823321 T^{4}$$)($$1 - 30 T - 23489 T^{2} - 731670 T^{3} + 594823321 T^{4}$$)($$1 + 135 T - 6164 T^{2} + 3292515 T^{3} + 594823321 T^{4}$$)($$1 - 135 T - 6164 T^{2} - 3292515 T^{3} + 594823321 T^{4}$$)($$1 + 30 T - 23489 T^{2} + 731670 T^{3} + 594823321 T^{4}$$)($$1 + 210 T + 19711 T^{2} + 5121690 T^{3} + 594823321 T^{4}$$)($$1 - 96 T - 15173 T^{2} - 2341344 T^{3} + 594823321 T^{4}$$)($$1 + 324 T + 31277 T^{2} + 8074404 T^{3} + 2276459616 T^{4} + 196926639156 T^{5} + 18604289010917 T^{6} + 4700315296181556 T^{7} + 353814783205469041 T^{8}$$)($$1 - 324 T + 31277 T^{2} - 8074404 T^{3} + 2276459616 T^{4} - 196926639156 T^{5} + 18604289010917 T^{6} - 4700315296181556 T^{7} + 353814783205469041 T^{8}$$)
$31$ ($$1 + 205 T + 29791 T^{2}$$)($$1 - 308 T + 29791 T^{2}$$)($$1 - 308 T + 29791 T^{2}$$)($$1 + 205 T + 29791 T^{2}$$)($$1 + 248 T + 39966 T^{2} + 7388168 T^{3} + 887503681 T^{4}$$)($$1 - 211 T + 68586 T^{2} - 6285901 T^{3} + 887503681 T^{4}$$)($$1 - 211 T + 68586 T^{2} - 6285901 T^{3} + 887503681 T^{4}$$)($$1 + 248 T + 39966 T^{2} + 7388168 T^{3} + 887503681 T^{4}$$)($$1 + 20 T - 29391 T^{2} + 595820 T^{3} + 887503681 T^{4}$$)($$1 + 47 T - 27582 T^{2} + 1400177 T^{3} + 887503681 T^{4}$$)($$1 - 88 T - 22047 T^{2} - 2621608 T^{3} + 887503681 T^{4}$$)($$( 1 + 19 T + 29791 T^{2} )( 1 + 289 T + 29791 T^{2} )$$)($$( 1 + 19 T + 29791 T^{2} )( 1 + 289 T + 29791 T^{2} )$$)($$1 - 88 T - 22047 T^{2} - 2621608 T^{3} + 887503681 T^{4}$$)($$1 + 47 T - 27582 T^{2} + 1400177 T^{3} + 887503681 T^{4}$$)($$1 + 20 T - 29391 T^{2} + 595820 T^{3} + 887503681 T^{4}$$)($$1 - 248 T + 21538 T^{2} + 4864768 T^{3} - 1122488189 T^{4} + 144926303488 T^{5} + 19115054281378 T^{6} - 6557026295846408 T^{7} + 787662783788549761 T^{8}$$)($$1 - 248 T + 21538 T^{2} + 4864768 T^{3} - 1122488189 T^{4} + 144926303488 T^{5} + 19115054281378 T^{6} - 6557026295846408 T^{7} + 787662783788549761 T^{8}$$)
$37$ ($$1 + 166 T + 50653 T^{2}$$)($$1 + T + 50653 T^{2}$$)($$1 + T + 50653 T^{2}$$)($$1 + 166 T + 50653 T^{2}$$)($$1 + 434 T + 148287 T^{2} + 21983402 T^{3} + 2565726409 T^{4}$$)($$1 - 262 T + 72162 T^{2} - 13271086 T^{3} + 2565726409 T^{4}$$)($$1 - 262 T + 72162 T^{2} - 13271086 T^{3} + 2565726409 T^{4}$$)($$1 + 434 T + 148287 T^{2} + 21983402 T^{3} + 2565726409 T^{4}$$)($$( 1 + 169 T + 50653 T^{2} )^{2}$$)($$( 1 - 2 T + 50653 T^{2} )^{2}$$)($$( 1 - 254 T + 50653 T^{2} )^{2}$$)($$( 1 + T + 50653 T^{2} )^{2}$$)($$( 1 + T + 50653 T^{2} )^{2}$$)($$( 1 - 254 T + 50653 T^{2} )^{2}$$)($$( 1 - 2 T + 50653 T^{2} )^{2}$$)($$( 1 + 169 T + 50653 T^{2} )^{2}$$)($$( 1 + 434 T + 148287 T^{2} + 21983402 T^{3} + 2565726409 T^{4} )^{2}$$)($$( 1 + 434 T + 148287 T^{2} + 21983402 T^{3} + 2565726409 T^{4} )^{2}$$)
$41$ ($$1 - 261 T + 68921 T^{2}$$)($$1 + 42 T + 68921 T^{2}$$)($$1 - 42 T + 68921 T^{2}$$)($$1 + 261 T + 68921 T^{2}$$)($$1 + 192 T + 94786 T^{2} + 13232832 T^{3} + 4750104241 T^{4}$$)($$1 - 468 T + 191653 T^{2} - 32255028 T^{3} + 4750104241 T^{4}$$)($$1 + 468 T + 191653 T^{2} + 32255028 T^{3} + 4750104241 T^{4}$$)($$1 - 192 T + 94786 T^{2} - 13232832 T^{3} + 4750104241 T^{4}$$)($$1 + 192 T - 32057 T^{2} + 13232832 T^{3} + 4750104241 T^{4}$$)($$1 - 6 T - 68885 T^{2} - 413526 T^{3} + 4750104241 T^{4}$$)($$1 - 42 T - 67157 T^{2} - 2894682 T^{3} + 4750104241 T^{4}$$)($$1 + 42 T - 67157 T^{2} + 2894682 T^{3} + 4750104241 T^{4}$$)($$1 - 42 T - 67157 T^{2} - 2894682 T^{3} + 4750104241 T^{4}$$)($$1 + 42 T - 67157 T^{2} + 2894682 T^{3} + 4750104241 T^{4}$$)($$1 + 6 T - 68885 T^{2} + 413526 T^{3} + 4750104241 T^{4}$$)($$1 - 192 T - 32057 T^{2} - 13232832 T^{3} + 4750104241 T^{4}$$)($$1 + 192 T - 57922 T^{2} - 8266752 T^{3} + 1693577811 T^{4} - 569752814592 T^{5} - 275135537847202 T^{6} + 62857331403640512 T^{7} + 22563490300366186081 T^{8}$$)($$1 - 192 T - 57922 T^{2} + 8266752 T^{3} + 1693577811 T^{4} + 569752814592 T^{5} - 275135537847202 T^{6} - 62857331403640512 T^{7} + 22563490300366186081 T^{8}$$)
$43$ ($$1 + 43 T + 79507 T^{2}$$)($$1 - 20 T + 79507 T^{2}$$)($$1 - 20 T + 79507 T^{2}$$)($$1 + 43 T + 79507 T^{2}$$)($$1 + 608 T + 250458 T^{2} + 48340256 T^{3} + 6321363049 T^{4}$$)($$1 + 86 T + 24783 T^{2} + 6837602 T^{3} + 6321363049 T^{4}$$)($$1 + 86 T + 24783 T^{2} + 6837602 T^{3} + 6321363049 T^{4}$$)($$1 + 608 T + 250458 T^{2} + 48340256 T^{3} + 6321363049 T^{4}$$)($$1 + 488 T + 158637 T^{2} + 38799416 T^{3} + 6321363049 T^{4}$$)($$1 + 218 T - 31983 T^{2} + 17332526 T^{3} + 6321363049 T^{4}$$)($$1 - 52 T - 76803 T^{2} - 4134364 T^{3} + 6321363049 T^{4}$$)($$1 + 20 T - 79107 T^{2} + 1590140 T^{3} + 6321363049 T^{4}$$)($$1 + 20 T - 79107 T^{2} + 1590140 T^{3} + 6321363049 T^{4}$$)($$1 - 52 T - 76803 T^{2} - 4134364 T^{3} + 6321363049 T^{4}$$)($$1 + 218 T - 31983 T^{2} + 17332526 T^{3} + 6321363049 T^{4}$$)($$1 + 488 T + 158637 T^{2} + 38799416 T^{3} + 6321363049 T^{4}$$)($$1 - 608 T + 119206 T^{2} - 55597952 T^{3} + 27016971067 T^{4} - 4420426369664 T^{5} + 753544403619094 T^{6} - 305576308057600544 T^{7} + 39959630797262576401 T^{8}$$)($$1 - 608 T + 119206 T^{2} - 55597952 T^{3} + 27016971067 T^{4} - 4420426369664 T^{5} + 753544403619094 T^{6} - 305576308057600544 T^{7} + 39959630797262576401 T^{8}$$)
$47$ ($$1 + 177 T + 103823 T^{2}$$)($$1 - 84 T + 103823 T^{2}$$)($$1 + 84 T + 103823 T^{2}$$)($$1 - 177 T + 103823 T^{2}$$)($$1 - 384 T + 209518 T^{2} - 39868032 T^{3} + 10779215329 T^{4}$$)($$1 - 483 T + 212812 T^{2} - 50146509 T^{3} + 10779215329 T^{4}$$)($$1 + 483 T + 212812 T^{2} + 50146509 T^{3} + 10779215329 T^{4}$$)($$1 + 384 T + 209518 T^{2} + 39868032 T^{3} + 10779215329 T^{4}$$)($$1 + 204 T - 62207 T^{2} + 21179892 T^{3} + 10779215329 T^{4}$$)($$1 + 474 T + 120853 T^{2} + 49212102 T^{3} + 10779215329 T^{4}$$)($$1 + 96 T - 94607 T^{2} + 9967008 T^{3} + 10779215329 T^{4}$$)($$1 - 84 T - 96767 T^{2} - 8721132 T^{3} + 10779215329 T^{4}$$)($$1 + 84 T - 96767 T^{2} + 8721132 T^{3} + 10779215329 T^{4}$$)($$1 - 96 T - 94607 T^{2} - 9967008 T^{3} + 10779215329 T^{4}$$)($$1 - 474 T + 120853 T^{2} - 49212102 T^{3} + 10779215329 T^{4}$$)($$1 - 204 T - 62207 T^{2} - 21179892 T^{3} + 10779215329 T^{4}$$)($$1 - 384 T - 62062 T^{2} - 718848 T^{3} + 17809252707 T^{4} - 74632955904 T^{5} - 668979661748398 T^{6} - 429746101671462528 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$)($$1 + 384 T - 62062 T^{2} + 718848 T^{3} + 17809252707 T^{4} + 74632955904 T^{5} - 668979661748398 T^{6} + 429746101671462528 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$)
$53$ ($$1 + 114 T + 148877 T^{2}$$)($$1 + 174 T + 148877 T^{2}$$)($$1 - 174 T + 148877 T^{2}$$)($$1 - 114 T + 148877 T^{2}$$)($$1 - 408 T + 183418 T^{2} - 60741816 T^{3} + 22164361129 T^{4}$$)($$1 - 150 T + 257074 T^{2} - 22331550 T^{3} + 22164361129 T^{4}$$)($$1 + 150 T + 257074 T^{2} + 22331550 T^{3} + 22164361129 T^{4}$$)($$1 + 408 T + 183418 T^{2} + 60741816 T^{3} + 22164361129 T^{4}$$)($$( 1 - 360 T + 148877 T^{2} )^{2}$$)($$( 1 - 81 T + 148877 T^{2} )^{2}$$)($$( 1 + 198 T + 148877 T^{2} )^{2}$$)($$( 1 - 174 T + 148877 T^{2} )^{2}$$)($$( 1 + 174 T + 148877 T^{2} )^{2}$$)($$( 1 - 198 T + 148877 T^{2} )^{2}$$)($$( 1 + 81 T + 148877 T^{2} )^{2}$$)($$( 1 + 360 T + 148877 T^{2} )^{2}$$)($$( 1 + 408 T + 183418 T^{2} + 60741816 T^{3} + 22164361129 T^{4} )^{2}$$)($$( 1 - 408 T + 183418 T^{2} - 60741816 T^{3} + 22164361129 T^{4} )^{2}$$)
$59$ ($$1 + 159 T + 205379 T^{2}$$)($$1 - 504 T + 205379 T^{2}$$)($$1 + 504 T + 205379 T^{2}$$)($$1 - 159 T + 205379 T^{2}$$)($$1 - 1008 T + 637126 T^{2} - 207022032 T^{3} + 42180533641 T^{4}$$)($$1 - 168 T + 416869 T^{2} - 34503672 T^{3} + 42180533641 T^{4}$$)($$1 + 168 T + 416869 T^{2} + 34503672 T^{3} + 42180533641 T^{4}$$)($$1 + 1008 T + 637126 T^{2} + 207022032 T^{3} + 42180533641 T^{4}$$)($$1 + 156 T - 181043 T^{2} + 32039124 T^{3} + 42180533641 T^{4}$$)($$1 + 84 T - 198323 T^{2} + 17251836 T^{3} + 42180533641 T^{4}$$)($$1 + 660 T + 230221 T^{2} + 135550140 T^{3} + 42180533641 T^{4}$$)($$1 - 504 T + 48637 T^{2} - 103511016 T^{3} + 42180533641 T^{4}$$)($$1 + 504 T + 48637 T^{2} + 103511016 T^{3} + 42180533641 T^{4}$$)($$1 - 660 T + 230221 T^{2} - 135550140 T^{3} + 42180533641 T^{4}$$)($$1 - 84 T - 198323 T^{2} - 17251836 T^{3} + 42180533641 T^{4}$$)($$1 - 156 T - 181043 T^{2} - 32039124 T^{3} + 42180533641 T^{4}$$)($$1 - 1008 T + 378938 T^{2} - 228178944 T^{3} + 155070797979 T^{4} - 46863163339776 T^{5} + 15983807056853258 T^{6} - 8732299785204178512 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$)($$1 + 1008 T + 378938 T^{2} + 228178944 T^{3} + 155070797979 T^{4} + 46863163339776 T^{5} + 15983807056853258 T^{6} + 8732299785204178512 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$)
$61$ ($$1 - 191 T + 226981 T^{2}$$)($$1 + 385 T + 226981 T^{2}$$)($$1 + 385 T + 226981 T^{2}$$)($$1 - 191 T + 226981 T^{2}$$)($$1 - 742 T + 582855 T^{2} - 168419902 T^{3} + 51520374361 T^{4}$$)($$1 + 1049 T + 675906 T^{2} + 238103069 T^{3} + 51520374361 T^{4}$$)($$1 + 1049 T + 675906 T^{2} + 238103069 T^{3} + 51520374361 T^{4}$$)($$1 - 742 T + 582855 T^{2} - 168419902 T^{3} + 51520374361 T^{4}$$)($$1 + 83 T - 220092 T^{2} + 18839423 T^{3} + 51520374361 T^{4}$$)($$1 + 56 T - 223845 T^{2} + 12710936 T^{3} + 51520374361 T^{4}$$)($$1 - 538 T + 62463 T^{2} - 122115778 T^{3} + 51520374361 T^{4}$$)($$1 - 385 T - 78756 T^{2} - 87387685 T^{3} + 51520374361 T^{4}$$)($$1 - 385 T - 78756 T^{2} - 87387685 T^{3} + 51520374361 T^{4}$$)($$1 - 538 T + 62463 T^{2} - 122115778 T^{3} + 51520374361 T^{4}$$)($$1 + 56 T - 223845 T^{2} + 12710936 T^{3} + 51520374361 T^{4}$$)($$1 + 83 T - 220092 T^{2} + 18839423 T^{3} + 51520374361 T^{4}$$)($$1 + 742 T - 32291 T^{2} + 95638606 T^{3} + 163232009380 T^{4} + 21708146428486 T^{5} - 1663644408491051 T^{6} + 8677056400882932622 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$)($$1 + 742 T - 32291 T^{2} + 95638606 T^{3} + 163232009380 T^{4} + 21708146428486 T^{5} - 1663644408491051 T^{6} + 8677056400882932622 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$)
$67$ ($$1 + 421 T + 300763 T^{2}$$)($$1 - 272 T + 300763 T^{2}$$)($$1 - 272 T + 300763 T^{2}$$)($$1 + 421 T + 300763 T^{2}$$)($$1 + 104 T + 90042 T^{2} + 31279352 T^{3} + 90458382169 T^{4}$$)($$1 + 1166 T + 907395 T^{2} + 350689658 T^{3} + 90458382169 T^{4}$$)($$1 + 1166 T + 907395 T^{2} + 350689658 T^{3} + 90458382169 T^{4}$$)($$1 + 104 T + 90042 T^{2} + 31279352 T^{3} + 90458382169 T^{4}$$)($$1 + 47 T - 298554 T^{2} + 14135861 T^{3} + 90458382169 T^{4}$$)($$1 - 142 T - 280599 T^{2} - 42708346 T^{3} + 90458382169 T^{4}$$)($$1 + 884 T + 480693 T^{2} + 265874492 T^{3} + 90458382169 T^{4}$$)($$1 + 272 T - 226779 T^{2} + 81807536 T^{3} + 90458382169 T^{4}$$)($$1 + 272 T - 226779 T^{2} + 81807536 T^{3} + 90458382169 T^{4}$$)($$1 + 884 T + 480693 T^{2} + 265874492 T^{3} + 90458382169 T^{4}$$)($$1 - 142 T - 280599 T^{2} - 42708346 T^{3} + 90458382169 T^{4}$$)($$1 + 47 T - 298554 T^{2} + 14135861 T^{3} + 90458382169 T^{4}$$)($$1 - 104 T - 79226 T^{2} + 53194336 T^{3} - 85603873013 T^{4} + 15998888078368 T^{5} - 7166655785721194 T^{6} - 2829479577214674488 T^{7} +$$$$81\!\cdots\!61$$$$T^{8}$$)($$1 - 104 T - 79226 T^{2} + 53194336 T^{3} - 85603873013 T^{4} + 15998888078368 T^{5} - 7166655785721194 T^{6} - 2829479577214674488 T^{7} +$$$$81\!\cdots\!61$$$$T^{8}$$)
$71$ ($$1 + 156 T + 357911 T^{2}$$)($$1 + 888 T + 357911 T^{2}$$)($$1 - 888 T + 357911 T^{2}$$)($$1 - 156 T + 357911 T^{2}$$)($$1 - 1140 T + 1038994 T^{2} - 408018540 T^{3} + 128100283921 T^{4}$$)($$1 + 312 T + 498238 T^{2} + 111668232 T^{3} + 128100283921 T^{4}$$)($$1 - 312 T + 498238 T^{2} - 111668232 T^{3} + 128100283921 T^{4}$$)($$1 + 1140 T + 1038994 T^{2} + 408018540 T^{3} + 128100283921 T^{4}$$)($$( 1 - 216 T + 357911 T^{2} )^{2}$$)($$( 1 - 360 T + 357911 T^{2} )^{2}$$)($$( 1 + 792 T + 357911 T^{2} )^{2}$$)($$( 1 - 888 T + 357911 T^{2} )^{2}$$)($$( 1 + 888 T + 357911 T^{2} )^{2}$$)($$( 1 - 792 T + 357911 T^{2} )^{2}$$)($$( 1 + 360 T + 357911 T^{2} )^{2}$$)($$( 1 + 216 T + 357911 T^{2} )^{2}$$)($$( 1 + 1140 T + 1038994 T^{2} + 408018540 T^{3} + 128100283921 T^{4} )^{2}$$)($$( 1 - 1140 T + 1038994 T^{2} - 408018540 T^{3} + 128100283921 T^{4} )^{2}$$)
$73$ ($$1 - 182 T + 389017 T^{2}$$)($$1 - 371 T + 389017 T^{2}$$)($$1 - 371 T + 389017 T^{2}$$)($$1 - 182 T + 389017 T^{2}$$)($$1 - 850 T + 709827 T^{2} - 330664450 T^{3} + 151334226289 T^{4}$$)($$1 + 311 T + 698028 T^{2} + 120984287 T^{3} + 151334226289 T^{4}$$)($$1 + 311 T + 698028 T^{2} + 120984287 T^{3} + 151334226289 T^{4}$$)($$1 - 850 T + 709827 T^{2} - 330664450 T^{3} + 151334226289 T^{4}$$)($$( 1 + 511 T + 389017 T^{2} )^{2}$$)($$( 1 + 1159 T + 389017 T^{2} )^{2}$$)($$( 1 - 218 T + 389017 T^{2} )^{2}$$)($$( 1 - 371 T + 389017 T^{2} )^{2}$$)($$( 1 - 371 T + 389017 T^{2} )^{2}$$)($$( 1 - 218 T + 389017 T^{2} )^{2}$$)($$( 1 + 1159 T + 389017 T^{2} )^{2}$$)($$( 1 + 511 T + 389017 T^{2} )^{2}$$)($$( 1 - 850 T + 709827 T^{2} - 330664450 T^{3} + 151334226289 T^{4} )^{2}$$)($$( 1 - 850 T + 709827 T^{2} - 330664450 T^{3} + 151334226289 T^{4} )^{2}$$)
$79$ ($$1 - 1133 T + 493039 T^{2}$$)($$1 + 652 T + 493039 T^{2}$$)($$1 + 652 T + 493039 T^{2}$$)($$1 - 1133 T + 493039 T^{2}$$)($$1 + 440 T + 966978 T^{2} + 216937160 T^{3} + 243087455521 T^{4}$$)($$1 - 349 T + 976602 T^{2} - 172070611 T^{3} + 243087455521 T^{4}$$)($$1 - 349 T + 976602 T^{2} - 172070611 T^{3} + 243087455521 T^{4}$$)($$1 + 440 T + 966978 T^{2} + 216937160 T^{3} + 243087455521 T^{4}$$)($$1 - 529 T - 213198 T^{2} - 260817631 T^{3} + 243087455521 T^{4}$$)($$1 - 160 T - 467439 T^{2} - 78886240 T^{3} + 243087455521 T^{4}$$)($$1 - 520 T - 222639 T^{2} - 256380280 T^{3} + 243087455521 T^{4}$$)($$1 - 652 T - 67935 T^{2} - 321461428 T^{3} + 243087455521 T^{4}$$)($$1 - 652 T - 67935 T^{2} - 321461428 T^{3} + 243087455521 T^{4}$$)($$1 - 520 T - 222639 T^{2} - 256380280 T^{3} + 243087455521 T^{4}$$)($$1 - 160 T - 467439 T^{2} - 78886240 T^{3} + 243087455521 T^{4}$$)($$1 - 529 T - 213198 T^{2} - 260817631 T^{3} + 243087455521 T^{4}$$)($$1 - 440 T - 773378 T^{2} + 8404000 T^{3} + 596506646563 T^{4} + 4143499756000 T^{5} - 187998490175919938 T^{6} - 52734702232352060360 T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$)($$1 - 440 T - 773378 T^{2} + 8404000 T^{3} + 596506646563 T^{4} + 4143499756000 T^{5} - 187998490175919938 T^{6} - 52734702232352060360 T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$)
$83$ ($$1 - 1083 T + 571787 T^{2}$$)($$1 - 84 T + 571787 T^{2}$$)($$1 + 84 T + 571787 T^{2}$$)($$1 + 1083 T + 571787 T^{2}$$)($$1 + 264 T + 1063798 T^{2} + 150951768 T^{3} + 326940373369 T^{4}$$)($$1 - 1221 T + 1412098 T^{2} - 698151927 T^{3} + 326940373369 T^{4}$$)($$1 + 1221 T + 1412098 T^{2} + 698151927 T^{3} + 326940373369 T^{4}$$)($$1 - 264 T + 1063798 T^{2} - 150951768 T^{3} + 326940373369 T^{4}$$)($$1 - 1128 T + 700597 T^{2} - 644975736 T^{3} + 326940373369 T^{4}$$)($$1 + 735 T - 31562 T^{2} + 420263445 T^{3} + 326940373369 T^{4}$$)($$1 + 492 T - 329723 T^{2} + 281319204 T^{3} + 326940373369 T^{4}$$)($$1 - 84 T - 564731 T^{2} - 48030108 T^{3} + 326940373369 T^{4}$$)($$1 + 84 T - 564731 T^{2} + 48030108 T^{3} + 326940373369 T^{4}$$)($$1 - 492 T - 329723 T^{2} - 281319204 T^{3} + 326940373369 T^{4}$$)($$1 - 735 T - 31562 T^{2} - 420263445 T^{3} + 326940373369 T^{4}$$)($$1 + 1128 T + 700597 T^{2} + 644975736 T^{3} + 326940373369 T^{4}$$)($$1 + 264 T - 994102 T^{2} - 21060864 T^{3} + 764874544683 T^{4} - 12042328243968 T^{5} - 325012079046869638 T^{6} + 49352227390630666392 T^{7} +$$$$10\!\cdots\!61$$$$T^{8}$$)($$1 - 264 T - 994102 T^{2} + 21060864 T^{3} + 764874544683 T^{4} + 12042328243968 T^{5} - 325012079046869638 T^{6} - 49352227390630666392 T^{7} +$$$$10\!\cdots\!61$$$$T^{8}$$)
$89$ ($$1 - 1050 T + 704969 T^{2}$$)($$1 - 21 T + 704969 T^{2}$$)($$1 + 21 T + 704969 T^{2}$$)($$1 + 1050 T + 704969 T^{2}$$)($$1 + 768 T + 1343527 T^{2} + 541416192 T^{3} + 496981290961 T^{4}$$)($$1 + 492 T + 1092454 T^{2} + 346844748 T^{3} + 496981290961 T^{4}$$)($$1 - 492 T + 1092454 T^{2} - 346844748 T^{3} + 496981290961 T^{4}$$)($$1 - 768 T + 1343527 T^{2} - 541416192 T^{3} + 496981290961 T^{4}$$)($$( 1 - 36 T + 704969 T^{2} )^{2}$$)($$( 1 + 954 T + 704969 T^{2} )^{2}$$)($$( 1 + 810 T + 704969 T^{2} )^{2}$$)($$( 1 + 21 T + 704969 T^{2} )^{2}$$)($$( 1 - 21 T + 704969 T^{2} )^{2}$$)($$( 1 - 810 T + 704969 T^{2} )^{2}$$)($$( 1 - 954 T + 704969 T^{2} )^{2}$$)($$( 1 + 36 T + 704969 T^{2} )^{2}$$)($$( 1 - 768 T + 1343527 T^{2} - 541416192 T^{3} + 496981290961 T^{4} )^{2}$$)($$( 1 + 768 T + 1343527 T^{2} + 541416192 T^{3} + 496981290961 T^{4} )^{2}$$)
$97$ ($$1 + 901 T + 912673 T^{2}$$)($$1 + 1246 T + 912673 T^{2}$$)($$1 + 1246 T + 912673 T^{2}$$)($$1 + 901 T + 912673 T^{2}$$)($$1 + 764 T + 1886598 T^{2} + 697282172 T^{3} + 832972004929 T^{4}$$)($$1 + 128 T + 1715097 T^{2} + 116822144 T^{3} + 832972004929 T^{4}$$)($$1 + 128 T + 1715097 T^{2} + 116822144 T^{3} + 832972004929 T^{4}$$)($$1 + 764 T + 1886598 T^{2} + 697282172 T^{3} + 832972004929 T^{4}$$)($$1 + 605 T - 546648 T^{2} + 552167165 T^{3} + 832972004929 T^{4}$$)($$1 + 191 T - 876192 T^{2} + 174320543 T^{3} + 832972004929 T^{4}$$)($$1 + 1154 T + 419043 T^{2} + 1053224642 T^{3} + 832972004929 T^{4}$$)($$1 - 1246 T + 639843 T^{2} - 1137190558 T^{3} + 832972004929 T^{4}$$)($$1 - 1246 T + 639843 T^{2} - 1137190558 T^{3} + 832972004929 T^{4}$$)($$1 + 1154 T + 419043 T^{2} + 1053224642 T^{3} + 832972004929 T^{4}$$)($$1 + 191 T - 876192 T^{2} + 174320543 T^{3} + 832972004929 T^{4}$$)($$1 + 605 T - 546648 T^{2} + 552167165 T^{3} + 832972004929 T^{4}$$)($$1 - 764 T - 1302902 T^{2} - 46796528 T^{3} + 2193556429267 T^{4} - 42709927599344 T^{5} - 1085280891166003958 T^{6} -$$$$58\!\cdots\!88$$$$T^{7} +$$$$69\!\cdots\!41$$$$T^{8}$$)($$1 - 764 T - 1302902 T^{2} - 46796528 T^{3} + 2193556429267 T^{4} - 42709927599344 T^{5} - 1085280891166003958 T^{6} -$$$$58\!\cdots\!88$$$$T^{7} +$$$$69\!\cdots\!41$$$$T^{8}$$)