Properties

 Label 354.2 Level 354 Weight 2 Dimension 871 Nonzero newspaces 4 Newform subspaces 16 Sturm bound 13920 Trace bound 1

Defining parameters

 Level: $$N$$ = $$354 = 2 \cdot 3 \cdot 59$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newform subspaces: $$16$$ Sturm bound: $$13920$$ Trace bound: $$1$$

Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(354))$$.

Total New Old
Modular forms 3712 871 2841
Cusp forms 3249 871 2378
Eisenstein series 463 0 463

Trace form

 $$871q + q^{2} + q^{3} + q^{4} + 6q^{5} + q^{6} + 8q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$871q + q^{2} + q^{3} + q^{4} + 6q^{5} + q^{6} + 8q^{7} + q^{8} + q^{9} + 6q^{10} + 12q^{11} + q^{12} + 14q^{13} + 8q^{14} + 6q^{15} + q^{16} + 18q^{17} + q^{18} + 20q^{19} + 6q^{20} + 8q^{21} + 12q^{22} + 24q^{23} + q^{24} + 31q^{25} + 14q^{26} + q^{27} + 8q^{28} + 30q^{29} + 6q^{30} + 32q^{31} + q^{32} + 12q^{33} + 18q^{34} + 48q^{35} + q^{36} + 38q^{37} + 20q^{38} + 14q^{39} + 6q^{40} + 42q^{41} + 8q^{42} + 44q^{43} + 12q^{44} - 52q^{45} - 92q^{46} - 68q^{47} + q^{48} - 175q^{49} - 201q^{50} - 214q^{51} - 102q^{52} - 178q^{53} - 202q^{54} - 276q^{55} + 8q^{56} - 270q^{57} - 86q^{58} - 173q^{59} - 168q^{60} - 170q^{61} - 84q^{62} - 282q^{63} + q^{64} - 264q^{65} - 191q^{66} - 164q^{67} - 98q^{68} - 208q^{69} - 184q^{70} - 160q^{71} + q^{72} - 42q^{73} - 78q^{74} - 27q^{75} + 20q^{76} + 96q^{77} + 14q^{78} + 80q^{79} + 6q^{80} + q^{81} + 42q^{82} + 84q^{83} + 8q^{84} + 108q^{85} + 44q^{86} + 30q^{87} + 12q^{88} + 90q^{89} + 6q^{90} + 112q^{91} + 24q^{92} + 32q^{93} + 48q^{94} + 120q^{95} + q^{96} + 98q^{97} + 57q^{98} + 12q^{99} + O(q^{100})$$

Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(354))$$

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
354.2.a $$\chi_{354}(1, \cdot)$$ 354.2.a.a 1 1
354.2.a.b 1
354.2.a.c 1
354.2.a.d 1
354.2.a.e 1
354.2.a.f 1
354.2.a.g 2
354.2.a.h 3
354.2.c $$\chi_{354}(353, \cdot)$$ 354.2.c.a 10 1
354.2.c.b 10
354.2.e $$\chi_{354}(7, \cdot)$$ 354.2.e.a 56 28
354.2.e.b 56
354.2.e.c 84
354.2.e.d 84
354.2.g $$\chi_{354}(11, \cdot)$$ 354.2.g.a 280 28
354.2.g.b 280

Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(354))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(354)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(59))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(118))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(177))$$$$^{\oplus 2}$$

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + T$$)($$1 + T$$)($$1 + T$$)($$1 - T$$)($$1 - T$$)($$1 - T$$)($$( 1 + T )^{2}$$)($$( 1 - T )^{3}$$)($$( 1 + T )^{10}$$)($$( 1 - T )^{10}$$)
$3$ ($$1 + T$$)($$1 + T$$)($$1 - T$$)($$1 + T$$)($$1 + T$$)($$1 + T$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{3}$$)($$1 + T - T^{2} + 2 T^{3} + 2 T^{4} - 14 T^{5} + 6 T^{6} + 18 T^{7} - 27 T^{8} + 81 T^{9} + 243 T^{10}$$)($$1 + T - T^{2} + 2 T^{3} + 2 T^{4} - 14 T^{5} + 6 T^{6} + 18 T^{7} - 27 T^{8} + 81 T^{9} + 243 T^{10}$$)
$5$ ($$1 + 5 T^{2}$$)($$1 - 2 T + 5 T^{2}$$)($$1 + 5 T^{2}$$)($$1 + 4 T + 5 T^{2}$$)($$1 + 5 T^{2}$$)($$1 - 4 T + 5 T^{2}$$)($$1 - 2 T - 10 T^{3} + 25 T^{4}$$)($$1 - 2 T + 9 T^{2} - 12 T^{3} + 45 T^{4} - 50 T^{5} + 125 T^{6}$$)($$1 - 27 T^{2} + 395 T^{4} - 3938 T^{6} + 29068 T^{8} - 165022 T^{10} + 726700 T^{12} - 2461250 T^{14} + 6171875 T^{16} - 10546875 T^{18} + 9765625 T^{20}$$)($$1 - 27 T^{2} + 395 T^{4} - 3938 T^{6} + 29068 T^{8} - 165022 T^{10} + 726700 T^{12} - 2461250 T^{14} + 6171875 T^{16} - 10546875 T^{18} + 9765625 T^{20}$$)
$7$ ($$1 + T + 7 T^{2}$$)($$1 + 7 T^{2}$$)($$1 + T + 7 T^{2}$$)($$1 + T + 7 T^{2}$$)($$1 + 7 T^{2}$$)($$1 + 7 T^{2}$$)($$( 1 - 4 T + 7 T^{2} )^{2}$$)($$1 + T + 5 T^{2} + 30 T^{3} + 35 T^{4} + 49 T^{5} + 343 T^{6}$$)($$( 1 + T + 16 T^{2} + 17 T^{3} + 179 T^{4} + 148 T^{5} + 1253 T^{6} + 833 T^{7} + 5488 T^{8} + 2401 T^{9} + 16807 T^{10} )^{2}$$)($$( 1 + T + 16 T^{2} + 17 T^{3} + 179 T^{4} + 148 T^{5} + 1253 T^{6} + 833 T^{7} + 5488 T^{8} + 2401 T^{9} + 16807 T^{10} )^{2}$$)
$11$ ($$1 + 5 T + 11 T^{2}$$)($$1 - 4 T + 11 T^{2}$$)($$1 - 3 T + 11 T^{2}$$)($$1 + 3 T + 11 T^{2}$$)($$1 - 4 T + 11 T^{2}$$)($$1 + 4 T + 11 T^{2}$$)($$( 1 + 2 T + 11 T^{2} )^{2}$$)($$1 - T + T^{2} + 54 T^{3} + 11 T^{4} - 121 T^{5} + 1331 T^{6}$$)($$( 1 - 2 T + 30 T^{2} - 34 T^{3} + 439 T^{4} - 372 T^{5} + 4829 T^{6} - 4114 T^{7} + 39930 T^{8} - 29282 T^{9} + 161051 T^{10} )^{2}$$)($$( 1 + 2 T + 30 T^{2} + 34 T^{3} + 439 T^{4} + 372 T^{5} + 4829 T^{6} + 4114 T^{7} + 39930 T^{8} + 29282 T^{9} + 161051 T^{10} )^{2}$$)
$13$ ($$1 - T + 13 T^{2}$$)($$1 + 6 T + 13 T^{2}$$)($$1 - 5 T + 13 T^{2}$$)($$1 + T + 13 T^{2}$$)($$1 - 4 T + 13 T^{2}$$)($$1 + 13 T^{2}$$)($$1 + 2 T + 16 T^{2} + 26 T^{3} + 169 T^{4}$$)($$1 - T + 35 T^{2} - 24 T^{3} + 455 T^{4} - 169 T^{5} + 2197 T^{6}$$)($$1 - 52 T^{2} + 1144 T^{4} - 10254 T^{6} - 57459 T^{8} + 2166640 T^{10} - 9710571 T^{12} - 292864494 T^{14} + 5521869496 T^{16} - 42417997492 T^{18} + 137858491849 T^{20}$$)($$1 - 52 T^{2} + 1144 T^{4} - 10254 T^{6} - 57459 T^{8} + 2166640 T^{10} - 9710571 T^{12} - 292864494 T^{14} + 5521869496 T^{16} - 42417997492 T^{18} + 137858491849 T^{20}$$)
$17$ ($$1 - T + 17 T^{2}$$)($$1 - 2 T + 17 T^{2}$$)($$1 + 3 T + 17 T^{2}$$)($$1 + 7 T + 17 T^{2}$$)($$1 - 6 T + 17 T^{2}$$)($$1 + 2 T + 17 T^{2}$$)($$1 - 10 T^{2} + 289 T^{4}$$)($$1 + 3 T + 47 T^{2} + 98 T^{3} + 799 T^{4} + 867 T^{5} + 4913 T^{6}$$)($$1 - 108 T^{2} + 5426 T^{4} - 171278 T^{6} + 3938173 T^{8} - 73013740 T^{10} + 1138131997 T^{12} - 14305309838 T^{14} + 130970449394 T^{16} - 753381803628 T^{18} + 2015993900449 T^{20}$$)($$1 - 108 T^{2} + 5426 T^{4} - 171278 T^{6} + 3938173 T^{8} - 73013740 T^{10} + 1138131997 T^{12} - 14305309838 T^{14} + 130970449394 T^{16} - 753381803628 T^{18} + 2015993900449 T^{20}$$)
$19$ ($$1 + 19 T^{2}$$)($$1 - 4 T + 19 T^{2}$$)($$1 - 8 T + 19 T^{2}$$)($$1 + 4 T + 19 T^{2}$$)($$1 + 4 T + 19 T^{2}$$)($$1 - 4 T + 19 T^{2}$$)($$( 1 + 2 T + 19 T^{2} )^{2}$$)($$1 + 29 T^{2} - 16 T^{3} + 551 T^{4} + 6859 T^{6}$$)($$( 1 + 3 T + 77 T^{2} + 178 T^{3} + 2608 T^{4} + 4670 T^{5} + 49552 T^{6} + 64258 T^{7} + 528143 T^{8} + 390963 T^{9} + 2476099 T^{10} )^{2}$$)($$( 1 + 3 T + 77 T^{2} + 178 T^{3} + 2608 T^{4} + 4670 T^{5} + 49552 T^{6} + 64258 T^{7} + 528143 T^{8} + 390963 T^{9} + 2476099 T^{10} )^{2}$$)
$23$ ($$1 + 6 T + 23 T^{2}$$)($$1 - 8 T + 23 T^{2}$$)($$1 + 6 T + 23 T^{2}$$)($$1 - 2 T + 23 T^{2}$$)($$1 + 4 T + 23 T^{2}$$)($$1 - 4 T + 23 T^{2}$$)($$1 - 4 T + 6 T^{2} - 92 T^{3} + 529 T^{4}$$)($$1 + 10 T + 85 T^{2} + 444 T^{3} + 1955 T^{4} + 5290 T^{5} + 12167 T^{6}$$)($$( 1 + 4 T + 29 T^{2} + 188 T^{3} + 1084 T^{4} + 2688 T^{5} + 24932 T^{6} + 99452 T^{7} + 352843 T^{8} + 1119364 T^{9} + 6436343 T^{10} )^{2}$$)($$( 1 - 4 T + 29 T^{2} - 188 T^{3} + 1084 T^{4} - 2688 T^{5} + 24932 T^{6} - 99452 T^{7} + 352843 T^{8} - 1119364 T^{9} + 6436343 T^{10} )^{2}$$)
$29$ ($$1 + 10 T + 29 T^{2}$$)($$1 - 2 T + 29 T^{2}$$)($$1 - 6 T + 29 T^{2}$$)($$1 + 2 T + 29 T^{2}$$)($$1 + 29 T^{2}$$)($$1 - 4 T + 29 T^{2}$$)($$1 + 10 T + 72 T^{2} + 290 T^{3} + 841 T^{4}$$)($$1 + 8 T + 93 T^{2} + 420 T^{3} + 2697 T^{4} + 6728 T^{5} + 24389 T^{6}$$)($$1 - 155 T^{2} + 12907 T^{4} - 729682 T^{6} + 30825980 T^{8} - 1008918398 T^{10} + 25924649180 T^{12} - 516090214642 T^{14} + 7677384604147 T^{16} - 77538194008955 T^{18} + 420707233300201 T^{20}$$)($$1 - 155 T^{2} + 12907 T^{4} - 729682 T^{6} + 30825980 T^{8} - 1008918398 T^{10} + 25924649180 T^{12} - 516090214642 T^{14} + 7677384604147 T^{16} - 77538194008955 T^{18} + 420707233300201 T^{20}$$)
$31$ ($$1 + 8 T + 31 T^{2}$$)($$1 - 8 T + 31 T^{2}$$)($$1 - 8 T + 31 T^{2}$$)($$1 + 31 T^{2}$$)($$1 - 2 T + 31 T^{2}$$)($$1 + 10 T + 31 T^{2}$$)($$1 + 2 T + 52 T^{2} + 62 T^{3} + 961 T^{4}$$)($$1 + 2 T + 79 T^{2} + 92 T^{3} + 2449 T^{4} + 1922 T^{5} + 29791 T^{6}$$)($$1 - 66 T^{2} + 2027 T^{4} - 68420 T^{6} + 2683060 T^{8} - 85000756 T^{10} + 2578420660 T^{12} - 63187306820 T^{14} + 1798969961387 T^{16} - 56290808471106 T^{18} + 819628286980801 T^{20}$$)($$1 - 66 T^{2} + 2027 T^{4} - 68420 T^{6} + 2683060 T^{8} - 85000756 T^{10} + 2578420660 T^{12} - 63187306820 T^{14} + 1798969961387 T^{16} - 56290808471106 T^{18} + 819628286980801 T^{20}$$)
$37$ ($$1 - 9 T + 37 T^{2}$$)($$1 - 2 T + 37 T^{2}$$)($$1 - 5 T + 37 T^{2}$$)($$1 - 7 T + 37 T^{2}$$)($$1 + 8 T + 37 T^{2}$$)($$1 + 4 T + 37 T^{2}$$)($$1 + 2 T + 64 T^{2} + 74 T^{3} + 1369 T^{4}$$)($$1 - 9 T + 35 T^{2} - 40 T^{3} + 1295 T^{4} - 12321 T^{5} + 50653 T^{6}$$)($$1 - 188 T^{2} + 14592 T^{4} - 563614 T^{6} + 9281381 T^{8} - 55855104 T^{10} + 12706210589 T^{12} - 1056303377854 T^{14} + 37439079760128 T^{16} - 660346137337148 T^{18} + 4808584372417849 T^{20}$$)($$1 - 188 T^{2} + 14592 T^{4} - 563614 T^{6} + 9281381 T^{8} - 55855104 T^{10} + 12706210589 T^{12} - 1056303377854 T^{14} + 37439079760128 T^{16} - 660346137337148 T^{18} + 4808584372417849 T^{20}$$)
$41$ ($$1 + 5 T + 41 T^{2}$$)($$1 - 2 T + 41 T^{2}$$)($$1 + 9 T + 41 T^{2}$$)($$1 - 3 T + 41 T^{2}$$)($$1 - 6 T + 41 T^{2}$$)($$1 + 2 T + 41 T^{2}$$)($$1 - 8 T + 54 T^{2} - 328 T^{3} + 1681 T^{4}$$)($$1 + T + 31 T^{2} + 246 T^{3} + 1271 T^{4} + 1681 T^{5} + 68921 T^{6}$$)($$1 - 209 T^{2} + 23428 T^{4} - 1817599 T^{6} + 106670111 T^{8} - 4914350696 T^{10} + 179312456591 T^{12} - 5136100367839 T^{14} + 111285442158148 T^{16} - 1668849372886289 T^{18} + 13422659310152401 T^{20}$$)($$1 - 209 T^{2} + 23428 T^{4} - 1817599 T^{6} + 106670111 T^{8} - 4914350696 T^{10} + 179312456591 T^{12} - 5136100367839 T^{14} + 111285442158148 T^{16} - 1668849372886289 T^{18} + 13422659310152401 T^{20}$$)
$43$ ($$1 - 3 T + 43 T^{2}$$)($$1 + 43 T^{2}$$)($$1 + T + 43 T^{2}$$)($$1 - 5 T + 43 T^{2}$$)($$1 - 4 T + 43 T^{2}$$)($$1 + 12 T + 43 T^{2}$$)($$( 1 - 4 T + 43 T^{2} )^{2}$$)($$1 + 7 T + 113 T^{2} + 474 T^{3} + 4859 T^{4} + 12943 T^{5} + 79507 T^{6}$$)($$1 - 124 T^{2} + 8986 T^{4} - 558234 T^{6} + 29482965 T^{8} - 1340267732 T^{10} + 54514002285 T^{12} - 1908490957434 T^{14} + 56803768358314 T^{16} - 1449336834422524 T^{18} + 21611482313284249 T^{20}$$)($$1 - 124 T^{2} + 8986 T^{4} - 558234 T^{6} + 29482965 T^{8} - 1340267732 T^{10} + 54514002285 T^{12} - 1908490957434 T^{14} + 56803768358314 T^{16} - 1449336834422524 T^{18} + 21611482313284249 T^{20}$$)
$47$ ($$1 + 47 T^{2}$$)($$1 - 8 T + 47 T^{2}$$)($$1 + 47 T^{2}$$)($$1 - 12 T + 47 T^{2}$$)($$1 + 4 T + 47 T^{2}$$)($$1 - 4 T + 47 T^{2}$$)($$1 - 4 T + 54 T^{2} - 188 T^{3} + 2209 T^{4}$$)($$1 + 8 T + 101 T^{2} + 784 T^{3} + 4747 T^{4} + 17672 T^{5} + 103823 T^{6}$$)($$( 1 + 149 T^{2} + 36 T^{3} + 11260 T^{4} + 2520 T^{5} + 529220 T^{6} + 79524 T^{7} + 15469627 T^{8} + 229345007 T^{10} )^{2}$$)($$( 1 + 149 T^{2} - 36 T^{3} + 11260 T^{4} - 2520 T^{5} + 529220 T^{6} - 79524 T^{7} + 15469627 T^{8} + 229345007 T^{10} )^{2}$$)
$53$ ($$1 - 4 T + 53 T^{2}$$)($$1 + 6 T + 53 T^{2}$$)($$1 - 12 T + 53 T^{2}$$)($$1 + 8 T + 53 T^{2}$$)($$1 - 4 T + 53 T^{2}$$)($$1 + 53 T^{2}$$)($$1 + 14 T + 144 T^{2} + 742 T^{3} + 2809 T^{4}$$)($$1 + 18 T + 209 T^{2} + 1652 T^{3} + 11077 T^{4} + 50562 T^{5} + 148877 T^{6}$$)($$1 - 355 T^{2} + 60783 T^{4} - 6738486 T^{6} + 540355392 T^{8} - 32753382654 T^{10} + 1517858296128 T^{12} - 53169895751766 T^{14} + 1347216362504007 T^{16} - 22102190096033155 T^{18} + 174887470365513049 T^{20}$$)($$1 - 355 T^{2} + 60783 T^{4} - 6738486 T^{6} + 540355392 T^{8} - 32753382654 T^{10} + 1517858296128 T^{12} - 53169895751766 T^{14} + 1347216362504007 T^{16} - 22102190096033155 T^{18} + 174887470365513049 T^{20}$$)
$59$ ($$1 + T$$)($$1 - T$$)($$1 + T$$)($$1 - T$$)($$1 + T$$)($$1 + T$$)($$( 1 + T )^{2}$$)($$( 1 - T )^{3}$$)($$1 - 20 T + 71 T^{2} + 896 T^{3} - 6134 T^{4} + 9960 T^{5} - 361906 T^{6} + 3118976 T^{7} + 14581909 T^{8} - 242347220 T^{9} + 714924299 T^{10}$$)($$1 + 20 T + 71 T^{2} - 896 T^{3} - 6134 T^{4} - 9960 T^{5} - 361906 T^{6} - 3118976 T^{7} + 14581909 T^{8} + 242347220 T^{9} + 714924299 T^{10}$$)
$61$ ($$1 - 6 T + 61 T^{2}$$)($$1 - 10 T + 61 T^{2}$$)($$1 + 10 T + 61 T^{2}$$)($$1 + 14 T + 61 T^{2}$$)($$1 + 61 T^{2}$$)($$1 - 4 T + 61 T^{2}$$)($$1 + 2 T + 112 T^{2} + 122 T^{3} + 3721 T^{4}$$)($$1 + 29 T^{2} - 724 T^{3} + 1769 T^{4} + 226981 T^{6}$$)($$1 - 226 T^{2} + 28531 T^{4} - 2771220 T^{6} + 216253284 T^{8} - 14141970452 T^{10} + 804678469764 T^{12} - 38369871496020 T^{14} + 1469927800893691 T^{16} - 43325852737385506 T^{18} + 713342911662882601 T^{20}$$)($$1 - 226 T^{2} + 28531 T^{4} - 2771220 T^{6} + 216253284 T^{8} - 14141970452 T^{10} + 804678469764 T^{12} - 38369871496020 T^{14} + 1469927800893691 T^{16} - 43325852737385506 T^{18} + 713342911662882601 T^{20}$$)
$67$ ($$1 - 4 T + 67 T^{2}$$)($$1 + 8 T + 67 T^{2}$$)($$1 + 4 T + 67 T^{2}$$)($$1 + 4 T + 67 T^{2}$$)($$1 + 16 T + 67 T^{2}$$)($$1 + 8 T + 67 T^{2}$$)($$1 + 4 T + 94 T^{2} + 268 T^{3} + 4489 T^{4}$$)($$1 - 8 T + 161 T^{2} - 1104 T^{3} + 10787 T^{4} - 35912 T^{5} + 300763 T^{6}$$)($$1 - 226 T^{2} + 24469 T^{4} - 2099544 T^{6} + 180779250 T^{8} - 13661508812 T^{10} + 811518053250 T^{12} - 42308165188824 T^{14} + 2213426153293261 T^{16} - 91771295127800866 T^{18} + 1822837804551761449 T^{20}$$)($$1 - 226 T^{2} + 24469 T^{4} - 2099544 T^{6} + 180779250 T^{8} - 13661508812 T^{10} + 811518053250 T^{12} - 42308165188824 T^{14} + 2213426153293261 T^{16} - 91771295127800866 T^{18} + 1822837804551761449 T^{20}$$)
$71$ ($$1 - T + 71 T^{2}$$)($$1 + 12 T + 71 T^{2}$$)($$1 + 3 T + 71 T^{2}$$)($$1 + 15 T + 71 T^{2}$$)($$1 + 14 T + 71 T^{2}$$)($$1 - 6 T + 71 T^{2}$$)($$1 - 6 T + 140 T^{2} - 426 T^{3} + 5041 T^{4}$$)($$1 + 9 T + 201 T^{2} + 1172 T^{3} + 14271 T^{4} + 45369 T^{5} + 357911 T^{6}$$)($$1 - 404 T^{2} + 85792 T^{4} - 12232858 T^{6} + 1288692677 T^{8} - 103955631344 T^{10} + 6496299784757 T^{12} - 310857485214298 T^{14} + 10989979558150432 T^{16} - 260884426623287444 T^{18} + 3255243551009881201 T^{20}$$)($$1 - 404 T^{2} + 85792 T^{4} - 12232858 T^{6} + 1288692677 T^{8} - 103955631344 T^{10} + 6496299784757 T^{12} - 310857485214298 T^{14} + 10989979558150432 T^{16} - 260884426623287444 T^{18} + 3255243551009881201 T^{20}$$)
$73$ ($$1 + 73 T^{2}$$)($$1 + 14 T + 73 T^{2}$$)($$1 + 16 T + 73 T^{2}$$)($$1 + 4 T + 73 T^{2}$$)($$1 - 2 T + 73 T^{2}$$)($$1 + 14 T + 73 T^{2}$$)($$1 + 4 T - 26 T^{2} + 292 T^{3} + 5329 T^{4}$$)($$1 - 8 T - T^{2} + 768 T^{3} - 73 T^{4} - 42632 T^{5} + 389017 T^{6}$$)($$1 - 234 T^{2} + 40625 T^{4} - 4874480 T^{6} + 488701726 T^{8} - 38421788236 T^{10} + 2604291497854 T^{12} - 138426657789680 T^{14} + 6147952942990625 T^{16} - 188711661503214954 T^{18} + 4297625829703557649 T^{20}$$)($$1 - 234 T^{2} + 40625 T^{4} - 4874480 T^{6} + 488701726 T^{8} - 38421788236 T^{10} + 2604291497854 T^{12} - 138426657789680 T^{14} + 6147952942990625 T^{16} - 188711661503214954 T^{18} + 4297625829703557649 T^{20}$$)
$79$ ($$1 + 3 T + 79 T^{2}$$)($$1 + 16 T + 79 T^{2}$$)($$1 - 5 T + 79 T^{2}$$)($$1 - 5 T + 79 T^{2}$$)($$1 + 8 T + 79 T^{2}$$)($$1 - 8 T + 79 T^{2}$$)($$( 1 + 79 T^{2} )^{2}$$)($$1 + 19 T + 301 T^{2} + 2746 T^{3} + 23779 T^{4} + 118579 T^{5} + 493039 T^{6}$$)($$( 1 - 3 T + 192 T^{2} - 299 T^{3} + 24163 T^{4} - 44052 T^{5} + 1908877 T^{6} - 1866059 T^{7} + 94663488 T^{8} - 116850243 T^{9} + 3077056399 T^{10} )^{2}$$)($$( 1 - 3 T + 192 T^{2} - 299 T^{3} + 24163 T^{4} - 44052 T^{5} + 1908877 T^{6} - 1866059 T^{7} + 94663488 T^{8} - 116850243 T^{9} + 3077056399 T^{10} )^{2}$$)
$83$ ($$1 - 7 T + 83 T^{2}$$)($$1 - 4 T + 83 T^{2}$$)($$1 + 9 T + 83 T^{2}$$)($$1 - T + 83 T^{2}$$)($$1 + 4 T + 83 T^{2}$$)($$1 + 4 T + 83 T^{2}$$)($$( 1 + 4 T + 83 T^{2} )^{2}$$)($$1 + 7 T + 209 T^{2} + 890 T^{3} + 17347 T^{4} + 48223 T^{5} + 571787 T^{6}$$)($$( 1 - 6 T + 260 T^{2} - 1794 T^{3} + 33211 T^{4} - 211464 T^{5} + 2756513 T^{6} - 12358866 T^{7} + 148664620 T^{8} - 284749926 T^{9} + 3939040643 T^{10} )^{2}$$)($$( 1 + 6 T + 260 T^{2} + 1794 T^{3} + 33211 T^{4} + 211464 T^{5} + 2756513 T^{6} + 12358866 T^{7} + 148664620 T^{8} + 284749926 T^{9} + 3939040643 T^{10} )^{2}$$)
$89$ ($$1 - 16 T + 89 T^{2}$$)($$1 - 6 T + 89 T^{2}$$)($$1 + 89 T^{2}$$)($$1 - 4 T + 89 T^{2}$$)($$1 - 14 T + 89 T^{2}$$)($$1 + 18 T + 89 T^{2}$$)($$1 - 4 T + 138 T^{2} - 356 T^{3} + 7921 T^{4}$$)($$1 + 3 T^{2} + 736 T^{3} + 267 T^{4} + 704969 T^{6}$$)($$( 1 + 8 T + 201 T^{2} + 220 T^{3} + 15682 T^{4} - 31200 T^{5} + 1395698 T^{6} + 1742620 T^{7} + 141698769 T^{8} + 501937928 T^{9} + 5584059449 T^{10} )^{2}$$)($$( 1 - 8 T + 201 T^{2} - 220 T^{3} + 15682 T^{4} + 31200 T^{5} + 1395698 T^{6} - 1742620 T^{7} + 141698769 T^{8} - 501937928 T^{9} + 5584059449 T^{10} )^{2}$$)
$97$ ($$1 + 12 T + 97 T^{2}$$)($$1 - 2 T + 97 T^{2}$$)($$1 + 4 T + 97 T^{2}$$)($$1 + 4 T + 97 T^{2}$$)($$1 + 10 T + 97 T^{2}$$)($$1 - 14 T + 97 T^{2}$$)($$1 + 24 T + 294 T^{2} + 2328 T^{3} + 9409 T^{4}$$)($$1 - 28 T + 535 T^{2} - 6088 T^{3} + 51895 T^{4} - 263452 T^{5} + 912673 T^{6}$$)($$1 - 682 T^{2} + 227521 T^{4} - 48709872 T^{6} + 7388315838 T^{8} - 828434696780 T^{10} + 69516663719742 T^{12} - 4312249945762032 T^{14} + 189518623533451009 T^{16} - 5345129711365087402 T^{18} + 73742412689492826049 T^{20}$$)($$1 - 682 T^{2} + 227521 T^{4} - 48709872 T^{6} + 7388315838 T^{8} - 828434696780 T^{10} + 69516663719742 T^{12} - 4312249945762032 T^{14} + 189518623533451009 T^{16} - 5345129711365087402 T^{18} + 73742412689492826049 T^{20}$$)