Properties

Label 162.2
Level 162
Weight 2
Dimension 192
Nonzero newspaces 4
Newform subspaces 12
Sturm bound 2916
Trace bound 1

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Defining parameters

Level: \( N \) = \( 162\( 162 = 2 \cdot 3^{4} \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 12 \)
Sturm bound: \(2916\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(162))\).

Total New Old
Modular forms 837 192 645
Cusp forms 622 192 430
Eisenstein series 215 0 215

Trace form

\( 192q + 6q^{5} + 6q^{7} + 3q^{8} + O(q^{10}) \) \( 192q + 6q^{5} + 6q^{7} + 3q^{8} + 6q^{10} + 15q^{11} + 12q^{13} + 6q^{14} + 12q^{17} - 9q^{18} - 18q^{19} - 30q^{20} - 54q^{21} - 21q^{22} - 78q^{23} - 36q^{25} - 90q^{26} - 54q^{27} - 24q^{28} - 84q^{29} - 54q^{30} - 36q^{31} - 54q^{33} - 15q^{34} - 60q^{35} - 18q^{36} - 6q^{37} + 3q^{38} + 6q^{40} - 3q^{41} + 9q^{43} + 6q^{44} - 54q^{45} + 12q^{46} - 54q^{47} - 12q^{49} + 36q^{50} - 63q^{51} + 12q^{52} - 30q^{53} - 36q^{55} + 6q^{56} - 54q^{57} + 24q^{58} - 69q^{59} - 18q^{61} + 24q^{62} - 54q^{63} + 3q^{64} + 12q^{65} + 72q^{66} - 45q^{67} + 30q^{68} + 126q^{69} - 24q^{70} + 120q^{71} + 72q^{72} + 42q^{73} + 114q^{74} + 180q^{75} - 15q^{76} + 246q^{77} + 144q^{78} - 60q^{79} + 24q^{80} + 144q^{81} + 6q^{82} + 144q^{83} + 36q^{84} - 36q^{85} + 117q^{86} + 288q^{87} - 21q^{88} + 174q^{89} + 144q^{90} + 30q^{91} + 66q^{92} + 126q^{93} - 18q^{94} + 84q^{95} + 18q^{96} - 15q^{97} + 87q^{98} + 18q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(162))\)

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
162.2.a \(\chi_{162}(1, \cdot)\) 162.2.a.a 1 1
162.2.a.b 1
162.2.a.c 1
162.2.a.d 1
162.2.c \(\chi_{162}(55, \cdot)\) 162.2.c.a 2 2
162.2.c.b 2
162.2.c.c 2
162.2.c.d 2
162.2.e \(\chi_{162}(19, \cdot)\) 162.2.e.a 6 6
162.2.e.b 12
162.2.g \(\chi_{162}(7, \cdot)\) 162.2.g.a 72 18
162.2.g.b 90

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(162))\) into lower level spaces

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + T^{3} + T^{6} \))(\( ( 1 - T^{3} + T^{6} )^{2} \))
$3$ 1
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 9 T^{2} - 9 T^{3} + 36 T^{4} - 12 T^{5} + 109 T^{6} - 60 T^{7} + 900 T^{8} - 1125 T^{9} + 5625 T^{10} - 9375 T^{11} + 15625 T^{12} \))(\( 1 - 3 T + 9 T^{2} - 9 T^{3} - 18 T^{4} + 42 T^{5} - 142 T^{6} - 9 T^{7} - 171 T^{8} + 702 T^{9} - 603 T^{10} + 1791 T^{11} + 8049 T^{12} + 8955 T^{13} - 15075 T^{14} + 87750 T^{15} - 106875 T^{16} - 28125 T^{17} - 2218750 T^{18} + 3281250 T^{19} - 7031250 T^{20} - 17578125 T^{21} + 87890625 T^{22} - 146484375 T^{23} + 244140625 T^{24} \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + T + 7 T^{2} ) \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + T + 7 T^{2} ) \))(\( 1 - 3 T - 6 T^{2} + 50 T^{3} - 99 T^{4} - 207 T^{5} + 1401 T^{6} - 1449 T^{7} - 4851 T^{8} + 17150 T^{9} - 14406 T^{10} - 50421 T^{11} + 117649 T^{12} \))(\( 1 + 3 T + 24 T^{2} + 46 T^{3} + 321 T^{4} + 561 T^{5} + 3304 T^{6} + 4068 T^{7} + 25767 T^{8} + 26386 T^{9} + 193845 T^{10} + 149838 T^{11} + 1294315 T^{12} + 1048866 T^{13} + 9498405 T^{14} + 9050398 T^{15} + 61866567 T^{16} + 68370876 T^{17} + 388712296 T^{18} + 462007623 T^{19} + 1850501121 T^{20} + 1856265922 T^{21} + 6779405976 T^{22} + 5931980229 T^{23} + 13841287201 T^{24} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 3 T + 9 T^{2} + 9 T^{3} - 18 T^{4} + 114 T^{5} + 1225 T^{6} + 1254 T^{7} - 2178 T^{8} + 11979 T^{9} + 131769 T^{10} - 483153 T^{11} + 1771561 T^{12} \))(\( 1 - 12 T + 90 T^{2} - 504 T^{3} + 2628 T^{4} - 12720 T^{5} + 57872 T^{6} - 239643 T^{7} + 957330 T^{8} - 3660336 T^{9} + 13502205 T^{10} - 46654983 T^{11} + 157089783 T^{12} - 513204813 T^{13} + 1633766805 T^{14} - 4871907216 T^{15} + 14016268530 T^{16} - 38594744793 T^{17} + 102523778192 T^{18} - 247876815120 T^{19} + 563335139268 T^{20} - 1188405636264 T^{21} + 2334368214090 T^{22} - 3423740047332 T^{23} + 3138428376721 T^{24} \))
$13$ (\( 1 + T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \))(\( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \))(\( 1 + 12 T + 78 T^{2} + 386 T^{3} + 1566 T^{4} + 5886 T^{5} + 21843 T^{6} + 76518 T^{7} + 264654 T^{8} + 848042 T^{9} + 2227758 T^{10} + 4455516 T^{11} + 4826809 T^{12} \))(\( 1 - 12 T + 48 T^{2} - 74 T^{3} + 246 T^{4} - 1974 T^{5} + 8974 T^{6} - 33102 T^{7} + 94338 T^{8} - 225668 T^{9} + 896478 T^{10} - 3069618 T^{11} + 8300191 T^{12} - 39905034 T^{13} + 151504782 T^{14} - 495792596 T^{15} + 2694387618 T^{16} - 12290540886 T^{17} + 43315783966 T^{18} - 123865572558 T^{19} + 200669757366 T^{20} - 784732953602 T^{21} + 6617207608752 T^{22} - 21505924728444 T^{23} + 23298085122481 T^{24} \))
$17$ (\( 1 + 3 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( ( 1 - 3 T + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 3 T + 17 T^{2} )^{2} \))(\( 1 - 6 T + 12 T^{2} - 54 T^{3} - 102 T^{4} + 2082 T^{5} - 8345 T^{6} + 35394 T^{7} - 29478 T^{8} - 265302 T^{9} + 1002252 T^{10} - 8519142 T^{11} + 24137569 T^{12} \))(\( 1 - 6 T - 48 T^{2} + 252 T^{3} + 1722 T^{4} - 5946 T^{5} - 48874 T^{6} + 104832 T^{7} + 1071234 T^{8} - 1189944 T^{9} - 21101022 T^{10} + 6521976 T^{11} + 379698279 T^{12} + 110873592 T^{13} - 6098195358 T^{14} - 5846194872 T^{15} + 89470534914 T^{16} + 148846449024 T^{17} - 1179699547306 T^{18} - 2439873749658 T^{19} + 12012254313402 T^{20} + 29884144877244 T^{21} - 96767707221552 T^{22} - 205631377845798 T^{23} + 582622237229761 T^{24} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( 1 - 9 T + 36 T^{2} - 79 T^{3} - 297 T^{4} + 4806 T^{5} - 27429 T^{6} + 91314 T^{7} - 107217 T^{8} - 541861 T^{9} + 4691556 T^{10} - 22284891 T^{11} + 47045881 T^{12} \))(\( 1 + 9 T - 36 T^{2} - 419 T^{3} + 1611 T^{4} + 13806 T^{5} - 55610 T^{6} - 315819 T^{7} + 1580067 T^{8} + 4822864 T^{9} - 38744127 T^{10} - 34656777 T^{11} + 805313071 T^{12} - 658478763 T^{13} - 13986629847 T^{14} + 33080024176 T^{15} + 205915911507 T^{16} - 781999110081 T^{17} - 2616221442410 T^{18} + 12340793228634 T^{19} + 27360520059051 T^{20} - 135206145369401 T^{21} - 220718385280836 T^{22} + 1048412330083971 T^{23} + 2213314919066161 T^{24} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 6 T + 36 T^{2} - 180 T^{3} + 1386 T^{4} - 6954 T^{5} + 33589 T^{6} - 159942 T^{7} + 733194 T^{8} - 2190060 T^{9} + 10074276 T^{10} - 38618058 T^{11} + 148035889 T^{12} \))(\( 1 + 30 T + 414 T^{2} + 3222 T^{3} + 12348 T^{4} - 19608 T^{5} - 550618 T^{6} - 3222558 T^{7} - 7146234 T^{8} + 22769856 T^{9} + 232883658 T^{10} + 864182466 T^{11} + 2884098855 T^{12} + 19876196718 T^{13} + 123195455082 T^{14} + 277040837952 T^{15} - 1999809268794 T^{16} - 20741488625394 T^{17} - 81511225129402 T^{18} - 66761817364776 T^{19} + 966984046249788 T^{20} + 5803313875233786 T^{21} + 17150575642450686 T^{22} + 28584292737417810 T^{23} + 21914624432020321 T^{24} \))
$29$ (\( 1 - 9 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 9 T + 29 T^{2} \))(\( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} \))(\( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} \))(\( 1 + 15 T + 99 T^{2} + 387 T^{3} - 162 T^{4} - 17112 T^{5} - 132695 T^{6} - 496248 T^{7} - 136242 T^{8} + 9438543 T^{9} + 70020819 T^{10} + 307667235 T^{11} + 594823321 T^{12} \))(\( 1 + 15 T + 81 T^{2} - 45 T^{3} - 4518 T^{4} - 31692 T^{5} - 51964 T^{6} + 564705 T^{7} + 4930839 T^{8} + 14676930 T^{9} - 17187291 T^{10} - 338065677 T^{11} - 2175351603 T^{12} - 9803904633 T^{13} - 14454511731 T^{14} + 357955645770 T^{15} + 3487488738759 T^{16} + 11582748396045 T^{17} - 30909399052444 T^{18} - 546683079984828 T^{19} - 2260113293757798 T^{20} - 652821568914105 T^{21} + 34077285897316281 T^{22} + 183007646485587435 T^{23} + 353814783205469041 T^{24} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 + 18 T + 171 T^{2} + 1253 T^{3} + 7263 T^{4} + 37719 T^{5} + 206634 T^{6} + 1169289 T^{7} + 6979743 T^{8} + 37328123 T^{9} + 157922091 T^{10} + 515324718 T^{11} + 887503681 T^{12} \))(\( 1 + 81 T^{2} + 49 T^{3} + 4023 T^{4} - 2565 T^{5} + 163531 T^{6} - 464805 T^{7} + 4701186 T^{8} - 27482744 T^{9} + 112711716 T^{10} - 1250347077 T^{11} + 3229099084 T^{12} - 38760759387 T^{13} + 108315959076 T^{14} - 818738426504 T^{15} + 4341643995906 T^{16} - 13306972530555 T^{17} + 145134364457611 T^{18} - 70569855194715 T^{19} + 3431180643625143 T^{20} + 1295541485872879 T^{21} + 66389891245444881 T^{22} + 787662783788549761 T^{24} \))
$37$ (\( 1 + T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( ( 1 + T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + T + 37 T^{2} )^{2} \))(\( 1 - 15 T + 60 T^{2} - 289 T^{3} + 4725 T^{4} - 17730 T^{5} - 19395 T^{6} - 656010 T^{7} + 6468525 T^{8} - 14638717 T^{9} + 112449660 T^{10} - 1040159355 T^{11} + 2565726409 T^{12} \))(\( 1 + 15 T + 3 T^{2} - 578 T^{3} + 2505 T^{4} + 24945 T^{5} - 192167 T^{6} - 791163 T^{7} + 6816672 T^{8} + 1096072 T^{9} - 278194485 T^{10} + 271278831 T^{11} + 11925628483 T^{12} + 10037316747 T^{13} - 380848249965 T^{14} + 55519335016 T^{15} + 12775540812192 T^{16} - 54862373051991 T^{17} - 493047946838303 T^{18} + 2368075675082685 T^{19} + 8798761032072105 T^{20} - 75117885601554506 T^{21} + 14425753117253547 T^{22} + 2668764326691906195 T^{23} + 6582952005840035281 T^{24} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( 1 - 9 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} \))(\( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} \))(\( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} \))(\( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} \))(\( 1 - 3 T + 36 T^{2} + 72 T^{3} + 738 T^{4} - 1119 T^{5} + 93799 T^{6} - 45879 T^{7} + 1240578 T^{8} + 4962312 T^{9} + 101727396 T^{10} - 347568603 T^{11} + 4750104241 T^{12} \))(\( 1 - 12 T + 117 T^{2} - 630 T^{3} + 5031 T^{4} - 23943 T^{5} + 171524 T^{6} - 408330 T^{7} + 3551247 T^{8} + 22823100 T^{9} - 180203112 T^{10} + 2500776837 T^{11} - 9745515051 T^{12} + 102531850317 T^{13} - 302921431272 T^{14} + 1572990875100 T^{15} + 10034975273967 T^{16} - 47307562554330 T^{17} + 814756879833284 T^{18} - 4663001579532783 T^{19} + 40172158827707751 T^{20} - 206250618668195430 T^{21} + 1570451139287830917 T^{22} - 6603948380594981292 T^{23} + 22563490300366186081 T^{24} \))
$43$ (\( 1 - 8 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( ( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( ( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 + 18 T + 144 T^{2} + 740 T^{3} + 432 T^{4} - 23706 T^{5} - 185739 T^{6} - 1019358 T^{7} + 798768 T^{8} + 58835180 T^{9} + 492307344 T^{10} + 2646151974 T^{11} + 6321363049 T^{12} \))(\( 1 - 9 T + 36 T^{2} - 248 T^{3} + 1629 T^{4} - 13671 T^{5} + 99658 T^{6} - 1291248 T^{7} + 7996347 T^{8} - 44034284 T^{9} + 342067257 T^{10} - 2064609882 T^{11} + 16750938331 T^{12} - 88778224926 T^{13} + 632482358193 T^{14} - 3501033817988 T^{15} + 27337919119947 T^{16} - 189824358006864 T^{17} + 629974398737242 T^{18} - 3716032232443797 T^{19} + 19040078252212029 T^{20} - 124642967760337064 T^{21} + 778013363278232964 T^{22} - 8363643655241004363 T^{23} + 39959630797262576401 T^{24} \))
$47$ (\( 1 + 12 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 + 9 T + 9 T^{2} - 495 T^{3} - 3222 T^{4} + 3726 T^{5} + 123409 T^{6} + 175122 T^{7} - 7117398 T^{8} - 51392385 T^{9} + 43917129 T^{10} + 2064105063 T^{11} + 10779215329 T^{12} \))(\( 1 - 9 T + 99 T^{2} - 981 T^{3} + 6354 T^{4} - 60768 T^{5} + 331868 T^{6} - 2211183 T^{7} + 13193361 T^{8} - 82909494 T^{9} + 638454717 T^{10} - 3119341005 T^{11} + 27438934035 T^{12} - 146609027235 T^{13} + 1410346469853 T^{14} - 8607912395562 T^{15} + 64379392997841 T^{16} - 507123780613281 T^{17} + 3577276632804572 T^{18} - 30786473784295584 T^{19} + 151296915448829394 T^{20} - 1097866994113814427 T^{21} + 5207314091347174851 T^{22} - 22249432935756110727 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 9 T + 53 T^{2} )^{2} \))(\( ( 1 - 9 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 150 T^{2} - 639 T^{3} + 7950 T^{4} - 16854 T^{5} + 148877 T^{6} )^{2} \))(\( ( 1 - 6 T + 255 T^{2} - 1593 T^{3} + 29193 T^{4} - 168801 T^{5} + 1959550 T^{6} - 8946453 T^{7} + 82003137 T^{8} - 237161061 T^{9} + 2012072655 T^{10} - 2509172958 T^{11} + 22164361129 T^{12} )^{2} \))
$59$ (\( 1 + 59 T^{2} \))(\( 1 + 3 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 6 T + 36 T^{2} - 261 T^{3} - 639 T^{4} + 33681 T^{5} - 161243 T^{6} + 1987179 T^{7} - 2224359 T^{8} - 53603919 T^{9} + 436224996 T^{10} - 4289545794 T^{11} + 42180533641 T^{12} \))(\( 1 + 12 T + 9 T^{2} - 288 T^{3} - 7416 T^{4} - 73437 T^{5} + 128189 T^{6} + 3672837 T^{7} + 36794322 T^{8} + 361771704 T^{9} - 137989746 T^{10} - 18392855166 T^{11} - 121796603070 T^{12} - 1085178454794 T^{13} - 480342305826 T^{14} + 74300310795816 T^{15} + 445850082424242 T^{16} + 2625800417566263 T^{17} + 5407080426906149 T^{18} - 182759099090652903 T^{19} - 1088894525273644536 T^{20} - 2494942795772622432 T^{21} + 4600050779705772609 T^{22} + \)\(36\!\cdots\!08\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))
$61$ (\( 1 + T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + T + 61 T^{2} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 - 18 T + 153 T^{2} - 745 T^{3} - 3915 T^{4} + 107703 T^{5} - 1034862 T^{6} + 6569883 T^{7} - 14567715 T^{8} - 169100845 T^{9} + 2118413673 T^{10} - 15202733418 T^{11} + 51520374361 T^{12} \))(\( 1 + 36 T + 531 T^{2} + 3559 T^{3} + 3123 T^{4} - 91953 T^{5} - 447281 T^{6} - 4553469 T^{7} - 84400488 T^{8} - 593953490 T^{9} - 1210754052 T^{10} - 5462757351 T^{11} - 104608387268 T^{12} - 333228198411 T^{13} - 4505215827492 T^{14} - 134816157113690 T^{15} - 1168595737170408 T^{16} - 3845843074118169 T^{17} - 23044084564562441 T^{18} - 288984632000639013 T^{19} + 598701938490508563 T^{20} + 41619465944396707819 T^{21} + \)\(37\!\cdots\!31\)\( T^{22} + \)\(15\!\cdots\!96\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 - 5 T + 67 T^{2} \))(\( 1 - 5 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4} \))(\( 1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 9 T + 45 T^{2} + 281 T^{3} - 1836 T^{4} - 68094 T^{5} - 564675 T^{6} - 4562298 T^{7} - 8241804 T^{8} + 84514403 T^{9} + 906800445 T^{10} + 12151125963 T^{11} + 90458382169 T^{12} \))(\( 1 - 36 T + 630 T^{2} - 7592 T^{3} + 71910 T^{4} - 581724 T^{5} + 4963294 T^{6} - 52102143 T^{7} + 557562474 T^{8} - 5169740204 T^{9} + 40448438361 T^{10} - 276054423399 T^{11} + 1991438619445 T^{12} - 18495646367733 T^{13} + 181573039802529 T^{14} - 1554866572975652 T^{15} + 11235508878633354 T^{16} - 70344411392804301 T^{17} + 448971545469104686 T^{18} - 3525661397894916852 T^{19} + 29200326693098054310 T^{20} - \)\(20\!\cdots\!24\)\( T^{21} + \)\(11\!\cdots\!70\)\( T^{22} - \)\(43\!\cdots\!88\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))
$71$ (\( 1 + 12 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( 1 + 12 T - 24 T^{2} - 738 T^{3} - 228 T^{4} + 5556 T^{5} - 117857 T^{6} + 394476 T^{7} - 1149348 T^{8} - 264138318 T^{9} - 609880344 T^{10} + 21650752212 T^{11} + 128100283921 T^{12} \))(\( 1 + 12 T - 201 T^{2} - 2430 T^{3} + 25608 T^{4} + 247278 T^{5} - 2990308 T^{6} - 17740170 T^{7} + 310819581 T^{8} + 1005967566 T^{9} - 26322276303 T^{10} - 30298327380 T^{11} + 1913975417427 T^{12} - 2151181243980 T^{13} - 132690594843423 T^{14} + 360046857514626 T^{15} + 7898448040925661 T^{16} - 32007335405729670 T^{17} - 383059303811237668 T^{18} + 2249023122526609698 T^{19} + 16536456428141447688 T^{20} - \)\(11\!\cdots\!30\)\( T^{21} - \)\(65\!\cdots\!01\)\( T^{22} + \)\(27\!\cdots\!52\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))
$73$ (\( 1 - 11 T + 73 T^{2} \))(\( 1 - 11 T + 73 T^{2} \))(\( 1 - 11 T + 73 T^{2} \))(\( 1 - 11 T + 73 T^{2} \))(\( ( 1 - 11 T + 73 T^{2} )^{2} \))(\( ( 1 + 7 T + 73 T^{2} )^{2} \))(\( ( 1 + 7 T + 73 T^{2} )^{2} \))(\( ( 1 - 11 T + 73 T^{2} )^{2} \))(\( 1 - 3 T - 96 T^{2} + 23 T^{3} + 2853 T^{4} + 12258 T^{5} - 46191 T^{6} + 894834 T^{7} + 15203637 T^{8} + 8947391 T^{9} - 2726231136 T^{10} - 6219214779 T^{11} + 151334226289 T^{12} \))(\( 1 + 21 T - 48 T^{2} - 3887 T^{3} - 4845 T^{4} + 448266 T^{5} + 1641574 T^{6} - 22871205 T^{7} - 33688827 T^{8} + 739848004 T^{9} - 9072236607 T^{10} + 8731491045 T^{11} + 1415682555505 T^{12} + 637398846285 T^{13} - 48345948878703 T^{14} + 287813450972068 T^{15} - 956703428153307 T^{16} - 47413645383179565 T^{17} + 248426331186138886 T^{18} + 4952173144561535802 T^{19} - 3907299145226822445 T^{20} - \)\(22\!\cdots\!31\)\( T^{21} - \)\(20\!\cdots\!52\)\( T^{22} + \)\(65\!\cdots\!17\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))
$79$ (\( 1 + 16 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( 1 - 33 T + 510 T^{2} - 4168 T^{3} + 3429 T^{4} + 380187 T^{5} - 5136507 T^{6} + 30034773 T^{7} + 21400389 T^{8} - 2054986552 T^{9} + 19864541310 T^{10} - 101542861167 T^{11} + 243087455521 T^{12} \))(\( 1 - 39 T + 822 T^{2} - 11936 T^{3} + 137379 T^{4} - 1304607 T^{5} + 9780148 T^{6} - 42815682 T^{7} - 154130391 T^{8} + 6081344458 T^{9} - 88425769017 T^{10} + 1013814962454 T^{11} - 9774915393497 T^{12} + 80091382033866 T^{13} - 551865224435097 T^{14} + 2998339990227862 T^{15} - 6003391214011671 T^{16} - 131746268275649118 T^{17} + 2377431291938797108 T^{18} - 25053554090705934513 T^{19} + \)\(20\!\cdots\!19\)\( T^{20} - \)\(14\!\cdots\!84\)\( T^{21} + \)\(77\!\cdots\!22\)\( T^{22} - \)\(29\!\cdots\!81\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))
$83$ (\( 1 + 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \))(\( 1 + 3 T - 74 T^{2} + 249 T^{3} + 6889 T^{4} \))(\( 1 - 3 T - 74 T^{2} - 249 T^{3} + 6889 T^{4} \))(\( 1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4} \))(\( 1 - 18 T + 144 T^{2} - 720 T^{3} + 5580 T^{4} - 58968 T^{5} + 392545 T^{6} - 4894344 T^{7} + 38440620 T^{8} - 411686640 T^{9} + 6833998224 T^{10} - 70902731574 T^{11} + 326940373369 T^{12} \))(\( 1 + 18 T + 126 T^{2} - 1260 T^{3} - 41166 T^{4} - 525114 T^{5} - 2039866 T^{6} + 36031104 T^{7} + 749106684 T^{8} + 6189995052 T^{9} + 9354005928 T^{10} - 495774705924 T^{11} - 6862378086069 T^{12} - 41149300591692 T^{13} + 64439746837992 T^{14} + 3539358700797924 T^{15} + 35551345472517564 T^{16} + 141927983068159872 T^{17} - 666914551662728554 T^{18} - 14249520279366992478 T^{19} - 92717862028235761806 T^{20} - \)\(23\!\cdots\!80\)\( T^{21} + \)\(19\!\cdots\!74\)\( T^{22} + \)\(23\!\cdots\!06\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))
$89$ (\( 1 + 3 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 3 T + 89 T^{2} \))(\( ( 1 - 3 T + 89 T^{2} )^{2} \))(\( ( 1 - 18 T + 89 T^{2} )^{2} \))(\( ( 1 + 18 T + 89 T^{2} )^{2} \))(\( ( 1 + 3 T + 89 T^{2} )^{2} \))(\( 1 - 15 T - 78 T^{2} + 477 T^{3} + 27177 T^{4} - 70638 T^{5} - 2238167 T^{6} - 6286782 T^{7} + 215269017 T^{8} + 336270213 T^{9} - 4893894798 T^{10} - 83760891735 T^{11} + 496981290961 T^{12} \))(\( 1 + 12 T - 120 T^{2} - 3294 T^{3} - 12273 T^{4} + 248142 T^{5} + 2973695 T^{6} + 4697397 T^{7} - 153066366 T^{8} - 1115642430 T^{9} + 2098738827 T^{10} + 25141746861 T^{11} - 47336761416 T^{12} + 2237615470629 T^{13} + 16624110248667 T^{14} - 786493328234670 T^{15} - 9603726824566206 T^{16} + 26230544103554253 T^{17} + 1477870780024270895 T^{18} + 10975651903646357118 T^{19} - 48313754412381640113 T^{20} - \)\(11\!\cdots\!46\)\( T^{21} - \)\(37\!\cdots\!20\)\( T^{22} + \)\(33\!\cdots\!68\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( 1 - 5 T + 97 T^{2} \))(\( 1 - 5 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} \))(\( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} \))(\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 + 12 T + 51 T^{2} + 1277 T^{3} + 801 T^{4} - 56169 T^{5} + 617238 T^{6} - 5448393 T^{7} + 7536609 T^{8} + 1165483421 T^{9} + 4514993331 T^{10} + 103048083084 T^{11} + 832972004929 T^{12} \))(\( 1 - 39 T + 993 T^{2} - 19028 T^{3} + 304221 T^{4} - 4077399 T^{5} + 47593567 T^{6} - 476249436 T^{7} + 4125278700 T^{8} - 29493453512 T^{9} + 169040152338 T^{10} - 681702895626 T^{11} + 3451348757128 T^{12} - 66125180875722 T^{13} + 1590498793348242 T^{14} - 26917878697157576 T^{15} + 365207957235614700 T^{16} - 4089715954136345052 T^{17} + 39644108925712691743 T^{18} - \)\(32\!\cdots\!87\)\( T^{19} + \)\(23\!\cdots\!81\)\( T^{20} - \)\(14\!\cdots\!76\)\( T^{21} + \)\(73\!\cdots\!57\)\( T^{22} - \)\(27\!\cdots\!67\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))
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