Properties

Label 335.2.a
Level 335
Weight 2
Character orbit a
Rep. character \(\chi_{335}(1,\cdot)\)
Character field \(\Q\)
Dimension 23
Newform subspaces 5
Sturm bound 68
Trace bound 2

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 335 = 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 335.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(68\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(335))\).

Total New Old
Modular forms 36 23 13
Cusp forms 33 23 10
Eisenstein series 3 0 3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(5\)\(67\)FrickeDim.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(9\)
\(-\)\(+\)\(-\)\(11\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(3\)
Minus space\(-\)\(20\)

Trace form

\( 23q + 3q^{2} + 4q^{3} + 25q^{4} + q^{5} - 4q^{6} + 8q^{7} + 3q^{8} + 31q^{9} + O(q^{10}) \) \( 23q + 3q^{2} + 4q^{3} + 25q^{4} + q^{5} - 4q^{6} + 8q^{7} + 3q^{8} + 31q^{9} - 3q^{10} + 4q^{11} - 4q^{12} + 14q^{13} - 8q^{14} - 4q^{15} + 33q^{16} + 8q^{17} - 9q^{18} + 6q^{19} - q^{20} + 32q^{22} - 10q^{23} + 4q^{24} + 23q^{25} - 26q^{26} - 8q^{27} + 36q^{28} + 12q^{29} + 12q^{30} + 8q^{31} - 21q^{32} - 8q^{33} - 10q^{34} - 4q^{35} + 29q^{36} + 8q^{37} - 36q^{38} - 8q^{39} - 15q^{40} - 2q^{41} - 40q^{42} + 32q^{43} - 16q^{44} + 5q^{45} - 12q^{46} + 2q^{47} - 68q^{48} + 51q^{49} + 3q^{50} - 24q^{51} + 10q^{52} + 26q^{53} - 24q^{54} + 4q^{55} - 72q^{56} + 48q^{57} - 30q^{58} - 14q^{59} - 28q^{60} + 34q^{61} - 32q^{62} + 33q^{64} - 10q^{65} - 96q^{66} - 3q^{67} - 26q^{68} - 40q^{69} - 8q^{70} - 24q^{71} - 105q^{72} + 36q^{73} - 26q^{74} + 4q^{75} - 40q^{76} - 40q^{77} - 96q^{78} + 48q^{79} + 15q^{80} + 39q^{81} - 58q^{82} - 60q^{83} - 126q^{84} - 10q^{85} - 38q^{86} - 28q^{87} + 28q^{88} + 24q^{89} - 21q^{90} + 32q^{91} - 32q^{92} - 36q^{93} - 8q^{94} + 12q^{95} - 14q^{96} + 26q^{97} - q^{98} + 8q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(335))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 67
335.2.a.a \(1\) \(2.675\) \(\Q\) None \(0\) \(0\) \(1\) \(-2\) \(-\) \(-\) \(q-2q^{4}+q^{5}-2q^{7}-3q^{9}-2q^{11}+\cdots\)
335.2.a.b \(2\) \(2.675\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-2\) \(-4\) \(+\) \(+\) \(q+\beta q^{2}-\beta q^{3}-q^{5}-2q^{6}-2q^{7}+\cdots\)
335.2.a.c \(2\) \(2.675\) \(\Q(\sqrt{5}) \) None \(1\) \(0\) \(-2\) \(0\) \(+\) \(-\) \(q+\beta q^{2}+(-1+2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
335.2.a.d \(7\) \(2.675\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(2\) \(4\) \(-7\) \(10\) \(+\) \(-\) \(q+(\beta _{2}+\beta _{5})q^{2}+(1-\beta _{4})q^{3}+(1+\beta _{1}+\cdots)q^{4}+\cdots\)
335.2.a.e \(11\) \(2.675\) \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(0\) \(0\) \(11\) \(4\) \(-\) \(+\) \(q-\beta _{1}q^{2}+\beta _{3}q^{3}+(1+\beta _{2})q^{4}+q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(335))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(335)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} \))(\( 1 + 2 T^{2} + 4 T^{4} \))(\( 1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 3 T^{3} + 6 T^{4} - 4 T^{5} + 13 T^{6} - 30 T^{7} + 26 T^{8} - 16 T^{9} + 48 T^{10} - 48 T^{11} + 64 T^{12} - 128 T^{13} + 128 T^{14} \))(\( 1 + 4 T^{2} + 2 T^{3} + 10 T^{4} + 8 T^{5} + 18 T^{6} + 22 T^{7} + 32 T^{8} + 35 T^{9} + 54 T^{10} + 74 T^{11} + 108 T^{12} + 140 T^{13} + 256 T^{14} + 352 T^{15} + 576 T^{16} + 512 T^{17} + 1280 T^{18} + 512 T^{19} + 2048 T^{20} + 2048 T^{22} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 + 4 T^{2} + 9 T^{4} \))(\( 1 + T^{2} + 9 T^{4} \))(\( 1 - 4 T + 13 T^{2} - 26 T^{3} + 42 T^{4} - 24 T^{5} - 19 T^{6} + 110 T^{7} - 57 T^{8} - 216 T^{9} + 1134 T^{10} - 2106 T^{11} + 3159 T^{12} - 2916 T^{13} + 2187 T^{14} \))(\( 1 + 6 T^{2} + 2 T^{3} + 29 T^{4} + 6 T^{5} + 82 T^{6} + 38 T^{7} + 230 T^{8} - 24 T^{9} + 538 T^{10} + 44 T^{11} + 1614 T^{12} - 216 T^{13} + 6210 T^{14} + 3078 T^{15} + 19926 T^{16} + 4374 T^{17} + 63423 T^{18} + 13122 T^{19} + 118098 T^{20} + 177147 T^{22} \))
$5$ (\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{7} \))(\( ( 1 - T )^{11} \))
$7$ (\( 1 + 2 T + 7 T^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( 1 + 9 T^{2} + 49 T^{4} \))(\( 1 - 10 T + 61 T^{2} - 244 T^{3} + 692 T^{4} - 1286 T^{5} + 1315 T^{6} - 818 T^{7} + 9205 T^{8} - 63014 T^{9} + 237356 T^{10} - 585844 T^{11} + 1025227 T^{12} - 1176490 T^{13} + 823543 T^{14} \))(\( 1 - 4 T + 28 T^{2} - 96 T^{3} + 519 T^{4} - 1514 T^{5} + 6534 T^{6} - 17894 T^{7} + 67810 T^{8} - 165078 T^{9} + 564536 T^{10} - 1285228 T^{11} + 3951752 T^{12} - 8088822 T^{13} + 23258830 T^{14} - 42963494 T^{15} + 109816938 T^{16} - 178120586 T^{17} + 427418817 T^{18} - 553420896 T^{19} + 1129900996 T^{20} - 1129900996 T^{21} + 1977326743 T^{22} \))
$11$ (\( 1 + 2 T + 11 T^{2} \))(\( 1 + 20 T^{2} + 121 T^{4} \))(\( 1 + 6 T + 26 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - 6 T + 45 T^{2} - 212 T^{3} + 1021 T^{4} - 4202 T^{5} + 15209 T^{6} - 54040 T^{7} + 167299 T^{8} - 508442 T^{9} + 1358951 T^{10} - 3103892 T^{11} + 7247295 T^{12} - 10629366 T^{13} + 19487171 T^{14} \))(\( 1 - 6 T + 35 T^{2} - 124 T^{3} + 637 T^{4} - 1470 T^{5} + 4807 T^{6} - 272 T^{7} - 2750 T^{8} + 256244 T^{9} - 748698 T^{10} + 4031128 T^{11} - 8235678 T^{12} + 31005524 T^{13} - 3660250 T^{14} - 3982352 T^{15} + 774172157 T^{16} - 2604194670 T^{17} + 12413327927 T^{18} - 26580501244 T^{19} + 82528169185 T^{20} - 155624547606 T^{21} + 285311670611 T^{22} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2} \))(\( 1 - 4 T + 46 T^{2} - 178 T^{3} + 1263 T^{4} - 4188 T^{5} + 22931 T^{6} - 67078 T^{7} + 298103 T^{8} - 707772 T^{9} + 2774811 T^{10} - 5083858 T^{11} + 17079478 T^{12} - 19307236 T^{13} + 62748517 T^{14} \))(\( 1 - 4 T + 74 T^{2} - 190 T^{3} + 2429 T^{4} - 3860 T^{5} + 52231 T^{6} - 48014 T^{7} + 896430 T^{8} - 489200 T^{9} + 13321091 T^{10} - 5586616 T^{11} + 173174183 T^{12} - 82674800 T^{13} + 1969456710 T^{14} - 1371327854 T^{15} + 19393004683 T^{16} - 18631482740 T^{17} + 152416147793 T^{18} - 154988836990 T^{19} + 784732953602 T^{20} - 551433967396 T^{21} + 1792160394037 T^{22} \))
$17$ (\( 1 + 3 T + 17 T^{2} \))(\( 1 + 6 T + 41 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 + 2 T - 10 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 - 17 T + 197 T^{2} - 1646 T^{3} + 11467 T^{4} - 66639 T^{5} + 338863 T^{6} - 1487332 T^{7} + 5760671 T^{8} - 19258671 T^{9} + 56337371 T^{10} - 137475566 T^{11} + 279711829 T^{12} - 410338673 T^{13} + 410338673 T^{14} \))(\( 1 - 2 T + 64 T^{2} - 106 T^{3} + 1892 T^{4} - 1522 T^{5} + 27193 T^{6} + 31064 T^{7} + 51518 T^{8} + 1880500 T^{9} - 5626668 T^{10} + 42036516 T^{11} - 95653356 T^{12} + 543464500 T^{13} + 253107934 T^{14} + 2594496344 T^{15} + 38610171401 T^{16} - 36737380018 T^{17} + 776360769316 T^{18} - 739430288746 T^{19} + 7589624095808 T^{20} - 4031987800898 T^{21} + 34271896307633 T^{22} \))
$19$ (\( 1 + T + 19 T^{2} \))(\( ( 1 + T + 19 T^{2} )^{2} \))(\( 1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4} \))(\( 1 - 3 T + 70 T^{2} - 169 T^{3} + 2727 T^{4} - 5946 T^{5} + 71873 T^{6} - 132741 T^{7} + 1365587 T^{8} - 2146506 T^{9} + 18704493 T^{10} - 22024249 T^{11} + 173326930 T^{12} - 141137643 T^{13} + 893871739 T^{14} \))(\( 1 - 10 T + 151 T^{2} - 1254 T^{3} + 11345 T^{4} - 78256 T^{5} + 545753 T^{6} - 3190098 T^{7} + 18511191 T^{8} - 93499662 T^{9} + 464784761 T^{10} - 2045874144 T^{11} + 8830910459 T^{12} - 33753377982 T^{13} + 126968259069 T^{14} - 415736761458 T^{15} + 1351338457547 T^{16} - 3681622463536 T^{17} + 10140974878955 T^{18} - 21297388053414 T^{19} + 48725842364629 T^{20} - 61310662578010 T^{21} + 116490258898219 T^{22} \))
$23$ (\( 1 + T + 23 T^{2} \))(\( 1 + 6 T + 53 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 2 T + 42 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 - 5 T + 101 T^{2} - 302 T^{3} + 4593 T^{4} - 10491 T^{5} + 148477 T^{6} - 292996 T^{7} + 3414971 T^{8} - 5549739 T^{9} + 55883031 T^{10} - 84511982 T^{11} + 650070643 T^{12} - 740179445 T^{13} + 3404825447 T^{14} \))(\( 1 + 10 T + 106 T^{2} + 434 T^{3} + 1858 T^{4} - 7702 T^{5} - 61201 T^{6} - 577816 T^{7} - 823778 T^{8} + 365612 T^{9} + 77435136 T^{10} + 328317260 T^{11} + 1781008128 T^{12} + 193408748 T^{13} - 10022906926 T^{14} - 161696607256 T^{15} - 393910627943 T^{16} - 1140172417078 T^{17} + 6326165680526 T^{18} + 33986967611954 T^{19} + 190922182115078 T^{20} + 414265112136490 T^{21} + 952809757913927 T^{22} \))
$29$ (\( 1 + 9 T + 29 T^{2} \))(\( 1 - 6 T + 59 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 + 10 T + 63 T^{2} + 290 T^{3} + 841 T^{4} \))(\( 1 + T + 77 T^{2} + 22 T^{3} + 3426 T^{4} - 2082 T^{5} + 114307 T^{6} - 141241 T^{7} + 3314903 T^{8} - 1750962 T^{9} + 83556714 T^{10} + 15560182 T^{11} + 1579358473 T^{12} + 594823321 T^{13} + 17249876309 T^{14} \))(\( 1 - 26 T + 483 T^{2} - 6346 T^{3} + 69805 T^{4} - 644676 T^{5} + 5336880 T^{6} - 39717356 T^{7} + 274853464 T^{8} - 1757877530 T^{9} + 10549928760 T^{10} - 58691457986 T^{11} + 305947934040 T^{12} - 1478375002730 T^{13} + 6703401133496 T^{14} - 28091331269036 T^{15} + 109465540875120 T^{16} - 383468319288996 T^{17} + 1204127615749745 T^{18} - 3174563736650506 T^{19} + 7006951506344727 T^{20} - 10938388065805226 T^{21} + 12200509765705829 T^{22} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 - 4 T + 48 T^{2} - 124 T^{3} + 961 T^{4} \))(\( 1 - 12 T + 78 T^{2} - 372 T^{3} + 961 T^{4} \))(\( 1 + 33 T^{2} + 40 T^{3} + 1501 T^{4} + 3904 T^{5} + 38973 T^{6} + 198832 T^{7} + 1208163 T^{8} + 3751744 T^{9} + 44716291 T^{10} + 36940840 T^{11} + 944761983 T^{12} + 27512614111 T^{14} \))(\( 1 + 8 T + 207 T^{2} + 1228 T^{3} + 18245 T^{4} + 78664 T^{5} + 896859 T^{6} + 2424656 T^{7} + 27159066 T^{8} + 22974384 T^{9} + 618266038 T^{10} - 359149112 T^{11} + 19166247178 T^{12} + 22078383024 T^{13} + 809095735206 T^{14} + 2239220733776 T^{15} + 25676311736709 T^{16} + 69814589562184 T^{17} + 501967644455195 T^{18} + 1047350193977548 T^{19} + 5473001787258897 T^{20} + 6557026295846408 T^{21} + 25408476896404831 T^{22} \))
$37$ (\( 1 + 3 T + 37 T^{2} \))(\( 1 + 10 T + 81 T^{2} + 370 T^{3} + 1369 T^{4} \))(\( 1 + 6 T + 78 T^{2} + 222 T^{3} + 1369 T^{4} \))(\( 1 - 15 T + 151 T^{2} - 1034 T^{3} + 7321 T^{4} - 47449 T^{5} + 324199 T^{6} - 1904108 T^{7} + 11995363 T^{8} - 64957681 T^{9} + 370830613 T^{10} - 1937882474 T^{11} + 10470937507 T^{12} - 38485896135 T^{13} + 94931877133 T^{14} \))(\( 1 - 12 T + 264 T^{2} - 2780 T^{3} + 35700 T^{4} - 331372 T^{5} + 3177245 T^{6} - 25881584 T^{7} + 204234822 T^{8} - 1460083240 T^{9} + 9878161996 T^{10} - 61906160872 T^{11} + 365491993852 T^{12} - 1998853955560 T^{13} + 10345106438766 T^{14} - 48506255351024 T^{15} + 220322740658465 T^{16} - 850209891603148 T^{17} + 3389068013648100 T^{18} - 9764692881900380 T^{19} + 34309899305900328 T^{20} - 57703012469014188 T^{21} + 177917621779460413 T^{22} \))
$41$ (\( 1 + 2 T + 41 T^{2} \))(\( 1 + 80 T^{2} + 1681 T^{4} \))(\( 1 - 6 T + 86 T^{2} - 246 T^{3} + 1681 T^{4} \))(\( 1 + 20 T + 211 T^{2} + 1608 T^{3} + 10057 T^{4} + 44908 T^{5} + 137811 T^{6} + 451824 T^{7} + 5650251 T^{8} + 75490348 T^{9} + 693138497 T^{10} + 4543823688 T^{11} + 24445658411 T^{12} + 95002084820 T^{13} + 194754273881 T^{14} \))(\( 1 - 14 T + 377 T^{2} - 4108 T^{3} + 64501 T^{4} - 581766 T^{5} + 6809693 T^{6} - 52475120 T^{7} + 500762458 T^{8} - 3349939788 T^{9} + 27167544698 T^{10} - 158483464456 T^{11} + 1113869332618 T^{12} - 5631248783628 T^{13} + 34513049367818 T^{14} - 148282147566320 T^{15} + 788945160956293 T^{16} - 2763449143869606 T^{17} + 12561845419598381 T^{18} - 32802072841229068 T^{19} + 123422989266523297 T^{20} - 187917230342133614 T^{21} + 550329031716248441 T^{22} \))
$43$ (\( 1 - 6 T + 43 T^{2} \))(\( ( 1 + 6 T + 43 T^{2} )^{2} \))(\( 1 - 4 T + 85 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 - 20 T + 335 T^{2} - 3678 T^{3} + 37846 T^{4} - 311362 T^{5} + 2475607 T^{6} - 16483114 T^{7} + 106451101 T^{8} - 575708338 T^{9} + 3009021922 T^{10} - 12574350078 T^{11} + 49247828405 T^{12} - 126427260980 T^{13} + 271818611107 T^{14} \))(\( 1 - 14 T + 294 T^{2} - 2920 T^{3} + 38795 T^{4} - 318364 T^{5} + 3358980 T^{6} - 24058584 T^{7} + 218270200 T^{8} - 1404268446 T^{9} + 11395036930 T^{10} - 66651925504 T^{11} + 489986587990 T^{12} - 2596492356654 T^{13} + 17354008791400 T^{14} - 82251511037784 T^{15} + 493798419868140 T^{16} - 2012494425731836 T^{17} + 10545203017896065 T^{18} - 34129544810594920 T^{19} + 147762227909431842 T^{20} - 302560752385979486 T^{21} + 929293739471222707 T^{22} \))
$47$ (\( 1 - 9 T + 47 T^{2} \))(\( 1 - 6 T + 101 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( 1 - 19 T + 305 T^{2} - 3046 T^{3} + 30085 T^{4} - 243069 T^{5} + 2048533 T^{6} - 14193428 T^{7} + 96281051 T^{8} - 536939421 T^{9} + 3123514955 T^{10} - 14863508326 T^{11} + 69950227135 T^{12} - 204805091251 T^{13} + 506623120463 T^{14} \))(\( 1 + 16 T + 386 T^{2} + 4060 T^{3} + 56946 T^{4} + 431896 T^{5} + 4529743 T^{6} + 24774048 T^{7} + 224709270 T^{8} + 849971544 T^{9} + 8661617704 T^{10} + 27932602376 T^{11} + 407096032088 T^{12} + 1877587140696 T^{13} + 23329990539210 T^{14} + 120889451318688 T^{15} + 1038873940043201 T^{16} + 4655499983733784 T^{17} + 28850160217885998 T^{18} + 96673823846749660 T^{19} + 431984362617668062 T^{20} + 841586115773280784 T^{21} + 2472159215084012303 T^{22} \))
$53$ (\( 1 - 12 T + 53 T^{2} \))(\( 1 + 88 T^{2} + 2809 T^{4} \))(\( ( 1 + T + 53 T^{2} )^{2} \))(\( 1 - 18 T + 363 T^{2} - 4346 T^{3} + 53700 T^{4} - 489044 T^{5} + 4523561 T^{6} - 32799886 T^{7} + 239748733 T^{8} - 1373724596 T^{9} + 7994694900 T^{10} - 34292030426 T^{11} + 151804963959 T^{12} - 398958500322 T^{13} + 1174711139837 T^{14} \))(\( 1 + 2 T + 226 T^{2} + 930 T^{3} + 30303 T^{4} + 141378 T^{5} + 3051254 T^{6} + 14129762 T^{7} + 242404970 T^{8} + 1073305044 T^{9} + 15542235576 T^{10} + 64551597196 T^{11} + 823738485528 T^{12} + 3014913868596 T^{13} + 36088524718690 T^{14} + 111490618595522 T^{15} + 1276020670798222 T^{16} + 3133553047695762 T^{17} + 35597271670480611 T^{18} + 57901512082565730 T^{19} + 745746571747282058 T^{20} + 349774940731026098 T^{21} + 9269035929372191597 T^{22} \))
$59$ (\( 1 - 5 T + 59 T^{2} \))(\( ( 1 - 3 T + 59 T^{2} )^{2} \))(\( 1 - 12 T + 109 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 + 39 T + 999 T^{2} + 17908 T^{3} + 257608 T^{4} + 2999992 T^{5} + 29574113 T^{6} + 245281391 T^{7} + 1744872667 T^{8} + 10442972152 T^{9} + 52907273432 T^{10} + 216997700788 T^{11} + 714209374701 T^{12} + 1645040811999 T^{13} + 2488651484819 T^{14} \))(\( 1 - 2 T + 415 T^{2} - 346 T^{3} + 83873 T^{4} - 15224 T^{5} + 11188668 T^{6} + 1276308 T^{7} + 1095594928 T^{8} + 253746548 T^{9} + 82369727660 T^{10} + 20324729436 T^{11} + 4859813931940 T^{12} + 883291733588 T^{13} + 225012190717712 T^{14} + 15465484783188 T^{15} + 7999050626643732 T^{16} - 642156444150584 T^{17} + 208730665986223987 T^{18} - 50803331411095066 T^{19} + 3595143264741799685 T^{20} - 1022233506601282802 T^{21} + 30155888444737842659 T^{22} \))
$61$ (\( 1 + 61 T^{2} \))(\( 1 + 4 T + 108 T^{2} + 244 T^{3} + 3721 T^{4} \))(\( ( 1 - 12 T + 61 T^{2} )^{2} \))(\( 1 + 6 T + 231 T^{2} + 1996 T^{3} + 30121 T^{4} + 265034 T^{5} + 2622263 T^{6} + 20730664 T^{7} + 159958043 T^{8} + 986191514 T^{9} + 6836894701 T^{10} + 27636298636 T^{11} + 195101745531 T^{12} + 309122246166 T^{13} + 3142742836021 T^{14} \))(\( 1 - 20 T + 569 T^{2} - 8708 T^{3} + 148697 T^{4} - 1855476 T^{5} + 24030681 T^{6} - 253036176 T^{7} + 2690535802 T^{8} - 24346607896 T^{9} + 219960633754 T^{10} - 1722061094232 T^{11} + 13417598658994 T^{12} - 90593727981016 T^{13} + 610700506873762 T^{14} - 3503498660144016 T^{15} + 20296224283110981 T^{16} - 95594818137850836 T^{17} + 467316431487814637 T^{18} - 1669387281580322948 T^{19} + 6653969126822626229 T^{20} - 14266858233257652020 T^{21} + 43513917611435838661 T^{22} \))
$67$ (\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{7} \))(\( ( 1 + T )^{11} \))
$71$ (\( 1 + 4 T + 71 T^{2} \))(\( 1 - 12 T + 146 T^{2} - 852 T^{3} + 5041 T^{4} \))(\( 1 - 20 T + 222 T^{2} - 1420 T^{3} + 5041 T^{4} \))(\( 1 + 28 T + 766 T^{2} + 13072 T^{3} + 203615 T^{4} + 2444032 T^{5} + 26568885 T^{6} + 235303704 T^{7} + 1886390835 T^{8} + 12320365312 T^{9} + 72876048265 T^{10} + 332181494032 T^{11} + 1382039682866 T^{12} + 3586807949788 T^{13} + 9095120158391 T^{14} \))(\( 1 + 24 T + 806 T^{2} + 14620 T^{3} + 284965 T^{4} + 4133448 T^{5} + 59744017 T^{6} + 715969868 T^{7} + 8352710974 T^{8} + 84280505728 T^{9} + 822737470509 T^{10} + 7051803124976 T^{11} + 58414360406139 T^{12} + 424858029374848 T^{13} + 2989527137415314 T^{14} + 18193997891228108 T^{15} + 107791909018042967 T^{16} + 529495862372689608 T^{17} + 2591790915935891315 T^{18} + 9440916626813025820 T^{19} + 36953891579069918986 T^{20} + 78125845224237148824 T^{21} + \)\(23\!\cdots\!71\)\( T^{22} \))
$73$ (\( 1 + T + 73 T^{2} \))(\( 1 + 14 T + 177 T^{2} + 1022 T^{3} + 5329 T^{4} \))(\( 1 + 6 T + 150 T^{2} + 438 T^{3} + 5329 T^{4} \))(\( 1 - 41 T + 1003 T^{2} - 17142 T^{3} + 231601 T^{4} - 2586535 T^{5} + 25415699 T^{6} - 225623764 T^{7} + 1855346027 T^{8} - 13783645015 T^{9} + 90096726217 T^{10} - 486802647222 T^{11} + 2079290807779 T^{12} - 6204703277849 T^{13} + 11047398519097 T^{14} \))(\( 1 - 16 T + 608 T^{2} - 8324 T^{3} + 173572 T^{4} - 2049016 T^{5} + 31065297 T^{6} - 319256864 T^{7} + 3924866094 T^{8} - 35408346744 T^{9} + 371870632596 T^{10} - 2954746503608 T^{11} + 27146556179508 T^{12} - 188691079798776 T^{13} + 1526839633289598 T^{14} - 9066333364776224 T^{15} + 64400584738808121 T^{16} - 310086251013781624 T^{17} + 1917519055756704484 T^{18} - 6712973804926330244 T^{19} + 35793924718626891104 T^{20} - 68762013275256922384 T^{21} + \)\(31\!\cdots\!77\)\( T^{22} \))
$79$ (\( 1 + 4 T + 79 T^{2} \))(\( ( 1 - 4 T + 79 T^{2} )^{2} \))(\( 1 - 12 T + 114 T^{2} - 948 T^{3} + 6241 T^{4} \))(\( 1 - 24 T + 457 T^{2} - 6184 T^{3} + 65397 T^{4} - 581256 T^{5} + 4757453 T^{6} - 38403696 T^{7} + 375838787 T^{8} - 3627618696 T^{9} + 32243271483 T^{10} - 240867300904 T^{11} + 1406214774343 T^{12} - 5834098932504 T^{13} + 19203908986159 T^{14} \))(\( 1 - 8 T + 489 T^{2} - 4264 T^{3} + 129171 T^{4} - 1127112 T^{5} + 23185275 T^{6} - 192059296 T^{7} + 3079859802 T^{8} - 23390202928 T^{9} + 313367998506 T^{10} - 2124684991344 T^{11} + 24756071881974 T^{12} - 145978256473648 T^{13} + 1518490996918278 T^{14} - 7480725136002976 T^{15} + 71342398801324725 T^{16} - 273986788167185352 T^{17} + 2480588127651144189 T^{18} - 6468951965441576104 T^{19} + 58607430435500357991 T^{20} - 75746208661014777608 T^{21} + \)\(74\!\cdots\!79\)\( T^{22} \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( 1 + 38 T^{2} + 6889 T^{4} \))(\( 1 - 18 T + 202 T^{2} - 1494 T^{3} + 6889 T^{4} \))(\( 1 + 38 T + 1061 T^{2} + 20684 T^{3} + 334813 T^{4} + 4430346 T^{5} + 50689825 T^{6} + 494006632 T^{7} + 4207255475 T^{8} + 30520653594 T^{9} + 191441720831 T^{10} + 981627911564 T^{11} + 4179322122223 T^{12} + 12423734188022 T^{13} + 27136050989627 T^{14} \))(\( 1 + 36 T + 1185 T^{2} + 26112 T^{3} + 521695 T^{4} + 8520564 T^{5} + 127892663 T^{6} + 1677617792 T^{7} + 20465975050 T^{8} + 224865459464 T^{9} + 2316045000682 T^{10} + 21762488934144 T^{11} + 192231735056606 T^{12} + 1549098150247496 T^{13} + 11702178475914350 T^{14} + 79616923688047232 T^{15} + 503774397498502309 T^{16} + 2785716375474460116 T^{17} + 14156742121033457765 T^{18} + 58811854765614638592 T^{19} + \)\(22\!\cdots\!55\)\( T^{20} + \)\(55\!\cdots\!64\)\( T^{21} + \)\(12\!\cdots\!67\)\( T^{22} \))
$89$ (\( 1 - 3 T + 89 T^{2} \))(\( 1 - 18 T + 227 T^{2} - 1602 T^{3} + 7921 T^{4} \))(\( 1 + 10 T + 183 T^{2} + 890 T^{3} + 7921 T^{4} \))(\( 1 + T + 445 T^{2} + 442 T^{3} + 95718 T^{4} + 83766 T^{5} + 12760587 T^{6} + 9374795 T^{7} + 1135692243 T^{8} + 663510486 T^{9} + 67478222742 T^{10} + 27732070522 T^{11} + 2484906454805 T^{12} + 496981290961 T^{13} + 44231334895529 T^{14} \))(\( 1 - 14 T + 351 T^{2} - 4690 T^{3} + 84713 T^{4} - 1009512 T^{5} + 14202560 T^{6} - 154705492 T^{7} + 1897961660 T^{8} - 19000139010 T^{9} + 205109608404 T^{10} - 1857843010638 T^{11} + 18254755147956 T^{12} - 150500101098210 T^{13} + 1338004133488540 T^{14} - 9706569263087572 T^{15} + 79307939367989440 T^{16} - 501708577000621032 T^{17} + 3746969073004948177 T^{18} - 18462601498742759890 T^{19} + \)\(12\!\cdots\!59\)\( T^{20} - \)\(43\!\cdots\!14\)\( T^{21} + \)\(27\!\cdots\!89\)\( T^{22} \))
$97$ (\( 1 + 14 T + 97 T^{2} \))(\( 1 + 4 T + 36 T^{2} + 388 T^{3} + 9409 T^{4} \))(\( 1 - 6 T + 183 T^{2} - 582 T^{3} + 9409 T^{4} \))(\( 1 - 18 T + 541 T^{2} - 8044 T^{3} + 137990 T^{4} - 1631768 T^{5} + 21022429 T^{6} - 198030280 T^{7} + 2039175613 T^{8} - 15353305112 T^{9} + 125939747270 T^{10} - 712129536364 T^{11} + 4645751079037 T^{12} - 14993496088722 T^{13} + 80798284478113 T^{14} \))(\( 1 - 20 T + 868 T^{2} - 14570 T^{3} + 353427 T^{4} - 5010208 T^{5} + 89110856 T^{6} - 1082261340 T^{7} + 15576051840 T^{8} - 164142335972 T^{9} + 2001190046706 T^{10} - 18386760801608 T^{11} + 194115434530482 T^{12} - 1544415239160548 T^{13} + 14215841960968320 T^{14} - 95811818284296540 T^{15} + 765225241064529992 T^{16} - 4173363002871315232 T^{17} + 28556295288246043251 T^{18} - \)\(11\!\cdots\!70\)\( T^{19} + \)\(65\!\cdots\!56\)\( T^{20} - \)\(14\!\cdots\!80\)\( T^{21} + \)\(71\!\cdots\!53\)\( T^{22} \))
show more
show less