↑

Properties

Label 504.2.r
Level 504
Weight 2
Character orbit r
Rep. character \(\chi_{504}(169,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 36
Newform subspaces 6
Sturm bound 192
Trace bound 3

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 504.r (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 6 \)
Sturm bound: \(192\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(504, [\chi])\).

Total New Old
Modular forms 208 36 172
Cusp forms 176 36 140
Eisenstein series 32 0 32

Trace form

\( 36q - 2q^{3} + 6q^{9} + O(q^{10}) \) \( 36q - 2q^{3} + 6q^{9} + 14q^{11} + 20q^{15} + 12q^{17} - 12q^{19} + 4q^{23} - 18q^{25} - 20q^{27} - 12q^{29} - 18q^{33} - 24q^{35} - 28q^{39} - 6q^{41} + 6q^{43} + 36q^{45} - 12q^{47} - 18q^{49} + 38q^{51} + 48q^{53} + 30q^{57} + 14q^{59} - 4q^{63} - 16q^{65} + 6q^{67} - 16q^{69} + 24q^{71} + 36q^{73} - 22q^{75} - 8q^{77} + 6q^{81} + 16q^{83} - 24q^{85} + 40q^{87} - 48q^{89} - 44q^{93} + 12q^{95} - 42q^{97} + 36q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(504, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
504.2.r.a \(2\) \(4.024\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(1\) \(-1\) \(q+(-1+2\zeta_{6})q^{3}+(1-\zeta_{6})q^{5}-\zeta_{6}q^{7}+\cdots\)
504.2.r.b \(2\) \(4.024\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-2\) \(-1\) \(q+(2-\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{5}-\zeta_{6}q^{7}+\cdots\)
504.2.r.c \(6\) \(4.024\) \(\Q(\zeta_{18})\) None \(0\) \(0\) \(3\) \(-3\) \(q+(-\zeta_{18}^{2}-\zeta_{18}^{5})q^{3}+(1-\zeta_{18}+\zeta_{18}^{2}+\cdots)q^{5}+\cdots\)
504.2.r.d \(8\) \(4.024\) 8.0.508277025.1 None \(0\) \(-4\) \(4\) \(-4\) \(q+(-\beta _{4}-\beta _{6})q^{3}+(1+\beta _{1}+2\beta _{3}-\beta _{4}+\cdots)q^{5}+\cdots\)
504.2.r.e \(8\) \(4.024\) 8.0.2091141441.1 None \(0\) \(-1\) \(-3\) \(4\) \(q+\beta _{4}q^{3}+(\beta _{2}+\beta _{4}+\beta _{6})q^{5}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)
504.2.r.f \(10\) \(4.024\) 10.0.\(\cdots\).1 None \(0\) \(0\) \(-3\) \(5\) \(q+\beta _{6}q^{3}+(-\beta _{2}-\beta _{7})q^{5}+(1-\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(504, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(504, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + 9 T^{3} + 27 T^{6} \))(\( 1 + 4 T + 3 T^{2} - 11 T^{3} - 32 T^{4} - 33 T^{5} + 27 T^{6} + 108 T^{7} + 81 T^{8} \))(\( 1 + T + T^{2} - 3 T^{3} + 3 T^{4} - 9 T^{5} + 9 T^{6} + 27 T^{7} + 81 T^{8} \))(\( 1 + 3 T^{3} + 3 T^{4} - 18 T^{5} + 9 T^{6} + 27 T^{7} + 243 T^{10} \))
$5$ (\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 - 3 T - 6 T^{2} + 13 T^{3} + 63 T^{4} - 60 T^{5} - 259 T^{6} - 300 T^{7} + 1575 T^{8} + 1625 T^{9} - 3750 T^{10} - 9375 T^{11} + 15625 T^{12} \))(\( 1 - 4 T + 5 T^{2} + 18 T^{3} - 94 T^{4} + 232 T^{5} - 110 T^{6} - 1011 T^{7} + 3826 T^{8} - 5055 T^{9} - 2750 T^{10} + 29000 T^{11} - 58750 T^{12} + 56250 T^{13} + 78125 T^{14} - 312500 T^{15} + 390625 T^{16} \))(\( 1 + 3 T - 3 T^{2} - 36 T^{3} - 32 T^{4} + 204 T^{5} + 540 T^{6} - 597 T^{7} - 3831 T^{8} - 2985 T^{9} + 13500 T^{10} + 25500 T^{11} - 20000 T^{12} - 112500 T^{13} - 46875 T^{14} + 234375 T^{15} + 390625 T^{16} \))(\( 1 + 3 T - 3 T^{2} - 16 T^{3} - 25 T^{4} - 63 T^{5} - 19 T^{6} + 277 T^{7} + 775 T^{8} + 451 T^{9} - 1814 T^{10} + 2255 T^{11} + 19375 T^{12} + 34625 T^{13} - 11875 T^{14} - 196875 T^{15} - 390625 T^{16} - 1250000 T^{17} - 1171875 T^{18} + 5859375 T^{19} + 9765625 T^{20} \))
$7$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( ( 1 + T + T^{2} )^{3} \))(\( ( 1 + T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{5} \))
$11$ (\( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 - 24 T^{2} - 18 T^{3} + 312 T^{4} + 216 T^{5} - 3593 T^{6} + 2376 T^{7} + 37752 T^{8} - 23958 T^{9} - 351384 T^{10} + 1771561 T^{12} \))(\( 1 + 6 T + T^{2} - 24 T^{3} + 49 T^{4} - 306 T^{5} - 2153 T^{6} - 1236 T^{7} + 7063 T^{8} - 13596 T^{9} - 260513 T^{10} - 407286 T^{11} + 717409 T^{12} - 3865224 T^{13} + 1771561 T^{14} + 116923026 T^{15} + 214358881 T^{16} \))(\( 1 - 7 T - 2 T^{2} + 81 T^{3} + 203 T^{4} - 1468 T^{5} - 675 T^{6} + 3769 T^{7} + 21730 T^{8} + 41459 T^{9} - 81675 T^{10} - 1953908 T^{11} + 2972123 T^{12} + 13045131 T^{13} - 3543122 T^{14} - 136410197 T^{15} + 214358881 T^{16} \))(\( 1 - 4 T - 6 T^{2} - 10 T^{3} + 191 T^{4} + 240 T^{5} + 1529 T^{6} - 16178 T^{7} - 5837 T^{8} + 14002 T^{9} + 502605 T^{10} + 154022 T^{11} - 706277 T^{12} - 21532918 T^{13} + 22386089 T^{14} + 38652240 T^{15} + 338368151 T^{16} - 194871710 T^{17} - 1286153286 T^{18} - 9431790764 T^{19} + 25937424601 T^{20} \))
$13$ (\( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} \))(\( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} \))(\( 1 - 3 T - 21 T^{2} + 60 T^{3} + 285 T^{4} - 417 T^{5} - 3202 T^{6} - 5421 T^{7} + 48165 T^{8} + 131820 T^{9} - 599781 T^{10} - 1113879 T^{11} + 4826809 T^{12} \))(\( 1 + 3 T - 16 T^{2} + 39 T^{3} + 337 T^{4} - 720 T^{5} + 2222 T^{6} + 11778 T^{7} - 44096 T^{8} + 153114 T^{9} + 375518 T^{10} - 1581840 T^{11} + 9625057 T^{12} + 14480427 T^{13} - 77228944 T^{14} + 188245551 T^{15} + 815730721 T^{16} \))(\( 1 - 3 T - 32 T^{2} + 57 T^{3} + 673 T^{4} - 432 T^{5} - 11810 T^{6} + 2142 T^{7} + 166240 T^{8} + 27846 T^{9} - 1995890 T^{10} - 949104 T^{11} + 19221553 T^{12} + 21163701 T^{13} - 154457888 T^{14} - 188245551 T^{15} + 815730721 T^{16} \))(\( 1 + 3 T - 31 T^{2} - 106 T^{3} + 402 T^{4} + 1426 T^{5} - 5437 T^{6} - 15135 T^{7} + 85123 T^{8} + 101952 T^{9} - 1104756 T^{10} + 1325376 T^{11} + 14385787 T^{12} - 33251595 T^{13} - 155286157 T^{14} + 529463818 T^{15} + 1940377218 T^{16} - 6651342802 T^{17} - 25287652351 T^{18} + 31813498119 T^{19} + 137858491849 T^{20} \))
$17$ (\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 7 T + 17 T^{2} )^{2} \))(\( ( 1 - 6 T + 54 T^{2} - 203 T^{3} + 918 T^{4} - 1734 T^{5} + 4913 T^{6} )^{2} \))(\( ( 1 + 8 T + 35 T^{2} + 167 T^{3} + 886 T^{4} + 2839 T^{5} + 10115 T^{6} + 39304 T^{7} + 83521 T^{8} )^{2} \))(\( ( 1 - 3 T + 41 T^{2} - 189 T^{3} + 807 T^{4} - 3213 T^{5} + 11849 T^{6} - 14739 T^{7} + 83521 T^{8} )^{2} \))(\( ( 1 + 46 T^{2} - 9 T^{3} + 1261 T^{4} - 198 T^{5} + 21437 T^{6} - 2601 T^{7} + 225998 T^{8} + 1419857 T^{10} )^{2} \))
$19$ (\( ( 1 - 7 T + 19 T^{2} )^{2} \))(\( ( 1 - T + 19 T^{2} )^{2} \))(\( ( 1 + 9 T + 63 T^{2} + 323 T^{3} + 1197 T^{4} + 3249 T^{5} + 6859 T^{6} )^{2} \))(\( ( 1 + 2 T + 55 T^{2} + 41 T^{3} + 1309 T^{4} + 779 T^{5} + 19855 T^{6} + 13718 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 + 4 T + 49 T^{2} + 193 T^{3} + 1297 T^{4} + 3667 T^{5} + 17689 T^{6} + 27436 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 - T + 42 T^{2} - 66 T^{3} + 1281 T^{4} - 1209 T^{5} + 24339 T^{6} - 23826 T^{7} + 288078 T^{8} - 130321 T^{9} + 2476099 T^{10} )^{2} \))
$23$ (\( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \))(\( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 12 T^{2} + 130 T^{3} - 18 T^{4} - 3282 T^{5} - 7909 T^{6} - 75486 T^{7} - 9522 T^{8} + 1581710 T^{9} + 3358092 T^{10} + 38618058 T^{11} + 148035889 T^{12} \))(\( 1 + 5 T - 28 T^{2} - 177 T^{3} - 25 T^{4} - 494 T^{5} - 12137 T^{6} + 44331 T^{7} + 708646 T^{8} + 1019613 T^{9} - 6420473 T^{10} - 6010498 T^{11} - 6996025 T^{12} - 1139232711 T^{13} - 4145004892 T^{14} + 17024127235 T^{15} + 78310985281 T^{16} \))(\( 1 - 2 T - 59 T^{2} + 192 T^{3} + 1775 T^{4} - 6170 T^{5} - 33567 T^{6} + 71510 T^{7} + 655807 T^{8} + 1644730 T^{9} - 17756943 T^{10} - 75070390 T^{11} + 496717775 T^{12} + 1235777856 T^{13} - 8734117451 T^{14} - 6809650894 T^{15} + 78310985281 T^{16} \))(\( 1 - 8 T + 9 T^{2} - 222 T^{3} + 2650 T^{4} - 9138 T^{5} + 54826 T^{6} - 441030 T^{7} + 1653286 T^{8} - 8929410 T^{9} + 60424408 T^{10} - 205376430 T^{11} + 874588294 T^{12} - 5366012010 T^{13} + 15342562666 T^{14} - 58815302334 T^{15} + 392295105850 T^{16} - 755871249234 T^{17} + 704798867529 T^{18} - 14409221291704 T^{19} + 41426511213649 T^{20} \))
$29$ (\( 1 + 2 T - 25 T^{2} + 58 T^{3} + 841 T^{4} \))(\( 1 - 4 T - 13 T^{2} - 116 T^{3} + 841 T^{4} \))(\( 1 - 3 T - 42 T^{2} + 157 T^{3} + 657 T^{4} - 1554 T^{5} - 10195 T^{6} - 45066 T^{7} + 552537 T^{8} + 3829073 T^{9} - 29705802 T^{10} - 61533447 T^{11} + 594823321 T^{12} \))(\( 1 - T - 49 T^{2} + 294 T^{3} + 1136 T^{4} - 11204 T^{5} + 49096 T^{6} + 227337 T^{7} - 2198075 T^{8} + 6592773 T^{9} + 41289736 T^{10} - 273254356 T^{11} + 803471216 T^{12} + 6030277806 T^{13} - 29146342729 T^{14} - 17249876309 T^{15} + 500246412961 T^{16} \))(\( 1 + 9 T + 13 T^{2} + 72 T^{3} + 904 T^{4} + 2970 T^{5} + 29086 T^{6} + 17613 T^{7} - 1095047 T^{8} + 510777 T^{9} + 24461326 T^{10} + 72435330 T^{11} + 639382024 T^{12} + 1476802728 T^{13} + 7732703173 T^{14} + 155248886781 T^{15} + 500246412961 T^{16} \))(\( 1 + 9 T - 52 T^{2} - 657 T^{3} + 1944 T^{4} + 29007 T^{5} - 46389 T^{6} - 788049 T^{7} + 920775 T^{8} + 10497078 T^{9} - 9302769 T^{10} + 304415262 T^{11} + 774371775 T^{12} - 19219727061 T^{13} - 32810058309 T^{14} + 594966899043 T^{15} + 1156336536024 T^{16} - 11333168735013 T^{17} - 26012813473972 T^{18} + 130564313782821 T^{19} + 420707233300201 T^{20} \))
$31$ (\( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} \))(\( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} \))(\( 1 - 9 T - 18 T^{2} + 79 T^{3} + 2367 T^{4} - 1422 T^{5} - 86865 T^{6} - 44082 T^{7} + 2274687 T^{8} + 2353489 T^{9} - 16623378 T^{10} - 257662359 T^{11} + 887503681 T^{12} \))(\( 1 - 11 T - 39 T^{2} + 356 T^{3} + 5954 T^{4} - 25902 T^{5} - 215174 T^{6} + 3835 T^{7} + 11112081 T^{8} + 118885 T^{9} - 206782214 T^{10} - 771646482 T^{11} + 5498644034 T^{12} + 10191977756 T^{13} - 34612643559 T^{14} - 302638755221 T^{15} + 852891037441 T^{16} \))(\( 1 - 3 T - 41 T^{2} + 354 T^{3} - 128 T^{4} - 9180 T^{5} + 42568 T^{6} + 75933 T^{7} - 1254023 T^{8} + 2353923 T^{9} + 40907848 T^{10} - 273481380 T^{11} - 118210688 T^{12} + 10134719454 T^{13} - 36387650921 T^{14} - 82537842333 T^{15} + 852891037441 T^{16} \))(\( 1 + 3 T - 106 T^{2} - 157 T^{3} + 6498 T^{4} + 2989 T^{5} - 283525 T^{6} + 3627 T^{7} + 9910699 T^{8} - 942522 T^{9} - 316500783 T^{10} - 29218182 T^{11} + 9524181739 T^{12} + 108051957 T^{13} - 261841291525 T^{14} + 85572532339 T^{15} + 5766998919138 T^{16} - 4319480415427 T^{17} - 90406449968746 T^{18} + 79318866482013 T^{19} + 819628286980801 T^{20} \))
$37$ (\( ( 1 + 6 T + 37 T^{2} )^{2} \))(\( ( 1 + 6 T + 37 T^{2} )^{2} \))(\( ( 1 + 15 T + 165 T^{2} + 1167 T^{3} + 6105 T^{4} + 20535 T^{5} + 50653 T^{6} )^{2} \))(\( ( 1 - 27 T + 412 T^{2} - 4104 T^{3} + 29436 T^{4} - 151848 T^{5} + 564028 T^{6} - 1367631 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 - 3 T + 140 T^{2} - 324 T^{3} + 7620 T^{4} - 11988 T^{5} + 191660 T^{6} - 151959 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 + 3 T + 103 T^{2} + 197 T^{3} + 6112 T^{4} + 11312 T^{5} + 226144 T^{6} + 269693 T^{7} + 5217259 T^{8} + 5622483 T^{9} + 69343957 T^{10} )^{2} \))
$41$ (\( 1 - 8 T + 23 T^{2} - 328 T^{3} + 1681 T^{4} \))(\( 1 + 7 T + 8 T^{2} + 287 T^{3} + 1681 T^{4} \))(\( 1 + 6 T - 72 T^{2} - 298 T^{3} + 4398 T^{4} + 8034 T^{5} - 177169 T^{6} + 329394 T^{7} + 7393038 T^{8} - 20538458 T^{9} - 203454792 T^{10} + 695137206 T^{11} + 4750104241 T^{12} \))(\( 1 - 2 T - 37 T^{2} + 432 T^{3} - 841 T^{4} - 13042 T^{5} + 100489 T^{6} + 72294 T^{7} - 3776099 T^{8} + 2964054 T^{9} + 168922009 T^{10} - 898867682 T^{11} - 2376465001 T^{12} + 50049878832 T^{13} - 175753856917 T^{14} - 389508547762 T^{15} + 7984925229121 T^{16} \))(\( 1 - 9 T - 36 T^{2} + 381 T^{3} + 895 T^{4} + 852 T^{5} - 99027 T^{6} - 57981 T^{7} + 4458864 T^{8} - 2377221 T^{9} - 166464387 T^{10} + 58720692 T^{11} + 2529056095 T^{12} + 44141212581 T^{13} - 171003752676 T^{14} - 1752788464929 T^{15} + 7984925229121 T^{16} \))(\( 1 + 12 T - 30 T^{2} - 790 T^{3} - 277 T^{4} + 13968 T^{5} - 140563 T^{6} - 610562 T^{7} + 9173539 T^{8} + 24360490 T^{9} - 273097535 T^{10} + 998780090 T^{11} + 15420719059 T^{12} - 42080543602 T^{13} - 397197443443 T^{14} + 1618279415568 T^{15} - 1315778874757 T^{16} - 153855876365990 T^{17} - 239547756873630 T^{18} + 3928583212727532 T^{19} + 13422659310152401 T^{20} \))
$43$ (\( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( 1 + 11 T + 78 T^{2} + 473 T^{3} + 1849 T^{4} \))(\( 1 - 15 T + 42 T^{2} - 199 T^{3} + 6525 T^{4} - 24624 T^{5} - 76509 T^{6} - 1058832 T^{7} + 12064725 T^{8} - 15821893 T^{9} + 143589642 T^{10} - 2205126645 T^{11} + 6321363049 T^{12} \))(\( 1 + 11 T - 27 T^{2} - 596 T^{3} + 140 T^{4} + 2688 T^{5} - 152864 T^{6} + 330797 T^{7} + 13009029 T^{8} + 14224271 T^{9} - 282645536 T^{10} + 213714816 T^{11} + 478632140 T^{12} - 87617032028 T^{13} - 170676802323 T^{14} + 2990004722177 T^{15} + 11688200277601 T^{16} \))(\( 1 - 8 T - 73 T^{2} + 914 T^{3} + 2642 T^{4} - 51508 T^{5} + 63012 T^{6} + 1050915 T^{7} - 5977292 T^{8} + 45189345 T^{9} + 116509188 T^{10} - 4095246556 T^{11} + 9032472242 T^{12} + 134365716902 T^{13} - 461459502577 T^{14} - 2174548888856 T^{15} + 11688200277601 T^{16} \))(\( 1 + 5 T - 88 T^{2} + 275 T^{3} + 6540 T^{4} - 38663 T^{5} - 127543 T^{6} + 2251353 T^{7} - 2752001 T^{8} - 27649602 T^{9} + 456453449 T^{10} - 1188932886 T^{11} - 5088449849 T^{12} + 178998322971 T^{13} - 436044135943 T^{14} - 5683787431709 T^{15} + 41341714340460 T^{16} + 74750118054425 T^{17} - 1028561624428888 T^{18} + 2512963059684215 T^{19} + 21611482313284249 T^{20} \))
$47$ (\( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 + 9 T - 66 T^{2} - 371 T^{3} + 7077 T^{4} + 20028 T^{5} - 269497 T^{6} + 941316 T^{7} + 15633093 T^{8} - 38518333 T^{9} - 322058946 T^{10} + 2064105063 T^{11} + 10779215329 T^{12} \))(\( 1 - 7 T - 61 T^{2} - 174 T^{3} + 5108 T^{4} + 21898 T^{5} - 62918 T^{6} - 808359 T^{7} - 1538879 T^{8} - 37992873 T^{9} - 138985862 T^{10} + 2273516054 T^{11} + 24925410548 T^{12} - 39906031218 T^{13} - 657532135069 T^{14} - 3546361843241 T^{15} + 23811286661761 T^{16} \))(\( 1 - 3 T - 141 T^{2} + 486 T^{3} + 11422 T^{4} - 34152 T^{5} - 635364 T^{6} + 765849 T^{7} + 30578781 T^{8} + 35994903 T^{9} - 1403519076 T^{10} - 3545763096 T^{11} + 55735716382 T^{12} + 111461673402 T^{13} - 1519869361389 T^{14} - 1519869361389 T^{15} + 23811286661761 T^{16} \))(\( 1 - 3 T - 24 T^{2} + 751 T^{3} - 2770 T^{4} - 11169 T^{5} + 300305 T^{6} - 1566601 T^{7} - 499337 T^{8} + 58566752 T^{9} - 599985995 T^{10} + 2752637344 T^{11} - 1103035433 T^{12} - 162649215623 T^{13} + 1465392602705 T^{14} - 2561554383183 T^{15} - 29858426461330 T^{16} + 380473963467713 T^{17} - 571470879882264 T^{18} - 3357391419308301 T^{19} + 52599132235830049 T^{20} \))
$53$ (\( ( 1 - 2 T + 53 T^{2} )^{2} \))(\( ( 1 + 4 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 162 T^{2} + 617 T^{3} + 8586 T^{4} + 16854 T^{5} + 148877 T^{6} )^{2} \))(\( ( 1 + 4 T + 83 T^{2} - 197 T^{3} + 2722 T^{4} - 10441 T^{5} + 233147 T^{6} + 595508 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 - 6 T + 59 T^{2} - 621 T^{3} + 4704 T^{4} - 32913 T^{5} + 165731 T^{6} - 893262 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 - 30 T + 580 T^{2} - 7683 T^{3} + 79903 T^{4} - 646182 T^{5} + 4234859 T^{6} - 21581547 T^{7} + 86348660 T^{8} - 236714430 T^{9} + 418195493 T^{10} )^{2} \))
$59$ (\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 24 T^{2} + 369 T^{3} - 453 T^{4} - 5694 T^{5} + 333403 T^{6} - 335946 T^{7} - 1576893 T^{8} + 75784851 T^{9} - 290816664 T^{10} + 2144772897 T^{11} + 42180533641 T^{12} \))(\( 1 - 9 T - 119 T^{2} + 1116 T^{3} + 10336 T^{4} - 79866 T^{5} - 564434 T^{6} + 2176353 T^{7} + 30131725 T^{8} + 128404827 T^{9} - 1964794754 T^{10} - 16402799214 T^{11} + 125245043296 T^{12} + 797855517684 T^{13} - 5019483503279 T^{14} - 22397863363371 T^{15} + 146830437604321 T^{16} \))(\( 1 - 10 T - 71 T^{2} + 1536 T^{3} - 346 T^{4} - 106150 T^{5} + 583014 T^{6} + 3064183 T^{7} - 49833644 T^{8} + 180786797 T^{9} + 2029471734 T^{10} - 21800980850 T^{11} - 4192606906 T^{12} + 1098123723264 T^{13} - 2994817888511 T^{14} - 24886514848190 T^{15} + 146830437604321 T^{16} \))(\( 1 - 7 T - 84 T^{2} + 891 T^{3} - 548 T^{4} - 21051 T^{5} + 190525 T^{6} - 1336983 T^{7} + 4615039 T^{8} + 49687002 T^{9} - 884146943 T^{10} + 2931533118 T^{11} + 16064950759 T^{12} - 274588231557 T^{13} + 2308660204525 T^{14} - 15049871418249 T^{15} - 23114932435268 T^{16} + 2217388472973729 T^{17} - 12333756758762964 T^{18} - 60640970730584573 T^{19} + 511116753300641401 T^{20} \))
$61$ (\( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \))(\( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} \))(\( 1 - 12 T - 48 T^{2} + 478 T^{3} + 8460 T^{4} - 30636 T^{5} - 395685 T^{6} - 1868796 T^{7} + 31479660 T^{8} + 108496918 T^{9} - 664600368 T^{10} - 10135155612 T^{11} + 51520374361 T^{12} \))(\( 1 + 7 T - 144 T^{2} - 1315 T^{3} + 11783 T^{4} + 112692 T^{5} - 470051 T^{6} - 3262925 T^{7} + 21742764 T^{8} - 199038425 T^{9} - 1749059771 T^{10} + 25578942852 T^{11} + 163145544503 T^{12} - 1110644135815 T^{13} - 7418933907984 T^{14} + 21999199852147 T^{15} + 191707312997281 T^{16} \))(\( 1 - 20 T + 161 T^{2} - 562 T^{3} - 931 T^{4} + 2180 T^{5} - 139869 T^{6} + 5694174 T^{7} - 64208639 T^{8} + 347344614 T^{9} - 520452549 T^{10} + 494818580 T^{11} - 12890477971 T^{12} - 474663121162 T^{13} + 8294780272121 T^{14} - 62854856720420 T^{15} + 191707312997281 T^{16} \))(\( 1 + 14 T - 67 T^{2} - 1032 T^{3} + 10040 T^{4} + 70122 T^{5} - 675608 T^{6} - 3001836 T^{7} + 20748698 T^{8} - 6195900 T^{9} - 1395441908 T^{10} - 377949900 T^{11} + 77205905258 T^{12} - 681359737116 T^{13} - 9354360946328 T^{14} + 59224781818722 T^{15} + 517264558584440 T^{16} - 3243310606773672 T^{17} - 12844389970817827 T^{18} + 163718045299677974 T^{19} + 713342911662882601 T^{20} \))
$67$ (\( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} \))(\( 1 + 9 T + 14 T^{2} + 603 T^{3} + 4489 T^{4} \))(\( 1 - 12 T - 12 T^{2} + 690 T^{3} - 1488 T^{4} - 2856 T^{5} - 34441 T^{6} - 191352 T^{7} - 6679632 T^{8} + 207526470 T^{9} - 241813452 T^{10} - 16201501284 T^{11} + 90458382169 T^{12} \))(\( 1 + 12 T - 139 T^{2} - 1338 T^{3} + 22729 T^{4} + 133506 T^{5} - 1954153 T^{6} - 2368722 T^{7} + 170018875 T^{8} - 158704374 T^{9} - 8772192817 T^{10} + 40153665078 T^{11} + 458014829209 T^{12} - 1806467393166 T^{13} - 12573715121491 T^{14} + 72728539263876 T^{15} + 406067677556641 T^{16} \))(\( 1 - 11 T - 82 T^{2} + 689 T^{3} + 8039 T^{4} - 18292 T^{5} - 590799 T^{6} + 1428153 T^{7} + 17009218 T^{8} + 95686251 T^{9} - 2652096711 T^{10} - 5501556796 T^{11} + 161994861719 T^{12} + 930236198723 T^{13} - 7417587337858 T^{14} - 66667827658553 T^{15} + 406067677556641 T^{16} \))(\( 1 + 8 T - 142 T^{2} - 1462 T^{3} + 7779 T^{4} + 110116 T^{5} - 343663 T^{6} - 6410154 T^{7} + 10809007 T^{8} + 209041554 T^{9} + 61061861 T^{10} + 14005784118 T^{11} + 48521632423 T^{12} - 1927937147502 T^{13} - 6925194696223 T^{14} + 148670376282412 T^{15} + 703675754892651 T^{16} - 8860760366982226 T^{17} - 57661610213043022 T^{18} + 217652275170359576 T^{19} + 1822837804551761449 T^{20} \))
$71$ (\( ( 1 - 15 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 21 T + 249 T^{2} - 2115 T^{3} + 17679 T^{4} - 105861 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 + 12 T + 167 T^{2} + 591 T^{3} + 7587 T^{4} + 41961 T^{5} + 841847 T^{6} + 4294932 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 + 3 T + 276 T^{2} + 630 T^{3} + 29126 T^{4} + 44730 T^{5} + 1391316 T^{6} + 1073733 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 + 9 T + 306 T^{2} + 2174 T^{3} + 40411 T^{4} + 221049 T^{5} + 2869181 T^{6} + 10959134 T^{7} + 109520766 T^{8} + 228705129 T^{9} + 1804229351 T^{10} )^{2} \))
$73$ (\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 - 13 T + 73 T^{2} )^{2} \))(\( ( 1 - 3 T + 105 T^{2} - 475 T^{3} + 7665 T^{4} - 15987 T^{5} + 389017 T^{6} )^{2} \))(\( ( 1 - 13 T + 232 T^{2} - 1504 T^{3} + 18820 T^{4} - 109792 T^{5} + 1236328 T^{6} - 5057221 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 + 24 T + 425 T^{2} + 4923 T^{3} + 48495 T^{4} + 359379 T^{5} + 2264825 T^{6} + 9336408 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 - 15 T + 247 T^{2} - 2719 T^{3} + 26596 T^{4} - 243028 T^{5} + 1941508 T^{6} - 14489551 T^{7} + 96087199 T^{8} - 425973615 T^{9} + 2073071593 T^{10} )^{2} \))
$79$ (\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} \))(\( 1 - 15 T + 30 T^{2} - 59 T^{3} + 1125 T^{4} + 89100 T^{5} - 1368633 T^{6} + 7038900 T^{7} + 7021125 T^{8} - 29089301 T^{9} + 1168502430 T^{10} - 46155845985 T^{11} + 243087455521 T^{12} \))(\( 1 + 22 T + 237 T^{2} + 1616 T^{3} + 2882 T^{4} - 87342 T^{5} - 574370 T^{6} + 1050079 T^{7} + 29800944 T^{8} + 82956241 T^{9} - 3584643170 T^{10} - 43063012338 T^{11} + 112254133442 T^{12} + 4972523140784 T^{13} + 57611726958477 T^{14} + 422485997695498 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 21 T + 133 T^{2} + 252 T^{3} - 9896 T^{4} + 136080 T^{5} - 973784 T^{6} - 2121567 T^{7} + 81625693 T^{8} - 167603793 T^{9} - 6077385944 T^{10} + 67092747120 T^{11} - 385450001576 T^{12} + 775418212548 T^{13} + 32330631584293 T^{14} - 403282088709339 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 3 T - 271 T^{2} - 1532 T^{3} + 38715 T^{4} + 263753 T^{5} - 3446203 T^{6} - 24313053 T^{7} + 236347015 T^{8} + 845708523 T^{9} - 15972358530 T^{10} + 66810973317 T^{11} + 1475041720615 T^{12} - 11987283338067 T^{13} - 134229885992443 T^{14} + 811582856405447 T^{15} + 9411130840495515 T^{16} - 29420388566795588 T^{17} - 411136487484678031 T^{18} + 359554787947854957 T^{19} + 9468276082626847201 T^{20} \))
$83$ (\( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \))(\( 1 - 83 T^{2} + 6889 T^{4} \))(\( 1 - 6 T - 42 T^{2} + 2610 T^{3} - 9624 T^{4} - 83688 T^{5} + 2759119 T^{6} - 6946104 T^{7} - 66299736 T^{8} + 1492364070 T^{9} - 1993249482 T^{10} - 23634243858 T^{11} + 326940373369 T^{12} \))(\( 1 + 6 T - 113 T^{2} - 1704 T^{3} + 3151 T^{4} + 166296 T^{5} + 1416277 T^{6} - 7918434 T^{7} - 172589093 T^{8} - 657230022 T^{9} + 9756732253 T^{10} + 95085890952 T^{11} + 149541169471 T^{12} - 6712125255672 T^{13} - 36944262190697 T^{14} + 162816305937762 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 8 T - 259 T^{2} + 1146 T^{3} + 49577 T^{4} - 119878 T^{5} - 6175577 T^{6} + 3335418 T^{7} + 604903483 T^{8} + 276839694 T^{9} - 42543549953 T^{10} - 68544681986 T^{11} + 2352841180217 T^{12} + 4514140576878 T^{13} - 84677556702571 T^{14} - 217088407917016 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 20 T + 98 T^{2} - 318 T^{3} + 879 T^{4} + 169800 T^{5} - 2124519 T^{6} + 6011178 T^{7} + 51057123 T^{8} - 1398046658 T^{9} + 18343755493 T^{10} - 116037872614 T^{11} + 351732520347 T^{12} + 3437113435086 T^{13} - 100826104672599 T^{14} + 668849101181400 T^{15} + 287380588191351 T^{16} - 8629264214701386 T^{17} + 220724638749626018 T^{18} - 3738805105350808060 T^{19} + 15516041187205853449 T^{20} \))
$89$ (\( ( 1 - 4 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 18 T + 318 T^{2} + 3241 T^{3} + 28302 T^{4} + 142578 T^{5} + 704969 T^{6} )^{2} \))(\( ( 1 + 14 T + 323 T^{2} + 2963 T^{3} + 41152 T^{4} + 263707 T^{5} + 2558483 T^{6} + 9869566 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 6 T + 263 T^{2} - 1377 T^{3} + 32646 T^{4} - 122553 T^{5} + 2083223 T^{6} - 4229814 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 12 T + 292 T^{2} + 2463 T^{3} + 41383 T^{4} + 291726 T^{5} + 3683087 T^{6} + 19509423 T^{7} + 205850948 T^{8} + 752906892 T^{9} + 5584059449 T^{10} )^{2} \))
$97$ (\( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} \))(\( 1 + 7 T - 48 T^{2} + 679 T^{3} + 9409 T^{4} \))(\( 1 + 15 T + 168 T^{2} + 1477 T^{3} + 2385 T^{4} - 42462 T^{5} - 510111 T^{6} - 4118814 T^{7} + 22440465 T^{8} + 1348018021 T^{9} + 14872919208 T^{10} + 128810103855 T^{11} + 832972004929 T^{12} \))(\( 1 + T - 123 T^{2} - 856 T^{3} - 4138 T^{4} + 70242 T^{5} + 167026 T^{6} - 386291 T^{7} + 107246157 T^{8} - 37470227 T^{9} + 1571547634 T^{10} + 64107976866 T^{11} - 366334164778 T^{12} - 7350763259992 T^{13} - 102455556606267 T^{14} + 80798284478113 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 16 T - 175 T^{2} + 2086 T^{3} + 49490 T^{4} - 309968 T^{5} - 6453258 T^{6} + 6735477 T^{7} + 823825414 T^{8} + 653341269 T^{9} - 60718704522 T^{10} - 282899424464 T^{11} + 4381314116690 T^{12} + 17913191776102 T^{13} - 145770100862575 T^{14} - 1292772551649808 T^{15} + 7837433594376961 T^{16} \))(\( 1 + 37 T + 608 T^{2} + 4491 T^{3} - 11038 T^{4} - 630495 T^{5} - 6947105 T^{6} - 31739955 T^{7} + 128335391 T^{8} + 3636093960 T^{9} + 40545974467 T^{10} + 352701114120 T^{11} + 1207507693919 T^{12} - 28968199949715 T^{13} - 615022210681505 T^{14} - 5414275095337215 T^{15} - 9194344990406302 T^{16} + 362865095591205483 T^{17} + 4765159625381192288 T^{18} + 28128549170218913029 T^{19} + 73742412689492826049 T^{20} \))
show more
show less