Properties

Label 430.2.e
Level 430
Weight 2
Character orbit e
Rep. character \(\chi_{430}(221,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 32
Newform subspaces 7
Sturm bound 132
Trace bound 3

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Defining parameters

Level: \( N \) = \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 430.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 43 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 7 \)
Sturm bound: \(132\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(430, [\chi])\).

Total New Old
Modular forms 140 32 108
Cusp forms 124 32 92
Eisenstein series 16 0 16

Trace form

\( 32q - 4q^{2} + 4q^{3} + 32q^{4} + 2q^{5} + 2q^{6} + 8q^{7} - 4q^{8} - 18q^{9} + O(q^{10}) \) \( 32q - 4q^{2} + 4q^{3} + 32q^{4} + 2q^{5} + 2q^{6} + 8q^{7} - 4q^{8} - 18q^{9} - 4q^{11} + 4q^{12} + 4q^{14} + 32q^{16} + 2q^{17} + 10q^{18} - 10q^{19} + 2q^{20} + 8q^{21} + 4q^{22} + 12q^{23} + 2q^{24} - 16q^{25} + 8q^{26} - 32q^{27} + 8q^{28} - 6q^{29} + 2q^{30} - 8q^{31} - 4q^{32} - 4q^{33} - 18q^{34} + 8q^{35} - 18q^{36} + 12q^{37} - 10q^{38} - 8q^{39} - 32q^{41} + 2q^{43} - 4q^{44} + 8q^{45} - 4q^{46} - 16q^{47} + 4q^{48} - 16q^{49} + 2q^{50} - 40q^{51} - 4q^{53} - 28q^{54} + 8q^{55} + 4q^{56} - 32q^{57} + 28q^{58} + 44q^{59} - 8q^{61} + 12q^{62} + 36q^{63} + 32q^{64} - 8q^{65} + 4q^{66} + 14q^{67} + 2q^{68} + 36q^{69} - 32q^{70} - 12q^{71} + 10q^{72} + 46q^{73} - 20q^{74} - 8q^{75} - 10q^{76} - 12q^{77} - 16q^{78} - 20q^{79} + 2q^{80} - 24q^{81} - 20q^{82} - 20q^{83} + 8q^{84} - 16q^{86} + 112q^{87} + 4q^{88} - 20q^{89} + 16q^{90} - 4q^{91} + 12q^{92} - 44q^{93} + 20q^{94} - 12q^{95} + 2q^{96} + 12q^{97} + 10q^{98} + 46q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(430, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
430.2.e.a \(2\) \(3.434\) \(\Q(\sqrt{-3}) \) None \(-2\) \(-3\) \(-1\) \(0\) \(q-q^{2}+(-3+3\zeta_{6})q^{3}+q^{4}+(-1+\cdots)q^{5}+\cdots\)
430.2.e.b \(2\) \(3.434\) \(\Q(\sqrt{-3}) \) None \(-2\) \(-1\) \(-1\) \(-4\) \(q-q^{2}+(-1+\zeta_{6})q^{3}+q^{4}+(-1+\zeta_{6})q^{5}+\cdots\)
430.2.e.c \(2\) \(3.434\) \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(1\) \(-4\) \(q+q^{2}+q^{4}+(1-\zeta_{6})q^{5}-4\zeta_{6}q^{7}+\cdots\)
430.2.e.d \(4\) \(3.434\) \(\Q(\sqrt{-3}, \sqrt{10})\) None \(-4\) \(4\) \(-2\) \(0\) \(q-q^{2}-2\beta _{2}q^{3}+q^{4}+\beta _{2}q^{5}+2\beta _{2}q^{6}+\cdots\)
430.2.e.e \(6\) \(3.434\) 6.0.954288.1 None \(6\) \(1\) \(3\) \(4\) \(q+q^{2}+(-\beta _{4}+\beta _{5})q^{3}+q^{4}+\beta _{3}q^{5}+\cdots\)
430.2.e.f \(6\) \(3.434\) 6.0.27870912.1 None \(6\) \(2\) \(-3\) \(6\) \(q+q^{2}+(-\beta _{1}+\beta _{4})q^{3}+q^{4}-\beta _{4}q^{5}+\cdots\)
430.2.e.g \(10\) \(3.434\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-10\) \(1\) \(5\) \(6\) \(q-q^{2}+\beta _{1}q^{3}+q^{4}+(1-\beta _{5})q^{5}-\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(430, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(430, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(43, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(86, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(215, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{6} \))(\( ( 1 - T )^{6} \))(\( ( 1 + T )^{10} \))
$3$ (\( ( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( ( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} )^{2} \))(\( ( 1 - 2 T + T^{2} + 3 T^{4} - 18 T^{5} + 27 T^{6} )( 1 + T - 2 T^{2} - 3 T^{3} - 6 T^{4} + 9 T^{5} + 27 T^{6} ) \))(\( 1 - 2 T + 8 T^{3} - 12 T^{4} - 2 T^{5} + 34 T^{6} - 6 T^{7} - 108 T^{8} + 216 T^{9} - 486 T^{11} + 729 T^{12} \))(\( 1 - T - 3 T^{2} + 10 T^{3} - 2 T^{4} - 42 T^{5} + 35 T^{6} + 115 T^{7} - 245 T^{8} - 132 T^{9} + 1032 T^{10} - 396 T^{11} - 2205 T^{12} + 3105 T^{13} + 2835 T^{14} - 10206 T^{15} - 1458 T^{16} + 21870 T^{17} - 19683 T^{18} - 19683 T^{19} + 59049 T^{20} \))
$5$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{3} \))(\( ( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{5} \))
$7$ (\( 1 - 7 T^{2} + 49 T^{4} \))(\( ( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))(\( ( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))(\( 1 - 4 T^{2} - 33 T^{4} - 196 T^{6} + 2401 T^{8} \))(\( 1 - 4 T - T^{2} + 36 T^{3} - 38 T^{4} - 128 T^{5} + 583 T^{6} - 896 T^{7} - 1862 T^{8} + 12348 T^{9} - 2401 T^{10} - 67228 T^{11} + 117649 T^{12} \))(\( ( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} )^{3} \))(\( 1 - 6 T + 3 T^{2} + 50 T^{3} - 95 T^{4} - 16 T^{5} - 208 T^{6} + 348 T^{7} + 4257 T^{8} - 5530 T^{9} - 15197 T^{10} - 38710 T^{11} + 208593 T^{12} + 119364 T^{13} - 499408 T^{14} - 268912 T^{15} - 11176655 T^{16} + 41177150 T^{17} + 17294403 T^{18} - 242121642 T^{19} + 282475249 T^{20} \))
$11$ (\( ( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 - 4 T + 11 T^{2} )^{2} \))(\( ( 1 - 5 T + 11 T^{2} )^{2} \))(\( ( 1 + 2 T + 13 T^{2} + 22 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 4 T + 33 T^{2} + 82 T^{3} + 363 T^{4} + 484 T^{5} + 1331 T^{6} )^{2} \))(\( ( 1 + T + 27 T^{2} + 20 T^{3} + 297 T^{4} + 121 T^{5} + 1331 T^{6} )^{2} \))(\( ( 1 - 2 T + 18 T^{2} - 76 T^{3} + 371 T^{4} - 702 T^{5} + 4081 T^{6} - 9196 T^{7} + 23958 T^{8} - 29282 T^{9} + 161051 T^{10} )^{2} \))
$13$ (\( 1 - 13 T^{2} + 169 T^{4} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - 16 T^{2} + 87 T^{4} - 2704 T^{6} + 28561 T^{8} \))(\( 1 - 6 T + 21 T^{2} - 70 T^{3} - 30 T^{4} + 1176 T^{5} - 4515 T^{6} + 15288 T^{7} - 5070 T^{8} - 153790 T^{9} + 599781 T^{10} - 2227758 T^{11} + 4826809 T^{12} \))(\( 1 - 25 T^{2} + 32 T^{3} + 300 T^{4} - 400 T^{5} - 3475 T^{6} - 5200 T^{7} + 50700 T^{8} + 70304 T^{9} - 714025 T^{10} + 4826809 T^{12} \))(\( 1 + 6 T - 5 T^{2} + 18 T^{3} + 467 T^{4} - 554 T^{5} - 1746 T^{6} + 28202 T^{7} + 10477 T^{8} + 16676 T^{9} + 1420225 T^{10} + 216788 T^{11} + 1770613 T^{12} + 61959794 T^{13} - 49867506 T^{14} - 205696322 T^{15} + 2254119803 T^{16} + 1129473306 T^{17} - 4078653605 T^{18} + 63626996238 T^{19} + 137858491849 T^{20} \))
$17$ (\( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 2 T - 21 T^{2} - 18 T^{3} + 268 T^{4} - 306 T^{5} - 6069 T^{6} + 9826 T^{7} + 83521 T^{8} \))(\( ( 1 - 17 T^{2} + 289 T^{4} )^{3} \))(\( 1 + 5 T - 18 T^{2} - 93 T^{3} + 348 T^{4} + 757 T^{5} - 5116 T^{6} + 12869 T^{7} + 100572 T^{8} - 456909 T^{9} - 1503378 T^{10} + 7099285 T^{11} + 24137569 T^{12} \))(\( 1 - 4 T - 6 T^{2} + 280 T^{3} - 1081 T^{4} - 208 T^{5} + 32636 T^{6} - 139236 T^{7} + 180789 T^{8} + 1937608 T^{9} - 11672626 T^{10} + 32939336 T^{11} + 52248021 T^{12} - 684066468 T^{13} + 2725791356 T^{14} - 295330256 T^{15} - 26092712089 T^{16} + 114894828440 T^{17} - 41854544646 T^{18} - 474351505988 T^{19} + 2015993900449 T^{20} \))
$19$ (\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 + 2 T - 25 T^{2} - 18 T^{3} + 404 T^{4} - 342 T^{5} - 9025 T^{6} + 13718 T^{7} + 130321 T^{8} \))(\( 1 + 2 T - 49 T^{2} - 42 T^{3} + 1654 T^{4} + 616 T^{5} - 35621 T^{6} + 11704 T^{7} + 597094 T^{8} - 288078 T^{9} - 6385729 T^{10} + 4952198 T^{11} + 47045881 T^{12} \))(\( 1 + 7 T + 30 T^{2} + 117 T^{3} - 56 T^{4} - 2375 T^{5} - 10766 T^{6} - 45125 T^{7} - 20216 T^{8} + 802503 T^{9} + 3909630 T^{10} + 17332693 T^{11} + 47045881 T^{12} \))(\( 1 - 4 T - 30 T^{2} + 180 T^{3} + 97 T^{4} - 2250 T^{5} + 5972 T^{6} - 11828 T^{7} - 17043 T^{8} + 256830 T^{9} - 1468214 T^{10} + 4879770 T^{11} - 6152523 T^{12} - 81128252 T^{13} + 778277012 T^{14} - 5571222750 T^{15} + 4563450457 T^{16} + 160896913020 T^{17} - 509506891230 T^{18} - 1290750791116 T^{19} + 6131066257801 T^{20} \))
$23$ (\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \))(\( ( 1 - 23 T^{2} + 529 T^{4} )^{2} \))(\( 1 + 8 T - 17 T^{2} - 64 T^{3} + 1850 T^{4} + 3260 T^{5} - 33689 T^{6} + 74980 T^{7} + 978650 T^{8} - 778688 T^{9} - 4757297 T^{10} + 51490744 T^{11} + 148035889 T^{12} \))(\( 1 - 8 T - T^{2} + 120 T^{3} + 170 T^{4} - 1112 T^{5} - 4297 T^{6} - 25576 T^{7} + 89930 T^{8} + 1460040 T^{9} - 279841 T^{10} - 51490744 T^{11} + 148035889 T^{12} \))(\( 1 - 8 T - 27 T^{2} + 192 T^{3} + 1575 T^{4} - 5756 T^{5} - 23066 T^{6} + 48700 T^{7} - 54019 T^{8} + 249988 T^{9} + 6209083 T^{10} + 5749724 T^{11} - 28576051 T^{12} + 592532900 T^{13} - 6454812506 T^{14} - 37047590308 T^{15} + 233156525175 T^{16} + 653726485824 T^{17} - 2114396602587 T^{18} - 14409221291704 T^{19} + 41426511213649 T^{20} \))
$29$ (\( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} \))(\( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \))(\( 1 - 8 T + 35 T^{2} - 232 T^{3} + 841 T^{4} \))(\( 1 + 8 T + 30 T^{2} - 192 T^{3} - 1541 T^{4} - 5568 T^{5} + 25230 T^{6} + 195112 T^{7} + 707281 T^{8} \))(\( 1 + 7 T + 13 T^{2} + 46 T^{3} - 469 T^{4} - 5021 T^{5} - 19406 T^{6} - 145609 T^{7} - 394429 T^{8} + 1121894 T^{9} + 9194653 T^{10} + 143578043 T^{11} + 594823321 T^{12} \))(\( 1 - 10 T + 34 T^{2} - 76 T^{3} - 478 T^{4} + 10598 T^{5} - 76382 T^{6} + 307342 T^{7} - 401998 T^{8} - 1853564 T^{9} + 24047554 T^{10} - 205111490 T^{11} + 594823321 T^{12} \))(\( 1 + T - 95 T^{2} - 82 T^{3} + 4736 T^{4} + 3458 T^{5} - 162633 T^{6} - 116645 T^{7} + 4535747 T^{8} + 1675080 T^{9} - 123827076 T^{10} + 48577320 T^{11} + 3814563227 T^{12} - 2844854905 T^{13} - 115027230873 T^{14} + 70927553242 T^{15} + 2817083248256 T^{16} - 1414489857338 T^{17} - 47523409231295 T^{18} + 14507145975869 T^{19} + 420707233300201 T^{20} \))
$31$ (\( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} \))(\( 1 - 31 T^{2} + 961 T^{4} \))(\( 1 - 12 T + 56 T^{2} - 312 T^{3} + 2319 T^{4} - 9672 T^{5} + 53816 T^{6} - 357492 T^{7} + 923521 T^{8} \))(\( 1 - 8 T - 39 T^{2} + 156 T^{3} + 3496 T^{4} - 7070 T^{5} - 90701 T^{6} - 219170 T^{7} + 3359656 T^{8} + 4647396 T^{9} - 36017319 T^{10} - 229033208 T^{11} + 887503681 T^{12} \))(\( 1 + 6 T - 55 T^{2} - 190 T^{3} + 3252 T^{4} + 5402 T^{5} - 96301 T^{6} + 167462 T^{7} + 3125172 T^{8} - 5660290 T^{9} - 50793655 T^{10} + 171774906 T^{11} + 887503681 T^{12} \))(\( 1 + 8 T - 13 T^{2} + 388 T^{3} + 4345 T^{4} - 8230 T^{5} + 52964 T^{6} + 1316242 T^{7} - 980339 T^{8} + 3056850 T^{9} + 278176459 T^{10} + 94762350 T^{11} - 942105779 T^{12} + 39212165422 T^{13} + 48913366244 T^{14} - 235617912730 T^{15} + 3856203493945 T^{16} + 10674894275068 T^{17} - 11087583486733 T^{18} + 211516977285368 T^{19} + 819628286980801 T^{20} \))
$37$ (\( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( ( 1 - T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} )^{2} \))(\( 1 - 16 T + 65 T^{2} - 444 T^{3} + 8518 T^{4} - 45206 T^{5} + 106681 T^{6} - 1672622 T^{7} + 11661142 T^{8} - 22489932 T^{9} + 121820465 T^{10} - 1109503312 T^{11} + 2565726409 T^{12} \))(\( 1 + 10 T - 13 T^{2} - 102 T^{3} + 2428 T^{4} - 322 T^{5} - 128591 T^{6} - 11914 T^{7} + 3323932 T^{8} - 5166606 T^{9} - 24364093 T^{10} + 693439570 T^{11} + 2565726409 T^{12} \))(\( 1 - 14 T + 49 T^{2} + 226 T^{3} - 2811 T^{4} + 16774 T^{5} - 31644 T^{6} - 434806 T^{7} + 2812513 T^{8} - 2824952 T^{9} - 16711059 T^{10} - 104523224 T^{11} + 3850330297 T^{12} - 22024228318 T^{13} - 59305950684 T^{14} + 1163175534718 T^{15} - 7212256935699 T^{16} + 21454604232058 T^{17} + 172111493242129 T^{18} - 1819464357131078 T^{19} + 4808584372417849 T^{20} \))
$41$ (\( ( 1 - 5 T + 41 T^{2} )^{2} \))(\( ( 1 - 5 T + 41 T^{2} )^{2} \))(\( ( 1 + 7 T + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 67 T^{2} + 410 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + T + 78 T^{2} + 109 T^{3} + 3198 T^{4} + 1681 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 5 T + 30 T^{2} + 241 T^{3} + 1230 T^{4} + 8405 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 3 T + 163 T^{2} + 398 T^{3} + 11909 T^{4} + 22481 T^{5} + 488269 T^{6} + 669038 T^{7} + 11234123 T^{8} + 8477283 T^{9} + 115856201 T^{10} )^{2} \))
$43$ (\( 1 + 13 T + 43 T^{2} \))(\( 1 - 13 T + 43 T^{2} \))(\( 1 + 5 T + 43 T^{2} \))(\( 1 + 6 T + 55 T^{2} + 258 T^{3} + 1849 T^{4} \))(\( 1 - 5 T - 22 T^{2} + 163 T^{3} - 946 T^{4} - 9245 T^{5} + 79507 T^{6} \))(\( 1 + 7 T + 14 T^{2} - 167 T^{3} + 602 T^{4} + 12943 T^{5} + 79507 T^{6} \))(\( 1 - 15 T + 13 T^{2} + 604 T^{3} + 2365 T^{4} - 59039 T^{5} + 101695 T^{6} + 1116796 T^{7} + 1033591 T^{8} - 51282015 T^{9} + 147008443 T^{10} \))
$47$ (\( ( 1 + 3 T + 47 T^{2} )^{2} \))(\( ( 1 - T + 47 T^{2} )^{2} \))(\( ( 1 - 6 T + 47 T^{2} )^{2} \))(\( ( 1 - 4 T + 58 T^{2} - 188 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 9 T + 72 T^{2} + 297 T^{3} + 3384 T^{4} + 19881 T^{5} + 103823 T^{6} )^{2} \))(\( ( 1 - 4 T + 130 T^{2} - 350 T^{3} + 6110 T^{4} - 8836 T^{5} + 103823 T^{6} )^{2} \))(\( ( 1 + 11 T + 106 T^{2} + 41 T^{3} - 3215 T^{4} - 52196 T^{5} - 151105 T^{6} + 90569 T^{7} + 11005238 T^{8} + 53676491 T^{9} + 229345007 T^{10} )^{2} \))
$53$ (\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 + 10 T + 47 T^{2} + 530 T^{3} + 2809 T^{4} \))(\( 1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4} \))(\( 1 - 8 T - 18 T^{2} + 192 T^{3} + 523 T^{4} + 10176 T^{5} - 50562 T^{6} - 1191016 T^{7} + 7890481 T^{8} \))(\( 1 - 8 T - 71 T^{2} + 328 T^{3} + 5438 T^{4} - 3032 T^{5} - 352331 T^{6} - 160696 T^{7} + 15275342 T^{8} + 48831656 T^{9} - 560224151 T^{10} - 3345563944 T^{11} + 22164361129 T^{12} \))(\( 1 - 8 T - 51 T^{2} + 360 T^{3} + 2010 T^{4} + 1448 T^{5} - 173347 T^{6} + 76744 T^{7} + 5646090 T^{8} + 53595720 T^{9} - 402414531 T^{10} - 3345563944 T^{11} + 22164361129 T^{12} \))(\( 1 + 8 T + 31 T^{2} + 520 T^{3} + 3871 T^{4} - 2352 T^{5} + 98098 T^{6} + 852976 T^{7} - 7861235 T^{8} - 76503880 T^{9} - 119377183 T^{10} - 4054705640 T^{11} - 22082209115 T^{12} + 126988507952 T^{13} + 774040405138 T^{14} - 983595799536 T^{15} + 85798241930359 T^{16} + 610849792715240 T^{17} + 1930050402752191 T^{18} + 26398108734417064 T^{19} + 174887470365513049 T^{20} \))
$59$ (\( ( 1 - 2 T + 59 T^{2} )^{2} \))(\( ( 1 - 14 T + 59 T^{2} )^{2} \))(\( ( 1 + 3 T + 59 T^{2} )^{2} \))(\( ( 1 - 14 T + 157 T^{2} - 826 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 12 T + 153 T^{2} + 984 T^{3} + 9027 T^{4} + 41772 T^{5} + 205379 T^{6} )^{2} \))(\( ( 1 - 3 T + 97 T^{2} + 2 T^{3} + 5723 T^{4} - 10443 T^{5} + 205379 T^{6} )^{2} \))(\( ( 1 - 4 T + 232 T^{2} - 856 T^{3} + 24811 T^{4} - 72008 T^{5} + 1463849 T^{6} - 2979736 T^{7} + 47647928 T^{8} - 48469444 T^{9} + 714924299 T^{10} )^{2} \))
$61$ (\( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} \))(\( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( ( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 - 4 T - 100 T^{2} + 24 T^{3} + 8759 T^{4} + 1464 T^{5} - 372100 T^{6} - 907924 T^{7} + 13845841 T^{8} \))(\( 1 - 12 T - 63 T^{2} + 308 T^{3} + 13470 T^{4} - 30576 T^{5} - 746835 T^{6} - 1865136 T^{7} + 50121870 T^{8} + 69910148 T^{9} - 872287983 T^{10} - 10135155612 T^{11} + 51520374361 T^{12} \))(\( 1 + 4 T + 17 T^{2} + 1516 T^{3} + 3102 T^{4} + 16132 T^{5} + 971965 T^{6} + 984052 T^{7} + 11542542 T^{8} + 344103196 T^{9} + 235379297 T^{10} + 3378385204 T^{11} + 51520374361 T^{12} \))(\( 1 + 14 T + 45 T^{2} - 806 T^{3} - 8787 T^{4} - 17696 T^{5} + 772476 T^{6} + 8884756 T^{7} + 31411545 T^{8} - 248377590 T^{9} - 3246005175 T^{10} - 15151032990 T^{11} + 116882358945 T^{12} + 2016670801636 T^{13} + 10695579872316 T^{14} - 14945976142496 T^{15} - 452709529510107 T^{16} - 2533050725832926 T^{17} + 8626829084877645 T^{18} + 163718045299677974 T^{19} + 713342911662882601 T^{20} \))
$67$ (\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( 1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( 1 + 2 T + 29 T^{2} - 318 T^{3} - 4132 T^{4} - 21306 T^{5} + 130181 T^{6} + 601526 T^{7} + 20151121 T^{8} \))(\( 1 + 7 T - 27 T^{2} + 474 T^{3} + 481 T^{4} - 37661 T^{5} - 4586 T^{6} - 2523287 T^{7} + 2159209 T^{8} + 142561662 T^{9} - 544080267 T^{10} + 9450875749 T^{11} + 90458382169 T^{12} \))(\( 1 - 9 T - 21 T^{2} - 72 T^{3} - 237 T^{4} + 39465 T^{5} - 280222 T^{6} + 2644155 T^{7} - 1063893 T^{8} - 21654936 T^{9} - 423173541 T^{10} - 12151125963 T^{11} + 90458382169 T^{12} \))(\( 1 - 19 T + 54 T^{2} - 221 T^{3} + 14098 T^{4} - 32577 T^{5} - 397044 T^{6} - 4826483 T^{7} + 18752721 T^{8} + 110914612 T^{9} + 927211540 T^{10} + 7431279004 T^{11} + 84180964569 T^{12} - 1451627506529 T^{13} - 8000881686324 T^{14} - 43983025610739 T^{15} + 1275282271818562 T^{16} - 1339417264776383 T^{17} + 21927654588058614 T^{18} - 516924153529603993 T^{19} + 1822837804551761449 T^{20} \))
$71$ (\( 1 + 8 T - 7 T^{2} + 568 T^{3} + 5041 T^{4} \))(\( 1 + 10 T + 29 T^{2} + 710 T^{3} + 5041 T^{4} \))(\( 1 + 6 T - 35 T^{2} + 426 T^{3} + 5041 T^{4} \))(\( 1 + 16 T + 60 T^{2} + 864 T^{3} + 15199 T^{4} + 61344 T^{5} + 302460 T^{6} + 5726576 T^{7} + 25411681 T^{8} \))(\( 1 + 12 T - 99 T^{2} - 528 T^{3} + 21132 T^{4} + 64866 T^{5} - 1237277 T^{6} + 4605486 T^{7} + 106526412 T^{8} - 188977008 T^{9} - 2515756419 T^{10} + 21650752212 T^{11} + 128100283921 T^{12} \))(\( 1 - 4 T - 155 T^{2} + 580 T^{3} + 14572 T^{4} - 31108 T^{5} - 1041521 T^{6} - 2208668 T^{7} + 73457452 T^{8} + 207588380 T^{9} - 3938810555 T^{10} - 7216917404 T^{11} + 128100283921 T^{12} \))(\( 1 - 36 T + 567 T^{2} - 5968 T^{3} + 54917 T^{4} - 373762 T^{5} + 748760 T^{6} + 12952414 T^{7} - 202612107 T^{8} + 2525412690 T^{9} - 25391593021 T^{10} + 179304300990 T^{11} - 1021367631387 T^{12} + 4635811447154 T^{13} + 19027250265560 T^{14} - 674352370688462 T^{15} + 7034883292089557 T^{16} - 54279677105277488 T^{17} + 366142252216346487 T^{18} - 1650546025864165116 T^{19} + 3255243551009881201 T^{20} \))
$73$ (\( 1 + 8 T - 9 T^{2} + 584 T^{3} + 5329 T^{4} \))(\( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} \))(\( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \))(\( 1 + 6 T - 109 T^{2} - 6 T^{3} + 13068 T^{4} - 438 T^{5} - 580861 T^{6} + 2334102 T^{7} + 28398241 T^{8} \))(\( 1 + 14 T + 11 T^{2} - 702 T^{3} - 2324 T^{4} + 976 T^{5} - 55559 T^{6} + 71248 T^{7} - 12384596 T^{8} - 273089934 T^{9} + 312380651 T^{10} + 29023002302 T^{11} + 151334226289 T^{12} \))(\( 1 - 21 T + 132 T^{2} - 473 T^{3} + 9756 T^{4} - 93987 T^{5} + 505896 T^{6} - 6861051 T^{7} + 51989724 T^{8} - 184005041 T^{9} + 3748567812 T^{10} - 43534503453 T^{11} + 151334226289 T^{12} \))(\( 1 - 28 T + 130 T^{2} + 788 T^{3} + 40731 T^{4} - 647242 T^{5} - 299136 T^{6} - 781424 T^{7} + 698093137 T^{8} - 3419792278 T^{9} - 15252313914 T^{10} - 249644836294 T^{11} + 3720138327073 T^{12} - 303987220208 T^{13} - 8494936219776 T^{14} - 1341779003996506 T^{15} + 6163994370977259 T^{16} + 8705350033048436 T^{17} + 104839811946230530 T^{18} - 1648404427831501564 T^{19} + 4297625829703557649 T^{20} \))
$79$ (\( 1 - 79 T^{2} + 6241 T^{4} \))(\( 1 + 16 T + 177 T^{2} + 1264 T^{3} + 6241 T^{4} \))(\( 1 - 6 T - 43 T^{2} - 474 T^{3} + 6241 T^{4} \))(\( 1 - 148 T^{2} + 15663 T^{4} - 923668 T^{6} + 38950081 T^{8} \))(\( 1 + 4 T - 45 T^{2} + 132 T^{3} - 1322 T^{4} - 10172 T^{5} + 595399 T^{6} - 803588 T^{7} - 8250602 T^{8} + 65081148 T^{9} - 1752753645 T^{10} + 12308225596 T^{11} + 243087455521 T^{12} \))(\( 1 - 4 T - 125 T^{2} + 444 T^{3} + 6726 T^{4} - 6916 T^{5} - 430361 T^{6} - 546364 T^{7} + 41976966 T^{8} + 218909316 T^{9} - 4868760125 T^{10} - 12308225596 T^{11} + 243087455521 T^{12} \))(\( 1 + 10 T + 3 T^{2} - 790 T^{3} - 11175 T^{4} + 7900 T^{5} - 88976 T^{6} - 5523320 T^{7} - 28654919 T^{8} + 457995090 T^{9} + 11536331579 T^{10} + 36181612110 T^{11} - 178835349479 T^{12} - 2723212169480 T^{13} - 3465622407056 T^{14} + 24308745552100 T^{15} - 2716502315447175 T^{16} - 15171088099065610 T^{17} + 4551326429719683 T^{18} + 1198515959826183190 T^{19} + 9468276082626847201 T^{20} \))
$83$ (\( 1 + 11 T + 38 T^{2} + 913 T^{3} + 6889 T^{4} \))(\( 1 - 15 T + 142 T^{2} - 1245 T^{3} + 6889 T^{4} \))(\( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \))(\( ( 1 + 6 T - 47 T^{2} + 498 T^{3} + 6889 T^{4} )^{2} \))(\( 1 - 15 T + 3 T^{2} + 1164 T^{3} - 2847 T^{4} - 49389 T^{5} + 551986 T^{6} - 4099287 T^{7} - 19612983 T^{8} + 665560068 T^{9} + 142374963 T^{10} - 59085609645 T^{11} + 326940373369 T^{12} \))(\( 1 - 10 T - 112 T^{2} + 816 T^{3} + 12668 T^{4} - 28186 T^{5} - 1120654 T^{6} - 2339438 T^{7} + 87269852 T^{8} + 466578192 T^{9} - 5315331952 T^{10} - 39390406430 T^{11} + 326940373369 T^{12} \))(\( 1 + 25 T + 191 T^{2} + 240 T^{3} + 2762 T^{4} + 155288 T^{5} + 1413133 T^{6} - 7730059 T^{7} - 182939905 T^{8} - 397129048 T^{9} + 6416763512 T^{10} - 32961710984 T^{11} - 1260273005545 T^{12} - 4419947245433 T^{13} + 67064919529693 T^{14} + 611685743370184 T^{15} + 903009311245178 T^{16} + 6512652237510480 T^{17} + 430187816338556831 T^{18} + 4673506381688510075 T^{19} + 15516041187205853449 T^{20} \))
$89$ (\( 1 + 13 T + 80 T^{2} + 1157 T^{3} + 7921 T^{4} \))(\( 1 - 3 T - 80 T^{2} - 267 T^{3} + 7921 T^{4} \))(\( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \))(\( ( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} )^{2} \))(\( 1 + 15 T + 75 T^{2} - 876 T^{3} - 14295 T^{4} - 47235 T^{5} + 68182 T^{6} - 4203915 T^{7} - 113230695 T^{8} - 617552844 T^{9} + 4705668075 T^{10} + 83760891735 T^{11} + 496981290961 T^{12} \))(\( 1 - 7 T - 169 T^{2} + 792 T^{3} + 20009 T^{4} - 36481 T^{5} - 1864546 T^{6} - 3246809 T^{7} + 158491289 T^{8} + 558335448 T^{9} - 10603438729 T^{10} - 39088416143 T^{11} + 496981290961 T^{12} \))(\( 1 - 19 T + 62 T^{2} - 165 T^{3} + 5234 T^{4} + 112771 T^{5} - 1360484 T^{6} - 1913717 T^{7} + 39369557 T^{8} - 1124166758 T^{9} + 20476807868 T^{10} - 100050841462 T^{11} + 311846260997 T^{12} - 1349111159773 T^{13} - 85359815004644 T^{14} + 629719968123179 T^{15} + 2601200076889874 T^{16} - 7298170257762285 T^{17} + 244068505953529022 T^{18} - 6656771670442218971 T^{19} + 31181719929966183601 T^{20} \))
$97$ (\( ( 1 + 8 T + 97 T^{2} )^{2} \))(\( ( 1 + 12 T + 97 T^{2} )^{2} \))(\( ( 1 - 7 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 233 T^{2} - 1358 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 10 T + 239 T^{2} + 1612 T^{3} + 23183 T^{4} + 94090 T^{5} + 912673 T^{6} )^{2} \))(\( ( 1 - 5 T + 155 T^{2} - 538 T^{3} + 15035 T^{4} - 47045 T^{5} + 912673 T^{6} )^{2} \))(\( ( 1 - 10 T + 318 T^{2} - 2434 T^{3} + 47465 T^{4} - 293384 T^{5} + 4604105 T^{6} - 22901506 T^{7} + 290230014 T^{8} - 885292810 T^{9} + 8587340257 T^{10} )^{2} \))
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