Properties

Label 231.2.i
Level 231
Weight 2
Character orbit i
Rep. character \(\chi_{231}(67,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 28
Newform subspaces 6
Sturm bound 64
Trace bound 2

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Defining parameters

Level: \( N \) = \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 231.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 6 \)
Sturm bound: \(64\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(231, [\chi])\).

Total New Old
Modular forms 72 28 44
Cusp forms 56 28 28
Eisenstein series 16 0 16

Trace form

\( 28q + 2q^{3} - 16q^{4} + 2q^{7} + 24q^{8} - 14q^{9} + O(q^{10}) \) \( 28q + 2q^{3} - 16q^{4} + 2q^{7} + 24q^{8} - 14q^{9} - 12q^{10} + 4q^{12} + 12q^{13} - 4q^{14} - 28q^{16} - 6q^{19} - 32q^{20} - 8q^{21} + 8q^{23} - 10q^{25} - 4q^{26} - 4q^{27} - 8q^{28} - 16q^{30} + 22q^{31} - 44q^{32} + 4q^{33} + 24q^{34} + 48q^{35} + 32q^{36} - 2q^{37} + 12q^{38} + 2q^{39} - 12q^{40} - 32q^{41} + 12q^{42} - 12q^{43} - 8q^{44} - 32q^{46} - 28q^{47} + 16q^{48} + 22q^{49} + 64q^{50} + 8q^{51} - 8q^{52} + 36q^{53} - 32q^{55} + 24q^{56} + 12q^{57} + 4q^{58} - 32q^{59} - 8q^{60} + 8q^{61} - 88q^{62} + 2q^{63} + 88q^{64} + 40q^{65} + 8q^{66} - 22q^{67} + 36q^{68} - 24q^{69} + 24q^{70} + 16q^{71} - 12q^{72} - 30q^{73} + 48q^{74} + 6q^{75} + 120q^{76} + 4q^{77} - 8q^{78} + 14q^{79} - 16q^{80} - 14q^{81} - 32q^{83} - 52q^{84} + 8q^{85} - 36q^{86} - 12q^{87} + 12q^{88} - 20q^{89} + 24q^{90} - 10q^{91} + 80q^{92} + 2q^{93} - 52q^{94} + 16q^{95} - 20q^{96} - 128q^{97} + 12q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(231, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
231.2.i.a \(2\) \(1.845\) \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(-4\) \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
231.2.i.b \(2\) \(1.845\) \(\Q(\sqrt{-3}) \) None \(1\) \(1\) \(-4\) \(-4\) \(q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
231.2.i.c \(2\) \(1.845\) \(\Q(\sqrt{-3}) \) None \(2\) \(1\) \(0\) \(5\) \(q+2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
231.2.i.d \(4\) \(1.845\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(2\) \(-2\) \(4\) \(4\) \(q+(1+\beta _{1}+\beta _{2})q^{2}+\beta _{2}q^{3}+(2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
231.2.i.e \(8\) \(1.845\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-2\) \(-4\) \(-4\) \(2\) \(q-\beta _{1}q^{2}+(-1+\beta _{4})q^{3}+(\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
231.2.i.f \(10\) \(1.845\) \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(-2\) \(5\) \(4\) \(-1\) \(q-\beta _{3}q^{2}+(1+\beta _{2})q^{3}+(-2-2\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(231, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(231, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 - T - T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} \))(\( 1 - 2 T + T^{2} + 2 T^{3} - 3 T^{4} + 4 T^{5} + 4 T^{6} - 16 T^{7} + 16 T^{8} \))(\( 1 + 2 T - 8 T^{3} - 11 T^{4} + 4 T^{5} + 20 T^{6} + 8 T^{7} - 23 T^{8} + 16 T^{9} + 80 T^{10} + 32 T^{11} - 176 T^{12} - 256 T^{13} + 256 T^{15} + 256 T^{16} \))(\( 1 + 2 T + 2 T^{2} - 3 T^{4} - 6 T^{5} + 6 T^{6} + 18 T^{7} + 9 T^{8} - 34 T^{9} - 62 T^{10} - 68 T^{11} + 36 T^{12} + 144 T^{13} + 96 T^{14} - 192 T^{15} - 192 T^{16} + 512 T^{18} + 1024 T^{19} + 1024 T^{20} \))
$3$ (\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{5} \))
$5$ (\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( ( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} )^{2} \))(\( 1 + 4 T - 16 T^{3} - 2 T^{4} + 56 T^{5} + 32 T^{6} - 380 T^{7} - 1421 T^{8} - 1900 T^{9} + 800 T^{10} + 7000 T^{11} - 1250 T^{12} - 50000 T^{13} + 312500 T^{15} + 390625 T^{16} \))(\( 1 - 4 T - T^{2} + 36 T^{3} - 69 T^{4} - 36 T^{5} + 246 T^{6} - 132 T^{7} - 279 T^{8} - 400 T^{9} + 2929 T^{10} - 2000 T^{11} - 6975 T^{12} - 16500 T^{13} + 153750 T^{14} - 112500 T^{15} - 1078125 T^{16} + 2812500 T^{17} - 390625 T^{18} - 7812500 T^{19} + 9765625 T^{20} \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( 1 - 2 T + 4 T^{2} - 10 T^{3} - 22 T^{4} - 70 T^{5} + 196 T^{6} - 686 T^{7} + 2401 T^{8} \))(\( 1 + T - 15 T^{2} - 36 T^{3} + 60 T^{4} + 402 T^{5} + 420 T^{6} - 1764 T^{7} - 5145 T^{8} + 2401 T^{9} + 16807 T^{10} \))
$11$ (\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{4} \))(\( ( 1 + T + T^{2} )^{5} \))
$13$ (\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 22 T^{2} - 52 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 2 T + 44 T^{2} + 62 T^{3} + 802 T^{4} + 806 T^{5} + 7436 T^{6} + 4394 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 - 5 T + 27 T^{2} - 180 T^{3} + 732 T^{4} - 2514 T^{5} + 9516 T^{6} - 30420 T^{7} + 59319 T^{8} - 142805 T^{9} + 371293 T^{10} )^{2} \))
$17$ (\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - 7 T + 32 T^{2} - 119 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 6 T - 5 T^{2} - 42 T^{3} + 780 T^{4} - 714 T^{5} - 1445 T^{6} - 29478 T^{7} + 83521 T^{8} \))(\( 1 + 2 T - 24 T^{2} + 100 T^{3} + 439 T^{4} - 2546 T^{5} + 6524 T^{6} + 38396 T^{7} - 127700 T^{8} + 652732 T^{9} + 1885436 T^{10} - 12508498 T^{11} + 36665719 T^{12} + 141985700 T^{13} - 579301656 T^{14} + 820677346 T^{15} + 6975757441 T^{16} \))(\( 1 + 2 T - 73 T^{2} - 90 T^{3} + 3192 T^{4} + 2424 T^{5} - 98319 T^{6} - 38142 T^{7} + 2348199 T^{8} + 280406 T^{9} - 44440892 T^{10} + 4766902 T^{11} + 678629511 T^{12} - 187391646 T^{13} - 8211701199 T^{14} + 3441733368 T^{15} + 77047120248 T^{16} - 36930480570 T^{17} - 509230293193 T^{18} + 237175752994 T^{19} + 2015993900449 T^{20} \))
$19$ (\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 + 6 T + 7 T^{2} - 54 T^{3} - 204 T^{4} - 1026 T^{5} + 2527 T^{6} + 41154 T^{7} + 130321 T^{8} \))(\( 1 - 36 T^{2} - 244 T^{3} + 739 T^{4} + 6710 T^{5} + 20824 T^{6} - 98942 T^{7} - 570724 T^{8} - 1879898 T^{9} + 7517464 T^{10} + 46023890 T^{11} + 96307219 T^{12} - 604168156 T^{13} - 1693651716 T^{14} + 16983563041 T^{16} \))(\( 1 - 3 T - 44 T^{2} + 81 T^{3} + 990 T^{4} - 603 T^{5} - 16476 T^{6} + 16119 T^{7} + 196221 T^{8} - 324198 T^{9} - 1776456 T^{10} - 6159762 T^{11} + 70835781 T^{12} + 110560221 T^{13} - 2147168796 T^{14} - 1493087697 T^{15} + 46575422190 T^{16} + 72403610859 T^{17} - 747276773804 T^{18} - 968063093337 T^{19} + 6131066257801 T^{20} \))
$23$ (\( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \))(\( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \))(\( ( 1 - 7 T + 26 T^{2} - 161 T^{3} + 529 T^{4} )^{2} \))(\( 1 + 4 T - 34 T^{2} - 200 T^{3} - 7 T^{4} + 944 T^{5} - 10850 T^{6} + 35868 T^{7} + 780852 T^{8} + 824964 T^{9} - 5739650 T^{10} + 11485648 T^{11} - 1958887 T^{12} - 1287268600 T^{13} - 5033220226 T^{14} + 13619301788 T^{15} + 78310985281 T^{16} \))(\( 1 + 16 T + 71 T^{2} - 72 T^{3} + 24 T^{4} + 14832 T^{5} + 74109 T^{6} + 71904 T^{7} + 239211 T^{8} + 5124568 T^{9} + 32231056 T^{10} + 117865064 T^{11} + 126542619 T^{12} + 874855968 T^{13} + 20738736669 T^{14} + 95463839376 T^{15} + 3552861336 T^{16} - 245147432184 T^{17} + 5560079954951 T^{18} + 28818442583408 T^{19} + 41426511213649 T^{20} \))
$29$ (\( ( 1 + 5 T + 29 T^{2} )^{2} \))(\( ( 1 - T + 29 T^{2} )^{2} \))(\( ( 1 + 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} + 58 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 8 T + 128 T^{2} - 682 T^{3} + 5745 T^{4} - 19778 T^{5} + 107648 T^{6} - 195112 T^{7} + 707281 T^{8} )^{2} \))(\( ( 1 + 49 T^{2} + 24 T^{3} + 1765 T^{4} + 102 T^{5} + 51185 T^{6} + 20184 T^{7} + 1195061 T^{8} + 20511149 T^{10} )^{2} \))
$31$ (\( 1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4} \))(\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 - 30 T^{2} - 61 T^{4} - 28830 T^{6} + 923521 T^{8} \))(\( 1 - 12 T + 28 T^{2} - 160 T^{3} + 2190 T^{4} + 2424 T^{5} - 76128 T^{6} + 348684 T^{7} - 2030189 T^{8} + 10809204 T^{9} - 73159008 T^{10} + 72213384 T^{11} + 2022510990 T^{12} - 4580664160 T^{13} + 24850103068 T^{14} - 330151369332 T^{15} + 852891037441 T^{16} \))(\( 1 + 5 T - 122 T^{2} - 441 T^{3} + 9747 T^{4} + 24042 T^{5} - 546756 T^{6} - 692730 T^{7} + 24442245 T^{8} + 9807311 T^{9} - 844500050 T^{10} + 304026641 T^{11} + 23488997445 T^{12} - 20637119430 T^{13} - 504940647876 T^{14} + 688302048342 T^{15} + 8650498378707 T^{16} - 12133062822951 T^{17} - 104052706567802 T^{18} + 132198110803355 T^{19} + 819628286980801 T^{20} \))
$37$ (\( ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) \))(\( 1 - 3 T - 28 T^{2} - 111 T^{3} + 1369 T^{4} \))(\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( 1 - 2 T + T^{2} + 142 T^{3} - 1508 T^{4} + 5254 T^{5} + 1369 T^{6} - 101306 T^{7} + 1874161 T^{8} \))(\( 1 - 4 T - 122 T^{2} + 336 T^{3} + 9793 T^{4} - 17900 T^{5} - 522282 T^{6} + 256928 T^{7} + 22257092 T^{8} + 9506336 T^{9} - 715004058 T^{10} - 906688700 T^{11} + 18353658673 T^{12} + 23299569552 T^{13} - 313018621898 T^{14} - 379727508532 T^{15} + 3512479453921 T^{16} \))(\( 1 + 15 T + 10 T^{2} - 603 T^{3} + 1314 T^{4} + 25845 T^{5} - 126420 T^{6} - 536415 T^{7} + 9212421 T^{8} + 1856814 T^{9} - 461695068 T^{10} + 68702118 T^{11} + 12611804349 T^{12} - 27171028995 T^{13} - 236931433620 T^{14} + 1792194568665 T^{15} + 3371364501426 T^{16} - 57243921911199 T^{17} + 35124794539210 T^{18} + 1949426096926155 T^{19} + 4808584372417849 T^{20} \))
$41$ (\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 4 T + 41 T^{2} )^{2} \))(\( ( 1 + 8 T + 90 T^{2} + 328 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 2 T + 120 T^{2} - 310 T^{3} + 6538 T^{4} - 12710 T^{5} + 201720 T^{6} - 137842 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + 22 T + 329 T^{2} + 3256 T^{3} + 27018 T^{4} + 181492 T^{5} + 1107738 T^{6} + 5473336 T^{7} + 22675009 T^{8} + 62166742 T^{9} + 115856201 T^{10} )^{2} \))
$43$ (\( ( 1 + 3 T + 43 T^{2} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 + 9 T + 43 T^{2} )^{2} \))(\( ( 1 + 14 T + 117 T^{2} + 602 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 18 T + 232 T^{2} - 2014 T^{3} + 14869 T^{4} - 86602 T^{5} + 428968 T^{6} - 1431126 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 - 3 T + 107 T^{2} - 114 T^{3} + 5299 T^{4} - 1683 T^{5} + 227857 T^{6} - 210786 T^{7} + 8507249 T^{8} - 10256403 T^{9} + 147008443 T^{10} )^{2} \))
$47$ (\( 1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4} \))(\( 1 + 7 T + 2 T^{2} + 329 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 + 14 T + 61 T^{2} + 574 T^{3} + 6804 T^{4} + 26978 T^{5} + 134749 T^{6} + 1453522 T^{7} + 4879681 T^{8} \))(\( 1 + 12 T - 82 T^{2} - 752 T^{3} + 13457 T^{4} + 68796 T^{5} - 783266 T^{6} - 399704 T^{7} + 54619956 T^{8} - 18786088 T^{9} - 1730234594 T^{10} + 7142607108 T^{11} + 65665867217 T^{12} - 172467445264 T^{13} - 883895656978 T^{14} + 6079477445556 T^{15} + 23811286661761 T^{16} \))(\( 1 - 2 T - 37 T^{2} - 354 T^{3} - 1920 T^{4} + 14694 T^{5} + 211017 T^{6} + 985758 T^{7} + 2945283 T^{8} - 50876312 T^{9} - 605046656 T^{10} - 2391186664 T^{11} + 6506130147 T^{12} + 102344352834 T^{13} + 1029695645577 T^{14} + 3369995532858 T^{15} - 20696093431680 T^{16} - 179344584643902 T^{17} - 881017606485157 T^{18} - 2238260946205534 T^{19} + 52599132235830049 T^{20} \))
$53$ (\( 1 - 8 T + 11 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} \))(\( 1 - 20 T + 202 T^{2} - 1840 T^{3} + 15195 T^{4} - 97520 T^{5} + 567418 T^{6} - 2977540 T^{7} + 7890481 T^{8} \))(\( 1 - 12 T + 56 T^{2} + 832 T^{3} - 13546 T^{4} + 105984 T^{5} - 91520 T^{6} - 5351324 T^{7} + 65244435 T^{8} - 283620172 T^{9} - 257079680 T^{10} + 15778579968 T^{11} - 106884455626 T^{12} + 347938650176 T^{13} + 1241204223224 T^{14} - 14096533678044 T^{15} + 62259690411361 T^{16} \))(\( 1 + 6 T - 133 T^{2} - 354 T^{3} + 10587 T^{4} - 5544 T^{5} - 677754 T^{6} + 507696 T^{7} + 35054169 T^{8} + 3665190 T^{9} - 1656194019 T^{10} + 194255070 T^{11} + 98467160721 T^{12} + 75584257392 T^{13} - 5347805059674 T^{14} - 2318475813192 T^{15} + 234654091272723 T^{16} - 415847743502298 T^{17} - 8280538824711013 T^{18} + 19798581550812798 T^{19} + 174887470365513049 T^{20} \))
$59$ (\( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \))(\( 1 + 7 T - 10 T^{2} + 413 T^{3} + 3481 T^{4} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 59 T^{2} + 138 T^{3} + 3420 T^{4} + 8142 T^{5} - 205379 T^{6} - 1232274 T^{7} + 12117361 T^{8} \))(\( 1 + 12 T - 74 T^{2} - 1208 T^{3} + 5569 T^{4} + 73360 T^{5} - 324634 T^{6} - 1264412 T^{7} + 27329972 T^{8} - 74600308 T^{9} - 1130050954 T^{10} + 15066603440 T^{11} + 67481583409 T^{12} - 863628553192 T^{13} - 3121359489434 T^{14} + 29863817817828 T^{15} + 146830437604321 T^{16} \))(\( 1 + 16 T - 109 T^{2} - 1800 T^{3} + 25080 T^{4} + 228672 T^{5} - 2449527 T^{6} - 10271040 T^{7} + 253584315 T^{8} + 417790408 T^{9} - 15321919856 T^{10} + 24649634072 T^{11} + 882727000515 T^{12} - 2109455924160 T^{13} - 29681802938247 T^{14} + 163483169300928 T^{15} + 1057887783716280 T^{16} - 4479572672674200 T^{17} - 16004517698870989 T^{18} + 138607933098479024 T^{19} + 511116753300641401 T^{20} \))
$61$ (\( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} \))(\( ( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} )^{2} \))(\( 1 + 2 T - 216 T^{2} - 228 T^{3} + 28314 T^{4} + 16182 T^{5} - 2563912 T^{6} - 387746 T^{7} + 178127415 T^{8} - 23652506 T^{9} - 9540316552 T^{10} + 3673006542 T^{11} + 392031142074 T^{12} - 192567956628 T^{13} - 11128400861976 T^{14} + 6285485672042 T^{15} + 191707312997281 T^{16} \))(\( 1 + 12 T + 19 T^{2} - 900 T^{3} - 12441 T^{4} - 72288 T^{5} + 130650 T^{6} + 4443912 T^{7} + 34513281 T^{8} + 10060452 T^{9} - 1324169655 T^{10} + 613687572 T^{11} + 128423918601 T^{12} + 1008683589672 T^{13} + 1808959126650 T^{14} - 61054177406688 T^{15} - 640964977425201 T^{16} - 2828468552418900 T^{17} + 3642438946948339 T^{18} + 140329753114009692 T^{19} + 713342911662882601 T^{20} \))
$67$ (\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} \))(\( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \))(\( 1 - 12 T - 18 T^{2} - 336 T^{3} + 12107 T^{4} - 22512 T^{5} - 80802 T^{6} - 3609156 T^{7} + 20151121 T^{8} \))(\( 1 + 28 T + 324 T^{2} + 1888 T^{3} + 6510 T^{4} + 29080 T^{5} + 472480 T^{6} + 8872436 T^{7} + 99414595 T^{8} + 594453212 T^{9} + 2120962720 T^{10} + 8746188040 T^{11} + 131183797710 T^{12} + 2549036202016 T^{13} + 29308515822756 T^{14} + 169699924949044 T^{15} + 406067677556641 T^{16} \))(\( 1 + 7 T - 194 T^{2} - 1203 T^{3} + 21423 T^{4} + 103134 T^{5} - 1828416 T^{6} - 6223746 T^{7} + 129539373 T^{8} + 171080449 T^{9} - 8649139910 T^{10} + 11462390083 T^{11} + 581502245397 T^{12} - 1871872518198 T^{13} - 36844632054336 T^{14} + 139243802785338 T^{15} + 1937889921206487 T^{16} - 7291036061203569 T^{17} - 78777129445988354 T^{18} + 190445740774064629 T^{19} + 1822837804551761449 T^{20} \))
$71$ (\( ( 1 + 7 T + 71 T^{2} )^{2} \))(\( ( 1 + 15 T + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 14 T + 183 T^{2} + 994 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 12 T + 286 T^{2} - 2260 T^{3} + 29831 T^{4} - 160460 T^{5} + 1441726 T^{6} - 4294932 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 - 24 T + 421 T^{2} - 5436 T^{3} + 57793 T^{4} - 524712 T^{5} + 4103303 T^{6} - 27402876 T^{7} + 150680531 T^{8} - 609880344 T^{9} + 1804229351 T^{10} )^{2} \))
$73$ (\( 1 + 4 T - 57 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 - 5 T - 48 T^{2} - 365 T^{3} + 5329 T^{4} \))(\( 1 + 12 T - 6 T^{2} + 48 T^{3} + 6659 T^{4} + 3504 T^{5} - 31974 T^{6} + 4668204 T^{7} + 28398241 T^{8} \))(\( 1 + 6 T + 16 T^{2} - 1084 T^{3} - 12630 T^{4} - 57030 T^{5} + 123528 T^{6} + 6885378 T^{7} + 59722615 T^{8} + 502632594 T^{9} + 658280712 T^{10} - 22185639510 T^{11} - 358669783830 T^{12} - 2247209606812 T^{13} + 2421347620624 T^{14} + 66284391114582 T^{15} + 806460091894081 T^{16} \))(\( 1 + 17 T + 22 T^{2} + 51 T^{3} + 12429 T^{4} + 38334 T^{5} + 91686 T^{6} + 7567200 T^{7} + 6227061 T^{8} + 177295901 T^{9} + 7596523564 T^{10} + 12942600773 T^{11} + 33184008069 T^{12} + 2943769442400 T^{13} + 2603721124326 T^{14} + 79469126446062 T^{15} + 1880933098545981 T^{16} + 563417324473947 T^{17} + 17742122021669782 T^{18} + 1000816974040554521 T^{19} + 4297625829703557649 T^{20} \))
$79$ (\( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4} \))(\( 1 + 4 T - 18 T^{2} - 496 T^{3} - 6349 T^{4} - 39184 T^{5} - 112338 T^{6} + 1972156 T^{7} + 38950081 T^{8} \))(\( 1 + 2 T - 164 T^{2} - 1476 T^{3} + 11962 T^{4} + 165982 T^{5} + 347064 T^{6} - 7874110 T^{7} - 67195297 T^{8} - 622054690 T^{9} + 2166026424 T^{10} + 81835599298 T^{11} + 465920868922 T^{12} - 4541735244924 T^{13} - 39866342705444 T^{14} + 38407817972318 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 7 T - 92 T^{2} + 1035 T^{3} + 8313 T^{4} - 163986 T^{5} + 174174 T^{6} + 5014596 T^{7} - 64520235 T^{8} + 8158399 T^{9} + 3747754354 T^{10} + 644513521 T^{11} - 402670786635 T^{12} + 2472391397244 T^{13} + 6784091408094 T^{14} - 504594170646414 T^{15} + 2020786017746073 T^{16} + 19876045800674565 T^{17} - 139574010511403612 T^{18} + 838961171878328233 T^{19} + 9468276082626847201 T^{20} \))
$83$ (\( ( 1 - 8 T + 83 T^{2} )^{2} \))(\( ( 1 - 4 T + 83 T^{2} )^{2} \))(\( ( 1 + 4 T + 83 T^{2} )^{2} \))(\( ( 1 + 158 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 12 T + 236 T^{2} + 2044 T^{3} + 27990 T^{4} + 169652 T^{5} + 1625804 T^{6} + 6861444 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 + 12 T + 295 T^{2} + 3168 T^{3} + 40450 T^{4} + 363240 T^{5} + 3357350 T^{6} + 21824352 T^{7} + 168677165 T^{8} + 569499852 T^{9} + 3939040643 T^{10} )^{2} \))
$89$ (\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 + 12 T + 55 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 22 T^{2} - 7437 T^{4} + 174262 T^{6} + 62742241 T^{8} \))(\( 1 + 8 T - 60 T^{2} - 848 T^{3} - 10262 T^{4} - 96488 T^{5} - 110704 T^{6} + 10006904 T^{7} + 164456179 T^{8} + 890614456 T^{9} - 876886384 T^{10} - 68021048872 T^{11} - 643860877142 T^{12} - 4735282412752 T^{13} - 29818877457660 T^{14} + 353850679164232 T^{15} + 3936588805702081 T^{16} \))(\( 1 - 6 T - 241 T^{2} + 978 T^{3} + 28551 T^{4} - 40212 T^{5} - 3189150 T^{6} + 2254020 T^{7} + 350296557 T^{8} - 203364690 T^{9} - 32844883695 T^{10} - 18099457410 T^{11} + 2774699027997 T^{12} + 1589014225380 T^{13} - 200094417885150 T^{14} - 224546198563188 T^{15} + 14189312838227511 T^{16} + 43258245527827362 T^{17} - 948717902174201521 T^{18} - 2102138422244911254 T^{19} + 31181719929966183601 T^{20} \))
$97$ (\( ( 1 + T + 97 T^{2} )^{2} \))(\( ( 1 + T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 6 T + 131 T^{2} + 582 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 44 T + 1070 T^{2} + 17084 T^{3} + 197263 T^{4} + 1657148 T^{5} + 10067630 T^{6} + 40157612 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 + 14 T + 159 T^{2} + 2856 T^{3} + 33045 T^{4} + 251706 T^{5} + 3205365 T^{6} + 26872104 T^{7} + 145115007 T^{8} + 1239409934 T^{9} + 8587340257 T^{10} )^{2} \))
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