Properties

Label 70.3.l
Level 70
Weight 3
Character orbit l
Rep. character \(\chi_{70}(23,\cdot)\)
Character field \(\Q(\zeta_{12})\)
Dimension 32
Newform subspaces 3
Sturm bound 36
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 70.l (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 35 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 3 \)
Sturm bound: \(36\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(70, [\chi])\).

Total New Old
Modular forms 112 32 80
Cusp forms 80 32 48
Eisenstein series 32 0 32

Trace form

\( 32q + 4q^{5} - 16q^{6} - 4q^{7} + O(q^{10}) \) \( 32q + 4q^{5} - 16q^{6} - 4q^{7} - 16q^{10} + 32q^{11} - 72q^{15} + 64q^{16} + 92q^{17} - 32q^{18} - 24q^{21} - 96q^{22} + 72q^{23} - 4q^{25} - 32q^{26} - 288q^{27} - 64q^{28} - 96q^{30} - 248q^{31} - 44q^{33} + 232q^{35} + 288q^{36} - 28q^{37} + 32q^{38} + 112q^{41} + 272q^{42} + 280q^{43} + 68q^{45} + 72q^{46} + 220q^{47} - 192q^{50} + 120q^{51} + 204q^{53} - 8q^{55} - 112q^{56} - 792q^{57} + 160q^{58} + 32q^{60} - 528q^{61} + 336q^{62} - 256q^{63} + 20q^{65} - 80q^{67} - 184q^{68} + 144q^{70} + 432q^{71} - 64q^{72} - 268q^{73} + 464q^{75} - 160q^{76} + 676q^{77} - 64q^{78} - 16q^{80} + 232q^{81} + 128q^{82} + 72q^{83} - 128q^{85} - 8q^{86} + 388q^{87} - 96q^{88} - 1280q^{90} - 864q^{91} - 288q^{92} - 132q^{93} - 508q^{95} - 32q^{96} - 368q^{97} - 224q^{98} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(70, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
70.3.l.a \(8\) \(1.907\) 8.0.3317760000.2 None \(4\) \(-4\) \(12\) \(-4\) \(q+(1+\beta _{2}-\beta _{4})q^{2}+(-\beta _{2}+2\beta _{3}-\beta _{4}+\cdots)q^{3}+\cdots\)
70.3.l.b \(8\) \(1.907\) 8.0.303595776.1 None \(4\) \(2\) \(-6\) \(-12\) \(q+(1-\beta _{3}+\beta _{4})q^{2}+(-\beta _{1}+\beta _{2}-\beta _{4}+\cdots)q^{3}+\cdots\)
70.3.l.c \(16\) \(1.907\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-8\) \(2\) \(-2\) \(12\) \(q+(-1-\beta _{4}+\beta _{8})q^{2}+(-\beta _{3}-\beta _{10}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(70, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(70, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2} \))(\( ( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{4} \))
$3$ (\( 1 + 4 T + 8 T^{2} + 64 T^{3} + 265 T^{4} + 824 T^{5} + 3224 T^{6} + 10004 T^{7} + 28048 T^{8} + 90036 T^{9} + 261144 T^{10} + 600696 T^{11} + 1738665 T^{12} + 3779136 T^{13} + 4251528 T^{14} + 19131876 T^{15} + 43046721 T^{16} \))(\( 1 - 2 T + 2 T^{2} + 12 T^{3} - 19 T^{4} - 60 T^{5} + 230 T^{6} - 1550 T^{7} - 1580 T^{8} - 13950 T^{9} + 18630 T^{10} - 43740 T^{11} - 124659 T^{12} + 708588 T^{13} + 1062882 T^{14} - 9565938 T^{15} + 43046721 T^{16} \))(\( 1 - 2 T + 2 T^{2} + 28 T^{3} - 2 T^{4} - 86 T^{5} + 568 T^{6} - 522 T^{7} - 4623 T^{8} - 5994 T^{9} + 11452 T^{10} - 106106 T^{11} + 135058 T^{12} - 1589180 T^{13} + 3237770 T^{14} + 5097334 T^{15} - 9280004 T^{16} + 45876006 T^{17} + 262259370 T^{18} - 1158512220 T^{19} + 886115538 T^{20} - 6265453194 T^{21} + 6086062332 T^{22} - 28669116186 T^{23} - 199004991183 T^{24} - 202233495258 T^{25} + 1980493539768 T^{26} - 2698771126374 T^{27} - 564859072962 T^{28} + 71172243193212 T^{29} + 45753584909922 T^{30} - 411782264189298 T^{31} + 1853020188851841 T^{32} \))
$5$ (\( ( 1 - 6 T + 11 T^{2} - 150 T^{3} + 625 T^{4} )^{2} \))(\( 1 + 6 T + 21 T^{2} - 210 T^{3} - 1300 T^{4} - 5250 T^{5} + 13125 T^{6} + 93750 T^{7} + 390625 T^{8} \))(\( 1 + 2 T + 15 T^{2} + 102 T^{3} + 745 T^{4} + 1132 T^{5} - 12306 T^{6} + 14400 T^{7} - 187350 T^{8} + 360000 T^{9} - 7691250 T^{10} + 17687500 T^{11} + 291015625 T^{12} + 996093750 T^{13} + 3662109375 T^{14} + 12207031250 T^{15} + 152587890625 T^{16} \))
$7$ (\( 1 + 4 T + 8 T^{2} - 336 T^{3} - 3577 T^{4} - 16464 T^{5} + 19208 T^{6} + 470596 T^{7} + 5764801 T^{8} \))(\( 1 + 12 T + 72 T^{2} - 252 T^{3} - 4018 T^{4} - 12348 T^{5} + 172872 T^{6} + 1411788 T^{7} + 5764801 T^{8} \))(\( 1 - 12 T + 72 T^{2} - 468 T^{3} - 59 T^{4} + 23880 T^{5} - 172800 T^{6} + 1375080 T^{7} - 10586940 T^{8} + 67378920 T^{9} - 414892800 T^{10} + 2809458120 T^{11} - 340123259 T^{12} - 132198416532 T^{13} + 996572678472 T^{14} - 8138676874188 T^{15} + 33232930569601 T^{16} \))
$11$ (\( ( 1 - 4 T - 200 T^{2} + 104 T^{3} + 30079 T^{4} + 12584 T^{5} - 2928200 T^{6} - 7086244 T^{7} + 214358881 T^{8} )^{2} \))(\( ( 1 + 8 T - 183 T^{2} + 40 T^{3} + 38624 T^{4} + 4840 T^{5} - 2679303 T^{6} + 14172488 T^{7} + 214358881 T^{8} )^{2} \))(\( ( 1 - 20 T + 53 T^{2} + 1868 T^{3} - 19793 T^{4} + 23808 T^{5} + 812056 T^{6} + 3236448 T^{7} - 190136502 T^{8} + 391610208 T^{9} + 11889311896 T^{10} + 42177324288 T^{11} - 4242805331633 T^{12} + 48451109154668 T^{13} + 166336703966213 T^{14} - 7594996671664820 T^{15} + 45949729863572161 T^{16} )^{2} \))
$13$ (\( ( 1 - 47518 T^{4} + 815730721 T^{8} )^{2} \))(\( ( 1 + 8 T + 32 T^{2} - 168 T^{3} - 35218 T^{4} - 28392 T^{5} + 913952 T^{6} + 38614472 T^{7} + 815730721 T^{8} )^{2} \))(\( ( 1 - 8 T + 32 T^{2} - 1112 T^{3} + 33680 T^{4} - 119592 T^{5} + 497248 T^{6} - 10550648 T^{7} + 68709598 T^{8} - 1783059512 T^{9} + 14201900128 T^{10} - 577247741928 T^{11} + 27473810683280 T^{12} - 153298642936088 T^{13} + 745538723919392 T^{14} - 31499011085594312 T^{15} + 665416609183179841 T^{16} )^{2} \))
$17$ (\( 1 + 16 T + 128 T^{2} - 544 T^{3} + 25790 T^{4} + 1504976 T^{5} + 20926464 T^{6} + 422304816 T^{7} + 4437400963 T^{8} + 122046091824 T^{9} + 1747799199744 T^{10} + 36326462043344 T^{11} + 179904784403390 T^{12} - 1096700681844256 T^{13} + 74575646365409408 T^{14} + 2694045224950414864 T^{15} + 48661191875666868481 T^{16} \))(\( 1 - 62 T + 1922 T^{2} - 40796 T^{3} + 583477 T^{4} - 2167148 T^{5} - 154922810 T^{6} + 5388805206 T^{7} - 109634151516 T^{8} + 1557364704534 T^{9} - 12939308014010 T^{10} - 52309684383212 T^{11} + 4070194024402357 T^{12} - 82244487162717404 T^{13} + 1119799939955600642 T^{14} - 10439425246682857598 T^{15} + 48661191875666868481 T^{16} \))(\( 1 - 46 T + 1058 T^{2} - 15228 T^{3} + 49795 T^{4} + 3577952 T^{5} - 101322910 T^{6} + 1912395662 T^{7} - 35538003267 T^{8} + 729089246500 T^{9} - 13614614462848 T^{10} + 149191614412260 T^{11} + 396919406750154 T^{12} - 52349464424861728 T^{13} + 1230645459219027588 T^{14} - 18772373078273663580 T^{15} + \)\(27\!\cdots\!30\)\( T^{16} - \)\(54\!\cdots\!20\)\( T^{17} + \)\(10\!\cdots\!48\)\( T^{18} - \)\(12\!\cdots\!32\)\( T^{19} + \)\(27\!\cdots\!14\)\( T^{20} + \)\(30\!\cdots\!40\)\( T^{21} - \)\(79\!\cdots\!28\)\( T^{22} + \)\(12\!\cdots\!00\)\( T^{23} - \)\(17\!\cdots\!27\)\( T^{24} + \)\(26\!\cdots\!58\)\( T^{25} - \)\(41\!\cdots\!10\)\( T^{26} + \)\(42\!\cdots\!28\)\( T^{27} + \)\(16\!\cdots\!95\)\( T^{28} - \)\(14\!\cdots\!32\)\( T^{29} + \)\(29\!\cdots\!78\)\( T^{30} - \)\(37\!\cdots\!54\)\( T^{31} + \)\(23\!\cdots\!61\)\( T^{32} \))
$19$ (\( 1 + 896 T^{2} + 345790 T^{4} + 175960064 T^{6} + 87265417699 T^{8} + 22931291500544 T^{10} + 5872746263947390 T^{12} + 1983130167483280256 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \))(\( 1 + 1338 T^{2} + 1083625 T^{4} + 596717226 T^{6} + 249020459796 T^{8} + 77764785609546 T^{10} + 18403813500303625 T^{12} + 2961415361710523418 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \))(\( 1 + 962 T^{2} + 285903 T^{4} - 9599234 T^{6} - 15336277051 T^{8} + 2324255772132 T^{10} + 984840646865914 T^{12} - 821244130299946144 T^{14} - \)\(53\!\cdots\!82\)\( T^{16} - \)\(10\!\cdots\!24\)\( T^{18} + \)\(16\!\cdots\!74\)\( T^{20} + \)\(51\!\cdots\!52\)\( T^{22} - \)\(44\!\cdots\!31\)\( T^{24} - \)\(36\!\cdots\!34\)\( T^{26} + \)\(14\!\cdots\!63\)\( T^{28} + \)\(61\!\cdots\!42\)\( T^{30} + \)\(83\!\cdots\!61\)\( T^{32} \))
$23$ (\( 1 + 4 T + 8 T^{2} - 1216 T^{3} + 88025 T^{4} + 1058744 T^{5} + 4270104 T^{6} + 539371764 T^{7} - 65615257712 T^{8} + 285327663156 T^{9} + 1194950173464 T^{10} + 156732109263416 T^{11} + 6893324479360025 T^{12} - 50374637635797184 T^{13} + 175316995456162568 T^{14} + 46371345298154999236 T^{15} + \)\(61\!\cdots\!61\)\( T^{16} \))(\( 1 - 22 T + 242 T^{2} + 8756 T^{3} - 257435 T^{4} + 1661396 T^{5} + 64082326 T^{6} - 4078447362 T^{7} + 48340392708 T^{8} - 2157498654498 T^{9} + 17932862190166 T^{10} + 245946233841044 T^{11} - 20159988495814235 T^{12} + 362730532186710644 T^{13} + 5303339112548917682 T^{14} - \)\(25\!\cdots\!98\)\( T^{15} + \)\(61\!\cdots\!61\)\( T^{16} \))(\( 1 - 54 T + 1458 T^{2} - 41404 T^{3} + 1531542 T^{4} - 51187722 T^{5} + 1388294360 T^{6} - 34788210422 T^{7} + 978568184441 T^{8} - 28666106605246 T^{9} + 745448729152316 T^{10} - 17609421034026766 T^{11} + 432143271417763898 T^{12} - 11387865059627737708 T^{13} + \)\(28\!\cdots\!34\)\( T^{14} - \)\(63\!\cdots\!38\)\( T^{15} + \)\(13\!\cdots\!36\)\( T^{16} - \)\(33\!\cdots\!02\)\( T^{17} + \)\(80\!\cdots\!94\)\( T^{18} - \)\(16\!\cdots\!12\)\( T^{19} + \)\(33\!\cdots\!38\)\( T^{20} - \)\(72\!\cdots\!34\)\( T^{21} + \)\(16\!\cdots\!36\)\( T^{22} - \)\(33\!\cdots\!14\)\( T^{23} + \)\(60\!\cdots\!01\)\( T^{24} - \)\(11\!\cdots\!18\)\( T^{25} + \)\(23\!\cdots\!60\)\( T^{26} - \)\(46\!\cdots\!38\)\( T^{27} + \)\(73\!\cdots\!22\)\( T^{28} - \)\(10\!\cdots\!56\)\( T^{29} + \)\(19\!\cdots\!98\)\( T^{30} - \)\(38\!\cdots\!46\)\( T^{31} + \)\(37\!\cdots\!21\)\( T^{32} \))
$29$ (\( ( 1 - 626 T^{2} + 264051 T^{4} - 442757906 T^{6} + 500246412961 T^{8} )^{2} \))(\( ( 1 - 1756 T^{2} + 1903846 T^{4} - 1241985436 T^{6} + 500246412961 T^{8} )^{2} \))(\( ( 1 - 4830 T^{2} + 11400705 T^{4} - 16870918974 T^{6} + 17005870320740 T^{8} - 11932480442849694 T^{10} + 5703161781476537505 T^{12} - \)\(17\!\cdots\!30\)\( T^{14} + \)\(25\!\cdots\!21\)\( T^{16} )^{2} \))
$31$ (\( ( 1 + 16 T - 705 T^{2} + 15376 T^{3} + 923521 T^{4} )^{4} \))(\( ( 1 - 12 T - 923 T^{2} + 10260 T^{3} + 76584 T^{4} + 9859860 T^{5} - 852409883 T^{6} - 10650044172 T^{7} + 852891037441 T^{8} )^{2} \))(\( ( 1 + 104 T + 3335 T^{2} + 80216 T^{3} + 6958945 T^{4} + 299604368 T^{5} + 5519548642 T^{6} + 232310443200 T^{7} + 11482603425294 T^{8} + 223250335915200 T^{9} + 5097419081408482 T^{10} + 265899979443678608 T^{11} + 5935221820544859745 T^{12} + 65747302668451933016 T^{13} + \)\(26\!\cdots\!35\)\( T^{14} + \)\(78\!\cdots\!84\)\( T^{15} + \)\(72\!\cdots\!81\)\( T^{16} )^{2} \))
$37$ (\( 1 - 144 T + 10368 T^{2} - 421344 T^{3} + 8176990 T^{4} + 93049776 T^{5} - 9412816896 T^{6} + 169435939536 T^{7} - 564618587997 T^{8} + 231957801224784 T^{9} - 17641134326624256 T^{10} + 238740267634734384 T^{11} + 28721509369917477790 T^{12} - \)\(20\!\cdots\!56\)\( T^{13} + \)\(68\!\cdots\!08\)\( T^{14} - \)\(12\!\cdots\!16\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} \))(\( 1 + 134 T + 8978 T^{2} + 354028 T^{3} + 6883829 T^{4} - 119634932 T^{5} - 15166185258 T^{6} - 689629336926 T^{7} - 25693504224764 T^{8} - 944102562251694 T^{9} - 28423872929318538 T^{10} - 306950504471319188 T^{11} + 24179307926805543509 T^{12} + \)\(17\!\cdots\!72\)\( T^{13} + \)\(59\!\cdots\!18\)\( T^{14} + \)\(12\!\cdots\!26\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} \))(\( 1 + 38 T + 722 T^{2} + 36204 T^{3} + 921843 T^{4} + 54419040 T^{5} + 2057717682 T^{6} + 7359428170 T^{7} + 1748254042141 T^{8} - 98996461623124 T^{9} - 5173840906504960 T^{10} - 59928313854983156 T^{11} - 5578617333398806678 T^{12} + 6554354740117891808 T^{13} - \)\(20\!\cdots\!96\)\( T^{14} - \)\(42\!\cdots\!16\)\( T^{15} - \)\(83\!\cdots\!54\)\( T^{16} - \)\(57\!\cdots\!04\)\( T^{17} - \)\(39\!\cdots\!56\)\( T^{18} + \)\(16\!\cdots\!72\)\( T^{19} - \)\(19\!\cdots\!38\)\( T^{20} - \)\(28\!\cdots\!44\)\( T^{21} - \)\(34\!\cdots\!60\)\( T^{22} - \)\(89\!\cdots\!36\)\( T^{23} + \)\(21\!\cdots\!81\)\( T^{24} + \)\(12\!\cdots\!30\)\( T^{25} + \)\(47\!\cdots\!82\)\( T^{26} + \)\(17\!\cdots\!60\)\( T^{27} + \)\(39\!\cdots\!23\)\( T^{28} + \)\(21\!\cdots\!36\)\( T^{29} + \)\(58\!\cdots\!62\)\( T^{30} + \)\(42\!\cdots\!62\)\( T^{31} + \)\(15\!\cdots\!81\)\( T^{32} \))
$41$ (\( ( 1 + 34 T + 3531 T^{2} + 57154 T^{3} + 2825761 T^{4} )^{4} \))(\( ( 1 - 80 T + 3378 T^{2} - 134480 T^{3} + 2825761 T^{4} )^{4} \))(\( ( 1 + 18 T + 4237 T^{2} + 27182 T^{3} + 8225864 T^{4} + 45692942 T^{5} + 11972749357 T^{6} + 85501876338 T^{7} + 7984925229121 T^{8} )^{4} \))
$43$ (\( ( 1 - 60 T + 1800 T^{2} - 65040 T^{3} + 1764887 T^{4} - 120258960 T^{5} + 6153841800 T^{6} - 379281782940 T^{7} + 11688200277601 T^{8} )^{2} \))(\( ( 1 - 8 T + 32 T^{2} + 14888 T^{3} - 6837458 T^{4} + 27527912 T^{5} + 109401632 T^{6} - 50570904392 T^{7} + 11688200277601 T^{8} )^{2} \))(\( ( 1 - 72 T + 2592 T^{2} - 12288 T^{3} - 872859 T^{4} - 17006160 T^{5} + 3562391520 T^{6} + 141343265880 T^{7} - 11372920448860 T^{8} + 261343698612120 T^{9} + 12179107690967520 T^{10} - 107502111429381840 T^{11} - 10202150806106531259 T^{12} - \)\(26\!\cdots\!12\)\( T^{13} + \)\(10\!\cdots\!92\)\( T^{14} - \)\(53\!\cdots\!28\)\( T^{15} + \)\(13\!\cdots\!01\)\( T^{16} )^{2} \))
$47$ (\( 1 - 72 T + 2592 T^{2} - 86256 T^{3} + 4419742 T^{4} - 430225128 T^{5} + 23240286720 T^{6} - 1046121049944 T^{7} + 51328414956099 T^{8} - 2310881399326296 T^{9} + 113405185542136320 T^{10} - 4637489294658587112 T^{11} + \)\(10\!\cdots\!62\)\( T^{12} - \)\(45\!\cdots\!44\)\( T^{13} + \)\(30\!\cdots\!72\)\( T^{14} - \)\(18\!\cdots\!68\)\( T^{15} + \)\(56\!\cdots\!21\)\( T^{16} \))(\( 1 - 102 T + 5202 T^{2} + 157284 T^{3} - 29409683 T^{4} + 1814223612 T^{5} - 19692509130 T^{6} - 3140329970154 T^{7} + 264570468668724 T^{8} - 6936988904070186 T^{9} - 96093162643987530 T^{10} + 19555906968704148348 T^{11} - \)\(70\!\cdots\!63\)\( T^{12} + \)\(82\!\cdots\!16\)\( T^{13} + \)\(60\!\cdots\!82\)\( T^{14} - \)\(26\!\cdots\!38\)\( T^{15} + \)\(56\!\cdots\!21\)\( T^{16} \))(\( 1 - 46 T + 1058 T^{2} + 161412 T^{3} - 22698365 T^{4} + 1010022752 T^{5} - 9419259550 T^{6} - 2715134508658 T^{7} + 271485194118333 T^{8} - 10484618609225180 T^{9} + 53077017818847872 T^{10} + 26010553538653727460 T^{11} - \)\(20\!\cdots\!46\)\( T^{12} + \)\(72\!\cdots\!12\)\( T^{13} - \)\(91\!\cdots\!12\)\( T^{14} - \)\(16\!\cdots\!00\)\( T^{15} + \)\(11\!\cdots\!70\)\( T^{16} - \)\(36\!\cdots\!00\)\( T^{17} - \)\(44\!\cdots\!72\)\( T^{18} + \)\(78\!\cdots\!48\)\( T^{19} - \)\(49\!\cdots\!06\)\( T^{20} + \)\(13\!\cdots\!40\)\( T^{21} + \)\(61\!\cdots\!52\)\( T^{22} - \)\(26\!\cdots\!20\)\( T^{23} + \)\(15\!\cdots\!93\)\( T^{24} - \)\(34\!\cdots\!62\)\( T^{25} - \)\(26\!\cdots\!50\)\( T^{26} + \)\(61\!\cdots\!68\)\( T^{27} - \)\(30\!\cdots\!65\)\( T^{28} + \)\(48\!\cdots\!48\)\( T^{29} + \)\(69\!\cdots\!38\)\( T^{30} - \)\(66\!\cdots\!54\)\( T^{31} + \)\(32\!\cdots\!41\)\( T^{32} \))
$53$ (\( 1 - 76 T + 2888 T^{2} + 280744 T^{3} - 32097310 T^{4} + 1832302924 T^{5} - 7149394176 T^{6} - 4718024532996 T^{7} + 411450315196723 T^{8} - 13252930913185764 T^{9} - 56412158907238656 T^{10} + 40611823705258641196 T^{11} - \)\(19\!\cdots\!10\)\( T^{12} + \)\(49\!\cdots\!56\)\( T^{13} + \)\(14\!\cdots\!08\)\( T^{14} - \)\(10\!\cdots\!44\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} \))(\( 1 - 98 T + 4802 T^{2} - 358484 T^{3} + 21915517 T^{4} - 1279203212 T^{5} + 84378991270 T^{6} - 4129245378846 T^{7} + 191619264999924 T^{8} - 11599050269178414 T^{9} + 665790827415100870 T^{10} - 28352721948144746348 T^{11} + \)\(13\!\cdots\!37\)\( T^{12} - \)\(62\!\cdots\!16\)\( T^{13} + \)\(23\!\cdots\!82\)\( T^{14} - \)\(13\!\cdots\!62\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} \))(\( 1 - 30 T + 450 T^{2} + 53260 T^{3} - 13562801 T^{4} + 264802000 T^{5} - 422485750 T^{6} - 1346619990690 T^{7} + 94385910261585 T^{8} - 3394198091700460 T^{9} + 16961261878565400 T^{10} + 11461088832223742300 T^{11} - \)\(43\!\cdots\!94\)\( T^{12} + \)\(38\!\cdots\!80\)\( T^{13} - \)\(65\!\cdots\!00\)\( T^{14} - \)\(11\!\cdots\!00\)\( T^{15} + \)\(39\!\cdots\!74\)\( T^{16} - \)\(31\!\cdots\!00\)\( T^{17} - \)\(51\!\cdots\!00\)\( T^{18} + \)\(84\!\cdots\!20\)\( T^{19} - \)\(27\!\cdots\!34\)\( T^{20} + \)\(20\!\cdots\!00\)\( T^{21} + \)\(83\!\cdots\!00\)\( T^{22} - \)\(46\!\cdots\!40\)\( T^{23} + \)\(36\!\cdots\!85\)\( T^{24} - \)\(14\!\cdots\!10\)\( T^{25} - \)\(12\!\cdots\!50\)\( T^{26} + \)\(22\!\cdots\!00\)\( T^{27} - \)\(32\!\cdots\!81\)\( T^{28} + \)\(36\!\cdots\!40\)\( T^{29} + \)\(85\!\cdots\!50\)\( T^{30} - \)\(16\!\cdots\!70\)\( T^{31} + \)\(15\!\cdots\!41\)\( T^{32} \))
$59$ (\( 1 + 4064 T^{2} - 11259650 T^{4} + 14390721536 T^{6} + 424530130709539 T^{8} + 174377567902186496 T^{10} - \)\(16\!\cdots\!50\)\( T^{12} + \)\(72\!\cdots\!84\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \))(\( 1 + 9434 T^{2} + 45444361 T^{4} + 182276889482 T^{6} + 671844820055188 T^{8} + 2208714871810497002 T^{10} + \)\(66\!\cdots\!81\)\( T^{12} + \)\(16\!\cdots\!54\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \))(\( 1 + 6674 T^{2} - 7729537 T^{4} - 69884124210 T^{6} + 388809824859333 T^{8} + 851796830409747108 T^{10} - \)\(72\!\cdots\!98\)\( T^{12} - \)\(40\!\cdots\!08\)\( T^{14} + \)\(99\!\cdots\!50\)\( T^{16} - \)\(48\!\cdots\!88\)\( T^{18} - \)\(10\!\cdots\!58\)\( T^{20} + \)\(15\!\cdots\!48\)\( T^{22} + \)\(83\!\cdots\!53\)\( T^{24} - \)\(18\!\cdots\!10\)\( T^{26} - \)\(24\!\cdots\!57\)\( T^{28} + \)\(25\!\cdots\!54\)\( T^{30} + \)\(46\!\cdots\!81\)\( T^{32} \))
$61$ (\( ( 1 + 162 T + 12721 T^{2} + 985122 T^{3} + 71371764 T^{4} + 3665638962 T^{5} + 176132943361 T^{6} + 8346300646482 T^{7} + 191707312997281 T^{8} )^{2} \))(\( ( 1 + 42 T - 5019 T^{2} - 27678 T^{3} + 25599404 T^{4} - 102989838 T^{5} - 69492275979 T^{6} + 2163855723162 T^{7} + 191707312997281 T^{8} )^{2} \))(\( ( 1 + 60 T - 8926 T^{2} - 393256 T^{3} + 57653141 T^{4} + 1511546176 T^{5} - 266254549814 T^{6} - 2710779938676 T^{7} + 1000035050441884 T^{8} - 10086812151813396 T^{9} - 3686518162251223574 T^{10} + 77875424851457993536 T^{11} + \)\(11\!\cdots\!21\)\( T^{12} - \)\(28\!\cdots\!56\)\( T^{13} - \)\(23\!\cdots\!46\)\( T^{14} + \)\(59\!\cdots\!60\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} )^{2} \))
$67$ (\( 1 - 124 T + 7688 T^{2} + 603136 T^{3} - 112977415 T^{4} + 9268146136 T^{5} - 98793237096 T^{6} - 35532386900364 T^{7} + 4178772156136048 T^{8} - 159504884795733996 T^{9} - 1990794474703184616 T^{10} + \)\(83\!\cdots\!84\)\( T^{11} - \)\(45\!\cdots\!15\)\( T^{12} + \)\(10\!\cdots\!64\)\( T^{13} + \)\(62\!\cdots\!68\)\( T^{14} - \)\(45\!\cdots\!96\)\( T^{15} + \)\(16\!\cdots\!81\)\( T^{16} \))(\( 1 + 130 T + 8450 T^{2} - 101660 T^{3} - 49206683 T^{4} - 3311248460 T^{5} - 9498450650 T^{6} + 13933744675110 T^{7} + 1410770562904548 T^{8} + 62548579846568790 T^{9} - 191404428360678650 T^{10} - \)\(29\!\cdots\!40\)\( T^{11} - \)\(19\!\cdots\!03\)\( T^{12} - \)\(18\!\cdots\!40\)\( T^{13} + \)\(69\!\cdots\!50\)\( T^{14} + \)\(47\!\cdots\!70\)\( T^{15} + \)\(16\!\cdots\!81\)\( T^{16} \))(\( 1 + 74 T + 2738 T^{2} - 587900 T^{3} - 50988730 T^{4} - 3339904986 T^{5} + 65267378776 T^{6} + 10012501185306 T^{7} + 918173402551977 T^{8} - 2374991498059262 T^{9} + 329325818085928764 T^{10} + 1284143109513809266 T^{11} + \)\(25\!\cdots\!90\)\( T^{12} + \)\(13\!\cdots\!44\)\( T^{13} + \)\(44\!\cdots\!54\)\( T^{14} - \)\(11\!\cdots\!78\)\( T^{15} - \)\(84\!\cdots\!12\)\( T^{16} - \)\(50\!\cdots\!42\)\( T^{17} + \)\(88\!\cdots\!34\)\( T^{18} + \)\(12\!\cdots\!36\)\( T^{19} + \)\(10\!\cdots\!90\)\( T^{20} + \)\(23\!\cdots\!34\)\( T^{21} + \)\(26\!\cdots\!04\)\( T^{22} - \)\(87\!\cdots\!98\)\( T^{23} + \)\(15\!\cdots\!37\)\( T^{24} + \)\(74\!\cdots\!54\)\( T^{25} + \)\(21\!\cdots\!76\)\( T^{26} - \)\(49\!\cdots\!54\)\( T^{27} - \)\(34\!\cdots\!30\)\( T^{28} - \)\(17\!\cdots\!00\)\( T^{29} + \)\(36\!\cdots\!58\)\( T^{30} + \)\(44\!\cdots\!26\)\( T^{31} + \)\(27\!\cdots\!61\)\( T^{32} \))
$71$ (\( ( 1 - 4 T + 8616 T^{2} - 20164 T^{3} + 25411681 T^{4} )^{4} \))(\( ( 1 - 100 T + 12406 T^{2} - 504100 T^{3} + 25411681 T^{4} )^{4} \))(\( ( 1 - 4 T + 10238 T^{2} + 37292 T^{3} + 54945234 T^{4} + 187988972 T^{5} + 260164790078 T^{6} - 512401135684 T^{7} + 645753531245761 T^{8} )^{4} \))
$73$ (\( 1 - 32 T + 512 T^{2} - 51136 T^{3} + 34708862 T^{4} - 2067510688 T^{5} + 49696849920 T^{6} - 5991675606624 T^{7} + 642997612278979 T^{8} - 31929639307699296 T^{9} + 1411303120968990720 T^{10} - \)\(31\!\cdots\!32\)\( T^{11} + \)\(27\!\cdots\!22\)\( T^{12} - \)\(21\!\cdots\!64\)\( T^{13} + \)\(11\!\cdots\!52\)\( T^{14} - \)\(39\!\cdots\!88\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} \))(\( 1 + 246 T + 30258 T^{2} + 1881900 T^{3} + 11613493 T^{4} - 9211461636 T^{5} - 846646828650 T^{6} - 36781012225806 T^{7} - 1359453564456732 T^{8} - 196006014151320174 T^{9} - 24043280681888404650 T^{10} - \)\(13\!\cdots\!04\)\( T^{11} + \)\(93\!\cdots\!33\)\( T^{12} + \)\(80\!\cdots\!00\)\( T^{13} + \)\(69\!\cdots\!18\)\( T^{14} + \)\(30\!\cdots\!14\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} \))(\( 1 + 54 T + 1458 T^{2} + 376300 T^{3} - 9032525 T^{4} + 1427883664 T^{5} + 161075984306 T^{6} + 3449490767706 T^{7} + 2006002204707837 T^{8} + 28704897132082748 T^{9} + 2397705340658794944 T^{10} + \)\(54\!\cdots\!96\)\( T^{11} - \)\(52\!\cdots\!50\)\( T^{12} + \)\(41\!\cdots\!84\)\( T^{13} + \)\(17\!\cdots\!04\)\( T^{14} + \)\(42\!\cdots\!72\)\( T^{15} + \)\(19\!\cdots\!58\)\( T^{16} + \)\(22\!\cdots\!88\)\( T^{17} + \)\(49\!\cdots\!64\)\( T^{18} + \)\(62\!\cdots\!76\)\( T^{19} - \)\(42\!\cdots\!50\)\( T^{20} + \)\(23\!\cdots\!04\)\( T^{21} + \)\(54\!\cdots\!24\)\( T^{22} + \)\(35\!\cdots\!32\)\( T^{23} + \)\(13\!\cdots\!57\)\( T^{24} + \)\(11\!\cdots\!14\)\( T^{25} + \)\(29\!\cdots\!06\)\( T^{26} + \)\(14\!\cdots\!56\)\( T^{27} - \)\(47\!\cdots\!25\)\( T^{28} + \)\(10\!\cdots\!00\)\( T^{29} + \)\(21\!\cdots\!98\)\( T^{30} + \)\(42\!\cdots\!46\)\( T^{31} + \)\(42\!\cdots\!21\)\( T^{32} \))
$79$ (\( 1 + 13024 T^{2} + 67570270 T^{4} + 314583571456 T^{6} + 2291081588444419 T^{8} + 12253055589480487936 T^{10} + \)\(10\!\cdots\!70\)\( T^{12} + \)\(76\!\cdots\!84\)\( T^{14} + \)\(23\!\cdots\!21\)\( T^{16} \))(\( 1 + 5010 T^{2} - 52175183 T^{4} - 3130643790 T^{6} + 3476709624715428 T^{8} - 121938829202646990 T^{10} - \)\(79\!\cdots\!63\)\( T^{12} + \)\(29\!\cdots\!10\)\( T^{14} + \)\(23\!\cdots\!21\)\( T^{16} \))(\( 1 + 42538 T^{2} + 982751207 T^{4} + 15945124184102 T^{6} + 201431875816036549 T^{8} + \)\(20\!\cdots\!92\)\( T^{10} + \)\(18\!\cdots\!34\)\( T^{12} + \)\(14\!\cdots\!24\)\( T^{14} + \)\(93\!\cdots\!90\)\( T^{16} + \)\(54\!\cdots\!44\)\( T^{18} + \)\(27\!\cdots\!74\)\( T^{20} + \)\(12\!\cdots\!72\)\( T^{22} + \)\(46\!\cdots\!29\)\( T^{24} + \)\(14\!\cdots\!02\)\( T^{26} + \)\(34\!\cdots\!67\)\( T^{28} + \)\(57\!\cdots\!18\)\( T^{30} + \)\(52\!\cdots\!41\)\( T^{32} \))
$83$ (\( ( 1 - 84 T + 3528 T^{2} - 500304 T^{3} + 70077383 T^{4} - 3446594256 T^{5} + 167432956488 T^{6} - 27462991362996 T^{7} + 2252292232139041 T^{8} )^{2} \))(\( ( 1 + 80 T + 3200 T^{2} + 317680 T^{3} + 23022958 T^{4} + 2188497520 T^{5} + 151866627200 T^{6} + 26155229869520 T^{7} + 2252292232139041 T^{8} )^{2} \))(\( ( 1 - 32 T + 512 T^{2} + 252712 T^{3} - 42882475 T^{4} - 1371597568 T^{5} + 97778626848 T^{6} - 7242029223592 T^{7} + 506959325886468 T^{8} - 49890339321325288 T^{9} + 4640409459891602208 T^{10} - \)\(44\!\cdots\!92\)\( T^{11} - \)\(96\!\cdots\!75\)\( T^{12} + \)\(39\!\cdots\!88\)\( T^{13} + \)\(54\!\cdots\!32\)\( T^{14} - \)\(23\!\cdots\!28\)\( T^{15} + \)\(50\!\cdots\!81\)\( T^{16} )^{2} \))
$89$ (\( 1 + 22466 T^{2} + 258050305 T^{4} + 2722572965954 T^{6} + 25242583519811524 T^{8} + \)\(17\!\cdots\!14\)\( T^{10} + \)\(10\!\cdots\!05\)\( T^{12} + \)\(55\!\cdots\!86\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} \))(\( 1 + 24902 T^{2} + 348507721 T^{4} + 3638615519702 T^{6} + 30923927822861044 T^{8} + \)\(22\!\cdots\!82\)\( T^{10} + \)\(13\!\cdots\!01\)\( T^{12} + \)\(61\!\cdots\!42\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} \))(\( 1 + 32552 T^{2} + 612475958 T^{4} + 7153003757936 T^{6} + 52449920499191449 T^{8} + \)\(12\!\cdots\!92\)\( T^{10} - \)\(24\!\cdots\!66\)\( T^{12} - \)\(42\!\cdots\!64\)\( T^{14} - \)\(40\!\cdots\!12\)\( T^{16} - \)\(26\!\cdots\!24\)\( T^{18} - \)\(95\!\cdots\!46\)\( T^{20} + \)\(31\!\cdots\!32\)\( T^{22} + \)\(81\!\cdots\!89\)\( T^{24} + \)\(69\!\cdots\!36\)\( T^{26} + \)\(37\!\cdots\!78\)\( T^{28} + \)\(12\!\cdots\!12\)\( T^{30} + \)\(24\!\cdots\!21\)\( T^{32} \))
$97$ (\( ( 1 + 36 T + 648 T^{2} + 128556 T^{3} - 8578162 T^{4} + 1209583404 T^{5} + 57366974088 T^{6} + 29986992177444 T^{7} + 7837433594376961 T^{8} )^{2} \))(\( ( 1 + 216 T + 23328 T^{2} + 3287304 T^{3} + 429479822 T^{4} + 30930243336 T^{5} + 2065211067168 T^{6} + 179921953064664 T^{7} + 7837433594376961 T^{8} )^{2} \))(\( ( 1 - 68 T + 2312 T^{2} + 258100 T^{3} - 260409124 T^{4} + 11295624444 T^{5} - 132728762504 T^{6} - 78472536305132 T^{7} + 30681693645016390 T^{8} - 738348094094986988 T^{9} - 11750381912498879624 T^{10} + \)\(94\!\cdots\!76\)\( T^{11} - \)\(20\!\cdots\!64\)\( T^{12} + \)\(19\!\cdots\!00\)\( T^{13} + \)\(16\!\cdots\!92\)\( T^{14} - \)\(44\!\cdots\!92\)\( T^{15} + \)\(61\!\cdots\!21\)\( T^{16} )^{2} \))
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