Properties

Label 112.2.m
Level 112
Weight 2
Character orbit m
Rep. character \(\chi_{112}(29,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 24
Newform subspaces 4
Sturm bound 32
Trace bound 3

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Defining parameters

Level: \( N \) = \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 112.m (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 16 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 4 \)
Sturm bound: \(32\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(112, [\chi])\).

Total New Old
Modular forms 36 24 12
Cusp forms 28 24 4
Eisenstein series 8 0 8

Trace form

\( 24q - 2q^{4} - 12q^{6} + O(q^{10}) \) \( 24q - 2q^{4} - 12q^{6} + 8q^{10} - 4q^{11} + 4q^{12} - 2q^{14} - 16q^{15} + 2q^{16} + 10q^{18} - 16q^{19} - 28q^{20} + 18q^{22} - 24q^{24} - 12q^{26} + 24q^{27} - 8q^{29} - 24q^{30} + 12q^{34} - 8q^{37} + 20q^{40} - 28q^{43} - 26q^{44} + 52q^{48} - 24q^{49} + 18q^{50} + 16q^{51} + 16q^{52} + 8q^{53} + 32q^{54} + 14q^{56} + 26q^{58} - 24q^{59} + 32q^{60} + 32q^{61} + 36q^{62} + 20q^{63} - 50q^{64} - 16q^{65} - 28q^{66} + 20q^{67} + 32q^{69} + 24q^{70} + 58q^{72} - 34q^{74} - 24q^{75} + 32q^{76} + 8q^{77} - 64q^{78} - 8q^{79} + 12q^{80} - 24q^{81} - 36q^{82} - 40q^{83} - 32q^{85} + 62q^{86} - 54q^{88} - 56q^{90} + 52q^{92} - 48q^{93} + 44q^{94} + 64q^{95} - 8q^{96} + 44q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(112, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
112.2.m.a \(2\) \(0.894\) \(\Q(\sqrt{-1}) \) None \(-2\) \(-4\) \(-4\) \(0\) \(q+(-1+i)q^{2}+(-2+2i)q^{3}-2iq^{4}+\cdots\)
112.2.m.b \(2\) \(0.894\) \(\Q(\sqrt{-1}) \) None \(-2\) \(0\) \(4\) \(0\) \(q+(-1+i)q^{2}-2iq^{4}+(2+2i)q^{5}+\cdots\)
112.2.m.c \(8\) \(0.894\) 8.0.214798336.3 None \(2\) \(0\) \(-4\) \(0\) \(q-\beta _{3}q^{2}+(-\beta _{1}-\beta _{4})q^{3}+(1+\beta _{4}+\cdots)q^{4}+\cdots\)
112.2.m.d \(12\) \(0.894\) 12.0.\(\cdots\).1 None \(2\) \(4\) \(4\) \(0\) \(q+\beta _{2}q^{2}+(-\beta _{2}+\beta _{10})q^{3}+\beta _{1}q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(112, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(112, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 + 2 T + 2 T^{2} \))(\( 1 - 2 T + 4 T^{3} - 6 T^{4} + 8 T^{5} - 16 T^{7} + 16 T^{8} \))(\( 1 - 2 T + 5 T^{2} - 6 T^{3} + 6 T^{4} - 4 T^{5} + 2 T^{6} - 8 T^{7} + 24 T^{8} - 48 T^{9} + 80 T^{10} - 64 T^{11} + 64 T^{12} \))
$3$ (\( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4} \))(\( 1 + 9 T^{4} \))(\( 1 - 4 T^{3} - 16 T^{4} + 4 T^{5} + 8 T^{6} + 16 T^{7} + 130 T^{8} + 48 T^{9} + 72 T^{10} + 108 T^{11} - 1296 T^{12} - 972 T^{13} + 6561 T^{16} \))(\( 1 - 4 T + 8 T^{2} - 16 T^{3} + 34 T^{4} - 48 T^{5} + 48 T^{6} - 52 T^{7} + 83 T^{8} - 96 T^{9} + 72 T^{10} - 480 T^{11} + 1540 T^{12} - 1440 T^{13} + 648 T^{14} - 2592 T^{15} + 6723 T^{16} - 12636 T^{17} + 34992 T^{18} - 104976 T^{19} + 223074 T^{20} - 314928 T^{21} + 472392 T^{22} - 708588 T^{23} + 531441 T^{24} \))
$5$ (\( 1 + 4 T + 8 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 8 T^{2} - 20 T^{3} + 25 T^{4} \))(\( 1 + 4 T + 8 T^{2} + 8 T^{3} + 32 T^{4} + 168 T^{5} + 448 T^{6} + 692 T^{7} + 994 T^{8} + 3460 T^{9} + 11200 T^{10} + 21000 T^{11} + 20000 T^{12} + 25000 T^{13} + 125000 T^{14} + 312500 T^{15} + 390625 T^{16} \))(\( 1 - 4 T + 8 T^{2} - 8 T^{3} - 30 T^{4} + 96 T^{5} - 112 T^{6} - 52 T^{7} + 723 T^{8} - 552 T^{9} - 1912 T^{10} + 7928 T^{11} - 25276 T^{12} + 39640 T^{13} - 47800 T^{14} - 69000 T^{15} + 451875 T^{16} - 162500 T^{17} - 1750000 T^{18} + 7500000 T^{19} - 11718750 T^{20} - 15625000 T^{21} + 78125000 T^{22} - 195312500 T^{23} + 244140625 T^{24} \))
$7$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{6} \))
$11$ (\( 1 + 6 T + 18 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + 32 T^{3} - 28 T^{4} - 288 T^{5} + 512 T^{6} + 192 T^{7} - 21722 T^{8} + 2112 T^{9} + 61952 T^{10} - 383328 T^{11} - 409948 T^{12} + 5153632 T^{13} + 214358881 T^{16} \))(\( 1 + 32 T^{3} - 206 T^{4} - 160 T^{5} + 512 T^{6} - 3328 T^{7} + 26575 T^{8} + 13056 T^{9} + 11776 T^{10} + 286144 T^{11} - 3400484 T^{12} + 3147584 T^{13} + 1424896 T^{14} + 17377536 T^{15} + 389084575 T^{16} - 535977728 T^{17} + 907039232 T^{18} - 3117947360 T^{19} - 44157929486 T^{20} + 75454326112 T^{21} + 3138428376721 T^{24} \))
$13$ (\( 1 + 169 T^{4} \))(\( 1 + 169 T^{4} \))(\( 1 - 4 T^{3} - 336 T^{4} + 44 T^{5} + 8 T^{6} + 576 T^{7} + 70370 T^{8} + 7488 T^{9} + 1352 T^{10} + 96668 T^{11} - 9596496 T^{12} - 1485172 T^{13} + 815730721 T^{16} \))(\( 1 + 20 T^{3} - 174 T^{4} + 596 T^{5} + 200 T^{6} + 2744 T^{7} + 55763 T^{8} - 14088 T^{9} + 267288 T^{10} + 722392 T^{11} - 4753628 T^{12} + 9391096 T^{13} + 45171672 T^{14} - 30951336 T^{15} + 1592647043 T^{16} + 1018827992 T^{17} + 965361800 T^{18} + 37398116132 T^{19} - 141937145454 T^{20} + 212089987460 T^{21} + 23298085122481 T^{24} \))
$17$ (\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 12 T + 108 T^{2} - 652 T^{3} + 3102 T^{4} - 11084 T^{5} + 31212 T^{6} - 58956 T^{7} + 83521 T^{8} )^{2} \))(\( ( 1 + 4 T + 58 T^{2} + 140 T^{3} + 1383 T^{4} + 1648 T^{5} + 23228 T^{6} + 28016 T^{7} + 399687 T^{8} + 687820 T^{9} + 4844218 T^{10} + 5679428 T^{11} + 24137569 T^{12} )^{2} \))
$19$ (\( 1 + 8 T + 32 T^{2} + 152 T^{3} + 361 T^{4} \))(\( 1 - 4 T + 8 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 + 12 T + 72 T^{2} + 392 T^{3} + 1840 T^{4} + 6024 T^{5} + 16640 T^{6} + 25556 T^{7} - 36606 T^{8} + 485564 T^{9} + 6007040 T^{10} + 41318616 T^{11} + 239790640 T^{12} + 970630808 T^{13} + 3387303432 T^{14} + 10726460868 T^{15} + 16983563041 T^{16} \))(\( 1 - 76 T^{3} - 254 T^{4} + 308 T^{5} + 2888 T^{6} - 3600 T^{7} + 64723 T^{8} - 135088 T^{9} + 1054584 T^{10} - 4406416 T^{11} - 13090364 T^{12} - 83721904 T^{13} + 380704824 T^{14} - 926568592 T^{15} + 8434766083 T^{16} - 8913956400 T^{17} + 135868504328 T^{18} + 275312495612 T^{19} - 4313825012414 T^{20} - 24524265031204 T^{21} + 2213314919066161 T^{24} \))
$23$ (\( 1 - 42 T^{2} + 529 T^{4} \))(\( 1 - 10 T^{2} + 529 T^{4} \))(\( 1 - 104 T^{2} + 5156 T^{4} - 166136 T^{6} + 4169926 T^{8} - 87885944 T^{10} + 1442860196 T^{12} - 15395732456 T^{14} + 78310985281 T^{16} \))(\( 1 - 140 T^{2} + 10730 T^{4} - 569180 T^{6} + 22853823 T^{8} - 724561688 T^{10} + 18511773292 T^{12} - 383293132952 T^{14} + 6395436682143 T^{16} - 84259067301020 T^{18} + 840276872065130 T^{20} - 5799711569910860 T^{22} + 21914624432020321 T^{24} \))
$29$ (\( 1 + 2 T + 2 T^{2} + 58 T^{3} + 841 T^{4} \))(\( ( 1 - 10 T + 29 T^{2} )( 1 - 4 T + 29 T^{2} ) \))(\( 1 + 16 T + 128 T^{2} + 688 T^{3} + 4924 T^{4} + 42960 T^{5} + 293760 T^{6} + 1412336 T^{7} + 6689574 T^{8} + 40957744 T^{9} + 247052160 T^{10} + 1047751440 T^{11} + 3482651644 T^{12} + 14111670512 T^{13} + 76137385088 T^{14} + 275998020944 T^{15} + 500246412961 T^{16} \))(\( 1 + 4 T + 8 T^{2} - 100 T^{3} - 454 T^{4} + 2204 T^{5} + 17448 T^{6} - 18780 T^{7} - 48257 T^{8} + 1942504 T^{9} + 20384272 T^{10} + 8166104 T^{11} - 307305556 T^{12} + 236817016 T^{13} + 17143172752 T^{14} + 47375730056 T^{15} - 34131259217 T^{16} - 385199378220 T^{17} + 10378477304808 T^{18} + 38018727385036 T^{19} - 227111871484294 T^{20} - 1450714597586900 T^{21} + 3365657866401608 T^{22} + 48802039062823316 T^{23} + 353814783205469041 T^{24} \))
$31$ (\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 92 T^{2} - 560 T^{3} + 4270 T^{4} - 17360 T^{5} + 88412 T^{6} - 238328 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 + 4 T + 170 T^{2} + 548 T^{3} + 12407 T^{4} + 31840 T^{5} + 502140 T^{6} + 987040 T^{7} + 11923127 T^{8} + 16325468 T^{9} + 156998570 T^{10} + 114516604 T^{11} + 887503681 T^{12} )^{2} \))
$37$ (\( 1 - 6 T + 18 T^{2} - 222 T^{3} + 1369 T^{4} \))(\( ( 1 - 2 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 - 16 T + 128 T^{2} - 656 T^{3} + 796 T^{4} + 3920 T^{5} + 50560 T^{6} - 1167536 T^{7} + 10588326 T^{8} - 43198832 T^{9} + 69216640 T^{10} + 198559760 T^{11} + 1491832156 T^{12} - 45489635792 T^{13} + 328412980352 T^{14} - 1518910034128 T^{15} + 3512479453921 T^{16} \))(\( 1 + 20 T + 200 T^{2} + 1676 T^{3} + 13818 T^{4} + 99436 T^{5} + 629608 T^{6} + 3767444 T^{7} + 20529119 T^{8} + 106029960 T^{9} + 572847248 T^{10} + 3054508600 T^{11} + 16972881580 T^{12} + 113016818200 T^{13} + 784227882512 T^{14} + 5370735563880 T^{15} + 38474874194159 T^{16} + 261249474735908 T^{17} + 1615401872917672 T^{18} + 9439646134596988 T^{19} + 48535441094280378 T^{20} + 217815875896549052 T^{21} + 961716874483569800 T^{22} + 3558352435589208260 T^{23} + 6582952005840035281 T^{24} \))
$41$ (\( 1 - 78 T^{2} + 1681 T^{4} \))(\( ( 1 - 8 T + 41 T^{2} )( 1 + 8 T + 41 T^{2} ) \))(\( 1 - 40 T^{2} + 3564 T^{4} - 174616 T^{6} + 7396902 T^{8} - 293529496 T^{10} + 10071012204 T^{12} - 190004169640 T^{14} + 7984925229121 T^{16} \))(\( 1 - 372 T^{2} + 66450 T^{4} - 7559588 T^{6} + 611181263 T^{8} - 37032380712 T^{10} + 1725045735036 T^{12} - 62251431976872 T^{14} + 1727052176916143 T^{16} - 35908831019012708 T^{18} + 530598281475090450 T^{20} - 4993229263376693172 T^{22} + 22563490300366186081 T^{24} \))
$43$ (\( 1 + 10 T + 50 T^{2} + 430 T^{3} + 1849 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 86 T^{3} + 1849 T^{4} \))(\( 1 + 32 T + 512 T^{2} + 5632 T^{3} + 45796 T^{4} + 275136 T^{5} + 1216512 T^{6} + 3773920 T^{7} + 12376614 T^{8} + 162278560 T^{9} + 2249330688 T^{10} + 21875237952 T^{11} + 156567410596 T^{12} + 827951550976 T^{13} + 3236537881088 T^{14} + 8698195555424 T^{15} + 11688200277601 T^{16} \))(\( 1 - 16 T + 128 T^{2} - 1456 T^{3} + 14962 T^{4} - 80464 T^{5} + 432256 T^{6} - 3242224 T^{7} + 554063 T^{8} + 149895008 T^{9} - 978748672 T^{10} + 10447834784 T^{11} - 101492614052 T^{12} + 449256895712 T^{13} - 1809706294528 T^{14} + 11917702401056 T^{15} + 1894231138463 T^{16} - 476634302097232 T^{17} + 2732447106108544 T^{18} - 21871612724113648 T^{19} + 174878852553466162 T^{20} - 731774842980043408 T^{21} + 2766269736100383872 T^{22} - 14868699831539563312 T^{23} + 39959630797262576401 T^{24} \))
$47$ (\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 12 T + 47 T^{2} )^{2} \))(\( ( 1 - 12 T + 180 T^{2} - 1460 T^{3} + 12126 T^{4} - 68620 T^{5} + 397620 T^{6} - 1245876 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 - 8 T + 178 T^{2} - 848 T^{3} + 12375 T^{4} - 40136 T^{5} + 604268 T^{6} - 1886392 T^{7} + 27336375 T^{8} - 88041904 T^{9} + 868583218 T^{10} - 1834760056 T^{11} + 10779215329 T^{12} )^{2} \))
$53$ (\( 1 - 14 T + 98 T^{2} - 742 T^{3} + 2809 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 106 T^{3} + 2809 T^{4} \))(\( 1 + 8 T + 32 T^{2} + 408 T^{3} + 7260 T^{4} + 38376 T^{5} + 157920 T^{6} + 2094520 T^{7} + 27778214 T^{8} + 111009560 T^{9} + 443597280 T^{10} + 5713303752 T^{11} + 57284892060 T^{12} + 170623761144 T^{13} + 709259556128 T^{14} + 9397689118696 T^{15} + 62259690411361 T^{16} \))(\( 1 - 4 T + 8 T^{2} + 36 T^{3} + 186 T^{4} - 11260 T^{5} + 44200 T^{6} - 717636 T^{7} - 3727201 T^{8} + 56691992 T^{9} - 167612656 T^{10} + 731526824 T^{11} + 19642794028 T^{12} + 38770921672 T^{13} - 470823950704 T^{14} + 8440133692984 T^{15} - 29409408673681 T^{16} - 300112140814548 T^{17} + 979664761901800 T^{18} - 13227247434564620 T^{19} + 11580302416513146 T^{20} + 118791489304876788 T^{21} + 1399099762924104392 T^{22} - 37076143717488766388 T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))
$59$ (\( 1 - 4 T + 8 T^{2} - 236 T^{3} + 3481 T^{4} \))(\( 1 - 16 T + 128 T^{2} - 944 T^{3} + 3481 T^{4} \))(\( 1 + 28 T + 392 T^{2} + 4480 T^{3} + 42352 T^{4} + 296688 T^{5} + 1740480 T^{6} + 9230564 T^{7} + 53169090 T^{8} + 544603276 T^{9} + 6058610880 T^{10} + 60933484752 T^{11} + 513194473072 T^{12} + 3202860859520 T^{13} + 16534769187272 T^{14} + 69682241574932 T^{15} + 146830437604321 T^{16} \))(\( 1 + 16 T + 128 T^{2} + 1068 T^{3} + 9666 T^{4} + 81916 T^{5} + 643720 T^{6} + 5865072 T^{7} + 56589491 T^{8} + 406593184 T^{9} + 2658851000 T^{10} + 18843939424 T^{11} + 133096304068 T^{12} + 1111792426016 T^{13} + 9255460331000 T^{14} + 83505701536736 T^{15} + 685715291253251 T^{16} + 4193082488184528 T^{17} + 27152453115384520 T^{18} + 203860375030433204 T^{19} + 1419263009883366786 T^{20} + 9252079534323474852 T^{21} + 65422944422482099328 T^{22} + \)\(48\!\cdots\!44\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))
$61$ (\( 1 - 12 T + 72 T^{2} - 732 T^{3} + 3721 T^{4} \))(\( 1 - 12 T + 72 T^{2} - 732 T^{3} + 3721 T^{4} \))(\( 1 - 28 T + 392 T^{2} - 5048 T^{3} + 67840 T^{4} - 727768 T^{5} + 6525376 T^{6} - 59801612 T^{7} + 515000994 T^{8} - 3647898332 T^{9} + 24280924096 T^{10} - 165189508408 T^{11} + 939301853440 T^{12} - 4263522127448 T^{13} + 20195986749512 T^{14} - 87996799408588 T^{15} + 191707312997281 T^{16} \))(\( 1 + 20 T + 200 T^{2} + 1392 T^{3} + 16706 T^{4} + 219096 T^{5} + 2009552 T^{6} + 12282692 T^{7} + 94261875 T^{8} + 1070054824 T^{9} + 10076181640 T^{10} + 58881056712 T^{11} + 342120057220 T^{12} + 3591744459432 T^{13} + 37493471882440 T^{14} + 242882114006344 T^{15} + 1305134933611875 T^{16} + 10373916229522292 T^{17} + 103532871337896272 T^{18} + 688562384400857016 T^{19} + 3202662370932576386 T^{20} + 16278251361225124272 T^{21} + \)\(14\!\cdots\!00\)\( T^{22} + \)\(87\!\cdots\!20\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))
$67$ (\( 1 + 10 T + 50 T^{2} + 670 T^{3} + 4489 T^{4} \))(\( 1 - 6 T + 18 T^{2} - 402 T^{3} + 4489 T^{4} \))(\( 1 - 1024 T^{3} + 1508 T^{4} + 58368 T^{5} + 524288 T^{6} - 1406976 T^{7} - 49663130 T^{8} - 94267392 T^{9} + 2353528832 T^{10} + 17554934784 T^{11} + 30387890468 T^{12} - 1382528109568 T^{13} + 406067677556641 T^{16} \))(\( 1 - 24 T + 288 T^{2} - 2680 T^{3} + 35026 T^{4} - 481096 T^{5} + 5050016 T^{6} - 44472552 T^{7} + 431611055 T^{8} - 4488820656 T^{9} + 41458971456 T^{10} - 331005366640 T^{11} + 2603870224156 T^{12} - 22177359564880 T^{13} + 186109322865984 T^{14} - 1350071166960528 T^{15} + 8697446594242655 T^{16} - 60043509027563064 T^{17} + 456816277287564704 T^{18} - 2915784110474474008 T^{19} + 14222926474098907666 T^{20} - 72913512182070457960 T^{21} + \)\(52\!\cdots\!12\)\( T^{22} - \)\(29\!\cdots\!92\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))
$71$ (\( 1 - 78 T^{2} + 5041 T^{4} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( 1 - 72 T^{2} + 15132 T^{4} - 769400 T^{6} + 103336198 T^{8} - 3878545400 T^{10} + 384529556892 T^{12} - 9223220442312 T^{14} + 645753531245761 T^{16} \))(\( 1 - 420 T^{2} + 86946 T^{4} - 12437204 T^{6} + 1395640463 T^{8} - 128112122952 T^{10} + 9864369161052 T^{12} - 645813211801032 T^{14} + 35465570236448303 T^{16} - 1593209363583396884 T^{18} + 56145686527693935906 T^{20} - \)\(13\!\cdots\!20\)\( T^{22} + \)\(16\!\cdots\!41\)\( T^{24} \))
$73$ (\( 1 - 46 T^{2} + 5329 T^{4} \))(\( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \))(\( 1 - 312 T^{2} + 42076 T^{4} - 3442952 T^{6} + 240206086 T^{8} - 18347491208 T^{10} + 1194884388316 T^{12} - 47216278602168 T^{14} + 806460091894081 T^{16} \))(\( 1 - 260 T^{2} + 36418 T^{4} - 3914036 T^{6} + 381955535 T^{8} - 34248715528 T^{10} + 2684648528412 T^{12} - 182511405048712 T^{14} + 10846865334213935 T^{16} - 592327609727292404 T^{18} + 29369663626598641858 T^{20} - \)\(11\!\cdots\!40\)\( T^{22} + \)\(22\!\cdots\!21\)\( T^{24} \))
$79$ (\( ( 1 + 14 T + 79 T^{2} )^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{2} \))(\( ( 1 + 12 T + 156 T^{2} + 1132 T^{3} + 10006 T^{4} + 89428 T^{5} + 973596 T^{6} + 5916468 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 - 12 T + 430 T^{2} - 4164 T^{3} + 79599 T^{4} - 615800 T^{5} + 8193220 T^{6} - 48648200 T^{7} + 496777359 T^{8} - 2053014396 T^{9} + 16748534830 T^{10} - 36924676788 T^{11} + 243087455521 T^{12} )^{2} \))
$83$ (\( 1 + 6889 T^{4} \))(\( 1 + 20 T + 200 T^{2} + 1660 T^{3} + 6889 T^{4} \))(\( 1 - 4 T^{3} + 24944 T^{4} + 964 T^{5} + 8 T^{6} + 52496 T^{7} + 249837570 T^{8} + 4357168 T^{9} + 55112 T^{10} + 551202668 T^{11} + 1183800359024 T^{12} - 15756162572 T^{13} + 2252292232139041 T^{16} \))(\( 1 + 20 T + 200 T^{2} + 2776 T^{3} + 14978 T^{4} - 154136 T^{5} - 2225232 T^{6} - 36575116 T^{7} - 391023149 T^{8} - 799427344 T^{9} + 5892039048 T^{10} + 195956951952 T^{11} + 3311983427012 T^{12} + 16264427012016 T^{13} + 40590257001672 T^{14} - 457102162743728 T^{15} - 18557302123672829 T^{16} - 144070868446439588 T^{17} - 727518180912646608 T^{18} - 4182642355337147272 T^{19} + 33734833052978556098 T^{20} + \)\(51\!\cdots\!28\)\( T^{21} + \)\(31\!\cdots\!00\)\( T^{22} + \)\(25\!\cdots\!40\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))
$89$ (\( 1 - 142 T^{2} + 7921 T^{4} \))(\( 1 + 18 T^{2} + 7921 T^{4} \))(\( 1 - 280 T^{2} + 52956 T^{4} - 7229224 T^{6} + 718461574 T^{8} - 57262683304 T^{10} + 3322578114396 T^{12} - 139154761469080 T^{14} + 3936588805702081 T^{16} \))(\( 1 - 516 T^{2} + 133186 T^{4} - 22283444 T^{6} + 2738419343 T^{8} - 274144745736 T^{10} + 24925487001372 T^{12} - 2171500530974856 T^{14} + 171814566377567663 T^{16} - 11074454766177149684 T^{18} + \)\(52\!\cdots\!66\)\( T^{20} - \)\(16\!\cdots\!16\)\( T^{22} + \)\(24\!\cdots\!21\)\( T^{24} \))
$97$ (\( ( 1 + 6 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 16 T + 364 T^{2} + 4008 T^{3} + 52510 T^{4} + 388776 T^{5} + 3424876 T^{6} + 14602768 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 - 24 T + 522 T^{2} - 9072 T^{3} + 126023 T^{4} - 1521976 T^{5} + 16411804 T^{6} - 147631672 T^{7} + 1185750407 T^{8} - 8279769456 T^{9} + 46212284682 T^{10} - 206096166168 T^{11} + 832972004929 T^{12} )^{2} \))
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