Properties

Label 79.2
Level 79
Weight 2
Dimension 222
Nonzero newspaces 4
Newform subspaces 5
Sturm bound 1040
Trace bound 1

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Defining parameters

Level: \( N \) = \( 79 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 5 \)
Sturm bound: \(1040\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(79))\).

Total New Old
Modular forms 299 299 0
Cusp forms 222 222 0
Eisenstein series 77 77 0

Trace form

\( 222q - 36q^{2} - 35q^{3} - 32q^{4} - 33q^{5} - 27q^{6} - 31q^{7} - 24q^{8} - 26q^{9} + O(q^{10}) \) \( 222q - 36q^{2} - 35q^{3} - 32q^{4} - 33q^{5} - 27q^{6} - 31q^{7} - 24q^{8} - 26q^{9} - 21q^{10} - 27q^{11} - 11q^{12} - 25q^{13} - 15q^{14} - 15q^{15} - 8q^{16} - 21q^{17} - 19q^{19} + 3q^{20} - 7q^{21} - 3q^{22} - 15q^{23} + 21q^{24} - 8q^{25} + 3q^{26} + q^{27} + 17q^{28} - 9q^{29} + 33q^{30} - 7q^{31} + 24q^{32} + 9q^{33} + 15q^{34} + 9q^{35} + 52q^{36} - q^{37} + 21q^{38} + 17q^{39} + 51q^{40} + 3q^{41} + 57q^{42} + 5q^{43} + 45q^{44} + 39q^{45} + 33q^{46} + 9q^{47} + 85q^{48} + 18q^{49} + 54q^{50} + 33q^{51} + 59q^{52} + 15q^{53} + 81q^{54} + 33q^{55} + 81q^{56} + 41q^{57} + 51q^{58} + 21q^{59} + 129q^{60} + 23q^{61} + 57q^{62} + 13q^{63} - 68q^{64} - 33q^{65} - 51q^{66} - 62q^{67} - 69q^{68} - 99q^{69} - 207q^{70} - 45q^{71} - 234q^{72} - 43q^{73} - 81q^{74} - 71q^{75} - 263q^{76} - 60q^{77} - 66q^{78} - 194q^{79} - 282q^{80} - 152q^{81} - 69q^{82} - 72q^{83} - 179q^{84} - 87q^{85} - 63q^{86} + 3q^{87} - 249q^{88} - 27q^{89} - 117q^{90} - 83q^{91} - 27q^{92} - 2q^{93} - 51q^{94} + 3q^{95} + 57q^{96} + 7q^{97} + 132q^{98} + 117q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(79))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
79.2.a \(\chi_{79}(1, \cdot)\) 79.2.a.a 1 1
79.2.a.b 5
79.2.c \(\chi_{79}(23, \cdot)\) 79.2.c.a 12 2
79.2.e \(\chi_{79}(8, \cdot)\) 79.2.e.a 60 12
79.2.g \(\chi_{79}(2, \cdot)\) 79.2.g.a 144 24

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 + 4 T^{2} + 12 T^{4} - T^{5} + 24 T^{6} + 32 T^{8} + 32 T^{10} \))(\( 1 + 2 T - 2 T^{2} - 4 T^{3} + 6 T^{4} + 5 T^{5} + 28 T^{7} + 2 T^{8} - 46 T^{9} + 87 T^{10} + 83 T^{11} - 145 T^{12} + 166 T^{13} + 348 T^{14} - 368 T^{15} + 32 T^{16} + 896 T^{17} + 640 T^{19} + 1536 T^{20} - 2048 T^{21} - 2048 T^{22} + 4096 T^{23} + 4096 T^{24} \))
$3$ (\( 1 + T + 3 T^{2} \))(\( 1 - T + 3 T^{2} - 4 T^{3} + 6 T^{4} - 22 T^{5} + 18 T^{6} - 36 T^{7} + 81 T^{8} - 81 T^{9} + 243 T^{10} \))(\( 1 - 5 T^{2} + 4 T^{3} + 2 T^{4} - 12 T^{5} + 55 T^{6} - 32 T^{7} - 97 T^{8} + 364 T^{9} - 252 T^{10} - 574 T^{11} + 1765 T^{12} - 1722 T^{13} - 2268 T^{14} + 9828 T^{15} - 7857 T^{16} - 7776 T^{17} + 40095 T^{18} - 26244 T^{19} + 13122 T^{20} + 78732 T^{21} - 295245 T^{22} + 531441 T^{24} \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 - 7 T + 34 T^{2} - 113 T^{3} + 320 T^{4} - 749 T^{5} + 1600 T^{6} - 2825 T^{7} + 4250 T^{8} - 4375 T^{9} + 3125 T^{10} \))(\( 1 + T - 11 T^{2} + 10 T^{3} + 100 T^{4} - 182 T^{5} - 253 T^{6} + 2172 T^{7} - 1122 T^{8} - 10453 T^{9} + 27068 T^{10} + 28824 T^{11} - 160241 T^{12} + 144120 T^{13} + 676700 T^{14} - 1306625 T^{15} - 701250 T^{16} + 6787500 T^{17} - 3953125 T^{18} - 14218750 T^{19} + 39062500 T^{20} + 19531250 T^{21} - 107421875 T^{22} + 48828125 T^{23} + 244140625 T^{24} \))
$7$ (\( 1 + T + 7 T^{2} \))(\( 1 + 5 T + 29 T^{2} + 88 T^{3} + 308 T^{4} + 726 T^{5} + 2156 T^{6} + 4312 T^{7} + 9947 T^{8} + 12005 T^{9} + 16807 T^{10} \))(\( 1 - 23 T^{2} + 16 T^{3} + 263 T^{4} - 319 T^{5} - 1744 T^{6} + 3137 T^{7} + 7077 T^{8} - 17677 T^{9} - 18224 T^{10} + 44431 T^{11} + 58479 T^{12} + 311017 T^{13} - 892976 T^{14} - 6063211 T^{15} + 16991877 T^{16} + 52723559 T^{17} - 205179856 T^{18} - 262710217 T^{19} + 1516142663 T^{20} + 645657712 T^{21} - 6496930727 T^{22} + 13841287201 T^{24} \))
$11$ (\( 1 + 2 T + 11 T^{2} \))(\( 1 - 2 T + 20 T^{2} - 54 T^{3} + 240 T^{4} - 598 T^{5} + 2640 T^{6} - 6534 T^{7} + 26620 T^{8} - 29282 T^{9} + 161051 T^{10} \))(\( 1 - 3 T - 12 T^{2} + 87 T^{3} - 181 T^{4} - 302 T^{5} + 3815 T^{6} - 10177 T^{7} - 3876 T^{8} + 98137 T^{9} - 230445 T^{10} - 115524 T^{11} + 2286681 T^{12} - 1270764 T^{13} - 27883845 T^{14} + 130620347 T^{15} - 56748516 T^{16} - 1639016027 T^{17} + 6758505215 T^{18} - 5885125642 T^{19} - 38798957461 T^{20} + 205141449117 T^{21} - 311249095212 T^{22} - 855935011833 T^{23} + 3138428376721 T^{24} \))
$13$ (\( 1 - 3 T + 13 T^{2} \))(\( 1 + 3 T + 42 T^{2} + 33 T^{3} + 596 T^{4} - 259 T^{5} + 7748 T^{6} + 5577 T^{7} + 92274 T^{8} + 85683 T^{9} + 371293 T^{10} \))(\( 1 + 3 T - 37 T^{2} - 66 T^{3} + 944 T^{4} + 1027 T^{5} - 11307 T^{6} + 1460 T^{7} + 44344 T^{8} - 84978 T^{9} + 1955441 T^{10} + 1386108 T^{11} - 36595095 T^{12} + 18019404 T^{13} + 330469529 T^{14} - 186696666 T^{15} + 1266508984 T^{16} + 542087780 T^{17} - 54576729363 T^{18} + 64442726959 T^{19} + 770049800624 T^{20} - 699896958618 T^{21} - 5100764198413 T^{22} + 5376481182111 T^{23} + 23298085122481 T^{24} \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( 1 - 10 T + 101 T^{2} - 592 T^{3} + 3482 T^{4} - 14316 T^{5} + 59194 T^{6} - 171088 T^{7} + 496213 T^{8} - 835210 T^{9} + 1419857 T^{10} \))(\( ( 1 + 11 T + 133 T^{2} + 933 T^{3} + 6363 T^{4} + 31651 T^{5} + 149291 T^{6} + 538067 T^{7} + 1838907 T^{8} + 4583829 T^{9} + 11108293 T^{10} + 15618427 T^{11} + 24137569 T^{12} )^{2} \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 48 T^{2} + 180 T^{3} + 1472 T^{4} + 4440 T^{5} + 27968 T^{6} + 64980 T^{7} + 329232 T^{8} + 521284 T^{9} + 2476099 T^{10} \))(\( 1 + 6 T - 36 T^{2} - 268 T^{3} + 655 T^{4} + 7037 T^{5} - 6408 T^{6} - 187305 T^{7} - 345038 T^{8} + 3008459 T^{9} + 14211694 T^{10} - 17371445 T^{11} - 282982917 T^{12} - 330057455 T^{13} + 5130421534 T^{14} + 20635020281 T^{15} - 44965697198 T^{16} - 463785723195 T^{17} - 301470005448 T^{18} + 6290175427343 T^{19} + 11124233791855 T^{20} - 86480303004772 T^{21} - 220718385280836 T^{22} + 698941553389314 T^{23} + 2213314919066161 T^{24} \))
$23$ (\( 1 - 2 T + 23 T^{2} \))(\( 1 - 2 T + 72 T^{2} - 78 T^{3} + 2500 T^{4} - 1614 T^{5} + 57500 T^{6} - 41262 T^{7} + 876024 T^{8} - 559682 T^{9} + 6436343 T^{10} \))(\( 1 + 10 T - 38 T^{2} - 476 T^{3} + 2179 T^{4} + 15129 T^{5} - 113978 T^{6} - 472575 T^{7} + 3691698 T^{8} + 8461029 T^{9} - 110346678 T^{10} - 50978787 T^{11} + 3002262611 T^{12} - 1172512101 T^{13} - 58373392662 T^{14} + 102945339843 T^{15} + 1033088460018 T^{16} - 3041654793225 T^{17} - 16872834556442 T^{18} + 51511604187663 T^{19} + 170639636927299 T^{20} - 857348666856388 T^{21} - 1574207426118662 T^{22} + 9528097579139270 T^{23} + 21914624432020321 T^{24} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 - 6 T + 93 T^{2} - 304 T^{3} + 3390 T^{4} - 7572 T^{5} + 98310 T^{6} - 255664 T^{7} + 2268177 T^{8} - 4243686 T^{9} + 20511149 T^{10} \))(\( 1 - 15 T + 19 T^{2} + 610 T^{3} - 824 T^{4} - 15515 T^{5} - 65951 T^{6} + 479840 T^{7} + 5955088 T^{8} - 23378510 T^{9} - 179132411 T^{10} + 393700240 T^{11} + 4244528865 T^{12} + 11417306960 T^{13} - 150650357651 T^{14} - 570178480390 T^{15} + 4211920595728 T^{16} + 9842069736160 T^{17} - 39229192843271 T^{18} - 267631830934135 T^{19} - 412203044279864 T^{20} + 8849359045280090 T^{21} + 7993437432703819 T^{22} - 183007646485587435 T^{23} + 353814783205469041 T^{24} \))
$31$ (\( 1 + 10 T + 31 T^{2} \))(\( 1 - 2 T + 92 T^{2} - 242 T^{3} + 4148 T^{4} - 10846 T^{5} + 128588 T^{6} - 232562 T^{7} + 2740772 T^{8} - 1847042 T^{9} + 28629151 T^{10} \))(\( 1 + 13 T - 6 T^{2} + 105 T^{3} + 8997 T^{4} + 19802 T^{5} - 113291 T^{6} + 2345901 T^{7} + 10933976 T^{8} - 28413821 T^{9} + 395542479 T^{10} + 2332176568 T^{11} - 3779034687 T^{12} + 72297473608 T^{13} + 380116322319 T^{14} - 846476141411 T^{15} + 10097756449496 T^{16} + 67161153960051 T^{17} - 100546179524171 T^{18} + 544804784626022 T^{19} + 7673460663856677 T^{20} + 2776160326870455 T^{21} - 4917769721884806 T^{22} + 330310199653262803 T^{23} + 787662783788549761 T^{24} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 + 101 T^{2} - 64 T^{3} + 5630 T^{4} - 2464 T^{5} + 208310 T^{6} - 87616 T^{7} + 5115953 T^{8} + 69343957 T^{10} \))(\( 1 - 14 T - 47 T^{2} + 1194 T^{3} + 3961 T^{4} - 77307 T^{5} - 265548 T^{6} + 3533271 T^{7} + 15386395 T^{8} - 112765175 T^{9} - 722464860 T^{10} + 1446842051 T^{11} + 31322080545 T^{12} + 53533155887 T^{13} - 989054393340 T^{14} - 5711894409275 T^{15} + 28836581439595 T^{16} + 245010992293347 T^{17} - 681323516457132 T^{18} - 7338898625520831 T^{19} + 13912931116981081 T^{20} + 155174317315321938 T^{21} - 226003465503638903 T^{22} - 2490846704912445782 T^{23} + 6582952005840035281 T^{24} \))
$41$ (\( 1 + 10 T + 41 T^{2} \))(\( 1 - 30 T + 541 T^{2} - 6672 T^{3} + 62394 T^{4} - 450116 T^{5} + 2558154 T^{6} - 11215632 T^{7} + 37286261 T^{8} - 84772830 T^{9} + 115856201 T^{10} \))(\( ( 1 + 4 T + 66 T^{2} + 268 T^{3} + 4919 T^{4} + 13879 T^{5} + 169483 T^{6} + 569039 T^{7} + 8268839 T^{8} + 18470828 T^{9} + 186500226 T^{10} + 463424804 T^{11} + 4750104241 T^{12} )^{2} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 + 14 T + 259 T^{2} + 2288 T^{3} + 23478 T^{4} + 144292 T^{5} + 1009554 T^{6} + 4230512 T^{7} + 20592313 T^{8} + 47863214 T^{9} + 147008443 T^{10} \))(\( 1 + 8 T - 83 T^{2} - 1164 T^{3} - 275 T^{4} + 51553 T^{5} + 267922 T^{6} + 104887 T^{7} - 6243265 T^{8} - 108352577 T^{9} - 708456496 T^{10} + 2943445745 T^{11} + 58021358695 T^{12} + 126568167035 T^{13} - 1309936061104 T^{14} - 8614788339539 T^{15} - 21344480625265 T^{16} + 15419274560941 T^{17} + 1693632230814178 T^{18} + 14013064858399171 T^{19} - 3214255076340275 T^{20} - 585017800294485252 T^{21} - 1793753032002592667 T^{22} + 7434349915769781656 T^{23} + 39959630797262576401 T^{24} \))
$47$ (\( 1 - 7 T + 47 T^{2} \))(\( 1 - 5 T + 99 T^{2} - 404 T^{3} + 7090 T^{4} - 29342 T^{5} + 333230 T^{6} - 892436 T^{7} + 10278477 T^{8} - 24398405 T^{9} + 229345007 T^{10} \))(\( 1 - 6 T - 94 T^{2} + 1248 T^{3} + 289 T^{4} - 70847 T^{5} + 517974 T^{6} - 505797 T^{7} - 23119974 T^{8} + 251683115 T^{9} - 645134058 T^{10} - 7219367853 T^{11} + 89685713123 T^{12} - 339310289091 T^{13} - 1425101134122 T^{14} + 26130496048645 T^{15} - 112818097848294 T^{16} - 116002016505579 T^{17} + 5583353280823446 T^{18} - 35892728215442161 T^{19} + 6881461845248929 T^{20} + 1396674830432253216 T^{21} - 4944318430168024606 T^{22} - 14832955290504073818 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))
$53$ (\( 1 - 8 T + 53 T^{2} \))(\( 1 - 2 T + 129 T^{2} - 664 T^{3} + 10258 T^{4} - 46796 T^{5} + 543674 T^{6} - 1865176 T^{7} + 19205133 T^{8} - 15780962 T^{9} + 418195493 T^{10} \))(\( 1 - 14 T - 143 T^{2} + 1970 T^{3} + 22555 T^{4} - 205869 T^{5} - 2416414 T^{6} + 13744085 T^{7} + 211395891 T^{8} - 637106719 T^{9} - 14772851458 T^{10} + 13105934051 T^{11} + 861048502517 T^{12} + 694614504703 T^{13} - 41496939745522 T^{14} - 94850537004563 T^{15} + 1668015261413571 T^{16} + 5747714402408905 T^{17} - 53558272533171406 T^{18} - 241836607647103353 T^{19} + 1404267317228247355 T^{20} + 6500534275850202010 T^{21} - 25008908262268366007 T^{22} - \)\(12\!\cdots\!58\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))
$59$ (\( 1 + 3 T + 59 T^{2} \))(\( 1 - 5 T + 225 T^{2} - 812 T^{3} + 23284 T^{4} - 65630 T^{5} + 1373756 T^{6} - 2826572 T^{7} + 46210275 T^{8} - 60586805 T^{9} + 714924299 T^{10} \))(\( 1 - 7 T - 196 T^{2} + 1615 T^{3} + 20010 T^{4} - 186823 T^{5} - 1302783 T^{6} + 14243225 T^{7} + 55123603 T^{8} - 705499450 T^{9} - 1523635316 T^{10} + 16089659900 T^{11} + 46929630401 T^{12} + 949289934100 T^{13} - 5303774534996 T^{14} - 144894771541550 T^{15} + 667952597171683 T^{16} + 10182827648624275 T^{17} - 54952082158422903 T^{18} - 464937336348340037 T^{19} + 2938077056462463210 T^{20} + 13990738247127726485 T^{21} - \)\(10\!\cdots\!96\)\( T^{22} - \)\(21\!\cdots\!13\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))
$61$ (\( 1 + 4 T + 61 T^{2} \))(\( 1 + 6 T + 109 T^{2} + 656 T^{3} + 11182 T^{4} + 53364 T^{5} + 682102 T^{6} + 2440976 T^{7} + 24740929 T^{8} + 83075046 T^{9} + 844596301 T^{10} \))(\( ( 1 + 10 T + 299 T^{2} + 2514 T^{3} + 40360 T^{4} + 275679 T^{5} + 3152855 T^{6} + 16816419 T^{7} + 150179560 T^{8} + 570630234 T^{9} + 4139906459 T^{10} + 8445963010 T^{11} + 51520374361 T^{12} )^{2} \))
$67$ (\( 1 - 8 T + 67 T^{2} \))(\( 1 + 16 T + 288 T^{2} + 3204 T^{3} + 36308 T^{4} + 289056 T^{5} + 2432636 T^{6} + 14382756 T^{7} + 86619744 T^{8} + 322417936 T^{9} + 1350125107 T^{10} \))(\( ( 1 - 10 T + 314 T^{2} - 2258 T^{3} + 41105 T^{4} - 229705 T^{5} + 3292501 T^{6} - 15390235 T^{7} + 184520345 T^{8} - 679122854 T^{9} + 6327451994 T^{10} - 13501251070 T^{11} + 90458382169 T^{12} )^{2} \))
$71$ (\( 1 - 15 T + 71 T^{2} \))(\( 1 - 3 T + 261 T^{2} - 920 T^{3} + 31596 T^{4} - 99546 T^{5} + 2243316 T^{6} - 4637720 T^{7} + 93414771 T^{8} - 76235043 T^{9} + 1804229351 T^{10} \))(\( ( 1 + 18 T + 374 T^{2} + 4540 T^{3} + 58752 T^{4} + 546848 T^{5} + 5365245 T^{6} + 38826208 T^{7} + 296168832 T^{8} + 1624915940 T^{9} + 9503968694 T^{10} + 32476128318 T^{11} + 128100283921 T^{12} )^{2} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 + 12 T + 396 T^{2} + 3528 T^{3} + 60080 T^{4} + 387190 T^{5} + 4385840 T^{6} + 18800712 T^{7} + 154050732 T^{8} + 340778892 T^{9} + 2073071593 T^{10} \))(\( 1 - 46 T + 840 T^{2} - 10054 T^{3} + 151805 T^{4} - 2286935 T^{5} + 22901081 T^{6} - 212862086 T^{7} + 2631112173 T^{8} - 25874997749 T^{9} + 188388109601 T^{10} - 1839659177726 T^{11} + 19035457941639 T^{12} - 134295119973998 T^{13} + 1003920236063729 T^{14} - 10065813999322733 T^{15} + 74718957586887693 T^{16} - 441278343713322998 T^{17} + 3465717374316718409 T^{18} - 25264682332271097695 T^{19} + \)\(12\!\cdots\!05\)\( T^{20} - \)\(59\!\cdots\!02\)\( T^{21} + \)\(36\!\cdots\!60\)\( T^{22} - \)\(14\!\cdots\!42\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))
$79$ (\( 1 + T \))(\( ( 1 - T )^{5} \))(\( 1 + 34 T + 578 T^{2} + 7073 T^{3} + 71253 T^{4} + 659391 T^{5} + 5968197 T^{6} + 52091889 T^{7} + 444689973 T^{8} + 3487264847 T^{9} + 22513146818 T^{10} + 104619917566 T^{11} + 243087455521 T^{12} \))
$83$ (\( 1 + 6 T + 83 T^{2} \))(\( 1 + 30 T + 695 T^{2} + 10600 T^{3} + 137074 T^{4} + 1346772 T^{5} + 11377142 T^{6} + 73023400 T^{7} + 397391965 T^{8} + 1423749630 T^{9} + 3939040643 T^{10} \))(\( 1 - 6 T - 358 T^{2} + 2936 T^{3} + 70337 T^{4} - 668844 T^{5} - 8541362 T^{6} + 93314302 T^{7} + 712096370 T^{8} - 7868770298 T^{9} - 42497374814 T^{10} + 283961177692 T^{11} + 2761341546177 T^{12} + 23568777748436 T^{13} - 292764415093646 T^{14} - 4499260562382526 T^{15} + 33794898110394770 T^{16} + 367568828151176186 T^{17} - 2792516081359788578 T^{18} - 18149784888106081188 T^{19} + \)\(15\!\cdots\!17\)\( T^{20} + \)\(54\!\cdots\!08\)\( T^{21} - \)\(55\!\cdots\!42\)\( T^{22} - \)\(77\!\cdots\!02\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))
$89$ (\( 1 + 7 T + 89 T^{2} \))(\( 1 - 47 T + 1262 T^{2} - 22913 T^{3} + 313856 T^{4} - 3327989 T^{5} + 27933184 T^{6} - 181493873 T^{7} + 889670878 T^{8} - 2948885327 T^{9} + 5584059449 T^{10} \))(\( ( 1 + 5 T + 218 T^{2} + 1049 T^{3} + 21430 T^{4} + 84395 T^{5} + 1688253 T^{6} + 7511155 T^{7} + 169747030 T^{8} + 739512481 T^{9} + 13677808538 T^{10} + 27920297245 T^{11} + 496981290961 T^{12} )^{2} \))
$97$ (\( 1 + 19 T + 97 T^{2} \))(\( 1 + T + 274 T^{2} - 109 T^{3} + 39536 T^{4} - 41757 T^{5} + 3834992 T^{6} - 1025581 T^{7} + 250072402 T^{8} + 88529281 T^{9} + 8587340257 T^{10} \))(\( ( 1 - 22 T + 693 T^{2} - 10289 T^{3} + 180443 T^{4} - 1953272 T^{5} + 23791033 T^{6} - 189467384 T^{7} + 1697788187 T^{8} - 9390492497 T^{9} + 61350791733 T^{10} - 188921485654 T^{11} + 832972004929 T^{12} )^{2} \))
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