Properties

Label 29.2
Level 29
Weight 2
Dimension 22
Nonzero newspaces 4
Newform subspaces 4
Sturm bound 140
Trace bound 2

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Defining parameters

Level: \( N \) = \( 29\( 29 \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 4 \)
Sturm bound: \(140\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(29))\).

Total New Old
Modular forms 49 49 0
Cusp forms 22 22 0
Eisenstein series 27 27 0

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(29))\)

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
29.2.a \(\chi_{29}(1, \cdot)\) 29.2.a.a 2 1
29.2.b \(\chi_{29}(28, \cdot)\) 29.2.b.a 2 1
29.2.d \(\chi_{29}(7, \cdot)\) 29.2.d.a 6 6
29.2.e \(\chi_{29}(4, \cdot)\) 29.2.e.a 12 6

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 + T^{2} + 4 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 7 T^{3} + 10 T^{4} + 13 T^{5} + 27 T^{6} + 26 T^{7} + 40 T^{8} + 56 T^{9} + 32 T^{10} + 64 T^{11} + 64 T^{12} \))(\( 1 + 7 T + 27 T^{2} + 70 T^{3} + 130 T^{4} + 161 T^{5} + 84 T^{6} - 147 T^{7} - 401 T^{8} - 301 T^{9} + 646 T^{10} + 2534 T^{11} + 4577 T^{12} + 5068 T^{13} + 2584 T^{14} - 2408 T^{15} - 6416 T^{16} - 4704 T^{17} + 5376 T^{18} + 20608 T^{19} + 33280 T^{20} + 35840 T^{21} + 27648 T^{22} + 14336 T^{23} + 4096 T^{24} \))
$3$ (\( 1 - 2 T + 5 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 - T^{2} + 9 T^{4} \))(\( 1 + 5 T + 8 T^{2} + 4 T^{3} + 3 T^{4} + 3 T^{5} - 8 T^{6} + 9 T^{7} + 27 T^{8} + 108 T^{9} + 648 T^{10} + 1215 T^{11} + 729 T^{12} \))(\( 1 + 7 T + 29 T^{2} + 91 T^{3} + 244 T^{4} + 546 T^{5} + 1012 T^{6} + 1491 T^{7} + 1497 T^{8} - 259 T^{9} - 5693 T^{10} - 16660 T^{11} - 32900 T^{12} - 49980 T^{13} - 51237 T^{14} - 6993 T^{15} + 121257 T^{16} + 362313 T^{17} + 737748 T^{18} + 1194102 T^{19} + 1600884 T^{20} + 1791153 T^{21} + 1712421 T^{22} + 1240029 T^{23} + 531441 T^{24} \))
$5$ (\( ( 1 + T + 5 T^{2} )^{2} \))(\( ( 1 + 3 T + 5 T^{2} )^{2} \))(\( 1 - T + 10 T^{2} - 12 T^{3} + 81 T^{4} - 91 T^{5} + 456 T^{6} - 455 T^{7} + 2025 T^{8} - 1500 T^{9} + 6250 T^{10} - 3125 T^{11} + 15625 T^{12} \))(\( 1 + T - 11 T^{2} - 11 T^{3} + 66 T^{4} + 76 T^{5} - 174 T^{6} - 69 T^{7} + 399 T^{8} - 1417 T^{9} - 1605 T^{10} + 5700 T^{11} + 14536 T^{12} + 28500 T^{13} - 40125 T^{14} - 177125 T^{15} + 249375 T^{16} - 215625 T^{17} - 2718750 T^{18} + 5937500 T^{19} + 25781250 T^{20} - 21484375 T^{21} - 107421875 T^{22} + 48828125 T^{23} + 244140625 T^{24} \))
$7$ (\( 1 + 6 T^{2} + 49 T^{4} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( 1 - T + 8 T^{2} - 22 T^{3} + 127 T^{4} - 169 T^{5} + 848 T^{6} - 1183 T^{7} + 6223 T^{8} - 7546 T^{9} + 19208 T^{10} - 16807 T^{11} + 117649 T^{12} \))(\( 1 + 11 T + 47 T^{2} + 123 T^{3} + 458 T^{4} + 2166 T^{5} + 7378 T^{6} + 20319 T^{7} + 61291 T^{8} + 197341 T^{9} + 577023 T^{10} + 1543704 T^{11} + 4058132 T^{12} + 10805928 T^{13} + 28274127 T^{14} + 67687963 T^{15} + 147159691 T^{16} + 341501433 T^{17} + 868014322 T^{18} + 1783794138 T^{19} + 2640278858 T^{20} + 4963493661 T^{21} + 13276336703 T^{22} + 21750594173 T^{23} + 13841287201 T^{24} \))
$11$ (\( 1 - 2 T + 21 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 - 17 T^{2} + 121 T^{4} \))(\( 1 + 11 T + 68 T^{2} + 354 T^{3} + 1683 T^{4} + 6723 T^{5} + 23296 T^{6} + 73953 T^{7} + 203643 T^{8} + 471174 T^{9} + 995588 T^{10} + 1771561 T^{11} + 1771561 T^{12} \))(\( 1 - 7 T + 17 T^{2} + 7 T^{3} - 14 T^{5} - 1448 T^{6} + 5383 T^{7} + 13177 T^{8} - 74137 T^{9} + 167363 T^{10} - 476952 T^{11} + 2425844 T^{12} - 5246472 T^{13} + 20250923 T^{14} - 98676347 T^{15} + 192924457 T^{16} + 866937533 T^{17} - 2565220328 T^{18} - 272820394 T^{19} + 16505633837 T^{21} + 440936218217 T^{22} - 1997181694277 T^{23} + 3138428376721 T^{24} \))
$13$ (\( 1 + 2 T + 19 T^{2} + 26 T^{3} + 169 T^{4} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( 1 + 5 T + 12 T^{2} - 40 T^{3} - 111 T^{4} + 875 T^{5} + 6308 T^{6} + 11375 T^{7} - 18759 T^{8} - 87880 T^{9} + 342732 T^{10} + 1856465 T^{11} + 4826809 T^{12} \))(\( 1 - 9 T - 9 T^{2} + 235 T^{3} + 440 T^{4} - 7052 T^{5} + 904 T^{6} + 93021 T^{7} + 10465 T^{8} - 1306603 T^{9} + 1744795 T^{10} + 3749024 T^{11} - 11467928 T^{12} + 48737312 T^{13} + 294870355 T^{14} - 2870606791 T^{15} + 298890865 T^{16} + 34538046153 T^{17} + 4363435336 T^{18} - 442502541884 T^{19} + 358921517240 T^{20} + 2492057352655 T^{21} - 1240726426641 T^{22} - 16129443546333 T^{23} + 23298085122481 T^{24} \))
$17$ (\( 1 + 4 T + 30 T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 14 T^{2} + 289 T^{4} \))(\( ( 1 - 4 T + 47 T^{2} - 128 T^{3} + 799 T^{4} - 1156 T^{5} + 4913 T^{6} )^{2} \))(\( 1 - 133 T^{2} + 8874 T^{4} - 389130 T^{6} + 12426957 T^{8} - 303621745 T^{10} + 5811079504 T^{12} - 87746684305 T^{14} + 1037911875597 T^{16} - 9392652224970 T^{18} + 61902871531434 T^{20} - 268127188759717 T^{22} + 582622237229761 T^{24} \))
$19$ (\( ( 1 - 6 T + 19 T^{2} )^{2} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( 1 - T - 18 T^{2} + 156 T^{3} + 67 T^{4} - 1127 T^{5} + 9612 T^{6} - 21413 T^{7} + 24187 T^{8} + 1070004 T^{9} - 2345778 T^{10} - 2476099 T^{11} + 47045881 T^{12} \))(\( 1 + 7 T + 31 T^{2} + 189 T^{3} + 1426 T^{4} + 8162 T^{5} + 37516 T^{6} + 183911 T^{7} + 989579 T^{8} + 5075581 T^{9} + 23151601 T^{10} + 103810854 T^{11} + 453309164 T^{12} + 1972406226 T^{13} + 8357727961 T^{14} + 34813410079 T^{15} + 128962924859 T^{16} + 455381843189 T^{17} + 1764973271596 T^{18} + 7295781133718 T^{19} + 24218560896466 T^{20} + 60987974880231 T^{21} + 190063053991831 T^{22} + 815431812287533 T^{23} + 2213314919066161 T^{24} \))
$23$ (\( 1 + 4 T + 18 T^{2} + 92 T^{3} + 529 T^{4} \))(\( ( 1 - 6 T + 23 T^{2} )^{2} \))(\( 1 + 7 T + 26 T^{2} + 84 T^{3} + 431 T^{4} - 175 T^{5} - 12020 T^{6} - 4025 T^{7} + 227999 T^{8} + 1022028 T^{9} + 7275866 T^{10} + 45054401 T^{11} + 148035889 T^{12} \))(\( 1 + 5 T - 43 T^{2} - 391 T^{3} - 308 T^{4} + 10696 T^{5} + 58590 T^{6} - 14345 T^{7} - 1422557 T^{8} - 5436135 T^{9} + 1966389 T^{10} + 79674994 T^{11} + 420307216 T^{12} + 1832524862 T^{13} + 1040219781 T^{14} - 66141454545 T^{15} - 398089773437 T^{16} - 92329340335 T^{17} + 8673422736510 T^{18} + 36418012981112 T^{19} - 24119783466548 T^{20} - 704250690632033 T^{21} - 1781339982186907 T^{22} + 4764048789569635 T^{23} + 21914624432020321 T^{24} \))
$29$ (\( ( 1 - T )^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 6 T - 13 T^{2} + 316 T^{3} - 377 T^{4} - 5046 T^{5} + 24389 T^{6} \))(\( 1 + 15 T + 126 T^{2} + 622 T^{3} + 1665 T^{4} - 4109 T^{5} - 52800 T^{6} - 119161 T^{7} + 1400265 T^{8} + 15169958 T^{9} + 89117406 T^{10} + 307667235 T^{11} + 594823321 T^{12} \))
$31$ (\( 1 - 6 T + 21 T^{2} - 186 T^{3} + 961 T^{4} \))(\( 1 - 17 T^{2} + 961 T^{4} \))(\( 1 - 5 T - 34 T^{2} + 388 T^{3} - 1649 T^{4} - 7283 T^{5} + 109220 T^{6} - 225773 T^{7} - 1584689 T^{8} + 11558908 T^{9} - 31399714 T^{10} - 143145755 T^{11} + 887503681 T^{12} \))(\( 1 + 21 T + 269 T^{2} + 2261 T^{3} + 14784 T^{4} + 77658 T^{5} + 355230 T^{6} + 1212127 T^{7} - 208667 T^{8} - 52988005 T^{9} - 588286643 T^{10} - 4352370134 T^{11} - 26495076348 T^{12} - 134923474154 T^{13} - 565343463923 T^{14} - 1578565656955 T^{15} - 192708356507 T^{16} + 34702166914177 T^{17} + 315267932601630 T^{18} + 2136574586632038 T^{19} + 12609141097527744 T^{20} + 59779985705277131 T^{21} + 220480009197835469 T^{22} + 533578014824501451 T^{23} + 787662783788549761 T^{24} \))
$37$ (\( ( 1 + 4 T + 37 T^{2} )^{2} \))(\( ( 1 - 37 T^{2} )^{2} \))(\( 1 - 11 T + 42 T^{2} - 328 T^{3} + 4049 T^{4} - 22869 T^{5} + 100360 T^{6} - 846153 T^{7} + 5543081 T^{8} - 16614184 T^{9} + 78714762 T^{10} - 762783527 T^{11} + 2565726409 T^{12} \))(\( 1 - 7 T - 3 T^{2} + 833 T^{3} - 5462 T^{4} - 9800 T^{5} + 411870 T^{6} - 2163693 T^{7} - 6602285 T^{8} + 140542059 T^{9} - 562585377 T^{10} - 2805854072 T^{11} + 36132816832 T^{12} - 103816600664 T^{13} - 770179381113 T^{14} + 7118876914527 T^{15} - 12373745057885 T^{16} - 150039034353201 T^{17} + 1056745736074830 T^{18} - 930332395903400 T^{19} - 19185162777316502 T^{20} + 108258129249299141 T^{21} - 14425753117253547 T^{22} - 1245423352456222891 T^{23} + 6582952005840035281 T^{24} \))
$41$ (\( 1 - 8 T + 26 T^{2} - 328 T^{3} + 1681 T^{4} \))(\( ( 1 - 12 T + 41 T^{2} )( 1 + 12 T + 41 T^{2} ) \))(\( ( 1 - 10 T + 147 T^{2} - 828 T^{3} + 6027 T^{4} - 16810 T^{5} + 68921 T^{6} )^{2} \))(\( 1 - 393 T^{2} + 73710 T^{4} - 8727326 T^{6} + 726625577 T^{8} - 44820888217 T^{10} + 2100378900880 T^{12} - 75343913092777 T^{14} + 2053270217089097 T^{16} - 41455708245189566 T^{18} + 588568838638508910 T^{20} - 5275105108889893593 T^{22} + 22563490300366186081 T^{24} \))
$43$ (\( 1 - 10 T + 109 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( 1 - 41 T^{2} + 1849 T^{4} \))(\( 1 - 13 T + 42 T^{2} - 50 T^{3} + 251 T^{4} + 17703 T^{5} - 219260 T^{6} + 761229 T^{7} + 464099 T^{8} - 3975350 T^{9} + 143589642 T^{10} - 1911109759 T^{11} + 6321363049 T^{12} \))(\( 1 - 7 T + 139 T^{2} - 525 T^{3} + 4746 T^{4} + 19698 T^{5} - 227814 T^{6} + 3326155 T^{7} - 15690613 T^{8} + 60933523 T^{9} + 454409879 T^{10} - 6181766668 T^{11} + 56674975708 T^{12} - 265815966724 T^{13} + 840203866271 T^{14} + 4844641613161 T^{15} - 53643083415013 T^{16} + 488972867726665 T^{17} - 1440095001644886 T^{18} + 5354283001585686 T^{19} + 55472198517494346 T^{20} - 263861121266842575 T^{21} + 3003996041546510611 T^{22} - 6505056176298558949 T^{23} + 39959630797262576401 T^{24} \))
$47$ (\( 1 - 2 T + 77 T^{2} - 94 T^{3} + 2209 T^{4} \))(\( 1 - 89 T^{2} + 2209 T^{4} \))(\( 1 - 11 T + 18 T^{2} + 410 T^{3} - 2409 T^{4} - 12679 T^{5} + 224300 T^{6} - 595913 T^{7} - 5321481 T^{8} + 42567430 T^{9} + 87834258 T^{10} - 2522795077 T^{11} + 10779215329 T^{12} \))(\( 1 + 7 T - 9 T^{2} - 497 T^{3} - 5250 T^{4} - 33334 T^{5} - 12592 T^{6} + 1824669 T^{7} + 19098315 T^{8} + 97990109 T^{9} + 127809129 T^{10} - 4616139010 T^{11} - 56042219764 T^{12} - 216958533470 T^{13} + 282330365961 T^{14} + 10173627086707 T^{15} + 93193684837515 T^{16} + 418478724577683 T^{17} - 135731879422768 T^{18} - 16887775097513642 T^{19} - 125009254974245250 T^{20} - 556207845132075199 T^{21} - 473392190122470441 T^{22} + 17305114505588086121 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))
$53$ (\( 1 - 2 T + 35 T^{2} - 106 T^{3} + 2809 T^{4} \))(\( ( 1 + 9 T + 53 T^{2} )^{2} \))(\( 1 - 3 T - 58 T^{2} + 648 T^{3} - 431 T^{4} - 23629 T^{5} + 179088 T^{6} - 1252337 T^{7} - 1210679 T^{8} + 96472296 T^{9} - 457647898 T^{10} - 1254586479 T^{11} + 22164361129 T^{12} \))(\( 1 + 10 T + 25 T^{2} - 278 T^{3} + 2512 T^{4} + 20294 T^{5} - 44832 T^{6} - 1725298 T^{7} + 3740845 T^{8} + 15939776 T^{9} - 71501305 T^{10} - 3189845416 T^{11} + 681878948 T^{12} - 169061807048 T^{13} - 200847165745 T^{14} + 2373066031552 T^{15} + 29517066396445 T^{16} - 721511847681914 T^{17} - 993672638135328 T^{18} + 23839587871852078 T^{19} + 156396342313338832 T^{20} - 917334278520992974 T^{21} + 4372186759137826225 T^{22} + 92690359293721915970 T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))
$59$ (\( 1 - 4 T + 90 T^{2} - 236 T^{3} + 3481 T^{4} \))(\( ( 1 - 6 T + 59 T^{2} )^{2} \))(\( ( 1 + 28 T + 429 T^{2} + 4032 T^{3} + 25311 T^{4} + 97468 T^{5} + 205379 T^{6} )^{2} \))(\( ( 1 - 22 T + 446 T^{2} - 6050 T^{3} + 72311 T^{4} - 688228 T^{5} + 5840260 T^{6} - 40605452 T^{7} + 251714591 T^{8} - 1242542950 T^{9} + 5404343006 T^{10} - 15728334578 T^{11} + 42180533641 T^{12} )^{2} \))
$61$ (\( 1 + 4 T + 118 T^{2} + 244 T^{3} + 3721 T^{4} \))(\( ( 1 - 8 T + 61 T^{2} )( 1 + 8 T + 61 T^{2} ) \))(\( 1 - 3 T - 24 T^{2} - 74 T^{3} + 4633 T^{4} - 24883 T^{5} - 67236 T^{6} - 1517863 T^{7} + 17239393 T^{8} - 16796594 T^{9} - 332300184 T^{10} - 2533788903 T^{11} + 51520374361 T^{12} \))(\( 1 + 7 T + 159 T^{2} + 105 T^{3} + 5012 T^{4} - 64120 T^{5} + 241116 T^{6} - 1456693 T^{7} + 53389705 T^{8} + 83034021 T^{9} + 2548004823 T^{10} - 10608027248 T^{11} + 41335695248 T^{12} - 647089662128 T^{13} + 9481125946383 T^{14} + 18847145120601 T^{15} + 739225366466905 T^{16} - 1230317519492593 T^{17} + 12422386584426876 T^{18} - 201512670645666520 T^{19} + 960837052742372372 T^{20} + 1227885339747584805 T^{21} + \)\(11\!\cdots\!59\)\( T^{22} + \)\(30\!\cdots\!27\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))
$67$ (\( 1 + 102 T^{2} + 4489 T^{4} \))(\( ( 1 - 8 T + 67 T^{2} )^{2} \))(\( 1 - 19 T + 168 T^{2} - 904 T^{3} + 1419 T^{4} + 3983 T^{5} + 31592 T^{6} + 266861 T^{7} + 6369891 T^{8} - 271889752 T^{9} + 3385388328 T^{10} - 25652377033 T^{11} + 90458382169 T^{12} \))(\( 1 + 37 T + 513 T^{2} + 3025 T^{3} - 450 T^{4} - 212780 T^{5} - 3203838 T^{6} - 28238209 T^{7} - 156569483 T^{8} - 254147545 T^{9} + 8082101273 T^{10} + 110845067132 T^{11} + 930583608704 T^{12} + 7426619497844 T^{13} + 36280552614497 T^{14} - 76438178076835 T^{15} - 3155050596840443 T^{16} - 38125114947613363 T^{17} - 289814002211564622 T^{18} - 1289598215380627940 T^{19} - 182730454900488450 T^{20} + 82299766548792214675 T^{21} + \)\(93\!\cdots\!37\)\( T^{22} + \)\(45\!\cdots\!71\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))
$71$ (\( 1 + 12 T + 170 T^{2} + 852 T^{3} + 5041 T^{4} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( 1 - 21 T + 118 T^{2} + 546 T^{3} - 4409 T^{4} - 107373 T^{5} + 1690660 T^{6} - 7623483 T^{7} - 22225769 T^{8} + 195419406 T^{9} + 2998578358 T^{10} - 37888816371 T^{11} + 128100283921 T^{12} \))(\( 1 + 21 T + 173 T^{2} + 721 T^{3} - 4778 T^{4} - 187446 T^{5} - 2306210 T^{6} - 16620177 T^{7} - 82259473 T^{8} + 61660641 T^{9} + 7528298205 T^{10} + 91816724300 T^{11} + 804638755620 T^{12} + 6518987425300 T^{13} + 37950151251405 T^{14} + 22069021680951 T^{15} - 2090351487104113 T^{16} - 29986611162215127 T^{17} - 295426155781449410 T^{18} - 1704843893209759386 T^{19} - 3085410372292246058 T^{20} + 33056769018001751351 T^{21} + \)\(56\!\cdots\!73\)\( T^{22} + \)\(48\!\cdots\!91\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))
$73$ (\( ( 1 - 4 T + 73 T^{2} )^{2} \))(\( ( 1 - 73 T^{2} )^{2} \))(\( 1 + 25 T + 300 T^{2} + 3400 T^{3} + 39069 T^{4} + 348943 T^{5} + 2806700 T^{6} + 25472839 T^{7} + 208198701 T^{8} + 1322657800 T^{9} + 8519472300 T^{10} + 51826789825 T^{11} + 151334226289 T^{12} \))(\( 1 - 14 T + 223 T^{2} - 1638 T^{3} + 14902 T^{4} - 24080 T^{5} - 61436 T^{6} + 7903616 T^{7} - 50738367 T^{8} + 656032846 T^{9} - 446390355 T^{10} + 13646495926 T^{11} + 186938570672 T^{12} + 996194202598 T^{13} - 2378814201795 T^{14} + 255207929652382 T^{15} - 1440880374012447 T^{16} + 16384761811580288 T^{17} - 9297369526291004 T^{18} - 266021356339855760 T^{19} + 12017868289405595062 T^{20} - 96431659028142841494 T^{21} + \)\(95\!\cdots\!27\)\( T^{22} - \)\(43\!\cdots\!78\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))
$79$ (\( 1 + 2 T + 157 T^{2} + 158 T^{3} + 6241 T^{4} \))(\( 1 - 113 T^{2} + 6241 T^{4} \))(\( 1 + 9 T + 2 T^{2} + 210 T^{3} - 977 T^{4} - 72339 T^{5} - 812260 T^{6} - 5714781 T^{7} - 6097457 T^{8} + 103538190 T^{9} + 77900162 T^{10} + 27693507591 T^{11} + 243087455521 T^{12} \))(\( 1 - 49 T + 1401 T^{2} - 29659 T^{3} + 510220 T^{4} - 7424550 T^{5} + 93952026 T^{6} - 1051207157 T^{7} + 10574913669 T^{8} - 97442137415 T^{9} + 846233526101 T^{10} - 7200350040954 T^{11} + 62681377749836 T^{12} - 568827653235366 T^{13} + 5281343436396341 T^{14} - 48042773988954185 T^{15} + 411893743975557189 T^{16} - 3234623709121447643 T^{17} + 22838558941382835546 T^{18} - \)\(14\!\cdots\!50\)\( T^{19} + \)\(77\!\cdots\!20\)\( T^{20} - \)\(35\!\cdots\!21\)\( T^{21} + \)\(13\!\cdots\!01\)\( T^{22} - \)\(36\!\cdots\!71\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))
$83$ (\( 1 - 4 T + 138 T^{2} - 332 T^{3} + 6889 T^{4} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( 1 - 17 T + 136 T^{2} - 1776 T^{3} + 27843 T^{4} - 251219 T^{5} + 1969848 T^{6} - 20851177 T^{7} + 191810427 T^{8} - 1015493712 T^{9} + 6454331656 T^{10} - 66963690931 T^{11} + 326940373369 T^{12} \))(\( 1 - 5 T + 19 T^{2} - 291 T^{3} + 6964 T^{4} - 112648 T^{5} + 680674 T^{6} - 2772355 T^{7} + 73849055 T^{8} - 851751547 T^{9} + 7543184631 T^{10} - 61831617966 T^{11} + 242863289344 T^{12} - 5132024291178 T^{13} + 51964998922959 T^{14} - 487020461804489 T^{15} + 3504752157736655 T^{16} - 10920419021824265 T^{17} + 222539811702570706 T^{18} - 3056821871879502296 T^{19} + 15684963104616281524 T^{20} - 54399614282854257273 T^{21} + \)\(29\!\cdots\!31\)\( T^{22} - \)\(64\!\cdots\!35\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))
$89$ (\( 1 + 8 T + 122 T^{2} + 712 T^{3} + 7921 T^{4} \))(\( 1 - 158 T^{2} + 7921 T^{4} \))(\( 1 - 7 T + 30 T^{2} + 532 T^{3} + 7389 T^{4} - 23261 T^{5} + 663944 T^{6} - 2070229 T^{7} + 58528269 T^{8} + 375043508 T^{9} + 1882267230 T^{10} - 39088416143 T^{11} + 496981290961 T^{12} \))(\( 1 - 7 T + 319 T^{2} - 2107 T^{3} + 41592 T^{4} - 268912 T^{5} + 2519080 T^{6} - 17316453 T^{7} - 32462455 T^{8} + 662290951 T^{9} - 28551782921 T^{10} + 290592922368 T^{11} - 3660702381280 T^{12} + 25862770090752 T^{13} - 226158672517241 T^{14} + 466894589435519 T^{15} - 2036767175061655 T^{16} - 96696102997814397 T^{17} + 1251935630434035880 T^{18} - 11894336729426494448 T^{19} + \)\(16\!\cdots\!52\)\( T^{20} - \)\(73\!\cdots\!63\)\( T^{21} + \)\(99\!\cdots\!19\)\( T^{22} - \)\(19\!\cdots\!23\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))
$97$ (\( 1 + 8 T + 138 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 - 14 T^{2} + 9409 T^{4} \))(\( 1 - T + 114 T^{2} + 382 T^{3} + 20669 T^{4} + 10611 T^{5} + 2196928 T^{6} + 1029267 T^{7} + 194474621 T^{8} + 348641086 T^{9} + 10092338034 T^{10} - 8587340257 T^{11} + 832972004929 T^{12} \))(\( 1 - 14 T + 441 T^{2} - 5068 T^{3} + 83194 T^{4} - 784252 T^{5} + 9216214 T^{6} - 73602242 T^{7} + 720487153 T^{8} - 5423464648 T^{9} + 47440764385 T^{10} - 407534937824 T^{11} + 3645124614984 T^{12} - 39530888968928 T^{13} + 446370152098465 T^{14} - 4949849750684104 T^{15} + 63784209624826993 T^{16} - 632047495732056194 T^{17} + 7676848253434718806 T^{18} - 63366216198529076476 T^{19} + \)\(65\!\cdots\!34\)\( T^{20} - \)\(38\!\cdots\!56\)\( T^{21} + \)\(32\!\cdots\!09\)\( T^{22} - \)\(10\!\cdots\!42\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))
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