Properties

Label 50.2
Level 50
Weight 2
Dimension 24
Nonzero newspaces 4
Newform subspaces 6
Sturm bound 300
Trace bound 4

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 6 \)
Sturm bound: \(300\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(50))\).

Total New Old
Modular forms 103 24 79
Cusp forms 48 24 24
Eisenstein series 55 0 55

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(50))\)

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
50.2.a \(\chi_{50}(1, \cdot)\) 50.2.a.a 1 1
50.2.a.b 1
50.2.b \(\chi_{50}(49, \cdot)\) 50.2.b.a 2 1
50.2.d \(\chi_{50}(11, \cdot)\) 50.2.d.a 4 4
50.2.d.b 8
50.2.e \(\chi_{50}(9, \cdot)\) 50.2.e.a 8 4

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(50))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(50)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T^{2} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))
$3$ (\( 1 - T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 - 5 T^{2} + 9 T^{4} \))(\( 1 + T + 3 T^{2} + 5 T^{3} + 16 T^{4} + 15 T^{5} + 27 T^{6} + 27 T^{7} + 81 T^{8} \))(\( ( 1 - T + 3 T^{2} - 5 T^{3} + 16 T^{4} - 15 T^{5} + 27 T^{6} - 27 T^{7} + 81 T^{8} )( 1 + 4 T + 3 T^{2} - 10 T^{3} - 29 T^{4} - 30 T^{5} + 27 T^{6} + 108 T^{7} + 81 T^{8} ) \))(\( ( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} )^{2}( 1 - 4 T + 13 T^{2} - 30 T^{3} + 61 T^{4} - 90 T^{5} + 117 T^{6} - 108 T^{7} + 81 T^{8} ) \))
$5$ (\( 1 - 5 T + 15 T^{2} - 25 T^{3} + 25 T^{4} \))(\( 1 - 15 T^{2} + 105 T^{4} - 375 T^{6} + 625 T^{8} \))(\( 1 + 10 T + 45 T^{2} + 130 T^{3} + 305 T^{4} + 650 T^{5} + 1125 T^{6} + 1250 T^{7} + 625 T^{8} \))
$7$ (\( 1 - 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( ( 1 + 3 T + 7 T^{2} )^{4} \))(\( ( 1 - 2 T + 17 T^{2} - 30 T^{3} + 156 T^{4} - 210 T^{5} + 833 T^{6} - 686 T^{7} + 2401 T^{8} )^{2} \))(\( 1 - 20 T^{2} + 146 T^{4} - 160 T^{6} - 2789 T^{8} - 7840 T^{10} + 350546 T^{12} - 2352980 T^{14} + 5764801 T^{16} \))
$11$ (\( 1 + 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( 1 - 3 T + 8 T^{2} - 51 T^{3} + 265 T^{4} - 561 T^{5} + 968 T^{6} - 3993 T^{7} + 14641 T^{8} \))(\( 1 - T - 25 T^{2} + 40 T^{3} + 160 T^{4} - 653 T^{5} + 2838 T^{6} + 3940 T^{7} - 61905 T^{8} + 43340 T^{9} + 343398 T^{10} - 869143 T^{11} + 2342560 T^{12} + 6442040 T^{13} - 44289025 T^{14} - 19487171 T^{15} + 214358881 T^{16} \))(\( 1 + 4 T - 25 T^{2} - 80 T^{3} + 260 T^{4} + 752 T^{5} + 1353 T^{6} - 2310 T^{7} - 44765 T^{8} - 25410 T^{9} + 163713 T^{10} + 1000912 T^{11} + 3806660 T^{12} - 12884080 T^{13} - 44289025 T^{14} + 77948684 T^{15} + 214358881 T^{16} \))
$13$ (\( 1 + 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( 1 + T - 12 T^{2} - 25 T^{3} + 131 T^{4} - 325 T^{5} - 2028 T^{6} + 2197 T^{7} + 28561 T^{8} \))(\( 1 + 13 T + 67 T^{2} + 164 T^{3} + 76 T^{4} - 427 T^{5} + 3190 T^{6} + 37102 T^{7} + 173189 T^{8} + 482326 T^{9} + 539110 T^{10} - 938119 T^{11} + 2170636 T^{12} + 60892052 T^{13} + 323396203 T^{14} + 815730721 T^{15} + 815730721 T^{16} \))(\( 1 - 5 T^{2} + 20 T^{3} + 96 T^{4} - 100 T^{5} + 25 T^{6} + 6470 T^{7} - 6509 T^{8} + 84110 T^{9} + 4225 T^{10} - 219700 T^{11} + 2741856 T^{12} + 7425860 T^{13} - 24134045 T^{14} + 815730721 T^{16} \))
$17$ (\( 1 + 3 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 - 25 T^{2} + 289 T^{4} \))(\( 1 - 3 T + 37 T^{2} - 45 T^{3} + 676 T^{4} - 765 T^{5} + 10693 T^{6} - 14739 T^{7} + 83521 T^{8} \))(\( 1 + 11 T + 38 T^{2} - 8 T^{3} - 474 T^{4} - 1639 T^{5} + 2235 T^{6} + 60196 T^{7} + 352544 T^{8} + 1023332 T^{9} + 645915 T^{10} - 8052407 T^{11} - 39588954 T^{12} - 11358856 T^{13} + 917227622 T^{14} + 4513725403 T^{15} + 6975757441 T^{16} \))(\( 1 - 10 T + 75 T^{2} - 420 T^{3} + 2171 T^{4} - 9960 T^{5} + 43585 T^{6} - 177290 T^{7} + 711536 T^{8} - 3013930 T^{9} + 12596065 T^{10} - 48933480 T^{11} + 181324091 T^{12} - 596339940 T^{13} + 1810317675 T^{14} - 4103386730 T^{15} + 6975757441 T^{16} \))
$19$ (\( 1 - 5 T + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( ( 1 + 5 T + 19 T^{2} )^{2} \))(\( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 1330 T^{5} + 7581 T^{6} + 68590 T^{7} + 130321 T^{8} \))(\( ( 1 - 15 T + 93 T^{2} - 285 T^{3} + 361 T^{4} )^{2}( 1 + 10 T + 41 T^{2} - 10 T^{3} - 509 T^{4} - 190 T^{5} + 14801 T^{6} + 68590 T^{7} + 130321 T^{8} ) \))(\( 1 - 10 T + 22 T^{2} - 30 T^{3} + 1023 T^{4} - 4210 T^{5} - 4736 T^{6} - 17200 T^{7} + 424105 T^{8} - 326800 T^{9} - 1709696 T^{10} - 28876390 T^{11} + 133318383 T^{12} - 74282970 T^{13} + 1035009382 T^{14} - 8938717390 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 10 T^{2} + 529 T^{4} \))(\( 1 - 9 T + 8 T^{2} + 195 T^{3} - 1259 T^{4} + 4485 T^{5} + 4232 T^{6} - 109503 T^{7} + 279841 T^{8} \))(\( 1 + 3 T + 17 T^{2} - 26 T^{3} + 866 T^{4} - 2257 T^{5} - 7690 T^{6} - 132948 T^{7} + 158519 T^{8} - 3057804 T^{9} - 4068010 T^{10} - 27460919 T^{11} + 242342306 T^{12} - 167344918 T^{13} + 2516610113 T^{14} + 10214476341 T^{15} + 78310985281 T^{16} \))(\( 1 - 10 T + 105 T^{2} - 790 T^{3} + 5156 T^{4} - 30260 T^{5} + 156735 T^{6} - 798320 T^{7} + 3801671 T^{8} - 18361360 T^{9} + 82912815 T^{10} - 368173420 T^{11} + 1442860196 T^{12} - 5084710970 T^{13} + 15543768345 T^{14} - 34048254470 T^{15} + 78310985281 T^{16} \))
$29$ (\( 1 + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( 1 - 15 T + 56 T^{2} + 315 T^{3} - 3629 T^{4} + 9135 T^{5} + 47096 T^{6} - 365835 T^{7} + 707281 T^{8} \))(\( 1 + 15 T + 47 T^{2} - 480 T^{3} - 4732 T^{4} - 16485 T^{5} + 2494 T^{6} + 526050 T^{7} + 4240405 T^{8} + 15255450 T^{9} + 2097454 T^{10} - 402052665 T^{11} - 3346853692 T^{12} - 9845351520 T^{13} + 27956696087 T^{14} + 258748144635 T^{15} + 500246412961 T^{16} \))(\( 1 - 10 T + 47 T^{2} - 470 T^{3} + 4888 T^{4} - 28460 T^{5} + 139689 T^{6} - 999650 T^{7} + 6593475 T^{8} - 28989850 T^{9} + 117478449 T^{10} - 694110940 T^{11} + 3457189528 T^{12} - 9640240030 T^{13} + 27956696087 T^{14} - 172498763090 T^{15} + 500246412961 T^{16} \))
$31$ (\( 1 - 2 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( ( 1 - 2 T + 31 T^{2} )^{2} \))(\( 1 - 3 T - 12 T^{2} - 131 T^{3} + 1365 T^{4} - 4061 T^{5} - 11532 T^{6} - 89373 T^{7} + 923521 T^{8} \))(\( 1 + 9 T + 25 T^{2} + 330 T^{3} + 2510 T^{4} + 7017 T^{5} + 86378 T^{6} + 461340 T^{7} + 675415 T^{8} + 14301540 T^{9} + 83009258 T^{10} + 209043447 T^{11} + 2318037710 T^{12} + 9447619830 T^{13} + 22187592025 T^{14} + 247613526999 T^{15} + 852891037441 T^{16} \))(\( 1 - 6 T + 5 T^{2} - 110 T^{3} + 1840 T^{4} - 4048 T^{5} - 9417 T^{6} - 113520 T^{7} + 1708895 T^{8} - 3519120 T^{9} - 9049737 T^{10} - 120593968 T^{11} + 1699278640 T^{12} - 3149206610 T^{13} + 4437518405 T^{14} - 165075684666 T^{15} + 852891037441 T^{16} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 + 17 T + 147 T^{2} + 1135 T^{3} + 7976 T^{4} + 41995 T^{5} + 201243 T^{6} + 861101 T^{7} + 1874161 T^{8} \))(\( 1 + 6 T - 27 T^{2} - 318 T^{3} - 99 T^{4} + 14106 T^{5} + 58255 T^{6} - 123234 T^{7} - 1122756 T^{8} - 4559658 T^{9} + 79751095 T^{10} + 714511218 T^{11} - 185541939 T^{12} - 22051378326 T^{13} - 69274613043 T^{14} + 569591262798 T^{15} + 3512479453921 T^{16} \))(\( 1 + 10 T + 65 T^{2} + 230 T^{3} + 1471 T^{4} + 9410 T^{5} + 48855 T^{6} - 117890 T^{7} - 967744 T^{8} - 4361930 T^{9} + 66882495 T^{10} + 476644730 T^{11} + 2756890831 T^{12} + 15949110110 T^{13} + 166772216585 T^{14} + 949318771330 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 + 3 T + 41 T^{2} \))(\( 1 + 3 T + 41 T^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( 1 - 13 T + 28 T^{2} + 169 T^{3} - 945 T^{4} + 6929 T^{5} + 47068 T^{6} - 895973 T^{7} + 2825761 T^{8} \))(\( 1 + 9 T - 55 T^{2} - 1030 T^{3} - 2470 T^{4} + 35497 T^{5} + 265658 T^{6} - 359550 T^{7} - 11328415 T^{8} - 14741550 T^{9} + 446571098 T^{10} + 2446488737 T^{11} - 6979629670 T^{12} - 119331887030 T^{13} - 261255733255 T^{14} + 1752788464929 T^{15} + 7984925229121 T^{16} \))(\( 1 + 14 T + 45 T^{2} + 130 T^{3} + 2680 T^{4} + 7652 T^{5} - 46977 T^{6} - 402900 T^{7} - 2944785 T^{8} - 16518900 T^{9} - 78968337 T^{10} + 527383492 T^{11} + 7573039480 T^{12} + 15061306130 T^{13} + 213754690845 T^{14} + 2726559834334 T^{15} + 7984925229121 T^{16} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( ( 1 + 8 T + 97 T^{2} + 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 6 T + 133 T^{2} - 710 T^{3} + 7916 T^{4} - 30530 T^{5} + 245917 T^{6} - 477042 T^{7} + 3418801 T^{8} )^{2} \))(\( 1 - 120 T^{2} + 10666 T^{4} - 636400 T^{6} + 31214171 T^{8} - 1176703600 T^{10} + 36464931466 T^{12} - 758563565880 T^{14} + 11688200277601 T^{16} \))
$47$ (\( 1 - 12 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 + 50 T^{2} + 2209 T^{4} \))(\( 1 - 23 T + 202 T^{2} - 925 T^{3} + 4101 T^{4} - 43475 T^{5} + 446218 T^{6} - 2387929 T^{7} + 4879681 T^{8} \))(\( 1 + T - 77 T^{2} - 108 T^{3} + 5436 T^{4} + 2341 T^{5} - 357450 T^{6} - 29584 T^{7} + 16743019 T^{8} - 1390448 T^{9} - 789607050 T^{10} + 243049643 T^{11} + 26525945916 T^{12} - 24769260756 T^{13} - 829999580333 T^{14} + 506623120463 T^{15} + 23811286661761 T^{16} \))(\( 1 + 30 T + 425 T^{2} + 3450 T^{3} + 13416 T^{4} - 40020 T^{5} - 992525 T^{6} - 8396100 T^{7} - 57415609 T^{8} - 394616700 T^{9} - 2192487725 T^{10} - 4154996460 T^{11} + 65465800296 T^{12} + 791240274150 T^{13} + 4581166514825 T^{14} + 15198693613890 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 + 16 T + 43 T^{2} - 700 T^{3} - 7959 T^{4} - 37100 T^{5} + 120787 T^{6} + 2382032 T^{7} + 7890481 T^{8} \))(\( 1 - 7 T + 2 T^{2} - 6 T^{3} - 794 T^{4} + 27393 T^{5} + 64385 T^{6} - 1339888 T^{7} + 6789964 T^{8} - 71014064 T^{9} + 180857465 T^{10} + 4078187661 T^{11} - 6265041914 T^{12} - 2509172958 T^{13} + 44328722258 T^{14} - 8222977978859 T^{15} + 62259690411361 T^{16} \))(\( 1 + 90 T^{2} + 900 T^{3} + 5291 T^{4} + 81000 T^{5} + 628380 T^{6} + 3853800 T^{7} + 44701381 T^{8} + 204251400 T^{9} + 1765119420 T^{10} + 12059037000 T^{11} + 41748534971 T^{12} + 376375943700 T^{13} + 1994792501610 T^{14} + 62259690411361 T^{16} \))
$59$ (\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( 1 + 10 T + T^{2} + 200 T^{3} + 5061 T^{4} + 11800 T^{5} + 3481 T^{6} + 2053790 T^{7} + 12117361 T^{8} \))(\( 1 - 10 T + 142 T^{2} - 730 T^{3} + 11283 T^{4} - 22410 T^{5} + 330384 T^{6} + 1700800 T^{7} + 16244505 T^{8} + 100347200 T^{9} + 1150066704 T^{10} - 4602543390 T^{11} + 136720184163 T^{12} - 521894738270 T^{13} + 5989635777022 T^{14} - 24886514848190 T^{15} + 146830437604321 T^{16} \))(\( 1 - 118 T^{2} + 3563 T^{4} + 324004 T^{6} - 38801675 T^{8} + 1127857924 T^{10} + 43174157243 T^{12} - 4977302969638 T^{14} + 146830437604321 T^{16} \))
$61$ (\( 1 - 2 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( ( 1 - 2 T + 61 T^{2} )^{2} \))(\( 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 25864 T^{5} + 11163 T^{6} + 453962 T^{7} + 13845841 T^{8} \))(\( 1 - 6 T - 140 T^{2} + 210 T^{3} + 7045 T^{4} + 27242 T^{5} + 433648 T^{6} - 1657580 T^{7} - 60459295 T^{8} - 101112380 T^{9} + 1613604208 T^{10} + 6183416402 T^{11} + 97543949845 T^{12} + 177365223210 T^{13} - 7212852410540 T^{14} - 18856457016126 T^{15} + 191707312997281 T^{16} \))(\( 1 + 14 T + 30 T^{2} - 30 T^{3} + 5215 T^{4} + 25262 T^{5} - 186192 T^{6} - 1508020 T^{7} - 4396815 T^{8} - 91989220 T^{9} - 692820432 T^{10} + 5733994022 T^{11} + 72206060815 T^{12} - 25337889030 T^{13} + 1545611230830 T^{14} + 43998399704294 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 + 13 T + 67 T^{2} \))(\( 1 - 13 T + 67 T^{2} \))(\( 1 + 35 T^{2} + 4489 T^{4} \))(\( 1 - 13 T + 12 T^{2} + 895 T^{3} - 9919 T^{4} + 59965 T^{5} + 53868 T^{6} - 3909919 T^{7} + 20151121 T^{8} \))(\( 1 + 11 T + 103 T^{2} + 1672 T^{3} + 14576 T^{4} + 139271 T^{5} + 1482950 T^{6} + 11696476 T^{7} + 89866399 T^{8} + 783663892 T^{9} + 6656962550 T^{10} + 41887563773 T^{11} + 293722739696 T^{12} + 2257409178904 T^{13} + 9317213363407 T^{14} + 66667827658553 T^{15} + 406067677556641 T^{16} \))(\( 1 - 10 T + 195 T^{2} - 2790 T^{3} + 29716 T^{4} - 356960 T^{5} + 3452465 T^{6} - 30345750 T^{7} + 284471411 T^{8} - 2033165250 T^{9} + 15498115385 T^{10} - 107360360480 T^{11} + 598810711636 T^{12} - 3766849048530 T^{13} + 17639384522955 T^{14} - 60607116053230 T^{15} + 406067677556641 T^{16} \))
$71$ (\( 1 - 12 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( 1 - 3 T - 62 T^{2} + 399 T^{3} + 3205 T^{4} + 28329 T^{5} - 312542 T^{6} - 1073733 T^{7} + 25411681 T^{8} \))(\( 1 + 9 T + 75 T^{2} + 1220 T^{3} + 18360 T^{4} + 165937 T^{5} + 1180298 T^{6} + 14013840 T^{7} + 141940835 T^{8} + 994982640 T^{9} + 5949882218 T^{10} + 59390677607 T^{11} + 466558463160 T^{12} + 2201159808220 T^{13} + 9607521294075 T^{14} + 81856081425519 T^{15} + 645753531245761 T^{16} \))(\( 1 + 34 T + 425 T^{2} + 2570 T^{3} + 16960 T^{4} + 219512 T^{5} + 2212323 T^{6} + 19311440 T^{7} + 171308535 T^{8} + 1371112240 T^{9} + 11152320243 T^{10} + 78565759432 T^{11} + 430982109760 T^{12} + 4636869432070 T^{13} + 54442620666425 T^{14} + 309234085385294 T^{15} + 645753531245761 T^{16} \))
$73$ (\( 1 - 11 T + 73 T^{2} \))(\( 1 + 11 T + 73 T^{2} \))(\( 1 - 25 T^{2} + 5329 T^{4} \))(\( 1 - 14 T + 63 T^{2} - 850 T^{3} + 12521 T^{4} - 62050 T^{5} + 335727 T^{6} - 5446238 T^{7} + 28398241 T^{8} \))(\( 1 + 8 T + 62 T^{2} + 1114 T^{3} + 11231 T^{4} + 71208 T^{5} + 863380 T^{6} + 9235272 T^{7} + 61248289 T^{8} + 674174856 T^{9} + 4600952020 T^{10} + 27701122536 T^{11} + 318940644671 T^{12} + 2309401754602 T^{13} + 9382722029918 T^{14} + 88379188152776 T^{15} + 806460091894081 T^{16} \))(\( 1 + 110 T^{2} + 840 T^{3} + 6291 T^{4} + 92400 T^{5} + 907660 T^{6} + 5561520 T^{7} + 91719461 T^{8} + 405990960 T^{9} + 4836920140 T^{10} + 35945170800 T^{11} + 178653334131 T^{12} + 1741380138120 T^{13} + 16646764891790 T^{14} + 806460091894081 T^{16} \))
$79$ (\( 1 + 10 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{2} \))(\( 1 - 10 T - 39 T^{2} + 790 T^{3} - 2839 T^{4} + 62410 T^{5} - 243399 T^{6} - 4930390 T^{7} + 38950081 T^{8} \))(\( 1 + 10 T - 38 T^{2} + 320 T^{3} + 15323 T^{4} + 59210 T^{5} - 186436 T^{6} + 4386400 T^{7} + 89172925 T^{8} + 346525600 T^{9} - 1163547076 T^{10} + 29192839190 T^{11} + 596832091163 T^{12} + 984658047680 T^{13} - 9237323309798 T^{14} + 192039089861590 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 138 T^{2} + 1500 T^{3} + 4403 T^{4} - 205200 T^{5} + 1430844 T^{6} + 7779600 T^{7} - 206126795 T^{8} + 614588400 T^{9} + 8929897404 T^{10} - 101171602800 T^{11} + 171497206643 T^{12} + 4615584598500 T^{13} - 33546068861898 T^{14} + 1517108809906561 T^{16} \))
$83$ (\( 1 + 9 T + 83 T^{2} \))(\( 1 - 9 T + 83 T^{2} \))(\( 1 - 85 T^{2} + 6889 T^{4} \))(\( 1 + T + 58 T^{2} + 335 T^{3} + 7041 T^{4} + 27805 T^{5} + 399562 T^{6} + 571787 T^{7} + 47458321 T^{8} \))(\( 1 - 27 T + 237 T^{2} - 126 T^{3} - 12774 T^{4} + 113973 T^{5} - 181210 T^{6} - 11072628 T^{7} + 166729299 T^{8} - 919028124 T^{9} - 1248355690 T^{10} + 65168279751 T^{11} - 606232592454 T^{12} - 496319121018 T^{13} + 77484868488453 T^{14} - 732673376719929 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 50 T + 1335 T^{2} - 23750 T^{3} + 305836 T^{4} - 2886800 T^{5} + 19447445 T^{6} - 89310250 T^{7} + 442867371 T^{8} - 7412750750 T^{9} + 133973448605 T^{10} - 1650634711600 T^{11} + 14514463061356 T^{12} - 93552215271250 T^{13} + 436465398447615 T^{14} - 1356802549481350 T^{15} + 2252292232139041 T^{16} \))
$89$ (\( 1 - 15 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( ( 1 + 15 T + 89 T^{2} )^{2} \))(\( 1 - 10 T - 29 T^{2} - 200 T^{3} + 10101 T^{4} - 17800 T^{5} - 229709 T^{6} - 7049690 T^{7} + 62742241 T^{8} \))(\( 1 + 15 T - 108 T^{2} - 1640 T^{3} + 7368 T^{4} + 158465 T^{5} + 2297399 T^{6} + 297300 T^{7} - 272740720 T^{8} + 26459700 T^{9} + 18197697479 T^{10} + 111712912585 T^{11} + 462284831688 T^{12} - 9157857496360 T^{13} - 53673979423788 T^{14} + 663470023432935 T^{15} + 3936588805702081 T^{16} \))(\( ( 1 - 10 T + 71 T^{2} - 1290 T^{3} + 19001 T^{4} - 114810 T^{5} + 562391 T^{6} - 7049690 T^{7} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 190 T^{2} + 9409 T^{4} \))(\( 1 + 22 T + 207 T^{2} + 2420 T^{3} + 31541 T^{4} + 234740 T^{5} + 1947663 T^{6} + 20078806 T^{7} + 88529281 T^{8} \))(\( ( 1 + 18 T + 27 T^{2} - 1060 T^{3} - 9699 T^{4} - 102820 T^{5} + 254043 T^{6} + 16428114 T^{7} + 88529281 T^{8} )^{2} \))(\( 1 + 20 T + 270 T^{2} + 3780 T^{3} + 51031 T^{4} + 591220 T^{5} + 6636980 T^{6} + 69655800 T^{7} + 673071701 T^{8} + 6756612600 T^{9} + 62447344820 T^{10} + 539590531060 T^{11} + 4517737738711 T^{12} + 32460146171460 T^{13} + 224902441330830 T^{14} + 1615965689562260 T^{15} + 7837433594376961 T^{16} \))
show more
show less