Properties

Label 139.2
Level 139
Weight 2
Dimension 737
Nonzero newspaces 4
Newform subspaces 6
Sturm bound 3220
Trace bound 1

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Defining parameters

Level: \( N \) = \( 139 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 6 \)
Sturm bound: \(3220\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(139))\).

Total New Old
Modular forms 874 874 0
Cusp forms 737 737 0
Eisenstein series 137 137 0

Trace form

\( 737q - 66q^{2} - 65q^{3} - 62q^{4} - 63q^{5} - 57q^{6} - 61q^{7} - 54q^{8} - 56q^{9} + O(q^{10}) \) \( 737q - 66q^{2} - 65q^{3} - 62q^{4} - 63q^{5} - 57q^{6} - 61q^{7} - 54q^{8} - 56q^{9} - 51q^{10} - 57q^{11} - 41q^{12} - 55q^{13} - 45q^{14} - 45q^{15} - 38q^{16} - 51q^{17} - 30q^{18} - 49q^{19} - 27q^{20} - 37q^{21} - 33q^{22} - 45q^{23} - 9q^{24} - 38q^{25} - 27q^{26} - 29q^{27} - 13q^{28} - 39q^{29} + 3q^{30} - 37q^{31} - 6q^{32} - 21q^{33} - 15q^{34} - 21q^{35} + 22q^{36} - 31q^{37} - 9q^{38} - 13q^{39} + 21q^{40} - 27q^{41} + 27q^{42} - 25q^{43} + 15q^{44} + 9q^{45} + 3q^{46} - 21q^{47} + 55q^{48} - 12q^{49} + 24q^{50} + 3q^{51} + 29q^{52} - 15q^{53} + 51q^{54} + 3q^{55} + 51q^{56} + 11q^{57} + 21q^{58} - 9q^{59} + 99q^{60} - 7q^{61} + 27q^{62} + 35q^{63} + 58q^{64} + 15q^{65} + 75q^{66} - q^{67} + 57q^{68} + 27q^{69} + 75q^{70} + 3q^{71} + 126q^{72} + 5q^{73} + 45q^{74} + 55q^{75} + 71q^{76} + 27q^{77} + 99q^{78} + 11q^{79} + 117q^{80} + 52q^{81} + 57q^{82} + 15q^{83} + 155q^{84} + 39q^{85} + 63q^{86} + 51q^{87} + 111q^{88} + 21q^{89} + 165q^{90} + 43q^{91} + 99q^{92} + 59q^{93} + 75q^{94} + 51q^{95} + 183q^{96} + 29q^{97} + 102q^{98} + 87q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(139))\)

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
139.2.a \(\chi_{139}(1, \cdot)\) 139.2.a.a 1 1
139.2.a.b 3
139.2.a.c 7
139.2.c \(\chi_{139}(42, \cdot)\) 139.2.c.a 22 2
139.2.e \(\chi_{139}(6, \cdot)\) 139.2.e.a 220 22
139.2.g \(\chi_{139}(4, \cdot)\) 139.2.g.a 484 44

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + 2 T^{2} \))(\( 1 + 2 T + 5 T^{2} + 7 T^{3} + 10 T^{4} + 8 T^{5} + 8 T^{6} \))(\( 1 - T + 3 T^{2} - 4 T^{3} + 9 T^{4} - 6 T^{5} + 18 T^{6} - 16 T^{7} + 36 T^{8} - 24 T^{9} + 72 T^{10} - 64 T^{11} + 96 T^{12} - 64 T^{13} + 128 T^{14} \))
$3$ (\( 1 - 2 T + 3 T^{2} \))(\( 1 + 2 T + 8 T^{2} + 11 T^{3} + 24 T^{4} + 18 T^{5} + 27 T^{6} \))(\( 1 + 2 T + 6 T^{2} + 11 T^{3} + 20 T^{4} + 22 T^{5} + 43 T^{6} + 26 T^{7} + 129 T^{8} + 198 T^{9} + 540 T^{10} + 891 T^{11} + 1458 T^{12} + 1458 T^{13} + 2187 T^{14} \))
$5$ (\( 1 + T + 5 T^{2} \))(\( 1 + 8 T + 34 T^{2} + 93 T^{3} + 170 T^{4} + 200 T^{5} + 125 T^{6} \))(\( 1 - 11 T + 71 T^{2} - 328 T^{3} + 1214 T^{4} - 3766 T^{5} + 10155 T^{6} - 24093 T^{7} + 50775 T^{8} - 94150 T^{9} + 151750 T^{10} - 205000 T^{11} + 221875 T^{12} - 171875 T^{13} + 78125 T^{14} \))
$7$ (\( 1 - 3 T + 7 T^{2} \))(\( 1 + 14 T^{2} + 7 T^{3} + 98 T^{4} + 343 T^{6} \))(\( 1 + 5 T + 41 T^{2} + 128 T^{3} + 594 T^{4} + 1270 T^{5} + 4799 T^{6} + 8663 T^{7} + 33593 T^{8} + 62230 T^{9} + 203742 T^{10} + 307328 T^{11} + 689087 T^{12} + 588245 T^{13} + 823543 T^{14} \))
$11$ (\( 1 - 5 T + 11 T^{2} \))(\( 1 + 7 T + 33 T^{2} + 105 T^{3} + 363 T^{4} + 847 T^{5} + 1331 T^{6} \))(\( 1 - 2 T + 41 T^{2} - 50 T^{3} + 747 T^{4} - 336 T^{5} + 8869 T^{6} - 387 T^{7} + 97559 T^{8} - 40656 T^{9} + 994257 T^{10} - 732050 T^{11} + 6603091 T^{12} - 3543122 T^{13} + 19487171 T^{14} \))
$13$ (\( 1 + 7 T + 13 T^{2} \))(\( 1 - T + 23 T^{2} - 39 T^{3} + 299 T^{4} - 169 T^{5} + 2197 T^{6} \))(\( 1 - 6 T + 89 T^{2} - 404 T^{3} + 3311 T^{4} - 11844 T^{5} + 69309 T^{6} - 197757 T^{7} + 901017 T^{8} - 2001636 T^{9} + 7274267 T^{10} - 11538644 T^{11} + 33045077 T^{12} - 28960854 T^{13} + 62748517 T^{14} \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( 1 + 3 T + 47 T^{2} + 89 T^{3} + 799 T^{4} + 867 T^{5} + 4913 T^{6} \))(\( 1 - 5 T + 77 T^{2} - 147 T^{3} + 1585 T^{4} + 3829 T^{5} + 3881 T^{6} + 165878 T^{7} + 65977 T^{8} + 1106581 T^{9} + 7787105 T^{10} - 12277587 T^{11} + 109328989 T^{12} - 120687845 T^{13} + 410338673 T^{14} \))
$19$ (\( 1 + 2 T + 19 T^{2} \))(\( 1 + 2 T + 14 T^{2} - 51 T^{3} + 266 T^{4} + 722 T^{5} + 6859 T^{6} \))(\( 1 + 10 T + 130 T^{2} + 927 T^{3} + 7094 T^{4} + 39234 T^{5} + 218745 T^{6} + 956346 T^{7} + 4156155 T^{8} + 14163474 T^{9} + 48657746 T^{10} + 120807567 T^{11} + 321892870 T^{12} + 470458810 T^{13} + 893871739 T^{14} \))
$23$ (\( 1 - 2 T + 23 T^{2} \))(\( 1 + 7 T + 55 T^{2} + 315 T^{3} + 1265 T^{4} + 3703 T^{5} + 12167 T^{6} \))(\( 1 + T + 113 T^{2} + 3 T^{3} + 5837 T^{4} - 3577 T^{5} + 189029 T^{6} - 144326 T^{7} + 4347667 T^{8} - 1892233 T^{9} + 71018779 T^{10} + 839523 T^{11} + 727306759 T^{12} + 148035889 T^{13} + 3404825447 T^{14} \))
$29$ (\( 1 - 9 T + 29 T^{2} \))(\( 1 + 15 T + 141 T^{2} + 883 T^{3} + 4089 T^{4} + 12615 T^{5} + 24389 T^{6} \))(\( 1 - 30 T + 503 T^{2} - 5736 T^{3} + 49929 T^{4} - 352118 T^{5} + 2152887 T^{6} - 11980823 T^{7} + 62433723 T^{8} - 296131238 T^{9} + 1217718381 T^{10} - 4056963816 T^{11} + 10317107947 T^{12} - 17844699630 T^{13} + 17249876309 T^{14} \))
$31$ (\( 1 - 9 T + 31 T^{2} \))(\( 1 - 3 T + 75 T^{2} - 173 T^{3} + 2325 T^{4} - 2883 T^{5} + 29791 T^{6} \))(\( 1 + 20 T + 313 T^{2} + 3540 T^{3} + 33819 T^{4} + 267438 T^{5} + 1851523 T^{6} + 10966915 T^{7} + 57397213 T^{8} + 257007918 T^{9} + 1007501829 T^{10} + 3269264340 T^{11} + 8960924263 T^{12} + 17750073620 T^{13} + 27512614111 T^{14} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 + 9 T + 89 T^{2} + 595 T^{3} + 3293 T^{4} + 12321 T^{5} + 50653 T^{6} \))(\( 1 - 6 T + 103 T^{2} - 897 T^{3} + 7857 T^{4} - 56685 T^{5} + 420206 T^{6} - 2498246 T^{7} + 15547622 T^{8} - 77601765 T^{9} + 397980621 T^{10} - 1681122417 T^{11} + 7142427571 T^{12} - 15394358454 T^{13} + 94931877133 T^{14} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( 1 + 8 T + 135 T^{2} + 648 T^{3} + 5535 T^{4} + 13448 T^{5} + 68921 T^{6} \))(\( 1 - 19 T + 184 T^{2} - 1087 T^{3} + 6724 T^{4} - 57933 T^{5} + 542627 T^{6} - 3906482 T^{7} + 22247707 T^{8} - 97385373 T^{9} + 463424804 T^{10} - 3071602207 T^{11} + 21317540984 T^{12} - 90251980579 T^{13} + 194754273881 T^{14} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 2 T + 100 T^{2} - 101 T^{3} + 4300 T^{4} - 3698 T^{5} + 79507 T^{6} \))(\( 1 + 12 T + 246 T^{2} + 1651 T^{3} + 18912 T^{4} + 65268 T^{5} + 704463 T^{6} + 1413290 T^{7} + 30291909 T^{8} + 120680532 T^{9} + 1503636384 T^{10} + 5644440451 T^{11} + 36164076978 T^{12} + 75856356588 T^{13} + 271818611107 T^{14} \))
$47$ (\( 1 - 8 T + 47 T^{2} \))(\( 1 - 10 T + 144 T^{2} - 927 T^{3} + 6768 T^{4} - 22090 T^{5} + 103823 T^{6} \))(\( 1 + 3 T + 109 T^{2} - 37 T^{3} + 9701 T^{4} + 5669 T^{5} + 602068 T^{6} - 199798 T^{7} + 28297196 T^{8} + 12522821 T^{9} + 1007186923 T^{10} - 180548197 T^{11} + 24998605763 T^{12} + 32337645987 T^{13} + 506623120463 T^{14} \))
$53$ (\( 1 + 53 T^{2} \))(\( 1 + 12 T + 144 T^{2} + 895 T^{3} + 7632 T^{4} + 33708 T^{5} + 148877 T^{6} \))(\( 1 - 38 T + 918 T^{2} - 15753 T^{3} + 214716 T^{4} - 2385562 T^{5} + 22273641 T^{6} - 175687038 T^{7} + 1180502973 T^{8} - 6701043658 T^{9} + 31966273932 T^{10} - 124298747193 T^{11} + 383903462574 T^{12} - 842245722902 T^{13} + 1174711139837 T^{14} \))
$59$ (\( 1 - 6 T + 59 T^{2} \))(\( 1 + 12 T + 218 T^{2} + 1445 T^{3} + 12862 T^{4} + 41772 T^{5} + 205379 T^{6} \))(\( 1 + 14 T + 358 T^{2} + 4141 T^{3} + 58224 T^{4} + 550458 T^{5} + 5494079 T^{6} + 41878902 T^{7} + 324150661 T^{8} + 1916144298 T^{9} + 11957986896 T^{10} + 50177991901 T^{11} + 255942899042 T^{12} + 590527470974 T^{13} + 2488651484819 T^{14} \))
$61$ (\( 1 - 4 T + 61 T^{2} \))(\( 1 + 4 T + 32 T^{2} - 45 T^{3} + 1952 T^{4} + 14884 T^{5} + 226981 T^{6} \))(\( 1 - 4 T + 168 T^{2} - 931 T^{3} + 16996 T^{4} - 101432 T^{5} + 1438575 T^{6} - 7223874 T^{7} + 87753075 T^{8} - 377428472 T^{9} + 3857769076 T^{10} - 12890477971 T^{11} + 141892178568 T^{12} - 206081497444 T^{13} + 3142742836021 T^{14} \))
$67$ (\( 1 - 5 T + 67 T^{2} \))(\( 1 - 16 T + 277 T^{2} - 2248 T^{3} + 18559 T^{4} - 71824 T^{5} + 300763 T^{6} \))(\( 1 - 9 T + 252 T^{2} - 2212 T^{3} + 36841 T^{4} - 280719 T^{5} + 3525326 T^{6} - 23240880 T^{7} + 236196842 T^{8} - 1260147591 T^{9} + 11080409683 T^{10} - 44574279652 T^{11} + 340231526964 T^{12} - 814125439521 T^{13} + 6060711605323 T^{14} \))
$71$ (\( 1 - 5 T + 71 T^{2} \))(\( 1 - 3 T + 69 T^{2} - 777 T^{3} + 4899 T^{4} - 15123 T^{5} + 357911 T^{6} \))(\( 1 - 24 T + 531 T^{2} - 7902 T^{3} + 110959 T^{4} - 1236210 T^{5} + 12915763 T^{6} - 111985073 T^{7} + 917019173 T^{8} - 6231734610 T^{9} + 39713446649 T^{10} - 200803103262 T^{11} + 958045785381 T^{12} - 3074406814104 T^{13} + 9095120158391 T^{14} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( 1 - 13 T + 133 T^{2} - 709 T^{3} + 9709 T^{4} - 69277 T^{5} + 389017 T^{6} \))(\( 1 + 5 T + 241 T^{2} + 1463 T^{3} + 29835 T^{4} + 233795 T^{5} + 2631931 T^{6} + 21987634 T^{7} + 192130963 T^{8} + 1245893555 T^{9} + 11606322195 T^{10} + 41546626583 T^{11} + 499610253913 T^{12} + 756671131445 T^{13} + 11047398519097 T^{14} \))
$79$ (\( 1 + 5 T + 79 T^{2} \))(\( 1 - 13 T + 249 T^{2} - 1831 T^{3} + 19671 T^{4} - 81133 T^{5} + 493039 T^{6} \))(\( 1 - 8 T + 291 T^{2} - 2542 T^{3} + 45327 T^{4} - 424734 T^{5} + 4781091 T^{6} - 42061795 T^{7} + 377706189 T^{8} - 2650764894 T^{9} + 22347978753 T^{10} - 99011105902 T^{11} + 895423412109 T^{12} - 1944699644168 T^{13} + 19203908986159 T^{14} \))
$83$ (\( 1 - 7 T + 83 T^{2} \))(\( 1 - 28 T + 466 T^{2} - 5145 T^{3} + 38678 T^{4} - 192892 T^{5} + 571787 T^{6} \))(\( 1 + 9 T + 265 T^{2} + 1492 T^{3} + 31458 T^{4} + 105574 T^{5} + 2627287 T^{6} + 6171795 T^{7} + 218064821 T^{8} + 727299286 T^{9} + 17987275446 T^{10} + 70807814932 T^{11} + 1043845770395 T^{12} + 2942463360321 T^{13} + 27136050989627 T^{14} \))
$89$ (\( 1 - 7 T + 89 T^{2} \))(\( 1 - 3 T + 123 T^{2} - 1025 T^{3} + 10947 T^{4} - 23763 T^{5} + 704969 T^{6} \))(\( 1 - 10 T + 385 T^{2} - 3364 T^{3} + 76259 T^{4} - 584084 T^{5} + 9889117 T^{6} - 63995331 T^{7} + 880131413 T^{8} - 4626529364 T^{9} + 53760230971 T^{10} - 211064898724 T^{11} + 2149862887865 T^{12} - 4969812909610 T^{13} + 44231334895529 T^{14} \))
$97$ (\( 1 + 12 T + 97 T^{2} \))(\( 1 + 7 T + 137 T^{2} + 567 T^{3} + 13289 T^{4} + 65863 T^{5} + 912673 T^{6} \))(\( 1 + 5 T + 513 T^{2} + 1695 T^{3} + 120449 T^{4} + 268555 T^{5} + 17291453 T^{6} + 29069346 T^{7} + 1677270941 T^{8} + 2526833995 T^{9} + 109930550177 T^{10} + 150057131295 T^{11} + 4405305551841 T^{12} + 4164860024645 T^{13} + 80798284478113 T^{14} \))
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