Properties

Label 15.4
Level 15
Weight 4
Dimension 14
Nonzero newspaces 3
Newform subspaces 4
Sturm bound 64
Trace bound 1

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Defining parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 4 \)
Sturm bound: \(64\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(15))\).

Total New Old
Modular forms 32 22 10
Cusp forms 16 14 2
Eisenstein series 16 8 8

Trace form

\( 14q + 4q^{2} - 6q^{3} - 24q^{4} + 6q^{5} - 20q^{7} - 36q^{8} - 18q^{9} + O(q^{10}) \) \( 14q + 4q^{2} - 6q^{3} - 24q^{4} + 6q^{5} - 20q^{7} - 36q^{8} - 18q^{9} - 96q^{10} - 56q^{11} + 108q^{12} + 164q^{13} + 264q^{14} + 174q^{15} + 304q^{16} + 40q^{17} - 204q^{18} - 256q^{19} - 436q^{20} - 588q^{21} - 520q^{22} - 288q^{23} - 288q^{24} + 86q^{25} + 232q^{26} + 702q^{27} + 696q^{28} + 788q^{29} + 1116q^{30} + 632q^{31} + 116q^{32} - 12q^{33} - 568q^{34} - 520q^{35} - 696q^{36} - 1260q^{37} - 392q^{38} - 624q^{39} - 152q^{40} - 364q^{41} + 288q^{42} + 452q^{43} + 728q^{44} + 126q^{45} + 392q^{46} + 232q^{47} - 780q^{48} + 150q^{49} - 596q^{50} - 444q^{51} + 144q^{52} + 112q^{53} - 216q^{54} - 24q^{55} + 996q^{57} + 56q^{58} + 496q^{59} + 576q^{60} + 1116q^{61} + 312q^{62} + 1392q^{63} + 1224q^{64} + 688q^{65} + 1296q^{66} + 124q^{67} + 152q^{68} + 144q^{69} - 1560q^{70} - 688q^{71} - 2124q^{72} - 2980q^{73} - 3376q^{74} - 3486q^{75} - 3920q^{76} - 1728q^{77} - 960q^{78} - 1520q^{79} + 1004q^{80} + 774q^{81} + 3912q^{82} + 1368q^{83} + 2496q^{84} + 4092q^{85} + 4384q^{86} - 756q^{87} + 2184q^{88} - 876q^{89} + 384q^{90} + 1592q^{91} + 1056q^{92} + 1944q^{93} + 3416q^{94} + 3304q^{95} + 2928q^{96} + 2484q^{97} + 404q^{98} + 1008q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(15))\)

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
15.4.a \(\chi_{15}(1, \cdot)\) 15.4.a.a 1 1
15.4.a.b 1
15.4.b \(\chi_{15}(4, \cdot)\) 15.4.b.a 4 1
15.4.e \(\chi_{15}(2, \cdot)\) 15.4.e.a 8 2

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(15))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(15)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + 8 T^{2} \))(\( 1 - 3 T + 8 T^{2} \))(\( 1 - 7 T^{2} + 48 T^{4} - 448 T^{6} + 4096 T^{8} \))(\( 1 - 79 T^{4} + 2496 T^{8} - 323584 T^{12} + 16777216 T^{16} \))
$3$ (\( 1 - 3 T \))(\( 1 + 3 T \))(\( ( 1 + 9 T^{2} )^{2} \))(\( 1 + 6 T + 18 T^{2} - 198 T^{3} - 1422 T^{4} - 5346 T^{5} + 13122 T^{6} + 118098 T^{7} + 531441 T^{8} \))
$5$ (\( 1 - 5 T \))(\( 1 + 5 T \))(\( 1 - 6 T - 110 T^{2} - 750 T^{3} + 15625 T^{4} \))(\( 1 + 110 T^{2} + 5250 T^{4} + 1718750 T^{6} + 244140625 T^{8} \))
$7$ (\( 1 + 24 T + 343 T^{2} \))(\( 1 - 20 T + 343 T^{2} \))(\( 1 - 616 T^{2} + 316878 T^{4} - 72471784 T^{6} + 13841287201 T^{8} \))(\( ( 1 + 8 T + 32 T^{2} + 744 T^{3} - 45202 T^{4} + 255192 T^{5} + 3764768 T^{6} + 322828856 T^{7} + 13841287201 T^{8} )^{2} \))
$11$ (\( 1 - 52 T + 1331 T^{2} \))(\( 1 + 24 T + 1331 T^{2} \))(\( ( 1 + 42 T + 2734 T^{2} + 55902 T^{3} + 1771561 T^{4} )^{2} \))(\( ( 1 - 3334 T^{2} + 6292986 T^{4} - 5906384374 T^{6} + 3138428376721 T^{8} )^{2} \))
$13$ (\( 1 - 22 T + 2197 T^{2} \))(\( 1 - 74 T + 2197 T^{2} \))(\( 1 - 5008 T^{2} + 13678638 T^{4} - 24172659472 T^{6} + 23298085122481 T^{8} \))(\( ( 1 - 34 T + 578 T^{2} - 77418 T^{3} + 10363058 T^{4} - 170087346 T^{5} + 2789895602 T^{6} - 360552978682 T^{7} + 23298085122481 T^{8} )^{2} \))
$17$ (\( 1 + 14 T + 4913 T^{2} \))(\( 1 - 54 T + 4913 T^{2} \))(\( 1 - 12400 T^{2} + 76051038 T^{4} - 299305855600 T^{6} + 582622237229761 T^{8} \))(\( 1 - 10588864 T^{4} - 447530075229954 T^{8} - \)\(61\!\cdots\!04\)\( T^{12} + \)\(33\!\cdots\!21\)\( T^{16} \))
$19$ (\( 1 + 20 T + 6859 T^{2} \))(\( 1 + 124 T + 6859 T^{2} \))(\( ( 1 + 56 T + 8598 T^{2} + 384104 T^{3} + 47045881 T^{4} )^{2} \))(\( ( 1 - 24664 T^{2} + 245125086 T^{4} - 1160339608984 T^{6} + 2213314919066161 T^{8} )^{2} \))
$23$ (\( 1 + 168 T + 12167 T^{2} \))(\( 1 + 120 T + 12167 T^{2} \))(\( 1 - 46204 T^{2} + 828262758 T^{4} - 6839850215356 T^{6} + 21914624432020321 T^{8} \))(\( 1 + 232150736 T^{4} + 54876630002867166 T^{8} + \)\(50\!\cdots\!56\)\( T^{12} + \)\(48\!\cdots\!41\)\( T^{16} \))
$29$ (\( 1 - 230 T + 24389 T^{2} \))(\( 1 + 78 T + 24389 T^{2} \))(\( ( 1 - 318 T + 55978 T^{2} - 7755702 T^{3} + 594823321 T^{4} )^{2} \))(\( ( 1 + 79166 T^{2} + 2712313506 T^{4} + 47089783030286 T^{6} + 353814783205469041 T^{8} )^{2} \))
$31$ (\( 1 + 288 T + 29791 T^{2} \))(\( 1 - 200 T + 29791 T^{2} \))(\( ( 1 - 52 T + 58782 T^{2} - 1549132 T^{3} + 887503681 T^{4} )^{2} \))(\( ( 1 - 154 T + 36486 T^{2} - 4587814 T^{3} + 887503681 T^{4} )^{4} \))
$37$ (\( 1 + 34 T + 50653 T^{2} \))(\( 1 + 70 T + 50653 T^{2} \))(\( 1 - 96016 T^{2} + 4637534478 T^{4} - 246350786886544 T^{6} + 6582952005840035281 T^{8} \))(\( ( 1 + 578 T + 167042 T^{2} + 53079474 T^{3} + 15170807378 T^{4} + 2688634596522 T^{5} + 428584070812178 T^{6} + 75117885601554506 T^{7} + 6582952005840035281 T^{8} )^{2} \))
$41$ (\( 1 - 122 T + 68921 T^{2} \))(\( 1 - 330 T + 68921 T^{2} \))(\( ( 1 + 408 T + 177982 T^{2} + 28119768 T^{3} + 4750104241 T^{4} )^{2} \))(\( ( 1 - 115444 T^{2} + 7614039366 T^{4} - 548371033998004 T^{6} + 22563490300366186081 T^{8} )^{2} \))
$43$ (\( 1 + 188 T + 79507 T^{2} \))(\( 1 - 92 T + 79507 T^{2} \))(\( 1 - 121900 T^{2} + 14997346998 T^{4} - 770574155673100 T^{6} + 39959630797262576401 T^{8} \))(\( ( 1 - 274 T + 37538 T^{2} - 17976318 T^{3} + 8415346898 T^{4} - 1429243115226 T^{5} + 237291326133362 T^{6} - 137710375670694982 T^{7} + 39959630797262576401 T^{8} )^{2} \))
$47$ (\( 1 - 256 T + 103823 T^{2} \))(\( 1 + 24 T + 103823 T^{2} \))(\( 1 - 225580 T^{2} + 31469120358 T^{4} - 2431575393915820 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 + 29984206736 T^{4} + \)\(42\!\cdots\!06\)\( T^{8} + \)\(34\!\cdots\!76\)\( T^{12} + \)\(13\!\cdots\!81\)\( T^{16} \))
$53$ (\( 1 + 338 T + 148877 T^{2} \))(\( 1 - 450 T + 148877 T^{2} \))(\( 1 - 376864 T^{2} + 68697431598 T^{4} - 8352949792519456 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} \))(\( 1 - 24371904064 T^{4} + \)\(31\!\cdots\!06\)\( T^{8} - \)\(11\!\cdots\!24\)\( T^{12} + \)\(24\!\cdots\!81\)\( T^{16} \))
$59$ (\( 1 - 100 T + 205379 T^{2} \))(\( 1 - 24 T + 205379 T^{2} \))(\( ( 1 - 186 T + 419038 T^{2} - 38200494 T^{3} + 42180533641 T^{4} )^{2} \))(\( ( 1 + 112106 T^{2} + 56049165066 T^{4} + 4728690904357946 T^{6} + \)\(17\!\cdots\!81\)\( T^{8} )^{2} \))
$61$ (\( 1 - 742 T + 226981 T^{2} \))(\( 1 + 322 T + 226981 T^{2} \))(\( ( 1 - 340 T + 388398 T^{2} - 77173540 T^{3} + 51520374361 T^{4} )^{2} \))(\( ( 1 - 2 T + 226981 T^{2} )^{8} \))
$67$ (\( 1 + 84 T + 300763 T^{2} \))(\( 1 + 196 T + 300763 T^{2} \))(\( 1 - 861340 T^{2} + 346621507638 T^{4} - 77915422897446460 T^{6} + \)\(81\!\cdots\!61\)\( T^{8} \))(\( ( 1 - 202 T + 20402 T^{2} - 3297246 T^{3} - 80373232942 T^{4} - 991689598698 T^{5} + 1845531913011938 T^{6} - 5495719948051579294 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} )^{2} \))
$71$ (\( 1 + 328 T + 357911 T^{2} \))(\( 1 + 288 T + 357911 T^{2} \))(\( ( 1 + 36 T + 384046 T^{2} + 12884796 T^{3} + 128100283921 T^{4} )^{2} \))(\( ( 1 - 863584 T^{2} + 377860054206 T^{4} - 110625355589632864 T^{6} + \)\(16\!\cdots\!41\)\( T^{8} )^{2} \))
$73$ (\( 1 + 38 T + 389017 T^{2} \))(\( 1 + 430 T + 389017 T^{2} \))(\( 1 - 429844 T^{2} + 136740794118 T^{4} - 65050109164968916 T^{6} + \)\(22\!\cdots\!21\)\( T^{8} \))(\( ( 1 + 1256 T + 788768 T^{2} + 733362072 T^{3} + 643873740638 T^{4} + 285290313163224 T^{5} + 119367595001521952 T^{6} + 73942712905584498728 T^{7} + \)\(22\!\cdots\!21\)\( T^{8} )^{2} \))
$79$ (\( 1 + 240 T + 493039 T^{2} \))(\( 1 + 520 T + 493039 T^{2} \))(\( ( 1 + 380 T + 99678 T^{2} + 187354820 T^{3} + 243087455521 T^{4} )^{2} \))(\( ( 1 - 1624084 T^{2} + 1116095863206 T^{4} - 394794447112367764 T^{6} + \)\(59\!\cdots\!41\)\( T^{8} )^{2} \))
$83$ (\( 1 - 1212 T + 571787 T^{2} \))(\( 1 - 156 T + 571787 T^{2} \))(\( 1 - 916108 T^{2} + 434315280918 T^{4} - 299512691566327852 T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 + 1005705244496 T^{4} + \)\(46\!\cdots\!26\)\( T^{8} + \)\(10\!\cdots\!56\)\( T^{12} + \)\(11\!\cdots\!21\)\( T^{16} \))
$89$ (\( 1 - 330 T + 704969 T^{2} \))(\( 1 - 1026 T + 704969 T^{2} \))(\( ( 1 + 1116 T + 1508758 T^{2} + 786745404 T^{3} + 496981290961 T^{4} )^{2} \))(\( ( 1 - 806584 T^{2} + 1118488022286 T^{4} - 400857157588487224 T^{6} + \)\(24\!\cdots\!21\)\( T^{8} )^{2} \))
$97$ (\( 1 - 866 T + 912673 T^{2} \))(\( 1 + 286 T + 912673 T^{2} \))(\( 1 - 2174980 T^{2} + 2500346420358 T^{4} - 1811697451280476420 T^{6} + \)\(69\!\cdots\!41\)\( T^{8} \))(\( ( 1 - 952 T + 453152 T^{2} - 121725576 T^{3} - 583228842562 T^{4} - 111095646624648 T^{5} + 377462929977586208 T^{6} - \)\(72\!\cdots\!84\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} )^{2} \))
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