Properties

Label 16.3
Level 16
Weight 3
Dimension 7
Nonzero newspaces 2
Newform subspaces 2
Sturm bound 48
Trace bound 1

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Defining parameters

Level: \( N \) = \( 16 = 2^{4} \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 2 \)
Sturm bound: \(48\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(16))\).

Total New Old
Modular forms 23 11 12
Cusp forms 9 7 2
Eisenstein series 14 4 10

Trace form

\( 7q - 2q^{2} - 2q^{3} - 8q^{4} - 8q^{5} - 8q^{6} - 4q^{7} + 4q^{8} + 9q^{9} + O(q^{10}) \) \( 7q - 2q^{2} - 2q^{3} - 8q^{4} - 8q^{5} - 8q^{6} - 4q^{7} + 4q^{8} + 9q^{9} + 36q^{10} - 18q^{11} + 52q^{12} + 8q^{13} + 12q^{14} - 40q^{16} - 34q^{17} - 74q^{18} + 30q^{19} - 84q^{20} - 20q^{21} - 52q^{22} + 60q^{23} + 48q^{24} + 11q^{25} + 96q^{26} + 64q^{27} + 56q^{28} + 24q^{29} + 52q^{30} + 8q^{32} - 4q^{33} - 76q^{34} - 100q^{35} - 52q^{36} - 24q^{37} + 40q^{38} - 196q^{39} + 40q^{40} + 18q^{41} - 24q^{42} - 114q^{43} + 20q^{44} + 12q^{45} + 28q^{46} - 24q^{48} + 3q^{49} + 46q^{50} + 156q^{51} + 100q^{52} + 168q^{53} + 32q^{54} + 252q^{55} - 168q^{56} - 176q^{58} + 206q^{59} - 160q^{60} + 8q^{61} - 144q^{62} + 64q^{64} - 48q^{65} + 196q^{66} - 226q^{67} + 112q^{68} - 116q^{69} - 16q^{70} - 260q^{71} + 52q^{72} - 110q^{73} - 92q^{74} - 238q^{75} - 188q^{76} - 212q^{77} - 84q^{78} + 232q^{80} + 167q^{81} + 304q^{82} + 318q^{83} + 232q^{84} - 32q^{85} + 268q^{86} + 444q^{87} - 8q^{88} - 78q^{89} - 160q^{90} + 188q^{91} - 168q^{92} - 32q^{93} + 48q^{94} - 80q^{96} + 126q^{97} + 10q^{98} - 226q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(16))\)

We only show spaces with odd 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
16.3.c \(\chi_{16}(15, \cdot)\) 16.3.c.a 1 1
16.3.d \(\chi_{16}(7, \cdot)\) None 0 1
16.3.f \(\chi_{16}(3, \cdot)\) 16.3.f.a 6 2

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(16))\) into lower level spaces

\( S_{3}^{\mathrm{old}}(\Gamma_1(16)) \cong \) \(S_{3}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 6 T^{2} + 8 T^{3} + 24 T^{4} + 32 T^{5} + 64 T^{6} \))
$3$ (\( ( 1 - 3 T )( 1 + 3 T ) \))(\( 1 + 2 T + 2 T^{2} - 14 T^{3} - 65 T^{4} + 124 T^{5} + 476 T^{6} + 1116 T^{7} - 5265 T^{8} - 10206 T^{9} + 13122 T^{10} + 118098 T^{11} + 531441 T^{12} \))
$5$ (\( 1 + 6 T + 25 T^{2} \))(\( 1 + 2 T + 2 T^{2} - 14 T^{3} - 369 T^{4} + 636 T^{5} + 2108 T^{6} + 15900 T^{7} - 230625 T^{8} - 218750 T^{9} + 781250 T^{10} + 19531250 T^{11} + 244140625 T^{12} \))
$7$ (\( ( 1 - 7 T )( 1 + 7 T ) \))(\( ( 1 + 2 T + 87 T^{2} + 332 T^{3} + 4263 T^{4} + 4802 T^{5} + 117649 T^{6} )^{2} \))
$11$ (\( ( 1 - 11 T )( 1 + 11 T ) \))(\( 1 + 18 T + 162 T^{2} + 2146 T^{3} + 17759 T^{4} + 65756 T^{5} + 609308 T^{6} + 7956476 T^{7} + 260009519 T^{8} + 3801769906 T^{9} + 34726138722 T^{10} + 466873642818 T^{11} + 3138428376721 T^{12} \))
$13$ (\( 1 - 10 T + 169 T^{2} \))(\( 1 + 2 T + 2 T^{2} + 1554 T^{3} - 7825 T^{4} - 453380 T^{5} + 316348 T^{6} - 76621220 T^{7} - 223489825 T^{8} + 7500861186 T^{9} + 1631461442 T^{10} + 275716983698 T^{11} + 23298085122481 T^{12} \))
$17$ (\( 1 + 30 T + 289 T^{2} \))(\( ( 1 + 2 T + 607 T^{2} - 388 T^{3} + 175423 T^{4} + 167042 T^{5} + 24137569 T^{6} )^{2} \))
$19$ (\( ( 1 - 19 T )( 1 + 19 T ) \))(\( 1 - 30 T + 450 T^{2} - 12014 T^{3} + 441215 T^{4} - 8004292 T^{5} + 113750108 T^{6} - 2889549412 T^{7} + 57499580015 T^{8} - 565209214334 T^{9} + 7642603368450 T^{10} - 183931987734030 T^{11} + 2213314919066161 T^{12} \))
$23$ (\( ( 1 - 23 T )( 1 + 23 T ) \))(\( ( 1 - 30 T + 1751 T^{2} - 30772 T^{3} + 926279 T^{4} - 8395230 T^{5} + 148035889 T^{6} )^{2} \))
$29$ (\( 1 - 42 T + 841 T^{2} \))(\( 1 + 18 T + 162 T^{2} - 4894 T^{3} + 124463 T^{4} + 24625372 T^{5} + 435069308 T^{6} + 20709937852 T^{7} + 88030315103 T^{8} - 2911065332974 T^{9} + 81039918899682 T^{10} + 7572730199403618 T^{11} + 353814783205469041 T^{12} \))
$31$ (\( ( 1 - 31 T )( 1 + 31 T ) \))(\( 1 - 3846 T^{2} + 7131791 T^{4} - 8361808916 T^{6} + 6586358756111 T^{8} - 3280218929998086 T^{10} + 787662783788549761 T^{12} \))
$37$ (\( 1 + 70 T + 1369 T^{2} \))(\( 1 - 46 T + 1058 T^{2} + 6594 T^{3} - 356337 T^{4} - 78343460 T^{5} + 4002544124 T^{6} - 107252196740 T^{7} - 667832908257 T^{8} + 16918399940946 T^{9} + 3716203262248418 T^{10} - 221194881131221054 T^{11} + 6582952005840035281 T^{12} \))
$41$ (\( 1 - 18 T + 1681 T^{2} \))(\( 1 - 5094 T^{2} + 15050223 T^{4} - 31243096276 T^{6} + 42528333194703 T^{8} - 40675209117142374 T^{10} + 22563490300366186081 T^{12} \))
$43$ (\( ( 1 - 43 T )( 1 + 43 T ) \))(\( 1 + 114 T + 6498 T^{2} + 241730 T^{3} + 12357983 T^{4} + 838941724 T^{5} + 44553879452 T^{6} + 1551203247676 T^{7} + 42249484638383 T^{8} + 1528063089834770 T^{9} + 75949925403851298 T^{10} + 2463708983714404386 T^{11} + 39959630797262576401 T^{12} \))
$47$ (\( ( 1 - 47 T )( 1 + 47 T ) \))(\( 1 - 4678 T^{2} + 12462287 T^{4} - 24905944212 T^{6} + 60811985090447 T^{8} - 111389199003717958 T^{10} + \)\(11\!\cdots\!41\)\( T^{12} \))
$53$ (\( 1 - 90 T + 2809 T^{2} \))(\( 1 - 78 T + 3042 T^{2} - 270110 T^{3} + 31648463 T^{4} - 1389102820 T^{5} + 48555101564 T^{6} - 3901989821380 T^{7} + 249721595980703 T^{8} - 5986815584554190 T^{9} + 189393978231360162 T^{10} - 13641222688510017822 T^{11} + \)\(49\!\cdots\!41\)\( T^{12} \))
$59$ (\( ( 1 - 59 T )( 1 + 59 T ) \))(\( 1 - 206 T + 21218 T^{2} - 1942462 T^{3} + 171214239 T^{4} - 11916831972 T^{5} + 708622973852 T^{6} - 41482492094532 T^{7} + 2074664742303279 T^{8} - 81934083737364142 T^{9} + 3115448225088482978 T^{10} - \)\(10\!\cdots\!06\)\( T^{11} + \)\(17\!\cdots\!81\)\( T^{12} \))
$61$ (\( 1 + 22 T + 3721 T^{2} \))(\( 1 - 30 T + 450 T^{2} - 111694 T^{3} + 33268655 T^{4} - 582006980 T^{5} + 8727089468 T^{6} - 2165647972580 T^{7} + 460632507413855 T^{8} - 5754516693877534 T^{9} + 86268290848776450 T^{10} - 21400287349886478030 T^{11} + \)\(26\!\cdots\!21\)\( T^{12} \))
$67$ (\( ( 1 - 67 T )( 1 + 67 T ) \))(\( 1 + 226 T + 25538 T^{2} + 2083538 T^{3} + 120508479 T^{4} + 5203289532 T^{5} + 268963196252 T^{6} + 23357566709148 T^{7} + 2428380941854959 T^{8} + 188473476667633922 T^{9} + 10370156349441497858 T^{10} + \)\(41\!\cdots\!74\)\( T^{11} + \)\(81\!\cdots\!61\)\( T^{12} \))
$71$ (\( ( 1 - 71 T )( 1 + 71 T ) \))(\( ( 1 + 130 T + 11575 T^{2} + 918796 T^{3} + 58349575 T^{4} + 3303518530 T^{5} + 128100283921 T^{6} )^{2} \))
$73$ (\( 1 + 110 T + 5329 T^{2} \))(\( 1 - 13126 T^{2} + 103571951 T^{4} - 653716749588 T^{6} + 2941261225338191 T^{8} - 10585595166201707206 T^{10} + \)\(22\!\cdots\!21\)\( T^{12} \))
$79$ (\( ( 1 - 79 T )( 1 + 79 T ) \))(\( 1 - 70 T^{2} + 84324175 T^{4} + 17226941804 T^{6} + 3284433446508175 T^{8} - 106197616693459270 T^{10} + \)\(59\!\cdots\!41\)\( T^{12} \))
$83$ (\( ( 1 - 83 T )( 1 + 83 T ) \))(\( 1 - 318 T + 50562 T^{2} - 6712846 T^{3} + 819490815 T^{4} - 81203275140 T^{5} + 6918697616348 T^{6} - 559409362439460 T^{7} + 38891658154821615 T^{8} - 2194700377608598174 T^{9} + \)\(11\!\cdots\!42\)\( T^{10} - \)\(49\!\cdots\!82\)\( T^{11} + \)\(10\!\cdots\!61\)\( T^{12} \))
$89$ (\( 1 + 78 T + 7921 T^{2} \))(\( 1 - 31238 T^{2} + 466178479 T^{4} - 4433595811988 T^{6} + 29249082478431439 T^{8} - \)\(12\!\cdots\!78\)\( T^{10} + \)\(24\!\cdots\!21\)\( T^{12} \))
$97$ (\( 1 - 130 T + 9409 T^{2} \))(\( ( 1 + 2 T + 10687 T^{2} + 557564 T^{3} + 100553983 T^{4} + 177058562 T^{5} + 832972004929 T^{6} )^{2} \))
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