Properties

Label 14.6
Level 14
Weight 6
Dimension 10
Nonzero newspaces 2
Newform subspaces 4
Sturm bound 72
Trace bound 1

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Defining parameters

Level: \( N \) = \( 14\( 14 = 2 \cdot 7 \) \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 4 \)
Sturm bound: \(72\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(14))\).

Total New Old
Modular forms 36 10 26
Cusp forms 24 10 14
Eisenstein series 12 0 12

Trace form

\( 10q + 18q^{3} - 32q^{4} + 66q^{5} + 216q^{6} + 232q^{7} - 1218q^{9} + O(q^{10}) \) \( 10q + 18q^{3} - 32q^{4} + 66q^{5} + 216q^{6} + 232q^{7} - 1218q^{9} - 744q^{10} - 444q^{11} + 288q^{12} + 3650q^{13} - 120q^{14} - 2568q^{15} - 512q^{16} - 3228q^{17} + 1488q^{18} + 3518q^{19} + 2400q^{20} + 8790q^{21} + 2832q^{22} - 2112q^{23} - 1920q^{24} - 38q^{25} - 11352q^{26} - 24372q^{27} - 6656q^{28} - 4248q^{29} + 6288q^{30} + 18956q^{31} + 8844q^{33} + 13872q^{34} - 10878q^{35} + 23520q^{36} + 27476q^{37} - 1224q^{38} - 12840q^{39} - 11904q^{40} - 6336q^{41} - 57384q^{42} - 29308q^{43} - 7104q^{44} + 26742q^{45} + 46320q^{46} + 38196q^{47} + 4608q^{48} + 58570q^{49} + 65616q^{50} - 40140q^{51} - 22240q^{52} - 90840q^{53} - 89568q^{54} - 50328q^{55} - 21120q^{56} - 99468q^{57} - 28368q^{58} - 31098q^{59} + 42624q^{60} + 110474q^{61} + 161904q^{62} + 229020q^{63} + 40960q^{64} + 14868q^{65} - 96000q^{66} + 65504q^{67} - 51648q^{68} - 33336q^{69} - 131208q^{70} - 135720q^{71} + 23808q^{72} - 46048q^{73} + 36720q^{74} + 110310q^{75} + 26720q^{76} + 4368q^{77} + 182016q^{78} - 82672q^{79} + 16896q^{80} - 103470q^{81} - 128208q^{82} - 64770q^{83} - 69792q^{84} - 157548q^{85} + 63696q^{86} + 220428q^{87} - 23040q^{88} - 29064q^{89} - 64392q^{90} - 134530q^{91} - 38016q^{92} + 25020q^{93} + 47328q^{94} - 93984q^{95} - 30720q^{96} - 100264q^{97} - 50016q^{98} + 667740q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(14))\)

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
14.6.a \(\chi_{14}(1, \cdot)\) 14.6.a.a 1 1
14.6.a.b 1
14.6.c \(\chi_{14}(9, \cdot)\) 14.6.c.a 4 2
14.6.c.b 4

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(14))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(14)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 4 T \))(\( 1 - 4 T \))(\( ( 1 + 4 T + 16 T^{2} )^{2} \))(\( ( 1 - 4 T + 16 T^{2} )^{2} \))
$3$ (\( 1 - 10 T + 243 T^{2} \))(\( 1 - 8 T + 243 T^{2} \))(\( 1 + 14 T - 23 T^{2} - 3738 T^{3} - 58716 T^{4} - 908334 T^{5} - 1358127 T^{6} + 200884698 T^{7} + 3486784401 T^{8} \))(\( 1 - 14 T + 181 T^{2} + 6594 T^{3} - 106452 T^{4} + 1602342 T^{5} + 10687869 T^{6} - 200884698 T^{7} + 3486784401 T^{8} \))
$5$ (\( 1 - 84 T + 3125 T^{2} \))(\( 1 - 10 T + 3125 T^{2} \))(\( 1 + 70 T + 2481 T^{2} - 268170 T^{3} - 19226564 T^{4} - 838031250 T^{5} + 24228515625 T^{6} + 2136230468750 T^{7} + 95367431640625 T^{8} \))(\( ( 1 - 21 T - 2684 T^{2} - 65625 T^{3} + 9765625 T^{4} )^{2} \))
$7$ (\( 1 - 49 T \))(\( 1 + 49 T \))(\( 1 - 28322 T^{2} + 282475249 T^{4} \))(\( 1 - 232 T + 28350 T^{2} - 3899224 T^{3} + 282475249 T^{4} \))
$11$ (\( 1 + 336 T + 161051 T^{2} \))(\( 1 + 340 T + 161051 T^{2} \))(\( 1 + 62 T - 303735 T^{2} - 900426 T^{3} + 68048336596 T^{4} - 145014507726 T^{5} - 7878103661184735 T^{6} + 258989386503770362 T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \))(\( 1 - 294 T - 27955 T^{2} + 61067034 T^{3} - 26772727956 T^{4} + 9834906892734 T^{5} - 725080704720955 T^{6} - 1228110961808201394 T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \))
$13$ (\( 1 - 584 T + 371293 T^{2} \))(\( 1 + 294 T + 371293 T^{2} \))(\( ( 1 - 1820 T + 1508750 T^{2} - 675753260 T^{3} + 137858491849 T^{4} )^{2} \))(\( ( 1 + 140 T + 728766 T^{2} + 51981020 T^{3} + 137858491849 T^{4} )^{2} \))
$17$ (\( 1 + 1458 T + 1419857 T^{2} \))(\( 1 - 1226 T + 1419857 T^{2} \))(\( 1 + 1694 T - 534543 T^{2} + 956203710 T^{3} + 5497968001540 T^{4} + 1357672531069470 T^{5} - 1077635427527709807 T^{6} + \)\(48\!\cdots\!42\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} \))(\( 1 + 1302 T - 1399831 T^{2} + 332427942 T^{3} + 5156300484948 T^{4} + 472000140444294 T^{5} - 2822050757659424119 T^{6} + \)\(37\!\cdots\!86\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} \))
$19$ (\( 1 - 470 T + 2476099 T^{2} \))(\( 1 - 2432 T + 2476099 T^{2} \))(\( 1 + 826 T - 3959855 T^{2} - 256115342 T^{3} + 13728931042036 T^{4} - 634166942210858 T^{5} - 24278133376284578855 T^{6} + \)\(12\!\cdots\!74\)\( T^{7} + \)\(37\!\cdots\!01\)\( T^{8} \))(\( 1 - 1442 T - 2040155 T^{2} + 1200723118 T^{3} + 5690659412524 T^{4} + 2973109311756682 T^{5} - 12508325481183999155 T^{6} - \)\(21\!\cdots\!58\)\( T^{7} + \)\(37\!\cdots\!01\)\( T^{8} \))
$23$ (\( 1 + 4200 T + 6436343 T^{2} \))(\( 1 - 2000 T + 6436343 T^{2} \))(\( 1 - 2734 T - 2791743 T^{2} + 7125315258 T^{3} + 15864397046044 T^{4} + 45860972983621494 T^{5} - \)\(11\!\cdots\!07\)\( T^{6} - \)\(72\!\cdots\!38\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} \))(\( 1 + 2646 T - 1888699 T^{2} - 10538147466 T^{3} - 7457015740812 T^{4} - 67827131675756838 T^{5} - 78242210302707652651 T^{6} + \)\(70\!\cdots\!22\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} \))
$29$ (\( 1 - 4866 T + 20511149 T^{2} \))(\( 1 + 6746 T + 20511149 T^{2} \))(\( ( 1 + 2852 T + 25156270 T^{2} + 58497796948 T^{3} + 420707233300201 T^{4} )^{2} \))(\( ( 1 - 1668 T + 40800574 T^{2} - 34212596532 T^{3} + 420707233300201 T^{4} )^{2} \))
$31$ (\( 1 + 7372 T + 28629151 T^{2} \))(\( 1 - 8856 T + 28629151 T^{2} \))(\( 1 - 2674 T - 50939695 T^{2} - 2223882906 T^{3} + 2350110112504364 T^{4} - 63667879522192806 T^{5} - \)\(41\!\cdots\!95\)\( T^{6} - \)\(62\!\cdots\!74\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \))(\( 1 - 14798 T + 109902301 T^{2} - 766835334398 T^{3} + 4809257025922804 T^{4} - 21953844580615836098 T^{5} + \)\(90\!\cdots\!01\)\( T^{6} - \)\(34\!\cdots\!98\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \))
$37$ (\( 1 - 14330 T + 69343957 T^{2} \))(\( 1 - 9182 T + 69343957 T^{2} \))(\( 1 - 9146 T - 70934111 T^{2} - 145380361898 T^{3} + 13286876958863068 T^{4} - 10081249564099350386 T^{5} - \)\(34\!\cdots\!39\)\( T^{6} - \)\(30\!\cdots\!78\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} \))(\( 1 + 5182 T - 44248391 T^{2} - 350232719618 T^{3} - 1615260813094292 T^{4} - 24286522649183648426 T^{5} - \)\(21\!\cdots\!59\)\( T^{6} + \)\(17\!\cdots\!26\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} \))
$41$ (\( 1 - 6222 T + 115856201 T^{2} \))(\( 1 + 14574 T + 115856201 T^{2} \))(\( ( 1 - 6132 T + 240555334 T^{2} - 710430224532 T^{3} + 13422659310152401 T^{4} )^{2} \))(\( ( 1 + 5124 T + 23950966 T^{2} + 593647173924 T^{3} + 13422659310152401 T^{4} )^{2} \))
$43$ (\( 1 - 3704 T + 147008443 T^{2} \))(\( 1 - 8108 T + 147008443 T^{2} \))(\( ( 1 + 16040 T + 215636742 T^{2} + 2358015425720 T^{3} + 21611482313284249 T^{4} )^{2} \))(\( ( 1 + 4520 T + 240418566 T^{2} + 664478162360 T^{3} + 21611482313284249 T^{4} )^{2} \))
$47$ (\( 1 + 1812 T + 229345007 T^{2} \))(\( 1 + 312 T + 229345007 T^{2} \))(\( 1 - 25326 T + 24756497 T^{2} - 4000489008390 T^{3} + 180554490845854860 T^{4} - \)\(91\!\cdots\!30\)\( T^{5} + \)\(13\!\cdots\!53\)\( T^{6} - \)\(30\!\cdots\!18\)\( T^{7} + \)\(27\!\cdots\!01\)\( T^{8} \))(\( 1 - 14994 T - 202236067 T^{2} + 474318861534 T^{3} + 78216430143556308 T^{4} + \)\(10\!\cdots\!38\)\( T^{5} - \)\(10\!\cdots\!83\)\( T^{6} - \)\(18\!\cdots\!42\)\( T^{7} + \)\(27\!\cdots\!01\)\( T^{8} \))
$53$ (\( 1 + 37242 T + 418195493 T^{2} \))(\( 1 + 14634 T + 418195493 T^{2} \))(\( 1 + 14958 T - 320194663 T^{2} - 4374535293522 T^{3} + 27411568951343628 T^{4} - \)\(18\!\cdots\!46\)\( T^{5} - \)\(55\!\cdots\!87\)\( T^{6} + \)\(10\!\cdots\!06\)\( T^{7} + \)\(30\!\cdots\!01\)\( T^{8} \))(\( 1 + 24006 T - 359228239 T^{2} + 2379601687734 T^{3} + 459302171099340828 T^{4} + \)\(99\!\cdots\!62\)\( T^{5} - \)\(62\!\cdots\!11\)\( T^{6} + \)\(17\!\cdots\!42\)\( T^{7} + \)\(30\!\cdots\!01\)\( T^{8} \))
$59$ (\( 1 - 34302 T + 714924299 T^{2} \))(\( 1 + 27656 T + 714924299 T^{2} \))(\( 1 - 1106 T - 1399148487 T^{2} + 32601423750 T^{3} + 1449049688137535764 T^{4} + 23307550020870701250 T^{5} - \)\(71\!\cdots\!87\)\( T^{6} - \)\(40\!\cdots\!94\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} \))(\( 1 + 38850 T + 184375157 T^{2} - 4075413756750 T^{3} + 165317378144690748 T^{4} - \)\(29\!\cdots\!50\)\( T^{5} + \)\(94\!\cdots\!57\)\( T^{6} + \)\(14\!\cdots\!50\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} \))
$61$ (\( 1 - 24476 T + 844596301 T^{2} \))(\( 1 - 34338 T + 844596301 T^{2} \))(\( 1 - 28042 T - 1098758623 T^{2} - 5493982610970 T^{3} + 2176154319606128204 T^{4} - \)\(46\!\cdots\!70\)\( T^{5} - \)\(78\!\cdots\!23\)\( T^{6} - \)\(16\!\cdots\!42\)\( T^{7} + \)\(50\!\cdots\!01\)\( T^{8} \))(\( 1 - 23618 T - 1176467639 T^{2} - 1064816608898 T^{3} + 1823251555390845244 T^{4} - \)\(89\!\cdots\!98\)\( T^{5} - \)\(83\!\cdots\!39\)\( T^{6} - \)\(14\!\cdots\!18\)\( T^{7} + \)\(50\!\cdots\!01\)\( T^{8} \))
$67$ (\( 1 + 17452 T + 1350125107 T^{2} \))(\( 1 - 12316 T + 1350125107 T^{2} \))(\( 1 - 102642 T + 5270792585 T^{2} - 263208715818330 T^{3} + 11668963301203142244 T^{4} - \)\(35\!\cdots\!10\)\( T^{5} + \)\(96\!\cdots\!65\)\( T^{6} - \)\(25\!\cdots\!06\)\( T^{7} + \)\(33\!\cdots\!01\)\( T^{8} \))(\( 1 + 32002 T - 115710491 T^{2} - 49936295831438 T^{3} - 1906306940850092852 T^{4} - \)\(67\!\cdots\!66\)\( T^{5} - \)\(21\!\cdots\!59\)\( T^{6} + \)\(78\!\cdots\!86\)\( T^{7} + \)\(33\!\cdots\!01\)\( T^{8} \))
$71$ (\( 1 - 28224 T + 1804229351 T^{2} \))(\( 1 - 36920 T + 1804229351 T^{2} \))(\( ( 1 + 11056 T + 3430664782 T^{2} + 19947559704656 T^{3} + 3255243551009881201 T^{4} )^{2} \))(\( ( 1 + 89376 T + 5425689166 T^{2} + 161254802474976 T^{3} + 3255243551009881201 T^{4} )^{2} \))
$73$ (\( 1 - 3602 T + 2073071593 T^{2} \))(\( 1 + 61718 T + 2073071593 T^{2} \))(\( 1 + 35070 T - 2438841295 T^{2} - 16742312474370 T^{3} + 6612391903341914676 T^{4} - \)\(34\!\cdots\!10\)\( T^{5} - \)\(10\!\cdots\!55\)\( T^{6} + \)\(31\!\cdots\!90\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} \))(\( 1 - 47138 T - 2210081903 T^{2} - 13478157074018 T^{3} + 10739298264608894068 T^{4} - \)\(27\!\cdots\!74\)\( T^{5} - \)\(94\!\cdots\!47\)\( T^{6} - \)\(41\!\cdots\!66\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} \))
$79$ (\( 1 - 42872 T + 3077056399 T^{2} \))(\( 1 + 64752 T + 3077056399 T^{2} \))(\( 1 + 101762 T + 4989405761 T^{2} - 80189872018230 T^{3} - 12537730690187416468 T^{4} - \)\(24\!\cdots\!70\)\( T^{5} + \)\(47\!\cdots\!61\)\( T^{6} + \)\(29\!\cdots\!38\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} \))(\( 1 - 40970 T - 4344609803 T^{2} + 5365517032150 T^{3} + 21645103291094327908 T^{4} + \)\(16\!\cdots\!50\)\( T^{5} - \)\(41\!\cdots\!03\)\( T^{6} - \)\(11\!\cdots\!30\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} \))
$83$ (\( 1 + 35202 T + 3939040643 T^{2} \))(\( 1 + 77056 T + 3939040643 T^{2} \))(\( ( 1 + 44632 T + 2269443286 T^{2} + 175807261978376 T^{3} + 15516041187205853449 T^{4} )^{2} \))(\( ( 1 - 68376 T + 8908447510 T^{2} - 269335843005768 T^{3} + 15516041187205853449 T^{4} )^{2} \))
$89$ (\( 1 - 26730 T + 5584059449 T^{2} \))(\( 1 + 8166 T + 5584059449 T^{2} \))(\( 1 - 75474 T - 3554016127 T^{2} + 144742383942030 T^{3} + 22578469410090837684 T^{4} + \)\(80\!\cdots\!70\)\( T^{5} - \)\(11\!\cdots\!27\)\( T^{6} - \)\(13\!\cdots\!26\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} \))(\( 1 + 123102 T + 1602880625 T^{2} + 293364730856862 T^{3} + 67832483053507925844 T^{4} + \)\(16\!\cdots\!38\)\( T^{5} + \)\(49\!\cdots\!25\)\( T^{6} + \)\(21\!\cdots\!98\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} \))
$97$ (\( 1 + 16978 T + 8587340257 T^{2} \))(\( 1 - 20650 T + 8587340257 T^{2} \))(\( ( 1 + 8316 T + 16824750934 T^{2} + 71412321577212 T^{3} + 73742412689492826049 T^{4} )^{2} \))(\( ( 1 + 43652 T + 8867162790 T^{2} + 374854576898564 T^{3} + 73742412689492826049 T^{4} )^{2} \))
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