# Properties

 Label 24.3 Level 24 Weight 3 Dimension 12 Nonzero newspaces 3 Newform subspaces 5 Sturm bound 96 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$24\( 24 = 2^{3} \cdot 3$$ \) Weight: $$k$$ = $$3$$ Nonzero newspaces: $$3$$ Newform subspaces: $$5$$ Sturm bound: $$96$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(24))$$.

Total New Old
Modular forms 44 16 28
Cusp forms 20 12 8
Eisenstein series 24 4 20

## Trace form

 $$12q + 2q^{2} + 2q^{3} - 12q^{4} - 14q^{6} - 16q^{7} - 4q^{8} - 4q^{9} + O(q^{10})$$ $$12q + 2q^{2} + 2q^{3} - 12q^{4} - 14q^{6} - 16q^{7} - 4q^{8} - 4q^{9} - 12q^{10} - 32q^{11} + 16q^{12} + 20q^{13} + 36q^{14} + 12q^{15} + 32q^{16} - 8q^{17} + 62q^{18} + 36q^{19} + 72q^{20} - 12q^{21} + 48q^{22} - 52q^{24} - 72q^{25} - 96q^{26} - 46q^{27} - 176q^{28} - 172q^{30} + 16q^{31} - 88q^{32} + 20q^{33} - 68q^{34} + 96q^{35} + 108q^{36} - 12q^{37} + 40q^{38} + 132q^{39} + 248q^{40} + 40q^{41} + 220q^{42} + 196q^{43} + 64q^{44} + 64q^{45} + 152q^{46} - 72q^{48} - 100q^{49} - 118q^{50} - 224q^{51} - 160q^{52} - 202q^{54} - 296q^{55} - 168q^{56} + 12q^{57} + 4q^{58} - 128q^{59} + 48q^{60} - 172q^{61} + 204q^{62} - 176q^{63} + 240q^{64} + 96q^{65} + 216q^{66} - 252q^{67} + 112q^{68} - 64q^{69} + 40q^{70} - 124q^{72} + 440q^{73} - 120q^{74} + 178q^{75} - 72q^{76} - 160q^{78} + 400q^{79} - 96q^{80} + 108q^{81} - 348q^{82} + 160q^{83} + 88q^{84} + 256q^{85} + 88q^{86} + 492q^{87} - 48q^{88} - 200q^{89} + 44q^{90} + 168q^{91} + 96q^{92} - 44q^{93} - 168q^{94} - 40q^{96} - 40q^{97} + 170q^{98} - 32q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(24))$$

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
24.3.b $$\chi_{24}(19, \cdot)$$ 24.3.b.a 4 1
24.3.e $$\chi_{24}(17, \cdot)$$ 24.3.e.a 2 1
24.3.g $$\chi_{24}(7, \cdot)$$ None 0 1
24.3.h $$\chi_{24}(5, \cdot)$$ 24.3.h.a 1 1
24.3.h.b 1
24.3.h.c 4

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(24))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(24)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 - 2 T + 6 T^{2} - 8 T^{3} + 16 T^{4}$$)($$1 + 2 T$$)($$1 - 2 T$$)($$1 + 6 T^{2} + 16 T^{4}$$)
$3$ ($$( 1 - 3 T^{2} )^{2}$$)($$1 - 2 T + 9 T^{2}$$)($$1 - 3 T$$)($$1 + 3 T$$)($$1 + 10 T^{2} + 81 T^{4}$$)
$5$ ($$1 - 28 T^{2} + 678 T^{4} - 17500 T^{6} + 390625 T^{8}$$)($$1 - 18 T^{2} + 625 T^{4}$$)($$1 - 2 T + 25 T^{2}$$)($$1 + 2 T + 25 T^{2}$$)($$( 1 + 18 T^{2} + 625 T^{4} )^{2}$$)
$7$ ($$1 - 76 T^{2} + 3174 T^{4} - 182476 T^{6} + 5764801 T^{8}$$)($$( 1 + 6 T + 49 T^{2} )^{2}$$)($$1 + 10 T + 49 T^{2}$$)($$1 + 10 T + 49 T^{2}$$)($$( 1 - 4 T + 49 T^{2} )^{4}$$)
$11$ ($$( 1 + 8 T + 121 T^{2} )^{4}$$)($$1 - 210 T^{2} + 14641 T^{4}$$)($$1 + 10 T + 121 T^{2}$$)($$1 - 10 T + 121 T^{2}$$)($$( 1 + 170 T^{2} + 14641 T^{4} )^{2}$$)
$13$ ($$1 - 292 T^{2} + 75366 T^{4} - 8339812 T^{6} + 815730721 T^{8}$$)($$( 1 - 10 T + 169 T^{2} )^{2}$$)($$( 1 - 13 T )( 1 + 13 T )$$)($$( 1 - 13 T )( 1 + 13 T )$$)($$( 1 - 226 T^{2} + 28561 T^{4} )^{2}$$)
$17$ ($$( 1 + 4 T + 390 T^{2} + 1156 T^{3} + 83521 T^{4} )^{2}$$)($$1 - 66 T^{2} + 83521 T^{4}$$)($$( 1 - 17 T )( 1 + 17 T )$$)($$( 1 - 17 T )( 1 + 17 T )$$)($$( 1 - 354 T^{2} + 83521 T^{4} )^{2}$$)
$19$ ($$( 1 - 16 T + 738 T^{2} - 5776 T^{3} + 130321 T^{4} )^{2}$$)($$( 1 - 2 T + 361 T^{2} )^{2}$$)($$( 1 - 19 T )( 1 + 19 T )$$)($$( 1 - 19 T )( 1 + 19 T )$$)($$( 1 - 694 T^{2} + 130321 T^{4} )^{2}$$)
$23$ ($$1 - 1636 T^{2} + 1179654 T^{4} - 457819876 T^{6} + 78310985281 T^{8}$$)($$1 - 930 T^{2} + 279841 T^{4}$$)($$( 1 - 23 T )( 1 + 23 T )$$)($$( 1 - 23 T )( 1 + 23 T )$$)($$( 1 - 162 T^{2} + 279841 T^{4} )^{2}$$)
$29$ ($$1 - 1756 T^{2} + 1539558 T^{4} - 1241985436 T^{6} + 500246412961 T^{8}$$)($$1 - 1394 T^{2} + 707281 T^{4}$$)($$1 - 50 T + 841 T^{2}$$)($$1 + 50 T + 841 T^{2}$$)($$( 1 + 1394 T^{2} + 707281 T^{4} )^{2}$$)
$31$ ($$1 - 460 T^{2} - 683610 T^{4} - 424819660 T^{6} + 852891037441 T^{8}$$)($$( 1 + 22 T + 961 T^{2} )^{2}$$)($$1 - 38 T + 961 T^{2}$$)($$1 - 38 T + 961 T^{2}$$)($$( 1 + 4 T + 961 T^{2} )^{4}$$)
$37$ ($$1 - 4612 T^{2} + 8955366 T^{4} - 8643630532 T^{6} + 3512479453921 T^{8}$$)($$( 1 + 6 T + 1369 T^{2} )^{2}$$)($$( 1 - 37 T )( 1 + 37 T )$$)($$( 1 - 37 T )( 1 + 37 T )$$)($$( 1 + 62 T^{2} + 1874161 T^{4} )^{2}$$)
$41$ ($$( 1 - 20 T + 1734 T^{2} - 33620 T^{3} + 2825761 T^{4} )^{2}$$)($$1 - 2210 T^{2} + 2825761 T^{4}$$)($$( 1 - 41 T )( 1 + 41 T )$$)($$( 1 - 41 T )( 1 + 41 T )$$)($$( 1 - 2466 T^{2} + 2825761 T^{4} )^{2}$$)
$43$ ($$( 1 - 16 T + 3330 T^{2} - 29584 T^{3} + 3418801 T^{4} )^{2}$$)($$( 1 - 82 T + 1849 T^{2} )^{2}$$)($$( 1 - 43 T )( 1 + 43 T )$$)($$( 1 - 43 T )( 1 + 43 T )$$)($$( 1 - 3670 T^{2} + 3418801 T^{4} )^{2}$$)
$47$ ($$1 - 5284 T^{2} + 13593798 T^{4} - 25784234404 T^{6} + 23811286661761 T^{8}$$)($$1 + 190 T^{2} + 4879681 T^{4}$$)($$( 1 - 47 T )( 1 + 47 T )$$)($$( 1 - 47 T )( 1 + 47 T )$$)($$( 1 - 47 T )^{4}( 1 + 47 T )^{4}$$)
$53$ ($$1 - 9436 T^{2} + 38033574 T^{4} - 74454578716 T^{6} + 62259690411361 T^{8}$$)($$1 - 1746 T^{2} + 7890481 T^{4}$$)($$1 + 94 T + 2809 T^{2}$$)($$1 - 94 T + 2809 T^{2}$$)($$( 1 + 3026 T^{2} + 7890481 T^{4} )^{2}$$)
$59$ ($$( 1 + 64 T + 7554 T^{2} + 222784 T^{3} + 12117361 T^{4} )^{2}$$)($$1 - 1554 T^{2} + 12117361 T^{4}$$)($$1 + 10 T + 3481 T^{2}$$)($$1 - 10 T + 3481 T^{2}$$)($$( 1 + 4650 T^{2} + 12117361 T^{4} )^{2}$$)
$61$ ($$1 - 11332 T^{2} + 56649510 T^{4} - 156901070212 T^{6} + 191707312997281 T^{8}$$)($$( 1 + 86 T + 3721 T^{2} )^{2}$$)($$( 1 - 61 T )( 1 + 61 T )$$)($$( 1 - 61 T )( 1 + 61 T )$$)($$( 1 + 1630 T^{2} + 13845841 T^{4} )^{2}$$)
$67$ ($$( 1 + 128 T + 12642 T^{2} + 574592 T^{3} + 20151121 T^{4} )^{2}$$)($$( 1 - 2 T + 4489 T^{2} )^{2}$$)($$( 1 - 67 T )( 1 + 67 T )$$)($$( 1 - 67 T )( 1 + 67 T )$$)($$( 1 - 6710 T^{2} + 20151121 T^{4} )^{2}$$)
$71$ ($$1 - 11236 T^{2} + 75307398 T^{4} - 285525647716 T^{6} + 645753531245761 T^{8}$$)($$1 + 5406 T^{2} + 25411681 T^{4}$$)($$( 1 - 71 T )( 1 + 71 T )$$)($$( 1 - 71 T )( 1 + 71 T )$$)($$( 1 - 110 T + 5041 T^{2} )^{2}( 1 + 110 T + 5041 T^{2} )^{2}$$)
$73$ ($$( 1 - 100 T + 10086 T^{2} - 532900 T^{3} + 28398241 T^{4} )^{2}$$)($$( 1 - 82 T + 5329 T^{2} )^{2}$$)($$1 - 50 T + 5329 T^{2}$$)($$1 - 50 T + 5329 T^{2}$$)($$( 1 + 6 T + 5329 T^{2} )^{4}$$)
$79$ ($$1 - 17548 T^{2} + 151931046 T^{4} - 683496021388 T^{6} + 1517108809906561 T^{8}$$)($$( 1 - 10 T + 6241 T^{2} )^{2}$$)($$1 + 58 T + 6241 T^{2}$$)($$1 + 58 T + 6241 T^{2}$$)($$( 1 - 124 T + 6241 T^{2} )^{4}$$)
$83$ ($$( 1 - 80 T + 14610 T^{2} - 551120 T^{3} + 47458321 T^{4} )^{2}$$)($$1 - 8370 T^{2} + 47458321 T^{4}$$)($$1 - 134 T + 6889 T^{2}$$)($$1 + 134 T + 6889 T^{2}$$)($$( 1 + 13770 T^{2} + 47458321 T^{4} )^{2}$$)
$89$ ($$( 1 + 100 T + 11430 T^{2} + 792100 T^{3} + 62742241 T^{4} )^{2}$$)($$1 - 14690 T^{2} + 62742241 T^{4}$$)($$( 1 - 89 T )( 1 + 89 T )$$)($$( 1 - 89 T )( 1 + 89 T )$$)($$( 1 - 4866 T^{2} + 62742241 T^{4} )^{2}$$)
$97$ ($$( 1 - 28 T + 12102 T^{2} - 263452 T^{3} + 88529281 T^{4} )^{2}$$)($$( 1 + 94 T + 9409 T^{2} )^{2}$$)($$1 + 190 T + 9409 T^{2}$$)($$1 + 190 T + 9409 T^{2}$$)($$( 1 - 118 T + 9409 T^{2} )^{4}$$)