# Properties

 Label 15.3 Level 15 Weight 3 Dimension 8 Nonzero newspaces 3 Newform subspaces 4 Sturm bound 48 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$15\( 15 = 3 \cdot 5$$ \) Weight: $$k$$ = $$3$$ Nonzero newspaces: $$3$$ Newform subspaces: $$4$$ Sturm bound: $$48$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(15))$$.

Total New Old
Modular forms 24 12 12
Cusp forms 8 8 0
Eisenstein series 16 4 12

## Trace form

 $$8q - 4q^{2} - 4q^{3} - 8q^{4} - 4q^{5} - 8q^{6} - 8q^{7} + 12q^{8} + 16q^{9} + O(q^{10})$$ $$8q - 4q^{2} - 4q^{3} - 8q^{4} - 4q^{5} - 8q^{6} - 8q^{7} + 12q^{8} + 16q^{9} + 24q^{10} + 16q^{11} + 28q^{12} - 16q^{15} - 48q^{16} - 40q^{17} - 52q^{18} - 48q^{19} - 36q^{20} + 40q^{22} + 56q^{23} + 72q^{24} + 56q^{25} + 88q^{26} + 44q^{27} + 56q^{28} - 44q^{30} - 48q^{31} - 76q^{32} - 56q^{33} - 8q^{34} - 40q^{35} - 40q^{36} + 32q^{37} - 96q^{38} - 64q^{39} + 8q^{40} - 56q^{41} - 48q^{42} + 24q^{43} + 76q^{45} - 8q^{46} + 128q^{47} + 124q^{48} + 72q^{49} + 164q^{50} + 136q^{51} - 112q^{52} + 56q^{53} + 16q^{54} - 144q^{55} - 64q^{57} - 152q^{58} + 16q^{60} + 128q^{61} + 88q^{62} + 24q^{63} - 56q^{64} - 112q^{65} - 16q^{66} - 152q^{67} - 104q^{68} - 264q^{69} - 120q^{70} - 272q^{71} - 156q^{72} - 72q^{73} + 44q^{75} + 448q^{76} + 88q^{77} + 280q^{78} + 472q^{79} + 164q^{80} - 32q^{81} + 408q^{82} - 16q^{83} - 24q^{84} + 72q^{85} - 224q^{86} + 56q^{87} + 72q^{88} - 16q^{90} - 208q^{91} + 104q^{92} - 248q^{94} + 144q^{95} - 352q^{96} - 352q^{97} - 188q^{98} + 80q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(15))$$

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
15.3.c $$\chi_{15}(11, \cdot)$$ 15.3.c.a 2 1
15.3.d $$\chi_{15}(14, \cdot)$$ 15.3.d.a 1 1
15.3.d.b 1
15.3.f $$\chi_{15}(7, \cdot)$$ 15.3.f.a 4 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 - 3 T^{2} + 16 T^{4}$$)($$1 + T + 4 T^{2}$$)($$1 - T + 4 T^{2}$$)($$1 + 4 T + 8 T^{2} + 12 T^{3} + 17 T^{4} + 48 T^{5} + 128 T^{6} + 256 T^{7} + 256 T^{8}$$)
$3$ ($$1 + 4 T + 9 T^{2}$$)($$1 - 3 T$$)($$1 + 3 T$$)($$1 + 9 T^{4}$$)
$5$ ($$1 + 5 T^{2}$$)($$1 + 5 T$$)($$1 - 5 T$$)($$1 + 4 T + 100 T^{3} + 625 T^{4}$$)
$7$ ($$( 1 + 6 T + 49 T^{2} )^{2}$$)($$( 1 - 7 T )( 1 + 7 T )$$)($$( 1 - 7 T )( 1 + 7 T )$$)($$1 - 4 T + 8 T^{2} - 156 T^{3} + 2942 T^{4} - 7644 T^{5} + 19208 T^{6} - 470596 T^{7} + 5764801 T^{8}$$)
$11$ ($$1 - 222 T^{2} + 14641 T^{4}$$)($$( 1 - 11 T )( 1 + 11 T )$$)($$( 1 - 11 T )( 1 + 11 T )$$)($$( 1 - 8 T + 204 T^{2} - 968 T^{3} + 14641 T^{4} )^{2}$$)
$13$ ($$( 1 - 16 T + 169 T^{2} )^{2}$$)($$( 1 - 13 T )( 1 + 13 T )$$)($$( 1 - 13 T )( 1 + 13 T )$$)($$1 + 32 T + 512 T^{2} + 9120 T^{3} + 148994 T^{4} + 1541280 T^{5} + 14623232 T^{6} + 154457888 T^{7} + 815730721 T^{8}$$)
$17$ ($$1 - 558 T^{2} + 83521 T^{4}$$)($$1 - 14 T + 289 T^{2}$$)($$1 + 14 T + 289 T^{2}$$)($$1 + 40 T + 800 T^{2} + 15240 T^{3} + 281858 T^{4} + 4404360 T^{5} + 66816800 T^{6} + 965502760 T^{7} + 6975757441 T^{8}$$)
$19$ ($$( 1 + 2 T + 361 T^{2} )^{2}$$)($$1 + 22 T + 361 T^{2}$$)($$1 + 22 T + 361 T^{2}$$)($$1 - 940 T^{2} + 450438 T^{4} - 122501740 T^{6} + 16983563041 T^{8}$$)
$23$ ($$( 1 - 44 T + 529 T^{2} )( 1 + 44 T + 529 T^{2} )$$)($$1 + 34 T + 529 T^{2}$$)($$1 - 34 T + 529 T^{2}$$)($$1 - 56 T + 1568 T^{2} - 50904 T^{3} + 1508162 T^{4} - 26928216 T^{5} + 438790688 T^{6} - 8290009784 T^{7} + 78310985281 T^{8}$$)
$29$ ($$1 - 702 T^{2} + 707281 T^{4}$$)($$( 1 - 29 T )( 1 + 29 T )$$)($$( 1 - 29 T )( 1 + 29 T )$$)($$1 - 2128 T^{2} + 2165634 T^{4} - 1505093968 T^{6} + 500246412961 T^{8}$$)
$31$ ($$( 1 + 18 T + 961 T^{2} )^{2}$$)($$1 - 2 T + 961 T^{2}$$)($$1 - 2 T + 961 T^{2}$$)($$( 1 + 8 T + 1722 T^{2} + 7688 T^{3} + 923521 T^{4} )^{2}$$)
$37$ ($$( 1 + 16 T + 1369 T^{2} )^{2}$$)($$( 1 - 37 T )( 1 + 37 T )$$)($$( 1 - 37 T )( 1 + 37 T )$$)($$1 - 64 T + 2048 T^{2} - 58176 T^{3} + 1440962 T^{4} - 79642944 T^{5} + 3838281728 T^{6} - 164206490176 T^{7} + 3512479453921 T^{8}$$)
$41$ ($$1 + 558 T^{2} + 2825761 T^{4}$$)($$( 1 - 41 T )( 1 + 41 T )$$)($$( 1 - 41 T )( 1 + 41 T )$$)($$( 1 + 28 T + 3342 T^{2} + 47068 T^{3} + 2825761 T^{4} )^{2}$$)
$43$ ($$( 1 - 16 T + 1849 T^{2} )^{2}$$)($$( 1 - 43 T )( 1 + 43 T )$$)($$( 1 - 43 T )( 1 + 43 T )$$)($$1 + 8 T + 32 T^{2} + 5256 T^{3} - 557566 T^{4} + 9718344 T^{5} + 109401632 T^{6} + 50570904392 T^{7} + 11688200277601 T^{8}$$)
$47$ ($$1 - 1998 T^{2} + 4879681 T^{4}$$)($$1 - 14 T + 2209 T^{2}$$)($$1 + 14 T + 2209 T^{2}$$)($$1 - 128 T + 8192 T^{2} - 506496 T^{3} + 28260194 T^{4} - 1118849664 T^{5} + 39974346752 T^{6} - 1379739562112 T^{7} + 23811286661761 T^{8}$$)
$53$ ($$1 - 5598 T^{2} + 7890481 T^{4}$$)($$1 - 86 T + 2809 T^{2}$$)($$1 + 86 T + 2809 T^{2}$$)($$1 - 56 T + 1568 T^{2} - 155064 T^{3} + 15333122 T^{4} - 435574776 T^{5} + 12372274208 T^{6} - 1241204223224 T^{7} + 62259690411361 T^{8}$$)
$59$ ($$1 - 6942 T^{2} + 12117361 T^{4}$$)($$( 1 - 59 T )( 1 + 59 T )$$)($$( 1 - 59 T )( 1 + 59 T )$$)($$1 + 200 T^{2} - 5646222 T^{4} + 2423472200 T^{6} + 146830437604321 T^{8}$$)
$61$ ($$( 1 - 82 T + 3721 T^{2} )^{2}$$)($$1 + 118 T + 3721 T^{2}$$)($$1 + 118 T + 3721 T^{2}$$)($$( 1 - 100 T + 7998 T^{2} - 372100 T^{3} + 13845841 T^{4} )^{2}$$)
$67$ ($$( 1 - 24 T + 4489 T^{2} )^{2}$$)($$( 1 - 67 T )( 1 + 67 T )$$)($$( 1 - 67 T )( 1 + 67 T )$$)($$1 + 200 T + 20000 T^{2} + 1888200 T^{3} + 153742658 T^{4} + 8476129800 T^{5} + 403022420000 T^{6} + 18091676433800 T^{7} + 406067677556641 T^{8}$$)
$71$ ($$1 + 5598 T^{2} + 25411681 T^{4}$$)($$( 1 - 71 T )( 1 + 71 T )$$)($$( 1 - 71 T )( 1 + 71 T )$$)($$( 1 + 68 T + 5041 T^{2} )^{4}$$)
$73$ ($$( 1 + 74 T + 5329 T^{2} )^{2}$$)($$( 1 - 73 T )( 1 + 73 T )$$)($$( 1 - 73 T )( 1 + 73 T )$$)($$1 - 76 T + 2888 T^{2} + 65436 T^{3} - 36833458 T^{4} + 348708444 T^{5} + 82014120008 T^{6} - 11501401197964 T^{7} + 806460091894081 T^{8}$$)
$79$ ($$( 1 - 138 T + 6241 T^{2} )^{2}$$)($$1 - 98 T + 6241 T^{2}$$)($$1 - 98 T + 6241 T^{2}$$)($$( 1 - 11882 T^{2} + 38950081 T^{4} )^{2}$$)
$83$ ($$1 - 4958 T^{2} + 47458321 T^{4}$$)($$1 + 154 T + 6889 T^{2}$$)($$1 - 154 T + 6889 T^{2}$$)($$1 + 16 T + 128 T^{2} + 101328 T^{3} + 79904642 T^{4} + 698048592 T^{5} + 6074665088 T^{6} + 5231045973904 T^{7} + 2252292232139041 T^{8}$$)
$89$ ($$( 1 - 142 T + 7921 T^{2} )( 1 + 142 T + 7921 T^{2} )$$)($$( 1 - 89 T )( 1 + 89 T )$$)($$( 1 - 89 T )( 1 + 89 T )$$)($$1 - 16060 T^{2} + 188845638 T^{4} - 1007640390460 T^{6} + 3936588805702081 T^{8}$$)
$97$ ($$( 1 + 166 T + 9409 T^{2} )^{2}$$)($$( 1 - 97 T )( 1 + 97 T )$$)($$( 1 - 97 T )( 1 + 97 T )$$)($$1 + 20 T + 200 T^{2} + 173820 T^{3} + 150551438 T^{4} + 1635472380 T^{5} + 17705856200 T^{6} + 16659440098580 T^{7} + 7837433594376961 T^{8}$$)