# Properties

 Label 15.9 Level 15 Weight 9 Dimension 40 Nonzero newspaces 3 Newform subspaces 5 Sturm bound 144 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$15 = 3 \cdot 5$$ Weight: $$k$$ = $$9$$ Nonzero newspaces: $$3$$ Newform subspaces: $$5$$ Sturm bound: $$144$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(\Gamma_1(15))$$.

Total New Old
Modular forms 72 44 28
Cusp forms 56 40 16
Eisenstein series 16 4 12

## Trace form

 $$40 q - 112 q^{3} + 984 q^{4} - 444 q^{5} - 7808 q^{6} + 11696 q^{7} + 17460 q^{8} - 5780 q^{9} + O(q^{10})$$ $$40 q - 112 q^{3} + 984 q^{4} - 444 q^{5} - 7808 q^{6} + 11696 q^{7} + 17460 q^{8} - 5780 q^{9} - 48276 q^{10} - 23616 q^{11} + 18868 q^{12} + 77956 q^{13} - 75136 q^{15} + 76544 q^{16} + 573300 q^{17} - 419800 q^{18} - 363320 q^{19} - 863436 q^{20} - 150528 q^{21} + 895240 q^{22} + 651480 q^{23} + 1695528 q^{24} - 576464 q^{25} - 448848 q^{26} - 335512 q^{27} - 6908664 q^{28} + 6063316 q^{30} + 5577544 q^{31} + 641460 q^{32} - 2548640 q^{33} - 3054632 q^{34} + 841080 q^{35} - 5027488 q^{36} + 1065276 q^{37} + 8139840 q^{38} + 5607656 q^{39} - 5496592 q^{40} - 14740104 q^{41} - 15036720 q^{42} + 2925856 q^{43} + 9585976 q^{45} + 7311496 q^{46} + 26529600 q^{47} + 26074588 q^{48} + 14661516 q^{49} - 38452896 q^{50} - 33390872 q^{51} + 5202936 q^{52} + 16612140 q^{53} - 43192496 q^{54} - 28324224 q^{55} + 10752000 q^{56} + 1857896 q^{57} + 59211880 q^{58} - 2447144 q^{60} - 53442552 q^{61} - 35190840 q^{62} + 18759576 q^{63} + 62748552 q^{64} + 125689188 q^{65} + 130603856 q^{66} + 190736 q^{67} - 197811840 q^{68} - 47411256 q^{69} - 58334760 q^{70} - 85681968 q^{71} + 15990540 q^{72} - 180800684 q^{73} - 27564136 q^{75} + 53978336 q^{76} + 97175880 q^{77} + 192097600 q^{78} + 443479040 q^{79} + 339741204 q^{80} + 14471320 q^{81} + 21335160 q^{82} - 208234800 q^{83} - 725704584 q^{84} - 463414188 q^{85} - 187512576 q^{86} - 242620600 q^{87} - 346568280 q^{88} + 453645764 q^{90} + 382700992 q^{91} + 652331400 q^{92} + 623199456 q^{93} + 725214088 q^{94} - 74686896 q^{95} + 460479584 q^{96} - 397196244 q^{97} - 50186520 q^{98} - 674325280 q^{99} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(\Gamma_1(15))$$

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
15.9.c $$\chi_{15}(11, \cdot)$$ 15.9.c.a 10 1
15.9.d $$\chi_{15}(14, \cdot)$$ 15.9.d.a 1 1
15.9.d.b 1
15.9.d.c 12
15.9.f $$\chi_{15}(7, \cdot)$$ 15.9.f.a 16 2

## Decomposition of $$S_{9}^{\mathrm{old}}(\Gamma_1(15))$$ into lower level spaces

$$S_{9}^{\mathrm{old}}(\Gamma_1(15)) \cong$$ $$S_{9}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 2}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 2}$$