## Defining parameters

 Level: $$N$$ = $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$28$$ Sturm bound: $$9216$$ Trace bound: $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(288))$$.

Total New Old
Modular forms 2560 1071 1489
Cusp forms 2049 981 1068
Eisenstein series 511 90 421

## Trace form

 $$981q - 12q^{2} - 12q^{3} - 12q^{4} - 14q^{5} - 16q^{6} - 10q^{7} - 12q^{8} - 24q^{9} + O(q^{10})$$ $$981q - 12q^{2} - 12q^{3} - 12q^{4} - 14q^{5} - 16q^{6} - 10q^{7} - 12q^{8} - 24q^{9} - 28q^{10} - 6q^{11} - 16q^{12} - 6q^{13} + 4q^{14} - 6q^{15} + 8q^{16} + 14q^{17} - 16q^{18} - 12q^{19} + 4q^{20} + 18q^{23} - 16q^{24} + 15q^{25} - 32q^{26} - 56q^{28} + 10q^{29} - 32q^{30} + 26q^{31} - 32q^{32} - 44q^{33} - 24q^{34} + 12q^{35} - 40q^{36} - 70q^{37} - 116q^{38} - 18q^{39} - 128q^{40} - 70q^{41} - 96q^{42} - 34q^{43} - 148q^{44} - 64q^{45} - 132q^{46} - 66q^{47} - 120q^{48} - 75q^{49} - 180q^{50} - 44q^{51} - 132q^{52} - 78q^{53} - 104q^{54} - 72q^{55} - 160q^{56} - 72q^{57} - 144q^{58} - 74q^{59} - 72q^{60} - 6q^{61} - 48q^{62} - 66q^{63} - 72q^{64} - 112q^{65} - 16q^{66} - 34q^{67} - 56q^{68} - 80q^{69} - 24q^{70} - 68q^{71} - 16q^{72} - 10q^{73} + 4q^{74} - 124q^{75} - 12q^{76} - 28q^{77} + 8q^{78} - 70q^{79} + 112q^{80} - 88q^{81} + 124q^{82} - 186q^{83} + 96q^{84} + 20q^{85} + 176q^{86} - 170q^{87} + 152q^{88} - 78q^{89} + 128q^{90} - 192q^{91} + 256q^{92} - 16q^{93} + 160q^{94} - 248q^{95} + 120q^{96} - 74q^{97} + 208q^{98} - 154q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(288))$$

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
288.2.a $$\chi_{288}(1, \cdot)$$ 288.2.a.a 1 1
288.2.a.b 1
288.2.a.c 1
288.2.a.d 1
288.2.a.e 1
288.2.c $$\chi_{288}(287, \cdot)$$ 288.2.c.a 4 1
288.2.d $$\chi_{288}(145, \cdot)$$ 288.2.d.a 2 1
288.2.d.b 2
288.2.f $$\chi_{288}(143, \cdot)$$ 288.2.f.a 4 1
288.2.i $$\chi_{288}(97, \cdot)$$ 288.2.i.a 2 2
288.2.i.b 2
288.2.i.c 4
288.2.i.d 4
288.2.i.e 4
288.2.i.f 8
288.2.k $$\chi_{288}(73, \cdot)$$ None 0 2
288.2.l $$\chi_{288}(71, \cdot)$$ None 0 2
288.2.p $$\chi_{288}(47, \cdot)$$ 288.2.p.a 4 2
288.2.p.b 16
288.2.r $$\chi_{288}(49, \cdot)$$ 288.2.r.a 4 2
288.2.r.b 16
288.2.s $$\chi_{288}(95, \cdot)$$ 288.2.s.a 24 2
288.2.v $$\chi_{288}(37, \cdot)$$ 288.2.v.a 4 4
288.2.v.b 8
288.2.v.c 32
288.2.v.d 32
288.2.w $$\chi_{288}(35, \cdot)$$ 288.2.w.a 32 4
288.2.w.b 32
288.2.y $$\chi_{288}(23, \cdot)$$ None 0 4
288.2.bb $$\chi_{288}(25, \cdot)$$ None 0 4
288.2.bc $$\chi_{288}(13, \cdot)$$ 288.2.bc.a 368 8
288.2.bf $$\chi_{288}(11, \cdot)$$ 288.2.bf.a 368 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(288))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(288)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(96))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(144))$$$$^{\oplus 2}$$