## 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

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

## 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

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