## Defining parameters

 Level: $$N$$ = $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$14$$ Newform subspaces: $$41$$ Sturm bound: $$18432$$ Trace bound: $$12$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(320))$$.

Total New Old
Modular forms 6432 3234 3198
Cusp forms 5856 3102 2754
Eisenstein series 576 132 444

## Trace form

 $$3102q - 16q^{2} - 12q^{3} - 16q^{4} - 24q^{5} - 48q^{6} - 16q^{7} - 16q^{8} - 38q^{9} + O(q^{10})$$ $$3102q - 16q^{2} - 12q^{3} - 16q^{4} - 24q^{5} - 48q^{6} - 16q^{7} - 16q^{8} - 38q^{9} - 24q^{10} - 68q^{11} - 16q^{12} - 48q^{13} - 16q^{14} - 16q^{15} - 48q^{16} + 4q^{17} - 16q^{18} + 52q^{19} - 24q^{20} + 120q^{21} + 128q^{22} + 112q^{23} + 544q^{24} + 66q^{25} + 352q^{26} + 120q^{27} + 224q^{28} + 48q^{29} + 56q^{30} - 16q^{31} - 96q^{32} - 176q^{33} - 256q^{34} - 116q^{35} - 848q^{36} - 368q^{37} - 576q^{38} - 392q^{39} - 384q^{40} - 572q^{41} - 896q^{42} - 236q^{43} - 224q^{44} - 172q^{45} - 48q^{46} - 8q^{47} - 16q^{48} + 134q^{49} - 336q^{50} + 1048q^{51} - 1072q^{52} + 272q^{53} - 1168q^{54} + 748q^{55} - 832q^{56} + 528q^{57} - 736q^{58} + 916q^{59} - 312q^{60} + 176q^{61} - 48q^{62} + 368q^{63} + 176q^{64} + 288q^{65} + 464q^{66} - 524q^{67} + 464q^{68} + 536q^{69} + 648q^{70} - 1064q^{71} + 1280q^{72} - 148q^{73} + 1216q^{74} - 624q^{75} + 1616q^{76} + 472q^{77} + 2240q^{78} - 1040q^{79} + 1344q^{80} + 486q^{81} + 2064q^{82} + 308q^{83} + 2448q^{84} - 136q^{85} + 1824q^{86} - 16q^{87} + 1104q^{88} + 428q^{89} + 696q^{90} - 424q^{91} + 896q^{92} - 640q^{93} + 176q^{94} - 776q^{95} - 320q^{96} - 1004q^{97} - 832q^{98} - 1252q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(320))$$

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
320.3.b $$\chi_{320}(191, \cdot)$$ 320.3.b.a 4 1
320.3.b.b 4
320.3.b.c 4
320.3.b.d 4
320.3.e $$\chi_{320}(159, \cdot)$$ 320.3.e.a 8 1
320.3.e.b 16
320.3.g $$\chi_{320}(31, \cdot)$$ 320.3.g.a 8 1
320.3.g.b 8
320.3.h $$\chi_{320}(319, \cdot)$$ 320.3.h.a 1 1
320.3.h.b 1
320.3.h.c 2
320.3.h.d 2
320.3.h.e 4
320.3.h.f 6
320.3.h.g 6
320.3.i $$\chi_{320}(177, \cdot)$$ 320.3.i.a 44 2
320.3.k $$\chi_{320}(79, \cdot)$$ 320.3.k.a 44 2
320.3.m $$\chi_{320}(33, \cdot)$$ 320.3.m.a 8 2
320.3.m.b 8
320.3.m.c 16
320.3.m.d 16
320.3.p $$\chi_{320}(193, \cdot)$$ 320.3.p.a 2 2
320.3.p.b 2
320.3.p.c 2
320.3.p.d 2
320.3.p.e 2
320.3.p.f 2
320.3.p.g 2
320.3.p.h 2
320.3.p.i 4
320.3.p.j 4
320.3.p.k 4
320.3.p.l 4
320.3.p.m 6
320.3.p.n 6
320.3.r $$\chi_{320}(111, \cdot)$$ 320.3.r.a 32 2
320.3.t $$\chi_{320}(17, \cdot)$$ 320.3.t.a 44 2
320.3.v $$\chi_{320}(57, \cdot)$$ None 0 4
320.3.w $$\chi_{320}(71, \cdot)$$ None 0 4
320.3.y $$\chi_{320}(39, \cdot)$$ None 0 4
320.3.bb $$\chi_{320}(137, \cdot)$$ None 0 4
320.3.bc $$\chi_{320}(53, \cdot)$$ 320.3.bc.a 752 8
320.3.bg $$\chi_{320}(11, \cdot)$$ 320.3.bg.a 512 8
320.3.bh $$\chi_{320}(19, \cdot)$$ 320.3.bh.a 752 8
320.3.bi $$\chi_{320}(13, \cdot)$$ 320.3.bi.a 752 8

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(320))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(320)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(160))$$$$^{\oplus 2}$$