## Defining parameters

 Level: $$N$$ = $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$12$$ Newform subspaces: $$29$$ Sturm bound: $$17280$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(304))$$.

Total New Old
Modular forms 6012 3299 2713
Cusp forms 5508 3145 2363
Eisenstein series 504 154 350

## Trace form

 $$3145q - 32q^{2} - 23q^{3} - 20q^{4} - 29q^{5} - 20q^{6} - 19q^{7} - 44q^{8} - 27q^{9} + O(q^{10})$$ $$3145q - 32q^{2} - 23q^{3} - 20q^{4} - 29q^{5} - 20q^{6} - 19q^{7} - 44q^{8} - 27q^{9} - 108q^{10} + 9q^{11} - 140q^{12} - 61q^{13} - 60q^{14} - 27q^{15} + 44q^{16} - 13q^{17} + 112q^{18} - 57q^{19} + 96q^{20} - 5q^{21} + 68q^{22} - 147q^{23} - 132q^{24} - 31q^{25} - 228q^{26} - 155q^{27} - 148q^{28} - 93q^{29} - 140q^{30} - 27q^{31} - 52q^{32} - 73q^{33} + 116q^{34} + 173q^{35} + 68q^{36} + 12q^{37} - 76q^{38} + 338q^{39} - 116q^{40} - 45q^{41} + 12q^{42} + 201q^{43} - 76q^{44} - 69q^{45} - 92q^{46} - 27q^{47} + 12q^{48} - 87q^{49} - 128q^{50} - 339q^{51} - 236q^{52} - 381q^{53} - 100q^{54} - 531q^{55} + 300q^{56} - 9q^{57} + 280q^{58} - 439q^{59} + 284q^{60} + 443q^{61} + 252q^{62} + 837q^{63} - 164q^{64} + 735q^{65} - 428q^{66} + 1289q^{67} - 260q^{68} + 619q^{69} - 4q^{70} + 709q^{71} - 140q^{72} + 427q^{73} + 148q^{74} + 440q^{75} + 152q^{76} + 118q^{77} + 132q^{78} - 531q^{79} - 500q^{80} - 1063q^{81} - 644q^{82} - 1095q^{83} - 500q^{84} - 701q^{85} - 572q^{86} - 2211q^{87} - 20q^{88} - 861q^{89} + 284q^{90} - 1555q^{91} + 300q^{92} - 629q^{93} - 132q^{94} - 27q^{95} + 88q^{96} - 333q^{97} - 56q^{98} + 425q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(304))$$

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
304.3.d $$\chi_{304}(191, \cdot)$$ 304.3.d.a 6 1
304.3.d.b 12
304.3.e $$\chi_{304}(113, \cdot)$$ 304.3.e.a 1 1
304.3.e.b 2
304.3.e.c 2
304.3.e.d 2
304.3.e.e 2
304.3.e.f 2
304.3.e.g 8
304.3.f $$\chi_{304}(39, \cdot)$$ None 0 1
304.3.g $$\chi_{304}(265, \cdot)$$ None 0 1
304.3.j $$\chi_{304}(37, \cdot)$$ 304.3.j.a 156 2
304.3.l $$\chi_{304}(115, \cdot)$$ 304.3.l.a 144 2
304.3.o $$\chi_{304}(7, \cdot)$$ None 0 2
304.3.p $$\chi_{304}(217, \cdot)$$ None 0 2
304.3.q $$\chi_{304}(159, \cdot)$$ 304.3.q.a 12 2
304.3.q.b 12
304.3.q.c 16
304.3.r $$\chi_{304}(65, \cdot)$$ 304.3.r.a 4 2
304.3.r.b 6
304.3.r.c 8
304.3.r.d 20
304.3.w $$\chi_{304}(69, \cdot)$$ 304.3.w.a 312 4
304.3.y $$\chi_{304}(11, \cdot)$$ 304.3.y.a 312 4
304.3.z $$\chi_{304}(33, \cdot)$$ 304.3.z.a 12 6
304.3.z.b 18
304.3.z.c 24
304.3.z.d 60
304.3.ba $$\chi_{304}(41, \cdot)$$ None 0 6
304.3.bc $$\chi_{304}(23, \cdot)$$ None 0 6
304.3.bf $$\chi_{304}(47, \cdot)$$ 304.3.bf.a 36 6
304.3.bf.b 42
304.3.bf.c 42
304.3.bh $$\chi_{304}(35, \cdot)$$ 304.3.bh.a 936 12
304.3.bj $$\chi_{304}(13, \cdot)$$ 304.3.bj.a 936 12

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(304))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(304)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(76))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(152))$$$$^{\oplus 2}$$