## Defining parameters

 Level: $$N$$ = $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ = $$7$$ Nonzero newspaces: $$12$$ Sturm bound: $$40320$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{7}(\Gamma_1(304))$$.

Total New Old
Modular forms 17532 9743 7789
Cusp forms 17028 9589 7439
Eisenstein series 504 154 350

## Trace form

 $$9589q - 32q^{2} - 23q^{3} - 212q^{4} + 91q^{5} + 988q^{6} - 19q^{7} - 1964q^{8} - 1311q^{9} + O(q^{10})$$ $$9589q - 32q^{2} - 23q^{3} - 212q^{4} + 91q^{5} + 988q^{6} - 19q^{7} - 1964q^{8} - 1311q^{9} + 2212q^{10} - 2743q^{11} + 4660q^{12} + 7483q^{13} - 15164q^{14} - 27q^{15} + 15916q^{16} - 27925q^{17} + 3712q^{18} - 3961q^{19} + 33056q^{20} + 70459q^{21} - 50540q^{22} + 26221q^{23} + 65148q^{24} - 105771q^{25} - 117940q^{26} - 71579q^{27} - 90388q^{28} + 173147q^{29} - 16940q^{30} - 27q^{31} + 79308q^{32} - 101449q^{33} + 105140q^{34} + 224813q^{35} + 276932q^{36} + 149188q^{37} + 187244q^{38} - 508846q^{39} - 330356q^{40} - 98613q^{41} - 707028q^{42} + 535945q^{43} - 761484q^{44} - 430029q^{45} - 496348q^{46} - 27q^{47} + 921612q^{48} + 444741q^{49} + 1439312q^{50} - 603603q^{51} + 1569748q^{52} + 62651q^{53} + 1095068q^{54} + 465389q^{55} - 732884q^{56} + 118263q^{57} - 2883896q^{58} - 78327q^{59} - 4811236q^{60} - 1465373q^{61} - 365700q^{62} - 3149307q^{63} + 637276q^{64} - 769745q^{65} + 6785428q^{66} + 600649q^{67} + 5008764q^{68} + 4464139q^{69} + 2373596q^{70} + 2874357q^{71} - 3497228q^{72} + 1686483q^{73} - 7220828q^{74} + 550520q^{75} - 4246024q^{76} - 6272058q^{77} - 6358140q^{78} - 7017219q^{79} + 5797900q^{80} - 4157083q^{81} + 10009212q^{82} - 941863q^{83} + 15205516q^{84} + 5930579q^{85} + 6480916q^{86} + 4583229q^{87} - 6616340q^{88} + 3012027q^{89} - 20546836q^{90} + 2626541q^{91} - 14507348q^{92} - 4610501q^{93} - 7718148q^{94} - 27q^{95} + 11378776q^{96} - 697429q^{97} + 18353864q^{98} + 543017q^{99} + O(q^{100})$$

## Decomposition of $$S_{7}^{\mathrm{new}}(\Gamma_1(304))$$

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
304.7.d $$\chi_{304}(191, \cdot)$$ 304.7.d.a 18 1
304.7.d.b 36
304.7.e $$\chi_{304}(113, \cdot)$$ 304.7.e.a 1 1
304.7.e.b 2
304.7.e.c 8
304.7.e.d 8
304.7.e.e 10
304.7.e.f 30
304.7.f $$\chi_{304}(39, \cdot)$$ None 0 1
304.7.g $$\chi_{304}(265, \cdot)$$ None 0 1
304.7.j $$\chi_{304}(37, \cdot)$$ n/a 476 2
304.7.l $$\chi_{304}(115, \cdot)$$ n/a 432 2
304.7.o $$\chi_{304}(7, \cdot)$$ None 0 2
304.7.p $$\chi_{304}(217, \cdot)$$ None 0 2
304.7.q $$\chi_{304}(159, \cdot)$$ n/a 120 2
304.7.r $$\chi_{304}(65, \cdot)$$ n/a 118 2
304.7.w $$\chi_{304}(69, \cdot)$$ n/a 952 4
304.7.y $$\chi_{304}(11, \cdot)$$ n/a 952 4
304.7.z $$\chi_{304}(33, \cdot)$$ n/a 354 6
304.7.ba $$\chi_{304}(41, \cdot)$$ None 0 6
304.7.bc $$\chi_{304}(23, \cdot)$$ None 0 6
304.7.bf $$\chi_{304}(47, \cdot)$$ n/a 360 6
304.7.bh $$\chi_{304}(35, \cdot)$$ n/a 2856 12
304.7.bj $$\chi_{304}(13, \cdot)$$ n/a 2856 12

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{7}^{\mathrm{old}}(\Gamma_1(304))$$ into lower level spaces

$$S_{7}^{\mathrm{old}}(\Gamma_1(304)) \cong$$ $$S_{7}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 6}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 5}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 4}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(76))$$$$^{\oplus 3}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(152))$$$$^{\oplus 2}$$