# Properties

 Label 350.3 Level 350 Weight 3 Dimension 2120 Nonzero newspaces 12 Newform subspaces 35 Sturm bound 21600 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$12$$ Newform subspaces: $$35$$ Sturm bound: $$21600$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(350))$$.

Total New Old
Modular forms 7536 2120 5416
Cusp forms 6864 2120 4744
Eisenstein series 672 0 672

## Trace form

 $$2120 q - 8 q^{2} - 22 q^{3} - 4 q^{4} + 32 q^{6} + 16 q^{7} + 16 q^{8} + O(q^{10})$$ $$2120 q - 8 q^{2} - 22 q^{3} - 4 q^{4} + 32 q^{6} + 16 q^{7} + 16 q^{8} - 20 q^{10} - 46 q^{11} - 20 q^{12} + 24 q^{13} - 36 q^{14} + 40 q^{15} - 40 q^{16} + 306 q^{17} + 284 q^{18} + 406 q^{19} + 80 q^{20} + 142 q^{21} + 248 q^{22} + 94 q^{23} + 24 q^{24} - 4 q^{25} + 280 q^{26} + 488 q^{27} + 172 q^{28} + 128 q^{29} - 192 q^{30} + 102 q^{31} - 48 q^{32} - 442 q^{33} - 500 q^{34} - 272 q^{35} - 208 q^{36} - 398 q^{37} - 556 q^{38} - 388 q^{39} - 40 q^{40} - 416 q^{41} - 1048 q^{42} - 728 q^{43} - 444 q^{44} - 132 q^{45} - 508 q^{46} - 434 q^{47} + 64 q^{48} - 20 q^{49} + 100 q^{50} - 410 q^{51} - 24 q^{52} - 14 q^{53} - 36 q^{54} - 144 q^{55} + 80 q^{56} + 204 q^{57} + 344 q^{58} + 682 q^{59} + 480 q^{60} + 1286 q^{61} + 1024 q^{62} + 1468 q^{63} + 32 q^{64} + 1868 q^{65} + 560 q^{66} + 1942 q^{67} + 668 q^{68} + 1400 q^{69} + 280 q^{70} + 40 q^{71} - 32 q^{72} - 666 q^{73} + 96 q^{74} - 696 q^{75} + 160 q^{76} - 1570 q^{77} - 608 q^{78} - 2610 q^{79} - 2262 q^{81} - 1336 q^{82} - 3080 q^{83} - 588 q^{84} - 1688 q^{85} + 120 q^{86} - 840 q^{87} + 616 q^{88} + 322 q^{89} - 596 q^{90} + 2208 q^{91} + 280 q^{92} + 2446 q^{93} + 1908 q^{94} + 1648 q^{95} + 208 q^{96} + 1592 q^{97} + 1248 q^{98} + 2064 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(350))$$

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
350.3.b $$\chi_{350}(251, \cdot)$$ 350.3.b.a 4 1
350.3.b.b 4
350.3.b.c 8
350.3.b.d 8
350.3.d $$\chi_{350}(349, \cdot)$$ 350.3.d.a 8 1
350.3.d.b 16
350.3.f $$\chi_{350}(43, \cdot)$$ 350.3.f.a 4 2
350.3.f.b 4
350.3.f.c 4
350.3.f.d 8
350.3.f.e 8
350.3.f.f 8
350.3.i $$\chi_{350}(199, \cdot)$$ 350.3.i.a 8 2
350.3.i.b 16
350.3.i.c 24
350.3.k $$\chi_{350}(101, \cdot)$$ 350.3.k.a 4 2
350.3.k.b 8
350.3.k.c 12
350.3.k.d 12
350.3.k.e 16
350.3.l $$\chi_{350}(69, \cdot)$$ 350.3.l.a 160 4
350.3.n $$\chi_{350}(41, \cdot)$$ 350.3.n.a 160 4
350.3.p $$\chi_{350}(93, \cdot)$$ 350.3.p.a 8 4
350.3.p.b 8
350.3.p.c 8
350.3.p.d 8
350.3.p.e 16
350.3.p.f 24
350.3.p.g 24
350.3.s $$\chi_{350}(113, \cdot)$$ 350.3.s.a 112 8
350.3.s.b 128
350.3.t $$\chi_{350}(31, \cdot)$$ 350.3.t.a 320 8
350.3.v $$\chi_{350}(19, \cdot)$$ 350.3.v.a 320 8
350.3.w $$\chi_{350}(23, \cdot)$$ 350.3.w.a 320 16
350.3.w.b 320

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(350))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(350)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(175))$$$$^{\oplus 2}$$