## Defining parameters

 Level: $$N$$ = $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Sturm bound: $$115200$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1050))$$.

Total New Old
Modular forms 30144 6362 23782
Cusp forms 27457 6362 21095
Eisenstein series 2687 0 2687

## Trace form

 $$6362q - 6q^{2} - 16q^{3} - 10q^{4} - 20q^{5} - 20q^{6} - 40q^{7} - 6q^{8} - 14q^{9} + O(q^{10})$$ $$6362q - 6q^{2} - 16q^{3} - 10q^{4} - 20q^{5} - 20q^{6} - 40q^{7} - 6q^{8} - 14q^{9} - 4q^{10} - 32q^{11} + 16q^{12} - 36q^{13} + 8q^{14} + 8q^{15} - 6q^{16} + 56q^{17} + 58q^{18} + 88q^{19} + 16q^{20} + 42q^{21} + 120q^{22} + 100q^{23} + 32q^{24} + 220q^{25} + 80q^{26} + 128q^{27} + 100q^{28} + 244q^{29} + 112q^{30} + 184q^{31} + 14q^{32} + 256q^{33} + 224q^{34} + 152q^{35} + 82q^{36} + 280q^{37} + 132q^{38} + 320q^{39} + 12q^{40} + 132q^{41} + 122q^{42} + 264q^{43} + 32q^{44} + 212q^{45} + 96q^{46} + 124q^{47} + 16q^{48} - 34q^{49} - 36q^{50} + 72q^{51} - 12q^{52} + 12q^{53} + 4q^{54} + 96q^{55} - 20q^{56} + 108q^{57} - 104q^{58} + 104q^{59} - 64q^{60} + 84q^{61} - 120q^{62} - 20q^{63} - 10q^{64} - 4q^{65} - 96q^{66} + 144q^{67} - 24q^{68} - 192q^{69} - 48q^{70} + 72q^{71} - 50q^{72} + 112q^{73} - 104q^{74} - 232q^{75} - 8q^{76} + 84q^{77} - 260q^{78} + 136q^{79} - 20q^{80} - 94q^{81} - 196q^{82} + 32q^{83} - 94q^{84} + 28q^{85} - 100q^{86} - 280q^{87} - 52q^{88} - 104q^{89} - 204q^{90} - 112q^{91} - 72q^{92} - 324q^{93} - 232q^{94} - 32q^{95} - 24q^{96} - 160q^{97} - 78q^{98} - 424q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1050))$$

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
1050.2.a $$\chi_{1050}(1, \cdot)$$ 1050.2.a.a 1 1
1050.2.a.b 1
1050.2.a.c 1
1050.2.a.d 1
1050.2.a.e 1
1050.2.a.f 1
1050.2.a.g 1
1050.2.a.h 1
1050.2.a.i 1
1050.2.a.j 1
1050.2.a.k 1
1050.2.a.l 1
1050.2.a.m 1
1050.2.a.n 1
1050.2.a.o 1
1050.2.a.p 1
1050.2.a.q 1
1050.2.a.r 1
1050.2.b $$\chi_{1050}(251, \cdot)$$ 1050.2.b.a 4 1
1050.2.b.b 4
1050.2.b.c 4
1050.2.b.d 12
1050.2.b.e 12
1050.2.b.f 16
1050.2.d $$\chi_{1050}(1049, \cdot)$$ 1050.2.d.a 4 1
1050.2.d.b 4
1050.2.d.c 4
1050.2.d.d 4
1050.2.d.e 4
1050.2.d.f 4
1050.2.d.g 12
1050.2.d.h 12
1050.2.g $$\chi_{1050}(799, \cdot)$$ 1050.2.g.a 2 1
1050.2.g.b 2
1050.2.g.c 2
1050.2.g.d 2
1050.2.g.e 2
1050.2.g.f 2
1050.2.g.g 2
1050.2.g.h 2
1050.2.g.i 2
1050.2.g.j 2
1050.2.i $$\chi_{1050}(151, \cdot)$$ 1050.2.i.a 2 2
1050.2.i.b 2
1050.2.i.c 2
1050.2.i.d 2
1050.2.i.e 2
1050.2.i.f 2
1050.2.i.g 2
1050.2.i.h 2
1050.2.i.i 2
1050.2.i.j 2
1050.2.i.k 2
1050.2.i.l 2
1050.2.i.m 2
1050.2.i.n 2
1050.2.i.o 2
1050.2.i.p 2
1050.2.i.q 2
1050.2.i.r 2
1050.2.i.s 2
1050.2.i.t 2
1050.2.i.u 6
1050.2.i.v 6
1050.2.j $$\chi_{1050}(407, \cdot)$$ 1050.2.j.a 8 2
1050.2.j.b 8
1050.2.j.c 12
1050.2.j.d 12
1050.2.j.e 16
1050.2.j.f 16
1050.2.m $$\chi_{1050}(307, \cdot)$$ 1050.2.m.a 8 2
1050.2.m.b 8
1050.2.m.c 8
1050.2.m.d 8
1050.2.m.e 8
1050.2.m.f 8
1050.2.n $$\chi_{1050}(211, \cdot)$$ n/a 128 4
1050.2.o $$\chi_{1050}(499, \cdot)$$ 1050.2.o.a 4 2
1050.2.o.b 4
1050.2.o.c 4
1050.2.o.d 4
1050.2.o.e 4
1050.2.o.f 4
1050.2.o.g 4
1050.2.o.h 4
1050.2.o.i 4
1050.2.o.j 4
1050.2.o.k 4
1050.2.o.l 4
1050.2.s $$\chi_{1050}(101, \cdot)$$ 1050.2.s.a 4 2
1050.2.s.b 4
1050.2.s.c 4
1050.2.s.d 8
1050.2.s.e 8
1050.2.s.f 12
1050.2.s.g 12
1050.2.s.h 16
1050.2.s.i 16
1050.2.s.j 16
1050.2.u $$\chi_{1050}(299, \cdot)$$ 1050.2.u.a 4 2
1050.2.u.b 4
1050.2.u.c 4
1050.2.u.d 4
1050.2.u.e 12
1050.2.u.f 12
1050.2.u.g 12
1050.2.u.h 12
1050.2.u.i 16
1050.2.u.j 16
1050.2.w $$\chi_{1050}(169, \cdot)$$ n/a 112 4
1050.2.z $$\chi_{1050}(209, \cdot)$$ n/a 320 4
1050.2.bb $$\chi_{1050}(41, \cdot)$$ n/a 320 4
1050.2.bc $$\chi_{1050}(157, \cdot)$$ 1050.2.bc.a 8 4
1050.2.bc.b 8
1050.2.bc.c 8
1050.2.bc.d 8
1050.2.bc.e 16
1050.2.bc.f 16
1050.2.bc.g 16
1050.2.bc.h 16
1050.2.bf $$\chi_{1050}(107, \cdot)$$ n/a 192 4
1050.2.bg $$\chi_{1050}(121, \cdot)$$ n/a 320 8
1050.2.bh $$\chi_{1050}(13, \cdot)$$ n/a 320 8
1050.2.bk $$\chi_{1050}(113, \cdot)$$ n/a 480 8
1050.2.bl $$\chi_{1050}(59, \cdot)$$ n/a 640 8
1050.2.bn $$\chi_{1050}(131, \cdot)$$ n/a 640 8
1050.2.br $$\chi_{1050}(79, \cdot)$$ n/a 320 8
1050.2.bs $$\chi_{1050}(23, \cdot)$$ n/a 1280 16
1050.2.bv $$\chi_{1050}(73, \cdot)$$ n/a 640 16

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1050))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1050)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(105))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(175))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(210))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(350))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(525))$$$$^{\oplus 2}$$