## Defining parameters

 Level: $$N$$ = $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$18$$ Sturm bound: $$194400$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1350))$$.

Total New Old
Modular forms 50280 11798 38482
Cusp forms 46921 11798 35123
Eisenstein series 3359 0 3359

## Trace form

 $$11798 q + q^{2} + q^{4} - 6 q^{6} - 4 q^{7} - 5 q^{8} - 12 q^{9} + O(q^{10})$$ $$11798 q + q^{2} + q^{4} - 6 q^{6} - 4 q^{7} - 5 q^{8} - 12 q^{9} - 8 q^{10} - 50 q^{11} - 3 q^{12} - 42 q^{13} - 36 q^{14} + q^{16} - 54 q^{17} + 6 q^{18} - 30 q^{19} - 8 q^{20} + 24 q^{21} - 16 q^{22} - 30 q^{23} - 16 q^{25} + 32 q^{26} + 27 q^{27} + 2 q^{28} - 12 q^{29} - 36 q^{31} + q^{32} + 75 q^{33} + 58 q^{34} + 96 q^{35} + 54 q^{36} + 60 q^{37} + 227 q^{38} + 198 q^{39} + 32 q^{40} + 298 q^{41} + 168 q^{42} + 228 q^{43} + 80 q^{44} + 120 q^{45} + 96 q^{46} + 378 q^{47} + 42 q^{48} + 213 q^{49} + 160 q^{50} + 192 q^{51} + 54 q^{52} + 336 q^{53} + 108 q^{54} + 32 q^{55} + 76 q^{56} + 135 q^{57} + 46 q^{58} + 65 q^{59} + 6 q^{61} + 56 q^{62} - 6 q^{63} - 5 q^{64} + 144 q^{65} + 114 q^{67} + 57 q^{68} + 18 q^{69} + 128 q^{70} + 16 q^{71} + 24 q^{72} + 134 q^{73} + 146 q^{74} + 72 q^{75} + 39 q^{76} + 548 q^{77} + 36 q^{78} + 364 q^{79} + 192 q^{81} + 126 q^{82} + 580 q^{83} + 208 q^{85} + 34 q^{86} + 276 q^{87} + 11 q^{88} + 505 q^{89} + 206 q^{91} - 8 q^{92} + 282 q^{93} - 34 q^{94} + 136 q^{95} + 6 q^{96} + 84 q^{97} - 105 q^{98} + 204 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1350))$$

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
1350.2.a $$\chi_{1350}(1, \cdot)$$ 1350.2.a.a 1 1
1350.2.a.b 1
1350.2.a.c 1
1350.2.a.d 1
1350.2.a.e 1
1350.2.a.f 1
1350.2.a.g 1
1350.2.a.h 1
1350.2.a.i 1
1350.2.a.j 1
1350.2.a.k 1
1350.2.a.l 1
1350.2.a.m 1
1350.2.a.n 1
1350.2.a.o 1
1350.2.a.p 1
1350.2.a.q 1
1350.2.a.r 1
1350.2.a.s 1
1350.2.a.t 1
1350.2.a.u 1
1350.2.a.v 1
1350.2.a.w 2
1350.2.a.x 2
1350.2.c $$\chi_{1350}(649, \cdot)$$ 1350.2.c.a 2 1
1350.2.c.b 2
1350.2.c.c 2
1350.2.c.d 2
1350.2.c.e 2
1350.2.c.f 2
1350.2.c.g 2
1350.2.c.h 2
1350.2.c.i 2
1350.2.c.j 2
1350.2.c.k 2
1350.2.c.l 2
1350.2.e $$\chi_{1350}(451, \cdot)$$ 1350.2.e.a 2 2
1350.2.e.b 2
1350.2.e.c 2
1350.2.e.d 2
1350.2.e.e 2
1350.2.e.f 2
1350.2.e.g 2
1350.2.e.h 2
1350.2.e.i 2
1350.2.e.j 4
1350.2.e.k 4
1350.2.e.l 4
1350.2.e.m 4
1350.2.e.n 4
1350.2.f $$\chi_{1350}(107, \cdot)$$ 1350.2.f.a 8 2
1350.2.f.b 8
1350.2.f.c 8
1350.2.f.d 8
1350.2.f.e 8
1350.2.f.f 8
1350.2.h $$\chi_{1350}(271, \cdot)$$ n/a 160 4
1350.2.j $$\chi_{1350}(199, \cdot)$$ 1350.2.j.a 4 2
1350.2.j.b 4
1350.2.j.c 4
1350.2.j.d 4
1350.2.j.e 4
1350.2.j.f 8
1350.2.j.g 8
1350.2.l $$\chi_{1350}(151, \cdot)$$ n/a 342 6
1350.2.m $$\chi_{1350}(109, \cdot)$$ n/a 160 4
1350.2.q $$\chi_{1350}(143, \cdot)$$ 1350.2.q.a 8 4
1350.2.q.b 8
1350.2.q.c 8
1350.2.q.d 8
1350.2.q.e 8
1350.2.q.f 8
1350.2.q.g 8
1350.2.q.h 16
1350.2.r $$\chi_{1350}(91, \cdot)$$ n/a 240 8
1350.2.u $$\chi_{1350}(49, \cdot)$$ n/a 324 6
1350.2.w $$\chi_{1350}(53, \cdot)$$ n/a 320 8
1350.2.z $$\chi_{1350}(19, \cdot)$$ n/a 240 8
1350.2.bb $$\chi_{1350}(257, \cdot)$$ n/a 648 12
1350.2.bc $$\chi_{1350}(31, \cdot)$$ n/a 2160 24
1350.2.bd $$\chi_{1350}(17, \cdot)$$ n/a 480 16
1350.2.bf $$\chi_{1350}(79, \cdot)$$ n/a 2160 24
1350.2.bi $$\chi_{1350}(23, \cdot)$$ n/a 4320 48

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1350))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1350)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(225))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(270))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(450))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(675))$$$$^{\oplus 2}$$