## Defining parameters

 Level: $$N$$ = $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$18$$ Sturm bound: $$388800$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1350))$$.

Total New Old
Modular forms 147480 35392 112088
Cusp forms 144120 35392 108728
Eisenstein series 3360 0 3360

## Trace form

 $$35392 q - 2 q^{2} + 4 q^{4} + 12 q^{6} + 56 q^{7} + 40 q^{8} + 96 q^{9} + O(q^{10})$$ $$35392 q - 2 q^{2} + 4 q^{4} + 12 q^{6} + 56 q^{7} + 40 q^{8} + 96 q^{9} - 48 q^{10} - 26 q^{11} - 12 q^{12} + 242 q^{13} + 216 q^{14} + 16 q^{16} + 666 q^{17} + 276 q^{18} - 172 q^{19} - 128 q^{20} + 204 q^{21} - 624 q^{22} - 1740 q^{23} + 144 q^{25} - 544 q^{26} - 1377 q^{27} + 80 q^{28} + 978 q^{29} + 1100 q^{31} - 32 q^{32} + 4359 q^{33} + 3324 q^{34} + 5496 q^{35} + 1944 q^{36} + 146 q^{37} - 130 q^{38} - 2664 q^{39} - 768 q^{40} - 9662 q^{41} - 6384 q^{42} - 9724 q^{43} - 2944 q^{44} - 5280 q^{45} - 3360 q^{46} - 15930 q^{47} - 1344 q^{48} - 6543 q^{49} - 2240 q^{50} - 3570 q^{51} + 200 q^{52} + 7356 q^{53} + 5616 q^{54} + 3792 q^{55} + 6112 q^{56} + 12987 q^{57} + 8628 q^{58} + 19961 q^{59} + 2810 q^{61} + 9056 q^{62} + 3936 q^{63} - 1088 q^{64} + 3024 q^{65} + 4320 q^{66} + 2276 q^{67} - 1428 q^{68} + 6120 q^{69} - 6912 q^{70} - 1784 q^{71} - 192 q^{72} - 334 q^{73} - 12820 q^{74} + 8712 q^{75} - 3412 q^{76} - 2632 q^{77} - 5688 q^{78} - 9964 q^{79} - 9420 q^{81} - 3900 q^{82} - 23696 q^{83} - 2160 q^{84} - 912 q^{85} - 5948 q^{86} - 22098 q^{87} - 3144 q^{88} - 30527 q^{89} - 19862 q^{91} + 7216 q^{92} - 12480 q^{93} + 6612 q^{94} - 10904 q^{95} + 960 q^{96} + 8834 q^{97} + 8706 q^{98} + 6522 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1350))$$

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
1350.4.a $$\chi_{1350}(1, \cdot)$$ 1350.4.a.a 1 1
1350.4.a.b 1
1350.4.a.c 1
1350.4.a.d 1
1350.4.a.e 1
1350.4.a.f 1
1350.4.a.g 1
1350.4.a.h 1
1350.4.a.i 1
1350.4.a.j 1
1350.4.a.k 1
1350.4.a.l 1
1350.4.a.m 1
1350.4.a.n 1
1350.4.a.o 1
1350.4.a.p 1
1350.4.a.q 1
1350.4.a.r 1
1350.4.a.s 1
1350.4.a.t 1
1350.4.a.u 1
1350.4.a.v 1
1350.4.a.w 1
1350.4.a.x 1
1350.4.a.y 1
1350.4.a.z 1
1350.4.a.ba 1
1350.4.a.bb 1
1350.4.a.bc 2
1350.4.a.bd 2
1350.4.a.be 2
1350.4.a.bf 2
1350.4.a.bg 2
1350.4.a.bh 2
1350.4.a.bi 2
1350.4.a.bj 2
1350.4.a.bk 2
1350.4.a.bl 2
1350.4.a.bm 2
1350.4.a.bn 2
1350.4.a.bo 2
1350.4.a.bp 2
1350.4.a.bq 3
1350.4.a.br 3
1350.4.a.bs 3
1350.4.a.bt 3
1350.4.a.bu 4
1350.4.a.bv 4
1350.4.c $$\chi_{1350}(649, \cdot)$$ 1350.4.c.a 2 1
1350.4.c.b 2
1350.4.c.c 2
1350.4.c.d 2
1350.4.c.e 2
1350.4.c.f 2
1350.4.c.g 2
1350.4.c.h 2
1350.4.c.i 2
1350.4.c.j 2
1350.4.c.k 2
1350.4.c.l 2
1350.4.c.m 2
1350.4.c.n 2
1350.4.c.o 2
1350.4.c.p 2
1350.4.c.q 2
1350.4.c.r 2
1350.4.c.s 2
1350.4.c.t 2
1350.4.c.u 4
1350.4.c.v 4
1350.4.c.w 4
1350.4.c.x 4
1350.4.c.y 4
1350.4.c.z 4
1350.4.c.ba 4
1350.4.c.bb 4
1350.4.e $$\chi_{1350}(451, \cdot)$$ n/a 114 2
1350.4.f $$\chi_{1350}(107, \cdot)$$ n/a 144 2
1350.4.h $$\chi_{1350}(271, \cdot)$$ n/a 480 4
1350.4.j $$\chi_{1350}(199, \cdot)$$ n/a 108 2
1350.4.l $$\chi_{1350}(151, \cdot)$$ n/a 1026 6
1350.4.m $$\chi_{1350}(109, \cdot)$$ n/a 480 4
1350.4.q $$\chi_{1350}(143, \cdot)$$ n/a 216 4
1350.4.r $$\chi_{1350}(91, \cdot)$$ n/a 720 8
1350.4.u $$\chi_{1350}(49, \cdot)$$ n/a 972 6
1350.4.w $$\chi_{1350}(53, \cdot)$$ n/a 960 8
1350.4.z $$\chi_{1350}(19, \cdot)$$ n/a 720 8
1350.4.bb $$\chi_{1350}(257, \cdot)$$ n/a 1944 12
1350.4.bc $$\chi_{1350}(31, \cdot)$$ n/a 6480 24
1350.4.bd $$\chi_{1350}(17, \cdot)$$ n/a 1440 16
1350.4.bf $$\chi_{1350}(79, \cdot)$$ n/a 6480 24
1350.4.bi $$\chi_{1350}(23, \cdot)$$ n/a 12960 48

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1350))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1350)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(225))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(270))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(450))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(675))$$$$^{\oplus 2}$$