# Properties

 Label 1575.4 Level 1575 Weight 4 Dimension 180323 Nonzero newspaces 60 Sturm bound 691200 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$60$$ Sturm bound: $$691200$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1575))$$.

Total New Old
Modular forms 261888 182167 79721
Cusp forms 256512 180323 76189
Eisenstein series 5376 1844 3532

## Trace form

 $$180323 q - 83 q^{2} - 102 q^{3} - 15 q^{4} - 102 q^{5} - 190 q^{6} - 72 q^{7} - 93 q^{8} + 58 q^{9} + O(q^{10})$$ $$180323 q - 83 q^{2} - 102 q^{3} - 15 q^{4} - 102 q^{5} - 190 q^{6} - 72 q^{7} - 93 q^{8} + 58 q^{9} - 52 q^{10} + 19 q^{11} - 260 q^{12} - 728 q^{13} - 411 q^{14} - 656 q^{15} - 1475 q^{16} - 703 q^{17} - 876 q^{18} + 325 q^{19} + 1596 q^{20} + 543 q^{21} + 3066 q^{22} + 2027 q^{23} + 938 q^{24} - 1386 q^{25} - 1398 q^{26} - 1056 q^{27} - 2017 q^{28} - 5120 q^{29} - 2448 q^{30} - 2983 q^{31} - 6425 q^{32} - 4208 q^{33} + 918 q^{34} - 1016 q^{35} - 4346 q^{36} + 3001 q^{37} + 6688 q^{38} + 3258 q^{39} + 7752 q^{40} + 7144 q^{41} + 4456 q^{42} - 616 q^{43} + 11226 q^{44} + 4392 q^{45} - 242 q^{46} + 12623 q^{47} + 16058 q^{48} + 3410 q^{49} - 3400 q^{50} + 238 q^{51} - 3162 q^{52} - 8225 q^{53} - 14624 q^{54} - 564 q^{55} + 645 q^{56} - 17214 q^{57} - 3816 q^{58} - 16771 q^{59} - 28072 q^{60} - 14479 q^{61} - 26320 q^{62} - 3519 q^{63} - 23411 q^{64} - 10418 q^{65} + 4354 q^{66} - 5757 q^{67} + 10634 q^{68} + 16234 q^{69} + 2166 q^{70} + 8312 q^{71} + 19968 q^{72} + 26227 q^{73} + 20438 q^{74} + 12992 q^{75} + 21554 q^{76} - 1266 q^{77} - 10704 q^{78} + 7983 q^{79} + 32680 q^{80} - 1154 q^{81} + 14842 q^{82} + 24226 q^{83} - 938 q^{84} + 10298 q^{85} + 16156 q^{86} - 3518 q^{87} - 66348 q^{88} - 26289 q^{89} - 18512 q^{90} - 22236 q^{91} - 30212 q^{92} + 19294 q^{93} - 30636 q^{94} - 4772 q^{95} + 42568 q^{96} - 12520 q^{97} + 47433 q^{98} + 33414 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1575))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
1575.4.a $$\chi_{1575}(1, \cdot)$$ 1575.4.a.a 1 1
1575.4.a.b 1
1575.4.a.c 1
1575.4.a.d 1
1575.4.a.e 1
1575.4.a.f 1
1575.4.a.g 1
1575.4.a.h 1
1575.4.a.i 1
1575.4.a.j 1
1575.4.a.k 1
1575.4.a.l 1
1575.4.a.m 2
1575.4.a.n 2
1575.4.a.o 2
1575.4.a.p 2
1575.4.a.q 2
1575.4.a.r 2
1575.4.a.s 2
1575.4.a.t 2
1575.4.a.u 2
1575.4.a.v 2
1575.4.a.w 2
1575.4.a.x 2
1575.4.a.y 2
1575.4.a.z 2
1575.4.a.ba 3
1575.4.a.bb 3
1575.4.a.bc 3
1575.4.a.bd 3
1575.4.a.be 3
1575.4.a.bf 4
1575.4.a.bg 4
1575.4.a.bh 4
1575.4.a.bi 4
1575.4.a.bj 4
1575.4.a.bk 4
1575.4.a.bl 4
1575.4.a.bm 4
1575.4.a.bn 5
1575.4.a.bo 5
1575.4.a.bp 5
1575.4.a.bq 5
1575.4.a.br 8
1575.4.a.bs 8
1575.4.a.bt 10
1575.4.a.bu 10
1575.4.b $$\chi_{1575}(251, \cdot)$$ n/a 152 1
1575.4.d $$\chi_{1575}(1324, \cdot)$$ n/a 134 1
1575.4.g $$\chi_{1575}(1574, \cdot)$$ n/a 144 1
1575.4.i $$\chi_{1575}(526, \cdot)$$ n/a 684 2
1575.4.j $$\chi_{1575}(226, \cdot)$$ n/a 374 2
1575.4.k $$\chi_{1575}(1201, \cdot)$$ n/a 900 2
1575.4.l $$\chi_{1575}(151, \cdot)$$ n/a 900 2
1575.4.m $$\chi_{1575}(1268, \cdot)$$ n/a 216 2
1575.4.p $$\chi_{1575}(118, \cdot)$$ n/a 356 2
1575.4.q $$\chi_{1575}(316, \cdot)$$ n/a 896 4
1575.4.s $$\chi_{1575}(499, \cdot)$$ n/a 856 2
1575.4.u $$\chi_{1575}(101, \cdot)$$ n/a 900 2
1575.4.v $$\chi_{1575}(299, \cdot)$$ n/a 856 2
1575.4.ba $$\chi_{1575}(524, \cdot)$$ n/a 856 2
1575.4.bc $$\chi_{1575}(899, \cdot)$$ n/a 288 2
1575.4.bf $$\chi_{1575}(551, \cdot)$$ n/a 900 2
1575.4.bg $$\chi_{1575}(424, \cdot)$$ n/a 356 2
1575.4.bi $$\chi_{1575}(274, \cdot)$$ n/a 648 2
1575.4.bk $$\chi_{1575}(26, \cdot)$$ n/a 304 2
1575.4.bm $$\chi_{1575}(776, \cdot)$$ n/a 900 2
1575.4.bp $$\chi_{1575}(949, \cdot)$$ n/a 856 2
1575.4.br $$\chi_{1575}(824, \cdot)$$ n/a 856 2
1575.4.bu $$\chi_{1575}(314, \cdot)$$ n/a 960 4
1575.4.bx $$\chi_{1575}(64, \cdot)$$ n/a 904 4
1575.4.bz $$\chi_{1575}(566, \cdot)$$ n/a 960 4
1575.4.ca $$\chi_{1575}(418, \cdot)$$ n/a 1712 4
1575.4.cd $$\chi_{1575}(893, \cdot)$$ n/a 1712 4
1575.4.cf $$\chi_{1575}(32, \cdot)$$ n/a 1712 4
1575.4.ch $$\chi_{1575}(82, \cdot)$$ n/a 712 4
1575.4.cj $$\chi_{1575}(643, \cdot)$$ n/a 1712 4
1575.4.ck $$\chi_{1575}(218, \cdot)$$ n/a 1296 4
1575.4.cm $$\chi_{1575}(107, \cdot)$$ n/a 576 4
1575.4.co $$\chi_{1575}(157, \cdot)$$ n/a 1712 4
1575.4.cq $$\chi_{1575}(121, \cdot)$$ n/a 5728 8
1575.4.cr $$\chi_{1575}(16, \cdot)$$ n/a 5728 8
1575.4.cs $$\chi_{1575}(46, \cdot)$$ n/a 2384 8
1575.4.ct $$\chi_{1575}(106, \cdot)$$ n/a 4320 8
1575.4.cu $$\chi_{1575}(433, \cdot)$$ n/a 2384 8
1575.4.cx $$\chi_{1575}(8, \cdot)$$ n/a 1440 8
1575.4.cz $$\chi_{1575}(164, \cdot)$$ n/a 5728 8
1575.4.db $$\chi_{1575}(4, \cdot)$$ n/a 5728 8
1575.4.de $$\chi_{1575}(41, \cdot)$$ n/a 5728 8
1575.4.dg $$\chi_{1575}(206, \cdot)$$ n/a 1920 8
1575.4.di $$\chi_{1575}(169, \cdot)$$ n/a 4320 8
1575.4.dk $$\chi_{1575}(109, \cdot)$$ n/a 2384 8
1575.4.dl $$\chi_{1575}(236, \cdot)$$ n/a 5728 8
1575.4.do $$\chi_{1575}(89, \cdot)$$ n/a 1920 8
1575.4.dq $$\chi_{1575}(104, \cdot)$$ n/a 5728 8
1575.4.dv $$\chi_{1575}(59, \cdot)$$ n/a 5728 8
1575.4.dw $$\chi_{1575}(131, \cdot)$$ n/a 5728 8
1575.4.dy $$\chi_{1575}(184, \cdot)$$ n/a 5728 8
1575.4.eb $$\chi_{1575}(187, \cdot)$$ n/a 11456 16
1575.4.ed $$\chi_{1575}(53, \cdot)$$ n/a 3840 16
1575.4.ef $$\chi_{1575}(92, \cdot)$$ n/a 8640 16
1575.4.eg $$\chi_{1575}(13, \cdot)$$ n/a 11456 16
1575.4.ei $$\chi_{1575}(73, \cdot)$$ n/a 4768 16
1575.4.ek $$\chi_{1575}(2, \cdot)$$ n/a 11456 16
1575.4.em $$\chi_{1575}(23, \cdot)$$ n/a 11456 16
1575.4.ep $$\chi_{1575}(52, \cdot)$$ n/a 11456 16

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1575))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1575)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 18}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(105))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(175))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(225))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(315))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(525))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(1575))$$$$^{\oplus 1}$$