# Properties

 Label 1575.2 Level 1575 Weight 2 Dimension 59493 Nonzero newspaces 60 Sturm bound 345600 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$60$$ Sturm bound: $$345600$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1575))$$.

Total New Old
Modular forms 89088 61337 27751
Cusp forms 83713 59493 24220
Eisenstein series 5375 1844 3531

## Trace form

 $$59493q$$ $$\mathstrut -\mathstrut 85q^{2}$$ $$\mathstrut -\mathstrut 108q^{3}$$ $$\mathstrut -\mathstrut 105q^{4}$$ $$\mathstrut -\mathstrut 102q^{5}$$ $$\mathstrut -\mathstrut 172q^{6}$$ $$\mathstrut -\mathstrut 115q^{7}$$ $$\mathstrut -\mathstrut 195q^{8}$$ $$\mathstrut -\mathstrut 92q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$59493q$$ $$\mathstrut -\mathstrut 85q^{2}$$ $$\mathstrut -\mathstrut 108q^{3}$$ $$\mathstrut -\mathstrut 105q^{4}$$ $$\mathstrut -\mathstrut 102q^{5}$$ $$\mathstrut -\mathstrut 172q^{6}$$ $$\mathstrut -\mathstrut 115q^{7}$$ $$\mathstrut -\mathstrut 195q^{8}$$ $$\mathstrut -\mathstrut 92q^{9}$$ $$\mathstrut -\mathstrut 282q^{10}$$ $$\mathstrut -\mathstrut 106q^{11}$$ $$\mathstrut -\mathstrut 32q^{12}$$ $$\mathstrut -\mathstrut 72q^{13}$$ $$\mathstrut -\mathstrut 57q^{14}$$ $$\mathstrut -\mathstrut 296q^{15}$$ $$\mathstrut -\mathstrut 65q^{16}$$ $$\mathstrut -\mathstrut 8q^{17}$$ $$\mathstrut +\mathstrut 12q^{18}$$ $$\mathstrut -\mathstrut 200q^{19}$$ $$\mathstrut +\mathstrut 36q^{20}$$ $$\mathstrut -\mathstrut 162q^{21}$$ $$\mathstrut -\mathstrut 44q^{22}$$ $$\mathstrut +\mathstrut 46q^{23}$$ $$\mathstrut +\mathstrut 104q^{24}$$ $$\mathstrut -\mathstrut 6q^{25}$$ $$\mathstrut -\mathstrut 24q^{26}$$ $$\mathstrut +\mathstrut 6q^{27}$$ $$\mathstrut -\mathstrut 95q^{28}$$ $$\mathstrut +\mathstrut 2q^{29}$$ $$\mathstrut -\mathstrut 48q^{30}$$ $$\mathstrut -\mathstrut 26q^{31}$$ $$\mathstrut +\mathstrut 257q^{32}$$ $$\mathstrut -\mathstrut 20q^{33}$$ $$\mathstrut +\mathstrut 190q^{34}$$ $$\mathstrut -\mathstrut 76q^{35}$$ $$\mathstrut -\mathstrut 428q^{36}$$ $$\mathstrut -\mathstrut 68q^{37}$$ $$\mathstrut +\mathstrut 98q^{38}$$ $$\mathstrut -\mathstrut 168q^{39}$$ $$\mathstrut +\mathstrut 42q^{40}$$ $$\mathstrut -\mathstrut 124q^{41}$$ $$\mathstrut -\mathstrut 122q^{42}$$ $$\mathstrut -\mathstrut 164q^{43}$$ $$\mathstrut -\mathstrut 18q^{44}$$ $$\mathstrut -\mathstrut 168q^{45}$$ $$\mathstrut -\mathstrut 184q^{46}$$ $$\mathstrut -\mathstrut 86q^{47}$$ $$\mathstrut -\mathstrut 328q^{48}$$ $$\mathstrut -\mathstrut 35q^{49}$$ $$\mathstrut -\mathstrut 110q^{50}$$ $$\mathstrut -\mathstrut 332q^{51}$$ $$\mathstrut +\mathstrut 284q^{52}$$ $$\mathstrut +\mathstrut 8q^{53}$$ $$\mathstrut -\mathstrut 122q^{54}$$ $$\mathstrut -\mathstrut 124q^{55}$$ $$\mathstrut -\mathstrut 105q^{56}$$ $$\mathstrut -\mathstrut 216q^{57}$$ $$\mathstrut +\mathstrut 352q^{58}$$ $$\mathstrut +\mathstrut 148q^{59}$$ $$\mathstrut -\mathstrut 232q^{60}$$ $$\mathstrut +\mathstrut 74q^{61}$$ $$\mathstrut +\mathstrut 88q^{62}$$ $$\mathstrut -\mathstrut 180q^{63}$$ $$\mathstrut -\mathstrut 531q^{64}$$ $$\mathstrut -\mathstrut 178q^{65}$$ $$\mathstrut -\mathstrut 38q^{66}$$ $$\mathstrut -\mathstrut 12q^{67}$$ $$\mathstrut -\mathstrut 488q^{68}$$ $$\mathstrut -\mathstrut 140q^{69}$$ $$\mathstrut -\mathstrut 114q^{70}$$ $$\mathstrut -\mathstrut 152q^{71}$$ $$\mathstrut -\mathstrut 594q^{72}$$ $$\mathstrut -\mathstrut 186q^{73}$$ $$\mathstrut -\mathstrut 440q^{74}$$ $$\mathstrut -\mathstrut 448q^{75}$$ $$\mathstrut -\mathstrut 138q^{76}$$ $$\mathstrut -\mathstrut 162q^{77}$$ $$\mathstrut -\mathstrut 756q^{78}$$ $$\mathstrut -\mathstrut 108q^{79}$$ $$\mathstrut -\mathstrut 1030q^{80}$$ $$\mathstrut -\mathstrut 344q^{81}$$ $$\mathstrut -\mathstrut 406q^{82}$$ $$\mathstrut -\mathstrut 514q^{83}$$ $$\mathstrut -\mathstrut 644q^{84}$$ $$\mathstrut -\mathstrut 282q^{85}$$ $$\mathstrut -\mathstrut 634q^{86}$$ $$\mathstrut -\mathstrut 506q^{87}$$ $$\mathstrut -\mathstrut 450q^{88}$$ $$\mathstrut -\mathstrut 690q^{89}$$ $$\mathstrut -\mathstrut 752q^{90}$$ $$\mathstrut -\mathstrut 704q^{91}$$ $$\mathstrut -\mathstrut 1312q^{92}$$ $$\mathstrut -\mathstrut 578q^{93}$$ $$\mathstrut -\mathstrut 230q^{94}$$ $$\mathstrut -\mathstrut 292q^{95}$$ $$\mathstrut -\mathstrut 1016q^{96}$$ $$\mathstrut -\mathstrut 136q^{97}$$ $$\mathstrut -\mathstrut 975q^{98}$$ $$\mathstrut -\mathstrut 600q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1575))$$

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 the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
1575.2.a $$\chi_{1575}(1, \cdot)$$ 1575.2.a.a 1 1
1575.2.a.b 1
1575.2.a.c 1
1575.2.a.d 1
1575.2.a.e 1
1575.2.a.f 1
1575.2.a.g 1
1575.2.a.h 1
1575.2.a.i 1
1575.2.a.j 1
1575.2.a.k 1
1575.2.a.l 2
1575.2.a.m 2
1575.2.a.n 2
1575.2.a.o 2
1575.2.a.p 2
1575.2.a.q 2
1575.2.a.r 2
1575.2.a.s 2
1575.2.a.t 2
1575.2.a.u 2
1575.2.a.v 2
1575.2.a.w 3
1575.2.a.x 3
1575.2.a.y 4
1575.2.a.z 4
1575.2.b $$\chi_{1575}(251, \cdot)$$ 1575.2.b.a 4 1
1575.2.b.b 4
1575.2.b.c 4
1575.2.b.d 4
1575.2.b.e 4
1575.2.b.f 8
1575.2.b.g 8
1575.2.b.h 16
1575.2.d $$\chi_{1575}(1324, \cdot)$$ 1575.2.d.a 2 1
1575.2.d.b 2
1575.2.d.c 2
1575.2.d.d 4
1575.2.d.e 4
1575.2.d.f 4
1575.2.d.g 4
1575.2.d.h 4
1575.2.d.i 4
1575.2.d.j 4
1575.2.d.k 4
1575.2.d.l 8
1575.2.g $$\chi_{1575}(1574, \cdot)$$ 1575.2.g.a 8 1
1575.2.g.b 8
1575.2.g.c 8
1575.2.g.d 8
1575.2.g.e 16
1575.2.i $$\chi_{1575}(526, \cdot)$$ n/a 228 2
1575.2.j $$\chi_{1575}(226, \cdot)$$ n/a 120 2
1575.2.k $$\chi_{1575}(1201, \cdot)$$ n/a 292 2
1575.2.l $$\chi_{1575}(151, \cdot)$$ n/a 292 2
1575.2.m $$\chi_{1575}(1268, \cdot)$$ 1575.2.m.a 8 2
1575.2.m.b 8
1575.2.m.c 12
1575.2.m.d 12
1575.2.m.e 16
1575.2.m.f 16
1575.2.p $$\chi_{1575}(118, \cdot)$$ n/a 116 2
1575.2.q $$\chi_{1575}(316, \cdot)$$ n/a 304 4
1575.2.s $$\chi_{1575}(499, \cdot)$$ n/a 280 2
1575.2.u $$\chi_{1575}(101, \cdot)$$ n/a 292 2
1575.2.v $$\chi_{1575}(299, \cdot)$$ n/a 280 2
1575.2.ba $$\chi_{1575}(524, \cdot)$$ n/a 280 2
1575.2.bc $$\chi_{1575}(899, \cdot)$$ 1575.2.bc.a 8 2
1575.2.bc.b 8
1575.2.bc.c 24
1575.2.bc.d 24
1575.2.bc.e 32
1575.2.bf $$\chi_{1575}(551, \cdot)$$ n/a 292 2
1575.2.bg $$\chi_{1575}(424, \cdot)$$ n/a 116 2
1575.2.bi $$\chi_{1575}(274, \cdot)$$ n/a 216 2
1575.2.bk $$\chi_{1575}(26, \cdot)$$ 1575.2.bk.a 4 2
1575.2.bk.b 4
1575.2.bk.c 4
1575.2.bk.d 8
1575.2.bk.e 12
1575.2.bk.f 12
1575.2.bk.g 16
1575.2.bk.h 16
1575.2.bk.i 24
1575.2.bm $$\chi_{1575}(776, \cdot)$$ n/a 292 2
1575.2.bp $$\chi_{1575}(949, \cdot)$$ n/a 280 2
1575.2.br $$\chi_{1575}(824, \cdot)$$ n/a 280 2
1575.2.bu $$\chi_{1575}(314, \cdot)$$ n/a 320 4
1575.2.bx $$\chi_{1575}(64, \cdot)$$ n/a 296 4
1575.2.bz $$\chi_{1575}(566, \cdot)$$ n/a 320 4
1575.2.ca $$\chi_{1575}(418, \cdot)$$ n/a 560 4
1575.2.cd $$\chi_{1575}(893, \cdot)$$ n/a 560 4
1575.2.cf $$\chi_{1575}(32, \cdot)$$ n/a 560 4
1575.2.ch $$\chi_{1575}(82, \cdot)$$ n/a 232 4
1575.2.cj $$\chi_{1575}(643, \cdot)$$ n/a 560 4
1575.2.ck $$\chi_{1575}(218, \cdot)$$ n/a 432 4
1575.2.cm $$\chi_{1575}(107, \cdot)$$ n/a 192 4
1575.2.co $$\chi_{1575}(157, \cdot)$$ n/a 560 4
1575.2.cq $$\chi_{1575}(121, \cdot)$$ n/a 1888 8
1575.2.cr $$\chi_{1575}(16, \cdot)$$ n/a 1888 8
1575.2.cs $$\chi_{1575}(46, \cdot)$$ n/a 784 8
1575.2.ct $$\chi_{1575}(106, \cdot)$$ n/a 1440 8
1575.2.cu $$\chi_{1575}(433, \cdot)$$ n/a 784 8
1575.2.cx $$\chi_{1575}(8, \cdot)$$ n/a 480 8
1575.2.cz $$\chi_{1575}(164, \cdot)$$ n/a 1888 8
1575.2.db $$\chi_{1575}(4, \cdot)$$ n/a 1888 8
1575.2.de $$\chi_{1575}(41, \cdot)$$ n/a 1888 8
1575.2.dg $$\chi_{1575}(206, \cdot)$$ n/a 640 8
1575.2.di $$\chi_{1575}(169, \cdot)$$ n/a 1440 8
1575.2.dk $$\chi_{1575}(109, \cdot)$$ n/a 784 8
1575.2.dl $$\chi_{1575}(236, \cdot)$$ n/a 1888 8
1575.2.do $$\chi_{1575}(89, \cdot)$$ n/a 640 8
1575.2.dq $$\chi_{1575}(104, \cdot)$$ n/a 1888 8
1575.2.dv $$\chi_{1575}(59, \cdot)$$ n/a 1888 8
1575.2.dw $$\chi_{1575}(131, \cdot)$$ n/a 1888 8
1575.2.dy $$\chi_{1575}(184, \cdot)$$ n/a 1888 8
1575.2.eb $$\chi_{1575}(187, \cdot)$$ n/a 3776 16
1575.2.ed $$\chi_{1575}(53, \cdot)$$ n/a 1280 16
1575.2.ef $$\chi_{1575}(92, \cdot)$$ n/a 2880 16
1575.2.eg $$\chi_{1575}(13, \cdot)$$ n/a 3776 16
1575.2.ei $$\chi_{1575}(73, \cdot)$$ n/a 1568 16
1575.2.ek $$\chi_{1575}(2, \cdot)$$ n/a 3776 16
1575.2.em $$\chi_{1575}(23, \cdot)$$ n/a 3776 16
1575.2.ep $$\chi_{1575}(52, \cdot)$$ n/a 3776 16

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1575))$$ into lower level spaces

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