# Properties

 Label 735.2 Level 735 Weight 2 Dimension 12120 Nonzero newspaces 24 Newform subspaces 118 Sturm bound 75264 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Newform subspaces: $$118$$ Sturm bound: $$75264$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(735))$$.

Total New Old
Modular forms 19776 12704 7072
Cusp forms 17857 12120 5737
Eisenstein series 1919 584 1335

## Trace form

 $$12120 q - 4 q^{2} - 28 q^{3} - 36 q^{4} + 12 q^{5} - 56 q^{6} - 56 q^{7} + 60 q^{8} - 2 q^{9} + O(q^{10})$$ $$12120 q - 4 q^{2} - 28 q^{3} - 36 q^{4} + 12 q^{5} - 56 q^{6} - 56 q^{7} + 60 q^{8} - 2 q^{9} - 34 q^{10} + 56 q^{11} + 16 q^{12} + 4 q^{13} + 48 q^{14} - 71 q^{15} - 68 q^{16} + 32 q^{17} - 22 q^{18} + 4 q^{19} + 64 q^{20} - 94 q^{21} + 4 q^{22} + 72 q^{23} - 60 q^{24} - 54 q^{25} + 32 q^{26} - 76 q^{27} - 24 q^{28} + 16 q^{29} - 137 q^{30} - 132 q^{31} - 92 q^{32} - 74 q^{33} - 92 q^{34} - 30 q^{35} - 346 q^{36} - 180 q^{37} - 208 q^{38} - 192 q^{39} - 462 q^{40} - 200 q^{41} - 246 q^{42} - 204 q^{43} - 440 q^{44} - 141 q^{45} - 540 q^{46} - 40 q^{47} - 454 q^{48} - 300 q^{49} - 52 q^{50} - 218 q^{51} - 572 q^{52} - 40 q^{53} - 20 q^{54} - 256 q^{55} - 396 q^{56} - 58 q^{57} - 508 q^{58} - 88 q^{59} - 315 q^{60} - 456 q^{61} - 168 q^{62} - 24 q^{63} - 508 q^{64} - 124 q^{65} - 262 q^{66} - 212 q^{67} - 128 q^{68} - 162 q^{69} - 354 q^{70} - 32 q^{71} - 150 q^{72} - 308 q^{73} - 224 q^{74} - 241 q^{75} - 316 q^{76} - 386 q^{78} - 188 q^{79} - 230 q^{80} - 278 q^{81} - 496 q^{82} - 264 q^{83} - 424 q^{84} - 274 q^{85} - 412 q^{86} - 262 q^{87} - 768 q^{88} - 192 q^{89} - 226 q^{90} - 332 q^{91} - 84 q^{92} - 354 q^{93} - 320 q^{94} - 124 q^{95} - 166 q^{96} - 136 q^{97} - 876 q^{98} - 184 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(735))$$

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
735.2.a $$\chi_{735}(1, \cdot)$$ 735.2.a.a 1 1
735.2.a.b 1
735.2.a.c 1
735.2.a.d 1
735.2.a.e 1
735.2.a.f 1
735.2.a.g 2
735.2.a.h 2
735.2.a.i 2
735.2.a.j 2
735.2.a.k 2
735.2.a.l 2
735.2.a.m 2
735.2.a.n 4
735.2.a.o 4
735.2.b $$\chi_{735}(146, \cdot)$$ 735.2.b.a 2 1
735.2.b.b 2
735.2.b.c 8
735.2.b.d 8
735.2.b.e 16
735.2.b.f 16
735.2.d $$\chi_{735}(589, \cdot)$$ 735.2.d.a 2 1
735.2.d.b 6
735.2.d.c 8
735.2.d.d 8
735.2.d.e 8
735.2.d.f 8
735.2.g $$\chi_{735}(734, \cdot)$$ 735.2.g.a 16 1
735.2.g.b 24
735.2.g.c 32
735.2.i $$\chi_{735}(226, \cdot)$$ 735.2.i.a 2 2
735.2.i.b 2
735.2.i.c 2
735.2.i.d 2
735.2.i.e 2
735.2.i.f 2
735.2.i.g 4
735.2.i.h 4
735.2.i.i 4
735.2.i.j 4
735.2.i.k 4
735.2.i.l 4
735.2.i.m 8
735.2.i.n 8
735.2.j $$\chi_{735}(197, \cdot)$$ 735.2.j.a 8 2
735.2.j.b 8
735.2.j.c 16
735.2.j.d 16
735.2.j.e 24
735.2.j.f 24
735.2.j.g 24
735.2.j.h 24
735.2.m $$\chi_{735}(97, \cdot)$$ 735.2.m.a 24 2
735.2.m.b 24
735.2.m.c 32
735.2.p $$\chi_{735}(374, \cdot)$$ 735.2.p.a 8 2
735.2.p.b 8
735.2.p.c 8
735.2.p.d 16
735.2.p.e 16
735.2.p.f 24
735.2.p.g 64
735.2.q $$\chi_{735}(79, \cdot)$$ 735.2.q.a 4 2
735.2.q.b 4
735.2.q.c 8
735.2.q.d 8
735.2.q.e 12
735.2.q.f 12
735.2.q.g 16
735.2.q.h 16
735.2.s $$\chi_{735}(521, \cdot)$$ 735.2.s.a 2 2
735.2.s.b 2
735.2.s.c 2
735.2.s.d 2
735.2.s.e 2
735.2.s.f 2
735.2.s.g 4
735.2.s.h 4
735.2.s.i 4
735.2.s.j 4
735.2.s.k 8
735.2.s.l 8
735.2.s.m 32
735.2.s.n 32
735.2.u $$\chi_{735}(106, \cdot)$$ 735.2.u.a 48 6
735.2.u.b 48
735.2.u.c 60
735.2.u.d 60
735.2.v $$\chi_{735}(178, \cdot)$$ 735.2.v.a 32 4
735.2.v.b 32
735.2.v.c 48
735.2.v.d 48
735.2.y $$\chi_{735}(128, \cdot)$$ 735.2.y.a 8 4
735.2.y.b 8
735.2.y.c 8
735.2.y.d 8
735.2.y.e 32
735.2.y.f 32
735.2.y.g 48
735.2.y.h 48
735.2.y.i 48
735.2.y.j 48
735.2.ba $$\chi_{735}(104, \cdot)$$ 735.2.ba.a 648 6
735.2.bd $$\chi_{735}(64, \cdot)$$ 735.2.bd.a 336 6
735.2.bf $$\chi_{735}(41, \cdot)$$ 735.2.bf.a 228 6
735.2.bf.b 228
735.2.bg $$\chi_{735}(16, \cdot)$$ 735.2.bg.a 108 12
735.2.bg.b 108
735.2.bg.c 120
735.2.bg.d 120
735.2.bi $$\chi_{735}(13, \cdot)$$ 735.2.bi.a 672 12
735.2.bj $$\chi_{735}(8, \cdot)$$ 735.2.bj.a 1296 12
735.2.bm $$\chi_{735}(26, \cdot)$$ 735.2.bm.a 444 12
735.2.bm.b 444
735.2.bo $$\chi_{735}(4, \cdot)$$ 735.2.bo.a 672 12
735.2.bp $$\chi_{735}(59, \cdot)$$ 735.2.bp.a 1296 12
735.2.bt $$\chi_{735}(2, \cdot)$$ 735.2.bt.a 2592 24
735.2.bu $$\chi_{735}(52, \cdot)$$ 735.2.bu.a 1344 24

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(735))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(735)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(105))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(147))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(245))$$$$^{\oplus 2}$$