# Properties

 Label 560.4 Level 560 Weight 4 Dimension 13526 Nonzero newspaces 28 Sturm bound 73728 Trace bound 11

## Defining parameters

 Level: $$N$$ = $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$28$$ Sturm bound: $$73728$$ Trace bound: $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(560))$$.

Total New Old
Modular forms 28320 13798 14522
Cusp forms 26976 13526 13450
Eisenstein series 1344 272 1072

## Trace form

 $$13526 q - 16 q^{2} + 2 q^{3} + 24 q^{4} - 33 q^{5} - 168 q^{6} - 60 q^{7} - 208 q^{8} - 42 q^{9} + O(q^{10})$$ $$13526 q - 16 q^{2} + 2 q^{3} + 24 q^{4} - 33 q^{5} - 168 q^{6} - 60 q^{7} - 208 q^{8} - 42 q^{9} - 156 q^{10} + 218 q^{11} + 392 q^{12} + 260 q^{13} + 360 q^{14} - 282 q^{15} + 1080 q^{16} - 42 q^{17} + 688 q^{18} - 206 q^{19} - 412 q^{20} - 70 q^{21} - 2384 q^{22} - 202 q^{23} - 3400 q^{24} - 825 q^{25} - 1096 q^{26} - 400 q^{27} + 32 q^{28} - 1220 q^{29} - 812 q^{30} + 834 q^{31} + 2504 q^{32} + 730 q^{33} + 2888 q^{34} + 1253 q^{35} + 960 q^{36} + 3454 q^{37} - 792 q^{38} + 2672 q^{39} + 3548 q^{40} + 1572 q^{41} - 3920 q^{42} - 3932 q^{43} - 1800 q^{44} - 932 q^{45} - 96 q^{46} - 4586 q^{47} + 2600 q^{48} + 7226 q^{49} - 3184 q^{50} - 3994 q^{51} + 848 q^{52} - 890 q^{53} + 8840 q^{54} - 1258 q^{55} + 8208 q^{56} + 6180 q^{57} + 8256 q^{58} + 7302 q^{59} + 6500 q^{60} + 6366 q^{61} + 2480 q^{62} + 536 q^{63} + 6888 q^{64} - 5326 q^{65} - 1768 q^{66} - 2878 q^{67} - 9712 q^{68} - 2892 q^{69} - 6516 q^{70} + 4664 q^{71} - 13448 q^{72} + 5550 q^{73} - 10440 q^{74} + 12879 q^{75} - 13528 q^{76} - 4166 q^{77} - 26656 q^{78} - 4526 q^{79} - 17956 q^{80} - 1256 q^{81} - 14632 q^{82} - 2616 q^{83} - 5064 q^{84} + 2610 q^{85} + 10280 q^{86} + 4400 q^{87} + 4872 q^{88} - 1790 q^{89} + 13804 q^{90} - 12528 q^{91} + 19016 q^{92} - 19574 q^{93} + 29752 q^{94} - 21495 q^{95} + 37896 q^{96} - 4956 q^{97} + 17480 q^{98} - 8952 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(560))$$

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
560.4.a $$\chi_{560}(1, \cdot)$$ 560.4.a.a 1 1
560.4.a.b 1
560.4.a.c 1
560.4.a.d 1
560.4.a.e 1
560.4.a.f 1
560.4.a.g 1
560.4.a.h 1
560.4.a.i 1
560.4.a.j 1
560.4.a.k 1
560.4.a.l 1
560.4.a.m 1
560.4.a.n 1
560.4.a.o 1
560.4.a.p 1
560.4.a.q 1
560.4.a.r 2
560.4.a.s 2
560.4.a.t 3
560.4.a.u 3
560.4.a.v 3
560.4.a.w 3
560.4.a.x 3
560.4.b $$\chi_{560}(281, \cdot)$$ None 0 1
560.4.e $$\chi_{560}(559, \cdot)$$ 560.4.e.a 4 1
560.4.e.b 4
560.4.e.c 8
560.4.e.d 16
560.4.e.e 40
560.4.g $$\chi_{560}(449, \cdot)$$ 560.4.g.a 2 1
560.4.g.b 2
560.4.g.c 2
560.4.g.d 4
560.4.g.e 6
560.4.g.f 10
560.4.g.g 12
560.4.g.h 16
560.4.h $$\chi_{560}(391, \cdot)$$ None 0 1
560.4.k $$\chi_{560}(111, \cdot)$$ 560.4.k.a 16 1
560.4.k.b 32
560.4.l $$\chi_{560}(169, \cdot)$$ None 0 1
560.4.n $$\chi_{560}(279, \cdot)$$ None 0 1
560.4.q $$\chi_{560}(81, \cdot)$$ 560.4.q.a 2 2
560.4.q.b 2
560.4.q.c 2
560.4.q.d 2
560.4.q.e 2
560.4.q.f 2
560.4.q.g 2
560.4.q.h 4
560.4.q.i 4
560.4.q.j 4
560.4.q.k 4
560.4.q.l 4
560.4.q.m 6
560.4.q.n 10
560.4.q.o 10
560.4.q.p 12
560.4.q.q 12
560.4.q.r 12
560.4.r $$\chi_{560}(237, \cdot)$$ n/a 568 2
560.4.t $$\chi_{560}(43, \cdot)$$ n/a 432 2
560.4.w $$\chi_{560}(153, \cdot)$$ None 0 2
560.4.x $$\chi_{560}(127, \cdot)$$ n/a 108 2
560.4.bb $$\chi_{560}(29, \cdot)$$ n/a 432 2
560.4.bc $$\chi_{560}(251, \cdot)$$ n/a 384 2
560.4.bd $$\chi_{560}(141, \cdot)$$ n/a 288 2
560.4.be $$\chi_{560}(139, \cdot)$$ n/a 568 2
560.4.bi $$\chi_{560}(183, \cdot)$$ None 0 2
560.4.bj $$\chi_{560}(97, \cdot)$$ n/a 140 2
560.4.bl $$\chi_{560}(267, \cdot)$$ n/a 432 2
560.4.bn $$\chi_{560}(13, \cdot)$$ n/a 568 2
560.4.bq $$\chi_{560}(199, \cdot)$$ None 0 2
560.4.bs $$\chi_{560}(31, \cdot)$$ 560.4.bs.a 32 2
560.4.bs.b 32
560.4.bs.c 32
560.4.bv $$\chi_{560}(9, \cdot)$$ None 0 2
560.4.bw $$\chi_{560}(289, \cdot)$$ n/a 140 2
560.4.bz $$\chi_{560}(311, \cdot)$$ None 0 2
560.4.cb $$\chi_{560}(121, \cdot)$$ None 0 2
560.4.cc $$\chi_{560}(159, \cdot)$$ n/a 144 2
560.4.cf $$\chi_{560}(107, \cdot)$$ n/a 1136 4
560.4.ch $$\chi_{560}(117, \cdot)$$ n/a 1136 4
560.4.ci $$\chi_{560}(17, \cdot)$$ n/a 280 4
560.4.cl $$\chi_{560}(23, \cdot)$$ None 0 4
560.4.co $$\chi_{560}(19, \cdot)$$ n/a 1136 4
560.4.cp $$\chi_{560}(221, \cdot)$$ n/a 768 4
560.4.cq $$\chi_{560}(131, \cdot)$$ n/a 768 4
560.4.cr $$\chi_{560}(109, \cdot)$$ n/a 1136 4
560.4.cu $$\chi_{560}(207, \cdot)$$ n/a 288 4
560.4.cx $$\chi_{560}(73, \cdot)$$ None 0 4
560.4.cz $$\chi_{560}(157, \cdot)$$ n/a 1136 4
560.4.db $$\chi_{560}(67, \cdot)$$ n/a 1136 4

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(560))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(560)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 20}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(140))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(280))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(560))$$$$^{\oplus 1}$$