# Properties

 Label 840.2 Level 840 Weight 2 Dimension 6712 Nonzero newspaces 36 Newform subspaces 110 Sturm bound 73728 Trace bound 12

## Defining parameters

 Level: $$N$$ = $$840 = 2^{3} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$36$$ Newform subspaces: $$110$$ Sturm bound: $$73728$$ Trace bound: $$12$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(840))$$.

Total New Old
Modular forms 19584 6952 12632
Cusp forms 17281 6712 10569
Eisenstein series 2303 240 2063

## Trace form

 $$6712 q - 8 q^{2} - 16 q^{3} - 24 q^{4} - 8 q^{5} - 12 q^{6} - 32 q^{7} + 16 q^{8} - 36 q^{9} + O(q^{10})$$ $$6712 q - 8 q^{2} - 16 q^{3} - 24 q^{4} - 8 q^{5} - 12 q^{6} - 32 q^{7} + 16 q^{8} - 36 q^{9} - 12 q^{10} - 24 q^{11} + 52 q^{12} - 40 q^{13} + 32 q^{14} - 12 q^{15} + 24 q^{16} - 24 q^{17} + 28 q^{18} + 56 q^{19} + 100 q^{20} + 16 q^{21} + 136 q^{22} + 120 q^{23} + 76 q^{24} - 20 q^{25} + 152 q^{26} + 104 q^{27} + 168 q^{28} + 48 q^{29} + 38 q^{30} + 168 q^{31} + 72 q^{32} + 116 q^{33} + 88 q^{34} + 84 q^{35} - 32 q^{36} + 104 q^{37} - 8 q^{38} + 112 q^{39} - 104 q^{40} - 116 q^{42} + 48 q^{43} - 208 q^{44} + 18 q^{45} - 320 q^{46} + 112 q^{47} - 276 q^{48} + 48 q^{49} - 128 q^{50} - 224 q^{52} + 40 q^{53} - 276 q^{54} + 116 q^{55} - 184 q^{56} + 48 q^{57} - 176 q^{58} + 192 q^{59} - 264 q^{60} + 48 q^{61} - 232 q^{62} + 56 q^{63} - 312 q^{64} + 176 q^{65} - 356 q^{66} + 168 q^{67} - 232 q^{68} + 48 q^{69} - 156 q^{70} + 64 q^{71} - 384 q^{72} + 208 q^{73} - 112 q^{74} - 84 q^{75} - 136 q^{76} + 72 q^{77} - 336 q^{78} - 56 q^{79} - 160 q^{80} - 36 q^{81} - 280 q^{82} - 160 q^{83} - 300 q^{84} + 152 q^{85} + 32 q^{86} - 240 q^{87} + 72 q^{88} + 32 q^{89} - 204 q^{90} - 224 q^{91} - 8 q^{92} - 68 q^{93} + 88 q^{94} - 168 q^{95} - 100 q^{96} + 168 q^{97} - 40 q^{98} - 416 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(840))$$

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
840.2.a $$\chi_{840}(1, \cdot)$$ 840.2.a.a 1 1
840.2.a.b 1
840.2.a.c 1
840.2.a.d 1
840.2.a.e 1
840.2.a.f 1
840.2.a.g 1
840.2.a.h 1
840.2.a.i 1
840.2.a.j 1
840.2.a.k 2
840.2.d $$\chi_{840}(391, \cdot)$$ None 0 1
840.2.e $$\chi_{840}(491, \cdot)$$ 840.2.e.a 4 1
840.2.e.b 4
840.2.e.c 44
840.2.e.d 44
840.2.f $$\chi_{840}(41, \cdot)$$ 840.2.f.a 16 1
840.2.f.b 16
840.2.g $$\chi_{840}(421, \cdot)$$ 840.2.g.a 8 1
840.2.g.b 12
840.2.g.c 12
840.2.g.d 16
840.2.j $$\chi_{840}(589, \cdot)$$ 840.2.j.a 2 1
840.2.j.b 2
840.2.j.c 2
840.2.j.d 2
840.2.j.e 32
840.2.j.f 32
840.2.k $$\chi_{840}(209, \cdot)$$ 840.2.k.a 24 1
840.2.k.b 24
840.2.p $$\chi_{840}(659, \cdot)$$ 840.2.p.a 72 1
840.2.p.b 72
840.2.q $$\chi_{840}(559, \cdot)$$ None 0 1
840.2.t $$\chi_{840}(169, \cdot)$$ 840.2.t.a 2 1
840.2.t.b 2
840.2.t.c 4
840.2.t.d 6
840.2.t.e 6
840.2.u $$\chi_{840}(629, \cdot)$$ 840.2.u.a 4 1
840.2.u.b 4
840.2.u.c 8
840.2.u.d 8
840.2.u.e 160
840.2.v $$\chi_{840}(239, \cdot)$$ None 0 1
840.2.w $$\chi_{840}(139, \cdot)$$ 840.2.w.a 48 1
840.2.w.b 48
840.2.z $$\chi_{840}(811, \cdot)$$ 840.2.z.a 4 1
840.2.z.b 4
840.2.z.c 28
840.2.z.d 28
840.2.ba $$\chi_{840}(71, \cdot)$$ None 0 1
840.2.bf $$\chi_{840}(461, \cdot)$$ 840.2.bf.a 8 1
840.2.bf.b 120
840.2.bg $$\chi_{840}(121, \cdot)$$ 840.2.bg.a 2 2
840.2.bg.b 2
840.2.bg.c 2
840.2.bg.d 2
840.2.bg.e 2
840.2.bg.f 2
840.2.bg.g 4
840.2.bg.h 4
840.2.bg.i 6
840.2.bg.j 6
840.2.bj $$\chi_{840}(13, \cdot)$$ 840.2.bj.a 8 2
840.2.bj.b 184
840.2.bk $$\chi_{840}(113, \cdot)$$ 840.2.bk.a 4 2
840.2.bk.b 4
840.2.bk.c 32
840.2.bk.d 32
840.2.bl $$\chi_{840}(127, \cdot)$$ None 0 2
840.2.bm $$\chi_{840}(83, \cdot)$$ 840.2.bm.a 368 2
840.2.br $$\chi_{840}(43, \cdot)$$ 840.2.br.a 72 2
840.2.br.b 72
840.2.bs $$\chi_{840}(167, \cdot)$$ None 0 2
840.2.bt $$\chi_{840}(97, \cdot)$$ 840.2.bt.a 24 2
840.2.bt.b 24
840.2.bu $$\chi_{840}(197, \cdot)$$ 840.2.bu.a 288 2
840.2.bz $$\chi_{840}(19, \cdot)$$ 840.2.bz.a 96 2
840.2.bz.b 96
840.2.ca $$\chi_{840}(359, \cdot)$$ None 0 2
840.2.cb $$\chi_{840}(269, \cdot)$$ 840.2.cb.a 8 2
840.2.cb.b 8
840.2.cb.c 352
840.2.cc $$\chi_{840}(289, \cdot)$$ 840.2.cc.a 4 2
840.2.cc.b 20
840.2.cc.c 24
840.2.cf $$\chi_{840}(101, \cdot)$$ 840.2.cf.a 256 2
840.2.ck $$\chi_{840}(191, \cdot)$$ None 0 2
840.2.cl $$\chi_{840}(451, \cdot)$$ 840.2.cl.a 64 2
840.2.cl.b 64
840.2.co $$\chi_{840}(541, \cdot)$$ 840.2.co.a 4 2
840.2.co.b 4
840.2.co.c 60
840.2.co.d 60
840.2.cp $$\chi_{840}(521, \cdot)$$ 840.2.cp.a 32 2
840.2.cp.b 32
840.2.cq $$\chi_{840}(11, \cdot)$$ 840.2.cq.a 128 2
840.2.cq.b 128
840.2.cr $$\chi_{840}(31, \cdot)$$ None 0 2
840.2.cu $$\chi_{840}(199, \cdot)$$ None 0 2
840.2.cv $$\chi_{840}(179, \cdot)$$ 840.2.cv.a 368 2
840.2.da $$\chi_{840}(89, \cdot)$$ 840.2.da.a 48 2
840.2.da.b 48
840.2.db $$\chi_{840}(109, \cdot)$$ 840.2.db.a 4 2
840.2.db.b 4
840.2.db.c 4
840.2.db.d 4
840.2.db.e 88
840.2.db.f 88
840.2.dc $$\chi_{840}(53, \cdot)$$ 840.2.dc.a 8 4
840.2.dc.b 8
840.2.dc.c 8
840.2.dc.d 8
840.2.dc.e 704
840.2.dd $$\chi_{840}(73, \cdot)$$ 840.2.dd.a 48 4
840.2.dd.b 48
840.2.di $$\chi_{840}(47, \cdot)$$ None 0 4
840.2.dj $$\chi_{840}(67, \cdot)$$ 840.2.dj.a 384 4
840.2.dk $$\chi_{840}(227, \cdot)$$ 840.2.dk.a 736 4
840.2.dl $$\chi_{840}(247, \cdot)$$ None 0 4
840.2.dq $$\chi_{840}(137, \cdot)$$ 840.2.dq.a 192 4
840.2.dr $$\chi_{840}(157, \cdot)$$ 840.2.dr.a 384 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(840))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(840)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(105))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(140))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(168))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(210))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(280))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(420))$$$$^{\oplus 2}$$