# Properties

 Label 840.1 Level 840 Weight 1 Dimension 48 Nonzero newspaces 3 Newform subspaces 8 Sturm bound 36864 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$840 = 2^{3} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$3$$ Newform subspaces: $$8$$ Sturm bound: $$36864$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(840))$$.

Total New Old
Modular forms 1288 168 1120
Cusp forms 136 48 88
Eisenstein series 1152 120 1032

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 48 0 0 0

## Trace form

 $$48q + 4q^{4} - 8q^{6} + 4q^{9} + O(q^{10})$$ $$48q + 4q^{4} - 8q^{6} + 4q^{9} - 10q^{10} - 4q^{15} - 20q^{16} + 8q^{18} - 4q^{24} - 2q^{25} - 4q^{31} - 12q^{33} - 16q^{36} - 2q^{40} - 4q^{49} - 4q^{54} + 12q^{55} + 8q^{57} - 12q^{58} - 10q^{60} + 4q^{63} - 8q^{64} - 6q^{70} - 8q^{72} - 8q^{78} + 4q^{79} - 4q^{81} + 12q^{87} + 12q^{88} + 4q^{90} + 4q^{96} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(840))$$

We only show spaces with odd 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
840.1.b $$\chi_{840}(419, \cdot)$$ None 0 1
840.1.c $$\chi_{840}(799, \cdot)$$ None 0 1
840.1.h $$\chi_{840}(349, \cdot)$$ None 0 1
840.1.i $$\chi_{840}(449, \cdot)$$ None 0 1
840.1.l $$\chi_{840}(281, \cdot)$$ None 0 1
840.1.m $$\chi_{840}(181, \cdot)$$ None 0 1
840.1.n $$\chi_{840}(631, \cdot)$$ None 0 1
840.1.o $$\chi_{840}(251, \cdot)$$ None 0 1
840.1.r $$\chi_{840}(701, \cdot)$$ None 0 1
840.1.s $$\chi_{840}(601, \cdot)$$ None 0 1
840.1.x $$\chi_{840}(211, \cdot)$$ None 0 1
840.1.y $$\chi_{840}(671, \cdot)$$ None 0 1
840.1.bb $$\chi_{840}(839, \cdot)$$ None 0 1
840.1.bc $$\chi_{840}(379, \cdot)$$ None 0 1
840.1.bd $$\chi_{840}(769, \cdot)$$ None 0 1
840.1.be $$\chi_{840}(29, \cdot)$$ None 0 1
840.1.bh $$\chi_{840}(323, \cdot)$$ None 0 2
840.1.bi $$\chi_{840}(223, \cdot)$$ None 0 2
840.1.bn $$\chi_{840}(377, \cdot)$$ None 0 2
840.1.bo $$\chi_{840}(253, \cdot)$$ None 0 2
840.1.bp $$\chi_{840}(293, \cdot)$$ 840.1.bp.a 4 2
840.1.bp.b 4
840.1.bp.c 8
840.1.bp.d 8
840.1.bq $$\chi_{840}(337, \cdot)$$ None 0 2
840.1.bv $$\chi_{840}(407, \cdot)$$ None 0 2
840.1.bw $$\chi_{840}(307, \cdot)$$ None 0 2
840.1.bx $$\chi_{840}(311, \cdot)$$ None 0 2
840.1.by $$\chi_{840}(331, \cdot)$$ None 0 2
840.1.cd $$\chi_{840}(241, \cdot)$$ None 0 2
840.1.ce $$\chi_{840}(221, \cdot)$$ None 0 2
840.1.cg $$\chi_{840}(149, \cdot)$$ 840.1.cg.a 4 2
840.1.cg.b 4
840.1.ch $$\chi_{840}(409, \cdot)$$ None 0 2
840.1.ci $$\chi_{840}(499, \cdot)$$ None 0 2
840.1.cj $$\chi_{840}(479, \cdot)$$ None 0 2
840.1.cm $$\chi_{840}(569, \cdot)$$ None 0 2
840.1.cn $$\chi_{840}(229, \cdot)$$ None 0 2
840.1.cs $$\chi_{840}(79, \cdot)$$ None 0 2
840.1.ct $$\chi_{840}(59, \cdot)$$ None 0 2
840.1.cw $$\chi_{840}(131, \cdot)$$ None 0 2
840.1.cx $$\chi_{840}(151, \cdot)$$ None 0 2
840.1.cy $$\chi_{840}(61, \cdot)$$ None 0 2
840.1.cz $$\chi_{840}(401, \cdot)$$ None 0 2
840.1.de $$\chi_{840}(187, \cdot)$$ None 0 4
840.1.df $$\chi_{840}(23, \cdot)$$ None 0 4
840.1.dg $$\chi_{840}(193, \cdot)$$ None 0 4
840.1.dh $$\chi_{840}(173, \cdot)$$ 840.1.dh.a 8 4
840.1.dh.b 8
840.1.dm $$\chi_{840}(37, \cdot)$$ None 0 4
840.1.dn $$\chi_{840}(17, \cdot)$$ None 0 4
840.1.do $$\chi_{840}(103, \cdot)$$ None 0 4
840.1.dp $$\chi_{840}(107, \cdot)$$ None 0 4

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(840))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(840)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(140))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(168))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(280))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(420))$$$$^{\oplus 2}$$