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Properties

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

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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

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

Display 100 coefficients 10 coefficients

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 available 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

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