Properties

Label 800.1
Level 800
Weight 1
Dimension 26
Nonzero newspaces 5
Newforms 8
Sturm bound 38400
Trace bound 19

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Defining parameters

Level: \( N \) = \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 5 \)
Newforms: \( 8 \)
Sturm bound: \(38400\)
Trace bound: \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(800))\).

Total New Old
Modular forms 968 231 737
Cusp forms 72 26 46
Eisenstein series 896 205 691

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 18 0 0 8

Trace form

\( 26q - 2q^{5} + O(q^{10}) \) \( 26q - 2q^{5} + 4q^{11} + 10q^{13} + 4q^{17} - 4q^{21} - 6q^{29} - 2q^{37} - 4q^{41} - 2q^{45} - 4q^{49} - 4q^{51} - 2q^{53} - 4q^{57} - 2q^{61} - 6q^{65} + 2q^{69} - 6q^{73} + 2q^{81} - 8q^{85} - 10q^{89} - 8q^{93} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(800))\)

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
800.1.b \(\chi_{800}(351, \cdot)\) None 0 1
800.1.e \(\chi_{800}(399, \cdot)\) 800.1.e.a 2 1
800.1.g \(\chi_{800}(751, \cdot)\) 800.1.g.a 1 1
800.1.g.b 1
800.1.h \(\chi_{800}(799, \cdot)\) None 0 1
800.1.i \(\chi_{800}(57, \cdot)\) None 0 2
800.1.k \(\chi_{800}(199, \cdot)\) None 0 2
800.1.m \(\chi_{800}(593, \cdot)\) None 0 2
800.1.p \(\chi_{800}(193, \cdot)\) 800.1.p.a 2 2
800.1.p.b 2
800.1.p.c 2
800.1.r \(\chi_{800}(151, \cdot)\) None 0 2
800.1.t \(\chi_{800}(457, \cdot)\) None 0 2
800.1.w \(\chi_{800}(93, \cdot)\) None 0 4
800.1.x \(\chi_{800}(51, \cdot)\) None 0 4
800.1.z \(\chi_{800}(99, \cdot)\) None 0 4
800.1.bc \(\chi_{800}(157, \cdot)\) None 0 4
800.1.bd \(\chi_{800}(111, \cdot)\) None 0 4
800.1.bf \(\chi_{800}(159, \cdot)\) None 0 4
800.1.bh \(\chi_{800}(31, \cdot)\) 800.1.bh.a 8 4
800.1.bi \(\chi_{800}(79, \cdot)\) None 0 4
800.1.bk \(\chi_{800}(137, \cdot)\) None 0 8
800.1.bn \(\chi_{800}(39, \cdot)\) None 0 8
800.1.bo \(\chi_{800}(33, \cdot)\) 800.1.bo.a 8 8
800.1.br \(\chi_{800}(17, \cdot)\) None 0 8
800.1.bs \(\chi_{800}(71, \cdot)\) None 0 8
800.1.bv \(\chi_{800}(73, \cdot)\) None 0 8
800.1.bw \(\chi_{800}(13, \cdot)\) None 0 16
800.1.bz \(\chi_{800}(19, \cdot)\) None 0 16
800.1.cb \(\chi_{800}(11, \cdot)\) None 0 16
800.1.cc \(\chi_{800}(53, \cdot)\) None 0 16

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(800))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(800)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(160))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(200))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(400))\)\(^{\oplus 2}\)