Properties

Label 768.1
Level 768
Weight 1
Dimension 12
Nonzero newspaces 2
Newform subspaces 5
Sturm bound 32768
Trace bound 1

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Defining parameters

Level: \( N \) = \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 5 \)
Sturm bound: \(32768\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(768))\).

Total New Old
Modular forms 732 116 616
Cusp forms 28 12 16
Eisenstein series 704 104 600

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 12 0 0 0

Trace form

\( 12q + O(q^{10}) \) \( 12q + 4q^{25} - 4q^{33} - 12q^{49} - 4q^{57} - 4q^{81} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(768))\)

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
768.1.b \(\chi_{768}(127, \cdot)\) None 0 1
768.1.e \(\chi_{768}(257, \cdot)\) 768.1.e.a 1 1
768.1.e.b 1
768.1.e.c 2
768.1.g \(\chi_{768}(511, \cdot)\) None 0 1
768.1.h \(\chi_{768}(641, \cdot)\) None 0 1
768.1.i \(\chi_{768}(65, \cdot)\) 768.1.i.a 4 2
768.1.i.b 4
768.1.l \(\chi_{768}(319, \cdot)\) None 0 2
768.1.m \(\chi_{768}(31, \cdot)\) None 0 4
768.1.p \(\chi_{768}(161, \cdot)\) None 0 4
768.1.q \(\chi_{768}(17, \cdot)\) None 0 8
768.1.t \(\chi_{768}(79, \cdot)\) None 0 8
768.1.u \(\chi_{768}(7, \cdot)\) None 0 16
768.1.x \(\chi_{768}(41, \cdot)\) None 0 16
768.1.y \(\chi_{768}(5, \cdot)\) None 0 32
768.1.bb \(\chi_{768}(19, \cdot)\) None 0 32

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(768))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(768)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(128))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(192))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(256))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(384))\)\(^{\oplus 2}\)