Properties

Label 309.1
Level 309
Weight 1
Dimension 20
Nonzero newspaces 2
Newform subspaces 2
Sturm bound 7072
Trace bound 1

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Defining parameters

Level: \( N \) = \( 309 = 3 \cdot 103 \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 2 \)
Sturm bound: \(7072\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(309))\).

Total New Old
Modular forms 228 120 108
Cusp forms 24 20 4
Eisenstein series 204 100 104

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 16 4 0 0

Trace form

\( 20q - 5q^{3} - q^{4} + 3q^{9} + O(q^{10}) \) \( 20q - 5q^{3} - q^{4} + 3q^{9} + 4q^{10} - q^{12} - 2q^{13} + q^{16} - 4q^{19} - 4q^{21} - 4q^{22} - q^{25} - 5q^{27} - 2q^{28} - 4q^{30} - 2q^{31} - 4q^{34} - q^{36} - 2q^{37} - 2q^{39} - 2q^{40} - 4q^{43} + 4q^{46} - 3q^{48} - 3q^{49} - 2q^{52} - 2q^{55} - 2q^{58} - 2q^{61} - 5q^{64} + 4q^{66} - 4q^{67} + 2q^{70} - 2q^{73} - q^{75} - 2q^{76} + 6q^{79} + 3q^{81} + 2q^{82} + 15q^{84} - 2q^{85} + 2q^{88} + 4q^{90} + 13q^{91} - 2q^{93} - 4q^{94} + 17q^{97} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(309))\)

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
309.1.b \(\chi_{309}(104, \cdot)\) None 0 1
309.1.d \(\chi_{309}(205, \cdot)\) None 0 1
309.1.f \(\chi_{309}(160, \cdot)\) None 0 2
309.1.h \(\chi_{309}(56, \cdot)\) 309.1.h.a 4 2
309.1.j \(\chi_{309}(10, \cdot)\) None 0 16
309.1.l \(\chi_{309}(8, \cdot)\) 309.1.l.a 16 16
309.1.n \(\chi_{309}(2, \cdot)\) None 0 32
309.1.p \(\chi_{309}(40, \cdot)\) None 0 32

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(309))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(309)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(103))\)\(^{\oplus 2}\)