Properties

Label 600.1
Level 600
Weight 1
Dimension 30
Nonzero newspaces 4
Newforms 7
Sturm bound 19200
Trace bound 4

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Defining parameters

Level: \( N \) = \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 4 \)
Newforms: \( 7 \)
Sturm bound: \(19200\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(600))\).

Total New Old
Modular forms 746 112 634
Cusp forms 74 30 44
Eisenstein series 672 82 590

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 30 0 0 0

Trace form

\(30q \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(30q \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut -\mathstrut 4q^{10} \) \(\mathstrut -\mathstrut 4q^{15} \) \(\mathstrut -\mathstrut 14q^{16} \) \(\mathstrut -\mathstrut 4q^{22} \) \(\mathstrut +\mathstrut 14q^{24} \) \(\mathstrut +\mathstrut 16q^{28} \) \(\mathstrut -\mathstrut 4q^{31} \) \(\mathstrut -\mathstrut 4q^{33} \) \(\mathstrut -\mathstrut 2q^{36} \) \(\mathstrut -\mathstrut 4q^{42} \) \(\mathstrut -\mathstrut 10q^{49} \) \(\mathstrut -\mathstrut 12q^{51} \) \(\mathstrut -\mathstrut 6q^{54} \) \(\mathstrut -\mathstrut 4q^{55} \) \(\mathstrut -\mathstrut 4q^{58} \) \(\mathstrut -\mathstrut 4q^{63} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut +\mathstrut 12q^{66} \) \(\mathstrut -\mathstrut 4q^{70} \) \(\mathstrut -\mathstrut 4q^{73} \) \(\mathstrut -\mathstrut 12q^{79} \) \(\mathstrut -\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 4q^{87} \) \(\mathstrut -\mathstrut 4q^{88} \) \(\mathstrut -\mathstrut 2q^{96} \) \(\mathstrut +\mathstrut 16q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(600))\)

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
600.1.c \(\chi_{600}(449, \cdot)\) None 0 1
600.1.e \(\chi_{600}(151, \cdot)\) None 0 1
600.1.g \(\chi_{600}(451, \cdot)\) None 0 1
600.1.i \(\chi_{600}(149, \cdot)\) None 0 1
600.1.j \(\chi_{600}(199, \cdot)\) None 0 1
600.1.l \(\chi_{600}(401, \cdot)\) None 0 1
600.1.n \(\chi_{600}(101, \cdot)\) 600.1.n.a 1 1
600.1.n.b 1
600.1.p \(\chi_{600}(499, \cdot)\) None 0 1
600.1.q \(\chi_{600}(107, \cdot)\) 600.1.q.a 4 2
600.1.q.b 8
600.1.t \(\chi_{600}(157, \cdot)\) None 0 2
600.1.u \(\chi_{600}(193, \cdot)\) None 0 2
600.1.x \(\chi_{600}(143, \cdot)\) None 0 2
600.1.z \(\chi_{600}(29, \cdot)\) 600.1.z.a 8 4
600.1.bb \(\chi_{600}(91, \cdot)\) None 0 4
600.1.bd \(\chi_{600}(31, \cdot)\) None 0 4
600.1.bf \(\chi_{600}(89, \cdot)\) None 0 4
600.1.bh \(\chi_{600}(19, \cdot)\) None 0 4
600.1.bj \(\chi_{600}(221, \cdot)\) 600.1.bj.a 4 4
600.1.bj.b 4
600.1.bl \(\chi_{600}(41, \cdot)\) None 0 4
600.1.bn \(\chi_{600}(79, \cdot)\) None 0 4
600.1.bo \(\chi_{600}(23, \cdot)\) None 0 8
600.1.br \(\chi_{600}(73, \cdot)\) None 0 8
600.1.bs \(\chi_{600}(13, \cdot)\) None 0 8
600.1.bv \(\chi_{600}(83, \cdot)\) None 0 8

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(600))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(600)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(300))\)\(^{\oplus 2}\)