Properties

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

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Defining parameters

Level: \( N \) = \( 600\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 7 \)
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

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

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T^{4} \))(\( ( 1 + T^{4} )^{2} \))(\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))
$3$ (\( 1 - T \))(\( 1 + T \))(\( 1 + T^{4} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))
$5$ (\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))
$7$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))
$11$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \))(\( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \))
$13$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))
$17$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))
$19$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 - T^{2} + T^{4} )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))
$23$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))
$29$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T )^{8}( 1 + T )^{8} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))
$31$ (\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T )^{8}( 1 + T )^{8} \))(\( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \))
$37$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))
$41$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))
$43$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 + T^{2} )^{8} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))
$47$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))
$53$ (\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))
$59$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{8} \))(\( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \))(\( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \))
$61$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T )^{8}( 1 + T )^{8} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))
$67$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))
$71$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{8} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))
$73$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))
$79$ (\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{8} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))
$83$ (\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))(\( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \))(\( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \))
$89$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 - T^{2} + T^{4} )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))
$97$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 - T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \))
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