Properties

Label 1200.1
Level 1200
Weight 1
Dimension 28
Nonzero newspaces 6
Newform subspaces 10
Sturm bound 76800
Trace bound 9

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Defining parameters

Level: \( N \) = \( 1200\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \) \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 10 \)
Sturm bound: \(76800\)
Trace bound: \(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(1200))\).

Total New Old
Modular forms 1750 213 1537
Cusp forms 182 28 154
Eisenstein series 1568 185 1383

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 28 0 0 0

Trace form

\( 28q + O(q^{10}) \) \( 28q + 4q^{16} + 4q^{19} - 4q^{21} - 4q^{24} + 4q^{31} + 4q^{34} - 12q^{36} - 12q^{46} + 4q^{49} - 4q^{51} - 4q^{54} - 8q^{61} - 4q^{69} - 4q^{76} - 8q^{79} - 4q^{81} - 4q^{91} - 4q^{94} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(1200))\)

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
1200.1.c \(\chi_{1200}(449, \cdot)\) 1200.1.c.a 2 1
1200.1.e \(\chi_{1200}(751, \cdot)\) None 0 1
1200.1.g \(\chi_{1200}(151, \cdot)\) None 0 1
1200.1.i \(\chi_{1200}(1049, \cdot)\) None 0 1
1200.1.j \(\chi_{1200}(799, \cdot)\) None 0 1
1200.1.l \(\chi_{1200}(401, \cdot)\) 1200.1.l.a 1 1
1200.1.l.b 1
1200.1.n \(\chi_{1200}(1001, \cdot)\) None 0 1
1200.1.p \(\chi_{1200}(199, \cdot)\) None 0 1
1200.1.q \(\chi_{1200}(499, \cdot)\) None 0 2
1200.1.r \(\chi_{1200}(101, \cdot)\) 1200.1.r.a 4 2
1200.1.u \(\chi_{1200}(407, \cdot)\) None 0 2
1200.1.x \(\chi_{1200}(457, \cdot)\) None 0 2
1200.1.z \(\chi_{1200}(107, \cdot)\) 1200.1.z.a 2 2
1200.1.z.b 2
1200.1.ba \(\chi_{1200}(493, \cdot)\) None 0 2
1200.1.bd \(\chi_{1200}(443, \cdot)\) 1200.1.bd.a 2 2
1200.1.bd.b 2
1200.1.be \(\chi_{1200}(157, \cdot)\) None 0 2
1200.1.bg \(\chi_{1200}(193, \cdot)\) None 0 2
1200.1.bj \(\chi_{1200}(143, \cdot)\) 1200.1.bj.a 4 2
1200.1.bj.b 8
1200.1.bm \(\chi_{1200}(149, \cdot)\) None 0 2
1200.1.bn \(\chi_{1200}(451, \cdot)\) None 0 2
1200.1.bp \(\chi_{1200}(89, \cdot)\) None 0 4
1200.1.br \(\chi_{1200}(391, \cdot)\) None 0 4
1200.1.bt \(\chi_{1200}(31, \cdot)\) None 0 4
1200.1.bv \(\chi_{1200}(209, \cdot)\) None 0 4
1200.1.bx \(\chi_{1200}(439, \cdot)\) None 0 4
1200.1.bz \(\chi_{1200}(41, \cdot)\) None 0 4
1200.1.cb \(\chi_{1200}(161, \cdot)\) None 0 4
1200.1.cd \(\chi_{1200}(79, \cdot)\) None 0 4
1200.1.cg \(\chi_{1200}(221, \cdot)\) None 0 8
1200.1.ch \(\chi_{1200}(19, \cdot)\) None 0 8
1200.1.ci \(\chi_{1200}(47, \cdot)\) None 0 8
1200.1.cl \(\chi_{1200}(97, \cdot)\) None 0 8
1200.1.cn \(\chi_{1200}(133, \cdot)\) None 0 8
1200.1.co \(\chi_{1200}(203, \cdot)\) None 0 8
1200.1.cr \(\chi_{1200}(13, \cdot)\) None 0 8
1200.1.cs \(\chi_{1200}(83, \cdot)\) None 0 8
1200.1.cu \(\chi_{1200}(73, \cdot)\) None 0 8
1200.1.cx \(\chi_{1200}(23, \cdot)\) None 0 8
1200.1.cy \(\chi_{1200}(91, \cdot)\) None 0 8
1200.1.cz \(\chi_{1200}(29, \cdot)\) None 0 8

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(1200))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(1200)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(300))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(600))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

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