Properties

Label 360.1
Level 360
Weight 1
Dimension 6
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 6912
Trace bound 1

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Defining parameters

Level: \( N \) = \( 360\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \) \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(6912\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(360))\).

Total New Old
Modular forms 418 60 358
Cusp forms 34 6 28
Eisenstein series 384 54 330

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 6 0 0 0

Trace form

\( 6q + 2q^{4} - 4q^{7} + O(q^{10}) \) \( 6q + 2q^{4} - 4q^{7} + 2q^{10} - 2q^{16} - 4q^{19} + 4q^{22} + 2q^{25} - 4q^{28} - 2q^{40} - 4q^{46} - 2q^{49} - 4q^{55} - 4q^{58} + 2q^{64} - 4q^{70} + 4q^{73} - 4q^{76} + 4q^{88} + 4q^{94} - 4q^{97} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(360))\)

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
360.1.c \(\chi_{360}(89, \cdot)\) None 0 1
360.1.e \(\chi_{360}(271, \cdot)\) None 0 1
360.1.g \(\chi_{360}(91, \cdot)\) None 0 1
360.1.i \(\chi_{360}(269, \cdot)\) None 0 1
360.1.j \(\chi_{360}(199, \cdot)\) None 0 1
360.1.l \(\chi_{360}(161, \cdot)\) None 0 1
360.1.n \(\chi_{360}(341, \cdot)\) None 0 1
360.1.p \(\chi_{360}(19, \cdot)\) 360.1.p.a 1 1
360.1.p.b 1
360.1.r \(\chi_{360}(107, \cdot)\) None 0 2
360.1.u \(\chi_{360}(37, \cdot)\) 360.1.u.a 4 2
360.1.v \(\chi_{360}(73, \cdot)\) None 0 2
360.1.y \(\chi_{360}(143, \cdot)\) None 0 2
360.1.z \(\chi_{360}(139, \cdot)\) None 0 2
360.1.ba \(\chi_{360}(101, \cdot)\) None 0 2
360.1.bc \(\chi_{360}(41, \cdot)\) None 0 2
360.1.be \(\chi_{360}(79, \cdot)\) None 0 2
360.1.bh \(\chi_{360}(29, \cdot)\) None 0 2
360.1.bj \(\chi_{360}(211, \cdot)\) None 0 2
360.1.bl \(\chi_{360}(31, \cdot)\) None 0 2
360.1.bn \(\chi_{360}(209, \cdot)\) None 0 2
360.1.bp \(\chi_{360}(97, \cdot)\) None 0 4
360.1.bq \(\chi_{360}(23, \cdot)\) None 0 4
360.1.bt \(\chi_{360}(83, \cdot)\) None 0 4
360.1.bu \(\chi_{360}(13, \cdot)\) None 0 4

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(360))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(360)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(180))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

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