Properties

Label 224.1
Level 224
Weight 1
Dimension 9
Nonzero newspaces 3
Newform subspaces 3
Sturm bound 3072
Trace bound 5

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Defining parameters

Level: \( N \) = \( 224\( 224 = 2^{5} \cdot 7 \) \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 3 \)
Sturm bound: \(3072\)
Trace bound: \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(224))\).

Total New Old
Modular forms 208 59 149
Cusp forms 16 9 7
Eisenstein series 192 50 142

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 5 4 0 0

Trace form

\( 9q - 2q^{5} + q^{7} - q^{9} + O(q^{10}) \) \( 9q - 2q^{5} + q^{7} - q^{9} - 4q^{16} - 2q^{17} - 4q^{18} - 2q^{21} + 4q^{22} - 6q^{23} - q^{25} - 2q^{33} + 2q^{37} - 4q^{43} + 4q^{44} - 3q^{49} - 2q^{53} + 4q^{56} - 4q^{57} + 2q^{61} + 3q^{63} + 4q^{67} + 4q^{69} + 2q^{71} + 2q^{73} + 4q^{74} - 2q^{77} + 2q^{79} + 3q^{81} + 4q^{85} + 2q^{89} - 4q^{92} - 2q^{93} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(224))\)

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
224.1.c \(\chi_{224}(97, \cdot)\) None 0 1
224.1.d \(\chi_{224}(127, \cdot)\) None 0 1
224.1.g \(\chi_{224}(15, \cdot)\) None 0 1
224.1.h \(\chi_{224}(209, \cdot)\) 224.1.h.a 1 1
224.1.k \(\chi_{224}(71, \cdot)\) None 0 2
224.1.l \(\chi_{224}(41, \cdot)\) None 0 2
224.1.n \(\chi_{224}(17, \cdot)\) None 0 2
224.1.o \(\chi_{224}(79, \cdot)\) None 0 2
224.1.r \(\chi_{224}(95, \cdot)\) 224.1.r.a 4 2
224.1.s \(\chi_{224}(33, \cdot)\) None 0 2
224.1.v \(\chi_{224}(13, \cdot)\) 224.1.v.a 4 4
224.1.w \(\chi_{224}(43, \cdot)\) None 0 4
224.1.y \(\chi_{224}(23, \cdot)\) None 0 4
224.1.bb \(\chi_{224}(73, \cdot)\) None 0 4
224.1.bc \(\chi_{224}(5, \cdot)\) None 0 8
224.1.bf \(\chi_{224}(11, \cdot)\) None 0 8

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(224))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(224)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(112))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

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