Properties

Label 56.1
Level 56
Weight 1
Dimension 1
Nonzero newspaces 1
Newform subspaces 1
Sturm bound 192
Trace bound 0

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Defining parameters

Level: \( N \) = \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 1 \)
Sturm bound: \(192\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(56))\).

Total New Old
Modular forms 37 11 26
Cusp forms 1 1 0
Eisenstein series 36 10 26

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 1 0 0 0

Trace form

\( q - q^{2} + q^{4} - q^{7} - q^{8} - q^{9} + O(q^{10}) \) \( q - q^{2} + q^{4} - q^{7} - q^{8} - q^{9} + q^{14} + q^{16} + q^{18} + 2q^{23} - q^{25} - q^{28} - q^{32} - q^{36} - 2q^{46} + q^{49} + q^{50} + q^{56} + q^{63} + q^{64} - 2q^{71} + q^{72} - 2q^{79} + q^{81} + 2q^{92} - q^{98} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(56))\)

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
56.1.c \(\chi_{56}(41, \cdot)\) None 0 1
56.1.d \(\chi_{56}(15, \cdot)\) None 0 1
56.1.g \(\chi_{56}(43, \cdot)\) None 0 1
56.1.h \(\chi_{56}(13, \cdot)\) 56.1.h.a 1 1
56.1.j \(\chi_{56}(5, \cdot)\) None 0 2
56.1.k \(\chi_{56}(11, \cdot)\) None 0 2
56.1.n \(\chi_{56}(23, \cdot)\) None 0 2
56.1.o \(\chi_{56}(17, \cdot)\) None 0 2

Hecke characteristic polynomials

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