Properties

Label 56.2.a
Level 56
Weight 2
Character orbit a
Rep. character \(\chi_{56}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newform subspaces 2
Sturm bound 16
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 56.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(16\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(56))\).

Total New Old
Modular forms 12 2 10
Cusp forms 5 2 3
Eisenstein series 7 0 7

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(7\)FrickeDim.
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(2\)

Trace form

\( 2q + 2q^{3} - 2q^{5} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{5} - 2q^{9} - 4q^{11} + 2q^{13} - 8q^{15} - 8q^{17} + 6q^{19} + 2q^{21} + 8q^{23} + 10q^{25} - 4q^{27} + 8q^{29} + 12q^{31} - 6q^{35} - 8q^{37} + 4q^{43} - 10q^{45} - 12q^{47} + 2q^{49} - 4q^{51} - 4q^{53} - 8q^{55} - 4q^{57} + 6q^{59} - 2q^{61} + 4q^{63} + 4q^{65} - 16q^{67} + 16q^{69} - 8q^{71} - 4q^{73} + 22q^{75} + 4q^{77} + 8q^{79} - 2q^{81} + 14q^{83} - 4q^{85} + 4q^{87} + 4q^{89} - 2q^{91} + 8q^{93} + 24q^{95} - 8q^{97} + 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(56))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 7
56.2.a.a \(1\) \(0.447\) \(\Q\) None \(0\) \(0\) \(2\) \(-1\) \(-\) \(+\) \(q+2q^{5}-q^{7}-3q^{9}-4q^{11}+2q^{13}+\cdots\)
56.2.a.b \(1\) \(0.447\) \(\Q\) None \(0\) \(2\) \(-4\) \(1\) \(+\) \(-\) \(q+2q^{3}-4q^{5}+q^{7}+q^{9}-8q^{15}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(56))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(56)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))
$7$ (\( 1 + T \))(\( 1 - T \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))
$19$ (\( 1 - 8 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))
$31$ (\( 1 - 8 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))
$41$ (\( 1 - 2 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))
$47$ (\( 1 + 8 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))
$59$ (\( 1 + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))
$61$ (\( 1 + 6 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))
$71$ (\( 1 + 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))
$73$ (\( 1 - 10 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))
$79$ (\( 1 - 16 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))
$83$ (\( 1 - 8 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))
$97$ (\( 1 + 6 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))
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