Properties

Label 56.2.a
Level 56
Weight 2
Character orbit a
Rep. character \(\chi_{56}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newforms 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: \(\Q\)
Newforms: \( 2 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(56))\) into irreducible Hecke orbits

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

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