Properties

Label 56.2.a
Level $56$
Weight $2$
Character orbit 56.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

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

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 7
56.2.a.a 56.a 1.a $1$ $0.447$ \(\Q\) None 56.2.a.a \(0\) \(0\) \(2\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{5}-q^{7}-3q^{9}-4q^{11}+2q^{13}+\cdots\)
56.2.a.b 56.a 1.a $1$ $0.447$ \(\Q\) None 56.2.a.b \(0\) \(2\) \(-4\) \(1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}-4q^{5}+q^{7}+q^{9}-8q^{15}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(\Gamma_0(56)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 2}\)