Properties

Label 52.2.a
Level $52$
Weight $2$
Character orbit 52.a
Rep. character $\chi_{52}(1,\cdot)$
Character field $\Q$
Dimension $1$
Newform subspaces $1$
Sturm bound $14$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 52 = 2^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 52.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(14\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(52))\).

Total New Old
Modular forms 10 1 9
Cusp forms 5 1 4
Eisenstein series 5 0 5

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

\(2\)\(13\)FrickeDim.
\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(1\)

Trace form

\( q + 2 q^{5} - 2 q^{7} - 3 q^{9} + O(q^{10}) \) \( q + 2 q^{5} - 2 q^{7} - 3 q^{9} - 2 q^{11} - q^{13} + 6 q^{17} - 6 q^{19} + 8 q^{23} - q^{25} + 2 q^{29} + 10 q^{31} - 4 q^{35} - 6 q^{37} - 6 q^{41} + 4 q^{43} - 6 q^{45} - 2 q^{47} - 3 q^{49} + 6 q^{53} - 4 q^{55} - 10 q^{59} - 2 q^{61} + 6 q^{63} - 2 q^{65} + 10 q^{67} + 10 q^{71} + 2 q^{73} + 4 q^{77} - 4 q^{79} + 9 q^{81} - 6 q^{83} + 12 q^{85} - 6 q^{89} + 2 q^{91} - 12 q^{95} + 2 q^{97} + 6 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(52))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 13
52.2.a.a 52.a 1.a $1$ $0.415$ \(\Q\) None \(0\) \(0\) \(2\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{5}-2q^{7}-3q^{9}-2q^{11}-q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(52))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(52)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 2}\)