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Properties

Label 53.2.a
Level $53$
Weight $2$
Character orbit 53.a
Rep. character $\chi_{53}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $2$
Sturm bound $9$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 53.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(9\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(53))\).

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

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

\(53\)Dim
\(+\)\(1\)
\(-\)\(3\)

Trace form

\( 4 q - 2 q^{2} - 2 q^{5} + 8 q^{9} + O(q^{10}) \) \( 4 q - 2 q^{2} - 2 q^{5} + 8 q^{9} - 4 q^{10} - 4 q^{11} - 2 q^{12} - 2 q^{14} - 8 q^{15} - 4 q^{16} - 8 q^{17} - 6 q^{18} + 6 q^{19} + 10 q^{20} + 10 q^{21} + 10 q^{22} + 10 q^{23} - 4 q^{24} - 8 q^{25} + 2 q^{26} + 6 q^{27} + 16 q^{28} - 12 q^{29} + 2 q^{30} + 2 q^{31} - 2 q^{32} - 14 q^{33} + 18 q^{34} + 8 q^{35} - 28 q^{36} + 8 q^{38} + 12 q^{39} - 4 q^{40} - 4 q^{41} - 16 q^{42} + 16 q^{43} - 24 q^{45} - 24 q^{46} - 12 q^{47} + 16 q^{48} + 4 q^{49} + 2 q^{50} + 4 q^{52} + 2 q^{53} + 16 q^{54} + 4 q^{55} - 22 q^{56} + 24 q^{57} - 4 q^{58} + 20 q^{60} - 18 q^{61} + 28 q^{62} - 44 q^{63} - 4 q^{64} - 2 q^{65} + 4 q^{66} - 6 q^{67} - 8 q^{68} - 28 q^{69} - 16 q^{70} - 4 q^{71} + 38 q^{72} + 2 q^{73} + 32 q^{74} + 28 q^{75} + 4 q^{76} - 4 q^{77} - 12 q^{78} - 8 q^{79} - 10 q^{80} + 44 q^{81} - 16 q^{82} + 26 q^{83} - 2 q^{84} - 6 q^{85} - 30 q^{86} + 42 q^{87} - 6 q^{88} - 16 q^{89} + 20 q^{90} + 16 q^{91} + 20 q^{92} - 16 q^{93} - 12 q^{94} - 12 q^{95} + 8 q^{96} - 30 q^{98} - 24 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(53))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 53
53.2.a.a 53.a 1.a $1$ $0.423$ \(\Q\) None \(-1\) \(-3\) \(0\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}-q^{4}+3q^{6}-4q^{7}+3q^{8}+\cdots\)
53.2.a.b 53.a 1.a $3$ $0.423$ 3.3.148.1 None \(-1\) \(3\) \(-2\) \(4\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1-\beta _{2})q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)