Properties

Label 50.2.a
Level $50$
Weight $2$
Character orbit 50.a
Rep. character $\chi_{50}(1,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $2$
Sturm bound $15$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(15\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(50))\).

Total New Old
Modular forms 13 2 11
Cusp forms 2 2 0
Eisenstein series 11 0 11

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

\(2\)\(5\)FrickeDim.
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(2\)

Trace form

\( 2 q + 2 q^{4} - 2 q^{6} - 4 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{4} - 2 q^{6} - 4 q^{9} - 6 q^{11} - 4 q^{14} + 2 q^{16} + 10 q^{19} + 4 q^{21} - 2 q^{24} + 8 q^{26} + 4 q^{31} + 6 q^{34} - 4 q^{36} - 8 q^{39} - 6 q^{41} - 6 q^{44} - 12 q^{46} - 6 q^{49} - 6 q^{51} + 10 q^{54} - 4 q^{56} + 4 q^{61} + 2 q^{64} + 6 q^{66} + 12 q^{69} + 24 q^{71} - 4 q^{74} + 10 q^{76} - 20 q^{79} + 2 q^{81} + 4 q^{84} + 8 q^{86} + 30 q^{89} - 16 q^{91} - 24 q^{94} - 2 q^{96} + 12 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(50))\) 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 5
50.2.a.a 50.a 1.a $1$ $0.399$ \(\Q\) None \(-1\) \(1\) \(0\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots\)
50.2.a.b 50.a 1.a $1$ $0.399$ \(\Q\) None \(1\) \(-1\) \(0\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)