Space of modular forms of level 50, weight 2, and trivial character

• 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: $2$

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.

\(2\)	\(5\)	Fricke	Dim
\(+\)	\(-\)	$-$	\(1\)
\(-\)	\(+\)	$-$	\(1\)
Plus space		\(+\)	\(0\)
Minus space		\(-\)	\(2\)

Trace form

\( 2 q + 2 q^{4} - 2 q^{6} - 4 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(50))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5
50.2.a.a	$50$	$2$	50.a	1.a	$1$	$1$	$1$	$0.399$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots\)
50.2.a.b	$50$	$2$	50.a	1.a	$1$	$1$	$1$	$0.399$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)