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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(2\)	\(0\)	\(2\)		\(0\)	\(0\)	\(0\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(4\)	\(1\)	\(3\)		\(1\)	\(1\)	\(0\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(4\)	\(1\)	\(3\)		\(1\)	\(1\)	\(0\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(3\)	\(0\)	\(3\)		\(0\)	\(0\)	\(0\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(5\)	\(0\)	\(5\)		\(0\)	\(0\)	\(0\)		\(5\)	\(0\)	\(5\)
Minus space		\(-\)		\(8\)	\(2\)	\(6\)		\(2\)	\(2\)	\(0\)		\(6\)	\(0\)	\(6\)

Trace form

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

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	Twist minimal	Largest	Maximal	Minimal twist	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	✓	✓	✓	✓	50.2.a.a	$1$	$0$	\(-1\)	\(1\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots\)
50.2.a.b	$50$	$2$	50.a	1.a	$1$	$1$	$1$	$0.399$	\(\Q\)	None	✓	✓	✓	✓	50.2.a.a	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)