Space of modular forms of level 250, weight 2, and trivial character

• Label: 250.2.a
• Level: $250$
• Weight: $2$
• Character orbit: 250.a
• Rep. character: $\chi_{250}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $4$
• Sturm bound: $75$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 250 = 2 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	250.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(75\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(250))\).

	Total	New	Old
Modular forms	47	8	39
Cusp forms	28	8	20
Eisenstein series	19	0	19

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
\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(-\)	$-$	\(2\)
\(-\)	\(+\)	$-$	\(4\)
Plus space		\(+\)	\(2\)
Minus space		\(-\)	\(6\)

Trace form

\( 8 q + 8 q^{4} + 2 q^{6} + 14 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(250))\) 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
250.2.a.a	$250$	$2$	250.a	1.a	$1$	$2$	$2$	$1.996$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-2\)	\(-3\)	\(0\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1-\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots\)
250.2.a.b	$250$	$2$	250.a	1.a	$1$	$2$	$2$	$1.996$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-2\)	\(2\)	\(0\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2\beta q^{3}+q^{4}-2\beta q^{6}-\beta q^{7}+\cdots\)
250.2.a.c	$250$	$2$	250.a	1.a	$1$	$2$	$2$	$1.996$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(2\)	\(-2\)	\(0\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2\beta q^{3}+q^{4}-2\beta q^{6}+\beta q^{7}+\cdots\)
250.2.a.d	$250$	$2$	250.a	1.a	$1$	$2$	$2$	$1.996$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(2\)	\(3\)	\(0\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(250))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(250)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 2}\)