Space of modular forms of level 500, weight 2, and trivial character

• Label: 500.2.a
• Level: $500$
• Weight: $2$
• Character orbit: 500.a
• Rep. character: $\chi_{500}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $3$
• Sturm bound: $150$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	500.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(150\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(500))\).

	Total	New	Old
Modular forms	90	8	82
Cusp forms	61	8	53
Eisenstein series	29	0	29

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
\(+\)	\(+\)	\(+\)		\(21\)	\(0\)	\(21\)		\(12\)	\(0\)	\(12\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(-\)		\(26\)	\(0\)	\(26\)		\(16\)	\(0\)	\(16\)		\(10\)	\(0\)	\(10\)
\(-\)	\(+\)	\(-\)		\(24\)	\(6\)	\(18\)		\(19\)	\(6\)	\(13\)		\(5\)	\(0\)	\(5\)
\(-\)	\(-\)	\(+\)		\(19\)	\(2\)	\(17\)		\(14\)	\(2\)	\(12\)		\(5\)	\(0\)	\(5\)
Plus space		\(+\)		\(40\)	\(2\)	\(38\)		\(26\)	\(2\)	\(24\)		\(14\)	\(0\)	\(14\)
Minus space		\(-\)		\(50\)	\(6\)	\(44\)		\(35\)	\(6\)	\(29\)		\(15\)	\(0\)	\(15\)

Trace form

\( 8 q + 8 q^{9} + 18 q^{19} + 18 q^{21} - 2 q^{29} + 18 q^{31} + 18 q^{39} - 2 q^{41} + 26 q^{49} - 54 q^{51} - 54 q^{59} + 16 q^{61} + 4 q^{69} + 4 q^{71} - 36 q^{79} + 12 q^{81} + 2 q^{89} - 18 q^{91} - 50 q^{99}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(500))\) 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
500.2.a.a	$500$	$2$	500.a	1.a	$1$	$2$	$2$	$3.993$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	500.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-3+2\beta )q^{7}+(-2+\beta )q^{9}+\cdots\)
500.2.a.b	$500$	$2$	500.a	1.a	$1$	$2$	$2$	$3.993$	\(\Q(\sqrt{5}) \)	None	✓	✓			500.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(3-2\beta )q^{7}+(-2+\beta )q^{9}+\cdots\)
500.2.a.c	$500$	$2$	500.a	1.a	$1$	$4$	$4$	$3.993$	\(\Q(\sqrt{6 + \sqrt{5}})\)	None	✓	✓	✓		500.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+(-\beta _{1}-\beta _{3})q^{7}+(2-3\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(500))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(500)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(250))\)\(^{\oplus 2}\)