Space of modular forms of level 500, weight 4, and trivial character

• Label: 500.4.a
• Level: $500$
• Weight: $4$
• Character orbit: 500.a
• Rep. character: $\chi_{500}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $4$
• Sturm bound: $300$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	500.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(300\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(500))\).

	Total	New	Old
Modular forms	240	24	216
Cusp forms	210	24	186
Eisenstein series	30	0	30

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
\(+\)	\(+\)	\(+\)		\(64\)	\(0\)	\(64\)		\(54\)	\(0\)	\(54\)		\(10\)	\(0\)	\(10\)
\(+\)	\(-\)	\(-\)		\(59\)	\(0\)	\(59\)		\(49\)	\(0\)	\(49\)		\(10\)	\(0\)	\(10\)
\(-\)	\(+\)	\(-\)		\(56\)	\(10\)	\(46\)		\(51\)	\(10\)	\(41\)		\(5\)	\(0\)	\(5\)
\(-\)	\(-\)	\(+\)		\(61\)	\(14\)	\(47\)		\(56\)	\(14\)	\(42\)		\(5\)	\(0\)	\(5\)
Plus space		\(+\)		\(125\)	\(14\)	\(111\)		\(110\)	\(14\)	\(96\)		\(15\)	\(0\)	\(15\)
Minus space		\(-\)		\(115\)	\(10\)	\(105\)		\(100\)	\(10\)	\(90\)		\(15\)	\(0\)	\(15\)

Trace form

24 * q + 246 * q^9 - 10 * q^11 - 254 * q^19 - 196 * q^21 - 34 * q^29 + 224 * q^31 + 556 * q^39 + 244 * q^41 + 2232 * q^49 + 2248 * q^51 + 1282 * q^59 + 1898 * q^61 - 52 * q^69 - 128 * q^71 + 2608 * q^79 + 1696 * q^81 + 104 * q^89 + 3896 * q^91 - 4290 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(500))\) 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
500.4.a.a	$500$	$4$	500.a	1.a	$1$	$4$	$4$	$29.501$	\(\Q(\sqrt{94 -6 \sqrt{5}})\)	None	✓	✓			500.4.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(2\beta _{2}-\beta _{3})q^{7}+(-2-3\beta _{1}+\cdots)q^{9}+\cdots\)
500.4.a.b	$500$	$4$	500.a	1.a	$1$	$6$	$6$	$29.501$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			500.4.a.b	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(43\)		$-$	$-$	$+$	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(7+\beta _{1}-\beta _{2}+\beta _{3}+\beta _{4}+\cdots)q^{7}+\cdots\)
500.4.a.c	$500$	$4$	500.a	1.a	$1$	$6$	$6$	$29.501$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		500.4.a.b	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-43\)		$-$	$+$	$-$	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(-7-\beta _{1}+\beta _{2}-\beta _{3}-\beta _{4}+\cdots)q^{7}+\cdots\)
500.4.a.d	$500$	$4$	500.a	1.a	$1$	$8$	$8$	$29.501$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		500.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{8}\cdot 5^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(-\beta _{2}-\beta _{5})q^{7}+(2^{4}+\beta _{4}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(500))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(500)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(250))\)\(^{\oplus 2}\)