Space of modular forms of level 500, weight 4, and Character 500.i

• Label: 500.4.i
• Level: $500$
• Weight: $4$
• Character orbit: 500.i
• Rep. character: $\chi_{500}(49,\cdot)$
• Character field: $\Q(\zeta_{10})$
• Dimension: $88$
• Newform subspaces: $2$
• Sturm bound: $300$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	500.i (of order \(10\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 25 \)
Character field:			\(\Q(\zeta_{10})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(300\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(500, [\chi])\).

	Total	New	Old
Modular forms	960	88	872
Cusp forms	840	88	752
Eisenstein series	120	0	120

Trace form

\( 88 q + 148 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(500, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
500.4.i.a	$500$	$4$	500.i	25.e	$10$	$32$	$8$	$29.501$		None					100.4.i.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{10}]$
500.4.i.b	$500$	$4$	500.i	25.e	$10$	$56$	$14$	$29.501$		None			✓		100.4.g.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{10}]$

Decomposition of \(S_{4}^{\mathrm{old}}(500, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(500, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(125, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(250, [\chi])\)\(^{\oplus 2}\)