Space of modular forms of level 500, weight 2, and Character 500.i

• Label: 500.2.i
• Level: $500$
• Weight: $2$
• Character orbit: 500.i
• Rep. character: $\chi_{500}(49,\cdot)$
• Character field: $\Q(\zeta_{10})$
• Dimension: $32$
• Newform subspaces: $2$
• Sturm bound: $150$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
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:			\(150\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(500, [\chi])\).

	Total	New	Old
Modular forms	360	32	328
Cusp forms	240	32	208
Eisenstein series	120	0	120

Trace form

32 * q + 8 * q^9 - 5 * q^11 + 5 * q^17 + 8 * q^19 + 2 * q^21 + 20 * q^23 + 28 * q^29 + 12 * q^31 - 15 * q^33 + 10 * q^37 - 22 * q^39 - 33 * q^41 - 45 * q^47 - 54 * q^49 + 14 * q^51 - 30 * q^53 - 9 * q^59 - 36 * q^61 - 20 * q^63 + 5 * q^67 + 74 * q^69 + 41 * q^71 + 60 * q^73 + 20 * q^77 + 24 * q^79 + 33 * q^81 + 10 * q^83 + 55 * q^87 - 33 * q^89 - 32 * q^91 - 25 * q^97 - 60 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(500, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
500.2.i.a	$500$	$2$	500.i	25.e	$10$	$8$	$2$	$3.993$	8.0.58140625.2	None					100.2.i.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+(\beta _{1}+\beta _{5}+\beta _{7})q^{3}+(-\beta _{2}-\beta _{3}+\beta _{4}+\cdots)q^{7}+\cdots\)
500.2.i.b	$500$	$2$	500.i	25.e	$10$	$24$	$6$	$3.993$		None			✓		100.2.g.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{10}]$

Decomposition of \(S_{2}^{\mathrm{old}}(500, [\chi])\) into lower level spaces

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