Space of modular forms of level 513, weight 2, and Character 513.g

• Label: 513.2.g
• Level: $513$
• Weight: $2$
• Character orbit: 513.g
• Rep. character: $\chi_{513}(64,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $36$
• Newform subspaces: $3$
• Sturm bound: $120$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 513 = 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	513.g (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 171 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(120\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(513, [\chi])\).

	Total	New	Old
Modular forms	132	44	88
Cusp forms	108	36	72
Eisenstein series	24	8	16

Trace form

36 * q - 3 * q^2 - 15 * q^4 + 2 * q^5 - 3 * q^7 + 24 * q^8 - 6 * q^10 + q^11 + 10 * q^14 - 9 * q^16 + 5 * q^17 - 9 * q^19 + q^20 - 5 * q^23 + 18 * q^25 - 4 * q^26 - 6 * q^28 + 32 * q^29 - 6 * q^31 - 27 * q^32 + 30 * q^34 + 3 * q^35 - 6 * q^37 - 30 * q^38 - 12 * q^40 + 32 * q^41 - 9 * q^43 - 28 * q^44 - 42 * q^47 - 9 * q^49 - q^50 + 15 * q^52 - 14 * q^53 + 3 * q^55 + 18 * q^56 - 6 * q^58 + 58 * q^59 + 12 * q^61 - 50 * q^62 - 36 * q^64 - 42 * q^65 - 3 * q^67 + 4 * q^68 - 48 * q^70 - 21 * q^71 - 2 * q^74 + 21 * q^76 - 32 * q^77 + 22 * q^80 - 3 * q^82 + 5 * q^83 - 9 * q^85 - 33 * q^86 - 15 * q^88 - 40 * q^89 + 15 * q^91 + 17 * q^92 - 6 * q^94 + 22 * q^95 + 12 * q^97 - 14 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(513, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
513.2.g.a	$513$	$2$	513.g	171.g	$3$	$2$	$1$	$4.096$	\(\Q(\sqrt{-3}) \)	None					171.2.g.a	$2$	$1$	\(-1\)	\(0\)	\(-6\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}-3q^{5}+(-1+\cdots)q^{7}+\cdots\)
513.2.g.b	$513$	$2$	513.g	171.g	$3$	$2$	$1$	$4.096$	\(\Q(\sqrt{-3}) \)	None					171.2.g.b	$2$	$0$	\(-1\)	\(0\)	\(2\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}+q^{5}+(-3+3\zeta_{6})q^{7}+\cdots\)
513.2.g.c	$513$	$2$	513.g	171.g	$3$	$32$	$16$	$4.096$		None			✓		171.2.g.c	$2$	$0$	\(-1\)	\(0\)	\(6\)	\(1\)		$\mathrm{SU}(2)[C_{3}]$

Decomposition of \(S_{2}^{\mathrm{old}}(513, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(513, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(171, [\chi])\)\(^{\oplus 2}\)