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

• Label: 513.2.ba
• Level: $513$
• Weight: $2$
• Character orbit: 513.ba
• Rep. character: $\chi_{513}(58,\cdot)$
• Character field: $\Q(\zeta_{9})$
• Dimension: $324$
• Newform subspaces: $2$
• Sturm bound: $120$
• Trace bound: $8$

Defining parameters

Level:	\( N \)	\(=\)	\( 513 = 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	513.ba (of order \(9\) and degree \(6\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 27 \)
Character field:			\(\Q(\zeta_{9})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(120\)
Trace bound:			\(8\)
Distinguishing \(T_p\):			\(2\)

Dimensions

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

	Total	New	Old
Modular forms	372	324	48
Cusp forms	348	324	24
Eisenstein series	24	0	24

Trace form

324 * q - 6 * q^5 - 12 * q^6 - 12 * q^9 - 12 * q^11 - 18 * q^12 - 42 * q^14 - 6 * q^15 - 24 * q^17 + 48 * q^20 - 24 * q^21 + 36 * q^23 + 42 * q^24 - 18 * q^25 + 36 * q^26 - 24 * q^27 - 36 * q^29 + 48 * q^30 - 18 * q^31 + 12 * q^32 + 30 * q^33 - 48 * q^35 + 12 * q^36 - 78 * q^39 + 24 * q^41 - 36 * q^44 - 36 * q^45 + 12 * q^47 - 78 * q^48 - 36 * q^49 + 156 * q^50 - 18 * q^51 - 54 * q^52 - 60 * q^53 + 150 * q^54 + 72 * q^56 - 54 * q^58 - 42 * q^59 + 96 * q^60 - 36 * q^61 - 48 * q^62 - 66 * q^63 - 162 * q^64 - 18 * q^65 - 42 * q^66 + 36 * q^67 - 18 * q^68 - 48 * q^69 + 72 * q^70 - 60 * q^72 - 18 * q^74 - 6 * q^75 - 90 * q^77 - 60 * q^78 + 72 * q^79 - 12 * q^80 + 12 * q^81 - 96 * q^83 + 6 * q^84 + 72 * q^85 + 36 * q^86 - 36 * q^87 + 30 * q^89 - 174 * q^90 - 186 * q^92 + 90 * q^93 + 18 * q^94 + 108 * q^96 + 36 * q^97 + 138 * q^98 + 108 * q^99

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.ba.a	$513$	$2$	513.ba	27.e	$9$	$162$	$27$	$4.096$		None		✓			513.2.ba.a	$2$	$0$	\(0\)	\(0\)	\(-3\)	\(0\)		$\mathrm{SU}(2)[C_{9}]$
513.2.ba.b	$513$	$2$	513.ba	27.e	$9$	$162$	$27$	$4.096$		None		✓			513.2.ba.b	$2$	$0$	\(0\)	\(0\)	\(-3\)	\(0\)		$\mathrm{SU}(2)[C_{9}]$

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

\( S_{2}^{\mathrm{old}}(513, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 2}\)