Space of modular forms of level 1323, weight 2, and Character 1323.f

• Label: 1323.2.f
• Level: $1323$
• Weight: $2$
• Character orbit: 1323.f
• Rep. character: $\chi_{1323}(442,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $72$
• Newform subspaces: $8$
• Sturm bound: $336$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.f (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(336\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1323, [\chi])\).

	Total	New	Old
Modular forms	384	92	292
Cusp forms	288	72	216
Eisenstein series	96	20	76

Trace form

72 * q - 2 * q^2 - 30 * q^4 + 2 * q^5 + 24 * q^8 - 6 * q^11 - 18 * q^16 - 12 * q^17 + 12 * q^19 + 22 * q^20 - 12 * q^22 - 16 * q^23 - 18 * q^25 - 8 * q^26 - 6 * q^31 - 26 * q^32 + 6 * q^34 + 12 * q^37 - 14 * q^38 + 12 * q^40 + 22 * q^41 - 12 * q^43 + 44 * q^44 + 6 * q^47 + 18 * q^50 - 18 * q^52 - 16 * q^53 + 12 * q^55 - 18 * q^58 + 12 * q^59 - 96 * q^62 - 24 * q^64 - 24 * q^65 - 12 * q^67 - 12 * q^68 + 60 * q^71 + 36 * q^73 + 22 * q^74 - 6 * q^76 - 8 * q^80 + 30 * q^83 + 18 * q^85 - 58 * q^86 - 6 * q^88 + 20 * q^89 + 4 * q^92 + 6 * q^94 + 26 * q^95

Decomposition of \(S_{2}^{\mathrm{new}}(1323, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1323.2.f.a	$1323$	$2$	1323.f	9.c	$3$	$2$	$1$	$10.564$	\(\Q(\sqrt{-3}) \)	None					63.2.g.a	$2$	$0$	\(1\)	\(0\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+\zeta_{6}q^{4}-\zeta_{6}q^{5}+3q^{8}+\cdots\)
1323.2.f.b	$1323$	$2$	1323.f	9.c	$3$	$2$	$1$	$10.564$	\(\Q(\sqrt{-3}) \)	None					63.2.g.a	$2$	$0$	\(1\)	\(0\)	\(1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+\zeta_{6}q^{4}+\zeta_{6}q^{5}+3q^{8}+\cdots\)
1323.2.f.c	$1323$	$2$	1323.f	9.c	$3$	$6$	$3$	$10.564$	6.0.309123.1	None					63.2.f.b	$2$	$0$	\(-1\)	\(0\)	\(5\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}-\beta _{5})q^{2}+(-1+\beta _{2}+\beta _{4}+\beta _{5})q^{4}+\cdots\)
1323.2.f.d	$1323$	$2$	1323.f	9.c	$3$	$6$	$3$	$10.564$	\(\Q(\zeta_{18})\)	None					63.2.f.a	$2$	$0$	\(3\)	\(0\)	\(-3\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta_{5}-\beta_{4}-\beta_{3}+\beta_1)q^{2}+(2\beta_{5}-\beta_{3}+\beta_{2}+\cdots-1)q^{4}+\cdots\)
1323.2.f.e	$1323$	$2$	1323.f	9.c	$3$	$10$	$5$	$10.564$	10.0.\(\cdots\).1	None					63.2.g.b	$2$	$0$	\(-2\)	\(0\)	\(-4\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(-\beta _{3}+\beta _{6}+\beta _{7})q^{4}+(\beta _{6}+\cdots)q^{5}+\cdots\)
1323.2.f.f	$1323$	$2$	1323.f	9.c	$3$	$10$	$5$	$10.564$	10.0.\(\cdots\).1	None					63.2.g.b	$2$	$0$	\(-2\)	\(0\)	\(4\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(-\beta _{3}+\beta _{6}+\beta _{7})q^{4}+(-\beta _{6}+\cdots)q^{5}+\cdots\)
1323.2.f.g	$1323$	$2$	1323.f	9.c	$3$	$12$	$6$	$10.564$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					441.2.f.g	$4$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$3^{5}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{6}q^{2}+(-\beta _{1}+\beta _{4}-\beta _{5}+\beta _{6}-\beta _{7}+\cdots)q^{4}+\cdots\)
1323.2.f.h	$1323$	$2$	1323.f	9.c	$3$	$24$	$12$	$10.564$		None			✓		441.2.f.h	$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{3}]$

Decomposition of \(S_{2}^{\mathrm{old}}(1323, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1323, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(441, [\chi])\)\(^{\oplus 2}\)