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

• Label: 1323.2.h
• Level: $1323$
• Weight: $2$
• Character orbit: 1323.h
• Rep. character: $\chi_{1323}(226,\cdot)$
• 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.h (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 63 \)
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	88	296
Cusp forms	288	72	216
Eisenstein series	96	16	80

Trace form

72 * q - 2 * q^2 + 66 * q^4 + 5 * q^5 + 12 * q^8 + 6 * q^10 - 3 * q^11 + 3 * q^13 + 54 * q^16 + 9 * q^17 + 4 * q^20 - 4 * q^23 - 21 * q^25 + 16 * q^26 + 18 * q^29 - 6 * q^31 + 82 * q^32 + 3 * q^37 + 19 * q^38 + 6 * q^40 + 10 * q^41 + 11 * q^44 - 12 * q^46 - 54 * q^47 + 45 * q^50 + 15 * q^52 - 16 * q^53 + 6 * q^55 + 9 * q^58 - 60 * q^59 - 12 * q^62 + 12 * q^64 + 60 * q^65 - 12 * q^67 + 30 * q^68 - 6 * q^71 - 12 * q^73 - 41 * q^74 - 6 * q^76 - 36 * q^79 + 19 * q^80 + 18 * q^83 + 3 * q^85 - 25 * q^86 - 9 * q^88 + 41 * q^89 + 52 * q^92 - 6 * q^94 - 34 * q^95 + 3 * q^97

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.h.a	$1323$	$2$	1323.h	63.h	$3$	$2$	$1$	$10.564$	\(\Q(\sqrt{-3}) \)	None					63.2.g.a	$2$	$0$	\(-2\)	\(0\)	\(1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-q^{2}-q^{4}+(1-\zeta_{6})q^{5}+3q^{8}+(-1+\cdots)q^{10}+\cdots\)
1323.2.h.b	$1323$	$2$	1323.h	63.h	$3$	$6$	$3$	$10.564$	\(\Q(\zeta_{18})\)	None					63.2.f.a	$2$	$0$	\(-6\)	\(0\)	\(-3\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta_{4}+\beta_{3}-1)q^{2}+(-2\beta_{4}-\beta_{3}+1)q^{4}+\cdots\)
1323.2.h.c	$1323$	$2$	1323.h	63.h	$3$	$6$	$3$	$10.564$	\(\Q(\zeta_{18})\)	None					63.2.f.a	$2$	$0$	\(-6\)	\(0\)	\(3\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta_{4}+\beta_{3}-1)q^{2}+(-2\beta_{4}-\beta_{3}+1)q^{4}+\cdots\)
1323.2.h.d	$1323$	$2$	1323.h	63.h	$3$	$6$	$3$	$10.564$	6.0.309123.1	None					63.2.f.b	$2$	$0$	\(2\)	\(0\)	\(-5\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(1-\beta _{1}+\beta _{3})q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots\)
1323.2.h.e	$1323$	$2$	1323.h	63.h	$3$	$6$	$3$	$10.564$	6.0.309123.1	None					63.2.f.b	$2$	$0$	\(2\)	\(0\)	\(5\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(1-\beta _{1}+\beta _{3})q^{4}+(\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots\)
1323.2.h.f	$1323$	$2$	1323.h	63.h	$3$	$10$	$5$	$10.564$	10.0.\(\cdots\).1	None					63.2.g.b	$2$	$0$	\(4\)	\(0\)	\(4\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}-\beta _{5})q^{2}+(1+\beta _{3})q^{4}+(-\beta _{6}+\cdots)q^{5}+\cdots\)
1323.2.h.g	$1323$	$2$	1323.h	63.h	$3$	$12$	$6$	$10.564$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					441.2.f.g	$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$3^{5}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{5})q^{4}+\beta _{2}q^{5}+\cdots\)
1323.2.h.h	$1323$	$2$	1323.h	63.h	$3$	$24$	$12$	$10.564$		None			✓		441.2.f.h	$4$	$0$	\(8\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{3}]$

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

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