Space of modular forms of level 728, weight 2, and Character 728.r

• Label: 728.2.r
• Level: $728$
• Weight: $2$
• Character orbit: 728.r
• Rep. character: $\chi_{728}(417,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $48$
• Newform subspaces: $7$
• Sturm bound: $224$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 728 = 2^{3} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	728.r (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(224\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(728, [\chi])\).

	Total	New	Old
Modular forms	240	48	192
Cusp forms	208	48	160
Eisenstein series	32	0	32

Trace form

\( 48 q - 4 q^{5} - 4 q^{7} - 20 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(728, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
728.2.r.a	$728$	$2$	728.r	7.c	$3$	$2$	$1$	$5.813$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(1\)	\(-3\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}-3\zeta_{6}q^{5}+(3-2\zeta_{6})q^{7}+\cdots\)
728.2.r.b	$728$	$2$	728.r	7.c	$3$	$2$	$1$	$5.813$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(1\)	\(-1\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)
728.2.r.c	$728$	$2$	728.r	7.c	$3$	$2$	$1$	$5.813$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(1\)	\(3\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+3\zeta_{6}q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)
728.2.r.d	$728$	$2$	728.r	7.c	$3$	$4$	$2$	$5.813$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\beta _{2}q^{3}-\beta _{1}q^{7}+(-1-\beta _{2})q^{9}+\cdots\)
728.2.r.e	$728$	$2$	728.r	7.c	$3$	$8$	$4$	$5.813$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(1\)	\(-1\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-\beta _{6})q^{3}+(-\beta _{1}-\beta _{5})q^{5}+\cdots\)
728.2.r.f	$728$	$2$	728.r	7.c	$3$	$14$	$7$	$5.813$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None		$2$	$0$	\(0\)	\(2\)	\(-2\)	\(-2\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{3}-\beta _{9}q^{5}-\beta _{4}q^{7}+(-1-\beta _{3}+\cdots)q^{9}+\cdots\)
728.2.r.g	$728$	$2$	728.r	7.c	$3$	$16$	$8$	$5.813$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(1\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}-\beta _{2})q^{3}-\beta _{10}q^{5}+(\beta _{6}-\beta _{13}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(728, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(728, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(91, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(182, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(364, [\chi])\)\(^{\oplus 2}\)