Space of modular forms of level 1728, weight 2, and Character 1728.p

• Label: 1728.2.p
• Level: $1728$
• Weight: $2$
• Character orbit: 1728.p
• Rep. character: $\chi_{1728}(287,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $48$
• Newform subspaces: $3$
• Sturm bound: $576$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.p (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 72 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(576\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1728, [\chi])\).

	Total	New	Old
Modular forms	648	48	600
Cusp forms	504	48	456
Eisenstein series	144	0	144

Trace form

\( 48 q - 24 q^{25} - 72 q^{41} + 24 q^{49}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1728, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1728.2.p.a	$1728$	$2$	1728.p	72.l	$6$	$16$	$8$	$13.798$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None					576.2.p.a	$4$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{2}+\beta _{3}+\beta _{5})q^{5}+(\beta _{10}+\beta _{12}+\beta _{14}+\cdots)q^{7}+\cdots\)
1728.2.p.b	$1728$	$2$	1728.p	72.l	$6$	$16$	$8$	$13.798$	16.0.\(\cdots\).3	None					576.2.p.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{5}+\beta _{14}q^{7}+(-\beta _{7}-\beta _{10})q^{11}+\cdots\)
1728.2.p.c	$1728$	$2$	1728.p	72.l	$6$	$16$	$8$	$13.798$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None					576.2.p.a	$4$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1+\beta _{3}+\beta _{5})q^{5}+(-\beta _{14}+\beta _{15})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1728, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1728, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(864, [\chi])\)\(^{\oplus 2}\)