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

• Label: 1728.2.f
• Level: $1728$
• Weight: $2$
• Character orbit: 1728.f
• Rep. character: $\chi_{1728}(863,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $7$
• Sturm bound: $576$
• Trace bound: $49$

Defining parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 24 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(576\)
Trace bound:			\(49\)
Distinguishing \(T_p\):			\(5\), \(7\), \(23\)

Dimensions

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

	Total	New	Old
Modular forms	324	32	292
Cusp forms	252	32	220
Eisenstein series	72	0	72

Trace form

\( 32 q + O(q^{10}) \)

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	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1728.2.f.a	$1728$	$2$	1728.f	24.f	$2$	$4$	$4$	$13.798$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{12}^{3}q^{5}+\zeta_{12}q^{7}+\zeta_{12}^{2}q^{11}+\cdots\)
1728.2.f.b	$1728$	$2$	1728.f	24.f	$2$	$4$	$4$	$13.798$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\zeta_{12}^{3}q^{5}+\zeta_{12}q^{7}+2\zeta_{12}^{2}q^{11}+\cdots\)
1728.2.f.c	$1728$	$2$	1728.f	24.f	$2$	$4$	$4$	$13.798$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+5\zeta_{12}q^{7}+\zeta_{12}^{2}q^{13}-5\zeta_{12}^{3}q^{19}+\cdots\)
1728.2.f.d	$1728$	$2$	1728.f	24.f	$2$	$4$	$4$	$13.798$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\zeta_{12}q^{7}+\zeta_{12}^{2}q^{13}+\zeta_{12}^{3}q^{19}+\cdots\)
1728.2.f.e	$1728$	$2$	1728.f	24.f	$2$	$4$	$4$	$13.798$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{12}^{3}q^{5}+\zeta_{12}q^{7}+\zeta_{12}^{2}q^{11}+\cdots\)
1728.2.f.f	$1728$	$2$	1728.f	24.f	$2$	$4$	$4$	$13.798$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\zeta_{12}^{3}q^{5}+\zeta_{12}q^{7}+2\zeta_{12}^{2}q^{11}+\cdots\)
1728.2.f.g	$1728$	$2$	1728.f	24.f	$2$	$8$	$8$	$13.798$	\(\Q(\zeta_{24})\)	\(\Q(\sqrt{-6}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q-\zeta_{24}q^{5}+\zeta_{24}^{3}q^{7}-\zeta_{24}^{6}q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1728, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 12}\)