Space of modular forms of level 1728, weight 4, and Character 1728.f

• Label: 1728.4.f
• Level: $1728$
• Weight: $4$
• Character orbit: 1728.f
• Rep. character: $\chi_{1728}(863,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $10$
• Sturm bound: $1152$
• Trace bound: $49$

Defining parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 24 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(1152\)
Trace bound:			\(49\)
Distinguishing \(T_p\):			\(5\), \(7\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(1728, [\chi])\).

	Total	New	Old
Modular forms	900	96	804
Cusp forms	828	96	732
Eisenstein series	72	0	72

Trace form

\( 96 q + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(1728, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1728.4.f.a	$1728$	$4$	1728.f	24.f	$2$	$4$	$4$	$101.955$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-37\zeta_{12}q^{7}+53\zeta_{12}^{2}q^{13}+17\zeta_{12}^{3}q^{19}+\cdots\)
1728.4.f.b	$1728$	$4$	1728.f	24.f	$2$	$4$	$4$	$101.955$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+17\zeta_{12}q^{7}+17\zeta_{12}^{2}q^{13}-73\zeta_{12}^{3}q^{19}+\cdots\)
1728.4.f.c	$1728$	$4$	1728.f	24.f	$2$	$8$	$8$	$101.955$	8.0.\(\cdots\).3	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{5}+(-2\beta _{3}+\beta _{7})q^{7}+(-8\beta _{1}+\cdots)q^{11}+\cdots\)
1728.4.f.d	$1728$	$4$	1728.f	24.f	$2$	$8$	$8$	$101.955$	8.0.\(\cdots\).3	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{5}-13\beta _{3}q^{7}+13\beta _{1}q^{11}+\beta _{5}q^{13}+\cdots\)
1728.4.f.e	$1728$	$4$	1728.f	24.f	$2$	$8$	$8$	$101.955$	8.0.12960000.1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{5}-13\beta _{3}q^{7}+\beta _{6}q^{11}-13\beta _{1}q^{13}+\cdots\)
1728.4.f.f	$1728$	$4$	1728.f	24.f	$2$	$8$	$8$	$101.955$	\(\Q(\zeta_{24})\)	\(\Q(\sqrt{-6}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{8}$	$\mathrm{U}(1)[D_{2}]$	\(q+(2\zeta_{24}^{2}-\zeta_{24}^{5})q^{5}+(-8\zeta_{24}+\zeta_{24}^{4}+\cdots)q^{7}+\cdots\)
1728.4.f.g	$1728$	$4$	1728.f	24.f	$2$	$8$	$8$	$101.955$	8.0.\(\cdots\).3	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{5}+(-2\beta _{3}+\beta _{7})q^{7}+(-8\beta _{1}+\cdots)q^{11}+\cdots\)
1728.4.f.h	$1728$	$4$	1728.f	24.f	$2$	$16$	$16$	$101.955$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{30}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{9}q^{5}+(3\beta _{5}-\beta _{13})q^{7}+(2\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
1728.4.f.i	$1728$	$4$	1728.f	24.f	$2$	$16$	$16$	$101.955$	16.0.\(\cdots\).1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{48}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{5}+(-\beta _{6}-4\beta _{7})q^{7}-\beta _{10}q^{11}+\cdots\)
1728.4.f.j	$1728$	$4$	1728.f	24.f	$2$	$16$	$16$	$101.955$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{30}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{9}q^{5}+(-3\beta _{5}+\beta _{13})q^{7}+(-2\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(1728, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(1728, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 12}\)