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

• Label: 1728.2.r
• Level: $1728$
• Weight: $2$
• Character orbit: 1728.r
• Rep. character: $\chi_{1728}(289,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $48$
• Newform subspaces: $6$
• Sturm bound: $576$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.r (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 72 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(576\)
Trace bound:			\(17\)
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

\( 48 q + O(q^{10}) \)

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.r.a	$1728$	$2$	1728.r	72.n	$6$	$4$	$2$	$13.798$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(-12\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2-2\zeta_{12}^{2})q^{5}+(-3\zeta_{12}+3\zeta_{12}^{3})q^{11}+\cdots\)
1728.2.r.b	$1728$	$2$	1728.r	72.n	$6$	$4$	$2$	$13.798$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(12\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(4-2\zeta_{12}^{2})q^{5}+3\zeta_{12}q^{11}+(-4+\cdots)q^{13}+\cdots\)
1728.2.r.c	$1728$	$2$	1728.r	72.n	$6$	$8$	$4$	$13.798$	\(\Q(\zeta_{24})\)	\(\Q(\sqrt{-2}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}$	$\mathrm{U}(1)[D_{6}]$	\(q+(-\zeta_{24}-\zeta_{24}^{3})q^{11}+(-3+\zeta_{24}^{5}+\cdots)q^{17}+\cdots\)
1728.2.r.d	$1728$	$2$	1728.r	72.n	$6$	$8$	$4$	$13.798$	8.0.12960000.1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{4}q^{5}+(-\beta _{5}+\beta _{7})q^{7}-3\beta _{1}q^{11}+\cdots\)
1728.2.r.e	$1728$	$2$	1728.r	72.n	$6$	$12$	$6$	$13.798$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(-18\)	\(0\)	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2-\beta _{4})q^{5}+(-\beta _{2}-\beta _{11})q^{7}+\cdots\)
1728.2.r.f	$1728$	$2$	1728.r	72.n	$6$	$12$	$6$	$13.798$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(18\)	\(0\)	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2+\beta _{4})q^{5}+(\beta _{2}+\beta _{11})q^{7}+(-\beta _{5}+\cdots)q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1728, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(864, [\chi])\)\(^{\oplus 2}\)