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

• Label: 1728.2.s
• Level: $1728$
• Weight: $2$
• Character orbit: 1728.s
• Rep. character: $\chi_{1728}(575,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $44$
• Newform subspaces: $7$
• Sturm bound: $576$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.s (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 36 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(576\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	648	52	596
Cusp forms	504	44	460
Eisenstein series	144	8	136

Trace form

\( 44 q - 6 q^{5} + O(q^{10}) \)

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.s.a	$1728$	$2$	1728.s	36.h	$6$	$2$	$1$	$13.798$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-6\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-4+2\zeta_{6})q^{5}+(-2-2\zeta_{6})q^{7}+\cdots\)
1728.2.s.b	$1728$	$2$	1728.s	36.h	$6$	$2$	$1$	$13.798$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-6\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-4+2\zeta_{6})q^{5}+(2+2\zeta_{6})q^{7}+(3+\cdots)q^{11}+\cdots\)
1728.2.s.c	$1728$	$2$	1728.s	36.h	$6$	$2$	$1$	$13.798$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(3\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2-\zeta_{6})q^{5}+(-1-\zeta_{6})q^{7}+(3-3\zeta_{6})q^{11}+\cdots\)
1728.2.s.d	$1728$	$2$	1728.s	36.h	$6$	$2$	$1$	$13.798$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(3\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2-\zeta_{6})q^{5}+(1+\zeta_{6})q^{7}+(-3+3\zeta_{6})q^{11}+\cdots\)
1728.2.s.e	$1728$	$2$	1728.s	36.h	$6$	$4$	$2$	$13.798$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2-\zeta_{12}^{2})q^{5}-\zeta_{12}q^{7}+(-\zeta_{12}+\cdots)q^{11}+\cdots\)
1728.2.s.f	$1728$	$2$	1728.s	36.h	$6$	$8$	$4$	$13.798$	8.0.170772624.1	None		$4$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{2})q^{5}+(\beta _{1}-\beta _{6})q^{7}+(-\beta _{3}+\cdots)q^{11}+\cdots\)
1728.2.s.g	$1728$	$2$	1728.s	36.h	$6$	$24$	$12$	$13.798$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

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

\( S_{2}^{\mathrm{old}}(1728, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(864, [\chi])\)\(^{\oplus 2}\)