Space of modular forms of level 1728, weight 3, and Character 1728.o

• Label: 1728.3.o
• Level: $1728$
• Weight: $3$
• Character orbit: 1728.o
• Rep. character: $\chi_{1728}(127,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $92$
• Newform subspaces: $9$
• Sturm bound: $864$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.o (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 36 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(864\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1728, [\chi])\).

	Total	New	Old
Modular forms	1224	100	1124
Cusp forms	1080	92	988
Eisenstein series	144	8	136

Trace form

\( 92 q - 2 q^{5} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1728, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1728.3.o.a	$1728$	$3$	1728.o	36.f	$6$	$2$	$1$	$47.085$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-4+4\zeta_{6})q^{5}+(-4+2\zeta_{6})q^{7}+\cdots\)
1728.3.o.b	$1728$	$3$	1728.o	36.f	$6$	$2$	$1$	$47.085$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-4+4\zeta_{6})q^{5}+(4-2\zeta_{6})q^{7}+(14+\cdots)q^{11}+\cdots\)
1728.3.o.c	$1728$	$3$	1728.o	36.f	$6$	$4$	$2$	$47.085$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(-14\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-7+7\zeta_{12}^{2})q^{5}+(-5\zeta_{12}+5\zeta_{12}^{3})q^{7}+\cdots\)
1728.3.o.d	$1728$	$3$	1728.o	36.f	$6$	$8$	$4$	$47.085$	8.0.121550625.1	None		$4$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$2^{4}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2\beta _{2}-\beta _{4}+\beta _{6})q^{5}-\beta _{3}q^{7}+(\beta _{3}+\cdots)q^{11}+\cdots\)
1728.3.o.e	$1728$	$3$	1728.o	36.f	$6$	$8$	$4$	$47.085$	8.0.856615824.2	None		$2$	$0$	\(0\)	\(0\)	\(3\)	\(-3\)	$2^{2}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\beta _{1}+\beta _{4})q^{5}+(\beta _{4}+\beta _{5}+\beta _{7})q^{7}+\cdots\)
1728.3.o.f	$1728$	$3$	1728.o	36.f	$6$	$8$	$4$	$47.085$	8.0.856615824.2	None		$2$	$0$	\(0\)	\(0\)	\(3\)	\(3\)	$2^{2}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\beta _{1}+\beta _{4})q^{5}+(-\beta _{4}-\beta _{5}-\beta _{7})q^{7}+\cdots\)
1728.3.o.g	$1728$	$3$	1728.o	36.f	$6$	$16$	$8$	$47.085$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$2^{16}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\beta _{1}+\beta _{6})q^{5}+(\beta _{3}-\beta _{8}-\beta _{9}+\cdots)q^{7}+\cdots\)
1728.3.o.h	$1728$	$3$	1728.o	36.f	$6$	$20$	$10$	$47.085$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(14\)	\(0\)	$2^{26}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1+\beta _{2}+\beta _{3}-\beta _{6})q^{5}+(-\beta _{10}+\beta _{12}+\cdots)q^{7}+\cdots\)
1728.3.o.i	$1728$	$3$	1728.o	36.f	$6$	$24$	$12$	$47.085$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{3}^{\mathrm{old}}(1728, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1728, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(864, [\chi])\)\(^{\oplus 2}\)