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

• Label: 1728.3.g
• Level: $1728$
• Weight: $3$
• Character orbit: 1728.g
• Rep. character: $\chi_{1728}(703,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $14$
• Sturm bound: $864$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(864\)
Trace bound:			\(13\)
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	612	64	548
Cusp forms	540	64	476
Eisenstein series	72	0	72

Trace form

\( 64 q + O(q^{10}) \)

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.g.a	$1728$	$3$	1728.g	4.b	$2$	$2$	$2$	$47.085$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-14\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-7q^{5}-5\zeta_{6}q^{7}-5\zeta_{6}q^{11}-20q^{13}+\cdots\)
1728.3.g.b	$1728$	$3$	1728.g	4.b	$2$	$2$	$2$	$47.085$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-3q^{5}-\zeta_{6}q^{7}+3\zeta_{6}q^{11}-4q^{13}+\cdots\)
1728.3.g.c	$1728$	$3$	1728.g	4.b	$2$	$2$	$2$	$47.085$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 3$	$\mathrm{U}(1)[D_{2}]$	\(q+\zeta_{6}q^{7}-q^{13}-7\zeta_{6}q^{19}-5^{2}q^{25}+\cdots\)
1728.3.g.d	$1728$	$3$	1728.g	4.b	$2$	$2$	$2$	$47.085$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-5\zeta_{6}q^{7}+23q^{13}-5\zeta_{6}q^{19}-5^{2}q^{25}+\cdots\)
1728.3.g.e	$1728$	$3$	1728.g	4.b	$2$	$2$	$2$	$47.085$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+3q^{5}-\zeta_{6}q^{7}-3\zeta_{6}q^{11}-4q^{13}+\cdots\)
1728.3.g.f	$1728$	$3$	1728.g	4.b	$2$	$2$	$2$	$47.085$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(14\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+7q^{5}-5\zeta_{6}q^{7}+5\zeta_{6}q^{11}-20q^{13}+\cdots\)
1728.3.g.g	$1728$	$3$	1728.g	4.b	$2$	$4$	$4$	$47.085$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{5}+\beta _{2}q^{7}-\beta _{3}q^{11}-13q^{13}+\cdots\)
1728.3.g.h	$1728$	$3$	1728.g	4.b	$2$	$4$	$4$	$47.085$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{5}-\beta _{2}q^{7}+\beta _{3}q^{11}+8q^{13}+\cdots\)
1728.3.g.i	$1728$	$3$	1728.g	4.b	$2$	$4$	$4$	$47.085$	\(\Q(\sqrt{-3}, \sqrt{13})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{5}-\beta _{3}q^{7}-7\beta _{1}q^{11}+2^{4}q^{13}+\cdots\)
1728.3.g.j	$1728$	$3$	1728.g	4.b	$2$	$8$	$8$	$47.085$	8.0.56070144.2	None		$2$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{12}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{3})q^{5}+(\beta _{2}+\beta _{6})q^{7}+(\beta _{2}+\cdots)q^{11}+\cdots\)
1728.3.g.k	$1728$	$3$	1728.g	4.b	$2$	$8$	$8$	$47.085$	8.0.22581504.2	None		$2$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{14}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{2})q^{5}+\beta _{7}q^{7}+(\beta _{3}+\beta _{6}+\cdots)q^{11}+\cdots\)
1728.3.g.l	$1728$	$3$	1728.g	4.b	$2$	$8$	$8$	$47.085$	8.0.207360000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{5}+\beta _{7}q^{7}+\beta _{3}q^{11}+(1-\beta _{6}+\cdots)q^{13}+\cdots\)
1728.3.g.m	$1728$	$3$	1728.g	4.b	$2$	$8$	$8$	$47.085$	8.0.56070144.2	None		$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{12}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{3})q^{5}+(\beta _{2}+\beta _{6})q^{7}+(-\beta _{2}+\cdots)q^{11}+\cdots\)
1728.3.g.n	$1728$	$3$	1728.g	4.b	$2$	$8$	$8$	$47.085$	8.0.22581504.2	None		$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{14}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{2})q^{5}+\beta _{6}q^{7}+(\beta _{4}+\beta _{5}+\beta _{6}+\cdots)q^{11}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(1728, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 15}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(864, [\chi])\)\(^{\oplus 2}\)