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

• Label: 1728.2.c
• Level: $1728$
• Weight: $2$
• Character orbit: 1728.c
• Rep. character: $\chi_{1728}(1727,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $7$
• Sturm bound: $576$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 12 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(576\)
Trace bound:			\(13\)
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	324	32	292
Cusp forms	252	32	220
Eisenstein series	72	0	72

Trace form

\( 32 q + 8 q^{13} - 32 q^{25} - 8 q^{37} - 32 q^{49} - 24 q^{61} + 16 q^{85} + 16 q^{97}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1728.2.c.a	$1728$	$2$	1728.c	12.b	$2$	$2$	$2$	$13.798$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					432.2.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta q^{7}-7 q^{13}+5\beta q^{19}+5 q^{25}+\cdots\)
1728.2.c.b	$1728$	$2$	1728.c	12.b	$2$	$2$	$2$	$13.798$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					432.2.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 3$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta q^{7}+5 q^{13}+\beta q^{19}+5 q^{25}+\cdots\)
1728.2.c.c	$1728$	$2$	1728.c	12.b	$2$	$4$	$4$	$13.798$	\(\Q(\sqrt{3}, \sqrt{-5})\)	None					108.2.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{5}-\beta _{2}q^{7}-\beta _{3}q^{11}-2q^{13}+\cdots\)
1728.2.c.d	$1728$	$2$	1728.c	12.b	$2$	$4$	$4$	$13.798$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					108.2.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{5}+\beta _{2}q^{7}+\beta _{3}q^{11}+q^{13}+\cdots\)
1728.2.c.e	$1728$	$2$	1728.c	12.b	$2$	$4$	$4$	$13.798$	\(\Q(\zeta_{12})\)	None					432.2.c.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{5}-\beta_{2} q^{7}+\beta_{3} q^{11}+2 q^{13}+\cdots\)
1728.2.c.f	$1728$	$2$	1728.c	12.b	$2$	$8$	$8$	$13.798$	\(\Q(\zeta_{24})\)	None					864.2.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{2} q^{5}+(\beta_{3}+\beta_1)q^{7}-\beta_{6} q^{11}+\cdots\)
1728.2.c.g	$1728$	$2$	1728.c	12.b	$2$	$8$	$8$	$13.798$	\(\Q(\zeta_{24})\)	None					864.2.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{3} q^{5}-\beta_{7} q^{7}+(\beta_{5}+\beta_1)q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1728, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(864, [\chi])\)\(^{\oplus 2}\)