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

• Label: 1728.2.i
• Level: $1728$
• Weight: $2$
• Character orbit: 1728.i
• Rep. character: $\chi_{1728}(577,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $44$
• Newform subspaces: $14$
• Sturm bound: $576$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 14 \)
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

\( 44 q - 2 q^{5} + O(q^{10}) \)

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.i.a	$1728$	$2$	1728.i	9.c	$3$	$2$	$1$	$13.798$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(-5+5\zeta_{6})q^{11}+\cdots\)
1728.2.i.b	$1728$	$2$	1728.i	9.c	$3$	$2$	$1$	$13.798$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\zeta_{6}q^{5}+(2-2\zeta_{6})q^{7}+(5-5\zeta_{6})q^{11}+\cdots\)
1728.2.i.c	$1728$	$2$	1728.i	9.c	$3$	$2$	$1$	$13.798$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(0\)	\(0\)	\(-3\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-3\zeta_{6}q^{5}+(-1+\zeta_{6})q^{7}+(3-3\zeta_{6})q^{11}+\cdots\)
1728.2.i.d	$1728$	$2$	1728.i	9.c	$3$	$2$	$1$	$13.798$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(0\)	\(0\)	\(-3\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-3\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+(-3+3\zeta_{6})q^{11}+\cdots\)
1728.2.i.e	$1728$	$2$	1728.i	9.c	$3$	$2$	$1$	$13.798$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{7}+(3-3\zeta_{6})q^{11}+2\zeta_{6}q^{13}+\cdots\)
1728.2.i.f	$1728$	$2$	1728.i	9.c	$3$	$2$	$1$	$13.798$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{7}+(-3+3\zeta_{6})q^{11}+2\zeta_{6}q^{13}+\cdots\)
1728.2.i.g	$1728$	$2$	1728.i	9.c	$3$	$2$	$1$	$13.798$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(1\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+(5-5\zeta_{6})q^{11}+\cdots\)
1728.2.i.h	$1728$	$2$	1728.i	9.c	$3$	$2$	$1$	$13.798$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(1\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{5}+(3-3\zeta_{6})q^{7}+(-5+5\zeta_{6})q^{11}+\cdots\)
1728.2.i.i	$1728$	$2$	1728.i	9.c	$3$	$4$	$2$	$13.798$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		$2$	$0$	\(0\)	\(0\)	\(1\)	\(-3\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{3}q^{5}+(-1+2\beta _{1}+\beta _{2}-\beta _{3})q^{7}+\cdots\)
1728.2.i.j	$1728$	$2$	1728.i	9.c	$3$	$4$	$2$	$13.798$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		$2$	$0$	\(0\)	\(0\)	\(1\)	\(3\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{3}q^{5}+(1-2\beta _{1}-\beta _{2}+\beta _{3})q^{7}+\cdots\)
1728.2.i.k	$1728$	$2$	1728.i	9.c	$3$	$4$	$2$	$13.798$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{1})q^{5}+(-\beta _{1}+\beta _{2})q^{7}+(\beta _{1}+\cdots)q^{11}+\cdots\)
1728.2.i.l	$1728$	$2$	1728.i	9.c	$3$	$4$	$2$	$13.798$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{12})q^{5}-\zeta_{12}^{2}q^{7}+\zeta_{12}^{2}q^{11}+\cdots\)
1728.2.i.m	$1728$	$2$	1728.i	9.c	$3$	$4$	$2$	$13.798$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(2\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{1})q^{5}+(\beta _{1}-\beta _{2})q^{7}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
1728.2.i.n	$1728$	$2$	1728.i	9.c	$3$	$8$	$4$	$13.798$	8.0.170772624.1	None		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{2}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{6}q^{5}+(\beta _{1}-\beta _{7})q^{7}+\beta _{3}q^{11}+(2\beta _{4}+\cdots)q^{13}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1728, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(864, [\chi])\)\(^{\oplus 2}\)