Space of modular forms of level 1089, weight 3, and Character 1089.c

• Label: 1089.3.c
• Level: $1089$
• Weight: $3$
• Character orbit: 1089.c
• Rep. character: $\chi_{1089}(604,\cdot)$
• Character field: $\Q$
• Dimension: $86$
• Newform subspaces: $13$
• Sturm bound: $396$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 11 \)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(396\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1089, [\chi])\).

	Total	New	Old
Modular forms	288	94	194
Cusp forms	240	86	154
Eisenstein series	48	8	40

Trace form

\( 86 q - 168 q^{4} + 6 q^{5} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1089, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1089.3.c.a	$1089$	$3$	1089.c	11.b	$2$	$2$	$2$	$29.673$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(14\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}+2q^{4}+7q^{5}-5\beta q^{7}+6\beta q^{8}+\cdots\)
1089.3.c.b	$1089$	$3$	1089.c	11.b	$2$	$4$	$4$	$29.673$	\(\Q(\sqrt{-2}, \sqrt{-11})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}-7q^{4}-2\beta _{1}q^{5}+7\beta _{2}q^{7}+\cdots\)
1089.3.c.c	$1089$	$3$	1089.c	11.b	$2$	$4$	$4$	$29.673$	\(\Q(\sqrt{-2}, \sqrt{3})\)	None		$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{1}q^{2}+(-4+4\beta _{2})q^{4}+(2-2\beta _{2}+\cdots)q^{5}+\cdots\)
1089.3.c.d	$1089$	$3$	1089.c	11.b	$2$	$4$	$4$	$29.673$	\(\Q(\sqrt{-2}, \sqrt{3})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2\beta _{1}+\beta _{3})q^{2}+(-2+3\beta _{2})q^{4}+\cdots\)
1089.3.c.e	$1089$	$3$	1089.c	11.b	$2$	$4$	$4$	$29.673$	\(\Q(\zeta_{10})\)	None		$2$	$0$	\(0\)	\(0\)	\(-16\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\zeta_{10}-\zeta_{10}^{3})q^{2}+(1+4\zeta_{10}^{2})q^{4}+\cdots\)
1089.3.c.f	$1089$	$3$	1089.c	11.b	$2$	$4$	$4$	$29.673$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+q^{4}-2\beta _{3}q^{5}-5\beta _{1}q^{7}+\cdots\)
1089.3.c.g	$1089$	$3$	1089.c	11.b	$2$	$4$	$4$	$29.673$	\(\Q(\sqrt{-2}, \sqrt{3})\)	None		$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(2+3\beta _{2})q^{5}+\cdots\)
1089.3.c.h	$1089$	$3$	1089.c	11.b	$2$	$4$	$4$	$29.673$	\(\Q(\sqrt{-2}, \sqrt{3})\)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$11^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+4q^{4}+(-3\beta _{1}-\beta _{2})q^{7}+(-5\beta _{1}+\cdots)q^{13}+\cdots\)
1089.3.c.i	$1089$	$3$	1089.c	11.b	$2$	$8$	$8$	$29.673$	8.0.\(\cdots\).2	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{2}+(-4+\beta _{2})q^{4}+(-\beta _{3}+\beta _{7})q^{5}+\cdots\)
1089.3.c.j	$1089$	$3$	1089.c	11.b	$2$	$8$	$8$	$29.673$	8.0.\(\cdots\).15	None		$2$	$0$	\(0\)	\(0\)	\(-16\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{4})q^{2}+(-3-\beta _{6})q^{4}+(-2+\cdots)q^{5}+\cdots\)
1089.3.c.k	$1089$	$3$	1089.c	11.b	$2$	$8$	$8$	$29.673$	8.0.\(\cdots\).1	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$2\cdot 11$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-3+\beta _{4}+\beta _{5})q^{4}+(\beta _{3}+\cdots)q^{5}+\cdots\)
1089.3.c.l	$1089$	$3$	1089.c	11.b	$2$	$16$	$16$	$29.673$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 11^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+(-3+\beta _{2})q^{4}+\beta _{8}q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
1089.3.c.m	$1089$	$3$	1089.c	11.b	$2$	$16$	$16$	$29.673$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$2^{8}\cdot 11^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{2}+(-1+\beta _{2}-\beta _{3})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1089, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1089, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(11, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(33, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(99, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(121, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(363, [\chi])\)\(^{\oplus 2}\)