Space of modular forms of level 1089, weight 2, and Character 1089.e

• Label: 1089.2.e
• Level: $1089$
• Weight: $2$
• Character orbit: 1089.e
• Rep. character: $\chi_{1089}(364,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $200$
• Newform subspaces: $16$
• Sturm bound: $264$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1089.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(264\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1089, [\chi])\).

	Total	New	Old
Modular forms	288	236	52
Cusp forms	240	200	40
Eisenstein series	48	36	12

Trace form

200 * q - 90 * q^4 + 4 * q^5 - 2 * q^6 + 2 * q^7 + 12 * q^8 + 4 * q^9 - 22 * q^12 + 2 * q^13 - 6 * q^14 - 10 * q^15 - 70 * q^16 + 12 * q^17 - 4 * q^18 - 4 * q^19 - 2 * q^20 + 4 * q^21 - 6 * q^23 - 18 * q^24 - 60 * q^25 - 8 * q^26 + 30 * q^27 - 16 * q^28 - 12 * q^29 + 22 * q^30 + 8 * q^31 - 12 * q^32 + 2 * q^34 + 2 * q^36 - 4 * q^37 + 40 * q^38 + 40 * q^39 + 12 * q^40 - 12 * q^41 - 6 * q^42 + 8 * q^43 - 50 * q^45 - 2 * q^47 + 80 * q^48 - 50 * q^49 + 6 * q^50 + 34 * q^51 - 10 * q^52 - 8 * q^53 + 28 * q^54 - 18 * q^56 - 18 * q^57 - 18 * q^58 + 28 * q^59 - 62 * q^60 + 2 * q^61 + 60 * q^62 - 52 * q^63 + 72 * q^64 - 18 * q^65 + 4 * q^67 + 18 * q^68 + 14 * q^69 - 32 * q^70 - 76 * q^71 - 84 * q^72 - 28 * q^73 - 54 * q^74 - 2 * q^75 + 20 * q^76 - 4 * q^78 + 2 * q^79 + 176 * q^80 - 20 * q^81 - 12 * q^82 - 22 * q^84 + 24 * q^85 - 28 * q^86 - 8 * q^89 + 128 * q^90 - 44 * q^91 + 36 * q^92 - 18 * q^93 - 30 * q^94 - 36 * q^95 + 68 * q^96 + 26 * q^97 - 36 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(1089, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1089.2.e.a	$1089$	$2$	1089.e	9.c	$3$	$2$	$1$	$8.696$	\(\Q(\sqrt{-3}) \)	None					99.2.e.c	$2$	$0$	\(-2\)	\(0\)	\(2\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}+(1-2\zeta_{6})q^{3}-2\zeta_{6}q^{4}+\cdots\)
1089.2.e.b	$1089$	$2$	1089.e	9.c	$3$	$2$	$1$	$8.696$	\(\Q(\sqrt{-3}) \)	None					99.2.e.b	$2$	$0$	\(0\)	\(-3\)	\(3\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\zeta_{6})q^{3}+2\zeta_{6}q^{4}+3\zeta_{6}q^{5}+\cdots\)
1089.2.e.c	$1089$	$2$	1089.e	9.c	$3$	$2$	$1$	$8.696$	\(\Q(\sqrt{-3}) \)	None					99.2.e.a	$2$	$0$	\(1\)	\(-3\)	\(-1\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
1089.2.e.d	$1089$	$2$	1089.e	9.c	$3$	$4$	$2$	$8.696$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None					99.2.m.a	$2$	$1$	\(-3\)	\(-6\)	\(-2\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{1}-\beta _{3})q^{2}+(-2-\beta _{3})q^{3}+\cdots\)
1089.2.e.e	$1089$	$2$	1089.e	9.c	$3$	$4$	$2$	$8.696$	\(\Q(\zeta_{12})\)	None		✓			1089.2.e.e	$4$	$0$	\(0\)	\(-6\)	\(6\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta_{2} q^{2}+(-\beta_1-1)q^{3}+(\beta_1-1)q^{4}+\cdots\)
1089.2.e.f	$1089$	$2$	1089.e	9.c	$3$	$4$	$2$	$8.696$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	\(\Q(\sqrt{-11}) \)		✓			1089.2.e.f	$4$	$0$	\(0\)	\(1\)	\(-3\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q-\beta _{3}q^{3}+2\beta _{2}q^{4}+(1-2\beta _{1}-2\beta _{2}+\cdots)q^{5}+\cdots\)
1089.2.e.g	$1089$	$2$	1089.e	9.c	$3$	$4$	$2$	$8.696$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None					99.2.m.a	$2$	$0$	\(3\)	\(-6\)	\(-2\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}+\beta _{3})q^{2}+(-2-\beta _{3})q^{3}+\cdots\)
1089.2.e.h	$1089$	$2$	1089.e	9.c	$3$	$6$	$3$	$8.696$	\(\Q(\zeta_{18})\)	None					99.2.e.d	$2$	$0$	\(0\)	\(0\)	\(-3\)	\(-3\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta_{5}+\beta_{4}+\cdots-\beta_{2})q^{2}+\cdots+(\beta_{4}+\beta_{2})q^{3}+\cdots\)
1089.2.e.i	$1089$	$2$	1089.e	9.c	$3$	$8$	$4$	$8.696$	8.0.508277025.1	None					99.2.e.e	$2$	$0$	\(1\)	\(5\)	\(-4\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{2}+\beta _{7})q^{2}+(1+\beta _{7})q^{3}+(-2+\beta _{1}+\cdots)q^{4}+\cdots\)
1089.2.e.j	$1089$	$2$	1089.e	9.c	$3$	$12$	$6$	$8.696$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			1089.2.e.j	$4$	$0$	\(0\)	\(4\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{7}q^{2}+(1+\beta _{2}-\beta _{6})q^{3}+(-1-2\beta _{4}+\cdots)q^{4}+\cdots\)
1089.2.e.k	$1089$	$2$	1089.e	9.c	$3$	$16$	$8$	$8.696$	16.0.\(\cdots\).1	None		✓			1089.2.e.k	$4$	$0$	\(0\)	\(-2\)	\(4\)	\(0\)	$3^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-\beta _{2}-\beta _{6})q^{2}-\beta _{13}q^{3}+(2\beta _{5}+\cdots)q^{4}+\cdots\)
1089.2.e.l	$1089$	$2$	1089.e	9.c	$3$	$20$	$10$	$8.696$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		✓			1089.2.e.l	$2$	$0$	\(0\)	\(-1\)	\(-3\)	\(1\)	$3^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{3}q^{2}-\beta _{16}q^{3}+(-\beta _{12}+\beta _{17}+\cdots)q^{4}+\cdots\)
1089.2.e.m	$1089$	$2$	1089.e	9.c	$3$	$20$	$10$	$8.696$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		✓			1089.2.e.l	$2$	$0$	\(0\)	\(-1\)	\(-3\)	\(-1\)	$3^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{3}q^{2}-\beta _{16}q^{3}+(-\beta _{12}+\beta _{17}+\cdots)q^{4}+\cdots\)
1089.2.e.n	$1089$	$2$	1089.e	9.c	$3$	$24$	$12$	$8.696$		None		✓			1089.2.e.n	$4$	$0$	\(0\)	\(0\)	\(6\)	\(0\)		$\mathrm{SU}(2)[C_{3}]$
1089.2.e.o	$1089$	$2$	1089.e	9.c	$3$	$36$	$18$	$8.696$		None					99.2.m.b	$2$	$0$	\(-2\)	\(9\)	\(1\)	\(1\)		$\mathrm{SU}(2)[C_{3}]$
1089.2.e.p	$1089$	$2$	1089.e	9.c	$3$	$36$	$18$	$8.696$		None					99.2.m.b	$2$	$0$	\(2\)	\(9\)	\(1\)	\(-1\)		$\mathrm{SU}(2)[C_{3}]$

Decomposition of \(S_{2}^{\mathrm{old}}(1089, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1089, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(99, [\chi])\)\(^{\oplus 2}\)