Space of modular forms of level 1089, weight 2, and trivial character

• Label: 1089.2.a
• Level: $1089$
• Weight: $2$
• Character orbit: 1089.a
• Rep. character: $\chi_{1089}(1,\cdot)$
• Character field: $\Q$
• Dimension: $41$
• Newform subspaces: $23$
• Sturm bound: $264$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(264\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1089))\).

	Total	New	Old
Modular forms	156	50	106
Cusp forms	109	41	68
Eisenstein series	47	9	38

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(6\)
\(+\)	\(-\)	$-$	\(12\)
\(-\)	\(+\)	$-$	\(13\)
\(-\)	\(-\)	$+$	\(10\)
Plus space		\(+\)	\(16\)
Minus space		\(-\)	\(25\)

Trace form

\( 41 q - q^{2} + 41 q^{4} - 2 q^{5} + 2 q^{7} - 3 q^{8} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1089))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	11
1089.2.a.a	$1089$	$2$	1089.a	1.a	$1$	$1$	$1$	$8.696$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-4\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-4q^{5}-q^{7}+8q^{10}+\cdots\)
1089.2.a.b	$1089$	$2$	1089.a	1.a	$1$	$1$	$1$	$8.696$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-1\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-q^{5}+2q^{7}+2q^{10}+\cdots\)
1089.2.a.c	$1089$	$2$	1089.a	1.a	$1$	$1$	$1$	$8.696$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-q^{5}-2q^{7}+3q^{8}+q^{10}+\cdots\)
1089.2.a.d	$1089$	$2$	1089.a	1.a	$1$	$1$	$1$	$8.696$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(4\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+4q^{5}+2q^{7}+3q^{8}-4q^{10}+\cdots\)
1089.2.a.e	$1089$	$2$	1089.a	1.a	$1$	$1$	$1$	$8.696$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-5\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}-5q^{7}-2q^{13}+4q^{16}+7q^{19}+\cdots\)
1089.2.a.f	$1089$	$2$	1089.a	1.a	$1$	$1$	$1$	$8.696$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(5\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+5q^{7}+2q^{13}+4q^{16}-7q^{19}+\cdots\)
1089.2.a.g	$1089$	$2$	1089.a	1.a	$1$	$1$	$1$	$8.696$	\(\Q\)	\(\Q(\sqrt{-11}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(3\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+3q^{5}+4q^{16}-6q^{20}+9q^{23}+\cdots\)
1089.2.a.h	$1089$	$2$	1089.a	1.a	$1$	$1$	$1$	$8.696$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(-4\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-4q^{5}+2q^{7}-3q^{8}-4q^{10}+\cdots\)
1089.2.a.i	$1089$	$2$	1089.a	1.a	$1$	$1$	$1$	$8.696$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-q^{5}+2q^{7}-3q^{8}-q^{10}+\cdots\)
1089.2.a.j	$1089$	$2$	1089.a	1.a	$1$	$1$	$1$	$8.696$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(2\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+2q^{5}-4q^{7}-3q^{8}+2q^{10}+\cdots\)
1089.2.a.k	$1089$	$2$	1089.a	1.a	$1$	$1$	$1$	$8.696$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(0\)	\(-4\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-4q^{5}+q^{7}-8q^{10}+\cdots\)
1089.2.a.l	$1089$	$2$	1089.a	1.a	$1$	$2$	$2$	$8.696$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(-1\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+3\beta q^{4}+(-1+\beta )q^{5}+\cdots\)
1089.2.a.m	$1089$	$2$	1089.a	1.a	$1$	$2$	$2$	$8.696$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(3\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+(2-\beta )q^{5}+3q^{7}+\cdots\)
1089.2.a.n	$1089$	$2$	1089.a	1.a	$1$	$2$	$2$	$8.696$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-3}) \)	✓	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}-\beta q^{7}+4\beta q^{13}+4q^{16}-3\beta q^{19}+\cdots\)
1089.2.a.o	$1089$	$2$	1089.a	1.a	$1$	$2$	$2$	$8.696$	\(\Q(\sqrt{3}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(6\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{4}+3q^{5}+2\beta q^{7}-\beta q^{8}+\cdots\)
1089.2.a.p	$1089$	$2$	1089.a	1.a	$1$	$2$	$2$	$8.696$	\(\Q(\sqrt{5}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+3q^{4}-2q^{5}-2\beta q^{7}-\beta q^{8}+\cdots\)
1089.2.a.q	$1089$	$2$	1089.a	1.a	$1$	$2$	$2$	$8.696$	\(\Q(\sqrt{7}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+5q^{4}+\beta q^{5}-2q^{7}+3\beta q^{8}+\cdots\)
1089.2.a.r	$1089$	$2$	1089.a	1.a	$1$	$2$	$2$	$8.696$	\(\Q(\sqrt{7}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+5q^{4}-\beta q^{5}+2q^{7}+3\beta q^{8}+\cdots\)
1089.2.a.s	$1089$	$2$	1089.a	1.a	$1$	$2$	$2$	$8.696$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(1\)	\(0\)	\(3\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(2-\beta )q^{5}-3q^{7}+\cdots\)
1089.2.a.t	$1089$	$2$	1089.a	1.a	$1$	$2$	$2$	$8.696$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(3\)	\(0\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+3\beta q^{4}+(-1+\beta )q^{5}+\cdots\)
1089.2.a.u	$1089$	$2$	1089.a	1.a	$1$	$4$	$4$	$8.696$	\(\Q(\sqrt{3}, \sqrt{11})\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{3})q^{4}+\beta _{3}q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
1089.2.a.v	$1089$	$2$	1089.a	1.a	$1$	$4$	$4$	$8.696$	4.4.4400.1	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-8\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}-\beta _{1}q^{5}+(-3+\cdots)q^{7}+\cdots\)
1089.2.a.w	$1089$	$2$	1089.a	1.a	$1$	$4$	$4$	$8.696$	4.4.4400.1	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+\beta _{1}q^{5}+(3+2\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1089))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1089)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(363))\)\(^{\oplus 2}\)