Space of modular forms of level 1089, weight 6, and trivial character

• Label: 1089.6.a
• Level: $1089$
• Weight: $6$
• Character orbit: 1089.a
• Rep. character: $\chi_{1089}(1,\cdot)$
• Character field: $\Q$
• Dimension: $222$
• Newform subspaces: $41$
• Sturm bound: $792$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 41 \)
Sturm bound:			\(792\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(1089))\).

	Total	New	Old
Modular forms	684	231	453
Cusp forms	636	222	414
Eisenstein series	48	9	39

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
\(+\)	\(+\)	$+$	\(42\)
\(+\)	\(-\)	$-$	\(48\)
\(-\)	\(+\)	$-$	\(67\)
\(-\)	\(-\)	$+$	\(65\)
Plus space		\(+\)	\(107\)
Minus space		\(-\)	\(115\)

Trace form

\( 222 q - 2 q^{2} + 3456 q^{4} + 72 q^{5} + 18 q^{7} - 312 q^{8} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(1089))\) 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
1089.6.a.a	$1089$	$6$	1089.a	1.a	$1$	$1$	$1$	$174.658$	\(\Q\)	None	✓	$1$	$0$	\(-9\)	\(0\)	\(-24\)	\(-72\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{2}+7^{2}q^{4}-24q^{5}-72q^{7}-153q^{8}+\cdots\)
1089.6.a.b	$1089$	$6$	1089.a	1.a	$1$	$1$	$1$	$174.658$	\(\Q\)	None	✓	$1$	$1$	\(-6\)	\(0\)	\(-6\)	\(40\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-6q^{2}+4q^{4}-6q^{5}+40q^{7}+168q^{8}+\cdots\)
1089.6.a.c	$1089$	$6$	1089.a	1.a	$1$	$1$	$1$	$174.658$	\(\Q\)	None	✓	$1$	$1$	\(-4\)	\(0\)	\(19\)	\(-10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-2^{4}q^{4}+19q^{5}-10q^{7}+192q^{8}+\cdots\)
1089.6.a.d	$1089$	$6$	1089.a	1.a	$1$	$1$	$1$	$174.658$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-46\)	\(-148\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-28q^{4}-46q^{5}-148q^{7}+\cdots\)
1089.6.a.e	$1089$	$6$	1089.a	1.a	$1$	$1$	$1$	$174.658$	\(\Q\)	\(\Q(\sqrt{-11}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(-57\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2^{5}q^{4}-57q^{5}+2^{10}q^{16}+1824q^{20}+\cdots\)
1089.6.a.f	$1089$	$6$	1089.a	1.a	$1$	$1$	$1$	$174.658$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-25\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2^{5}q^{4}-5^{2}q^{7}+1202q^{13}+2^{10}q^{16}+\cdots\)
1089.6.a.g	$1089$	$6$	1089.a	1.a	$1$	$1$	$1$	$174.658$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(25\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2^{5}q^{4}+5^{2}q^{7}-1202q^{13}+2^{10}q^{16}+\cdots\)
1089.6.a.h	$1089$	$6$	1089.a	1.a	$1$	$1$	$1$	$174.658$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(92\)	\(26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-31q^{4}+92q^{5}+26q^{7}-63q^{8}+\cdots\)
1089.6.a.i	$1089$	$6$	1089.a	1.a	$1$	$1$	$1$	$174.658$	\(\Q\)	None	✓	$1$	$0$	\(9\)	\(0\)	\(-24\)	\(72\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+9q^{2}+7^{2}q^{4}-24q^{5}+72q^{7}+153q^{8}+\cdots\)
1089.6.a.j	$1089$	$6$	1089.a	1.a	$1$	$2$	$2$	$174.658$	\(\Q(\sqrt{177}) \)	None	✓	$1$	$1$	\(-5\)	\(0\)	\(-58\)	\(286\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{2}+(2^{4}+5\beta )q^{4}+(-24+\cdots)q^{5}+\cdots\)
1089.6.a.k	$1089$	$6$	1089.a	1.a	$1$	$2$	$2$	$174.658$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-3}) \)	✓	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2^{5}q^{4}-149\beta q^{7}+116\beta q^{13}+2^{10}q^{16}+\cdots\)
1089.6.a.l	$1089$	$6$	1089.a	1.a	$1$	$2$	$2$	$174.658$	\(\Q(\sqrt{5}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(196\)	\(0\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-12q^{4}+98q^{5}+53\beta q^{7}+\cdots\)
1089.6.a.m	$1089$	$6$	1089.a	1.a	$1$	$2$	$2$	$174.658$	\(\Q(\sqrt{38}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(38\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+6q^{4}+19q^{5}+8\beta q^{7}-26\beta q^{8}+\cdots\)
1089.6.a.n	$1089$	$6$	1089.a	1.a	$1$	$2$	$2$	$174.658$	\(\Q(\sqrt{3}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(-48\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4\beta q^{2}+2^{4}q^{4}-24q^{5}-\beta q^{7}-2^{6}\beta q^{8}+\cdots\)
1089.6.a.o	$1089$	$6$	1089.a	1.a	$1$	$2$	$2$	$174.658$	\(\Q(\sqrt{313}) \)	None	✓	$1$	$1$	\(1\)	\(0\)	\(38\)	\(18\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(46+\beta )q^{4}+(24-10\beta )q^{5}+\cdots\)
1089.6.a.p	$1089$	$6$	1089.a	1.a	$1$	$2$	$2$	$174.658$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$1$	\(13\)	\(0\)	\(-58\)	\(-146\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{2}+(5^{2}-13\beta )q^{4}+(-24+\cdots)q^{5}+\cdots\)
1089.6.a.q	$1089$	$6$	1089.a	1.a	$1$	$3$	$3$	$174.658$	3.3.193425.1	None	✓	$1$	$1$	\(-1\)	\(0\)	\(58\)	\(117\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(6+\beta _{1}+2\beta _{2})q^{4}+(17+6\beta _{1}+\cdots)q^{5}+\cdots\)
1089.6.a.r	$1089$	$6$	1089.a	1.a	$1$	$3$	$3$	$174.658$	3.3.54492.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-24\)	\(-84\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(28-2\beta _{1}-4\beta _{2})q^{4}+(-8+\cdots)q^{5}+\cdots\)
1089.6.a.s	$1089$	$6$	1089.a	1.a	$1$	$3$	$3$	$174.658$	3.3.193425.1	None	✓	$1$	$1$	\(1\)	\(0\)	\(58\)	\(-117\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(6+\beta _{1}+2\beta _{2})q^{4}+(17+6\beta _{1}+\cdots)q^{5}+\cdots\)
1089.6.a.t	$1089$	$6$	1089.a	1.a	$1$	$4$	$4$	$174.658$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(-9\)	\(0\)	\(-42\)	\(-14\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}+(12+6\beta _{1}+\beta _{2})q^{4}+\cdots\)
1089.6.a.u	$1089$	$6$	1089.a	1.a	$1$	$4$	$4$	$174.658$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(9\)	\(0\)	\(-42\)	\(14\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{2}+(12+6\beta _{1}+\beta _{2})q^{4}+\cdots\)
1089.6.a.v	$1089$	$6$	1089.a	1.a	$1$	$5$	$5$	$174.658$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(0\)	\(-29\)	\(-102\)		$-$	$-$	$+$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(21-\beta _{1}+\beta _{3})q^{4}+\cdots\)
1089.6.a.w	$1089$	$6$	1089.a	1.a	$1$	$5$	$5$	$174.658$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(-4\)	\(0\)	\(100\)	\(18\)		$+$	$-$	$-$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(21-3\beta _{1}+\beta _{4})q^{4}+\cdots\)
1089.6.a.x	$1089$	$6$	1089.a	1.a	$1$	$5$	$5$	$174.658$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(0\)	\(-100\)	\(18\)		$+$	$-$	$-$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(21-3\beta _{1}+\beta _{4})q^{4}+\cdots\)
1089.6.a.y	$1089$	$6$	1089.a	1.a	$1$	$5$	$5$	$174.658$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(4\)	\(0\)	\(-29\)	\(102\)		$-$	$-$	$+$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(21-\beta _{1}+\beta _{3})q^{4}+(-6+\cdots)q^{5}+\cdots\)
1089.6.a.z	$1089$	$6$	1089.a	1.a	$1$	$6$	$6$	$174.658$	6.6.\(\cdots\).1	None	✓	$2$	$0$	\(0\)	\(0\)	\(50\)	\(0\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}+3\beta _{2})q^{2}+(14+\beta _{3}-7\beta _{4}+\cdots)q^{4}+\cdots\)
1089.6.a.ba	$1089$	$6$	1089.a	1.a	$1$	$6$	$6$	$174.658$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(48\)	\(0\)		$-$	$+$	$-$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(28+\beta _{2})q^{4}+(8+\beta _{2}+\beta _{4}+\cdots)q^{5}+\cdots\)
1089.6.a.bb	$1089$	$6$	1089.a	1.a	$1$	$8$	$8$	$174.658$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-8\)	\(0\)	\(-70\)	\(292\)		$-$	$-$	$+$	$2^{2}\cdot 11^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(11-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1089.6.a.bc	$1089$	$6$	1089.a	1.a	$1$	$8$	$8$	$174.658$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{11}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{2}+(19-\beta _{1})q^{4}-\beta _{2}q^{5}+\beta _{7}q^{7}+\cdots\)
1089.6.a.bd	$1089$	$6$	1089.a	1.a	$1$	$8$	$8$	$174.658$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(256\)	\(0\)		$-$	$+$	$-$	$2^{4}\cdot 3\cdot 11$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(19-2\beta _{2}+\beta _{6})q^{4}+(33+\cdots)q^{5}+\cdots\)
1089.6.a.be	$1089$	$6$	1089.a	1.a	$1$	$8$	$8$	$174.658$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-106\)		$+$	$-$	$-$	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5^{2}+\beta _{2})q^{4}+(-3\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
1089.6.a.bf	$1089$	$6$	1089.a	1.a	$1$	$8$	$8$	$174.658$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(106\)		$+$	$-$	$-$	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5^{2}+\beta _{2})q^{4}+(3\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
1089.6.a.bg	$1089$	$6$	1089.a	1.a	$1$	$8$	$8$	$174.658$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(8\)	\(0\)	\(-70\)	\(-292\)		$-$	$+$	$-$	$2^{2}\cdot 11^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(11-\beta _{1}+\beta _{2})q^{4}+(-9+\cdots)q^{5}+\cdots\)
1089.6.a.bh	$1089$	$6$	1089.a	1.a	$1$	$10$	$10$	$174.658$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(-9\)	\(0\)	\(-11\)	\(470\)		$-$	$+$	$-$	$2^{2}\cdot 11^{4}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(19-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1089.6.a.bi	$1089$	$6$	1089.a	1.a	$1$	$10$	$10$	$174.658$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(-7\)	\(0\)	\(33\)	\(-78\)		$-$	$-$	$+$	$2^{2}\cdot 11^{4}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(15-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
1089.6.a.bj	$1089$	$6$	1089.a	1.a	$1$	$10$	$10$	$174.658$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(-198\)	\(0\)		$-$	$+$	$-$	$2^{6}\cdot 11^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{6}q^{2}+(18+\beta _{1})q^{4}+(-20-\beta _{1}+\cdots)q^{5}+\cdots\)
1089.6.a.bk	$1089$	$6$	1089.a	1.a	$1$	$10$	$10$	$174.658$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(7\)	\(0\)	\(33\)	\(78\)		$-$	$+$	$-$	$2^{2}\cdot 11^{4}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(15-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
1089.6.a.bl	$1089$	$6$	1089.a	1.a	$1$	$10$	$10$	$174.658$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(9\)	\(0\)	\(-11\)	\(-470\)		$-$	$-$	$+$	$2^{2}\cdot 11^{4}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(19-\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1089.6.a.bm	$1089$	$6$	1089.a	1.a	$1$	$12$	$12$	$174.658$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{6}q^{2}+(14+\beta _{3})q^{4}+(\beta _{1}-\beta _{2})q^{5}+\cdots\)
1089.6.a.bn	$1089$	$6$	1089.a	1.a	$1$	$20$	$20$	$174.658$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-472\)		$+$	$+$	$+$	$2^{8}\cdot 3^{8}\cdot 5^{2}\cdot 11^{9}\cdot 31^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(15+\beta _{2})q^{4}+(-\beta _{1}-\beta _{8}+\cdots)q^{5}+\cdots\)
1089.6.a.bo	$1089$	$6$	1089.a	1.a	$1$	$20$	$20$	$174.658$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(472\)		$+$	$-$	$-$	$2^{8}\cdot 3^{8}\cdot 5^{2}\cdot 11^{9}\cdot 31^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(15+\beta _{2})q^{4}+(\beta _{1}+\beta _{8})q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(1089))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(1089)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(363))\)\(^{\oplus 2}\)