Space of modular forms of level 9009, weight 2, and trivial character

• Label: 9009.2.a
• Level: $9009$
• Weight: $2$
• Character orbit: 9009.a
• Rep. character: $\chi_{9009}(1,\cdot)$
• Character field: $\Q$
• Dimension: $300$
• Newform subspaces: $53$
• Sturm bound: $2688$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 9009 = 3^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9009.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 53 \)
Sturm bound:			\(2688\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(5\), \(17\), \(19\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(9009))\).

	Total	New	Old
Modular forms	1360	300	1060
Cusp forms	1329	300	1029
Eisenstein series	31	0	31

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

\(3\)	\(7\)	\(11\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)	\(17\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	\(13\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	\(17\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)	\(13\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)	\(13\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	\(17\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)	\(13\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)	\(17\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(25\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(22\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(18\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(25\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(22\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(25\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(25\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(18\)
Plus space				\(+\)	\(140\)
Minus space				\(-\)	\(160\)

Trace form

\( 300 q + 284 q^{4} - 4 q^{5} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(9009))\) 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					A-L signs				Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	7	11	13
9009.2.a.a	$9009$	$2$	9009.a	1.a	$1$	$1$	$1$	$71.937$	\(\Q\)	None	✓				3003.2.a.i	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+q^{7}+q^{11}+q^{13}+\cdots\)
9009.2.a.b	$9009$	$2$	9009.a	1.a	$1$	$1$	$1$	$71.937$	\(\Q\)	None	✓				3003.2.a.h	$1$	$2$	\(-1\)	\(0\)	\(-2\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-2q^{5}-q^{7}+3q^{8}+2q^{10}+\cdots\)
9009.2.a.c	$9009$	$2$	9009.a	1.a	$1$	$1$	$1$	$71.937$	\(\Q\)	None	✓				3003.2.a.g	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+q^{7}+3q^{8}+q^{11}-q^{13}+\cdots\)
9009.2.a.d	$9009$	$2$	9009.a	1.a	$1$	$1$	$1$	$71.937$	\(\Q\)	None	✓				3003.2.a.f	$1$	$0$	\(-1\)	\(0\)	\(2\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+2q^{5}+q^{7}+3q^{8}-2q^{10}+\cdots\)
9009.2.a.e	$9009$	$2$	9009.a	1.a	$1$	$1$	$1$	$71.937$	\(\Q\)	None	✓				3003.2.a.d	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-3q^{5}+q^{7}+q^{11}-q^{13}+\cdots\)
9009.2.a.f	$9009$	$2$	9009.a	1.a	$1$	$1$	$1$	$71.937$	\(\Q\)	None	✓	✓			9009.2.a.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{7}-q^{11}+q^{13}+4q^{16}+\cdots\)
9009.2.a.g	$9009$	$2$	9009.a	1.a	$1$	$1$	$1$	$71.937$	\(\Q\)	None	✓	✓			9009.2.a.f	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{7}+q^{11}+q^{13}+4q^{16}+\cdots\)
9009.2.a.h	$9009$	$2$	9009.a	1.a	$1$	$1$	$1$	$71.937$	\(\Q\)	None	✓				1001.2.a.c	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-1\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{5}-q^{7}+q^{11}-q^{13}+4q^{16}+\cdots\)
9009.2.a.i	$9009$	$2$	9009.a	1.a	$1$	$1$	$1$	$71.937$	\(\Q\)	None	✓				3003.2.a.c	$1$	$1$	\(0\)	\(0\)	\(1\)	\(1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{5}+q^{7}-q^{11}-q^{13}+4q^{16}+\cdots\)
9009.2.a.j	$9009$	$2$	9009.a	1.a	$1$	$1$	$1$	$71.937$	\(\Q\)	None	✓				3003.2.a.e	$1$	$0$	\(0\)	\(0\)	\(1\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{5}+q^{7}+q^{11}-q^{13}+4q^{16}+\cdots\)
9009.2.a.k	$9009$	$2$	9009.a	1.a	$1$	$1$	$1$	$71.937$	\(\Q\)	None	✓				1001.2.a.b	$1$	$0$	\(1\)	\(0\)	\(2\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+2q^{5}-q^{7}-3q^{8}+2q^{10}+\cdots\)
9009.2.a.l	$9009$	$2$	9009.a	1.a	$1$	$1$	$1$	$71.937$	\(\Q\)	None	✓				3003.2.a.b	$1$	$0$	\(2\)	\(0\)	\(-4\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-4q^{5}-q^{7}-8q^{10}+\cdots\)
9009.2.a.m	$9009$	$2$	9009.a	1.a	$1$	$1$	$1$	$71.937$	\(\Q\)	None	✓				3003.2.a.a	$1$	$0$	\(2\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+q^{7}+q^{11}-q^{13}+\cdots\)
9009.2.a.n	$9009$	$2$	9009.a	1.a	$1$	$1$	$1$	$71.937$	\(\Q\)	None	✓				1001.2.a.a	$1$	$0$	\(2\)	\(0\)	\(3\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+3q^{5}-q^{7}+6q^{10}+\cdots\)
9009.2.a.o	$9009$	$2$	9009.a	1.a	$1$	$2$	$2$	$71.937$	\(\Q(\sqrt{21}) \)	None	✓				1001.2.a.e	$1$	$0$	\(-2\)	\(0\)	\(1\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+(1-\beta )q^{5}+q^{7}+3q^{8}+\cdots\)
9009.2.a.p	$9009$	$2$	9009.a	1.a	$1$	$2$	$2$	$71.937$	\(\Q(\sqrt{5}) \)	None	✓				1001.2.a.d	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta )q^{2}+3q^{4}+(-2+\beta )q^{5}+\cdots\)
9009.2.a.q	$9009$	$2$	9009.a	1.a	$1$	$2$	$2$	$71.937$	\(\Q(\sqrt{7}) \)	None	✓	✓			9009.2.a.q	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-2\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+5q^{4}+(-1-\beta )q^{5}-q^{7}+\cdots\)
9009.2.a.r	$9009$	$2$	9009.a	1.a	$1$	$2$	$2$	$71.937$	\(\Q(\sqrt{7}) \)	None	✓	✓			9009.2.a.q	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+5q^{4}+(1-\beta )q^{5}-q^{7}+3\beta q^{8}+\cdots\)
9009.2.a.s	$9009$	$2$	9009.a	1.a	$1$	$3$	$3$	$71.937$	3.3.1101.1	None	✓				1001.2.a.f	$1$	$0$	\(-3\)	\(0\)	\(-1\)	\(-3\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+(-1+\beta _{1}-\beta _{2})q^{5}-q^{7}+\cdots\)
9009.2.a.t	$9009$	$2$	9009.a	1.a	$1$	$3$	$3$	$71.937$	3.3.4892.1	None	✓	✓			9009.2.a.t	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-3\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+\beta _{1}q^{5}-q^{7}-q^{11}-q^{13}+\cdots\)
9009.2.a.u	$9009$	$2$	9009.a	1.a	$1$	$3$	$3$	$71.937$	3.3.4892.1	None	✓	✓			9009.2.a.t	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-\beta _{1}q^{5}-q^{7}+q^{11}-q^{13}+\cdots\)
9009.2.a.v	$9009$	$2$	9009.a	1.a	$1$	$4$	$4$	$71.937$	4.4.1957.1	None	✓				1001.2.a.h	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(-4\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{2}+\beta _{3})q^{2}+(1+\beta _{1})q^{4}+(-1+\cdots)q^{5}+\cdots\)
9009.2.a.w	$9009$	$2$	9009.a	1.a	$1$	$4$	$4$	$71.937$	4.4.23301.1	None	✓				1001.2.a.g	$1$	$0$	\(2\)	\(0\)	\(5\)	\(4\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{3})q^{2}+(2-\beta _{2})q^{4}+(2-\beta _{1}+\cdots)q^{5}+\cdots\)
9009.2.a.x	$9009$	$2$	9009.a	1.a	$1$	$5$	$5$	$71.937$	5.5.70601.1	None	✓				3003.2.a.m	$1$	$0$	\(-2\)	\(0\)	\(-2\)	\(-5\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{2}+(1-\beta _{1}+\beta _{3}-\beta _{4})q^{4}+(-1+\cdots)q^{5}+\cdots\)
9009.2.a.y	$9009$	$2$	9009.a	1.a	$1$	$5$	$5$	$71.937$	5.5.24217.1	None	✓				3003.2.a.l	$1$	$1$	\(0\)	\(0\)	\(2\)	\(5\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{2})q^{2}+(-\beta _{1}+\beta _{3}+\beta _{4})q^{4}+\cdots\)
9009.2.a.z	$9009$	$2$	9009.a	1.a	$1$	$5$	$5$	$71.937$	5.5.81509.1	None	✓				1001.2.a.k	$1$	$1$	\(0\)	\(0\)	\(6\)	\(5\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{2}+(1-\beta _{2}+\beta _{3})q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
9009.2.a.ba	$9009$	$2$	9009.a	1.a	$1$	$5$	$5$	$71.937$	5.5.106069.1	None	✓				1001.2.a.i	$1$	$1$	\(2\)	\(0\)	\(0\)	\(5\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(-\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
9009.2.a.bb	$9009$	$2$	9009.a	1.a	$1$	$5$	$5$	$71.937$	5.5.216637.1	None	✓				1001.2.a.j	$1$	$1$	\(2\)	\(0\)	\(4\)	\(-5\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(1-\beta _{3}-\beta _{4})q^{5}+\cdots\)
9009.2.a.bc	$9009$	$2$	9009.a	1.a	$1$	$5$	$5$	$71.937$	5.5.240881.1	None	✓				3003.2.a.k	$1$	$1$	\(2\)	\(0\)	\(0\)	\(5\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{3}-\beta _{4})q^{5}+\cdots\)
9009.2.a.bd	$9009$	$2$	9009.a	1.a	$1$	$5$	$5$	$71.937$	5.5.65657.1	None	✓				3003.2.a.j	$1$	$0$	\(4\)	\(0\)	\(10\)	\(5\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots\)
9009.2.a.be	$9009$	$2$	9009.a	1.a	$1$	$6$	$6$	$71.937$	6.6.1178892857.1	None	✓				3003.2.a.q	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(6\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(3-\beta _{1}+\beta _{2})q^{4}+\cdots\)
9009.2.a.bf	$9009$	$2$	9009.a	1.a	$1$	$6$	$6$	$71.937$	6.6.26054921.1	None	✓				3003.2.a.p	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-6\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{2}q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots\)
9009.2.a.bg	$9009$	$2$	9009.a	1.a	$1$	$6$	$6$	$71.937$	6.6.22848881.1	None	✓				3003.2.a.o	$1$	$0$	\(1\)	\(0\)	\(4\)	\(-6\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1-\beta _{1}-\beta _{5})q^{5}+\cdots\)
9009.2.a.bh	$9009$	$2$	9009.a	1.a	$1$	$6$	$6$	$71.937$	6.6.30320177.1	None	✓				3003.2.a.n	$1$	$0$	\(3\)	\(0\)	\(6\)	\(6\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1+\beta _{2})q^{4}+(\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
9009.2.a.bi	$9009$	$2$	9009.a	1.a	$1$	$7$	$7$	$71.937$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		3003.2.a.s	$1$	$1$	\(-2\)	\(0\)	\(-10\)	\(7\)		$-$	$-$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{2}+(1-\beta _{4}-\beta _{6})q^{4}+(-1+\beta _{2}+\cdots)q^{5}+\cdots\)
9009.2.a.bj	$9009$	$2$	9009.a	1.a	$1$	$7$	$7$	$71.937$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		3003.2.a.r	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-7\)		$-$	$+$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{5}q^{5}-q^{7}+\cdots\)
9009.2.a.bk	$9009$	$2$	9009.a	1.a	$1$	$7$	$7$	$71.937$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		1001.2.a.l	$1$	$0$	\(0\)	\(0\)	\(-9\)	\(-7\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{5}q^{2}+(4+\beta _{1}-\beta _{2}+\beta _{4})q^{4}+(-1+\cdots)q^{5}+\cdots\)
9009.2.a.bl	$9009$	$2$	9009.a	1.a	$1$	$8$	$8$	$71.937$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				3003.2.a.u	$1$	$1$	\(-2\)	\(0\)	\(7\)	\(-8\)		$-$	$+$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{4})q^{5}+\cdots\)
9009.2.a.bm	$9009$	$2$	9009.a	1.a	$1$	$8$	$8$	$71.937$	8.8.3212905625.1	None	✓				1001.2.a.m	$1$	$0$	\(-2\)	\(0\)	\(5\)	\(-8\)		$-$	$+$	$-$	$-$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{7}q^{2}+(2-\beta _{3})q^{4}+(1+\beta _{4})q^{5}+\cdots\)
9009.2.a.bn	$9009$	$2$	9009.a	1.a	$1$	$8$	$8$	$71.937$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				3003.2.a.t	$1$	$0$	\(-1\)	\(0\)	\(-7\)	\(-8\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{7})q^{5}+\cdots\)
9009.2.a.bo	$9009$	$2$	9009.a	1.a	$1$	$9$	$9$	$71.937$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				3003.2.a.x	$1$	$0$	\(-3\)	\(0\)	\(-4\)	\(9\)		$-$	$-$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{4}q^{5}+q^{7}+\cdots\)
9009.2.a.bp	$9009$	$2$	9009.a	1.a	$1$	$9$	$9$	$71.937$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		✓		3003.2.a.w	$1$	$1$	\(-2\)	\(0\)	\(-9\)	\(-9\)		$-$	$+$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{7})q^{5}+\cdots\)
9009.2.a.bq	$9009$	$2$	9009.a	1.a	$1$	$9$	$9$	$71.937$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		✓		3003.2.a.v	$1$	$0$	\(5\)	\(0\)	\(9\)	\(-9\)		$-$	$+$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
9009.2.a.br	$9009$	$2$	9009.a	1.a	$1$	$11$	$11$	$71.937$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		3003.2.a.y	$1$	$1$	\(-2\)	\(0\)	\(-7\)	\(11\)		$-$	$-$	$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{4})q^{5}+\cdots\)
9009.2.a.bs	$9009$	$2$	9009.a	1.a	$1$	$11$	$11$	$71.937$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		1001.2.a.n	$1$	$0$	\(1\)	\(0\)	\(-7\)	\(11\)		$-$	$-$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+\beta _{4})q^{5}+\cdots\)
9009.2.a.bt	$9009$	$2$	9009.a	1.a	$1$	$12$	$12$	$71.937$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓		9009.2.a.bt	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-12\)		$+$	$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(\beta _{3}+\beta _{4})q^{5}+\cdots\)
9009.2.a.bu	$9009$	$2$	9009.a	1.a	$1$	$12$	$12$	$71.937$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓		9009.2.a.bt	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)		$+$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{3}-\beta _{4}+\cdots)q^{5}+\cdots\)
9009.2.a.bv	$9009$	$2$	9009.a	1.a	$1$	$13$	$13$	$71.937$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	✓	✓	✓	9009.2.a.bv	$1$	$1$	\(-4\)	\(0\)	\(-2\)	\(-13\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}-\beta _{9}q^{5}-q^{7}+\cdots\)
9009.2.a.bw	$9009$	$2$	9009.a	1.a	$1$	$13$	$13$	$71.937$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	✓	✓	✓	9009.2.a.bw	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(13\)		$+$	$-$	$-$	$+$	$+$	$3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{5}q^{5}+q^{7}+\cdots\)
9009.2.a.bx	$9009$	$2$	9009.a	1.a	$1$	$13$	$13$	$71.937$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	✓	✓	✓	9009.2.a.bw	$1$	$0$	\(0\)	\(0\)	\(6\)	\(13\)		$+$	$-$	$+$	$+$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{5}q^{5}+q^{7}+\cdots\)
9009.2.a.by	$9009$	$2$	9009.a	1.a	$1$	$13$	$13$	$71.937$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	✓	✓	✓	9009.2.a.bv	$1$	$0$	\(4\)	\(0\)	\(2\)	\(-13\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+\beta _{9}q^{5}-q^{7}+\cdots\)
9009.2.a.bz	$9009$	$2$	9009.a	1.a	$1$	$16$	$16$	$71.937$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	✓	✓		9009.2.a.bz	$1$	$1$	\(-4\)	\(0\)	\(-10\)	\(16\)		$+$	$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{12}+\cdots)q^{5}+\cdots\)
9009.2.a.ca	$9009$	$2$	9009.a	1.a	$1$	$16$	$16$	$71.937$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	✓	✓		9009.2.a.bz	$1$	$0$	\(4\)	\(0\)	\(10\)	\(16\)		$+$	$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{12})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(9009))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(9009)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(143))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(231))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(273))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(429))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(693))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(819))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1001))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1287))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3003))\)\(^{\oplus 2}\)