Space of modular forms of level 8325, weight 2, and trivial character

• Label: 8325.2.a
• Level: $8325$
• Weight: $2$
• Character orbit: 8325.a
• Rep. character: $\chi_{8325}(1,\cdot)$
• Character field: $\Q$
• Dimension: $285$
• Newform subspaces: $76$
• Sturm bound: $2280$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8325.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 76 \)
Sturm bound:			\(2280\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(2\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8325))\).

	Total	New	Old
Modular forms	1164	285	879
Cusp forms	1117	285	832
Eisenstein series	47	0	47

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

\(3\)	\(5\)	\(37\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(141\)	\(24\)	\(117\)		\(136\)	\(24\)	\(112\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(147\)	\(30\)	\(117\)		\(141\)	\(30\)	\(111\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(149\)	\(32\)	\(117\)		\(143\)	\(32\)	\(111\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(143\)	\(28\)	\(115\)		\(137\)	\(28\)	\(109\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(150\)	\(42\)	\(108\)		\(144\)	\(42\)	\(102\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(144\)	\(39\)	\(105\)		\(138\)	\(39\)	\(99\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(142\)	\(44\)	\(98\)		\(136\)	\(44\)	\(92\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(148\)	\(46\)	\(102\)		\(142\)	\(46\)	\(96\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(570\)	\(135\)	\(435\)		\(547\)	\(135\)	\(412\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(594\)	\(150\)	\(444\)		\(570\)	\(150\)	\(420\)		\(24\)	\(0\)	\(24\)

Trace form

285 * q - q^2 + 289 * q^4 + 2 * q^7 - 15 * q^8 + 2 * q^11 + 8 * q^13 - 2 * q^14 + 297 * q^16 + 14 * q^19 + 2 * q^22 - 4 * q^23 - 18 * q^26 + 4 * q^28 + 22 * q^29 + 12 * q^31 - 11 * q^32 - 10 * q^34 + q^37 + 4 * q^38 - 4 * q^41 + 18 * q^43 - 20 * q^44 - 28 * q^46 + 14 * q^47 + 291 * q^49 + 10 * q^52 + 12 * q^53 - 28 * q^56 - 10 * q^58 - 44 * q^59 - 2 * q^61 - 32 * q^62 + 301 * q^64 + 16 * q^67 + 46 * q^68 + 2 * q^71 + 48 * q^73 + 5 * q^74 + 4 * q^76 - 46 * q^77 - 18 * q^79 + 4 * q^82 + 6 * q^83 + 8 * q^86 - 36 * q^88 + 32 * q^89 - 34 * q^91 - 72 * q^92 - 94 * q^94 + 54 * q^97 - 7 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8325))\) 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	5	37
8325.2.a.a	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				1665.2.a.a	$1$	$2$	\(-2\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-4q^{7}-q^{13}+8q^{14}+\cdots\)
8325.2.a.b	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓	✓			8325.2.a.b	$1$	$2$	\(-2\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-q^{7}-6q^{11}-7q^{13}+\cdots\)
8325.2.a.c	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				2775.2.a.b	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-q^{7}+5q^{13}+2q^{14}+\cdots\)
8325.2.a.d	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				555.2.c.a	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+2q^{11}+q^{13}-4q^{16}+\cdots\)
8325.2.a.e	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				37.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+q^{7}+5q^{11}+2q^{13}+\cdots\)
8325.2.a.f	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓	✓			8325.2.a.b	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+q^{7}+6q^{11}+7q^{13}+\cdots\)
8325.2.a.g	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				185.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(5\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+5q^{7}-3q^{11}+2q^{13}+\cdots\)
8325.2.a.h	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				333.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+4q^{7}+3q^{8}+4q^{11}+\cdots\)
8325.2.a.i	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				2775.2.a.c	$1$	$2$	\(0\)	\(0\)	\(0\)	\(-3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-3q^{7}-4q^{11}-5q^{13}+4q^{16}+\cdots\)
8325.2.a.j	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓	✓			8325.2.a.j	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-3q^{7}-4q^{11}+q^{13}+4q^{16}+\cdots\)
8325.2.a.k	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓	✓			8325.2.a.j	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-3q^{7}+4q^{11}+q^{13}+4q^{16}+\cdots\)
8325.2.a.l	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				1665.2.c.b	$1$	$2$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-2q^{7}-6q^{11}-q^{13}+4q^{16}+\cdots\)
8325.2.a.m	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				555.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-2q^{7}+q^{13}+4q^{16}+6q^{17}+\cdots\)
8325.2.a.n	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				1665.2.c.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-2q^{7}+6q^{11}-q^{13}+4q^{16}+\cdots\)
8325.2.a.o	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				2775.2.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-q^{7}+2q^{11}+q^{13}+4q^{16}+\cdots\)
8325.2.a.p	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				37.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{7}-3q^{11}+4q^{13}+4q^{16}+\cdots\)
8325.2.a.q	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				2775.2.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{7}+2q^{11}-q^{13}+4q^{16}+\cdots\)
8325.2.a.r	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				1665.2.c.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+2q^{7}-6q^{11}+q^{13}+4q^{16}+\cdots\)
8325.2.a.s	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				555.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+2q^{7}-4q^{11}-5q^{13}+4q^{16}+\cdots\)
8325.2.a.t	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				1665.2.c.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+2q^{7}+6q^{11}+q^{13}+4q^{16}+\cdots\)
8325.2.a.u	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓	✓			8325.2.a.j	$1$	$1$	\(0\)	\(0\)	\(0\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+3q^{7}-4q^{11}-q^{13}+4q^{16}+\cdots\)
8325.2.a.v	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				2775.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+3q^{7}-4q^{11}+5q^{13}+4q^{16}+\cdots\)
8325.2.a.w	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓	✓			8325.2.a.j	$1$	$1$	\(0\)	\(0\)	\(0\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+3q^{7}+4q^{11}-q^{13}+4q^{16}+\cdots\)
8325.2.a.x	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				185.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+3q^{7}+5q^{11}-4q^{13}+4q^{16}+\cdots\)
8325.2.a.y	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				185.2.a.c	$1$	$1$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+2q^{7}-3q^{8}+2q^{13}+\cdots\)
8325.2.a.z	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				333.2.a.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+4q^{7}-3q^{8}-4q^{11}+\cdots\)
8325.2.a.ba	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				1665.2.a.a	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-4q^{7}-q^{13}-8q^{14}+\cdots\)
8325.2.a.bb	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓	✓			8325.2.a.b	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-q^{7}+6q^{11}-7q^{13}+\cdots\)
8325.2.a.bc	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				555.2.c.a	$1$	$1$	\(2\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+2q^{11}-q^{13}-4q^{16}+\cdots\)
8325.2.a.bd	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓	✓			8325.2.a.b	$1$	$0$	\(2\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+q^{7}-6q^{11}+7q^{13}+\cdots\)
8325.2.a.be	$8325$	$2$	8325.a	1.a	$1$	$1$	$1$	$66.475$	\(\Q\)	None	✓				2775.2.a.b	$1$	$1$	\(2\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+q^{7}-5q^{13}+2q^{14}+\cdots\)
8325.2.a.bf	$8325$	$2$	8325.a	1.a	$1$	$2$	$2$	$66.475$	\(\Q(\sqrt{5}) \)	None	✓				555.2.a.c	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+3\beta q^{4}+(3-2\beta )q^{7}+\cdots\)
8325.2.a.bg	$8325$	$2$	8325.a	1.a	$1$	$2$	$2$	$66.475$	\(\Q(\sqrt{2}) \)	None	✓				2775.2.a.k	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+2q^{7}+\cdots\)
8325.2.a.bh	$8325$	$2$	8325.a	1.a	$1$	$2$	$2$	$66.475$	\(\Q(\sqrt{5}) \)	None	✓				555.2.a.e	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+q^{7}+(-1+2\beta )q^{8}+\cdots\)
8325.2.a.bi	$8325$	$2$	8325.a	1.a	$1$	$2$	$2$	$66.475$	\(\Q(\sqrt{13}) \)	None	✓				555.2.a.d	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(6\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(1+\beta )q^{4}+3q^{7}-3q^{8}+(-1+\cdots)q^{11}+\cdots\)
8325.2.a.bj	$8325$	$2$	8325.a	1.a	$1$	$2$	$2$	$66.475$	\(\Q(\sqrt{7}) \)	None	✓	✓			8325.2.a.bj	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+5q^{4}-2q^{7}+3\beta q^{8}-2q^{13}+\cdots\)
8325.2.a.bk	$8325$	$2$	8325.a	1.a	$1$	$2$	$2$	$66.475$	\(\Q(\sqrt{7}) \)	None	✓	✓			8325.2.a.bj	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+5q^{4}+2q^{7}+3\beta q^{8}+2q^{13}+\cdots\)
8325.2.a.bl	$8325$	$2$	8325.a	1.a	$1$	$2$	$2$	$66.475$	\(\Q(\sqrt{5}) \)	None	✓				555.2.a.f	$1$	$1$	\(1\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(-1+2\beta )q^{7}+\cdots\)
8325.2.a.bm	$8325$	$2$	8325.a	1.a	$1$	$2$	$2$	$66.475$	\(\Q(\sqrt{13}) \)	None	✓				555.2.a.g	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1+\beta )q^{4}-q^{7}+3q^{8}+(-1+\cdots)q^{11}+\cdots\)
8325.2.a.bn	$8325$	$2$	8325.a	1.a	$1$	$2$	$2$	$66.475$	\(\Q(\sqrt{2}) \)	None	✓				2775.2.a.k	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1+2\beta )q^{4}-2q^{7}+(3+\cdots)q^{8}+\cdots\)
8325.2.a.bo	$8325$	$2$	8325.a	1.a	$1$	$3$	$3$	$66.475$	3.3.257.1	None	✓				2775.2.a.u	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(-2+\beta _{2})q^{7}+\cdots\)
8325.2.a.bp	$8325$	$2$	8325.a	1.a	$1$	$3$	$3$	$66.475$	3.3.257.1	None	✓				2775.2.a.u	$1$	$1$	\(0\)	\(0\)	\(0\)	\(6\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(2-\beta _{2})q^{7}+\cdots\)
8325.2.a.bq	$8325$	$2$	8325.a	1.a	$1$	$3$	$3$	$66.475$	3.3.469.1	None	✓				555.2.a.h	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-2+\beta _{1}-\beta _{2})q^{7}+\cdots\)
8325.2.a.br	$8325$	$2$	8325.a	1.a	$1$	$3$	$3$	$66.475$	3.3.229.1	None	✓				555.2.a.i	$1$	$0$	\(2\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+(2+\beta _{1})q^{4}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
8325.2.a.bs	$8325$	$2$	8325.a	1.a	$1$	$3$	$3$	$66.475$	3.3.148.1	None	✓				111.2.a.a	$1$	$0$	\(3\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+(2+\cdots)q^{7}+\cdots\)
8325.2.a.bt	$8325$	$2$	8325.a	1.a	$1$	$4$	$4$	$66.475$	\(\Q(\sqrt{14 +2 \sqrt{5}})\)	None	✓				555.2.c.b	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+\beta _{2}q^{4}+(-2-\beta _{3})q^{7}+\cdots\)
8325.2.a.bu	$8325$	$2$	8325.a	1.a	$1$	$4$	$4$	$66.475$	\(\Q(\sqrt{3 + \sqrt{6}})\)	None	✓				333.2.a.f	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-8\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-2q^{7}+(\beta _{1}+\beta _{3})q^{8}+\cdots\)
8325.2.a.bv	$8325$	$2$	8325.a	1.a	$1$	$4$	$4$	$66.475$	4.4.6224.1	None	✓				111.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(-2+2\beta _{3})q^{7}+\cdots\)
8325.2.a.bw	$8325$	$2$	8325.a	1.a	$1$	$4$	$4$	$66.475$	\(\Q(\sqrt{14 +2 \sqrt{5}})\)	None	✓				555.2.c.b	$1$	$0$	\(2\)	\(0\)	\(0\)	\(8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+\beta _{2}q^{4}+(2-\beta _{3})q^{7}+\cdots\)
8325.2.a.bx	$8325$	$2$	8325.a	1.a	$1$	$5$	$5$	$66.475$	5.5.600268.1	None	✓				555.2.a.j	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{3})q^{2}+(2-\beta _{3}+\beta _{4})q^{4}+\cdots\)
8325.2.a.by	$8325$	$2$	8325.a	1.a	$1$	$5$	$5$	$66.475$	5.5.457904.1	None	✓				1665.2.a.o	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(\beta _{2}-\beta _{4})q^{7}+\cdots\)
8325.2.a.bz	$8325$	$2$	8325.a	1.a	$1$	$5$	$5$	$66.475$	5.5.176684.1	None	✓				2775.2.a.ba	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(\beta _{2}-\beta _{4})q^{4}+(1-\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
8325.2.a.ca	$8325$	$2$	8325.a	1.a	$1$	$5$	$5$	$66.475$	5.5.528933.1	None	✓				2775.2.a.z	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}+(2-\beta _{2}+\beta _{4})q^{4}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
8325.2.a.cb	$8325$	$2$	8325.a	1.a	$1$	$5$	$5$	$66.475$	5.5.65657.1	None	✓				925.2.a.g	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+\beta _{1}q^{4}+(-\beta _{2}-\beta _{3}-\beta _{4})q^{7}+\cdots\)
8325.2.a.cc	$8325$	$2$	8325.a	1.a	$1$	$5$	$5$	$66.475$	5.5.368464.1	None	✓				185.2.a.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(2-\beta _{1}-\beta _{4})q^{4}+(-1-\beta _{4})q^{7}+\cdots\)
8325.2.a.cd	$8325$	$2$	8325.a	1.a	$1$	$5$	$5$	$66.475$	5.5.65657.1	None	✓				925.2.a.g	$1$	$0$	\(1\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+\beta _{1}q^{4}+(\beta _{2}+\beta _{3}+\beta _{4})q^{7}+\cdots\)
8325.2.a.ce	$8325$	$2$	8325.a	1.a	$1$	$5$	$5$	$66.475$	5.5.176684.1	None	✓				2775.2.a.ba	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(\beta _{2}-\beta _{4})q^{4}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
8325.2.a.cf	$8325$	$2$	8325.a	1.a	$1$	$5$	$5$	$66.475$	5.5.457904.1	None	✓				1665.2.a.o	$1$	$1$	\(2\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(\beta _{2}-\beta _{4})q^{7}+\cdots\)
8325.2.a.cg	$8325$	$2$	8325.a	1.a	$1$	$5$	$5$	$66.475$	5.5.528933.1	None	✓				2775.2.a.z	$1$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}+(2-\beta _{2}+\beta _{4})q^{4}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
8325.2.a.ch	$8325$	$2$	8325.a	1.a	$1$	$5$	$5$	$66.475$	5.5.973904.1	None	✓				185.2.a.e	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-11\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{2}+(2+\beta _{3})q^{4}+(-2-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
8325.2.a.ci	$8325$	$2$	8325.a	1.a	$1$	$6$	$6$	$66.475$	6.6.297869800.1	None	✓		✓		2775.2.a.be	$1$	$0$	\(-4\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(2+\beta _{2})q^{4}-\beta _{3}q^{7}+\cdots\)
8325.2.a.cj	$8325$	$2$	8325.a	1.a	$1$	$6$	$6$	$66.475$	6.6.95034688.1	None	✓				1665.2.a.t	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(1-\beta _{3}+\cdots)q^{7}+\cdots\)
8325.2.a.ck	$8325$	$2$	8325.a	1.a	$1$	$6$	$6$	$66.475$	6.6.95034688.1	None	✓				1665.2.a.t	$1$	$0$	\(2\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(1-\beta _{3}+\cdots)q^{7}+\cdots\)
8325.2.a.cl	$8325$	$2$	8325.a	1.a	$1$	$6$	$6$	$66.475$	6.6.297869800.1	None	✓				2775.2.a.be	$1$	$0$	\(4\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2+\beta _{2})q^{4}+\beta _{3}q^{7}+\cdots\)
8325.2.a.cm	$8325$	$2$	8325.a	1.a	$1$	$7$	$7$	$66.475$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		925.2.a.j	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{6}q^{7}+(-\beta _{1}+\cdots)q^{8}+\cdots\)
8325.2.a.cn	$8325$	$2$	8325.a	1.a	$1$	$7$	$7$	$66.475$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				925.2.a.j	$1$	$1$	\(1\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{6}q^{7}+(\beta _{1}+\cdots)q^{8}+\cdots\)
8325.2.a.co	$8325$	$2$	8325.a	1.a	$1$	$8$	$8$	$66.475$	8.8.\(\cdots\).1	None	✓	✓	✓		8325.2.a.co	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-6\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(1-\beta _{1})q^{4}+(-1-\beta _{4})q^{7}+\cdots\)
8325.2.a.cp	$8325$	$2$	8325.a	1.a	$1$	$8$	$8$	$66.475$	8.8.\(\cdots\).1	None	✓	✓			8325.2.a.co	$2$	$1$	\(0\)	\(0\)	\(0\)	\(6\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(1-\beta _{1})q^{4}+(1+\beta _{4})q^{7}+\cdots\)
8325.2.a.cq	$8325$	$2$	8325.a	1.a	$1$	$9$	$9$	$66.475$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				185.2.b.a	$1$	$1$	\(-5\)	\(0\)	\(0\)	\(8\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
8325.2.a.cr	$8325$	$2$	8325.a	1.a	$1$	$9$	$9$	$66.475$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				185.2.b.a	$1$	$0$	\(5\)	\(0\)	\(0\)	\(-8\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
8325.2.a.cs	$8325$	$2$	8325.a	1.a	$1$	$10$	$10$	$66.475$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		8325.2.a.cs	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{2}-\beta _{7}+\cdots)q^{7}+\cdots\)
8325.2.a.ct	$8325$	$2$	8325.a	1.a	$1$	$10$	$10$	$66.475$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓			8325.2.a.cs	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{2}+\beta _{7}+\cdots)q^{7}+\cdots\)
8325.2.a.cu	$8325$	$2$	8325.a	1.a	$1$	$13$	$13$	$66.475$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓		✓		555.2.c.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+\beta _{9})q^{7}+\cdots\)
8325.2.a.cv	$8325$	$2$	8325.a	1.a	$1$	$13$	$13$	$66.475$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓		✓		555.2.c.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1-\beta _{9})q^{7}+\cdots\)
8325.2.a.cw	$8325$	$2$	8325.a	1.a	$1$	$16$	$16$	$66.475$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓		✓		1665.2.c.g	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-20\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{8})q^{7}+\cdots\)
8325.2.a.cx	$8325$	$2$	8325.a	1.a	$1$	$16$	$16$	$66.475$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓		✓		1665.2.c.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(20\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{8})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8325))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8325)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(111))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(333))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(555))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(925))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1665))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2775))\)\(^{\oplus 2}\)