Space of modular forms of level 6075, weight 2, and trivial character

• Label: 6075.2.a
• Level: $6075$
• Weight: $2$
• Character orbit: 6075.a
• Rep. character: $\chi_{6075}(1,\cdot)$
• Character field: $\Q$
• Dimension: $228$
• Newform subspaces: $70$
• Sturm bound: $1620$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 6075 = 3^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6075.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 70 \)
Sturm bound:			\(1620\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(2\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6075))\).

	Total	New	Old
Modular forms	864	228	636
Cusp forms	757	228	529
Eisenstein series	107	0	107

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

\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(207\)	\(48\)	\(159\)		\(181\)	\(48\)	\(133\)		\(26\)	\(0\)	\(26\)
\(+\)	\(-\)	\(-\)		\(225\)	\(64\)	\(161\)		\(198\)	\(64\)	\(134\)		\(27\)	\(0\)	\(27\)
\(-\)	\(+\)	\(-\)		\(225\)	\(60\)	\(165\)		\(198\)	\(60\)	\(138\)		\(27\)	\(0\)	\(27\)
\(-\)	\(-\)	\(+\)		\(207\)	\(56\)	\(151\)		\(180\)	\(56\)	\(124\)		\(27\)	\(0\)	\(27\)
Plus space		\(+\)		\(414\)	\(104\)	\(310\)		\(361\)	\(104\)	\(257\)		\(53\)	\(0\)	\(53\)
Minus space		\(-\)		\(450\)	\(124\)	\(326\)		\(396\)	\(124\)	\(272\)		\(54\)	\(0\)	\(54\)

Trace form

228 * q + 228 * q^4 - 3 * q^7 - 3 * q^13 + 228 * q^16 - 3 * q^19 - 12 * q^28 - 12 * q^31 + 18 * q^34 - 3 * q^37 - 12 * q^43 + 18 * q^46 + 225 * q^49 - 12 * q^52 + 18 * q^58 + 24 * q^61 + 246 * q^64 + 24 * q^67 + 24 * q^73 + 6 * q^76 + 15 * q^79 - 72 * q^82 + 36 * q^88 + 12 * q^91 + 36 * q^94 - 75 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6075))\) 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
6075.2.a.a	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓	✓			6075.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-3q^{7}-4q^{11}+2q^{13}+\cdots\)
6075.2.a.b	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓	✓			6075.2.a.b	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-2q^{11}-2q^{13}-4q^{16}+\cdots\)
6075.2.a.c	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓				1215.2.a.a	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+2q^{11}-q^{13}-4q^{16}+\cdots\)
6075.2.a.d	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓	✓			6075.2.a.b	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+2q^{11}+2q^{13}-4q^{16}+\cdots\)
6075.2.a.e	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓	✓			6075.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+3q^{7}+4q^{11}-2q^{13}+\cdots\)
6075.2.a.f	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓				1215.2.a.d	$1$	$2$	\(-1\)	\(0\)	\(0\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-3q^{7}+3q^{8}-5q^{11}+\cdots\)
6075.2.a.g	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓	✓			6075.2.a.g	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-3q^{7}+3q^{8}+4q^{11}+\cdots\)
6075.2.a.h	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓				1215.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-3q^{7}+3q^{8}+4q^{11}+\cdots\)
6075.2.a.i	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓	✓			6075.2.a.i	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+3q^{8}-2q^{11}-q^{13}+\cdots\)
6075.2.a.j	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓	✓			6075.2.a.i	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+3q^{8}+2q^{11}+q^{13}+\cdots\)
6075.2.a.k	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓	✓			6075.2.a.g	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+3q^{7}+3q^{8}-4q^{11}+\cdots\)
6075.2.a.l	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓				1215.2.a.b	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+3q^{7}+3q^{8}+q^{11}-4q^{13}+\cdots\)
6075.2.a.m	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				243.2.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-5\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}-5q^{7}-2q^{13}+4q^{16}+8q^{19}+\cdots\)
6075.2.a.n	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			6075.2.a.n	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-5\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}-5q^{7}+7q^{13}+4q^{16}-7q^{19}+\cdots\)
6075.2.a.o	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			6075.2.a.o	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}-4q^{7}+2q^{13}+4q^{16}-7q^{19}+\cdots\)
6075.2.a.p	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			6075.2.a.p	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}-q^{7}-7q^{13}+4q^{16}+8q^{19}+\cdots\)
6075.2.a.q	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			6075.2.a.q	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}-q^{7}+2q^{13}+4q^{16}-q^{19}+\cdots\)
6075.2.a.r	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓				1215.2.a.e	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{7}-6q^{11}-2q^{13}+4q^{16}+\cdots\)
6075.2.a.s	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			6075.2.a.q	$2$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+q^{7}-2q^{13}+4q^{16}-q^{19}+\cdots\)
6075.2.a.t	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			6075.2.a.p	$2$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+q^{7}+7q^{13}+4q^{16}+8q^{19}+\cdots\)
6075.2.a.u	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓				1215.2.a.e	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{7}+6q^{11}-2q^{13}+4q^{16}+\cdots\)
6075.2.a.v	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			6075.2.a.o	$2$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+4q^{7}-2q^{13}+4q^{16}-7q^{19}+\cdots\)
6075.2.a.w	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				243.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+4q^{7}+7q^{13}+4q^{16}-q^{19}+\cdots\)
6075.2.a.x	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			6075.2.a.n	$2$	$0$	\(0\)	\(0\)	\(0\)	\(5\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+5q^{7}-7q^{13}+4q^{16}-7q^{19}+\cdots\)
6075.2.a.y	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓	✓			6075.2.a.g	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-3q^{7}-3q^{8}-4q^{11}+\cdots\)
6075.2.a.z	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓				1215.2.a.c	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-3q^{7}-3q^{8}-4q^{11}+\cdots\)
6075.2.a.ba	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓				1215.2.a.d	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-3q^{7}-3q^{8}+5q^{11}+\cdots\)
6075.2.a.bb	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓	✓			6075.2.a.i	$1$	$1$	\(1\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-3q^{8}-2q^{11}+q^{13}+\cdots\)
6075.2.a.bc	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓	✓			6075.2.a.i	$1$	$1$	\(1\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-3q^{8}+2q^{11}-q^{13}+\cdots\)
6075.2.a.bd	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓				1215.2.a.b	$1$	$1$	\(1\)	\(0\)	\(0\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+3q^{7}-3q^{8}-q^{11}-4q^{13}+\cdots\)
6075.2.a.be	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓	✓			6075.2.a.g	$1$	$0$	\(1\)	\(0\)	\(0\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+3q^{7}-3q^{8}+4q^{11}+\cdots\)
6075.2.a.bf	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓	✓			6075.2.a.a	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-3q^{7}+4q^{11}+2q^{13}+\cdots\)
6075.2.a.bg	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓				1215.2.a.a	$1$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-2q^{11}-q^{13}-4q^{16}+\cdots\)
6075.2.a.bh	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓	✓			6075.2.a.b	$1$	$1$	\(2\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-2q^{11}+2q^{13}-4q^{16}+\cdots\)
6075.2.a.bi	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓	✓			6075.2.a.b	$1$	$1$	\(2\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+2q^{11}-2q^{13}-4q^{16}+\cdots\)
6075.2.a.bj	$6075$	$2$	6075.a	1.a	$1$	$1$	$1$	$48.509$	\(\Q\)	None	✓	✓			6075.2.a.a	$1$	$1$	\(2\)	\(0\)	\(0\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+3q^{7}-4q^{11}-2q^{13}+\cdots\)
6075.2.a.bk	$6075$	$2$	6075.a	1.a	$1$	$2$	$2$	$48.509$	\(\Q(\sqrt{13}) \)	None	✓				1215.2.a.k	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(1+\beta )q^{4}+(1-\beta )q^{7}-3q^{8}+\cdots\)
6075.2.a.bl	$6075$	$2$	6075.a	1.a	$1$	$2$	$2$	$48.509$	\(\Q(\sqrt{21}) \)	None	✓				1215.2.a.l	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(3+\beta )q^{4}+(1+\beta )q^{7}+(-5+\cdots)q^{8}+\cdots\)
6075.2.a.bm	$6075$	$2$	6075.a	1.a	$1$	$2$	$2$	$48.509$	\(\Q(\sqrt{3}) \)	None	✓				243.2.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{4}+q^{7}-\beta q^{8}+2\beta q^{11}+\cdots\)
6075.2.a.bn	$6075$	$2$	6075.a	1.a	$1$	$2$	$2$	$48.509$	\(\Q(\sqrt{6}) \)	None	✓				243.2.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+4q^{4}-2q^{7}+2\beta q^{8}-\beta q^{11}+\cdots\)
6075.2.a.bo	$6075$	$2$	6075.a	1.a	$1$	$2$	$2$	$48.509$	\(\Q(\sqrt{13}) \)	None	✓				1215.2.a.k	$1$	$1$	\(1\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1+\beta )q^{4}+(1-\beta )q^{7}+3q^{8}+\cdots\)
6075.2.a.bp	$6075$	$2$	6075.a	1.a	$1$	$2$	$2$	$48.509$	\(\Q(\sqrt{21}) \)	None	✓				1215.2.a.l	$1$	$0$	\(1\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(3+\beta )q^{4}+(1+\beta )q^{7}+(5+2\beta )q^{8}+\cdots\)
6075.2.a.bq	$6075$	$2$	6075.a	1.a	$1$	$3$	$3$	$48.509$	\(\Q(\zeta_{18})^+\)	None	✓				243.2.a.e	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
6075.2.a.br	$6075$	$2$	6075.a	1.a	$1$	$3$	$3$	$48.509$	3.3.564.1	None	✓				1215.2.a.o	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(1-2\beta _{1})q^{7}+\cdots\)
6075.2.a.bs	$6075$	$2$	6075.a	1.a	$1$	$3$	$3$	$48.509$	\(\Q(\zeta_{18})^+\)	None	✓				1215.2.a.p	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{2}q^{4}+(1+\beta _{2})q^{7}+(-1+\cdots)q^{8}+\cdots\)
6075.2.a.bt	$6075$	$2$	6075.a	1.a	$1$	$3$	$3$	$48.509$	\(\Q(\zeta_{18})^+\)	None	✓				1215.2.a.p	$1$	$1$	\(0\)	\(0\)	\(0\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(1+\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{8}+\cdots\)
6075.2.a.bu	$6075$	$2$	6075.a	1.a	$1$	$3$	$3$	$48.509$	3.3.564.1	None	✓				1215.2.a.o	$1$	$0$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(1-2\beta _{1})q^{7}+\cdots\)
6075.2.a.bv	$6075$	$2$	6075.a	1.a	$1$	$3$	$3$	$48.509$	\(\Q(\zeta_{18})^+\)	None	✓				243.2.a.e	$1$	$0$	\(3\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{7}+\cdots\)
6075.2.a.bw	$6075$	$2$	6075.a	1.a	$1$	$4$	$4$	$48.509$	4.4.92160.2	None	✓	✓			6075.2.a.bw	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+\beta _{2}q^{7}+(2\beta _{1}+\cdots)q^{8}+\cdots\)
6075.2.a.bx	$6075$	$2$	6075.a	1.a	$1$	$4$	$4$	$48.509$	4.4.29952.1	None	✓	✓			6075.2.a.bx	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}-2\beta _{2}q^{7}+(\beta _{1}+\cdots)q^{8}+\cdots\)
6075.2.a.by	$6075$	$2$	6075.a	1.a	$1$	$4$	$4$	$48.509$	4.4.29952.1	None	✓	✓			6075.2.a.bx	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+2\beta _{2}q^{7}+(\beta _{1}+\cdots)q^{8}+\cdots\)
6075.2.a.bz	$6075$	$2$	6075.a	1.a	$1$	$4$	$4$	$48.509$	4.4.92160.2	None	✓	✓			6075.2.a.bw	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}-\beta _{2}q^{7}+(2\beta _{1}+\cdots)q^{8}+\cdots\)
6075.2.a.ca	$6075$	$2$	6075.a	1.a	$1$	$4$	$4$	$48.509$	4.4.25088.1	None	✓	✓			6075.2.a.ca	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)		$+$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(3-\beta _{2})q^{4}+(-2-\beta _{2})q^{7}+\cdots\)
6075.2.a.cb	$6075$	$2$	6075.a	1.a	$1$	$4$	$4$	$48.509$	4.4.25088.1	None	✓	✓			6075.2.a.ca	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(3-\beta _{2})q^{4}+(2+\beta _{2})q^{7}+\cdots\)
6075.2.a.cc	$6075$	$2$	6075.a	1.a	$1$	$6$	$6$	$48.509$	6.6.252315648.2	None	✓				1215.2.b.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{2})q^{7}+\cdots\)
6075.2.a.cd	$6075$	$2$	6075.a	1.a	$1$	$6$	$6$	$48.509$	6.6.839808000.1	None	✓				1215.2.b.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+\beta _{4})q^{7}+\cdots\)
6075.2.a.ce	$6075$	$2$	6075.a	1.a	$1$	$6$	$6$	$48.509$	6.6.89672832.1	None	✓				1215.2.b.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{5}q^{7}+\beta _{3}q^{8}+\cdots\)
6075.2.a.cf	$6075$	$2$	6075.a	1.a	$1$	$6$	$6$	$48.509$	6.6.89672832.1	None	✓				1215.2.b.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{5}q^{7}+\beta _{3}q^{8}+\cdots\)
6075.2.a.cg	$6075$	$2$	6075.a	1.a	$1$	$6$	$6$	$48.509$	6.6.839808000.1	None	✓				1215.2.b.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1-\beta _{4})q^{7}+\cdots\)
6075.2.a.ch	$6075$	$2$	6075.a	1.a	$1$	$6$	$6$	$48.509$	6.6.252315648.2	None	✓				1215.2.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(6\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{2})q^{7}+\cdots\)
6075.2.a.ci	$6075$	$2$	6075.a	1.a	$1$	$6$	$6$	$48.509$	6.6.820125.1	\(\Q(\sqrt{-15}) \)	✓				1215.2.b.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+(-\beta _{1}-\beta _{2})q^{2}+(2+2\beta _{1}+\beta _{3}-\beta _{5})q^{4}+\cdots\)
6075.2.a.cj	$6075$	$2$	6075.a	1.a	$1$	$6$	$6$	$48.509$	6.6.820125.1	\(\Q(\sqrt{-15}) \)	✓				1215.2.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+(-\beta _{2}-\beta _{5})q^{2}+(2-\beta _{1}-\beta _{2}-\beta _{5})q^{4}+\cdots\)
6075.2.a.ck	$6075$	$2$	6075.a	1.a	$1$	$9$	$9$	$48.509$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				1215.2.a.s	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-9\)		$+$	$+$	$+$	$3^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{8})q^{7}+\cdots\)
6075.2.a.cl	$6075$	$2$	6075.a	1.a	$1$	$9$	$9$	$48.509$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				1215.2.a.s	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-9\)		$-$	$+$	$-$	$3^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{8})q^{7}+\cdots\)
6075.2.a.cm	$6075$	$2$	6075.a	1.a	$1$	$12$	$12$	$48.509$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				1215.2.b.e	$2$	$1$	\(-6\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$3^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{5})q^{2}+(\beta _{1}-\beta _{4}+\beta _{8}+\beta _{9}+\cdots)q^{4}+\cdots\)
6075.2.a.cn	$6075$	$2$	6075.a	1.a	$1$	$12$	$12$	$48.509$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			6075.2.a.cn	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)		$+$	$-$	$-$	$3^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{1}+\beta _{8}+\cdots)q^{7}+\cdots\)
6075.2.a.co	$6075$	$2$	6075.a	1.a	$1$	$12$	$12$	$48.509$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			6075.2.a.cn	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-3\)		$-$	$-$	$+$	$3^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{1}+\beta _{8}+\cdots)q^{7}+\cdots\)
6075.2.a.cp	$6075$	$2$	6075.a	1.a	$1$	$12$	$12$	$48.509$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓		6075.2.a.cn	$1$	$1$	\(0\)	\(0\)	\(0\)	\(3\)		$+$	$+$	$+$	$3^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(\beta _{1}-\beta _{8})q^{7}+\cdots\)
6075.2.a.cq	$6075$	$2$	6075.a	1.a	$1$	$12$	$12$	$48.509$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓		6075.2.a.cn	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$-$	$3^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(\beta _{1}-\beta _{8})q^{7}+\cdots\)
6075.2.a.cr	$6075$	$2$	6075.a	1.a	$1$	$12$	$12$	$48.509$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				1215.2.b.e	$2$	$0$	\(6\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{5})q^{2}+(\beta _{1}-\beta _{4}+\beta _{8}+\beta _{9}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6075))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6075)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(243))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(405))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(675))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1215))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2025))\)\(^{\oplus 2}\)