Space of modular forms of level 6975, weight 2, and trivial character

• Label: 6975.2.a
• Level: $6975$
• Weight: $2$
• Character orbit: 6975.a
• Rep. character: $\chi_{6975}(1,\cdot)$
• Character field: $\Q$
• Dimension: $237$
• Newform subspaces: $63$
• Sturm bound: $1920$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 63 \)
Sturm bound:			\(1920\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(2\), \(7\), \(11\), \(13\), \(17\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6975))\).

	Total	New	Old
Modular forms	984	237	747
Cusp forms	937	237	700
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.

\(3\)	\(5\)	\(31\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(114\)	\(20\)	\(94\)		\(109\)	\(20\)	\(89\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(132\)	\(26\)	\(106\)		\(126\)	\(26\)	\(100\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(126\)	\(30\)	\(96\)		\(120\)	\(30\)	\(90\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(120\)	\(18\)	\(102\)		\(114\)	\(18\)	\(96\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(132\)	\(35\)	\(97\)		\(126\)	\(35\)	\(91\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(114\)	\(32\)	\(82\)		\(108\)	\(32\)	\(76\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(120\)	\(35\)	\(85\)		\(114\)	\(35\)	\(79\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(126\)	\(41\)	\(85\)		\(120\)	\(41\)	\(79\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(468\)	\(105\)	\(363\)		\(445\)	\(105\)	\(340\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(516\)	\(132\)	\(384\)		\(492\)	\(132\)	\(360\)		\(24\)	\(0\)	\(24\)

Trace form

237 * q + 238 * q^4 - 3 * q^8 - 7 * q^14 + 248 * q^16 - 8 * q^17 + 4 * q^19 + 6 * q^22 + 6 * q^23 + 36 * q^26 + 9 * q^28 + 32 * q^29 - 3 * q^31 - 6 * q^32 + 16 * q^34 - 2 * q^37 - 7 * q^38 + 30 * q^43 + 42 * q^44 + 26 * q^46 - 28 * q^47 + 211 * q^49 + 6 * q^52 + 6 * q^53 + 6 * q^56 + 10 * q^58 - 28 * q^59 + 12 * q^61 - 4 * q^62 + 317 * q^64 + 48 * q^67 + 18 * q^68 + 36 * q^71 + 14 * q^73 - 20 * q^74 + 9 * q^76 + 24 * q^77 - 18 * q^79 + 11 * q^82 - 12 * q^83 + 20 * q^86 + 28 * q^88 + 20 * q^89 - 14 * q^91 + 8 * q^92 - 40 * q^94 - 4 * q^97 + 7 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6975))\) 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	31
6975.2.a.a	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓				2325.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-4q^{7}+q^{11}-6q^{13}+\cdots\)
6975.2.a.b	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓				155.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+2q^{7}-2q^{11}+6q^{13}+\cdots\)
6975.2.a.c	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓				2325.2.a.b	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+4q^{7}+5q^{11}-6q^{13}+\cdots\)
6975.2.a.d	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓				155.2.a.b	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-4q^{7}+3q^{8}-4q^{11}+\cdots\)
6975.2.a.e	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓				2325.2.a.e	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-2q^{7}+3q^{8}-2q^{11}+\cdots\)
6975.2.a.f	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓				2325.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+2q^{7}+3q^{8}+2q^{11}+\cdots\)
6975.2.a.g	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓				465.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+4q^{7}+3q^{8}+4q^{11}+\cdots\)
6975.2.a.h	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓	✓			6975.2.a.h	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-5q^{11}+4q^{16}-5q^{17}-q^{19}+\cdots\)
6975.2.a.i	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓	✓			6975.2.a.h	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-5q^{11}+4q^{16}+5q^{17}-q^{19}+\cdots\)
6975.2.a.j	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓				2325.2.a.f	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+3q^{11}-4q^{13}+4q^{16}-7q^{17}+\cdots\)
6975.2.a.k	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓				2325.2.a.f	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+3q^{11}+4q^{13}+4q^{16}+7q^{17}+\cdots\)
6975.2.a.l	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓				155.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+4q^{11}+6q^{13}+4q^{16}+5q^{17}+\cdots\)
6975.2.a.m	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓	✓			6975.2.a.h	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+5q^{11}+4q^{16}-5q^{17}-q^{19}+\cdots\)
6975.2.a.n	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓	✓			6975.2.a.h	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+5q^{11}+4q^{16}+5q^{17}-q^{19}+\cdots\)
6975.2.a.o	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓				2325.2.a.c	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-2q^{7}-3q^{8}+2q^{11}+\cdots\)
6975.2.a.p	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓				2325.2.a.e	$1$	$1$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+2q^{7}-3q^{8}-2q^{11}+\cdots\)
6975.2.a.q	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓				465.2.a.b	$1$	$1$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+2q^{7}-3q^{8}+4q^{11}+\cdots\)
6975.2.a.r	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓				2325.2.a.b	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-4q^{7}+5q^{11}+6q^{13}+\cdots\)
6975.2.a.s	$6975$	$2$	6975.a	1.a	$1$	$1$	$1$	$55.696$	\(\Q\)	None	✓				2325.2.a.a	$1$	$0$	\(2\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+4q^{7}+q^{11}+6q^{13}+\cdots\)
6975.2.a.t	$6975$	$2$	6975.a	1.a	$1$	$2$	$2$	$55.696$	\(\Q(\sqrt{5}) \)	None	✓				93.2.a.a	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+3\beta q^{4}+(1+2\beta )q^{7}+\cdots\)
6975.2.a.u	$6975$	$2$	6975.a	1.a	$1$	$2$	$2$	$55.696$	\(\Q(\sqrt{2}) \)	None	✓				465.2.a.c	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+(2+\beta )q^{7}+\cdots\)
6975.2.a.v	$6975$	$2$	6975.a	1.a	$1$	$2$	$2$	$55.696$	\(\Q(\sqrt{3}) \)	None	✓				465.2.a.d	$1$	$0$	\(0\)	\(0\)	\(0\)	\(6\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{4}+(3+\beta )q^{7}-\beta q^{8}+(-2+\cdots)q^{11}+\cdots\)
6975.2.a.w	$6975$	$2$	6975.a	1.a	$1$	$2$	$2$	$55.696$	\(\Q(\sqrt{5}) \)	None	✓				1395.2.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+3q^{4}+2\beta q^{7}-\beta q^{8}+2\beta q^{13}+\cdots\)
6975.2.a.x	$6975$	$2$	6975.a	1.a	$1$	$2$	$2$	$55.696$	\(\Q(\sqrt{5}) \)	None	✓				1395.2.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+3q^{4}-2\beta q^{7}-\beta q^{8}-2\beta q^{13}+\cdots\)
6975.2.a.y	$6975$	$2$	6975.a	1.a	$1$	$2$	$2$	$55.696$	\(\Q(\sqrt{5}) \)	None	✓				31.2.a.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(3-2\beta )q^{7}+\cdots\)
6975.2.a.z	$6975$	$2$	6975.a	1.a	$1$	$3$	$3$	$55.696$	3.3.148.1	None	✓	✓			6975.2.a.z	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-8\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots\)
6975.2.a.ba	$6975$	$2$	6975.a	1.a	$1$	$3$	$3$	$55.696$	3.3.148.1	None	✓	✓			6975.2.a.z	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(8\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots\)
6975.2.a.bb	$6975$	$2$	6975.a	1.a	$1$	$3$	$3$	$55.696$	3.3.229.1	None	✓				93.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{1}+\beta _{2})q^{7}+\cdots\)
6975.2.a.bc	$6975$	$2$	6975.a	1.a	$1$	$3$	$3$	$55.696$	3.3.148.1	None	✓				2325.2.a.t	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(1-\beta _{1}-\beta _{2})q^{4}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
6975.2.a.bd	$6975$	$2$	6975.a	1.a	$1$	$3$	$3$	$55.696$	3.3.148.1	None	✓				2325.2.a.t	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(1-\beta _{1}-\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
6975.2.a.be	$6975$	$2$	6975.a	1.a	$1$	$3$	$3$	$55.696$	3.3.148.1	None	✓				465.2.a.f	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
6975.2.a.bf	$6975$	$2$	6975.a	1.a	$1$	$3$	$3$	$55.696$	3.3.564.1	None	✓				465.2.a.e	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-8\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-3+\beta _{1})q^{7}+\cdots\)
6975.2.a.bg	$6975$	$2$	6975.a	1.a	$1$	$3$	$3$	$55.696$	3.3.148.1	None	✓	✓			6975.2.a.z	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-8\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+(-3+\cdots)q^{7}+\cdots\)
6975.2.a.bh	$6975$	$2$	6975.a	1.a	$1$	$3$	$3$	$55.696$	3.3.148.1	None	✓	✓			6975.2.a.z	$1$	$0$	\(2\)	\(0\)	\(0\)	\(8\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+(3+\cdots)q^{7}+\cdots\)
6975.2.a.bi	$6975$	$2$	6975.a	1.a	$1$	$3$	$3$	$55.696$	3.3.148.1	None	✓				465.2.a.g	$1$	$0$	\(3\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+(-2\beta _{1}+\cdots)q^{7}+\cdots\)
6975.2.a.bj	$6975$	$2$	6975.a	1.a	$1$	$4$	$4$	$55.696$	4.4.20308.1	None	✓				155.2.a.d	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}-\beta _{2}q^{7}+\cdots\)
6975.2.a.bk	$6975$	$2$	6975.a	1.a	$1$	$4$	$4$	$55.696$	\(\Q(\zeta_{24})^+\)	None	✓				155.2.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{3})q^{2}+(-\beta _{1}+\beta _{3})q^{7}+(-2\beta _{1}+\cdots)q^{8}+\cdots\)
6975.2.a.bl	$6975$	$2$	6975.a	1.a	$1$	$4$	$4$	$55.696$	\(\Q(\sqrt{4 + \sqrt{3}})\)	None	✓	✓			6975.2.a.bl	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-1+\beta _{2})q^{7}+\cdots\)
6975.2.a.bm	$6975$	$2$	6975.a	1.a	$1$	$4$	$4$	$55.696$	\(\Q(\sqrt{4 + \sqrt{3}})\)	None	✓	✓			6975.2.a.bl	$2$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(1-\beta _{2})q^{7}+\cdots\)
6975.2.a.bn	$6975$	$2$	6975.a	1.a	$1$	$4$	$4$	$55.696$	4.4.8468.1	None	✓				155.2.a.e	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(1+\beta _{1}-\beta _{2}+\beta _{3})q^{4}+(-1+\cdots)q^{7}+\cdots\)
6975.2.a.bo	$6975$	$2$	6975.a	1.a	$1$	$4$	$4$	$55.696$	4.4.8468.1	None	✓				465.2.a.h	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(2-\beta _{1})q^{4}+(-1+\beta _{1}-\beta _{3})q^{7}+\cdots\)
6975.2.a.bp	$6975$	$2$	6975.a	1.a	$1$	$5$	$5$	$55.696$	5.5.144209.1	None	✓				775.2.a.h	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{3})q^{2}+(1-\beta _{1}-\beta _{3}-\beta _{4})q^{4}+\cdots\)
6975.2.a.bq	$6975$	$2$	6975.a	1.a	$1$	$5$	$5$	$55.696$	5.5.205225.1	None	✓				775.2.a.i	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(6\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
6975.2.a.br	$6975$	$2$	6975.a	1.a	$1$	$5$	$5$	$55.696$	5.5.223824.1	None	✓				1395.2.a.n	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(6\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots\)
6975.2.a.bs	$6975$	$2$	6975.a	1.a	$1$	$5$	$5$	$55.696$	5.5.126032.1	None	✓				465.2.c.a	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{3})q^{2}+(1-\beta _{1}+\beta _{3})q^{4}+\cdots\)
6975.2.a.bt	$6975$	$2$	6975.a	1.a	$1$	$5$	$5$	$55.696$	5.5.582992.1	None	✓				1395.2.a.o	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{1}+\beta _{4})q^{7}+\cdots\)
6975.2.a.bu	$6975$	$2$	6975.a	1.a	$1$	$5$	$5$	$55.696$	5.5.582992.1	None	✓				1395.2.a.o	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{1}+\beta _{4})q^{7}+\cdots\)
6975.2.a.bv	$6975$	$2$	6975.a	1.a	$1$	$5$	$5$	$55.696$	5.5.126032.1	None	✓				465.2.c.a	$1$	$1$	\(3\)	\(0\)	\(0\)	\(-8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{3})q^{2}+(1-\beta _{1}+\beta _{3})q^{4}+(-2+\cdots)q^{7}+\cdots\)
6975.2.a.bw	$6975$	$2$	6975.a	1.a	$1$	$5$	$5$	$55.696$	5.5.223824.1	None	✓				1395.2.a.n	$1$	$0$	\(3\)	\(0\)	\(0\)	\(6\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{7}+\cdots\)
6975.2.a.bx	$6975$	$2$	6975.a	1.a	$1$	$5$	$5$	$55.696$	5.5.205225.1	None	✓				775.2.a.i	$1$	$0$	\(4\)	\(0\)	\(0\)	\(-6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots\)
6975.2.a.by	$6975$	$2$	6975.a	1.a	$1$	$5$	$5$	$55.696$	5.5.144209.1	None	✓				775.2.a.h	$1$	$0$	\(4\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{3})q^{2}+(1-\beta _{1}-\beta _{3}-\beta _{4})q^{4}+\cdots\)
6975.2.a.bz	$6975$	$2$	6975.a	1.a	$1$	$6$	$6$	$55.696$	6.6.136751504.1	None	✓		✓		2325.2.a.z	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-6\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{3})q^{7}+\cdots\)
6975.2.a.ca	$6975$	$2$	6975.a	1.a	$1$	$6$	$6$	$55.696$	6.6.75968016.1	None	✓		✓		2325.2.a.y	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{4}+\beta _{5})q^{4}+(-\beta _{2}+\cdots)q^{7}+\cdots\)
6975.2.a.cb	$6975$	$2$	6975.a	1.a	$1$	$6$	$6$	$55.696$	6.6.361944768.1	None	✓		✓		279.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{3})q^{4}+(-1-\beta _{5})q^{7}+\cdots\)
6975.2.a.cc	$6975$	$2$	6975.a	1.a	$1$	$6$	$6$	$55.696$	6.6.75968016.1	None	✓				2325.2.a.y	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{4}+\beta _{5})q^{4}+(\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots\)
6975.2.a.cd	$6975$	$2$	6975.a	1.a	$1$	$6$	$6$	$55.696$	6.6.136751504.1	None	✓				2325.2.a.z	$1$	$0$	\(1\)	\(0\)	\(0\)	\(6\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{3})q^{7}+\cdots\)
6975.2.a.ce	$6975$	$2$	6975.a	1.a	$1$	$8$	$8$	$55.696$	8.8.3057647616.1	None	✓		✓		1395.2.c.d	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}-\beta _{1}q^{4}+\beta _{7}q^{7}+(-\beta _{2}-\beta _{5}+\cdots)q^{8}+\cdots\)
6975.2.a.cf	$6975$	$2$	6975.a	1.a	$1$	$8$	$8$	$55.696$	8.8.\(\cdots\).1	None	✓	✓	✓		6975.2.a.cf	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{5}q^{2}+(2+\beta _{1})q^{4}+(-1-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
6975.2.a.cg	$6975$	$2$	6975.a	1.a	$1$	$8$	$8$	$55.696$	8.8.\(\cdots\).1	None	✓	✓			6975.2.a.cf	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{5}q^{2}+(2+\beta _{1})q^{4}+(1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
6975.2.a.ch	$6975$	$2$	6975.a	1.a	$1$	$10$	$10$	$55.696$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		155.2.b.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{3}-\beta _{6}+\cdots)q^{7}+\cdots\)
6975.2.a.ci	$6975$	$2$	6975.a	1.a	$1$	$11$	$11$	$55.696$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓				465.2.c.b	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(8\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{10})q^{7}+\cdots\)
6975.2.a.cj	$6975$	$2$	6975.a	1.a	$1$	$11$	$11$	$55.696$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓				465.2.c.b	$1$	$0$	\(3\)	\(0\)	\(0\)	\(-8\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{10})q^{7}+\cdots\)
6975.2.a.ck	$6975$	$2$	6975.a	1.a	$1$	$16$	$16$	$55.696$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓		✓		1395.2.c.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$2^{6}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(\beta _{1}-\beta _{10}-\beta _{11})q^{4}+(-\beta _{9}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6975))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6975)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(93))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(279))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(465))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(775))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1395))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2325))\)\(^{\oplus 2}\)