Space of modular forms of level 8512, weight 2, and trivial character

• Label: 8512.2.a
• Level: $8512$
• Weight: $2$
• Character orbit: 8512.a
• Rep. character: $\chi_{8512}(1,\cdot)$
• Character field: $\Q$
• Dimension: $216$
• Newform subspaces: $64$
• Sturm bound: $2560$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 64 \)
Sturm bound:			\(2560\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\), \(11\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8512))\).

	Total	New	Old
Modular forms	1304	216	1088
Cusp forms	1257	216	1041
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.

\(2\)	\(7\)	\(19\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(158\)	\(26\)	\(132\)		\(153\)	\(26\)	\(127\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(168\)	\(28\)	\(140\)		\(162\)	\(28\)	\(134\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(162\)	\(27\)	\(135\)		\(156\)	\(27\)	\(129\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(164\)	\(25\)	\(139\)		\(158\)	\(25\)	\(133\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(168\)	\(28\)	\(140\)		\(162\)	\(28\)	\(134\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(158\)	\(26\)	\(132\)		\(152\)	\(26\)	\(126\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(164\)	\(27\)	\(137\)		\(158\)	\(27\)	\(131\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(162\)	\(29\)	\(133\)		\(156\)	\(29\)	\(127\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(644\)	\(104\)	\(540\)		\(621\)	\(104\)	\(517\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(660\)	\(112\)	\(548\)		\(636\)	\(112\)	\(524\)		\(24\)	\(0\)	\(24\)

Trace form

\( 216 q + 216 q^{9} + 16 q^{17} + 232 q^{25} + 16 q^{29} + 16 q^{37} + 16 q^{41} - 96 q^{45} + 216 q^{49} - 80 q^{53} - 96 q^{69} - 16 q^{73} + 16 q^{77} + 216 q^{81} - 16 q^{89} + 16 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8512))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	7	19
8512.2.a.a	$8512$	$2$	8512.a	1.a	$1$	$1$	$1$	$67.969$	\(\Q\)	None	✓				4256.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-q^{7}-3q^{9}+4q^{13}+6q^{17}+\cdots\)
8512.2.a.b	$8512$	$2$	8512.a	1.a	$1$	$1$	$1$	$67.969$	\(\Q\)	None	✓				4256.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+q^{7}-3q^{9}+4q^{13}+6q^{17}+\cdots\)
8512.2.a.c	$8512$	$2$	8512.a	1.a	$1$	$1$	$1$	$67.969$	\(\Q\)	None	✓				532.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-q^{7}-3q^{9}+4q^{11}-4q^{13}+\cdots\)
8512.2.a.d	$8512$	$2$	8512.a	1.a	$1$	$1$	$1$	$67.969$	\(\Q\)	None	✓				532.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+q^{7}-3q^{9}-4q^{11}-4q^{13}+\cdots\)
8512.2.a.e	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				1064.2.a.e	$1$	$1$	\(0\)	\(-3\)	\(-5\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(-2-\beta )q^{5}+q^{7}+\cdots\)
8512.2.a.f	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				133.2.a.c	$1$	$1$	\(0\)	\(-3\)	\(-2\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}-q^{5}+q^{7}+(-1+3\beta )q^{9}+\cdots\)
8512.2.a.g	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				4256.2.a.c	$1$	$1$	\(0\)	\(-3\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}-\beta q^{5}-q^{7}+(-1+3\beta )q^{9}+\cdots\)
8512.2.a.h	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				266.2.a.b	$1$	$1$	\(0\)	\(-3\)	\(-1\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(-2+3\beta )q^{5}+q^{7}+\cdots\)
8512.2.a.i	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				1064.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(-1+2\beta )q^{5}-q^{7}+\cdots\)
8512.2.a.j	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				133.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(1-2\beta )q^{5}+q^{7}+(-1+\cdots)q^{9}+\cdots\)
8512.2.a.k	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				532.2.a.b	$1$	$0$	\(0\)	\(-3\)	\(2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+q^{5}-q^{7}+(-1+3\beta )q^{9}+\cdots\)
8512.2.a.l	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{13}) \)	None	✓				133.2.a.b	$1$	$1$	\(0\)	\(-3\)	\(6\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+3q^{5}-q^{7}+(1+3\beta )q^{9}+\cdots\)
8512.2.a.m	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{21}) \)	None	✓				532.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(-6\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-3q^{5}-q^{7}+(2+\beta )q^{9}+(-1+\cdots)q^{11}+\cdots\)
8512.2.a.n	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{13}) \)	None	✓				266.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-1+\beta )q^{5}+q^{7}+\beta q^{9}+\cdots\)
8512.2.a.o	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				1064.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1-2\beta )q^{5}-q^{7}+(-2+\beta )q^{9}+\cdots\)
8512.2.a.p	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{29}) \)	None	✓				266.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1-\beta )q^{5}-q^{7}+(4+\beta )q^{9}+\cdots\)
8512.2.a.q	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				1064.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+q^{5}-q^{7}+(-2+\beta )q^{9}+(-1+\cdots)q^{11}+\cdots\)
8512.2.a.r	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{13}) \)	None	✓				1064.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+q^{5}+q^{7}+\beta q^{9}+(-1+\beta )q^{11}+\cdots\)
8512.2.a.s	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				532.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(4\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1+2\beta )q^{5}-q^{7}+(-2+\beta )q^{9}+\cdots\)
8512.2.a.t	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{21}) \)	None	✓				532.2.a.c	$1$	$0$	\(0\)	\(1\)	\(-6\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-3q^{5}+q^{7}+(2+\beta )q^{9}+(1+\cdots)q^{11}+\cdots\)
8512.2.a.u	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{13}) \)	None	✓				266.2.a.c	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1+\beta )q^{5}-q^{7}+\beta q^{9}+\cdots\)
8512.2.a.v	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				1064.2.a.c	$1$	$1$	\(0\)	\(1\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1-2\beta )q^{5}+q^{7}+(-2+\beta )q^{9}+\cdots\)
8512.2.a.w	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{29}) \)	None	✓				266.2.a.a	$1$	$0$	\(0\)	\(1\)	\(1\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1-\beta )q^{5}+q^{7}+(4+\beta )q^{9}+\cdots\)
8512.2.a.x	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{13}) \)	None	✓				1064.2.a.b	$1$	$1$	\(0\)	\(1\)	\(2\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}-q^{7}+\beta q^{9}+(1-\beta )q^{11}+\cdots\)
8512.2.a.y	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				1064.2.a.d	$1$	$1$	\(0\)	\(1\)	\(2\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}+q^{7}+(-2+\beta )q^{9}+(1+\cdots)q^{11}+\cdots\)
8512.2.a.z	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				532.2.a.d	$1$	$0$	\(0\)	\(1\)	\(4\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1+2\beta )q^{5}+q^{7}+(-2+\beta )q^{9}+\cdots\)
8512.2.a.ba	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				1064.2.a.e	$1$	$0$	\(0\)	\(3\)	\(-5\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-2-\beta )q^{5}-q^{7}+(-1+\cdots)q^{9}+\cdots\)
8512.2.a.bb	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				133.2.a.c	$1$	$0$	\(0\)	\(3\)	\(-2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-q^{5}-q^{7}+(-1+3\beta )q^{9}+\cdots\)
8512.2.a.bc	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				266.2.a.b	$1$	$0$	\(0\)	\(3\)	\(-1\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-2+3\beta )q^{5}-q^{7}+(-1+\cdots)q^{9}+\cdots\)
8512.2.a.bd	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				4256.2.a.c	$1$	$1$	\(0\)	\(3\)	\(-1\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-\beta q^{5}+q^{7}+(-1+3\beta )q^{9}+\cdots\)
8512.2.a.be	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				133.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1-2\beta )q^{5}-q^{7}+(-1+\cdots)q^{9}+\cdots\)
8512.2.a.bf	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				1064.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-1+2\beta )q^{5}+q^{7}+(-1+\cdots)q^{9}+\cdots\)
8512.2.a.bg	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{5}) \)	None	✓				532.2.a.b	$1$	$1$	\(0\)	\(3\)	\(2\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+q^{5}+q^{7}+(-1+3\beta )q^{9}+\cdots\)
8512.2.a.bh	$8512$	$2$	8512.a	1.a	$1$	$2$	$2$	$67.969$	\(\Q(\sqrt{13}) \)	None	✓				133.2.a.b	$1$	$0$	\(0\)	\(3\)	\(6\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+3q^{5}+q^{7}+(1+3\beta )q^{9}+\cdots\)
8512.2.a.bi	$8512$	$2$	8512.a	1.a	$1$	$3$	$3$	$67.969$	3.3.229.1	None	✓				133.2.a.d	$1$	$1$	\(0\)	\(-3\)	\(2\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+\cdots\)
8512.2.a.bj	$8512$	$2$	8512.a	1.a	$1$	$3$	$3$	$67.969$	3.3.469.1	None	✓				266.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(-5\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-2+\beta _{1})q^{5}+q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
8512.2.a.bk	$8512$	$2$	8512.a	1.a	$1$	$3$	$3$	$67.969$	3.3.1101.1	None	✓				1064.2.a.f	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{2})q^{5}-q^{7}+(4-\beta _{1}+\cdots)q^{9}+\cdots\)
8512.2.a.bl	$8512$	$2$	8512.a	1.a	$1$	$3$	$3$	$67.969$	3.3.733.1	None	✓				532.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{1}+\beta _{2})q^{5}+q^{7}+\cdots\)
8512.2.a.bm	$8512$	$2$	8512.a	1.a	$1$	$3$	$3$	$67.969$	3.3.469.1	None	✓				266.2.a.d	$1$	$0$	\(0\)	\(1\)	\(-5\)	\(-3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(-2+\beta _{1})q^{5}-q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
8512.2.a.bn	$8512$	$2$	8512.a	1.a	$1$	$3$	$3$	$67.969$	3.3.733.1	None	✓				532.2.a.e	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(-3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{1}+\beta _{2})q^{5}-q^{7}+\cdots\)
8512.2.a.bo	$8512$	$2$	8512.a	1.a	$1$	$3$	$3$	$67.969$	3.3.1101.1	None	✓				1064.2.a.f	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{2})q^{5}+q^{7}+(4-\beta _{1}+\cdots)q^{9}+\cdots\)
8512.2.a.bp	$8512$	$2$	8512.a	1.a	$1$	$3$	$3$	$67.969$	3.3.229.1	None	✓				133.2.a.d	$1$	$0$	\(0\)	\(3\)	\(2\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+q^{7}+\cdots\)
8512.2.a.bq	$8512$	$2$	8512.a	1.a	$1$	$4$	$4$	$67.969$	4.4.25857.1	None	✓				1064.2.a.h	$1$	$0$	\(0\)	\(-2\)	\(-1\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(-1+\beta _{1}-\beta _{2})q^{5}+\cdots\)
8512.2.a.br	$8512$	$2$	8512.a	1.a	$1$	$4$	$4$	$67.969$	4.4.2624.1	None	✓				4256.2.a.e	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1+\beta _{1}+\beta _{2}-\beta _{3})q^{5}+\cdots\)
8512.2.a.bs	$8512$	$2$	8512.a	1.a	$1$	$4$	$4$	$67.969$	4.4.18097.1	None	✓				1064.2.a.g	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+(1+\beta _{2}-\beta _{3})q^{5}-q^{7}+(2+\cdots)q^{9}+\cdots\)
8512.2.a.bt	$8512$	$2$	8512.a	1.a	$1$	$4$	$4$	$67.969$	4.4.18097.1	None	✓				1064.2.a.g	$1$	$1$	\(0\)	\(0\)	\(3\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}+(1+\beta _{2}-\beta _{3})q^{5}+q^{7}+(2+\cdots)q^{9}+\cdots\)
8512.2.a.bu	$8512$	$2$	8512.a	1.a	$1$	$4$	$4$	$67.969$	4.4.25857.1	None	✓				1064.2.a.h	$1$	$1$	\(0\)	\(2\)	\(-1\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(-1+\beta _{1}-\beta _{2})q^{5}+\cdots\)
8512.2.a.bv	$8512$	$2$	8512.a	1.a	$1$	$4$	$4$	$67.969$	4.4.2624.1	None	✓				4256.2.a.e	$1$	$1$	\(0\)	\(2\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1+\beta _{1}+\beta _{2}-\beta _{3})q^{5}+\cdots\)
8512.2.a.bw	$8512$	$2$	8512.a	1.a	$1$	$5$	$5$	$67.969$	5.5.1730752.1	None	✓				4256.2.a.g	$1$	$0$	\(0\)	\(-2\)	\(6\)	\(-5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{3})q^{5}-q^{7}+(\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
8512.2.a.bx	$8512$	$2$	8512.a	1.a	$1$	$5$	$5$	$67.969$	5.5.10463409.1	None	✓				1064.2.a.i	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-5\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{4})q^{5}-q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
8512.2.a.by	$8512$	$2$	8512.a	1.a	$1$	$5$	$5$	$67.969$	5.5.10463409.1	None	✓				1064.2.a.i	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{4})q^{5}+q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
8512.2.a.bz	$8512$	$2$	8512.a	1.a	$1$	$5$	$5$	$67.969$	5.5.1730752.1	None	✓				4256.2.a.g	$1$	$0$	\(0\)	\(2\)	\(6\)	\(5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{3})q^{5}+q^{7}+(\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
8512.2.a.ca	$8512$	$2$	8512.a	1.a	$1$	$6$	$6$	$67.969$	6.6.41027408.1	None	✓				4256.2.a.i	$1$	$1$	\(0\)	\(-3\)	\(-3\)	\(6\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{3}q^{5}+q^{7}+(\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{9}+\cdots\)
8512.2.a.cb	$8512$	$2$	8512.a	1.a	$1$	$6$	$6$	$67.969$	6.6.60663248.1	None	✓				4256.2.a.k	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-6\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{5}q^{5}-q^{7}+(1-\beta _{3}+\beta _{4}+\cdots)q^{9}+\cdots\)
8512.2.a.cc	$8512$	$2$	8512.a	1.a	$1$	$6$	$6$	$67.969$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				4256.2.a.j	$1$	$0$	\(0\)	\(-1\)	\(5\)	\(6\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{4})q^{5}+q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)
8512.2.a.cd	$8512$	$2$	8512.a	1.a	$1$	$6$	$6$	$67.969$	6.6.60663248.1	None	✓				4256.2.a.k	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(6\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{5}q^{5}+q^{7}+(1-\beta _{3}+\beta _{4}+\cdots)q^{9}+\cdots\)
8512.2.a.ce	$8512$	$2$	8512.a	1.a	$1$	$6$	$6$	$67.969$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				4256.2.a.j	$1$	$0$	\(0\)	\(1\)	\(5\)	\(-6\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{4})q^{5}-q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)
8512.2.a.cf	$8512$	$2$	8512.a	1.a	$1$	$6$	$6$	$67.969$	6.6.41027408.1	None	✓				4256.2.a.i	$1$	$1$	\(0\)	\(3\)	\(-3\)	\(-6\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{3}q^{5}-q^{7}+(\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{9}+\cdots\)
8512.2.a.cg	$8512$	$2$	8512.a	1.a	$1$	$7$	$7$	$67.969$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		4256.2.a.o	$1$	$1$	\(0\)	\(-7\)	\(-5\)	\(7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1-\beta _{3})q^{5}+q^{7}+\cdots\)
8512.2.a.ch	$8512$	$2$	8512.a	1.a	$1$	$7$	$7$	$67.969$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		4256.2.a.p	$1$	$0$	\(0\)	\(-3\)	\(5\)	\(-7\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{3})q^{5}-q^{7}+(1-\beta _{2}+\cdots)q^{9}+\cdots\)
8512.2.a.ci	$8512$	$2$	8512.a	1.a	$1$	$7$	$7$	$67.969$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		4256.2.a.p	$1$	$0$	\(0\)	\(3\)	\(5\)	\(7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{3})q^{5}+q^{7}+(1-\beta _{2}+\cdots)q^{9}+\cdots\)
8512.2.a.cj	$8512$	$2$	8512.a	1.a	$1$	$7$	$7$	$67.969$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		4256.2.a.o	$1$	$1$	\(0\)	\(7\)	\(-5\)	\(-7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1-\beta _{3})q^{5}-q^{7}+\cdots\)
8512.2.a.ck	$8512$	$2$	8512.a	1.a	$1$	$10$	$10$	$67.969$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		4256.2.a.s	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-10\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{5}q^{5}-q^{7}+(2+\beta _{2})q^{9}+\cdots\)
8512.2.a.cl	$8512$	$2$	8512.a	1.a	$1$	$10$	$10$	$67.969$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		4256.2.a.s	$1$	$0$	\(0\)	\(0\)	\(0\)	\(10\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{5}q^{5}+q^{7}+(2+\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8512))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8512)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 14}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(133))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(266))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(304))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(448))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(532))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(608))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1064))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1216))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2128))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4256))\)\(^{\oplus 2}\)