Space of modular forms of level 6800, weight 2, and trivial character

• Label: 6800.2.a
• Level: $6800$
• Weight: $2$
• Character orbit: 6800.a
• Rep. character: $\chi_{6800}(1,\cdot)$
• Character field: $\Q$
• Dimension: $152$
• Newform subspaces: $61$
• Sturm bound: $2160$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6800.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 61 \)
Sturm bound:			\(2160\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6800))\).

	Total	New	Old
Modular forms	1116	152	964
Cusp forms	1045	152	893
Eisenstein series	71	0	71

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

\(2\)	\(5\)	\(17\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(132\)	\(15\)	\(117\)		\(124\)	\(15\)	\(109\)		\(8\)	\(0\)	\(8\)
\(+\)	\(+\)	\(-\)	\(-\)		\(144\)	\(21\)	\(123\)		\(135\)	\(21\)	\(114\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(+\)	\(-\)		\(146\)	\(22\)	\(124\)		\(137\)	\(22\)	\(115\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(-\)	\(+\)		\(134\)	\(18\)	\(116\)		\(125\)	\(18\)	\(107\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(+\)	\(-\)		\(147\)	\(21\)	\(126\)		\(138\)	\(21\)	\(117\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(-\)	\(+\)		\(135\)	\(15\)	\(120\)		\(126\)	\(15\)	\(111\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(+\)	\(+\)		\(133\)	\(18\)	\(115\)		\(124\)	\(18\)	\(106\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(-\)	\(-\)		\(145\)	\(22\)	\(123\)		\(136\)	\(22\)	\(114\)		\(9\)	\(0\)	\(9\)
Plus space			\(+\)		\(534\)	\(66\)	\(468\)		\(499\)	\(66\)	\(433\)		\(35\)	\(0\)	\(35\)
Minus space			\(-\)		\(582\)	\(86\)	\(496\)		\(546\)	\(86\)	\(460\)		\(36\)	\(0\)	\(36\)

Trace form

152 * q + 2 * q^3 + 2 * q^7 + 152 * q^9 - 10 * q^11 + 4 * q^19 + 8 * q^21 - 10 * q^23 - 16 * q^27 + 16 * q^29 + 22 * q^31 + 8 * q^33 + 8 * q^37 + 8 * q^41 + 28 * q^43 + 176 * q^49 - 6 * q^51 + 8 * q^53 + 24 * q^57 - 12 * q^59 + 6 * q^63 - 16 * q^67 + 10 * q^71 + 16 * q^73 + 42 * q^79 + 160 * q^81 + 20 * q^83 + 4 * q^87 - 8 * q^89 + 56 * q^91 + 16 * q^93 - 8 * q^97 - 6 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6800))\) 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	5	17
6800.2.a.a	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				850.2.a.a	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{7}+6q^{9}+4q^{11}-3q^{13}+\cdots\)
6800.2.a.b	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				34.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-4q^{7}+q^{9}-6q^{11}-2q^{13}+\cdots\)
6800.2.a.c	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				170.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{7}+q^{9}+2q^{11}+6q^{13}+\cdots\)
6800.2.a.d	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				136.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{7}+q^{9}+6q^{11}-2q^{13}+\cdots\)
6800.2.a.e	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				170.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{7}+q^{9}-6q^{11}-2q^{13}+\cdots\)
6800.2.a.f	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				680.2.e.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}-2q^{9}-2q^{11}+5q^{13}+\cdots\)
6800.2.a.g	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				170.2.c.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}+6q^{11}+3q^{13}+q^{17}+\cdots\)
6800.2.a.h	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				1700.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}-2q^{9}-q^{13}-q^{17}+4q^{19}+\cdots\)
6800.2.a.i	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				425.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}-2q^{9}+4q^{11}+q^{13}+\cdots\)
6800.2.a.j	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				680.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}-2q^{9}-4q^{11}+q^{13}+\cdots\)
6800.2.a.k	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				850.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+5q^{7}-2q^{9}-4q^{11}+3q^{13}+\cdots\)
6800.2.a.l	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				340.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}-3q^{9}-2q^{11}+6q^{13}-q^{17}+\cdots\)
6800.2.a.m	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				680.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}+2q^{13}-q^{17}+4q^{19}-8q^{23}+\cdots\)
6800.2.a.n	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				17.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}-3q^{9}+2q^{13}-q^{17}+4q^{19}+\cdots\)
6800.2.a.o	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				850.2.a.c	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-5q^{7}-2q^{9}-4q^{11}-3q^{13}+\cdots\)
6800.2.a.p	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				1700.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}-2q^{9}+q^{13}+q^{17}+4q^{19}+\cdots\)
6800.2.a.q	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				425.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}-2q^{9}+4q^{11}-q^{13}+\cdots\)
6800.2.a.r	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				170.2.c.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}+6q^{11}-3q^{13}-q^{17}+\cdots\)
6800.2.a.s	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				170.2.a.c	$1$	$1$	\(0\)	\(1\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}-2q^{9}-5q^{13}+q^{17}+\cdots\)
6800.2.a.t	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				170.2.a.e	$1$	$1$	\(0\)	\(1\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}-2q^{9}+q^{13}+q^{17}+\cdots\)
6800.2.a.u	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				680.2.e.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{7}-2q^{9}-2q^{11}-5q^{13}+\cdots\)
6800.2.a.v	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				85.2.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
6800.2.a.w	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				136.2.a.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-2q^{11}+6q^{13}+q^{17}+\cdots\)
6800.2.a.x	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				680.2.a.c	$1$	$1$	\(0\)	\(2\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
6800.2.a.y	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				850.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{7}+6q^{9}+4q^{11}+3q^{13}+\cdots\)
6800.2.a.z	$6800$	$2$	6800.a	1.a	$1$	$1$	$1$	$54.298$	\(\Q\)	None	✓				170.2.a.d	$1$	$0$	\(0\)	\(3\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{7}+6q^{9}+4q^{11}+3q^{13}+\cdots\)
6800.2.a.ba	$6800$	$2$	6800.a	1.a	$1$	$2$	$2$	$54.298$	\(\Q(\sqrt{2}) \)	None	✓				85.2.a.b	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{3}+(-2-\beta )q^{7}+(3-4\beta )q^{9}+\cdots\)
6800.2.a.bb	$6800$	$2$	6800.a	1.a	$1$	$2$	$2$	$54.298$	\(\Q(\sqrt{3}) \)	None	✓				680.2.a.d	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-6\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(-3-\beta )q^{7}+(1-2\beta )q^{9}+\cdots\)
6800.2.a.bc	$6800$	$2$	6800.a	1.a	$1$	$2$	$2$	$54.298$	\(\Q(\sqrt{2}) \)	None	✓				850.2.a.m	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(-1+\beta )q^{7}-2\beta q^{9}+\cdots\)
6800.2.a.bd	$6800$	$2$	6800.a	1.a	$1$	$2$	$2$	$54.298$	\(\Q(\sqrt{5}) \)	None	✓				136.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(1+\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
6800.2.a.be	$6800$	$2$	6800.a	1.a	$1$	$2$	$2$	$54.298$	\(\Q(\sqrt{17}) \)	None	✓				170.2.a.f	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+2\beta q^{7}+(1+\beta )q^{9}+4q^{11}+\cdots\)
6800.2.a.bf	$6800$	$2$	6800.a	1.a	$1$	$2$	$2$	$54.298$	\(\Q(\sqrt{2}) \)	None	✓				680.2.a.e	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{7}-q^{9}+(2+3\beta )q^{11}+\cdots\)
6800.2.a.bg	$6800$	$2$	6800.a	1.a	$1$	$2$	$2$	$54.298$	\(\Q(\sqrt{3}) \)	None	✓				85.2.a.c	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-1-\beta )q^{7}+(1+2\beta )q^{9}+\cdots\)
6800.2.a.bh	$6800$	$2$	6800.a	1.a	$1$	$2$	$2$	$54.298$	\(\Q(\sqrt{3}) \)	None	✓				68.2.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-1-\beta )q^{7}+(1+2\beta )q^{9}+\cdots\)
6800.2.a.bi	$6800$	$2$	6800.a	1.a	$1$	$2$	$2$	$54.298$	\(\Q(\sqrt{2}) \)	None	✓				850.2.a.m	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1+\beta )q^{7}+2\beta q^{9}-2q^{11}+\cdots\)
6800.2.a.bj	$6800$	$2$	6800.a	1.a	$1$	$3$	$3$	$54.298$	3.3.564.1	None	✓				3400.2.a.m	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{1}-\beta _{2})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
6800.2.a.bk	$6800$	$2$	6800.a	1.a	$1$	$3$	$3$	$54.298$	3.3.568.1	None	✓				170.2.c.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-\beta _{1}-\beta _{2})q^{7}+(3-2\beta _{1}+\cdots)q^{9}+\cdots\)
6800.2.a.bl	$6800$	$2$	6800.a	1.a	$1$	$3$	$3$	$54.298$	3.3.148.1	None	✓				3400.2.a.o	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{2})q^{7}+(-1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
6800.2.a.bm	$6800$	$2$	6800.a	1.a	$1$	$3$	$3$	$54.298$	3.3.404.1	None	✓				340.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}-\beta _{2}q^{7}+(2-\beta _{1}+\beta _{2})q^{9}+\cdots\)
6800.2.a.bn	$6800$	$2$	6800.a	1.a	$1$	$3$	$3$	$54.298$	3.3.148.1	None	✓				3400.2.a.o	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{2})q^{7}+(-1+\beta _{1}+\cdots)q^{9}+\cdots\)
6800.2.a.bo	$6800$	$2$	6800.a	1.a	$1$	$3$	$3$	$54.298$	3.3.229.1	None	✓				680.2.a.f	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(\beta _{1}-\beta _{2})q^{7}+(1-\beta _{1})q^{9}+\cdots\)
6800.2.a.bp	$6800$	$2$	6800.a	1.a	$1$	$3$	$3$	$54.298$	3.3.568.1	None	✓				170.2.c.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(\beta _{1}+\beta _{2})q^{7}+(3-2\beta _{1}-\beta _{2})q^{9}+\cdots\)
6800.2.a.bq	$6800$	$2$	6800.a	1.a	$1$	$3$	$3$	$54.298$	3.3.564.1	None	✓				3400.2.a.m	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{1}+\beta _{2})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
6800.2.a.br	$6800$	$2$	6800.a	1.a	$1$	$3$	$3$	$54.298$	3.3.1016.1	None	✓				680.2.a.g	$1$	$0$	\(0\)	\(1\)	\(0\)	\(6\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(2+\beta _{2})q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
6800.2.a.bs	$6800$	$2$	6800.a	1.a	$1$	$3$	$3$	$54.298$	3.3.940.1	None	✓				680.2.a.h	$1$	$0$	\(0\)	\(3\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1-\beta _{2})q^{7}+(3-\beta _{1}+\cdots)q^{9}+\cdots\)
6800.2.a.bt	$6800$	$2$	6800.a	1.a	$1$	$4$	$4$	$54.298$	4.4.6224.1	None	✓				85.2.b.a	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(-10\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-2-\beta _{3})q^{7}+(1+\cdots)q^{9}+\cdots\)
6800.2.a.bu	$6800$	$2$	6800.a	1.a	$1$	$4$	$4$	$54.298$	4.4.11344.1	None	✓				340.2.e.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1-\beta _{2}-\beta _{3})q^{7}+\cdots\)
6800.2.a.bv	$6800$	$2$	6800.a	1.a	$1$	$4$	$4$	$54.298$	4.4.11344.1	None	✓				340.2.e.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1+\beta _{2}+\beta _{3})q^{7}+(1+\cdots)q^{9}+\cdots\)
6800.2.a.bw	$6800$	$2$	6800.a	1.a	$1$	$4$	$4$	$54.298$	4.4.6224.1	None	✓				85.2.b.a	$1$	$0$	\(0\)	\(4\)	\(0\)	\(10\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(2+\beta _{3})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
6800.2.a.bx	$6800$	$2$	6800.a	1.a	$1$	$5$	$5$	$54.298$	5.5.1981136.1	None	✓				3400.2.a.r	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1+\beta _{1}+\beta _{3}-\beta _{4})q^{7}+\cdots\)
6800.2.a.by	$6800$	$2$	6800.a	1.a	$1$	$5$	$5$	$54.298$	5.5.4153680.1	None	✓				3400.2.a.s	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{3})q^{7}+(2+\beta _{2}-\beta _{3}+\cdots)q^{9}+\cdots\)
6800.2.a.bz	$6800$	$2$	6800.a	1.a	$1$	$5$	$5$	$54.298$	5.5.1893456.1	None	✓		✓		425.2.a.i	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(\beta _{3}+\beta _{4})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
6800.2.a.ca	$6800$	$2$	6800.a	1.a	$1$	$5$	$5$	$54.298$	5.5.27977168.1	None	✓		✓		1700.2.a.h	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{4}q^{7}+(3+\beta _{1}+\beta _{3}+\beta _{4})q^{9}+\cdots\)
6800.2.a.cb	$6800$	$2$	6800.a	1.a	$1$	$5$	$5$	$54.298$	5.5.563792.1	None	✓				680.2.e.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(\beta _{1}+\beta _{3}-\beta _{4})q^{7}+(\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
6800.2.a.cc	$6800$	$2$	6800.a	1.a	$1$	$5$	$5$	$54.298$	5.5.563792.1	None	✓				680.2.e.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3}+\beta _{4})q^{7}+(\beta _{1}+\cdots)q^{9}+\cdots\)
6800.2.a.cd	$6800$	$2$	6800.a	1.a	$1$	$5$	$5$	$54.298$	5.5.1893456.1	None	✓		✓		425.2.a.i	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{3}-\beta _{4})q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
6800.2.a.ce	$6800$	$2$	6800.a	1.a	$1$	$5$	$5$	$54.298$	5.5.27977168.1	None	✓		✓		1700.2.a.h	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{4}q^{7}+(3+\beta _{1}+\beta _{3}+\beta _{4})q^{9}+\cdots\)
6800.2.a.cf	$6800$	$2$	6800.a	1.a	$1$	$5$	$5$	$54.298$	5.5.4153680.1	None	✓				3400.2.a.s	$1$	$0$	\(0\)	\(1\)	\(0\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{3})q^{7}+(2+\beta _{2}-\beta _{3}+\cdots)q^{9}+\cdots\)
6800.2.a.cg	$6800$	$2$	6800.a	1.a	$1$	$5$	$5$	$54.298$	5.5.1981136.1	None	✓				3400.2.a.r	$1$	$0$	\(0\)	\(3\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1-\beta _{1}-\beta _{3}+\beta _{4})q^{7}+\cdots\)
6800.2.a.ch	$6800$	$2$	6800.a	1.a	$1$	$6$	$6$	$54.298$	6.6.358395264.1	None	✓		✓		680.2.e.c	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}-\beta _{4}q^{7}+(3-\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots\)
6800.2.a.ci	$6800$	$2$	6800.a	1.a	$1$	$6$	$6$	$54.298$	6.6.358395264.1	None	✓		✓		680.2.e.c	$1$	$0$	\(0\)	\(4\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+\beta _{4}q^{7}+(3-\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6800))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6800)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(170))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(272))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(340))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(425))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(680))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(850))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1360))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1700))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3400))\)\(^{\oplus 2}\)