Space of modular forms of level 6840, weight 2, and trivial character

• Label: 6840.2.a
• Level: $6840$
• Weight: $2$
• Character orbit: 6840.a
• Rep. character: $\chi_{6840}(1,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $45$
• Sturm bound: $2880$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6840.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 45 \)
Sturm bound:			\(2880\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6840))\).

	Total	New	Old
Modular forms	1472	90	1382
Cusp forms	1409	90	1319
Eisenstein series	63	0	63

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

\(2\)	\(3\)	\(5\)	\(19\)	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(82\)	\(4\)	\(78\)		\(79\)	\(4\)	\(75\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(100\)	\(5\)	\(95\)		\(96\)	\(5\)	\(91\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(102\)	\(5\)	\(97\)		\(98\)	\(5\)	\(93\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(84\)	\(4\)	\(80\)		\(80\)	\(4\)	\(76\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(94\)	\(6\)	\(88\)		\(90\)	\(6\)	\(84\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(92\)	\(7\)	\(85\)		\(88\)	\(7\)	\(81\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(90\)	\(6\)	\(84\)		\(86\)	\(6\)	\(80\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(92\)	\(8\)	\(84\)		\(88\)	\(8\)	\(80\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(86\)	\(5\)	\(81\)		\(82\)	\(5\)	\(77\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(100\)	\(4\)	\(96\)		\(96\)	\(4\)	\(92\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(98\)	\(4\)	\(94\)		\(94\)	\(4\)	\(90\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(84\)	\(5\)	\(79\)		\(80\)	\(5\)	\(75\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(90\)	\(7\)	\(83\)		\(86\)	\(7\)	\(79\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(92\)	\(6\)	\(86\)		\(88\)	\(6\)	\(82\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(94\)	\(8\)	\(86\)		\(90\)	\(8\)	\(82\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(92\)	\(6\)	\(86\)		\(88\)	\(6\)	\(82\)		\(4\)	\(0\)	\(4\)
Plus space				\(+\)		\(728\)	\(42\)	\(686\)		\(697\)	\(42\)	\(655\)		\(31\)	\(0\)	\(31\)
Minus space				\(-\)		\(744\)	\(48\)	\(696\)		\(712\)	\(48\)	\(664\)		\(32\)	\(0\)	\(32\)

Trace form

90 * q + 2 * q^5 + 8 * q^17 + 90 * q^25 - 12 * q^29 + 8 * q^31 - 12 * q^35 + 4 * q^41 + 28 * q^43 + 36 * q^47 + 78 * q^49 + 16 * q^53 + 8 * q^55 + 16 * q^59 - 4 * q^61 + 8 * q^67 + 8 * q^71 - 48 * q^77 + 32 * q^79 - 60 * q^83 - 4 * q^85 - 36 * q^89 + 16 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6840))\) 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	3	5	19
6840.2.a.a	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓	✓			6840.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}-2q^{11}+6q^{17}-q^{19}+\cdots\)
6840.2.a.b	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓				2280.2.a.g	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}+2q^{11}-4q^{13}+2q^{17}+\cdots\)
6840.2.a.c	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓				760.2.a.e	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-q^{7}-4q^{11}+q^{13}+7q^{17}+\cdots\)
6840.2.a.d	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓				2280.2.a.h	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-4q^{11}-2q^{13}-2q^{17}-q^{19}+\cdots\)
6840.2.a.e	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓				2280.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-4q^{11}+2q^{13}-6q^{17}-q^{19}+\cdots\)
6840.2.a.f	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓	✓			6840.2.a.f	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{11}-q^{19}+4q^{23}+q^{25}+\cdots\)
6840.2.a.g	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓				2280.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{13}+6q^{17}+q^{19}-4q^{23}+\cdots\)
6840.2.a.h	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓				760.2.a.c	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{11}-6q^{13}+6q^{17}-q^{19}+\cdots\)
6840.2.a.i	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓				760.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{11}+4q^{13}+2q^{17}+q^{19}+\cdots\)
6840.2.a.j	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓				2280.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}-6q^{11}+4q^{13}+2q^{17}+\cdots\)
6840.2.a.k	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓				2280.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}-2q^{13}-4q^{17}+q^{19}+\cdots\)
6840.2.a.l	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓				760.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(4\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{7}-4q^{11}-4q^{13}+2q^{17}+\cdots\)
6840.2.a.m	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓				2280.2.a.j	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(4\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{7}+2q^{13}-2q^{17}-q^{19}+\cdots\)
6840.2.a.n	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓	✓			6840.2.a.n	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(4\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{7}+2q^{11}-4q^{13}-4q^{17}+\cdots\)
6840.2.a.o	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓				760.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(4\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{7}+4q^{11}-6q^{17}-q^{19}+\cdots\)
6840.2.a.p	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓				2280.2.a.i	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(4\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{7}+4q^{11}+2q^{13}+6q^{17}+\cdots\)
6840.2.a.q	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓	✓			6840.2.a.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(-2\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{7}+2q^{11}-6q^{17}-q^{19}+\cdots\)
6840.2.a.r	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓				2280.2.a.e	$1$	$0$	\(0\)	\(0\)	\(1\)	\(-2\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{7}+4q^{11}-2q^{13}+q^{19}+\cdots\)
6840.2.a.s	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓	✓			6840.2.a.f	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2q^{11}-q^{19}-4q^{23}+q^{25}+\cdots\)
6840.2.a.t	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓				2280.2.a.f	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{11}-2q^{13}+2q^{17}+q^{19}+\cdots\)
6840.2.a.u	$6840$	$2$	6840.a	1.a	$1$	$1$	$1$	$54.618$	\(\Q\)	None	✓	✓			6840.2.a.n	$1$	$0$	\(0\)	\(0\)	\(1\)	\(4\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{7}-2q^{11}-4q^{13}+4q^{17}+\cdots\)
6840.2.a.v	$6840$	$2$	6840.a	1.a	$1$	$2$	$2$	$54.618$	\(\Q(\sqrt{3}) \)	None	✓				2280.2.a.q	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-6\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+(-3+\beta )q^{7}+(1+3\beta )q^{11}+\cdots\)
6840.2.a.w	$6840$	$2$	6840.a	1.a	$1$	$2$	$2$	$54.618$	\(\Q(\sqrt{7}) \)	None	✓		✓		2280.2.a.n	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+(-1+\beta )q^{7}+(3-\beta )q^{11}+(1+\cdots)q^{13}+\cdots\)
6840.2.a.x	$6840$	$2$	6840.a	1.a	$1$	$2$	$2$	$54.618$	\(\Q(\sqrt{2}) \)	None	✓				2280.2.a.o	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-4\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(-2+\beta )q^{7}-\beta q^{11}-3\beta q^{13}+\cdots\)
6840.2.a.y	$6840$	$2$	6840.a	1.a	$1$	$2$	$2$	$54.618$	\(\Q(\sqrt{3}) \)	None	✓				760.2.a.g	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{11}+(1+\beta )q^{13}+(-4+2\beta )q^{17}+\cdots\)
6840.2.a.z	$6840$	$2$	6840.a	1.a	$1$	$2$	$2$	$54.618$	\(\Q(\sqrt{2}) \)	None	✓				760.2.a.f	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2\beta q^{7}+(-2-2\beta )q^{11}+(2+\cdots)q^{13}+\cdots\)
6840.2.a.ba	$6840$	$2$	6840.a	1.a	$1$	$2$	$2$	$54.618$	\(\Q(\sqrt{2}) \)	None	✓				2280.2.a.l	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{5}+\beta q^{7}+\beta q^{11}+(-2-\beta )q^{13}+\cdots\)
6840.2.a.bb	$6840$	$2$	6840.a	1.a	$1$	$2$	$2$	$54.618$	\(\Q(\sqrt{2}) \)	None	✓				2280.2.a.k	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+\beta q^{7}+(2-\beta )q^{11}+(2-\beta )q^{13}+\cdots\)
6840.2.a.bc	$6840$	$2$	6840.a	1.a	$1$	$2$	$2$	$54.618$	\(\Q(\sqrt{5}) \)	None	✓				2280.2.a.m	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{5}+(1+\beta )q^{7}+(1+\beta )q^{11}+(-1+\cdots)q^{13}+\cdots\)
6840.2.a.bd	$6840$	$2$	6840.a	1.a	$1$	$2$	$2$	$54.618$	\(\Q(\sqrt{5}) \)	None	✓				2280.2.a.p	$1$	$0$	\(0\)	\(0\)	\(2\)	\(6\)		$+$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{5}+(3+\beta )q^{7}+(-1+\beta )q^{11}+(3+\cdots)q^{13}+\cdots\)
6840.2.a.be	$6840$	$2$	6840.a	1.a	$1$	$3$	$3$	$54.618$	3.3.148.1	None	✓	✓	✓		6840.2.a.be	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-2\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+(-2\beta _{1}-\beta _{2})q^{7}+(1-\beta _{1})q^{11}+\cdots\)
6840.2.a.bf	$6840$	$2$	6840.a	1.a	$1$	$3$	$3$	$54.618$	3.3.316.1	None	✓		✓		2280.2.a.s	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(-2\)		$+$	$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{5}+(-1+\beta _{1})q^{7}+(1+\beta _{2})q^{11}+\cdots\)
6840.2.a.bg	$6840$	$2$	6840.a	1.a	$1$	$3$	$3$	$54.618$	3.3.229.1	None	✓		✓		760.2.a.j	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-1\)		$+$	$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{5}+\beta _{1}q^{7}+(2+\beta _{2})q^{13}+(-2+\cdots)q^{17}+\cdots\)
6840.2.a.bh	$6840$	$2$	6840.a	1.a	$1$	$3$	$3$	$54.618$	3.3.1772.1	None	✓		✓		2280.2.a.u	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{5}+\beta _{2}q^{7}+(-2-\beta _{2})q^{11}+(1+\cdots)q^{13}+\cdots\)
6840.2.a.bi	$6840$	$2$	6840.a	1.a	$1$	$3$	$3$	$54.618$	3.3.2804.1	None	✓	✓	✓		6840.2.a.bi	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+(1-\beta _{1})q^{7}+(-2-\beta _{2})q^{11}+\cdots\)
6840.2.a.bj	$6840$	$2$	6840.a	1.a	$1$	$3$	$3$	$54.618$	3.3.568.1	None	✓				760.2.a.h	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-5\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(-2-\beta _{2})q^{7}+(2-2\beta _{1})q^{11}+\cdots\)
6840.2.a.bk	$6840$	$2$	6840.a	1.a	$1$	$3$	$3$	$54.618$	3.3.568.1	None	✓		✓		2280.2.a.r	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-2\)		$-$	$-$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{5}+(-1-\beta _{2})q^{7}+(-1+\beta _{2})q^{11}+\cdots\)
6840.2.a.bl	$6840$	$2$	6840.a	1.a	$1$	$3$	$3$	$54.618$	3.3.148.1	None	✓	✓	✓		6840.2.a.be	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-2\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(-2\beta _{1}-\beta _{2})q^{7}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
6840.2.a.bm	$6840$	$2$	6840.a	1.a	$1$	$3$	$3$	$54.618$	3.3.316.1	None	✓		✓		760.2.a.i	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-1\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(-1+2\beta _{1}-\beta _{2})q^{7}-2\beta _{2}q^{11}+\cdots\)
6840.2.a.bn	$6840$	$2$	6840.a	1.a	$1$	$3$	$3$	$54.618$	3.3.1016.1	None	✓				2280.2.a.t	$1$	$0$	\(0\)	\(0\)	\(3\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{5}+\beta _{2}q^{7}+(-1+\beta _{1}+\beta _{2})q^{11}+\cdots\)
6840.2.a.bo	$6840$	$2$	6840.a	1.a	$1$	$3$	$3$	$54.618$	3.3.2804.1	None	✓	✓	✓		6840.2.a.bi	$1$	$0$	\(0\)	\(0\)	\(3\)	\(2\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(1-\beta _{1})q^{7}+(2+\beta _{2})q^{11}+(3+\cdots)q^{13}+\cdots\)
6840.2.a.bp	$6840$	$2$	6840.a	1.a	$1$	$4$	$4$	$54.618$	4.4.20308.1	None	✓	✓	✓	✓	6840.2.a.bp	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-4\)		$-$	$+$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{5}+(-1+\beta _{1})q^{7}+(1-\beta _{1}-\beta _{3})q^{11}+\cdots\)
6840.2.a.bq	$6840$	$2$	6840.a	1.a	$1$	$4$	$4$	$54.618$	4.4.20308.1	None	✓	✓	✓	✓	6840.2.a.bp	$1$	$1$	\(0\)	\(0\)	\(4\)	\(-4\)		$+$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{5}+(-1+\beta _{1})q^{7}+(-1+\beta _{1}+\beta _{3})q^{11}+\cdots\)
6840.2.a.br	$6840$	$2$	6840.a	1.a	$1$	$5$	$5$	$54.618$	5.5.4636128.1	None	✓	✓	✓	✓	6840.2.a.br	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(2\)		$+$	$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{5}+\beta _{4}q^{7}-\beta _{3}q^{11}+(\beta _{1}-\beta _{2}+\cdots)q^{13}+\cdots\)
6840.2.a.bs	$6840$	$2$	6840.a	1.a	$1$	$5$	$5$	$54.618$	5.5.4636128.1	None	✓	✓	✓	✓	6840.2.a.br	$1$	$0$	\(0\)	\(0\)	\(5\)	\(2\)		$-$	$+$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{5}+\beta _{4}q^{7}+\beta _{3}q^{11}+(\beta _{1}-\beta _{2}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6840))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6840)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(228))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(285))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(342))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(380))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(456))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(570))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(684))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(760))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(855))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1140))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1368))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1710))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2280))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3420))\)\(^{\oplus 2}\)