Space of modular forms of level 6768, weight 2, and trivial character

• Label: 6768.2.a
• Level: $6768$
• Weight: $2$
• Character orbit: 6768.a
• Rep. character: $\chi_{6768}(1,\cdot)$
• Character field: $\Q$
• Dimension: $115$
• Newform subspaces: $51$
• Sturm bound: $2304$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 6768 = 2^{4} \cdot 3^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6768.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 51 \)
Sturm bound:			\(2304\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6768))\).

	Total	New	Old
Modular forms	1176	115	1061
Cusp forms	1129	115	1014
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.

\(2\)	\(3\)	\(47\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(12\)
\(+\)	\(+\)	\(-\)	$-$	\(12\)
\(+\)	\(-\)	\(+\)	$-$	\(17\)
\(+\)	\(-\)	\(-\)	$+$	\(17\)
\(-\)	\(+\)	\(+\)	$-$	\(11\)
\(-\)	\(+\)	\(-\)	$+$	\(11\)
\(-\)	\(-\)	\(+\)	$+$	\(15\)
\(-\)	\(-\)	\(-\)	$-$	\(20\)
Plus space			\(+\)	\(55\)
Minus space			\(-\)	\(60\)

Trace form

\( 115 q - 2 q^{5} - 6 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6768))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	47
6768.2.a.a	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$2$	\(0\)	\(0\)	\(-3\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-3q^{7}-q^{11}-4q^{13}-4q^{17}+\cdots\)
6768.2.a.b	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-q^{7}-3q^{11}+4q^{19}+7q^{23}+\cdots\)
6768.2.a.c	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(5\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+5q^{7}+3q^{11}-6q^{13}-2q^{17}+\cdots\)
6768.2.a.d	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-6q^{11}+6q^{17}+4q^{19}-q^{25}+\cdots\)
6768.2.a.e	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+2q^{13}-2q^{17}-q^{25}-2q^{29}+\cdots\)
6768.2.a.f	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+4q^{11}-2q^{13}-2q^{17}-q^{25}+\cdots\)
6768.2.a.g	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(-3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{7}+q^{11}-2q^{13}-2q^{17}+\cdots\)
6768.2.a.h	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}+6q^{13}+6q^{17}-2q^{19}+4q^{23}+\cdots\)
6768.2.a.i	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{11}-4q^{13}+2q^{17}+2q^{19}+\cdots\)
6768.2.a.j	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}-4q^{11}+2q^{13}-2q^{17}+2q^{19}+\cdots\)
6768.2.a.k	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{7}-q^{11}-4q^{13}+4q^{17}+\cdots\)
6768.2.a.l	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(1\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{7}+3q^{11}-2q^{13}+6q^{17}+\cdots\)
6768.2.a.m	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+3q^{7}-3q^{11}-4q^{13}-8q^{17}+\cdots\)
6768.2.a.n	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+3q^{7}+q^{11}-2q^{13}-2q^{17}+\cdots\)
6768.2.a.o	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-2q^{11}+4q^{13}-2q^{17}-8q^{19}+\cdots\)
6768.2.a.p	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-3q^{7}+q^{11}-4q^{13}+4q^{17}+\cdots\)
6768.2.a.q	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-q^{7}+3q^{11}+4q^{19}-7q^{23}+\cdots\)
6768.2.a.r	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(3\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+q^{7}-3q^{11}-4q^{13}-2q^{19}+\cdots\)
6768.2.a.s	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+q^{7}+5q^{11}-4q^{13}+4q^{17}+\cdots\)
6768.2.a.t	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(3\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+3q^{7}-5q^{11}+2q^{13}+6q^{17}+\cdots\)
6768.2.a.u	$6768$	$2$	6768.a	1.a	$1$	$1$	$1$	$54.043$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(4\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}+4q^{7}-2q^{13}+6q^{17}-6q^{19}+\cdots\)
6768.2.a.v	$6768$	$2$	6768.a	1.a	$1$	$2$	$2$	$54.043$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{5}-2\beta q^{7}+2\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
6768.2.a.w	$6768$	$2$	6768.a	1.a	$1$	$2$	$2$	$54.043$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{5}+(2+2\beta )q^{7}+(4-\beta )q^{11}+\cdots\)
6768.2.a.x	$6768$	$2$	6768.a	1.a	$1$	$2$	$2$	$54.043$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{5}+q^{7}+q^{11}-\beta q^{13}+(4-\beta )q^{17}+\cdots\)
6768.2.a.y	$6768$	$2$	6768.a	1.a	$1$	$2$	$2$	$54.043$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{5}-\beta q^{7}+(4-\beta )q^{11}+(-2-2\beta )q^{13}+\cdots\)
6768.2.a.z	$6768$	$2$	6768.a	1.a	$1$	$2$	$2$	$54.043$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+\beta q^{7}+(-4+\beta )q^{11}+(2-2\beta )q^{13}+\cdots\)
6768.2.a.ba	$6768$	$2$	6768.a	1.a	$1$	$2$	$2$	$54.043$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-5\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{7}+2\beta q^{11}+2q^{13}+(3+\cdots)q^{17}+\cdots\)
6768.2.a.bb	$6768$	$2$	6768.a	1.a	$1$	$2$	$2$	$54.043$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-2q^{7}+\beta q^{11}+(2+\beta )q^{13}+\cdots\)
6768.2.a.bc	$6768$	$2$	6768.a	1.a	$1$	$2$	$2$	$54.043$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(1\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+\beta q^{7}+(4-\beta )q^{11}+(2-2\beta )q^{13}+\cdots\)
6768.2.a.bd	$6768$	$2$	6768.a	1.a	$1$	$2$	$2$	$54.043$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}-2\beta q^{7}+(-3+\beta )q^{11}+\cdots\)
6768.2.a.be	$6768$	$2$	6768.a	1.a	$1$	$2$	$2$	$54.043$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(2\)	\(2\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{5}+q^{7}-q^{11}+\beta q^{13}+(-4-\beta )q^{17}+\cdots\)
6768.2.a.bf	$6768$	$2$	6768.a	1.a	$1$	$2$	$2$	$54.043$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{5}+(2-\beta )q^{7}+(-2-2\beta )q^{11}+\cdots\)
6768.2.a.bg	$6768$	$2$	6768.a	1.a	$1$	$2$	$2$	$54.043$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{5}+(3+\beta )q^{7}+(-4+4\beta )q^{11}+\cdots\)
6768.2.a.bh	$6768$	$2$	6768.a	1.a	$1$	$2$	$2$	$54.043$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(4\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+(-2+3\beta )q^{7}-2q^{11}+(-2+\cdots)q^{13}+\cdots\)
6768.2.a.bi	$6768$	$2$	6768.a	1.a	$1$	$2$	$2$	$54.043$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+2\beta q^{7}+2\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
6768.2.a.bj	$6768$	$2$	6768.a	1.a	$1$	$3$	$3$	$54.043$	3.3.316.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(-3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{5}+(-1-\beta _{2})q^{7}+(1+\cdots)q^{11}+\cdots\)
6768.2.a.bk	$6768$	$2$	6768.a	1.a	$1$	$3$	$3$	$54.043$	3.3.229.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(1\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{2})q^{5}-\beta _{1}q^{7}+\beta _{1}q^{11}+\cdots\)
6768.2.a.bl	$6768$	$2$	6768.a	1.a	$1$	$3$	$3$	$54.043$	3.3.2700.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(-1-\beta _{2})q^{7}+(-1+\cdots)q^{11}+\cdots\)
6768.2.a.bm	$6768$	$2$	6768.a	1.a	$1$	$3$	$3$	$54.043$	3.3.148.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{5}+(\beta _{1}-\beta _{2})q^{7}+(-3+\cdots)q^{11}+\cdots\)
6768.2.a.bn	$6768$	$2$	6768.a	1.a	$1$	$3$	$3$	$54.043$	3.3.316.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}+(1+\beta _{2})q^{7}+(3-\beta _{2})q^{11}+\cdots\)
6768.2.a.bo	$6768$	$2$	6768.a	1.a	$1$	$3$	$3$	$54.043$	3.3.316.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(1\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+(1+\beta _{2})q^{7}+(-3+\beta _{2})q^{11}+\cdots\)
6768.2.a.bp	$6768$	$2$	6768.a	1.a	$1$	$3$	$3$	$54.043$	3.3.316.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(5\)	\(-3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{5}+(-1+\beta _{2})q^{7}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
6768.2.a.bq	$6768$	$2$	6768.a	1.a	$1$	$4$	$4$	$54.043$	4.4.7625.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(3\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{5}+(\beta _{1}-\beta _{3})q^{7}+(1+\beta _{2}+\cdots)q^{11}+\cdots\)
6768.2.a.br	$6768$	$2$	6768.a	1.a	$1$	$4$	$4$	$54.043$	4.4.115520.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(-1+\beta _{3})q^{7}+(2+\cdots)q^{11}+\cdots\)
6768.2.a.bs	$6768$	$2$	6768.a	1.a	$1$	$4$	$4$	$54.043$	4.4.11348.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-7\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+(-2+\beta _{2})q^{7}-\beta _{1}q^{11}+\cdots\)
6768.2.a.bt	$6768$	$2$	6768.a	1.a	$1$	$4$	$4$	$54.043$	4.4.13448.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-5\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}+\beta _{2}-\beta _{3})q^{5}+(-1+\beta _{2}+\cdots)q^{7}+\cdots\)
6768.2.a.bu	$6768$	$2$	6768.a	1.a	$1$	$4$	$4$	$54.043$	4.4.115520.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{5}+(-1+\beta _{3})q^{7}+(-2+\cdots)q^{11}+\cdots\)
6768.2.a.bv	$6768$	$2$	6768.a	1.a	$1$	$4$	$4$	$54.043$	4.4.1957.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-4\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}-\beta _{3})q^{5}+(-1+\beta _{1})q^{7}+\cdots\)
6768.2.a.bw	$6768$	$2$	6768.a	1.a	$1$	$5$	$5$	$54.043$	5.5.2324776.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(1\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{3})q^{5}+\beta _{4}q^{7}+(1-\beta _{2})q^{11}+\cdots\)
6768.2.a.bx	$6768$	$2$	6768.a	1.a	$1$	$8$	$8$	$54.043$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-2\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{5}-\beta _{3}q^{7}+(-1+\beta _{6})q^{11}+\cdots\)
6768.2.a.by	$6768$	$2$	6768.a	1.a	$1$	$8$	$8$	$54.043$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-2\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{5}-\beta _{3}q^{7}+(1-\beta _{6})q^{11}+(-\beta _{1}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6768))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6768)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(47))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(94))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(141))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(188))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(282))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(376))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(423))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(564))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(752))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(846))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1128))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1692))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2256))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3384))\)\(^{\oplus 2}\)