Space of modular forms of level 6848, weight 2, and trivial character

• Label: 6848.2.a
• Level: $6848$
• Weight: $2$
• Character orbit: 6848.a
• Rep. character: $\chi_{6848}(1,\cdot)$
• Character field: $\Q$
• Dimension: $212$
• Newform subspaces: $58$
• Sturm bound: $1728$
• Trace bound: $15$

Defining parameters

Level:	\( N \)	\(=\)	\( 6848 = 2^{6} \cdot 107 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6848.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 58 \)
Sturm bound:			\(1728\)
Trace bound:			\(15\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\), \(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6848))\).

	Total	New	Old
Modular forms	876	212	664
Cusp forms	853	212	641
Eisenstein series	23	0	23

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

\(2\)	\(107\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(210\)	\(49\)	\(161\)		\(205\)	\(49\)	\(156\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(228\)	\(58\)	\(170\)		\(222\)	\(58\)	\(164\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(228\)	\(57\)	\(171\)		\(222\)	\(57\)	\(165\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(210\)	\(48\)	\(162\)		\(204\)	\(48\)	\(156\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(420\)	\(97\)	\(323\)		\(409\)	\(97\)	\(312\)		\(11\)	\(0\)	\(11\)
Minus space		\(-\)		\(456\)	\(115\)	\(341\)		\(444\)	\(115\)	\(329\)		\(12\)	\(0\)	\(12\)

Trace form

212 * q + 212 * q^9 + 16 * q^13 - 8 * q^17 + 16 * q^21 + 204 * q^25 + 16 * q^37 - 8 * q^41 + 48 * q^45 + 212 * q^49 + 48 * q^53 + 16 * q^61 + 48 * q^69 + 8 * q^73 + 48 * q^77 + 212 * q^81 + 48 * q^85 + 8 * q^89 + 16 * q^93 - 8 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6848))\) 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	107
6848.2.a.a	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				3424.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(2\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+2q^{5}+2q^{7}+6q^{9}-3q^{11}+\cdots\)
6848.2.a.b	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				214.2.a.a	$1$	$2$	\(0\)	\(-2\)	\(1\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}-4q^{7}+q^{9}-6q^{11}+\cdots\)
6848.2.a.c	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				856.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(3\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+3q^{5}+q^{9}+6q^{11}-6q^{15}+\cdots\)
6848.2.a.d	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				214.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(3\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+3q^{5}+4q^{7}+q^{9}-2q^{11}+\cdots\)
6848.2.a.e	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				428.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+4q^{7}-2q^{9}-5q^{11}+\cdots\)
6848.2.a.f	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				428.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+4q^{7}-2q^{9}+3q^{11}+\cdots\)
6848.2.a.g	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				856.2.a.c	$1$	$2$	\(0\)	\(-1\)	\(0\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}-2q^{9}+3q^{11}-5q^{13}+\cdots\)
6848.2.a.h	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				856.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}+3q^{11}+3q^{13}-5q^{19}+\cdots\)
6848.2.a.i	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				214.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}-2q^{9}+3q^{11}+q^{13}+\cdots\)
6848.2.a.j	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				214.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}-2q^{7}-2q^{9}+3q^{11}+\cdots\)
6848.2.a.k	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				856.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}+2q^{7}-2q^{9}-5q^{11}+\cdots\)
6848.2.a.l	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				3424.2.a.b	$1$	$2$	\(0\)	\(0\)	\(-3\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}-2q^{7}-3q^{9}-2q^{13}-4q^{17}+\cdots\)
6848.2.a.m	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				3424.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+2q^{7}-3q^{9}-2q^{13}-4q^{17}+\cdots\)
6848.2.a.n	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				428.2.a.b	$1$	$2$	\(0\)	\(1\)	\(-2\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}-4q^{7}-2q^{9}-3q^{11}+\cdots\)
6848.2.a.o	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				428.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}-4q^{7}-2q^{9}+5q^{11}+\cdots\)
6848.2.a.p	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				214.2.a.d	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}-2q^{9}-3q^{11}+q^{13}+\cdots\)
6848.2.a.q	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				856.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}-3q^{11}+3q^{13}+5q^{19}+\cdots\)
6848.2.a.r	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				856.2.a.c	$1$	$0$	\(0\)	\(1\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{7}-2q^{9}-3q^{11}-5q^{13}+\cdots\)
6848.2.a.s	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				856.2.a.b	$1$	$0$	\(0\)	\(1\)	\(4\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}-2q^{7}-2q^{9}+5q^{11}+\cdots\)
6848.2.a.t	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				214.2.a.b	$1$	$0$	\(0\)	\(1\)	\(4\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}+2q^{7}-2q^{9}-3q^{11}+\cdots\)
6848.2.a.u	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				214.2.a.a	$1$	$0$	\(0\)	\(2\)	\(1\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+4q^{7}+q^{9}+6q^{11}+\cdots\)
6848.2.a.v	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				214.2.a.c	$1$	$1$	\(0\)	\(2\)	\(3\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+3q^{5}-4q^{7}+q^{9}+2q^{11}+\cdots\)
6848.2.a.w	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				856.2.a.d	$1$	$1$	\(0\)	\(2\)	\(3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+3q^{5}+q^{9}-6q^{11}+6q^{15}+\cdots\)
6848.2.a.x	$6848$	$2$	6848.a	1.a	$1$	$1$	$1$	$54.682$	\(\Q\)	None	✓				3424.2.a.a	$1$	$0$	\(0\)	\(3\)	\(2\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{5}-2q^{7}+6q^{9}+3q^{11}+\cdots\)
6848.2.a.y	$6848$	$2$	6848.a	1.a	$1$	$2$	$2$	$54.682$	\(\Q(\sqrt{13}) \)	None	✓				428.2.a.c	$1$	$1$	\(0\)	\(-3\)	\(1\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(1-\beta )q^{5}+q^{7}+(1+\cdots)q^{9}+\cdots\)
6848.2.a.z	$6848$	$2$	6848.a	1.a	$1$	$2$	$2$	$54.682$	\(\Q(\sqrt{5}) \)	None	✓				107.2.a.a	$1$	$0$	\(0\)	\(-3\)	\(3\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(1+\beta )q^{5}+(3-2\beta )q^{7}+\cdots\)
6848.2.a.ba	$6848$	$2$	6848.a	1.a	$1$	$2$	$2$	$54.682$	\(\Q(\sqrt{3}) \)	None	✓				214.2.a.e	$1$	$1$	\(0\)	\(-2\)	\(-4\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(-2-\beta )q^{5}+(1-\beta )q^{7}+\cdots\)
6848.2.a.bb	$6848$	$2$	6848.a	1.a	$1$	$2$	$2$	$54.682$	\(\Q(\sqrt{3}) \)	None	✓				214.2.a.f	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-\beta q^{5}+(-1+\beta )q^{7}+\cdots\)
6848.2.a.bc	$6848$	$2$	6848.a	1.a	$1$	$2$	$2$	$54.682$	\(\Q(\sqrt{3}) \)	None	✓				856.2.a.f	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+\beta q^{5}+(1-\beta )q^{7}+(1+\cdots)q^{9}+\cdots\)
6848.2.a.bd	$6848$	$2$	6848.a	1.a	$1$	$2$	$2$	$54.682$	\(\Q(\sqrt{17}) \)	None	✓				3424.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(-4\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-2q^{5}+2\beta q^{7}+(1+\beta )q^{9}+\cdots\)
6848.2.a.be	$6848$	$2$	6848.a	1.a	$1$	$2$	$2$	$54.682$	\(\Q(\sqrt{5}) \)	None	✓				856.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+\beta q^{5}+(1-2\beta )q^{7}+(-2+\beta )q^{9}+\cdots\)
6848.2.a.bf	$6848$	$2$	6848.a	1.a	$1$	$2$	$2$	$54.682$	\(\Q(\sqrt{6}) \)	None	✓				3424.2.a.f	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1-\beta )q^{5}-\beta q^{7}+3q^{9}+\cdots\)
6848.2.a.bg	$6848$	$2$	6848.a	1.a	$1$	$2$	$2$	$54.682$	\(\Q(\sqrt{6}) \)	None	✓				3424.2.a.f	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1+\beta )q^{5}-\beta q^{7}+3q^{9}+\cdots\)
6848.2.a.bh	$6848$	$2$	6848.a	1.a	$1$	$2$	$2$	$54.682$	\(\Q(\sqrt{17}) \)	None	✓				3424.2.a.e	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-2q^{5}-2\beta q^{7}+(1+\beta )q^{9}+\cdots\)
6848.2.a.bi	$6848$	$2$	6848.a	1.a	$1$	$2$	$2$	$54.682$	\(\Q(\sqrt{5}) \)	None	✓				856.2.a.e	$1$	$1$	\(0\)	\(1\)	\(1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+\beta q^{5}+(-1+2\beta )q^{7}+(-2+\cdots)q^{9}+\cdots\)
6848.2.a.bj	$6848$	$2$	6848.a	1.a	$1$	$2$	$2$	$54.682$	\(\Q(\sqrt{3}) \)	None	✓				214.2.a.e	$1$	$1$	\(0\)	\(2\)	\(-4\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-2+\beta )q^{5}+(-1-\beta )q^{7}+\cdots\)
6848.2.a.bk	$6848$	$2$	6848.a	1.a	$1$	$2$	$2$	$54.682$	\(\Q(\sqrt{3}) \)	None	✓				856.2.a.f	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-\beta q^{5}+(-1-\beta )q^{7}+(1+\cdots)q^{9}+\cdots\)
6848.2.a.bl	$6848$	$2$	6848.a	1.a	$1$	$2$	$2$	$54.682$	\(\Q(\sqrt{3}) \)	None	✓				214.2.a.f	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+\beta q^{5}+(1+\beta )q^{7}+(1+2\beta )q^{9}+\cdots\)
6848.2.a.bm	$6848$	$2$	6848.a	1.a	$1$	$2$	$2$	$54.682$	\(\Q(\sqrt{13}) \)	None	✓				428.2.a.c	$1$	$1$	\(0\)	\(3\)	\(1\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1-\beta )q^{5}-q^{7}+(1+3\beta )q^{9}+\cdots\)
6848.2.a.bn	$6848$	$2$	6848.a	1.a	$1$	$2$	$2$	$54.682$	\(\Q(\sqrt{5}) \)	None	✓				107.2.a.a	$1$	$0$	\(0\)	\(3\)	\(3\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1+\beta )q^{5}+(-3+2\beta )q^{7}+\cdots\)
6848.2.a.bo	$6848$	$2$	6848.a	1.a	$1$	$3$	$3$	$54.682$	3.3.229.1	None	✓				856.2.a.g	$1$	$1$	\(0\)	\(-2\)	\(-3\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(-1+\beta _{1})q^{5}+(1+\cdots)q^{7}+\cdots\)
6848.2.a.bp	$6848$	$2$	6848.a	1.a	$1$	$3$	$3$	$54.682$	3.3.229.1	None	✓				856.2.a.g	$1$	$1$	\(0\)	\(2\)	\(-3\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(-1+\beta _{1})q^{5}+(-1+\cdots)q^{7}+\cdots\)
6848.2.a.bq	$6848$	$2$	6848.a	1.a	$1$	$5$	$5$	$54.682$	5.5.1752165.1	None	✓				428.2.a.d	$1$	$0$	\(0\)	\(-5\)	\(1\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(\beta _{2}+\beta _{4})q^{5}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)
6848.2.a.br	$6848$	$2$	6848.a	1.a	$1$	$5$	$5$	$54.682$	5.5.1752165.1	None	✓				428.2.a.d	$1$	$0$	\(0\)	\(5\)	\(1\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(\beta _{2}+\beta _{4})q^{5}+(-1-\beta _{2}+\cdots)q^{7}+\cdots\)
6848.2.a.bs	$6848$	$2$	6848.a	1.a	$1$	$6$	$6$	$54.682$	6.6.199990501.1	None	✓				856.2.a.h	$1$	$0$	\(0\)	\(-7\)	\(2\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+\beta _{5}q^{5}+(1-\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
6848.2.a.bt	$6848$	$2$	6848.a	1.a	$1$	$6$	$6$	$54.682$	6.6.199990501.1	None	✓				856.2.a.h	$1$	$0$	\(0\)	\(7\)	\(2\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+\beta _{5}q^{5}+(-1+\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
6848.2.a.bu	$6848$	$2$	6848.a	1.a	$1$	$7$	$7$	$54.682$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				107.2.a.b	$1$	$1$	\(0\)	\(-3\)	\(-5\)	\(4\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{3}+(-1-\beta _{5})q^{5}+(1+\beta _{3}-\beta _{4}+\cdots)q^{7}+\cdots\)
6848.2.a.bv	$6848$	$2$	6848.a	1.a	$1$	$7$	$7$	$54.682$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				107.2.a.b	$1$	$1$	\(0\)	\(3\)	\(-5\)	\(-4\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{3}+(-1-\beta _{5})q^{5}+(-1-\beta _{3}+\cdots)q^{7}+\cdots\)
6848.2.a.bw	$6848$	$2$	6848.a	1.a	$1$	$10$	$10$	$54.682$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				3424.2.a.i	$1$	$1$	\(0\)	\(-9\)	\(6\)	\(-6\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(1+\beta _{9})q^{5}+(-1+\cdots)q^{7}+\cdots\)
6848.2.a.bx	$6848$	$2$	6848.a	1.a	$1$	$10$	$10$	$54.682$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				856.2.a.i	$1$	$0$	\(0\)	\(-1\)	\(-5\)	\(4\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{5}q^{5}-\beta _{6}q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)
6848.2.a.by	$6848$	$2$	6848.a	1.a	$1$	$10$	$10$	$54.682$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				856.2.a.i	$1$	$0$	\(0\)	\(1\)	\(-5\)	\(-4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{5}q^{5}+\beta _{6}q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)
6848.2.a.bz	$6848$	$2$	6848.a	1.a	$1$	$10$	$10$	$54.682$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				3424.2.a.i	$1$	$0$	\(0\)	\(9\)	\(6\)	\(6\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1+\beta _{9})q^{5}+(1-\beta _{4}+\cdots)q^{7}+\cdots\)
6848.2.a.ca	$6848$	$2$	6848.a	1.a	$1$	$12$	$12$	$54.682$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				3424.2.a.k	$1$	$1$	\(0\)	\(-5\)	\(8\)	\(-8\)		$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{5}q^{3}+(1+\beta _{8})q^{5}+(-1+\beta _{9})q^{7}+\cdots\)
6848.2.a.cb	$6848$	$2$	6848.a	1.a	$1$	$12$	$12$	$54.682$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				3424.2.a.l	$1$	$1$	\(0\)	\(-3\)	\(-8\)	\(-8\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{10})q^{5}+(-1+\beta _{3}+\cdots)q^{7}+\cdots\)
6848.2.a.cc	$6848$	$2$	6848.a	1.a	$1$	$12$	$12$	$54.682$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				3424.2.a.l	$1$	$0$	\(0\)	\(3\)	\(-8\)	\(8\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{10})q^{5}+(1-\beta _{3}+\cdots)q^{7}+\cdots\)
6848.2.a.cd	$6848$	$2$	6848.a	1.a	$1$	$12$	$12$	$54.682$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				3424.2.a.k	$1$	$0$	\(0\)	\(5\)	\(8\)	\(8\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{5}q^{3}+(1+\beta _{8})q^{5}+(1-\beta _{9})q^{7}+\cdots\)
6848.2.a.ce	$6848$	$2$	6848.a	1.a	$1$	$13$	$13$	$54.682$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓		✓		3424.2.a.o	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-12\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{2}q^{5}+(-1+\beta _{10})q^{7}+\cdots\)
6848.2.a.cf	$6848$	$2$	6848.a	1.a	$1$	$13$	$13$	$54.682$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓		✓		3424.2.a.o	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(12\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{2}q^{5}+(1-\beta _{10})q^{7}+(\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6848))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6848)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(107))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(214))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(428))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(856))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1712))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3424))\)\(^{\oplus 2}\)