Space of modular forms of level 968, weight 4, and trivial character

• Label: 968.4.a
• Level: $968$
• Weight: $4$
• Character orbit: 968.a
• Rep. character: $\chi_{968}(1,\cdot)$
• Character field: $\Q$
• Dimension: $82$
• Newform subspaces: $19$
• Sturm bound: $528$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	968.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(528\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(968))\).

	Total	New	Old
Modular forms	420	82	338
Cusp forms	372	82	290
Eisenstein series	48	0	48

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

\(2\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(22\)
\(+\)	\(-\)	$-$	\(19\)
\(-\)	\(+\)	$-$	\(20\)
\(-\)	\(-\)	$+$	\(21\)
Plus space		\(+\)	\(43\)
Minus space		\(-\)	\(39\)

Trace form

\( 82 q - 2 q^{3} - 2 q^{5} + 12 q^{7} + 748 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(968))\) 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	11
968.4.a.a	$968$	$4$	968.a	1.a	$1$	$1$	$1$	$57.114$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-4\)	\(-2\)	\(-24\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-2q^{5}-24q^{7}-11q^{9}-22q^{13}+\cdots\)
968.4.a.b	$968$	$4$	968.a	1.a	$1$	$1$	$1$	$57.114$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(13\)	\(-10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+13q^{5}-10q^{7}-23q^{9}+3^{3}q^{13}+\cdots\)
968.4.a.c	$968$	$4$	968.a	1.a	$1$	$1$	$1$	$57.114$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(13\)	\(10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+13q^{5}+10q^{7}-23q^{9}-3^{3}q^{13}+\cdots\)
968.4.a.d	$968$	$4$	968.a	1.a	$1$	$1$	$1$	$57.114$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-7\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-7q^{5}+6q^{7}-26q^{9}+40q^{13}+\cdots\)
968.4.a.e	$968$	$4$	968.a	1.a	$1$	$1$	$1$	$57.114$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(7\)	\(9\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}+9q^{5}-2q^{7}+22q^{9}+63q^{15}+\cdots\)
968.4.a.f	$968$	$4$	968.a	1.a	$1$	$2$	$2$	$57.114$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(-6\)	\(56\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(-3+4\beta )q^{5}+(28+\cdots)q^{7}+\cdots\)
968.4.a.g	$968$	$4$	968.a	1.a	$1$	$3$	$3$	$57.114$	3.3.1556.1	None	✓	$1$	$0$	\(0\)	\(2\)	\(-9\)	\(-6\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(-3+\beta _{2})q^{5}+(-2+\cdots)q^{7}+\cdots\)
968.4.a.h	$968$	$4$	968.a	1.a	$1$	$3$	$3$	$57.114$	3.3.1556.1	None	✓	$1$	$1$	\(0\)	\(2\)	\(-9\)	\(6\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(-3+\beta _{2})q^{5}+(2-\beta _{2})q^{7}+\cdots\)
968.4.a.i	$968$	$4$	968.a	1.a	$1$	$3$	$3$	$57.114$	3.3.11109.1	None	✓	$1$	$1$	\(0\)	\(2\)	\(8\)	\(-24\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(3+\beta _{2})q^{5}+(-8+\beta _{1}+\cdots)q^{7}+\cdots\)
968.4.a.j	$968$	$4$	968.a	1.a	$1$	$3$	$3$	$57.114$	3.3.3124.1	None	✓	$1$	$1$	\(0\)	\(6\)	\(-21\)	\(-2\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+(-7-\beta _{1})q^{5}+(-1+\cdots)q^{7}+\cdots\)
968.4.a.k	$968$	$4$	968.a	1.a	$1$	$3$	$3$	$57.114$	3.3.3124.1	None	✓	$1$	$0$	\(0\)	\(6\)	\(-21\)	\(2\)		$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+(-7-\beta _{1})q^{5}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
968.4.a.l	$968$	$4$	968.a	1.a	$1$	$4$	$4$	$57.114$	4.4.4166757.2	None	✓	$1$	$0$	\(0\)	\(-3\)	\(9\)	\(-8\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(2+\beta _{2})q^{5}+(-3+\cdots)q^{7}+\cdots\)
968.4.a.m	$968$	$4$	968.a	1.a	$1$	$4$	$4$	$57.114$	4.4.4166757.2	None	✓	$1$	$1$	\(0\)	\(-3\)	\(9\)	\(8\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(2+\beta _{2})q^{5}+(3+2\beta _{1}+\cdots)q^{7}+\cdots\)
968.4.a.n	$968$	$4$	968.a	1.a	$1$	$8$	$8$	$57.114$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-8\)	\(-13\)	\(-9\)		$-$	$+$	$-$	$2^{6}\cdot 11$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{3}+(-2-\beta _{5})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
968.4.a.o	$968$	$4$	968.a	1.a	$1$	$8$	$8$	$57.114$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-8\)	\(-13\)	\(9\)		$+$	$-$	$-$	$2^{6}\cdot 11$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{3}+(-2-\beta _{5})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
968.4.a.p	$968$	$4$	968.a	1.a	$1$	$8$	$8$	$57.114$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-6\)	\(6\)	\(-50\)		$-$	$+$	$-$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(1-2\beta _{3}+\beta _{7})q^{5}+\cdots\)
968.4.a.q	$968$	$4$	968.a	1.a	$1$	$8$	$8$	$57.114$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-6\)	\(6\)	\(50\)		$+$	$+$	$+$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(1-2\beta _{3}+\beta _{7})q^{5}+\cdots\)
968.4.a.r	$968$	$4$	968.a	1.a	$1$	$10$	$10$	$57.114$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(9\)	\(13\)	\(-3\)		$-$	$-$	$+$	$2^{8}\cdot 11^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1+\beta _{4})q^{5}+\beta _{3}q^{7}+\cdots\)
968.4.a.s	$968$	$4$	968.a	1.a	$1$	$10$	$10$	$57.114$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(9\)	\(13\)	\(3\)		$+$	$+$	$+$	$2^{8}\cdot 11^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1+\beta _{4})q^{5}-\beta _{3}q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(968))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(968)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(484))\)\(^{\oplus 2}\)