Space of modular forms of level 968, weight 2, and trivial character

• Label: 968.2.a
• Level: $968$
• Weight: $2$
• Character orbit: 968.a
• Rep. character: $\chi_{968}(1,\cdot)$
• Character field: $\Q$
• Dimension: $27$
• Newform subspaces: $14$
• Sturm bound: $264$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	968.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(264\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(968))\).

	Total	New	Old
Modular forms	156	27	129
Cusp forms	109	27	82
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.

\(2\)	\(11\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(36\)	\(5\)	\(31\)		\(25\)	\(5\)	\(20\)		\(11\)	\(0\)	\(11\)
\(+\)	\(-\)	\(-\)		\(41\)	\(9\)	\(32\)		\(29\)	\(9\)	\(20\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(42\)	\(7\)	\(35\)		\(30\)	\(7\)	\(23\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)		\(37\)	\(6\)	\(31\)		\(25\)	\(6\)	\(19\)		\(12\)	\(0\)	\(12\)
Plus space		\(+\)		\(73\)	\(11\)	\(62\)		\(50\)	\(11\)	\(39\)		\(23\)	\(0\)	\(23\)
Minus space		\(-\)		\(83\)	\(16\)	\(67\)		\(59\)	\(16\)	\(43\)		\(24\)	\(0\)	\(24\)

Trace form

27 * q + 2 * q^3 + 4 * q^7 + 21 * q^9 + 2 * q^13 - 8 * q^15 + 2 * q^17 + 4 * q^19 + 12 * q^21 - 8 * q^23 + 23 * q^25 - 4 * q^27 + 10 * q^29 + 16 * q^31 - 20 * q^35 + 16 * q^37 - 16 * q^39 - 10 * q^41 + 16 * q^45 - 12 * q^47 + 15 * q^49 - 20 * q^51 - 16 * q^53 + 16 * q^57 + 6 * q^59 + 2 * q^61 + 32 * q^63 + 20 * q^65 - 2 * q^67 - 12 * q^69 + 4 * q^71 - 18 * q^73 + 38 * q^75 + 12 * q^79 + 43 * q^81 - 8 * q^83 - 24 * q^85 - 40 * q^87 - 14 * q^89 + 40 * q^91 - 24 * q^93 + 24 * q^95 - 22 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(968))\) 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	11
968.2.a.a	$968$	$2$	968.a	1.a	$1$	$1$	$1$	$7.730$	\(\Q\)	None	✓				88.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(-3\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-3q^{5}+2q^{7}+6q^{9}+9q^{15}+\cdots\)
968.2.a.b	$968$	$2$	968.a	1.a	$1$	$1$	$1$	$7.730$	\(\Q\)	None	✓	✓			968.2.a.b	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-4q^{7}-3q^{9}-3q^{13}-3q^{17}+\cdots\)
968.2.a.c	$968$	$2$	968.a	1.a	$1$	$1$	$1$	$7.730$	\(\Q\)	None	✓	✓			968.2.a.b	$1$	$0$	\(0\)	\(0\)	\(3\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}+4q^{7}-3q^{9}+3q^{13}+3q^{17}+\cdots\)
968.2.a.d	$968$	$2$	968.a	1.a	$1$	$1$	$1$	$7.730$	\(\Q\)	None	✓	✓			968.2.a.d	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-4q^{7}-2q^{9}-4q^{13}+\cdots\)
968.2.a.e	$968$	$2$	968.a	1.a	$1$	$1$	$1$	$7.730$	\(\Q\)	None	✓	✓			968.2.a.d	$1$	$0$	\(0\)	\(1\)	\(1\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+4q^{7}-2q^{9}+4q^{13}+\cdots\)
968.2.a.f	$968$	$2$	968.a	1.a	$1$	$2$	$2$	$7.730$	\(\Q(\sqrt{5}) \)	None	✓	✓			968.2.a.f	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+\beta q^{5}+(-1-\beta )q^{7}+\cdots\)
968.2.a.g	$968$	$2$	968.a	1.a	$1$	$2$	$2$	$7.730$	\(\Q(\sqrt{5}) \)	None	✓	✓			968.2.a.f	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+\beta q^{5}+(1+\beta )q^{7}+(3+\cdots)q^{9}+\cdots\)
968.2.a.h	$968$	$2$	968.a	1.a	$1$	$2$	$2$	$7.730$	\(\Q(\sqrt{5}) \)	None	✓				88.2.i.a	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-1+\beta )q^{5}-\beta q^{7}+(-2+\cdots)q^{9}+\cdots\)
968.2.a.i	$968$	$2$	968.a	1.a	$1$	$2$	$2$	$7.730$	\(\Q(\sqrt{5}) \)	None	✓				88.2.i.a	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-1+\beta )q^{5}+\beta q^{7}+(-2+\cdots)q^{9}+\cdots\)
968.2.a.j	$968$	$2$	968.a	1.a	$1$	$2$	$2$	$7.730$	\(\Q(\sqrt{17}) \)	None	✓				88.2.a.b	$1$	$0$	\(0\)	\(1\)	\(3\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(2-\beta )q^{5}+2\beta q^{7}+(1+\beta )q^{9}+\cdots\)
968.2.a.k	$968$	$2$	968.a	1.a	$1$	$2$	$2$	$7.730$	\(\Q(\sqrt{3}) \)	None	✓	✓			968.2.a.k	$1$	$1$	\(0\)	\(2\)	\(-4\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-2-\beta )q^{5}+(-1-\beta )q^{7}+\cdots\)
968.2.a.l	$968$	$2$	968.a	1.a	$1$	$2$	$2$	$7.730$	\(\Q(\sqrt{3}) \)	None	✓	✓			968.2.a.k	$1$	$0$	\(0\)	\(2\)	\(-4\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-2-\beta )q^{5}+(1+\beta )q^{7}+\cdots\)
968.2.a.m	$968$	$2$	968.a	1.a	$1$	$4$	$4$	$7.730$	\(\Q(\sqrt{62 +10 \sqrt{5}})\)	None	✓		✓		88.2.i.b	$1$	$0$	\(0\)	\(2\)	\(1\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}+(-1+\beta _{1}-\beta _{2}+\beta _{3})q^{5}+\cdots\)
968.2.a.n	$968$	$2$	968.a	1.a	$1$	$4$	$4$	$7.730$	\(\Q(\sqrt{62 +10 \sqrt{5}})\)	None	✓		✓		88.2.i.b	$1$	$0$	\(0\)	\(2\)	\(1\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}+(-1+\beta _{1}-\beta _{2}+\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(968))\) into lower level spaces

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