Space of modular forms of level 68, weight 2, and trivial character

• Label: 68.2.a
• Level: $68$
• Weight: $2$
• Character orbit: 68.a
• Rep. character: $\chi_{68}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $1$
• Sturm bound: $18$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 68 = 2^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	68.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(18\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(68))\).

	Total	New	Old
Modular forms	12	2	10
Cusp forms	7	2	5
Eisenstein series	5	0	5

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

\(2\)	\(17\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(4\)	\(2\)	\(2\)		\(3\)	\(2\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(8\)	\(2\)	\(6\)		\(5\)	\(2\)	\(3\)		\(3\)	\(0\)	\(3\)

Trace form

2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 6 * q^11 + 4 * q^13 - 12 * q^15 - 2 * q^17 + 4 * q^19 - 8 * q^21 - 6 * q^23 + 14 * q^25 + 8 * q^27 - 2 * q^31 + 12 * q^35 + 16 * q^37 + 16 * q^39 - 12 * q^41 + 4 * q^43 - 24 * q^45 - 6 * q^49 - 2 * q^51 + 12 * q^53 - 12 * q^55 - 8 * q^57 + 12 * q^59 - 8 * q^61 - 14 * q^63 - 24 * q^65 + 16 * q^67 - 6 * q^71 + 4 * q^73 + 14 * q^75 - 14 * q^79 + 2 * q^81 - 12 * q^83 + 12 * q^87 + 12 * q^89 - 16 * q^91 + 16 * q^93 + 24 * q^95 + 4 * q^97 + 6 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(68))\) 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	17
68.2.a.a	$68$	$2$	68.a	1.a	$1$	$2$	$2$	$0.543$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	68.2.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-2\beta q^{5}+(-1-\beta )q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(68))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(68)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 2}\)