Space of modular forms of level 168, weight 2, and trivial character

• Label: 168.2.a
• Level: $168$
• Weight: $2$
• Character orbit: 168.a
• Rep. character: $\chi_{168}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $64$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	168.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(64\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(168))\).

	Total	New	Old
Modular forms	40	2	38
Cusp forms	25	2	23
Eisenstein series	15	0	15

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

\(2\)	\(3\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(6\)	\(1\)	\(5\)		\(4\)	\(1\)	\(3\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(5\)	\(1\)	\(4\)		\(3\)	\(1\)	\(2\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(7\)	\(0\)	\(7\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(18\)	\(0\)	\(18\)		\(11\)	\(0\)	\(11\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(22\)	\(2\)	\(20\)		\(14\)	\(2\)	\(12\)		\(8\)	\(0\)	\(8\)

Trace form

2 * q + 4 * q^5 + 2 * q^9 + 4 * q^13 + 4 * q^17 - 2 * q^21 - 8 * q^23 - 2 * q^25 - 4 * q^29 - 16 * q^31 - 4 * q^37 - 8 * q^39 - 12 * q^41 + 8 * q^43 + 4 * q^45 + 2 * q^49 + 8 * q^51 - 4 * q^53 - 8 * q^57 + 16 * q^59 - 12 * q^61 + 8 * q^65 + 24 * q^67 - 8 * q^71 - 12 * q^73 + 2 * q^81 + 16 * q^83 + 8 * q^85 + 16 * q^87 + 4 * q^89 + 8 * q^91 + 20 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(168))\) 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	3	7
168.2.a.a	$168$	$2$	168.a	1.a	$1$	$1$	$1$	$1.341$	\(\Q\)	None	✓	✓	✓	✓	168.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+q^{7}+q^{9}+6q^{13}-2q^{15}+\cdots\)
168.2.a.b	$168$	$2$	168.a	1.a	$1$	$1$	$1$	$1.341$	\(\Q\)	None	✓	✓	✓	✓	168.2.a.b	$1$	$0$	\(0\)	\(1\)	\(2\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}-q^{7}+q^{9}-2q^{13}+2q^{15}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(168))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(168)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 2}\)