Space of modular forms of level 168, weight 6, and trivial character

• Label: 168.6.a
• Level: $168$
• Weight: $6$
• Character orbit: 168.a
• Rep. character: $\chi_{168}(1,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $10$
• Sturm bound: $192$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	168.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(192\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(168))\).

	Total	New	Old
Modular forms	168	14	154
Cusp forms	152	14	138
Eisenstein series	16	0	16

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
\(+\)	\(+\)	\(+\)	\(+\)		\(19\)	\(1\)	\(18\)		\(17\)	\(1\)	\(16\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(22\)	\(2\)	\(20\)		\(20\)	\(2\)	\(18\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(21\)	\(2\)	\(19\)		\(19\)	\(2\)	\(17\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(22\)	\(1\)	\(21\)		\(20\)	\(1\)	\(19\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(23\)	\(2\)	\(21\)		\(21\)	\(2\)	\(19\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(20\)	\(2\)	\(18\)		\(18\)	\(2\)	\(16\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(21\)	\(2\)	\(19\)		\(19\)	\(2\)	\(17\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(20\)	\(2\)	\(18\)		\(18\)	\(2\)	\(16\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(82\)	\(6\)	\(76\)		\(74\)	\(6\)	\(68\)		\(8\)	\(0\)	\(8\)
Minus space			\(-\)		\(86\)	\(8\)	\(78\)		\(78\)	\(8\)	\(70\)		\(8\)	\(0\)	\(8\)

Trace form

14 * q - 196 * q^5 + 1134 * q^9 + 764 * q^13 - 1636 * q^17 - 882 * q^21 - 2152 * q^23 + 3010 * q^25 + 15796 * q^29 + 3888 * q^31 - 5516 * q^37 + 11160 * q^39 + 17484 * q^41 + 16792 * q^43 - 15876 * q^45 - 6432 * q^47 + 33614 * q^49 + 37656 * q^51 + 85492 * q^53 + 69088 * q^55 - 10440 * q^57 - 112336 * q^59 - 16916 * q^61 + 15928 * q^65 - 14776 * q^67 + 98168 * q^71 + 43340 * q^73 - 191776 * q^79 + 91854 * q^81 - 54736 * q^83 + 149096 * q^85 - 218448 * q^87 - 1348 * q^89 + 126616 * q^91 + 50544 * q^93 + 323808 * q^95 + 96268 * q^97

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(168))\) 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
168.6.a.a	$168$	$6$	168.a	1.a	$1$	$1$	$1$	$26.944$	\(\Q\)	None	✓	✓			168.6.a.a	$1$	$0$	\(0\)	\(-9\)	\(-34\)	\(49\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}-34q^{5}+7^{2}q^{7}+3^{4}q^{9}-756q^{11}+\cdots\)
168.6.a.b	$168$	$6$	168.a	1.a	$1$	$1$	$1$	$26.944$	\(\Q\)	None	✓	✓	✓	✓	168.6.a.b	$1$	$1$	\(0\)	\(-9\)	\(4\)	\(-49\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+4q^{5}-7^{2}q^{7}+3^{4}q^{9}+370q^{11}+\cdots\)
168.6.a.c	$168$	$6$	168.a	1.a	$1$	$1$	$1$	$26.944$	\(\Q\)	None	✓	✓			168.6.a.c	$1$	$0$	\(0\)	\(-9\)	\(74\)	\(49\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+74q^{5}+7^{2}q^{7}+3^{4}q^{9}+6^{3}q^{11}+\cdots\)
168.6.a.d	$168$	$6$	168.a	1.a	$1$	$1$	$1$	$26.944$	\(\Q\)	None	✓	✓	✓	✓	168.6.a.d	$1$	$1$	\(0\)	\(9\)	\(-64\)	\(49\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-2^{6}q^{5}+7^{2}q^{7}+3^{4}q^{9}-54q^{11}+\cdots\)
168.6.a.e	$168$	$6$	168.a	1.a	$1$	$1$	$1$	$26.944$	\(\Q\)	None	✓	✓			168.6.a.e	$1$	$1$	\(0\)	\(9\)	\(-38\)	\(-49\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-38q^{5}-7^{2}q^{7}+3^{4}q^{9}+600q^{11}+\cdots\)
168.6.a.f	$168$	$6$	168.a	1.a	$1$	$1$	$1$	$26.944$	\(\Q\)	None	✓	✓			168.6.a.f	$1$	$1$	\(0\)	\(9\)	\(14\)	\(-49\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}+14q^{5}-7^{2}q^{7}+3^{4}q^{9}-700q^{11}+\cdots\)
168.6.a.g	$168$	$6$	168.a	1.a	$1$	$2$	$2$	$26.944$	\(\Q(\sqrt{193}) \)	None	✓	✓	✓	✓	168.6.a.g	$1$	$0$	\(0\)	\(-18\)	\(-78\)	\(-98\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-39-5\beta )q^{5}-7^{2}q^{7}+3^{4}q^{9}+\cdots\)
168.6.a.h	$168$	$6$	168.a	1.a	$1$	$2$	$2$	$26.944$	\(\Q(\sqrt{249}) \)	None	✓	✓	✓	✓	168.6.a.h	$1$	$1$	\(0\)	\(-18\)	\(-64\)	\(98\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-2^{5}-\beta )q^{5}+7^{2}q^{7}+3^{4}q^{9}+\cdots\)
168.6.a.i	$168$	$6$	168.a	1.a	$1$	$2$	$2$	$26.944$	\(\Q(\sqrt{4281}) \)	None	✓	✓	✓	✓	168.6.a.i	$1$	$0$	\(0\)	\(18\)	\(-10\)	\(98\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+9q^{3}+(-5-\beta )q^{5}+7^{2}q^{7}+3^{4}q^{9}+\cdots\)
168.6.a.j	$168$	$6$	168.a	1.a	$1$	$2$	$2$	$26.944$	\(\Q(\sqrt{1129}) \)	None	✓	✓	✓	✓	168.6.a.j	$1$	$0$	\(0\)	\(18\)	\(0\)	\(-98\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+9q^{3}-\beta q^{5}-7^{2}q^{7}+3^{4}q^{9}+(50+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(168))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(168)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 2}\)