Space of modular forms of level 165, weight 6, and trivial character

• Label: 165.6.a
• Level: $165$
• Weight: $6$
• Character orbit: 165.a
• Rep. character: $\chi_{165}(1,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $8$
• Sturm bound: $144$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	165.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(144\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(165))\).

	Total	New	Old
Modular forms	124	32	92
Cusp forms	116	32	84
Eisenstein series	8	0	8

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

\(3\)	\(5\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(5\)
\(+\)	\(+\)	\(-\)	$-$	\(3\)
\(+\)	\(-\)	\(+\)	$-$	\(3\)
\(+\)	\(-\)	\(-\)	$+$	\(3\)
\(-\)	\(+\)	\(+\)	$-$	\(5\)
\(-\)	\(+\)	\(-\)	$+$	\(3\)
\(-\)	\(-\)	\(+\)	$+$	\(3\)
\(-\)	\(-\)	\(-\)	$-$	\(7\)
Plus space			\(+\)	\(14\)
Minus space			\(-\)	\(18\)

Trace form

\( 32 q + 36 q^{3} + 468 q^{4} - 180 q^{6} + 72 q^{7} + 2592 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(165))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	11
165.6.a.a	$165$	$6$	165.a	1.a	$1$	$3$	$3$	$26.463$	3.3.34253.1	None	✓	$1$	$1$	\(-7\)	\(27\)	\(75\)	\(-172\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}+9q^{3}+(7+4\beta _{1}+\beta _{2})q^{4}+\cdots\)
165.6.a.b	$165$	$6$	165.a	1.a	$1$	$3$	$3$	$26.463$	3.3.3368.1	None	✓	$1$	$1$	\(-2\)	\(27\)	\(-75\)	\(-232\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+9q^{3}+(8+4\beta _{2})q^{4}+\cdots\)
165.6.a.c	$165$	$6$	165.a	1.a	$1$	$3$	$3$	$26.463$	3.3.18257.1	None	✓	$1$	$1$	\(2\)	\(-27\)	\(75\)	\(-68\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-9q^{3}+(-14-\beta _{1}+\beta _{2})q^{4}+\cdots\)
165.6.a.d	$165$	$6$	165.a	1.a	$1$	$3$	$3$	$26.463$	3.3.788.1	None	✓	$1$	$0$	\(2\)	\(-27\)	\(75\)	\(152\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}-9q^{3}+(-8-4\beta _{1})q^{4}+\cdots\)
165.6.a.e	$165$	$6$	165.a	1.a	$1$	$3$	$3$	$26.463$	3.3.307532.1	None	✓	$1$	$0$	\(7\)	\(-27\)	\(-75\)	\(92\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{2}-9q^{3}+(24+2\beta _{1}+\beta _{2})q^{4}+\cdots\)
165.6.a.f	$165$	$6$	165.a	1.a	$1$	$5$	$5$	$26.463$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(-2\)	\(45\)	\(-125\)	\(184\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+9q^{3}+(2^{4}+\beta _{2})q^{4}-5^{2}q^{5}+\cdots\)
165.6.a.g	$165$	$6$	165.a	1.a	$1$	$5$	$5$	$26.463$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(-45\)	\(-125\)	\(116\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-9q^{3}+(5^{2}+\beta _{1}+\beta _{3})q^{4}+\cdots\)
165.6.a.h	$165$	$6$	165.a	1.a	$1$	$7$	$7$	$26.463$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(63\)	\(175\)	\(0\)		$-$	$-$	$-$	$-$	$2^{6}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+9q^{3}+(28+\beta _{2})q^{4}+5^{2}q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(165))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(165)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 2}\)