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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(14\)	\(5\)	\(9\)		\(13\)	\(5\)	\(8\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(17\)	\(3\)	\(14\)		\(16\)	\(3\)	\(13\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(15\)	\(3\)	\(12\)		\(14\)	\(3\)	\(11\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(16\)	\(3\)	\(13\)		\(15\)	\(3\)	\(12\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(15\)	\(5\)	\(10\)		\(14\)	\(5\)	\(9\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(16\)	\(3\)	\(13\)		\(15\)	\(3\)	\(12\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(14\)	\(3\)	\(11\)		\(13\)	\(3\)	\(10\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(17\)	\(7\)	\(10\)		\(16\)	\(7\)	\(9\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(60\)	\(14\)	\(46\)		\(56\)	\(14\)	\(42\)		\(4\)	\(0\)	\(4\)
Minus space			\(-\)		\(64\)	\(18\)	\(46\)		\(60\)	\(18\)	\(42\)		\(4\)	\(0\)	\(4\)

Trace form

32 * q + 36 * q^3 + 468 * q^4 - 180 * q^6 + 72 * q^7 + 2592 * q^9 - 100 * q^10 + 1728 * q^12 + 1024 * q^13 + 1240 * q^14 + 900 * q^15 + 6220 * q^16 + 1912 * q^17 + 1752 * q^19 - 3800 * q^20 - 4608 * q^21 + 1936 * q^22 - 2416 * q^23 - 540 * q^24 + 20000 * q^25 + 15472 * q^26 + 2916 * q^27 + 12936 * q^28 - 2128 * q^29 - 11488 * q^31 - 30040 * q^32 + 4356 * q^33 + 19792 * q^34 - 6200 * q^35 + 37908 * q^36 + 20032 * q^37 - 67624 * q^38 + 32112 * q^39 - 6300 * q^40 - 3584 * q^41 + 2304 * q^42 + 19880 * q^43 + 4840 * q^44 - 101872 * q^46 - 5792 * q^47 + 64512 * q^48 + 80864 * q^49 - 15336 * q^51 + 54400 * q^52 - 77312 * q^53 - 14580 * q^54 + 24200 * q^55 + 130344 * q^56 - 64440 * q^57 + 33384 * q^58 + 53888 * q^59 + 83700 * q^60 + 119104 * q^61 + 107384 * q^62 + 5832 * q^63 + 151156 * q^64 - 200 * q^65 + 2000 * q^67 - 88496 * q^68 - 32976 * q^69 + 105400 * q^70 - 22368 * q^71 - 192944 * q^73 - 15184 * q^74 + 22500 * q^75 - 60640 * q^76 - 59048 * q^77 - 61344 * q^78 + 76712 * q^79 - 88800 * q^80 + 209952 * q^81 - 107128 * q^82 - 150568 * q^83 - 91008 * q^84 - 128600 * q^85 - 462912 * q^86 - 64584 * q^87 + 92928 * q^88 + 254480 * q^89 - 8100 * q^90 - 125664 * q^91 - 93416 * q^92 + 354528 * q^93 + 269840 * q^94 - 111200 * q^95 + 9540 * q^96 - 113312 * q^97 - 5032 * q^98

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	11
165.6.a.a	$165$	$6$	165.a	1.a	$1$	$3$	$3$	$26.463$	3.3.34253.1	None	✓	✓	✓	✓	165.6.a.a	$1$	$1$	\(-7\)	\(27\)	\(75\)	\(-172\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}+9q^{3}+(7+4\beta _{1}+\beta _{2})q^{4}+\cdots\)
165.6.a.b	$165$	$6$	165.a	1.a	$1$	$3$	$3$	$26.463$	3.3.3368.1	None	✓	✓	✓	✓	165.6.a.b	$1$	$1$	\(-2\)	\(27\)	\(-75\)	\(-232\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+9q^{3}+(8+4\beta _{2})q^{4}+\cdots\)
165.6.a.c	$165$	$6$	165.a	1.a	$1$	$3$	$3$	$26.463$	3.3.18257.1	None	✓	✓	✓	✓	165.6.a.c	$1$	$1$	\(2\)	\(-27\)	\(75\)	\(-68\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-9q^{3}+(-14-\beta _{1}+\beta _{2})q^{4}+\cdots\)
165.6.a.d	$165$	$6$	165.a	1.a	$1$	$3$	$3$	$26.463$	3.3.788.1	None	✓	✓	✓	✓	165.6.a.d	$1$	$0$	\(2\)	\(-27\)	\(75\)	\(152\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}-9q^{3}+(-8-4\beta _{1})q^{4}+\cdots\)
165.6.a.e	$165$	$6$	165.a	1.a	$1$	$3$	$3$	$26.463$	3.3.307532.1	None	✓	✓	✓	✓	165.6.a.e	$1$	$0$	\(7\)	\(-27\)	\(-75\)	\(92\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{2}-9q^{3}+(24+2\beta _{1}+\beta _{2})q^{4}+\cdots\)
165.6.a.f	$165$	$6$	165.a	1.a	$1$	$5$	$5$	$26.463$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	165.6.a.f	$1$	$0$	\(-2\)	\(45\)	\(-125\)	\(184\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+9q^{3}+(2^{4}+\beta _{2})q^{4}-5^{2}q^{5}+\cdots\)
165.6.a.g	$165$	$6$	165.a	1.a	$1$	$5$	$5$	$26.463$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	165.6.a.g	$1$	$1$	\(-1\)	\(-45\)	\(-125\)	\(116\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-9q^{3}+(5^{2}+\beta _{1}+\beta _{3})q^{4}+\cdots\)
165.6.a.h	$165$	$6$	165.a	1.a	$1$	$7$	$7$	$26.463$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	165.6.a.h	$1$	$0$	\(1\)	\(63\)	\(175\)	\(0\)		$-$	$-$	$-$	$-$	$2^{6}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+9q^{3}+(28+\beta _{2})q^{4}+5^{2}q^{5}+\cdots\)

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

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