Space of modular forms of level 165, weight 2, and trivial character

• Label: 165.2.a
• Level: $165$
• Weight: $2$
• Character orbit: 165.a
• Rep. character: $\chi_{165}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $3$
• Sturm bound: $48$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	165.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(48\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(165))\).

	Total	New	Old
Modular forms	28	7	21
Cusp forms	21	7	14
Eisenstein series	7	0	7

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
\(+\)	\(+\)	\(+\)	\(+\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(3\)
Plus space			\(+\)	\(2\)
Minus space			\(-\)	\(5\)

Trace form

7 * q - 3 * q^2 + 3 * q^3 + 9 * q^4 - q^5 + q^6 - 15 * q^8 + 7 * q^9 + q^10 - q^11 + 5 * q^12 + 2 * q^13 - 16 * q^14 + 3 * q^15 + 9 * q^16 - 10 * q^17 - 3 * q^18 + 4 * q^19 + q^20 + 8 * q^21 + q^22 - 8 * q^23 - 3 * q^24 + 7 * q^25 + 14 * q^26 + 3 * q^27 + 8 * q^28 - 14 * q^29 - 3 * q^30 - 23 * q^32 + 3 * q^33 - 30 * q^34 + 9 * q^36 + 10 * q^37 + 4 * q^38 + 2 * q^39 - 3 * q^40 - 10 * q^41 - 8 * q^42 - 4 * q^43 + q^44 - q^45 + 8 * q^46 - 16 * q^47 - 3 * q^48 + 15 * q^49 - 3 * q^50 + 6 * q^51 - 58 * q^52 - 22 * q^53 + q^54 + 7 * q^55 + 20 * q^57 + 6 * q^58 + 4 * q^59 + 5 * q^60 - 14 * q^61 + 40 * q^62 + q^64 - 6 * q^65 - 3 * q^66 + 12 * q^67 + 50 * q^68 + 8 * q^69 - 8 * q^70 + 24 * q^71 - 15 * q^72 - 10 * q^73 - 2 * q^74 + 3 * q^75 + 28 * q^76 - 18 * q^78 - 8 * q^79 + 17 * q^80 + 7 * q^81 + 2 * q^82 + 4 * q^83 - 16 * q^84 + 6 * q^85 + 36 * q^86 - 6 * q^87 - 3 * q^88 - 26 * q^89 + q^90 - 16 * q^91 + 8 * q^92 + 16 * q^94 + 12 * q^95 - 35 * q^96 + 14 * q^97 + 61 * q^98 - q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(165))\) 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
165.2.a.a	$165$	$2$	165.a	1.a	$1$	$2$	$2$	$1.318$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	165.2.a.a	$1$	$1$	\(-2\)	\(-2\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}-q^{3}+(1-2\beta )q^{4}-q^{5}+\cdots\)
165.2.a.b	$165$	$2$	165.a	1.a	$1$	$2$	$2$	$1.318$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	165.2.a.b	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+q^{4}-q^{5}+\beta q^{6}+2q^{7}+\cdots\)
165.2.a.c	$165$	$2$	165.a	1.a	$1$	$3$	$3$	$1.318$	3.3.148.1	None	✓	✓	✓	✓	165.2.a.c	$1$	$0$	\(-1\)	\(3\)	\(3\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(165))\) into lower level spaces

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