Space of modular forms of level 69, weight 2, and trivial character

• Label: 69.2.a
• Level: $69$
• Weight: $2$
• Character orbit: 69.a
• Rep. character: $\chi_{69}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $16$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(16\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(69))\).

	Total	New	Old
Modular forms	10	3	7
Cusp forms	7	3	4
Eisenstein series	3	0	3

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

\(3\)	\(23\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(-\)	\(-\)		\(5\)	\(2\)	\(3\)		\(4\)	\(2\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(2\)	\(1\)	\(1\)		\(1\)	\(1\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
Minus space		\(-\)		\(7\)	\(3\)	\(4\)		\(5\)	\(3\)	\(2\)		\(2\)	\(0\)	\(2\)

Trace form

3 * q + q^2 - q^3 + 5 * q^4 - 2 * q^5 + q^6 - 3 * q^8 + 3 * q^9 - 10 * q^10 + 12 * q^11 - 7 * q^12 - 6 * q^13 - 12 * q^14 + 2 * q^15 - 3 * q^16 - 6 * q^17 + q^18 + 12 * q^19 - 6 * q^20 - 4 * q^21 + 4 * q^22 + q^23 - 3 * q^24 - 3 * q^25 + 14 * q^26 - q^27 + 8 * q^28 + 2 * q^29 + 10 * q^30 + 5 * q^32 - 4 * q^33 - 6 * q^34 + 8 * q^35 + 5 * q^36 + 2 * q^37 - 8 * q^38 - 6 * q^39 - 10 * q^40 - 2 * q^41 + 8 * q^42 + 12 * q^43 + 20 * q^44 - 2 * q^45 - q^46 - 8 * q^47 + q^48 - 5 * q^49 + 15 * q^50 + 14 * q^51 + 6 * q^52 - 18 * q^53 + q^54 - 8 * q^55 - 4 * q^56 - 8 * q^57 - 18 * q^58 - 4 * q^59 + 6 * q^60 - 6 * q^61 + 24 * q^62 - 19 * q^64 - 20 * q^65 + 4 * q^66 - 4 * q^67 - 34 * q^68 - 3 * q^69 - 8 * q^71 - 3 * q^72 - 18 * q^73 - 18 * q^74 - 7 * q^75 + 28 * q^76 - 26 * q^78 + 16 * q^79 + 2 * q^80 + 3 * q^81 + 42 * q^82 + 20 * q^83 - 4 * q^84 + 20 * q^85 + 40 * q^86 + 2 * q^87 - 12 * q^88 - 14 * q^89 - 10 * q^90 - 8 * q^91 + 7 * q^92 + 8 * q^93 + 5 * q^96 - 2 * q^97 - 23 * q^98 + 12 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(69))\) 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	23
69.2.a.a	$69$	$2$	69.a	1.a	$1$	$1$	$1$	$0.551$	\(\Q\)	None	✓	✓	✓	✓	69.2.a.a	$1$	$0$	\(1\)	\(1\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}+q^{6}-2q^{7}-3q^{8}+\cdots\)
69.2.a.b	$69$	$2$	69.a	1.a	$1$	$2$	$2$	$0.551$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	69.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(2\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+3q^{4}+(-1+\beta )q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(69))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(69)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 2}\)