Space of modular forms of level 169, weight 2, and trivial character

• Label: 169.2.a
• Level: $169$
• Weight: $2$
• Character orbit: 169.a
• Rep. character: $\chi_{169}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $3$
• Sturm bound: $30$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(30\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(169))\).

	Total	New	Old
Modular forms	21	19	2
Cusp forms	8	8	0
Eisenstein series	13	11	2

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

\(13\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(9\)	\(8\)	\(1\)		\(3\)	\(3\)	\(0\)		\(6\)	\(5\)	\(1\)
\(-\)		\(12\)	\(11\)	\(1\)		\(5\)	\(5\)	\(0\)		\(7\)	\(6\)	\(1\)

Trace form

8 * q + 2 * q^4 - 4 * q^9 + 4 * q^10 + 4 * q^12 - 10 * q^14 - 6 * q^16 + 2 * q^17 + 6 * q^22 + 2 * q^23 - 14 * q^25 - 6 * q^27 + 4 * q^29 - 14 * q^30 + 8 * q^35 - 12 * q^36 + 12 * q^38 + 16 * q^42 + 10 * q^43 - 18 * q^48 - 22 * q^49 + 10 * q^51 - 4 * q^53 + 12 * q^55 - 8 * q^56 + 10 * q^61 + 14 * q^62 - 20 * q^64 + 10 * q^66 - 36 * q^68 + 12 * q^69 + 14 * q^74 + 22 * q^75 + 16 * q^77 - 2 * q^79 - 24 * q^81 - 10 * q^82 - 24 * q^87 + 30 * q^88 - 30 * q^90 + 12 * q^92 + 22 * q^94 + 18 * q^95

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(169))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		13
169.2.a.a	$169$	$2$	169.a	1.a	$1$	$2$	$2$	$1.349$	\(\Q(\sqrt{3}) \)	None	✓				13.2.e.a	$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+2q^{3}+q^{4}-\beta q^{5}+2\beta q^{6}+\cdots\)
169.2.a.b	$169$	$2$	169.a	1.a	$1$	$3$	$3$	$1.349$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓	✓	169.2.a.b	$1$	$1$	\(-2\)	\(-2\)	\(-4\)	\(-3\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
169.2.a.c	$169$	$2$	169.a	1.a	$1$	$3$	$3$	$1.349$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		169.2.a.b	$1$	$0$	\(2\)	\(-2\)	\(4\)	\(3\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(-1-\beta _{2})q^{3}+(1-2\beta _{1}+\cdots)q^{4}+\cdots\)