Space of modular forms of level 169, weight 6, and Character 169.b

• Label: 169.6.b
• Level: $169$
• Weight: $6$
• Character orbit: 169.b
• Rep. character: $\chi_{169}(168,\cdot)$
• Character field: $\Q$
• Dimension: $58$
• Newform subspaces: $5$
• Sturm bound: $91$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	169.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(91\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(169, [\chi])\).

	Total	New	Old
Modular forms	84	68	16
Cusp forms	70	58	12
Eisenstein series	14	10	4

Trace form

\( 58 q - 16 q^{3} - 798 q^{4} + 3346 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(169, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
169.6.b.a	$169$	$6$	169.b	13.b	$2$	$4$	$4$	$27.105$	\(\Q(i, \sqrt{17})\)	None					13.6.a.a	$2$	$0$	\(0\)	\(-56\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+3\beta _{2})q^{2}+(-17+6\beta _{3})q^{3}+\cdots\)
169.6.b.b	$169$	$6$	169.b	13.b	$2$	$6$	$6$	$27.105$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None					13.6.a.b	$2$	$0$	\(0\)	\(16\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{2})q^{2}+(3+\beta _{3}+\beta _{4})q^{3}+(-40+\cdots)q^{4}+\cdots\)
169.6.b.c	$169$	$6$	169.b	13.b	$2$	$8$	$8$	$27.105$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					13.6.c.a	$2$	$0$	\(0\)	\(-16\)	\(0\)	\(0\)	$2^{2}\cdot 13^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(-2+\beta _{3})q^{3}+(-4+\beta _{4}+\cdots)q^{4}+\cdots\)
169.6.b.d	$169$	$6$	169.b	13.b	$2$	$10$	$10$	$27.105$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None					13.6.e.a	$2$	$0$	\(0\)	\(20\)	\(0\)	\(0\)	$2^{11}\cdot 3\cdot 13^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{2}+(2-\beta _{3})q^{3}+(-13+\beta _{1}+\cdots)q^{4}+\cdots\)
169.6.b.e	$169$	$6$	169.b	13.b	$2$	$30$	$30$	$27.105$		None		✓	✓		169.6.a.g	$2$	$0$	\(0\)	\(20\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{6}^{\mathrm{old}}(169, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(169, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 2}\)