Space of modular forms of level 169, weight 4, and Character 169.b

• Label: 169.4.b
• Level: $169$
• Weight: $4$
• Character orbit: 169.b
• Rep. character: $\chi_{169}(168,\cdot)$
• Character field: $\Q$
• Dimension: $34$
• Newform subspaces: $7$
• Sturm bound: $60$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(60\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(169, [\chi])\).

	Total	New	Old
Modular forms	52	44	8
Cusp forms	38	34	4
Eisenstein series	14	10	4

Trace form

\( 34 q + 2 q^{3} - 118 q^{4} + 268 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(169, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
169.4.b.a	$169$	$4$	169.b	13.b	$2$	$2$	$2$	$9.971$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-14\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5iq^{2}-7q^{3}-17q^{4}+7iq^{5}-35iq^{6}+\cdots\)
169.4.b.b	$169$	$4$	169.b	13.b	$2$	$2$	$2$	$9.971$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-14\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\zeta_{6}q^{2}-7q^{3}-4q^{4}-8\zeta_{6}q^{5}+\cdots\)
169.4.b.c	$169$	$4$	169.b	13.b	$2$	$2$	$2$	$9.971$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4iq^{2}+2q^{3}-8q^{4}+17iq^{5}+8iq^{6}+\cdots\)
169.4.b.d	$169$	$4$	169.b	13.b	$2$	$2$	$2$	$9.971$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{6}q^{2}+2q^{3}+5q^{4}-\zeta_{6}q^{5}-2\zeta_{6}q^{6}+\cdots\)
169.4.b.e	$169$	$4$	169.b	13.b	$2$	$4$	$4$	$9.971$	\(\Q(i, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(10\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+3\beta _{2})q^{2}+(1+3\beta _{3})q^{3}-5\beta _{3}q^{4}+\cdots\)
169.4.b.f	$169$	$4$	169.b	13.b	$2$	$4$	$4$	$9.971$	\(\Q(i, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(10\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(1+3\beta _{3})q^{3}+(3+\beta _{3})q^{4}+\cdots\)
169.4.b.g	$169$	$4$	169.b	13.b	$2$	$18$	$18$	$9.971$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$13^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{9}q^{2}-\beta _{2}q^{3}+(-4+\beta _{1})q^{4}+(-\beta _{9}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(169, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(169, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 2}\)