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

• Label: 169.4.c
• Level: $169$
• Weight: $4$
• Character orbit: 169.c
• Rep. character: $\chi_{169}(22,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $66$
• Newform subspaces: $12$
• Sturm bound: $60$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.c (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(60\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

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

	Total	New	Old
Modular forms	106	86	20
Cusp forms	78	66	12
Eisenstein series	28	20	8

Trace form

\( 66 q - q^{2} + 7 q^{3} - 107 q^{4} - 4 q^{5} - 30 q^{6} + 35 q^{7} + 30 q^{8} - 204 q^{9} + O(q^{10}) \)

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.c.a	$169$	$4$	169.c	13.c	$3$	$2$	$1$	$9.971$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-5\)	\(7\)	\(14\)	\(-13\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-5+5\zeta_{6})q^{2}+(7-7\zeta_{6})q^{3}-17\zeta_{6}q^{4}+\cdots\)
169.4.c.b	$169$	$4$	169.c	13.c	$3$	$2$	$1$	$9.971$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-3\)	\(1\)	\(-18\)	\(-15\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-3+3\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
169.4.c.c	$169$	$4$	169.c	13.c	$3$	$2$	$1$	$9.971$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(3\)	\(1\)	\(18\)	\(15\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(3-3\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
169.4.c.d	$169$	$4$	169.c	13.c	$3$	$2$	$1$	$9.971$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(4\)	\(-2\)	\(-34\)	\(20\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4-4\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}-8\zeta_{6}q^{4}+\cdots\)
169.4.c.e	$169$	$4$	169.c	13.c	$3$	$2$	$1$	$9.971$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(5\)	\(7\)	\(-14\)	\(13\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(5-5\zeta_{6})q^{2}+(7-7\zeta_{6})q^{3}-17\zeta_{6}q^{4}+\cdots\)
169.4.c.f	$169$	$4$	169.c	13.c	$3$	$4$	$2$	$9.971$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None		$2$	$0$	\(-5\)	\(-5\)	\(30\)	\(15\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+\beta _{1}+2\beta _{2}+\beta _{3})q^{2}+(-1+\cdots)q^{3}+\cdots\)
169.4.c.g	$169$	$4$	169.c	13.c	$3$	$4$	$2$	$9.971$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None		$2$	$0$	\(-1\)	\(-5\)	\(-6\)	\(9\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(3\beta _{1}-4\beta _{2})q^{3}+(4+\beta _{1}+\cdots)q^{4}+\cdots\)
169.4.c.h	$169$	$4$	169.c	13.c	$3$	$4$	$2$	$9.971$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{12}^{2}q^{2}-2\zeta_{12}q^{3}+(5-5\zeta_{12}+\cdots)q^{4}+\cdots\)
169.4.c.i	$169$	$4$	169.c	13.c	$3$	$4$	$2$	$9.971$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(14\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{12}^{2}q^{2}+7\zeta_{12}q^{3}+(-4+4\zeta_{12}+\cdots)q^{4}+\cdots\)
169.4.c.j	$169$	$4$	169.c	13.c	$3$	$4$	$2$	$9.971$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None		$2$	$0$	\(1\)	\(-5\)	\(6\)	\(-9\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+(3\beta _{1}-4\beta _{2})q^{3}+(4+\beta _{1}+\cdots)q^{4}+\cdots\)
169.4.c.k	$169$	$4$	169.c	13.c	$3$	$18$	$9$	$9.971$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		$2$	$0$	\(-5\)	\(-1\)	\(60\)	\(-38\)	$13^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}+\beta _{2}-\beta _{3})q^{2}+(\beta _{11}-\beta _{14}+\cdots)q^{3}+\cdots\)
169.4.c.l	$169$	$4$	169.c	13.c	$3$	$18$	$9$	$9.971$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		$2$	$0$	\(5\)	\(-1\)	\(-60\)	\(38\)	$13^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}-\beta _{2}+\beta _{3})q^{2}+(\beta _{11}-\beta _{14})q^{3}+\cdots\)

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

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