Space of modular forms of level 539, weight 2, and Character 539.f

• Label: 539.2.f
• Level: $539$
• Weight: $2$
• Character orbit: 539.f
• Rep. character: $\chi_{539}(148,\cdot)$
• Character field: $\Q(\zeta_{5})$
• Dimension: $144$
• Newform subspaces: $9$
• Sturm bound: $112$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.f (of order \(5\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 11 \)
Character field:			\(\Q(\zeta_{5})\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(112\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(539, [\chi])\).

	Total	New	Old
Modular forms	256	184	72
Cusp forms	192	144	48
Eisenstein series	64	40	24

Trace form

\( 144 q + 2 q^{2} + 6 q^{3} - 30 q^{4} + 2 q^{5} - 6 q^{6} - 12 q^{8} - 10 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(539, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
539.2.f.a	$539$	$2$	539.f	11.c	$5$	$4$	$1$	$4.304$	\(\Q(\zeta_{10})\)	None		$2$	$0$	\(2\)	\(-1\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(\zeta_{10}-\zeta_{10}^{2})q^{2}+\zeta_{10}^{2}q^{3}+(-1+\cdots)q^{4}+\cdots\)
539.2.f.b	$539$	$2$	539.f	11.c	$5$	$4$	$1$	$4.304$	\(\Q(\zeta_{10})\)	None		$2$	$0$	\(2\)	\(1\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(\zeta_{10}-\zeta_{10}^{2})q^{2}-\zeta_{10}^{2}q^{3}+(-1+\cdots)q^{4}+\cdots\)
539.2.f.c	$539$	$2$	539.f	11.c	$5$	$8$	$2$	$4.304$	8.0.37515625.1	\(\Q(\sqrt{-7}) \)		$4$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{5}]$	\(q+(-1+\beta _{1}+2\beta _{2}-\beta _{4}-\beta _{5}+\beta _{6}+\cdots)q^{2}+\cdots\)
539.2.f.d	$539$	$2$	539.f	11.c	$5$	$8$	$2$	$4.304$	8.0.159390625.1	None		$2$	$0$	\(-1\)	\(4\)	\(-3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+\beta _{4}q^{2}+(\beta _{2}+\beta _{3})q^{3}+(1-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
539.2.f.e	$539$	$2$	539.f	11.c	$5$	$16$	$4$	$4.304$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(-3\)	\(2\)	\(5\)	\(0\)	$5$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-\beta _{5}-\beta _{6})q^{2}+(-\beta _{9}+\beta _{11}-\beta _{13}+\cdots)q^{3}+\cdots\)
539.2.f.f	$539$	$2$	539.f	11.c	$5$	$16$	$4$	$4.304$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-1-\beta _{2}+\beta _{5}-\beta _{7})q^{2}+(\beta _{11}-\beta _{12}+\cdots)q^{3}+\cdots\)
539.2.f.g	$539$	$2$	539.f	11.c	$5$	$20$	$5$	$4.304$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		$2$	$0$	\(3\)	\(-4\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+\beta _{6}q^{2}+(-\beta _{1}-\beta _{7}-\beta _{18}+\beta _{19})q^{3}+\cdots\)
539.2.f.h	$539$	$2$	539.f	11.c	$5$	$20$	$5$	$4.304$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		$2$	$0$	\(3\)	\(4\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+\beta _{6}q^{2}+(\beta _{1}+\beta _{7}+\beta _{18}-\beta _{19})q^{3}+\cdots\)
539.2.f.i	$539$	$2$	539.f	11.c	$5$	$48$	$12$	$4.304$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{5}]$

Decomposition of \(S_{2}^{\mathrm{old}}(539, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(539, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 2}\)