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

• Label: 539.2.e
• Level: $539$
• Weight: $2$
• Character orbit: 539.e
• Rep. character: $\chi_{539}(67,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $68$
• Newform subspaces: $15$
• Sturm bound: $112$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 15 \)
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	128	68	60
Cusp forms	96	68	28
Eisenstein series	32	0	32

Trace form

68 * q + 2 * q^3 - 36 * q^4 + 4 * q^5 + 8 * q^6 + 12 * q^8 - 40 * q^9 - 6 * q^10 - 6 * q^12 + 16 * q^13 - 20 * q^15 - 44 * q^16 - 6 * q^17 + 6 * q^18 - 2 * q^19 - 40 * q^20 + 24 * q^23 + 8 * q^24 - 30 * q^25 + 10 * q^26 - 16 * q^27 + 16 * q^29 + 24 * q^30 + 6 * q^31 + 8 * q^32 + 4 * q^33 + 32 * q^34 + 52 * q^36 - 32 * q^37 + 8 * q^38 - 22 * q^39 - 8 * q^41 + 28 * q^43 - 4 * q^44 - 6 * q^45 - 26 * q^46 - 6 * q^47 + 16 * q^48 + 2 * q^52 - 2 * q^53 - 26 * q^54 - 16 * q^55 - 48 * q^57 - 18 * q^58 + 8 * q^59 + 68 * q^60 - 12 * q^61 - 20 * q^62 + 76 * q^64 + 18 * q^65 - 2 * q^66 - 36 * q^67 - 16 * q^68 - 36 * q^69 - 28 * q^71 + 94 * q^72 - 14 * q^73 + 36 * q^74 + 34 * q^75 + 96 * q^76 - 128 * q^78 - 16 * q^79 - 18 * q^80 - 2 * q^81 + 38 * q^82 + 52 * q^83 + 112 * q^85 + 6 * q^86 + 6 * q^87 + 24 * q^88 - 14 * q^89 + 52 * q^90 - 180 * q^92 - 32 * q^93 - 10 * q^94 + 28 * q^95 + 30 * q^96 - 108 * q^97

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
539.2.e.a	$539$	$2$	539.e	7.c	$3$	$2$	$1$	$4.304$	\(\Q(\sqrt{-3}) \)	None					77.2.a.c	$2$	$0$	\(-1\)	\(-2\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
539.2.e.b	$539$	$2$	539.e	7.c	$3$	$2$	$1$	$4.304$	\(\Q(\sqrt{-3}) \)	None					77.2.a.c	$2$	$0$	\(-1\)	\(2\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(2-2\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
539.2.e.c	$539$	$2$	539.e	7.c	$3$	$2$	$1$	$4.304$	\(\Q(\sqrt{-3}) \)	None					77.2.a.a	$2$	$0$	\(0\)	\(-3\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-3+3\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}-\zeta_{6}q^{5}+\cdots\)
539.2.e.d	$539$	$2$	539.e	7.c	$3$	$2$	$1$	$4.304$	\(\Q(\sqrt{-3}) \)	None					77.2.a.b	$2$	$0$	\(0\)	\(-1\)	\(-3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}-3\zeta_{6}q^{5}+\cdots\)
539.2.e.e	$539$	$2$	539.e	7.c	$3$	$2$	$1$	$4.304$	\(\Q(\sqrt{-3}) \)	None					77.2.a.b	$2$	$0$	\(0\)	\(1\)	\(3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots\)
539.2.e.f	$539$	$2$	539.e	7.c	$3$	$2$	$1$	$4.304$	\(\Q(\sqrt{-3}) \)	None					77.2.a.a	$2$	$0$	\(0\)	\(3\)	\(1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(3-3\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}+\zeta_{6}q^{5}+\cdots\)
539.2.e.g	$539$	$2$	539.e	7.c	$3$	$2$	$1$	$4.304$	\(\Q(\sqrt{-3}) \)	None					11.2.a.a	$2$	$0$	\(2\)	\(-1\)	\(1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
539.2.e.h	$539$	$2$	539.e	7.c	$3$	$2$	$1$	$4.304$	\(\Q(\sqrt{-3}) \)	None					11.2.a.a	$2$	$0$	\(2\)	\(1\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
539.2.e.i	$539$	$2$	539.e	7.c	$3$	$4$	$2$	$4.304$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None					77.2.a.d	$2$	$0$	\(0\)	\(-2\)	\(4\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{2}+(\beta _{1}-\beta _{2}-\beta _{3})q^{3}+3\beta _{1}q^{4}+\cdots\)
539.2.e.j	$539$	$2$	539.e	7.c	$3$	$4$	$2$	$4.304$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None					77.2.a.d	$2$	$0$	\(0\)	\(2\)	\(-4\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{2}+(-\beta _{1}+\beta _{2}+\beta _{3})q^{3}+3\beta _{1}q^{4}+\cdots\)
539.2.e.k	$539$	$2$	539.e	7.c	$3$	$4$	$2$	$4.304$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			539.2.a.e	$4$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{2}-\beta _{1}q^{3}+(1+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
539.2.e.l	$539$	$2$	539.e	7.c	$3$	$6$	$3$	$4.304$	6.0.1783323.2	None					77.2.e.b	$2$	$0$	\(0\)	\(-1\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{3}+\beta _{5})q^{2}-\beta _{1}q^{3}+(-1-\beta _{1}+\cdots)q^{4}+\cdots\)
539.2.e.m	$539$	$2$	539.e	7.c	$3$	$6$	$3$	$4.304$	\(\Q(\zeta_{18})\)	None					77.2.e.a	$2$	$0$	\(0\)	\(3\)	\(6\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta_{5}+\beta_{4}+\cdots-\beta_{2})q^{2}+\cdots+(-\beta_{3}+\beta_{2}-\beta_1+1)q^{3}+\cdots\)
539.2.e.n	$539$	$2$	539.e	7.c	$3$	$8$	$4$	$4.304$	8.0.6927565824.3	None		✓			539.2.a.k	$4$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{3}-\beta _{4}-\beta _{5})q^{2}+\beta _{2}q^{3}+\cdots\)
539.2.e.o	$539$	$2$	539.e	7.c	$3$	$20$	$10$	$4.304$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		✓	✓		539.2.a.l	$4$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{8}q^{2}+(-\beta _{1}+\beta _{10})q^{3}+(-2\beta _{2}+\cdots)q^{4}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(539, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 2}\)