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

• Label: 539.2.q
• Level: $539$
• Weight: $2$
• Character orbit: 539.q
• Rep. character: $\chi_{539}(214,\cdot)$
• Character field: $\Q(\zeta_{15})$
• Dimension: $288$
• 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.q (of order \(15\) and degree \(8\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 77 \)
Character field:			\(\Q(\zeta_{15})\)
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	512	352	160
Cusp forms	384	288	96
Eisenstein series	128	64	64

Trace form

288 * q + 7 * q^2 + 3 * q^3 + 39 * q^4 + q^5 + 12 * q^6 + 19 * q^9 - 4 * q^10 + 9 * q^11 + 16 * q^12 + 4 * q^13 + 27 * q^16 + 11 * q^17 - 57 * q^18 + 7 * q^19 + 40 * q^20 - 108 * q^22 + 2 * q^23 + 7 * q^24 - 15 * q^25 + 15 * q^26 + 6 * q^27 - 84 * q^29 + 41 * q^30 + 9 * q^31 - 8 * q^32 + 26 * q^33 - 72 * q^34 - 134 * q^36 - 15 * q^37 - 3 * q^38 - 53 * q^39 - 5 * q^40 - 62 * q^41 - 60 * q^43 - 45 * q^44 + 16 * q^45 + 17 * q^46 - 19 * q^47 - 146 * q^48 - 70 * q^50 - 55 * q^51 + 3 * q^52 - 33 * q^53 - 44 * q^54 - 4 * q^55 - 72 * q^57 + 57 * q^58 - 3 * q^59 + 137 * q^60 - 18 * q^61 + 80 * q^62 - 204 * q^64 + 22 * q^65 + 47 * q^66 + 104 * q^67 + 21 * q^68 + 126 * q^69 - 36 * q^71 - 31 * q^72 - 26 * q^73 + 47 * q^74 + 11 * q^75 + 144 * q^76 + 348 * q^78 - 48 * q^79 + 3 * q^80 + 65 * q^81 - 13 * q^82 + 28 * q^83 + 118 * q^85 + 145 * q^86 - 66 * q^87 - 101 * q^88 + 34 * q^89 + 18 * q^90 - 168 * q^92 - 68 * q^93 - 45 * q^94 - 98 * q^95 + 55 * q^96 + 68 * q^97 + 104 * q^99

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.q.a	$539$	$2$	539.q	77.m	$15$	$8$	$1$	$4.304$	\(\Q(\zeta_{15})\)	None					77.2.m.a	$4$	$0$	\(-2\)	\(-1\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{15}]$	\(q+(-1+\zeta_{15}-\zeta_{15}^{5}+\zeta_{15}^{7})q^{2}+\cdots\)
539.2.q.b	$539$	$2$	539.q	77.m	$15$	$16$	$2$	$4.304$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					77.2.f.a	$4$	$0$	\(1\)	\(-4\)	\(3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{15}]$	\(q+(\beta _{2}+\beta _{4}+\beta _{11}-\beta _{12})q^{2}+(\beta _{5}-\beta _{7}+\cdots)q^{3}+\cdots\)
539.2.q.c	$539$	$2$	539.q	77.m	$15$	$16$	$2$	$4.304$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					77.2.f.a	$4$	$0$	\(1\)	\(4\)	\(-3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{15}]$	\(q+(\beta _{2}+\beta _{4}+\beta _{11}-\beta _{12})q^{2}+(-\beta _{5}+\cdots)q^{3}+\cdots\)
539.2.q.d	$539$	$2$	539.q	77.m	$15$	$16$	$2$	$4.304$	16.0.\(\cdots\).1	\(\Q(\sqrt{-7}) \)		✓			539.2.f.c	$8$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{15}]$	\(q+(\beta _{1}-\beta _{4}-\beta _{5}+\beta _{6}+\beta _{7}-\beta _{9}-\beta _{10}+\cdots)q^{2}+\cdots\)
539.2.q.e	$539$	$2$	539.q	77.m	$15$	$32$	$4$	$4.304$		None		✓			539.2.f.f	$8$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{15}]$
539.2.q.f	$539$	$2$	539.q	77.m	$15$	$32$	$4$	$4.304$		None					77.2.f.b	$4$	$0$	\(3\)	\(-2\)	\(-5\)	\(0\)		$\mathrm{SU}(2)[C_{15}]$
539.2.q.g	$539$	$2$	539.q	77.m	$15$	$32$	$4$	$4.304$		None					77.2.f.b	$4$	$0$	\(3\)	\(2\)	\(5\)	\(0\)		$\mathrm{SU}(2)[C_{15}]$
539.2.q.h	$539$	$2$	539.q	77.m	$15$	$40$	$5$	$4.304$		None					77.2.m.b	$4$	$0$	\(-3\)	\(4\)	\(-4\)	\(0\)		$\mathrm{SU}(2)[C_{15}]$
539.2.q.i	$539$	$2$	539.q	77.m	$15$	$96$	$12$	$4.304$		None		✓	✓		539.2.f.i	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{15}]$

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

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