Space of modular forms of level 522, weight 2, and Character 522.e

• Label: 522.2.e
• Level: $522$
• Weight: $2$
• Character orbit: 522.e
• Rep. character: $\chi_{522}(175,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $56$
• Newform subspaces: $9$
• Sturm bound: $180$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	522.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(180\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(522, [\chi])\).

	Total	New	Old
Modular forms	188	56	132
Cusp forms	172	56	116
Eisenstein series	16	0	16

Trace form

56 * q + 2 * q^2 + 6 * q^3 - 28 * q^4 - 4 * q^5 + 2 * q^6 + 4 * q^7 - 4 * q^8 + 6 * q^9 + 2 * q^11 + 4 * q^13 + 8 * q^14 - 20 * q^15 - 28 * q^16 - 12 * q^17 + 12 * q^18 + 4 * q^19 - 4 * q^20 - 16 * q^21 - 6 * q^22 + 2 * q^24 - 40 * q^25 - 8 * q^26 - 8 * q^28 + 6 * q^29 - 10 * q^30 - 8 * q^31 + 2 * q^32 + 24 * q^33 - 6 * q^34 + 64 * q^35 - 6 * q^36 + 16 * q^37 - 8 * q^38 - 16 * q^39 - 2 * q^41 - 12 * q^42 - 2 * q^43 - 4 * q^44 + 48 * q^45 - 4 * q^47 - 6 * q^48 - 24 * q^49 - 10 * q^50 - 38 * q^51 + 4 * q^52 - 40 * q^53 + 2 * q^54 + 24 * q^55 + 8 * q^56 + 6 * q^57 + 14 * q^59 + 16 * q^60 + 28 * q^61 + 24 * q^62 - 20 * q^63 + 56 * q^64 - 42 * q^65 + 16 * q^66 + 10 * q^67 + 6 * q^68 + 4 * q^69 + 8 * q^71 - 6 * q^72 - 44 * q^73 - 54 * q^75 - 2 * q^76 - 8 * q^77 - 42 * q^78 + 4 * q^79 + 8 * q^80 + 54 * q^81 + 12 * q^82 + 60 * q^83 + 8 * q^84 - 30 * q^86 - 6 * q^88 - 96 * q^89 + 52 * q^90 - 64 * q^91 - 10 * q^93 - 4 * q^96 - 14 * q^97 - 20 * q^98 - 12 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(522, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
522.2.e.a	$522$	$2$	522.e	9.c	$3$	$2$	$1$	$4.168$	\(\Q(\sqrt{-3}) \)	None		✓			522.2.e.a	$2$	$1$	\(-1\)	\(-3\)	\(0\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
522.2.e.b	$522$	$2$	522.e	9.c	$3$	$2$	$1$	$4.168$	\(\Q(\sqrt{-3}) \)	None		✓			522.2.e.b	$2$	$0$	\(-1\)	\(3\)	\(-4\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
522.2.e.c	$522$	$2$	522.e	9.c	$3$	$2$	$1$	$4.168$	\(\Q(\sqrt{-3}) \)	None		✓			522.2.e.c	$2$	$0$	\(-1\)	\(3\)	\(2\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
522.2.e.d	$522$	$2$	522.e	9.c	$3$	$2$	$1$	$4.168$	\(\Q(\sqrt{-3}) \)	None		✓			522.2.e.d	$2$	$0$	\(1\)	\(3\)	\(0\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
522.2.e.e	$522$	$2$	522.e	9.c	$3$	$4$	$2$	$4.168$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		✓			522.2.e.e	$2$	$0$	\(2\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{2}q^{2}+(-\beta _{1}-\beta _{2}+\beta _{3})q^{3}+(-1+\cdots)q^{4}+\cdots\)
522.2.e.f	$522$	$2$	522.e	9.c	$3$	$6$	$3$	$4.168$	\(\Q(\zeta_{18})\)	None		✓			522.2.e.f	$2$	$0$	\(3\)	\(0\)	\(-6\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta_1+1)q^{2}+(\beta_{5}+\beta_{2})q^{3}+\cdots\)
522.2.e.g	$522$	$2$	522.e	9.c	$3$	$8$	$4$	$4.168$	8.0.2091141441.1	None		✓			522.2.e.g	$2$	$0$	\(-4\)	\(-1\)	\(2\)	\(8\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{2})q^{2}-\beta _{1}q^{3}+\beta _{2}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
522.2.e.h	$522$	$2$	522.e	9.c	$3$	$12$	$6$	$4.168$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			522.2.e.h	$2$	$0$	\(-6\)	\(2\)	\(-2\)	\(8\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{6}q^{2}+(\beta _{1}+\beta _{9})q^{3}+(-1+\beta _{6}+\cdots)q^{4}+\cdots\)
522.2.e.i	$522$	$2$	522.e	9.c	$3$	$18$	$9$	$4.168$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		✓	✓		522.2.e.i	$2$	$0$	\(9\)	\(1\)	\(4\)	\(-3\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{5}q^{2}-\beta _{7}q^{3}+(-1+\beta _{5})q^{4}+\beta _{3}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(522, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(522, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(261, [\chi])\)\(^{\oplus 2}\)