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

• Label: 522.2.k
• Level: $522$
• Weight: $2$
• Character orbit: 522.k
• Rep. character: $\chi_{522}(181,\cdot)$
• Character field: $\Q(\zeta_{7})$
• Dimension: $78$
• 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.k (of order \(7\) and degree \(6\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 29 \)
Character field:			\(\Q(\zeta_{7})\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(180\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

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

	Total	New	Old
Modular forms	588	78	510
Cusp forms	492	78	414
Eisenstein series	96	0	96

Trace form

78 * q - q^2 - 13 * q^4 + 4 * q^7 - q^8 + 2 * q^10 + 8 * q^11 - 24 * q^13 - 13 * q^16 + 2 * q^17 + 4 * q^19 + 7 * q^20 + 12 * q^22 - 32 * q^23 + q^25 + 11 * q^26 + 4 * q^28 + 25 * q^29 + 36 * q^31 - q^32 + 27 * q^34 + 36 * q^35 + 54 * q^37 + 10 * q^38 + 9 * q^40 - 10 * q^41 - 20 * q^43 - 20 * q^44 + 4 * q^46 + 58 * q^47 - 23 * q^49 + 5 * q^50 + 4 * q^52 + 19 * q^53 + 24 * q^55 + 15 * q^58 - 32 * q^59 - 74 * q^61 + 14 * q^62 - 13 * q^64 + 17 * q^65 + 28 * q^67 + 16 * q^68 - 52 * q^70 + 30 * q^71 + 5 * q^73 - 76 * q^74 - 24 * q^76 + 10 * q^77 - 28 * q^79 - 50 * q^82 - 8 * q^83 - 132 * q^85 - 68 * q^86 + 12 * q^88 - 100 * q^89 - 44 * q^91 - 32 * q^92 + 12 * q^94 - 176 * q^95 + 71 * q^97 - 25 * q^98

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.k.a	$522$	$2$	522.k	29.d	$7$	$6$	$1$	$4.168$	\(\Q(\zeta_{14})\)	None					174.2.g.b	$2$	$0$	\(-1\)	\(0\)	\(-4\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{7}]$	\(q+\zeta_{14}^{4}q^{2}-\zeta_{14}q^{4}+(-1+\zeta_{14}+\cdots)q^{5}+\cdots\)
522.2.k.b	$522$	$2$	522.k	29.d	$7$	$6$	$1$	$4.168$	\(\Q(\zeta_{14})\)	None		✓			522.2.k.b	$2$	$0$	\(-1\)	\(0\)	\(-3\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{7}]$	\(q+\zeta_{14}^{4}q^{2}-\zeta_{14}q^{4}+(-1+\zeta_{14}+\cdots)q^{5}+\cdots\)
522.2.k.c	$522$	$2$	522.k	29.d	$7$	$6$	$1$	$4.168$	\(\Q(\zeta_{14})\)	None					58.2.d.a	$2$	$0$	\(-1\)	\(0\)	\(4\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{7}]$	\(q+\zeta_{14}^{4}q^{2}-\zeta_{14}q^{4}+(\zeta_{14}^{3}-2\zeta_{14}^{4}+\cdots)q^{5}+\cdots\)
522.2.k.d	$522$	$2$	522.k	29.d	$7$	$6$	$1$	$4.168$	\(\Q(\zeta_{14})\)	None					174.2.g.a	$2$	$0$	\(1\)	\(0\)	\(1\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{7}]$	\(q-\zeta_{14}^{4}q^{2}-\zeta_{14}q^{4}+(-1+\zeta_{14}+\cdots)q^{5}+\cdots\)
522.2.k.e	$522$	$2$	522.k	29.d	$7$	$6$	$1$	$4.168$	\(\Q(\zeta_{14})\)	None		✓			522.2.k.b	$2$	$0$	\(1\)	\(0\)	\(3\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{7}]$	\(q-\zeta_{14}^{4}q^{2}-\zeta_{14}q^{4}+(1-\zeta_{14}+2\zeta_{14}^{4}+\cdots)q^{5}+\cdots\)
522.2.k.f	$522$	$2$	522.k	29.d	$7$	$12$	$2$	$4.168$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			522.2.k.f	$2$	$0$	\(-2\)	\(0\)	\(-2\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{7}]$	\(q+\beta _{9}q^{2}+\beta _{4}q^{4}+(\beta _{1}+\beta _{9})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
522.2.k.g	$522$	$2$	522.k	29.d	$7$	$12$	$2$	$4.168$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					174.2.g.c	$2$	$0$	\(-2\)	\(0\)	\(-1\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{7}]$	\(q+\beta _{5}q^{2}+\beta _{6}q^{4}+(\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4}+\cdots)q^{5}+\cdots\)
522.2.k.h	$522$	$2$	522.k	29.d	$7$	$12$	$2$	$4.168$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					58.2.d.b	$2$	$0$	\(2\)	\(0\)	\(0\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{7}]$	\(q-\beta _{2}q^{2}+\beta _{6}q^{4}+(-1+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
522.2.k.i	$522$	$2$	522.k	29.d	$7$	$12$	$2$	$4.168$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			522.2.k.f	$2$	$0$	\(2\)	\(0\)	\(2\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{7}]$	\(q-\beta _{9}q^{2}+\beta _{4}q^{4}+(-\beta _{1}-\beta _{9})q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(522, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(29, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(58, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(87, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(174, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(261, [\chi])\)\(^{\oplus 2}\)