Space of modular forms of level 63, weight 6, and Character 63.e

• Label: 63.6.e
• Level: $63$
• Weight: $6$
• Character orbit: 63.e
• Rep. character: $\chi_{63}(37,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $32$
• Newform subspaces: $6$
• Sturm bound: $48$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	63.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(48\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(63, [\chi])\).

	Total	New	Old
Modular forms	88	36	52
Cusp forms	72	32	40
Eisenstein series	16	4	12

Trace form

32 * q - 268 * q^4 - 60 * q^5 - 70 * q^7 + 72 * q^8 + 282 * q^10 - 42 * q^11 - 236 * q^13 - 1884 * q^14 - 3892 * q^16 - 498 * q^17 + 3982 * q^19 + 1968 * q^20 - 9204 * q^22 + 5220 * q^23 - 6478 * q^25 - 15882 * q^26 + 22712 * q^28 + 12624 * q^29 + 16426 * q^31 + 3288 * q^32 - 41940 * q^34 - 8262 * q^35 - 7424 * q^37 - 27612 * q^38 + 50004 * q^40 + 43812 * q^41 - 30716 * q^43 + 32580 * q^44 + 2862 * q^46 - 40074 * q^47 + 9398 * q^49 + 1308 * q^50 + 27436 * q^52 - 44610 * q^53 + 41472 * q^55 - 22992 * q^56 - 76128 * q^58 + 1956 * q^59 + 13102 * q^61 + 282144 * q^62 + 157544 * q^64 + 37206 * q^65 + 147586 * q^67 - 61152 * q^68 - 150822 * q^70 - 301764 * q^71 - 37352 * q^73 - 287940 * q^74 - 586736 * q^76 + 139206 * q^77 + 51130 * q^79 - 28860 * q^80 + 2712 * q^82 + 639396 * q^83 + 363228 * q^85 + 224106 * q^86 + 440364 * q^88 - 189786 * q^89 - 193502 * q^91 - 812544 * q^92 - 212502 * q^94 - 482982 * q^95 - 803528 * q^97 - 58050 * q^98

Decomposition of \(S_{6}^{\mathrm{new}}(63, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
63.6.e.a	$63$	$6$	63.e	7.c	$3$	$2$	$1$	$10.104$	\(\Q(\sqrt{-3}) \)	None					21.6.e.a	$2$	$0$	\(-2\)	\(0\)	\(11\)	\(259\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}+28\zeta_{6}q^{4}+(11-11\zeta_{6})q^{5}+\cdots\)
63.6.e.b	$63$	$6$	63.e	7.c	$3$	$2$	$1$	$10.104$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			63.6.e.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-211\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+2^{5}\zeta_{6}q^{4}+(-149+87\zeta_{6})q^{7}-427q^{13}+\cdots\)
63.6.e.c	$63$	$6$	63.e	7.c	$3$	$4$	$2$	$10.104$	\(\Q(\sqrt{-3}, \sqrt{-83})\)	None					21.6.e.b	$2$	$0$	\(-3\)	\(0\)	\(-33\)	\(-350\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2-2\beta _{1}-\beta _{3})q^{2}+(34\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots\)
63.6.e.d	$63$	$6$	63.e	7.c	$3$	$4$	$2$	$10.104$	\(\Q(\sqrt{-3}, \sqrt{37})\)	None					7.6.c.a	$2$	$0$	\(2\)	\(0\)	\(-38\)	\(-168\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+\beta _{2})q^{2}+(-6+6\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
63.6.e.e	$63$	$6$	63.e	7.c	$3$	$8$	$4$	$10.104$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					21.6.e.c	$2$	$0$	\(3\)	\(0\)	\(0\)	\(258\)	$3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{1}+\beta _{2})q^{2}+(-\beta _{1}+18\beta _{2}+\cdots)q^{4}+\cdots\)
63.6.e.f	$63$	$6$	63.e	7.c	$3$	$12$	$6$	$10.104$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓	✓		63.6.e.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(142\)	$2^{4}\cdot 3^{5}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+(30\beta _{2}-\beta _{5}-\beta _{8})q^{4}+(-2\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(63, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(63, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)