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

\( 32 q - 268 q^{4} - 60 q^{5} - 70 q^{7} + 72 q^{8} + O(q^{10}) \)

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

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

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