Space of modular forms of level 441, weight 3, and Character 441.q

• Label: 441.3.q
• Level: $441$
• Weight: $3$
• Character orbit: 441.q
• Rep. character: $\chi_{441}(116,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $52$
• Newform subspaces: $5$
• Sturm bound: $168$
• Trace bound: $10$

Defining parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	441.q (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(168\)
Trace bound:			\(10\)
Distinguishing \(T_p\):			\(2\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(441, [\chi])\).

	Total	New	Old
Modular forms	256	52	204
Cusp forms	192	52	140
Eisenstein series	64	0	64

Trace form

\( 52 q + 48 q^{4} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(441, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
441.3.q.a	$441$	$3$	441.q	21.h	$6$	$8$	$4$	$12.016$	8.0.\(\cdots\).5	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+(\beta _{3}+\beta _{6})q^{4}-2\beta _{1}q^{5}+(-2\beta _{1}+\cdots)q^{8}+\cdots\)
441.3.q.b	$441$	$3$	441.q	21.h	$6$	$8$	$4$	$12.016$	8.0.\(\cdots\).5	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+(\beta _{3}+\beta _{6})q^{4}+2\beta _{1}q^{5}+(-2\beta _{1}+\cdots)q^{8}+\cdots\)
441.3.q.c	$441$	$3$	441.q	21.h	$6$	$8$	$4$	$12.016$	8.0.\(\cdots\).5	\(\Q(\sqrt{-7}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{4}$	$\mathrm{U}(1)[D_{6}]$	\(q+\beta _{2}q^{2}+(4-4\beta _{3}+\beta _{5}+\beta _{7})q^{4}+\cdots\)
441.3.q.d	$441$	$3$	441.q	21.h	$6$	$12$	$6$	$12.016$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+(-3\beta _{6}+\beta _{9})q^{4}+\beta _{4}q^{5}+\cdots\)
441.3.q.e	$441$	$3$	441.q	21.h	$6$	$16$	$8$	$12.016$	16.0.\(\cdots\).1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{8}q^{2}+(2\beta _{1}+\beta _{10})q^{4}+\beta _{4}q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(441, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(441, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 2}\)