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

• Label: 441.3.r
• Level: $441$
• Weight: $3$
• Character orbit: 441.r
• Rep. character: $\chi_{441}(50,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $154$
• Newform subspaces: $9$
• Sturm bound: $168$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	441.r (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(168\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

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

	Total	New	Old
Modular forms	240	174	66
Cusp forms	208	154	54
Eisenstein series	32	20	12

Trace form

\( 154 q + 3 q^{2} + q^{3} + 145 q^{4} + 12 q^{5} + 5 q^{6} - 13 q^{9} + O(q^{10}) \)

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.r.a	$441$	$3$	441.r	9.d	$6$	$2$	$1$	$12.016$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-3\)	\(3\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
441.3.r.b	$441$	$3$	441.r	9.d	$6$	$6$	$3$	$12.016$	6.0.63369648.1	None		$2$	$0$	\(-3\)	\(-9\)	\(15\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{5})q^{2}+3\beta _{2}q^{3}+(\beta _{1}-4\beta _{2}+\cdots)q^{4}+\cdots\)
441.3.r.c	$441$	$3$	441.r	9.d	$6$	$6$	$3$	$12.016$	6.0.63369648.1	None		$2$	$0$	\(-3\)	\(9\)	\(-15\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{5})q^{2}-3\beta _{2}q^{3}+(\beta _{1}-4\beta _{2}+\cdots)q^{4}+\cdots\)
441.3.r.d	$441$	$3$	441.r	9.d	$6$	$8$	$4$	$12.016$	8.0.3317760000.3	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{5}q^{2}+\beta _{3}q^{3}+\beta _{4}q^{4}+(\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots\)
441.3.r.e	$441$	$3$	441.r	9.d	$6$	$16$	$8$	$12.016$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{4}q^{2}+\beta _{9}q^{3}+(-\beta _{4}+3\beta _{5}-\beta _{8}+\cdots)q^{4}+\cdots\)
441.3.r.f	$441$	$3$	441.r	9.d	$6$	$22$	$11$	$12.016$		None		$2$	$0$	\(6\)	\(-11\)	\(12\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$
441.3.r.g	$441$	$3$	441.r	9.d	$6$	$22$	$11$	$12.016$		None		$2$	$0$	\(6\)	\(11\)	\(-12\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$
441.3.r.h	$441$	$3$	441.r	9.d	$6$	$24$	$12$	$12.016$		None		$2$	$0$	\(0\)	\(-2\)	\(18\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$
441.3.r.i	$441$	$3$	441.r	9.d	$6$	$48$	$24$	$12.016$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

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

\( S_{3}^{\mathrm{old}}(441, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)