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

• Label: 441.3.d
• Level: $441$
• Weight: $3$
• Character orbit: 441.d
• Rep. character: $\chi_{441}(244,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $8$
• Sturm bound: $168$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	441.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(168\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

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

	Total	New	Old
Modular forms	128	36	92
Cusp forms	96	32	64
Eisenstein series	32	4	28

Trace form

\( 32 q + 60 q^{4} - 12 q^{8} + O(q^{10}) \)

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.d.a	$441$	$3$	441.d	7.b	$2$	$2$	$2$	$12.016$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-6\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-3q^{2}+5q^{4}-3\zeta_{6}q^{5}-3q^{8}+9\zeta_{6}q^{10}+\cdots\)
441.3.d.b	$441$	$3$	441.d	7.b	$2$	$2$	$2$	$12.016$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{2}-3q^{4}+3\zeta_{6}q^{5}+7q^{8}-3\zeta_{6}q^{10}+\cdots\)
441.3.d.c	$441$	$3$	441.d	7.b	$2$	$2$	$2$	$12.016$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 7$	$\mathrm{U}(1)[D_{2}]$	\(q-4q^{4}-\zeta_{6}q^{13}+2^{4}q^{16}+3\zeta_{6}q^{19}+\cdots\)
441.3.d.d	$441$	$3$	441.d	7.b	$2$	$2$	$2$	$12.016$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2q^{2}-2\zeta_{6}q^{5}-8q^{8}-4\zeta_{6}q^{10}+\cdots\)
441.3.d.e	$441$	$3$	441.d	7.b	$2$	$4$	$4$	$12.016$	4.0.2048.2	None		$2$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$7$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{2})q^{2}+(-1+2\beta _{2})q^{4}-\beta _{1}q^{5}+\cdots\)
441.3.d.f	$441$	$3$	441.d	7.b	$2$	$4$	$4$	$12.016$	\(\Q(\sqrt{-3}, \sqrt{13})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+9q^{4}+\beta _{3}q^{5}+5\beta _{2}q^{8}+\cdots\)
441.3.d.g	$441$	$3$	441.d	7.b	$2$	$8$	$8$	$12.016$	8.0.3288334336.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-1-\beta _{3})q^{4}+\beta _{6}q^{5}+(-4\beta _{1}+\cdots)q^{8}+\cdots\)
441.3.d.h	$441$	$3$	441.d	7.b	$2$	$8$	$8$	$12.016$	8.0.339738624.1	None		$2$	$0$	\(8\)	\(0\)	\(0\)	\(0\)	$2^{5}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{3})q^{2}+(5-\beta _{2})q^{4}+(-\beta _{5}-\beta _{7})q^{5}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(441, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 2}\)