Space of modular forms of level 448, weight 3, and Character 448.r

• Label: 448.3.r
• Level: $448$
• Weight: $3$
• Character orbit: 448.r
• Rep. character: $\chi_{448}(191,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $60$
• Newform subspaces: $8$
• Sturm bound: $192$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.r (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 28 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(192\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(448, [\chi])\).

	Total	New	Old
Modular forms	280	68	212
Cusp forms	232	60	172
Eisenstein series	48	8	40

Trace form

\( 60 q + 2 q^{5} + 76 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(448, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
448.3.r.a	$448$	$3$	448.r	28.g	$6$	$4$	$2$	$12.207$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		$4$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{3}-\beta _{2}q^{5}+(-3\beta _{1}+\beta _{3})q^{7}+\cdots\)
448.3.r.b	$448$	$3$	448.r	28.g	$6$	$4$	$2$	$12.207$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{3}+\zeta_{12}^{2}q^{5}+7\zeta_{12}^{3}q^{7}+\cdots\)
448.3.r.c	$448$	$3$	448.r	28.g	$6$	$4$	$2$	$12.207$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(18\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+5\zeta_{12}q^{3}+9\zeta_{12}^{2}q^{5}+(-8\zeta_{12}+\cdots)q^{7}+\cdots\)
448.3.r.d	$448$	$3$	448.r	28.g	$6$	$6$	$3$	$12.207$	6.0.259470000.1	None		$2$	$0$	\(0\)	\(-3\)	\(1\)	\(14\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{5})q^{3}+(-\beta _{1}+\beta _{3})q^{5}+(1+\cdots)q^{7}+\cdots\)
448.3.r.e	$448$	$3$	448.r	28.g	$6$	$6$	$3$	$12.207$	6.0.259470000.1	None		$2$	$0$	\(0\)	\(3\)	\(1\)	\(-14\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1+\beta _{5})q^{3}+(-\beta _{1}+\beta _{3})q^{5}+(-1+\cdots)q^{7}+\cdots\)
448.3.r.f	$448$	$3$	448.r	28.g	$6$	$12$	$6$	$12.207$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(-18\)	\(0\)	$2^{14}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{7}+\beta _{9}-\beta _{10})q^{3}+(3\beta _{1}-\beta _{4}+\beta _{5}+\cdots)q^{5}+\cdots\)
448.3.r.g	$448$	$3$	448.r	28.g	$6$	$12$	$6$	$12.207$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$2^{14}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{1}q^{3}+(-\beta _{5}-\beta _{11})q^{5}+(-\beta _{3}+\cdots)q^{7}+\cdots\)
448.3.r.h	$448$	$3$	448.r	28.g	$6$	$12$	$6$	$12.207$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{14}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{8}q^{3}+(\beta _{6}-\beta _{9})q^{5}+(-\beta _{3}+\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(448, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(448, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 3}\)