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

• Label: 448.3.c
• Level: $448$
• Weight: $3$
• Character orbit: 448.c
• Rep. character: $\chi_{448}(321,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $8$
• Sturm bound: $192$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(192\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	140	34	106
Cusp forms	116	30	86
Eisenstein series	24	4	20

Trace form

\( 30 q - 82 q^{9} + O(q^{10}) \)

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.c.a	$448$	$3$	448.c	7.b	$2$	$1$	$1$	$12.207$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-7\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-7q^{7}+9q^{9}+6q^{11}+18q^{23}+5^{2}q^{25}+\cdots\)
448.3.c.b	$448$	$3$	448.c	7.b	$2$	$1$	$1$	$12.207$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(7\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+7q^{7}+9q^{9}-6q^{11}-18q^{23}+5^{2}q^{25}+\cdots\)
448.3.c.c	$448$	$3$	448.c	7.b	$2$	$2$	$2$	$12.207$	\(\Q(\sqrt{-6}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-10\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+\beta q^{5}+(-5+\beta )q^{7}-15q^{9}+\cdots\)
448.3.c.d	$448$	$3$	448.c	7.b	$2$	$2$	$2$	$12.207$	\(\Q(\sqrt{-6}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(10\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-\beta q^{5}+(5+\beta )q^{7}-15q^{9}+\cdots\)
448.3.c.e	$448$	$3$	448.c	7.b	$2$	$4$	$4$	$12.207$	4.0.2048.2	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{5}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
448.3.c.f	$448$	$3$	448.c	7.b	$2$	$4$	$4$	$12.207$	4.0.2048.2	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{5}+(1-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
448.3.c.g	$448$	$3$	448.c	7.b	$2$	$8$	$8$	$12.207$	8.0.\(\cdots\).3	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{3}-\beta _{7}q^{5}+\beta _{2}q^{7}+(-7-\beta _{5}+\cdots)q^{9}+\cdots\)
448.3.c.h	$448$	$3$	448.c	7.b	$2$	$8$	$8$	$12.207$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+\beta _{5}q^{5}+(\beta _{2}-\beta _{6})q^{7}+(1+\cdots)q^{9}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(448, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 2}\)