Space of modular forms of level 448, weight 2, and Character 448.f

• Label: 448.2.f
• Level: $448$
• Weight: $2$
• Character orbit: 448.f
• Rep. character: $\chi_{448}(447,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $4$
• Sturm bound: $128$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 28 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(128\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(448, [\chi])\).

	Total	New	Old
Modular forms	76	18	58
Cusp forms	52	14	38
Eisenstein series	24	4	20

Trace form

\( 14 q + 6 q^{9} + 16 q^{21} - 10 q^{25} - 4 q^{29} + 12 q^{37} - 2 q^{49} + 12 q^{53} - 16 q^{57} + 16 q^{65} + 12 q^{77} - 50 q^{81} - 64 q^{85}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(448, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
448.2.f.a	$448$	$2$	448.f	28.d	$2$	$2$	$2$	$3.577$	\(\Q(\sqrt{-3}) \)	None					112.2.f.a	$2$	$1$	\(0\)	\(-4\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 q^{3}-2\beta q^{5}+(-\beta-2)q^{7}+q^{9}+\cdots\)
448.2.f.b	$448$	$2$	448.f	28.d	$2$	$2$	$2$	$3.577$	\(\Q(\sqrt{-7}) \)	\(\Q(\sqrt{-7}) \)					28.2.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta q^{7}-3q^{9}-2\beta q^{11}-2\beta q^{23}+\cdots\)
448.2.f.c	$448$	$2$	448.f	28.d	$2$	$2$	$2$	$3.577$	\(\Q(\sqrt{-3}) \)	None					112.2.f.a	$2$	$0$	\(0\)	\(4\)	\(0\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 q^{3}+2\beta q^{5}+(-\beta+2)q^{7}+q^{9}+\cdots\)
448.2.f.d	$448$	$2$	448.f	28.d	$2$	$8$	$8$	$3.577$	\(\Q(\zeta_{16})\)	None			✓		224.2.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{2} q^{3}-\beta_{5} q^{5}-\beta_{4} q^{7}+(\beta_{7}+1)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(448, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(448, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 2}\)