Space of modular forms of level 672, weight 4, and Character 672.i

• Label: 672.4.i
• Level: $672$
• Weight: $4$
• Character orbit: 672.i
• Rep. character: $\chi_{672}(209,\cdot)$
• Character field: $\Q$
• Dimension: $92$
• Newform subspaces: $3$
• Sturm bound: $512$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	672.i (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 168 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(512\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(672, [\chi])\).

	Total	New	Old
Modular forms	400	100	300
Cusp forms	368	92	276
Eisenstein series	32	8	24

Trace form

\( 92 q + 4 q^{7} - 4 q^{9} - 104 q^{15} - 1908 q^{25} + 112 q^{39} - 364 q^{49} - 112 q^{57} + 708 q^{63} + 488 q^{79} - 1236 q^{81}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(672, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
672.4.i.a	$672$	$4$	672.i	168.i	$2$	$4$	$4$	$39.649$	\(\Q(\sqrt{2}, \sqrt{-3})\)	\(\Q(\sqrt{-6}) \)					168.4.i.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(68\)	$2^{2}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{2}q^{3}+2\beta _{2}q^{5}+(17+\beta _{3})q^{7}-3^{3}q^{9}+\cdots\)
672.4.i.b	$672$	$4$	672.i	168.i	$2$	$8$	$8$	$39.649$	8.0.\(\cdots\).11	\(\Q(\sqrt{-14}) \)					168.4.i.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+(\beta _{3}+\beta _{5}+\beta _{6})q^{3}+(2\beta _{3}+3\beta _{5}-2\beta _{6}+\cdots)q^{5}+\cdots\)
672.4.i.c	$672$	$4$	672.i	168.i	$2$	$80$	$80$	$39.649$		None			✓		168.4.i.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(-64\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{4}^{\mathrm{old}}(672, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(672, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 3}\)