Space of modular forms of level 672, weight 2, and Character 672.i

• Label: 672.2.i
• Level: $672$
• Weight: $2$
• Character orbit: 672.i
• Rep. character: $\chi_{672}(209,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $5$
• Sturm bound: $256$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.i (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 168 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(256\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(672, [\chi])\).

	Total	New	Old
Modular forms	144	36	108
Cusp forms	112	28	84
Eisenstein series	32	8	24

Trace form

\( 28 q + 4 q^{7} - 4 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(672, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
672.2.i.a	$672$	$2$	672.i	168.i	$2$	$4$	$4$	$5.366$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{1})q^{3}+\beta _{1}q^{5}+(2-\beta _{2})q^{7}+\cdots\)
672.2.i.b	$672$	$2$	672.i	168.i	$2$	$4$	$4$	$5.366$	\(\Q(\sqrt{2}, \sqrt{-3})\)	\(\Q(\sqrt{-6}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{2}q^{3}+2\beta _{2}q^{5}+(-1+\beta _{3})q^{7}+\cdots\)
672.2.i.c	$672$	$2$	672.i	168.i	$2$	$4$	$4$	$5.366$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(4\)	\(0\)	\(8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{1})q^{3}+\beta _{1}q^{5}+(2-\beta _{2})q^{7}+\cdots\)
672.2.i.d	$672$	$2$	672.i	168.i	$2$	$8$	$8$	$5.366$	8.0.3317760000.1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{2}-\beta _{3})q^{3}-\beta _{3}q^{5}+(-1-\beta _{6}+\cdots)q^{7}+\cdots\)
672.2.i.e	$672$	$2$	672.i	168.i	$2$	$8$	$8$	$5.366$	8.0.\(\cdots\).11	\(\Q(\sqrt{-14}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}\cdot 3$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{3}+(\beta _{6}-\beta _{7})q^{5}+\beta _{3}q^{7}+(\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(672, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(672, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 3}\)