Space of modular forms of level 672, weight 3, and Character 672.d

• Label: 672.3.d
• Level: $672$
• Weight: $3$
• Character orbit: 672.d
• Rep. character: $\chi_{672}(449,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $4$
• Sturm bound: $384$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	672.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(384\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(672, [\chi])\).

	Total	New	Old
Modular forms	272	48	224
Cusp forms	240	48	192
Eisenstein series	32	0	32

Trace form

\( 48 q - 16 q^{9} - 240 q^{25} + 128 q^{33} + 64 q^{37} + 96 q^{45} + 336 q^{49} - 384 q^{57} - 320 q^{61} - 480 q^{69} + 96 q^{73} + 496 q^{81} + 480 q^{85} + 672 q^{93} - 480 q^{97}+O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(672, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
672.3.d.a	$672$	$3$	672.d	3.b	$2$	$4$	$4$	$18.311$	\(\Q(\sqrt{-2}, \sqrt{7})\)	None		✓			672.3.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{3})q^{3}+\beta _{2}q^{5}+\beta _{3}q^{7}+(5+\cdots)q^{9}+\cdots\)
672.3.d.b	$672$	$3$	672.d	3.b	$2$	$8$	$8$	$18.311$	8.0.157351936.1	None		✓			672.3.d.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta _{4}q^{3}+(\beta _{2}+4\beta _{3}+\beta _{7})q^{5}+\beta _{6}q^{7}+\cdots\)
672.3.d.c	$672$	$3$	672.d	3.b	$2$	$12$	$12$	$18.311$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			672.3.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+(\beta _{5}-\beta _{7}+\beta _{11})q^{5}+\beta _{1}q^{7}+\cdots\)
672.3.d.d	$672$	$3$	672.d	3.b	$2$	$24$	$24$	$18.311$		None		✓	✓		672.3.d.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{3}^{\mathrm{old}}(672, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(672, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)