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

• Label: 672.3.e
• Level: $672$
• Weight: $3$
• Character orbit: 672.e
• Rep. character: $\chi_{672}(335,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $6$
• Sturm bound: $384$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	672.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 168 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(384\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(13\), \(17\)

Dimensions

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

	Total	New	Old
Modular forms	272	68	204
Cusp forms	240	60	180
Eisenstein series	32	8	24

Trace form

\( 60 q - 4 q^{9} - 228 q^{25} + 8 q^{43} + 12 q^{49} - 120 q^{51} + 32 q^{57} - 120 q^{67} + 28 q^{81} + 296 q^{91} - 320 q^{99}+O(q^{100}) \)

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.e.a	$672$	$3$	672.e	168.e	$2$	$1$	$1$	$18.311$	\(\Q\)	\(\Q(\sqrt{-42}) \)	✓				168.3.e.a	$2$	$0$	\(0\)	\(-3\)	\(0\)	\(-7\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3q^{3}-7q^{7}+9q^{9}+2q^{13}-22q^{17}+\cdots\)
672.3.e.b	$672$	$3$	672.e	168.e	$2$	$1$	$1$	$18.311$	\(\Q\)	\(\Q(\sqrt{-42}) \)	✓				168.3.e.a	$2$	$0$	\(0\)	\(-3\)	\(0\)	\(7\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3q^{3}+7q^{7}+9q^{9}-2q^{13}-22q^{17}+\cdots\)
672.3.e.c	$672$	$3$	672.e	168.e	$2$	$1$	$1$	$18.311$	\(\Q\)	\(\Q(\sqrt{-42}) \)	✓				168.3.e.a	$2$	$0$	\(0\)	\(3\)	\(0\)	\(-7\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3q^{3}-7q^{7}+9q^{9}+2q^{13}+22q^{17}+\cdots\)
672.3.e.d	$672$	$3$	672.e	168.e	$2$	$1$	$1$	$18.311$	\(\Q\)	\(\Q(\sqrt{-42}) \)	✓				168.3.e.a	$2$	$0$	\(0\)	\(3\)	\(0\)	\(7\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3q^{3}+7q^{7}+9q^{9}-2q^{13}+22q^{17}+\cdots\)
672.3.e.e	$672$	$3$	672.e	168.e	$2$	$8$	$8$	$18.311$	8.0.\(\cdots\).2	None					168.3.e.e	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+(-\beta _{2}+\beta _{4}-\beta _{7})q^{5}+(\beta _{5}+\cdots)q^{7}+\cdots\)
672.3.e.f	$672$	$3$	672.e	168.e	$2$	$48$	$48$	$18.311$		None			✓		168.3.e.f	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

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

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