Space of modular forms of level 1008, weight 2, and Character 1008.bf

• Label: 1008.2.bf
• Level: $1008$
• Weight: $2$
• Character orbit: 1008.bf
• Rep. character: $\chi_{1008}(31,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $96$
• Newform subspaces: $9$
• Sturm bound: $384$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1008.bf (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 252 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(384\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1008, [\chi])\).

	Total	New	Old
Modular forms	408	96	312
Cusp forms	360	96	264
Eisenstein series	48	0	48

Trace form

\( 96 q - 24 q^{21} - 96 q^{25} - 12 q^{29} + 36 q^{45} - 24 q^{57} + 24 q^{65} - 24 q^{77} + 12 q^{81} + 36 q^{89} + 48 q^{93}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1008, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1008.2.bf.a	$1008$	$2$	1008.bf	252.n	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		✓			1008.2.bf.a	$2$	$1$	\(0\)	\(-3\)	\(0\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\zeta_{6})q^{3}+(-1+2\zeta_{6})q^{5}+(-3+\cdots)q^{7}+\cdots\)
1008.2.bf.b	$1008$	$2$	1008.bf	252.n	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		✓			1008.2.bf.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1+2\zeta_{6})q^{3}+(-1-2\zeta_{6})q^{7}+\cdots\)
1008.2.bf.c	$1008$	$2$	1008.bf	252.n	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		✓			1008.2.bf.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-2\zeta_{6})q^{3}+(1+2\zeta_{6})q^{7}-3q^{9}+\cdots\)
1008.2.bf.d	$1008$	$2$	1008.bf	252.n	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		✓			1008.2.bf.a	$2$	$0$	\(0\)	\(3\)	\(0\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2-\zeta_{6})q^{3}+(-1+2\zeta_{6})q^{5}+(3-\zeta_{6})q^{7}+\cdots\)
1008.2.bf.e	$1008$	$2$	1008.bf	252.n	$6$	$4$	$2$	$8.049$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			1008.2.bf.e	$2$	$0$	\(0\)	\(-6\)	\(0\)	\(2\)	$3$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2-\beta _{1})q^{3}+(1+2\beta _{1}-\beta _{2})q^{5}+\cdots\)
1008.2.bf.f	$1008$	$2$	1008.bf	252.n	$6$	$4$	$2$	$8.049$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			1008.2.bf.e	$2$	$0$	\(0\)	\(6\)	\(0\)	\(-2\)	$3$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2+\beta _{1})q^{3}+(1+2\beta _{1}-\beta _{2})q^{5}+(-1+\cdots)q^{7}+\cdots\)
1008.2.bf.g	$1008$	$2$	1008.bf	252.n	$6$	$24$	$12$	$8.049$		None		✓			1008.2.bf.g	$2$	$0$	\(0\)	\(-3\)	\(0\)	\(8\)		$\mathrm{SU}(2)[C_{6}]$
1008.2.bf.h	$1008$	$2$	1008.bf	252.n	$6$	$24$	$12$	$8.049$		None		✓			1008.2.bf.g	$2$	$0$	\(0\)	\(3\)	\(0\)	\(-8\)		$\mathrm{SU}(2)[C_{6}]$
1008.2.bf.i	$1008$	$2$	1008.bf	252.n	$6$	$32$	$16$	$8.049$		None		✓	✓		1008.2.bf.i	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{2}^{\mathrm{old}}(1008, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1008, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 3}\)