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

• Label: 1008.2.r
• Level: $1008$
• Weight: $2$
• Character orbit: 1008.r
• Rep. character: $\chi_{1008}(337,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $72$
• Newform subspaces: $14$
• 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.r (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(384\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	408	72	336
Cusp forms	360	72	288
Eisenstein series	48	0	48

Trace form

\( 72 q + 4 q^{9} + O(q^{10}) \)

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	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1008.2.r.a	$1008$	$2$	1008.r	9.c	$3$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-3\)	\(-2\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\zeta_{6})q^{3}-2\zeta_{6}q^{5}+(-1+\zeta_{6})q^{7}+\cdots\)
1008.2.r.b	$1008$	$2$	1008.r	9.c	$3$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-3\)	\(-2\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\zeta_{6})q^{3}-2\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+\cdots\)
1008.2.r.c	$1008$	$2$	1008.r	9.c	$3$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(1\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+2\zeta_{6})q^{3}+\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+\cdots\)
1008.2.r.d	$1008$	$2$	1008.r	9.c	$3$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(3\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-2\zeta_{6})q^{3}+3\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+\cdots\)
1008.2.r.e	$1008$	$2$	1008.r	9.c	$3$	$4$	$2$	$8.049$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$2$	$0$	\(0\)	\(-4\)	\(-2\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{3})q^{3}+(\beta _{1}-\beta _{2}+\beta _{3})q^{5}+\cdots\)
1008.2.r.f	$1008$	$2$	1008.r	9.c	$3$	$4$	$2$	$8.049$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		$2$	$0$	\(0\)	\(1\)	\(-3\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{2}+\beta _{3})q^{3}+(1-2\beta _{1}-2\beta _{2}+\beta _{3})q^{5}+\cdots\)
1008.2.r.g	$1008$	$2$	1008.r	9.c	$3$	$6$	$3$	$8.049$	6.0.309123.1	None		$2$	$0$	\(0\)	\(-2\)	\(3\)	\(3\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{2}+\beta _{3}+\beta _{4})q^{3}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
1008.2.r.h	$1008$	$2$	1008.r	9.c	$3$	$6$	$3$	$8.049$	\(\Q(\zeta_{18})\)	None		$2$	$0$	\(0\)	\(0\)	\(-3\)	\(3\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\zeta_{18}^{2}+\zeta_{18}^{4})q^{3}+(-1+\zeta_{18}+\zeta_{18}^{5})q^{5}+\cdots\)
1008.2.r.i	$1008$	$2$	1008.r	9.c	$3$	$6$	$3$	$8.049$	\(\Q(\zeta_{18})\)	None		$2$	$0$	\(0\)	\(0\)	\(3\)	\(3\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\zeta_{18}^{4}-\zeta_{18}^{5})q^{3}+(1-\zeta_{18}-\zeta_{18}^{2}+\cdots)q^{5}+\cdots\)
1008.2.r.j	$1008$	$2$	1008.r	9.c	$3$	$6$	$3$	$8.049$	6.0.309123.1	None		$2$	$0$	\(0\)	\(2\)	\(-1\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{1}-\beta _{2})q^{3}+(\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots\)
1008.2.r.k	$1008$	$2$	1008.r	9.c	$3$	$6$	$3$	$8.049$	6.0.309123.1	None		$2$	$0$	\(0\)	\(4\)	\(5\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{5})q^{3}+(2+\beta _{2}+2\beta _{4}-\beta _{5})q^{5}+\cdots\)
1008.2.r.l	$1008$	$2$	1008.r	9.c	$3$	$8$	$4$	$8.049$	8.0.2091141441.1	None		$2$	$0$	\(0\)	\(1\)	\(-3\)	\(-4\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}+\beta _{6})q^{3}+(-1+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
1008.2.r.m	$1008$	$2$	1008.r	9.c	$3$	$8$	$4$	$8.049$	8.0.508277025.1	None		$2$	$0$	\(0\)	\(4\)	\(4\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{2})q^{3}+(\beta _{2}-\beta _{3}+\beta _{4}+2\beta _{5}+\cdots)q^{5}+\cdots\)
1008.2.r.n	$1008$	$2$	1008.r	9.c	$3$	$10$	$5$	$8.049$	10.0.\(\cdots\).1	None		$2$	$0$	\(0\)	\(0\)	\(-3\)	\(-5\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{6}q^{3}+(-\beta _{2}-\beta _{7})q^{5}+(-1+\beta _{2}+\cdots)q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1008, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(504, [\chi])\)\(^{\oplus 2}\)