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

• Label: 1008.2.cs
• Level: $1008$
• Weight: $2$
• Character orbit: 1008.cs
• Rep. character: $\chi_{1008}(271,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $40$
• Newform subspaces: $17$
• 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.cs (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 28 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(384\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\), \(17\), \(19\)

Dimensions

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

	Total	New	Old
Modular forms	432	40	392
Cusp forms	336	40	296
Eisenstein series	96	0	96

Trace form

\( 40 q + O(q^{10}) \)

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.cs.a	$1008$	$2$	1008.cs	28.f	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-6\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-4+2\zeta_{6})q^{5}+(-2+3\zeta_{6})q^{7}+\cdots\)
1008.2.cs.b	$1008$	$2$	1008.cs	28.f	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-6\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-4+2\zeta_{6})q^{5}+(2-3\zeta_{6})q^{7}+(2+\cdots)q^{11}+\cdots\)
1008.2.cs.c	$1008$	$2$	1008.cs	28.f	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-3\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\zeta_{6})q^{5}+(-1-2\zeta_{6})q^{7}+(-1+\cdots)q^{11}+\cdots\)
1008.2.cs.d	$1008$	$2$	1008.cs	28.f	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-3\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\zeta_{6})q^{5}+(1-3\zeta_{6})q^{7}+(1+\zeta_{6})q^{11}+\cdots\)
1008.2.cs.e	$1008$	$2$	1008.cs	28.f	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-3\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\zeta_{6})q^{5}+(-1+3\zeta_{6})q^{7}+(-1+\cdots)q^{11}+\cdots\)
1008.2.cs.f	$1008$	$2$	1008.cs	28.f	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-3\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\zeta_{6})q^{5}+(1+2\zeta_{6})q^{7}+(1+\zeta_{6})q^{11}+\cdots\)
1008.2.cs.g	$1008$	$2$	1008.cs	28.f	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-5\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-2-\zeta_{6})q^{7}+(-3+6\zeta_{6})q^{13}+\cdots\)
1008.2.cs.h	$1008$	$2$	1008.cs	28.f	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-2+3\zeta_{6})q^{7}+(1-2\zeta_{6})q^{13}+7\zeta_{6}q^{19}+\cdots\)
1008.2.cs.i	$1008$	$2$	1008.cs	28.f	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(1\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(2-3\zeta_{6})q^{7}+(1-2\zeta_{6})q^{13}-7\zeta_{6}q^{19}+\cdots\)
1008.2.cs.j	$1008$	$2$	1008.cs	28.f	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(5\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(2+\zeta_{6})q^{7}+(-3+6\zeta_{6})q^{13}+\zeta_{6}q^{19}+\cdots\)
1008.2.cs.k	$1008$	$2$	1008.cs	28.f	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(3\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2-\zeta_{6})q^{5}+(-3+\zeta_{6})q^{7}+(3+3\zeta_{6})q^{11}+\cdots\)
1008.2.cs.l	$1008$	$2$	1008.cs	28.f	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(3\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2-\zeta_{6})q^{5}+(3-\zeta_{6})q^{7}+(-3-3\zeta_{6})q^{11}+\cdots\)
1008.2.cs.m	$1008$	$2$	1008.cs	28.f	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(6\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(4-2\zeta_{6})q^{5}+(-2+3\zeta_{6})q^{7}+(-2+\cdots)q^{11}+\cdots\)
1008.2.cs.n	$1008$	$2$	1008.cs	28.f	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(6\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(4-2\zeta_{6})q^{5}+(2-3\zeta_{6})q^{7}+(2+2\zeta_{6})q^{11}+\cdots\)
1008.2.cs.o	$1008$	$2$	1008.cs	28.f	$6$	$4$	$2$	$8.049$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-10\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{2}q^{5}+(-3-\beta _{1})q^{7}+(\beta _{2}-\beta _{3})q^{11}+\cdots\)
1008.2.cs.p	$1008$	$2$	1008.cs	28.f	$6$	$4$	$2$	$8.049$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(10\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{2}q^{5}+(3+\beta _{1})q^{7}+(-\beta _{2}+\beta _{3})q^{11}+\cdots\)
1008.2.cs.q	$1008$	$2$	1008.cs	28.f	$6$	$4$	$2$	$8.049$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None		$4$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\beta _{2})q^{5}+\beta _{3}q^{7}+(-\beta _{1}+\beta _{3})q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1008, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(504, [\chi])\)\(^{\oplus 2}\)