Space of modular forms of level 1008, weight 3, and Character 1008.cd

• Label: 1008.3.cd
• Level: $1008$
• Weight: $3$
• Character orbit: 1008.cd
• Rep. character: $\chi_{1008}(415,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $80$
• Newform subspaces: $16$
• Sturm bound: $576$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.cd (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 28 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(576\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1008, [\chi])\).

	Total	New	Old
Modular forms	816	80	736
Cusp forms	720	80	640
Eisenstein series	96	0	96

Trace form

\( 80 q - 16 q^{13} - 224 q^{25} + 144 q^{29} + 40 q^{37} - 48 q^{41} - 16 q^{49} - 24 q^{53} + 184 q^{61} - 24 q^{65} - 376 q^{73} - 144 q^{77} - 48 q^{85} + 408 q^{89} + 272 q^{97}+O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1008, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1008.3.cd.a	$1008$	$3$	1008.cd	28.g	$6$	$2$	$1$	$27.466$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			1008.3.cd.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-13\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-5-3\zeta_{6})q^{7}-23q^{13}+(21+21\zeta_{6})q^{19}+\cdots\)
1008.3.cd.b	$1008$	$3$	1008.cd	28.g	$6$	$2$	$1$	$27.466$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			1008.3.cd.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-11\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-3-5\zeta_{6})q^{7}+q^{13}+(-5-5\zeta_{6})q^{19}+\cdots\)
1008.3.cd.c	$1008$	$3$	1008.cd	28.g	$6$	$2$	$1$	$27.466$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			1008.3.cd.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(11\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(3+5\zeta_{6})q^{7}+q^{13}+(5+5\zeta_{6})q^{19}+\cdots\)
1008.3.cd.d	$1008$	$3$	1008.cd	28.g	$6$	$2$	$1$	$27.466$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			1008.3.cd.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(13\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(5+3\zeta_{6})q^{7}-23q^{13}+(-21-21\zeta_{6})q^{19}+\cdots\)
1008.3.cd.e	$1008$	$3$	1008.cd	28.g	$6$	$4$	$2$	$27.466$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None					112.3.r.a	$4$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{2}q^{5}+(3\beta _{1}-\beta _{3})q^{7}-5\beta _{1}q^{11}+\cdots\)
1008.3.cd.f	$1008$	$3$	1008.cd	28.g	$6$	$4$	$2$	$27.466$	\(\Q(\sqrt{-3}, \sqrt{-19})\)	None					336.3.be.a	$2$	$0$	\(0\)	\(0\)	\(-1\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1+2\beta _{1}-\beta _{3})q^{5}+(-3-3\beta _{1}+\cdots)q^{7}+\cdots\)
1008.3.cd.g	$1008$	$3$	1008.cd	28.g	$6$	$4$	$2$	$27.466$	\(\Q(\sqrt{-3}, \sqrt{-19})\)	None					336.3.be.a	$2$	$0$	\(0\)	\(0\)	\(-1\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1+2\beta _{1}-\beta _{3})q^{5}+(3+3\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
1008.3.cd.h	$1008$	$3$	1008.cd	28.g	$6$	$6$	$3$	$27.466$	6.0.1364138928.1	None					336.3.be.d	$2$	$0$	\(0\)	\(0\)	\(-1\)	\(-11\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{4}q^{5}+(-3+\beta _{1}+\beta _{2}-\beta _{3}+\beta _{4}+\cdots)q^{7}+\cdots\)
1008.3.cd.i	$1008$	$3$	1008.cd	28.g	$6$	$6$	$3$	$27.466$	6.0.1364138928.1	None					336.3.be.d	$2$	$0$	\(0\)	\(0\)	\(-1\)	\(11\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{4}q^{5}+(2-\beta _{2}+\beta _{3}-\beta _{4}+\beta _{5})q^{7}+\cdots\)
1008.3.cd.j	$1008$	$3$	1008.cd	28.g	$6$	$6$	$3$	$27.466$	6.0.259470000.1	None					112.3.r.b	$2$	$0$	\(0\)	\(0\)	\(1\)	\(-14\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{3}q^{5}+(-4-\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4}+\cdots)q^{7}+\cdots\)
1008.3.cd.k	$1008$	$3$	1008.cd	28.g	$6$	$6$	$3$	$27.466$	6.0.259470000.1	None					112.3.r.b	$2$	$0$	\(0\)	\(0\)	\(1\)	\(14\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{3}q^{5}+(4+\beta _{1}-\beta _{2}-\beta _{3}-\beta _{4}+\cdots)q^{7}+\cdots\)
1008.3.cd.l	$1008$	$3$	1008.cd	28.g	$6$	$6$	$3$	$27.466$	6.0.2682209403.3	None					336.3.be.c	$2$	$0$	\(0\)	\(0\)	\(2\)	\(-11\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\beta _{1}+\beta _{2}+\beta _{3})q^{5}+(-2+\beta _{1}+\cdots)q^{7}+\cdots\)
1008.3.cd.m	$1008$	$3$	1008.cd	28.g	$6$	$6$	$3$	$27.466$	6.0.2682209403.3	None					336.3.be.c	$2$	$0$	\(0\)	\(0\)	\(2\)	\(11\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\beta _{1}+\beta _{2}+\beta _{3})q^{5}+(2-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
1008.3.cd.n	$1008$	$3$	1008.cd	28.g	$6$	$8$	$4$	$27.466$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			1008.3.cd.n	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-24\)	$2^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{6}q^{5}+(-1-5\beta _{1}+\beta _{3}+3\beta _{4}+\cdots)q^{7}+\cdots\)
1008.3.cd.o	$1008$	$3$	1008.cd	28.g	$6$	$8$	$4$	$27.466$	8.0.\(\cdots\).8	None		✓			1008.3.cd.o	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{5}q^{5}+(2\beta _{3}+\beta _{4})q^{7}-\beta _{6}q^{11}+\cdots\)
1008.3.cd.p	$1008$	$3$	1008.cd	28.g	$6$	$8$	$4$	$27.466$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			1008.3.cd.n	$4$	$0$	\(0\)	\(0\)	\(0\)	\(24\)	$2^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{6}q^{5}+(1+5\beta _{1}-\beta _{3}-3\beta _{4})q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1008, [\chi])\) into lower level spaces

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