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

• Label: 1008.3.cg
• Level: $1008$
• Weight: $3$
• Character orbit: 1008.cg
• Rep. character: $\chi_{1008}(145,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $78$
• Newform subspaces: $17$
• Sturm bound: $576$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.cg (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(576\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\)

Dimensions

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

	Total	New	Old
Modular forms	816	82	734
Cusp forms	720	78	642
Eisenstein series	96	4	92

Trace form

78 * q + 3 * q^5 - 6 * q^7 - 9 * q^11 + 3 * q^17 - 45 * q^19 + 7 * q^23 + 182 * q^25 + 36 * q^29 - 45 * q^31 - 147 * q^35 - 25 * q^37 - 108 * q^43 - 147 * q^47 - 98 * q^49 - 47 * q^53 + 93 * q^59 + 69 * q^61 - 32 * q^65 + 33 * q^67 - 100 * q^71 - 123 * q^73 + 89 * q^77 - 71 * q^79 - 86 * q^85 + 75 * q^89 + 336 * q^91 + 117 * q^95

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.cg.a	$1008$	$3$	1008.cg	7.d	$6$	$2$	$1$	$27.466$	\(\Q(\sqrt{-3}) \)	None					21.3.f.a	$2$	$0$	\(0\)	\(0\)	\(-9\)	\(-13\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-6+3\zeta_{6})q^{5}+(-5-3\zeta_{6})q^{7}+\cdots\)
1008.3.cg.b	$1008$	$3$	1008.cg	7.d	$6$	$2$	$1$	$27.466$	\(\Q(\sqrt{-3}) \)	None					84.3.m.a	$2$	$0$	\(0\)	\(0\)	\(-3\)	\(-13\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\zeta_{6})q^{5}+(-5-3\zeta_{6})q^{7}+(3+\cdots)q^{11}+\cdots\)
1008.3.cg.c	$1008$	$3$	1008.cg	7.d	$6$	$2$	$1$	$27.466$	\(\Q(\sqrt{-3}) \)	None					28.3.h.a	$2$	$0$	\(0\)	\(0\)	\(-3\)	\(14\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\zeta_{6})q^{5}+7q^{7}+(-15+15\zeta_{6})q^{11}+\cdots\)
1008.3.cg.d	$1008$	$3$	1008.cg	7.d	$6$	$2$	$1$	$27.466$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					63.3.m.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-13\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-8+3\zeta_{6})q^{7}+(-7+14\zeta_{6})q^{13}+\cdots\)
1008.3.cg.e	$1008$	$3$	1008.cg	7.d	$6$	$2$	$1$	$27.466$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					252.3.z.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(11\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(8-5\zeta_{6})q^{7}+(-15+30\zeta_{6})q^{13}+\cdots\)
1008.3.cg.f	$1008$	$3$	1008.cg	7.d	$6$	$2$	$1$	$27.466$	\(\Q(\sqrt{-3}) \)	None					21.3.f.c	$2$	$0$	\(0\)	\(0\)	\(6\)	\(7\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(4-2\zeta_{6})q^{5}+7\zeta_{6}q^{7}+(-10+10\zeta_{6})q^{11}+\cdots\)
1008.3.cg.g	$1008$	$3$	1008.cg	7.d	$6$	$2$	$1$	$27.466$	\(\Q(\sqrt{-3}) \)	None					21.3.f.b	$2$	$0$	\(0\)	\(0\)	\(9\)	\(7\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(6-3\zeta_{6})q^{5}+(7-7\zeta_{6})q^{7}+(11-11\zeta_{6})q^{11}+\cdots\)
1008.3.cg.h	$1008$	$3$	1008.cg	7.d	$6$	$4$	$2$	$27.466$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					42.3.g.a	$2$	$0$	\(0\)	\(0\)	\(-12\)	\(10\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-4-2\beta _{1}+\beta _{3})q^{5}+(-5\beta _{1}-\beta _{3})q^{7}+\cdots\)
1008.3.cg.i	$1008$	$3$	1008.cg	7.d	$6$	$4$	$2$	$27.466$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					126.3.n.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-14\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{5}+7\beta _{1}q^{7}+(-2\beta _{2}-2\beta _{3})q^{11}+\cdots\)
1008.3.cg.j	$1008$	$3$	1008.cg	7.d	$6$	$4$	$2$	$27.466$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None					252.3.z.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-14\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{2}q^{5}+(-7-7\beta _{1})q^{7}+(2\beta _{2}-\beta _{3})q^{11}+\cdots\)
1008.3.cg.k	$1008$	$3$	1008.cg	7.d	$6$	$4$	$2$	$27.466$	\(\Q(\sqrt{-3}, \sqrt{13})\)	None					63.3.m.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(26\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{2}+\beta _{3})q^{5}+(8-3\beta _{1})q^{7}+\beta _{2}q^{11}+\cdots\)
1008.3.cg.l	$1008$	$3$	1008.cg	7.d	$6$	$4$	$2$	$27.466$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					14.3.d.a	$2$	$0$	\(0\)	\(0\)	\(6\)	\(-8\)	$3$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2+\beta _{1}+2\beta _{3})q^{5}+(-1+2\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
1008.3.cg.m	$1008$	$3$	1008.cg	7.d	$6$	$4$	$2$	$27.466$	\(\Q(\sqrt{-3}, \sqrt{65})\)	None					84.3.m.b	$2$	$0$	\(0\)	\(0\)	\(9\)	\(-2\)	$3$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2+\beta _{1}-\beta _{3})q^{5}+(-1+\beta _{2})q^{7}+\cdots\)
1008.3.cg.n	$1008$	$3$	1008.cg	7.d	$6$	$8$	$4$	$27.466$	8.0.\(\cdots\).2	None					168.3.z.a	$2$	$0$	\(0\)	\(0\)	\(-6\)	\(4\)	$2^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{4})q^{5}+(-\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
1008.3.cg.o	$1008$	$3$	1008.cg	7.d	$6$	$8$	$4$	$27.466$	8.0.\(\cdots\).2	None					56.3.o.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{5}q^{5}+(2-\beta _{1}-3\beta _{2}-\beta _{3}+\beta _{4}+\cdots)q^{7}+\cdots\)
1008.3.cg.p	$1008$	$3$	1008.cg	7.d	$6$	$8$	$4$	$27.466$	8.0.\(\cdots\).9	None					168.3.z.b	$2$	$0$	\(0\)	\(0\)	\(6\)	\(-8\)	$2^{4}\cdot 7$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\beta _{1}+\beta _{6})q^{5}+(-1-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
1008.3.cg.q	$1008$	$3$	1008.cg	7.d	$6$	$16$	$8$	$27.466$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None			✓		504.3.by.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{18}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{7}q^{5}+(-1+\beta _{2}-\beta _{5})q^{7}+(\beta _{4}+\cdots)q^{11}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(1008, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(504, [\chi])\)\(^{\oplus 2}\)