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

• Label: 1008.3.f
• Level: $1008$
• Weight: $3$
• Character orbit: 1008.f
• Rep. character: $\chi_{1008}(433,\cdot)$
• Character field: $\Q$
• Dimension: $39$
• Newform subspaces: $11$
• Sturm bound: $576$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(576\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	408	41	367
Cusp forms	360	39	321
Eisenstein series	48	2	46

Trace form

\( 39 q + 9 q^{7} + O(q^{10}) \)

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	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1008.3.f.a	$1008$	$3$	1008.f	7.b	$2$	$1$	$1$	$27.466$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(7\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+7q^{7}-6q^{11}+18q^{23}+5^{2}q^{25}+\cdots\)
1008.3.f.b	$1008$	$3$	1008.f	7.b	$2$	$2$	$2$	$27.466$	\(\Q(\sqrt{7}) \)	\(\Q(\sqrt{-7}) \)	✓	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-14\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q-7q^{7}+\beta q^{11}-2\beta q^{23}+5^{2}q^{25}+\cdots\)
1008.3.f.c	$1008$	$3$	1008.f	7.b	$2$	$2$	$2$	$27.466$	\(\Q(\sqrt{-6}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-10\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{5}+(-5+\beta )q^{7}-6q^{11}+\beta q^{13}+\cdots\)
1008.3.f.d	$1008$	$3$	1008.f	7.b	$2$	$2$	$2$	$27.466$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{6}q^{5}+(-1+\zeta_{6})q^{7}+10q^{11}+\cdots\)
1008.3.f.e	$1008$	$3$	1008.f	7.b	$2$	$2$	$2$	$27.466$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q+(1-\zeta_{6})q^{7}+2\zeta_{6}q^{13}-4\zeta_{6}q^{19}+\cdots\)
1008.3.f.f	$1008$	$3$	1008.f	7.b	$2$	$2$	$2$	$27.466$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(14\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{6}q^{5}+7q^{7}+18q^{11}+3\zeta_{6}q^{13}+\cdots\)
1008.3.f.g	$1008$	$3$	1008.f	7.b	$2$	$4$	$4$	$27.466$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+2\beta _{2})q^{5}+(-2-2\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
1008.3.f.h	$1008$	$3$	1008.f	7.b	$2$	$4$	$4$	$27.466$	4.0.2048.2	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{3})q^{5}+(-1+\beta _{1}-\beta _{2}-\beta _{3})q^{7}+\cdots\)
1008.3.f.i	$1008$	$3$	1008.f	7.b	$2$	$4$	$4$	$27.466$	\(\Q(\sqrt{2}, \sqrt{-33})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$2\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{5}+(4+\beta _{3})q^{7}+3\beta _{2}q^{11}+4\beta _{3}q^{13}+\cdots\)
1008.3.f.j	$1008$	$3$	1008.f	7.b	$2$	$8$	$8$	$27.466$	8.0.\(\cdots\).1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2^{11}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{5}+(1-\beta _{3}+\beta _{4})q^{7}+(-3-2\beta _{1}+\cdots)q^{11}+\cdots\)
1008.3.f.k	$1008$	$3$	1008.f	7.b	$2$	$8$	$8$	$27.466$	8.0.\(\cdots\).3	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2^{11}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{6}q^{5}+(1+\beta _{3})q^{7}+\beta _{5}q^{11}+(-1+\cdots)q^{13}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(1008, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 15}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(504, [\chi])\)\(^{\oplus 2}\)