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

• Label: 1008.2.df
• Level: $1008$
• Weight: $2$
• Character orbit: 1008.df
• Rep. character: $\chi_{1008}(689,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $92$
• Newform subspaces: $5$
• 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.df (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 63 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(384\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\)

Dimensions

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

	Total	New	Old
Modular forms	408	100	308
Cusp forms	360	92	268
Eisenstein series	48	8	40

Trace form

92 * q + 3 * q^3 - 6 * q^5 + q^7 - q^9 + 7 * q^15 + 6 * q^19 - 4 * q^21 + 74 * q^25 + 6 * q^29 - 15 * q^31 - 3 * q^33 + 15 * q^35 - 2 * q^37 + 19 * q^39 + 8 * q^43 - 15 * q^45 + 3 * q^47 - q^49 + 3 * q^51 - 2 * q^57 + 3 * q^59 - 3 * q^61 - 7 * q^63 - 27 * q^65 - q^67 + 9 * q^69 - 6 * q^73 + 18 * q^75 - 27 * q^77 - q^79 + 3 * q^81 + 30 * q^83 + 3 * q^85 + 66 * q^87 + 12 * q^89 + 12 * q^91 - 23 * q^93 + 87 * q^95 + 43 * q^99

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1008.2.df.a	$1008$	$2$	1008.df	63.s	$6$	$2$	$1$	$8.049$	\(\Q(\sqrt{-3}) \)	None					63.2.i.a	$2$	$0$	\(0\)	\(3\)	\(-6\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2-\zeta_{6})q^{3}-3q^{5}+(3-\zeta_{6})q^{7}+(3+\cdots)q^{9}+\cdots\)
1008.2.df.b	$1008$	$2$	1008.df	63.s	$6$	$10$	$5$	$8.049$	10.0.\(\cdots\).1	None					63.2.i.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}-\beta _{2}+\beta _{4}+\beta _{7}+\beta _{8}-\beta _{9})q^{3}+\cdots\)
1008.2.df.c	$1008$	$2$	1008.df	63.s	$6$	$16$	$8$	$8.049$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					126.2.l.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{3}q^{3}-\beta _{11}q^{5}+(1-\beta _{1}+\beta _{2}+\beta _{4}+\cdots)q^{7}+\cdots\)
1008.2.df.d	$1008$	$2$	1008.df	63.s	$6$	$16$	$8$	$8.049$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					252.2.w.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(1\)	$3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{3}-\beta _{12})q^{3}+(-\beta _{7}+\beta _{9})q^{5}+\cdots\)
1008.2.df.e	$1008$	$2$	1008.df	63.s	$6$	$48$	$24$	$8.049$		None			✓		504.2.bs.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

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

\( S_{2}^{\mathrm{old}}(1008, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(504, [\chi])\)\(^{\oplus 2}\)