Space of modular forms of level 1008, weight 5, and Character 1008.f

• Label: 1008.5.f
• Level: $1008$
• Weight: $5$
• Character orbit: 1008.f
• Rep. character: $\chi_{1008}(433,\cdot)$
• Character field: $\Q$
• Dimension: $79$
• Newform subspaces: $14$
• Sturm bound: $960$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1008.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(960\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(1008, [\chi])\).

	Total	New	Old
Modular forms	792	81	711
Cusp forms	744	79	665
Eisenstein series	48	2	46

Trace form

79 * q + 17 * q^7 - 50 * q^11 - 1154 * q^23 - 10113 * q^25 + 1490 * q^29 - 384 * q^35 - 914 * q^37 - 2798 * q^43 + 815 * q^49 + 1394 * q^53 - 8160 * q^65 - 7310 * q^67 + 13342 * q^71 - 1102 * q^77 + 14402 * q^79 + 3904 * q^85 - 8736 * q^91 - 22944 * q^95

Decomposition of \(S_{5}^{\mathrm{new}}(1008, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1008.5.f.a	$1008$	$5$	1008.f	7.b	$2$	$1$	$1$	$104.197$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				7.5.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-49\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-7^{2}q^{7}-206q^{11}-734q^{23}+5^{4}q^{25}+\cdots\)
1008.5.f.b	$1008$	$5$	1008.f	7.b	$2$	$2$	$2$	$104.197$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					252.5.d.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-94\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q+(\beta-47)q^{7}+22\beta q^{13}+52\beta q^{19}+\cdots\)
1008.5.f.c	$1008$	$5$	1008.f	7.b	$2$	$2$	$2$	$104.197$	\(\Q(\sqrt{-3}) \)	None					28.5.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(14\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta q^{5}+(-7\beta+7)q^{7}+18 q^{11}+\cdots\)
1008.5.f.d	$1008$	$5$	1008.f	7.b	$2$	$2$	$2$	$104.197$	\(\Q(\sqrt{7}) \)	\(\Q(\sqrt{-7}) \)	✓				63.5.d.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(98\)	$2^{4}\cdot 3$	$\mathrm{U}(1)[D_{2}]$	\(q+7^{2}q^{7}+\beta q^{11}+6\beta q^{23}+5^{4}q^{25}+\cdots\)
1008.5.f.e	$1008$	$5$	1008.f	7.b	$2$	$4$	$4$	$104.197$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					126.5.c.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-140\)	$2^{9}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{5}+(-35-7\beta _{1})q^{7}-3\beta _{2}q^{11}+\cdots\)
1008.5.f.f	$1008$	$5$	1008.f	7.b	$2$	$4$	$4$	$104.197$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					42.5.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-20\)	$2^{7}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2\beta _{1}+\beta _{2})q^{5}+(-5+\beta _{1}+5\beta _{2}+\cdots)q^{7}+\cdots\)
1008.5.f.g	$1008$	$5$	1008.f	7.b	$2$	$4$	$4$	$104.197$	\(\Q(\sqrt{10}, \sqrt{-106})\)	None					63.5.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{5}+(-4-\beta _{3})q^{7}-37\beta _{1}q^{11}+\cdots\)
1008.5.f.h	$1008$	$5$	1008.f	7.b	$2$	$4$	$4$	$104.197$	4.0.1308672.3	None					14.5.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(76\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{2})q^{5}+(19+\beta _{1}-\beta _{2}+3\beta _{3})q^{7}+\cdots\)
1008.5.f.i	$1008$	$5$	1008.f	7.b	$2$	$4$	$4$	$104.197$	\(\Q(\sqrt{-33}, \sqrt{-42})\)	None					252.5.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(112\)	$2\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{5}+(28+7\beta _{2})q^{7}-\beta _{3}q^{11}+\cdots\)
1008.5.f.j	$1008$	$5$	1008.f	7.b	$2$	$6$	$6$	$104.197$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None					84.5.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-42\)	$2^{6}\cdot 3^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{5}+(-7-\beta _{1}+\beta _{4})q^{7}+(-38+\cdots)q^{11}+\cdots\)
1008.5.f.k	$1008$	$5$	1008.f	7.b	$2$	$6$	$6$	$104.197$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None					21.5.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(22\)	$2^{6}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{5}+(3+2\beta _{1}+\beta _{2}+2\beta _{3}+\beta _{4}+\cdots)q^{7}+\cdots\)
1008.5.f.l	$1008$	$5$	1008.f	7.b	$2$	$8$	$8$	$104.197$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					56.5.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-56\)	$2^{20}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{5}+(-7+\beta _{5})q^{7}+(-18-\beta _{5}+\cdots)q^{11}+\cdots\)
1008.5.f.m	$1008$	$5$	1008.f	7.b	$2$	$16$	$16$	$104.197$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None					504.5.f.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(56\)	$2^{52}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{5}+(3-\beta _{8})q^{7}+\beta _{11}q^{11}+\beta _{9}q^{13}+\cdots\)
1008.5.f.n	$1008$	$5$	1008.f	7.b	$2$	$16$	$16$	$104.197$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					168.5.f.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(56\)	$2^{48}\cdot 3^{4}\cdot 7$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{5}+(3-\beta _{4})q^{7}+(18-\beta _{9})q^{11}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(1008, [\chi])\) into lower level spaces

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