Space of modular forms of level 320, weight 6, and Character 320.c

• Label: 320.6.c
• Level: $320$
• Weight: $6$
• Character orbit: 320.c
• Rep. character: $\chi_{320}(129,\cdot)$
• Character field: $\Q$
• Dimension: $58$
• Newform subspaces: $12$
• Sturm bound: $288$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(288\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(320, [\chi])\).

	Total	New	Old
Modular forms	252	62	190
Cusp forms	228	58	170
Eisenstein series	24	4	20

Trace form

\( 58 q + 2 q^{5} - 4378 q^{9} - 968 q^{21} - 3118 q^{25} + 4 q^{29} + 4948 q^{41} + 41790 q^{45} - 110450 q^{49} + 48084 q^{61} + 37776 q^{65} + 306424 q^{69} + 366210 q^{81} + 9568 q^{85} + 76548 q^{89}+O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(320, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
320.6.c.a	$320$	$6$	320.c	5.b	$2$	$2$	$2$	$51.323$	\(\Q(\sqrt{-1}) \)	None					10.6.b.a	$2$	$0$	\(0\)	\(0\)	\(-110\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+7\beta q^{3}+(-5\beta-55)q^{5}+79\beta q^{7}+\cdots\)
320.6.c.b	$320$	$6$	320.c	5.b	$2$	$2$	$2$	$51.323$	\(\Q(\sqrt{-1}) \)	None					10.6.b.a	$2$	$0$	\(0\)	\(0\)	\(-110\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+7\beta q^{3}+(5\beta-55)q^{5}+79\beta q^{7}+\cdots\)
320.6.c.c	$320$	$6$	320.c	5.b	$2$	$2$	$2$	$51.323$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					160.6.c.a	$4$	$0$	\(0\)	\(0\)	\(-82\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+(-19\beta-41)q^{5}+243 q^{9}+122\beta q^{13}+\cdots\)
320.6.c.d	$320$	$6$	320.c	5.b	$2$	$2$	$2$	$51.323$	\(\Q(\sqrt{-31}) \)	None					20.6.c.a	$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{3}+(5+5\beta )q^{5}+11\beta q^{7}+119q^{9}+\cdots\)
320.6.c.e	$320$	$6$	320.c	5.b	$2$	$2$	$2$	$51.323$	\(\Q(\sqrt{-31}) \)	None					20.6.c.a	$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{3}+(5-5\beta )q^{5}+11\beta q^{7}+119q^{9}+\cdots\)
320.6.c.f	$320$	$6$	320.c	5.b	$2$	$2$	$2$	$51.323$	\(\Q(\sqrt{-11}) \)	None					5.6.b.a	$2$	$0$	\(0\)	\(0\)	\(90\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta q^{3}+(45+5\beta )q^{5}+9\beta q^{7}-153q^{9}+\cdots\)
320.6.c.g	$320$	$6$	320.c	5.b	$2$	$2$	$2$	$51.323$	\(\Q(\sqrt{-11}) \)	None					5.6.b.a	$2$	$0$	\(0\)	\(0\)	\(90\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta q^{3}+(45-5\beta )q^{5}+9\beta q^{7}-153q^{9}+\cdots\)
320.6.c.h	$320$	$6$	320.c	5.b	$2$	$4$	$4$	$51.323$	\(\Q(i, \sqrt{5})\)	\(\Q(\sqrt{-5}) \)					160.6.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 5^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+(5\beta _{1}+\beta _{2})q^{3}-5\beta _{3}q^{5}+(-46\beta _{1}+\cdots)q^{7}+\cdots\)
320.6.c.i	$320$	$6$	320.c	5.b	$2$	$8$	$8$	$51.323$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					40.6.c.a	$2$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{31}\cdot 5^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(-1+\beta _{2})q^{5}+(-\beta _{2}-\beta _{6}+\cdots)q^{7}+\cdots\)
320.6.c.j	$320$	$6$	320.c	5.b	$2$	$8$	$8$	$51.323$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					40.6.c.a	$2$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{31}\cdot 5^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-1+\beta _{2})q^{5}+(\beta _{2}+\beta _{6}+\cdots)q^{7}+\cdots\)
320.6.c.k	$320$	$6$	320.c	5.b	$2$	$12$	$12$	$51.323$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					160.6.c.c	$4$	$0$	\(0\)	\(0\)	\(60\)	\(0\)	$2^{44}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{3}+(5+\beta _{5}+\beta _{7})q^{5}+\beta _{9}q^{7}+\cdots\)
320.6.c.l	$320$	$6$	320.c	5.b	$2$	$12$	$12$	$51.323$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					160.6.c.d	$4$	$0$	\(0\)	\(0\)	\(60\)	\(0\)	$2^{46}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(5-\beta _{5})q^{5}+(-5\beta _{2}-3\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(320, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(320, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 2}\)