Space of modular forms of level 320, weight 9, and Character 320.h

• Label: 320.9.h
• Level: $320$
• Weight: $9$
• Character orbit: 320.h
• Rep. character: $\chi_{320}(319,\cdot)$
• Character field: $\Q$
• Dimension: $94$
• Newform subspaces: $9$
• Sturm bound: $432$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	320.h (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(432\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(320, [\chi])\).

	Total	New	Old
Modular forms	396	98	298
Cusp forms	372	94	278
Eisenstein series	24	4	20

Trace form

\( 94 q + 2 q^{5} + 196826 q^{9} + O(q^{10}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(320, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
320.9.h.a	$320$	$9$	320.h	20.d	$2$	$1$	$1$	$130.361$	\(\Q\)	\(\Q(\sqrt{-5}) \)	✓	$2$	$0$	\(0\)	\(-158\)	\(-625\)	\(-1922\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-158q^{3}-5^{4}q^{5}-1922q^{7}+18403q^{9}+\cdots\)
320.9.h.b	$320$	$9$	320.h	20.d	$2$	$1$	$1$	$130.361$	\(\Q\)	\(\Q(\sqrt{-5}) \)	✓	$2$	$0$	\(0\)	\(158\)	\(-625\)	\(1922\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+158q^{3}-5^{4}q^{5}+1922q^{7}+18403q^{9}+\cdots\)
320.9.h.c	$320$	$9$	320.h	20.d	$2$	$2$	$2$	$130.361$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(-1054\)	\(0\)	$2^{4}\cdot 3\cdot 7$	$\mathrm{U}(1)[D_{2}]$	\(q+(-527+i)q^{5}-3^{8}q^{9}+170iq^{13}+\cdots\)
320.9.h.d	$320$	$9$	320.h	20.d	$2$	$2$	$2$	$130.361$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-5}) \)	✓	$4$	$0$	\(0\)	\(0\)	\(1250\)	\(0\)	$2^{5}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta q^{3}+5^{4}q^{5}+123\beta q^{7}-5281q^{9}+\cdots\)
320.9.h.e	$320$	$9$	320.h	20.d	$2$	$4$	$4$	$130.361$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(-100\)	\(0\)	$2^{10}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-5^{2}+5\beta _{2})q^{5}+7\beta _{1}q^{7}+\cdots\)
320.9.h.f	$320$	$9$	320.h	20.d	$2$	$16$	$16$	$130.361$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(-600\)	\(0\)	$2^{99}\cdot 3^{9}\cdot 5^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-37+\beta _{3})q^{5}+(-4\beta _{1}+\cdots)q^{7}+\cdots\)
320.9.h.g	$320$	$9$	320.h	20.d	$2$	$20$	$20$	$130.361$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(1420\)	\(0\)	$2^{140}\cdot 3^{4}\cdot 5^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(71-\beta _{1})q^{5}+(-3\beta _{2}-\beta _{7}+\cdots)q^{7}+\cdots\)
320.9.h.h	$320$	$9$	320.h	20.d	$2$	$24$	$24$	$130.361$		None		$2$	$0$	\(0\)	\(-32\)	\(168\)	\(-3936\)		$\mathrm{SU}(2)[C_{2}]$
320.9.h.i	$320$	$9$	320.h	20.d	$2$	$24$	$24$	$130.361$		None		$2$	$0$	\(0\)	\(32\)	\(168\)	\(3936\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{9}^{\mathrm{old}}(320, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(320, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 2}\)