Space of modular forms of level 320, weight 3, and Character 320.h

• Label: 320.3.h
• Level: $320$
• Weight: $3$
• Character orbit: 320.h
• Rep. character: $\chi_{320}(319,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $7$
• Sturm bound: $144$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	320.h (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(144\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(320, [\chi])\).

	Total	New	Old
Modular forms	108	26	82
Cusp forms	84	22	62
Eisenstein series	24	4	20

Trace form

\( 22 q + 2 q^{5} + 50 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(320, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
320.3.h.a	$320$	$3$	320.h	20.d	$2$	$1$	$1$	$8.719$	\(\Q\)	\(\Q(\sqrt{-5}) \)	✓	$2$	$0$	\(0\)	\(-4\)	\(5\)	\(-4\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-4q^{3}+5q^{5}-4q^{7}+7q^{9}-20q^{15}+\cdots\)
320.3.h.b	$320$	$3$	320.h	20.d	$2$	$1$	$1$	$8.719$	\(\Q\)	\(\Q(\sqrt{-5}) \)	✓	$2$	$0$	\(0\)	\(4\)	\(5\)	\(4\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+4q^{3}+5q^{5}+4q^{7}+7q^{9}+20q^{15}+\cdots\)
320.3.h.c	$320$	$3$	320.h	20.d	$2$	$2$	$2$	$8.719$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-5}) \)	✓	$4$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta q^{3}-5q^{5}-3\beta q^{7}+11q^{9}+5\beta q^{15}+\cdots\)
320.3.h.d	$320$	$3$	320.h	20.d	$2$	$2$	$2$	$8.719$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+(-3+i)q^{5}-9q^{9}-6iq^{13}-4iq^{17}+\cdots\)
320.3.h.e	$320$	$3$	320.h	20.d	$2$	$4$	$4$	$8.719$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(1+\beta _{2})q^{5}-3\beta _{1}q^{7}-q^{9}+\cdots\)
320.3.h.f	$320$	$3$	320.h	20.d	$2$	$6$	$6$	$8.719$	6.0.1827904.1	None		$2$	$0$	\(0\)	\(-4\)	\(2\)	\(12\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{1})q^{3}+\beta _{3}q^{5}+(2+\beta _{3}-\beta _{4}+\cdots)q^{7}+\cdots\)
320.3.h.g	$320$	$3$	320.h	20.d	$2$	$6$	$6$	$8.719$	6.0.1827904.1	None		$2$	$0$	\(0\)	\(4\)	\(2\)	\(-12\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{1})q^{3}+(1+\beta _{1}+\beta _{3})q^{5}+(-2+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(320, [\chi])\) into lower level spaces

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