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

• Label: 320.3.p
• Level: $320$
• Weight: $3$
• Character orbit: 320.p
• Rep. character: $\chi_{320}(193,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $44$
• Newform subspaces: $14$
• Sturm bound: $144$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	320.p (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(144\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\), \(13\)

Dimensions

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

	Total	New	Old
Modular forms	216	52	164
Cusp forms	168	44	124
Eisenstein series	48	8	40

Trace form

\( 44 q + 4 q^{5} + O(q^{10}) \)

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.p.a	$320$	$3$	320.p	5.c	$4$	$2$	$1$	$8.719$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2+2i)q^{3}+5iq^{5}+(-2-2i)q^{7}+\cdots\)
320.3.p.b	$320$	$3$	320.p	5.c	$4$	$2$	$1$	$8.719$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(-10\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+i)q^{3}-5q^{5}+(-3-3i)q^{7}+\cdots\)
320.3.p.c	$320$	$3$	320.p	5.c	$4$	$2$	$1$	$8.719$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(6\)	\(-14\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+i)q^{3}+(3-4i)q^{5}+(-7-7i)q^{7}+\cdots\)
320.3.p.d	$320$	$3$	320.p	5.c	$4$	$2$	$1$	$8.719$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-4-3i)q^{5}+9iq^{9}+(-7+7i)q^{13}+\cdots\)
320.3.p.e	$320$	$3$	320.p	5.c	$4$	$2$	$1$	$8.719$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(4-3i)q^{5}+9iq^{9}+(17-17i)q^{13}+\cdots\)
320.3.p.f	$320$	$3$	320.p	5.c	$4$	$2$	$1$	$8.719$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(-10\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-i)q^{3}-5q^{5}+(3+3i)q^{7}+7iq^{9}+\cdots\)
320.3.p.g	$320$	$3$	320.p	5.c	$4$	$2$	$1$	$8.719$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(6\)	\(14\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-i)q^{3}+(3-4i)q^{5}+(7+7i)q^{7}+\cdots\)
320.3.p.h	$320$	$3$	320.p	5.c	$4$	$2$	$1$	$8.719$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2-2i)q^{3}+5iq^{5}+(2+2i)q^{7}+\cdots\)
320.3.p.i	$320$	$3$	320.p	5.c	$4$	$4$	$2$	$8.719$	\(\Q(i, \sqrt{41})\)	None		$2$	$0$	\(0\)	\(-2\)	\(6\)	\(-14\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{2})q^{3}+(1+2\beta _{1}-\beta _{3})q^{5}+\cdots\)
320.3.p.j	$320$	$3$	320.p	5.c	$4$	$4$	$2$	$8.719$	\(\Q(i, \sqrt{7})\)	None		$4$	$0$	\(0\)	\(0\)	\(-12\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{3}+(-3-4\beta _{1})q^{5}+3\beta _{3}q^{7}+\cdots\)
320.3.p.k	$320$	$3$	320.p	5.c	$4$	$4$	$2$	$8.719$	\(\Q(i, \sqrt{15})\)	None		$4$	$0$	\(0\)	\(0\)	\(20\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{3}+5q^{5}-\beta _{3}q^{7}+21\beta _{1}q^{9}+\cdots\)
320.3.p.l	$320$	$3$	320.p	5.c	$4$	$4$	$2$	$8.719$	\(\Q(i, \sqrt{41})\)	None		$2$	$0$	\(0\)	\(2\)	\(6\)	\(14\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\beta _{1}+\beta _{3})q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+\cdots\)
320.3.p.m	$320$	$3$	320.p	5.c	$4$	$6$	$3$	$8.719$	6.0.3534400.1	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(-8\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{3}+(-1-\beta _{2}-\beta _{4})q^{5}+(-2+\cdots)q^{7}+\cdots\)
320.3.p.n	$320$	$3$	320.p	5.c	$4$	$6$	$3$	$8.719$	6.0.3534400.1	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(8\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{3}+(-1-\beta _{2}-\beta _{4})q^{5}+(2+2\beta _{3}+\cdots)q^{7}+\cdots\)

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

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