Space of modular forms of level 384, weight 5, and Character 384.h

• Label: 384.5.h
• Level: $384$
• Weight: $5$
• Character orbit: 384.h
• Rep. character: $\chi_{384}(65,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $8$
• Sturm bound: $320$
• Trace bound: $3$

Defining parameters

Level:	$N$	$=$	$384 = 2^{7} \cdot 3$
Weight:	$k$	$=$	$5$
Character orbit:	$[\chi]$	$=$	384.h (of order $2$ and degree $1$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$24$
Character field:			$\Q$
Newform subspaces:			$8$
Sturm bound:			$320$
Trace bound:			$3$
Distinguishing $T_p$:			$5$, $11$

Dimensions

The following table gives the dimensions of various subspaces of $M_{5}(384, [\chi])$.

	Total	New	Old
Modular forms	272	64	208
Cusp forms	240	64	176
Eisenstein series	32	0	32

Trace form

$64 q + 8000 q^{25} - 1984 q^{33} + 27968 q^{49} - 2240 q^{57} + 14720 q^{73} + 17728 q^{81} + 11264 q^{97}+O(q^{100})$

Decomposition of $S_{5}^{\mathrm{new}}(384, [\chi])$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
384.5.h.a	$384$	$5$	384.h	24.h	$2$	$2$	$2$	$39.694$	$\Q(\sqrt{6})$	$\Q(\sqrt{-6})$	✓	✓			384.5.h.a	$4$	$0$	$0$	$-18$	$0$	$0$	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	$q-9q^{3}+\beta q^{5}-5\beta q^{7}+3^{4}q^{9}-142q^{11}+\cdots$
384.5.h.b	$384$	$5$	384.h	24.h	$2$	$2$	$2$	$39.694$	$\Q(\sqrt{-2})$	$\Q(\sqrt{-2})$		✓			384.5.h.b	$4$	$0$	$0$	$-14$	$0$	$0$	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	$q+(-7-\beta )q^{3}+(17+14\beta )q^{9}+46q^{11}+\cdots$
384.5.h.c	$384$	$5$	384.h	24.h	$2$	$2$	$2$	$39.694$	$\Q(\sqrt{-2})$	$\Q(\sqrt{-2})$		✓			384.5.h.b	$4$	$0$	$0$	$14$	$0$	$0$	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	$q+(7+\beta )q^{3}+(17+14\beta )q^{9}-46q^{11}+\cdots$
384.5.h.d	$384$	$5$	384.h	24.h	$2$	$2$	$2$	$39.694$	$\Q(\sqrt{6})$	$\Q(\sqrt{-6})$	✓	✓			384.5.h.a	$4$	$0$	$0$	$18$	$0$	$0$	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	$q+9q^{3}+\beta q^{5}+5\beta q^{7}+3^{4}q^{9}+142q^{11}+\cdots$
384.5.h.e	$384$	$5$	384.h	24.h	$2$	$4$	$4$	$39.694$	$\Q(\sqrt{-2}, \sqrt{-5})$	None		✓			384.5.h.e	$4$	$0$	$0$	$-12$	$0$	$0$	$2^{8}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	$q+(-3-\beta _{1})q^{3}+\beta _{2}q^{5}+\beta _{2}q^{7}+(-63+\cdots)q^{9}+\cdots$
384.5.h.f	$384$	$5$	384.h	24.h	$2$	$4$	$4$	$39.694$	$\Q(\sqrt{-2}, \sqrt{-5})$	None		✓			384.5.h.e	$4$	$0$	$0$	$12$	$0$	$0$	$2^{8}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	$q+(3-\beta _{1})q^{3}-\beta _{2}q^{5}+\beta _{2}q^{7}+(-63+\cdots)q^{9}+\cdots$
384.5.h.g	$384$	$5$	384.h	24.h	$2$	$16$	$16$	$39.694$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	None		✓			384.5.h.g	$8$	$0$	$0$	$0$	$0$	$0$	$2^{70}\cdot 3^{4}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{4}q^{3}+\beta _{8}q^{5}+\beta _{7}q^{7}+(-7-\beta _{5}+\cdots)q^{9}+\cdots$
384.5.h.h	$384$	$5$	384.h	24.h	$2$	$32$	$32$	$39.694$		None		✓	✓		384.5.h.h	$8$	$0$	$0$	$0$	$0$	$0$		$\mathrm{SU}(2)[C_{2}]$

Decomposition of $S_{5}^{\mathrm{old}}(384, [\chi])$ into lower level spaces

$S_{5}^{\mathrm{old}}(384, [\chi]) \simeq$ $S_{5}^{\mathrm{new}}(24, [\chi])$$^{\oplus 5}$$\oplus$$S_{5}^{\mathrm{new}}(96, [\chi])$$^{\oplus 3}$$\oplus$$S_{5}^{\mathrm{new}}(192, [\chi])$$^{\oplus 2}$