Embedded newform 384.2.w.a.299.46

• Label: 384.2.w.a.299.46
• Level: $384$
• Weight: $2$
• Character: 384.299
• Analytic conductor: $3.066$
• Analytic rank: $0$
• Dimension: $992$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	384.w (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.06625543762\)
Analytic rank:	\(0\)
Dimension:	\(992\)
Relative dimension:	\(62\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			299.46
Character	\(\chi\)	\(=\)	384.299
Dual form			384.2.w.a.131.46

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00475 + 0.995229i) q^{2} +(1.26400 + 1.18419i) q^{3} +(0.0190404 + 1.99991i) q^{4} +(0.476468 + 1.57070i) q^{5} +(0.0914584 + 2.44778i) q^{6} +(0.623411 - 3.13410i) q^{7} +(-1.97124 + 2.02836i) q^{8} +(0.195379 + 2.99363i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{29}{32}\right)\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00475	+	0.995229i	0.710465	+	0.703733i
							\(3\)	1.26400	+	1.18419i	0.729769	+	0.683694i
\(5\)	0.476468	+	1.57070i	0.213083	+	0.702441i								0.996698	+	0.0811942i	\(0.0258734\pi\)
							−0.783615	+	0.621246i	\(0.786627\pi\)
\(7\)	0.623411	−	3.13410i	0.235627	−	1.18458i	−0.663936	−	0.747789i	\(-0.731115\pi\)
							0.899564	−	0.436789i	\(-0.143885\pi\)
\(11\)	−1.36278	−	0.134222i	−0.410894	−	0.0404695i	−0.109541	−	0.993982i	\(-0.534938\pi\)
							−0.301353	+	0.953513i	\(0.597438\pi\)
\(13\)	0.849128	−	2.79920i	0.235506	−	0.776359i	−0.756919	−	0.653508i	\(-0.773296\pi\)
							0.992425	−	0.122850i	\(-0.0392036\pi\)
\(17\)	1.79685	−	0.744278i	0.435799	−	0.180514i	−0.153988	−	0.988073i	\(-0.549212\pi\)
							0.589787	+	0.807559i	\(0.299212\pi\)
\(19\)	−0.623583	−	1.16664i	−0.143060	−	0.267646i	0.800179	−	0.599761i	\(-0.204738\pi\)
							−0.943239	+	0.332115i	\(0.892238\pi\)
\(23\)	0.359719	−	0.538358i	0.0750067	−	0.112255i	−0.792070	−	0.610431i	\(-0.790996\pi\)
							0.867076	+	0.498175i	\(0.165996\pi\)
\(29\)	−9.00674	+	0.887087i	−1.67251	+	0.164728i	−0.889392	−	0.457145i	\(-0.848872\pi\)
							−0.783118	+	0.621873i	\(0.786372\pi\)
\(31\)	6.19650	−	6.19650i	1.11292	−	1.11292i	0.120172	−	0.992753i	\(-0.461655\pi\)
							0.992753	−	0.120172i	\(-0.0383445\pi\)
\(37\)	−1.99830	+	3.73856i	−0.328519	+	0.614616i	−0.990908	−	0.134545i	\(-0.957043\pi\)
							0.662389	+	0.749160i	\(0.269543\pi\)
\(41\)	−4.03674	+	6.04140i	−0.630432	+	0.943509i	0.369466	+	0.929244i	\(0.379541\pi\)
							−0.999898	+	0.0142643i	\(0.995459\pi\)
\(43\)	−2.76154	+	2.26634i	−0.421131	+	0.345613i	−0.820969	−	0.570973i	\(-0.806566\pi\)
							0.399838	+	0.916586i	\(0.369066\pi\)
\(47\)	10.7796	−	4.46505i	1.57237	−	0.651295i	0.585184	−	0.810900i	\(-0.301022\pi\)
							0.987181	+	0.159605i	\(0.0510222\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.299.46	yes	992
3.2	odd	2	inner	384.2.w.a.299.17	yes	992
128.3	odd	32	inner	384.2.w.a.131.17	✓	992
384.131	even	32	inner	384.2.w.a.131.46	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.131.17	✓	992	128.3	odd	32	inner
384.2.w.a.131.46	yes	992	384.131	even	32	inner
384.2.w.a.299.17	yes	992	3.2	odd	2	inner
384.2.w.a.299.46	yes	992	1.1	even	1	trivial