Embedded newform 384.2.w.a.251.5

• Label: 384.2.w.a.251.5
• Level: $384$
• Weight: $2$
• Character: 384.251
• 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			251.5
Character	\(\chi\)	\(=\)	384.251
Dual form			384.2.w.a.179.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37600 + 0.326534i) q^{2} +(0.0779601 + 1.73030i) q^{3} +(1.78675 - 0.898623i) q^{4} +(-0.296184 + 3.00721i) q^{5} +(-0.672274 - 2.35543i) q^{6} +(0.110643 - 0.0739290i) q^{7} +(-2.16514 + 1.81994i) q^{8} +(-2.98784 + 0.269788i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37600	+	0.326534i	−0.972979	+	0.230895i
							\(3\)	0.0779601	+	1.73030i	0.0450103	+	0.998987i
\(5\)	−0.296184	+	3.00721i	−0.132457	+	1.34486i								0.667412	+	0.744689i	\(0.267402\pi\)
							−0.799869	+	0.600174i	\(0.795098\pi\)
\(7\)	0.110643	−	0.0739290i	0.0418189	−	0.0279425i	−0.534485	−	0.845178i	\(-0.679494\pi\)
							0.576304	+	0.817235i	\(0.304494\pi\)
\(11\)	−1.00961	+	1.88884i	−0.304408	+	0.569507i	−0.986890	−	0.161396i	\(-0.948400\pi\)
							0.682482	+	0.730902i	\(0.260900\pi\)
\(13\)	0.0663469	+	0.673631i	0.0184013	+	0.186832i	0.999991	−	0.00416855i	\(-0.00132690\pi\)
							−0.981590	+	0.191000i	\(0.938827\pi\)
\(17\)	−0.0442622	−	0.106858i	−0.0107352	−	0.0259170i	0.918421	−	0.395604i	\(-0.129465\pi\)
							−0.929156	+	0.369687i	\(0.879465\pi\)
\(19\)	2.33102	+	1.91302i	0.534773	+	0.438877i	0.862619	−	0.505854i	\(-0.168823\pi\)
							−0.327846	+	0.944731i	\(0.606323\pi\)
\(23\)	−1.27439	−	6.40681i	−0.265730	−	1.33591i	−0.851039	−	0.525103i	\(-0.824027\pi\)
							0.585309	−	0.810810i	\(-0.300973\pi\)
\(29\)	0.984464	+	1.84180i	0.182810	+	0.342014i	0.956588	−	0.291443i	\(-0.0941353\pi\)
							−0.773778	+	0.633457i	\(0.781635\pi\)
\(31\)	−4.49674	+	4.49674i	−0.807638	+	0.807638i	−0.984276	−	0.176638i	\(-0.943478\pi\)
							0.176638	+	0.984276i	\(0.443478\pi\)
\(37\)	−1.32920	+	1.09085i	−0.218519	+	0.179334i	−0.737284	−	0.675583i	\(-0.763892\pi\)
							0.518765	+	0.854917i	\(0.326392\pi\)
\(41\)	1.34140	+	6.74369i	0.209492	+	1.05319i	0.932175	+	0.362008i	\(0.117909\pi\)
							−0.722683	+	0.691180i	\(0.757091\pi\)
\(43\)	−2.62582	−	8.65616i	−0.400433	−	1.32005i	−0.893110	−	0.449839i	\(-0.851481\pi\)
							0.492676	−	0.870213i	\(-0.336019\pi\)
\(47\)	−4.07173	−	9.83003i	−0.593923	−	1.43386i	−0.879685	−	0.475556i	\(-0.842247\pi\)
							0.285762	−	0.958301i	\(-0.407753\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.251.5	yes	992
3.2	odd	2	inner	384.2.w.a.251.58	yes	992
128.51	odd	32	inner	384.2.w.a.179.58	yes	992
384.179	even	32	inner	384.2.w.a.179.5	✓	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.179.5	✓	992	384.179	even	32	inner
384.2.w.a.179.58	yes	992	128.51	odd	32	inner
384.2.w.a.251.5	yes	992	1.1	even	1	trivial
384.2.w.a.251.58	yes	992	3.2	odd	2	inner