Embedded newform 384.2.w.a.251.58

• Label: 384.2.w.a.251.58
• Level: $384$
• Weight: $2$
• Character: 384.251
• Analytic conductor: $3.066$
• Analytic rank: $0$
• Dimension: $992$
• 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			251.58
Character	\(\chi\)	\(=\)	384.251
Dual form			384.2.w.a.179.58

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.37600 - 0.326534i) q^{2} +(-1.02612 + 1.39538i) q^{3} +(1.78675 - 0.898623i) q^{4} +(0.296184 - 3.00721i) q^{5} +(-0.956305 + 2.25510i) q^{6} +(0.110643 - 0.0739290i) q^{7} +(2.16514 - 1.81994i) q^{8} +(-0.894147 - 2.86365i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.02612	+	1.39538i	−0.592432	+	0.805621i
\(5\)	0.296184	−	3.00721i	0.132457	−	1.34486i								−0.667412	−	0.744689i	\(-0.732598\pi\)
							0.799869	−	0.600174i	\(-0.204902\pi\)
\(7\)	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	−0.682482	−	0.730902i	\(-0.739100\pi\)
							0.986890	+	0.161396i	\(0.0515995\pi\)
\(13\)	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	0.929156	−	0.369687i	\(-0.120535\pi\)
							−0.918421	+	0.395604i	\(0.870535\pi\)
\(19\)	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	\(0.175973\pi\)
							−0.585309	+	0.810810i	\(0.699027\pi\)
\(29\)	−0.984464	−	1.84180i	−0.182810	−	0.342014i	0.773778	−	0.633457i	\(-0.218365\pi\)
							−0.956588	+	0.291443i	\(0.905865\pi\)
\(31\)	−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	\(-0.882091\pi\)
							0.722683	−	0.691180i	\(-0.242909\pi\)
\(43\)	−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	\(0.157753\pi\)
							−0.285762	+	0.958301i	\(0.592247\pi\)

(See \(a_n\) instead)

Twists

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

    

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