Embedded newform 384.2.w.a.35.17

• Label: 384.2.w.a.35.17
• Level: $384$
• Weight: $2$
• Character: 384.35
• 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			35.17
Character	\(\chi\)	\(=\)	384.35
Dual form			384.2.w.a.11.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.02490 + 0.974463i) q^{2} +(1.27873 - 1.16826i) q^{3} +(0.100842 - 1.99746i) q^{4} +(-1.18082 + 0.631161i) q^{5} +(-0.172143 + 2.44343i) q^{6} +(4.72705 - 0.940268i) q^{7} +(1.84309 + 2.14546i) q^{8} +(0.270316 - 2.98780i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.02490	+	0.974463i	−0.724714	+	0.689050i
							\(3\)	1.27873	−	1.16826i	0.738277	−	0.674498i
\(5\)	−1.18082	+	0.631161i	−0.528078	+	0.282264i								−0.713813	−	0.700336i	\(-0.753033\pi\)
							0.185735	+	0.982600i	\(0.440533\pi\)
\(7\)	4.72705	−	0.940268i	1.78666	−	0.355388i	0.812798	−	0.582546i	\(-0.197943\pi\)
							0.973858	+	0.227158i	\(0.0729434\pi\)
\(11\)	−2.00468	−	1.64520i	−0.604435	−	0.496047i	0.281682	−	0.959508i	\(-0.409108\pi\)
							−0.886118	+	0.463461i	\(0.846608\pi\)
\(13\)	−2.66435	−	1.42412i	−0.738957	−	0.394981i	0.0585550	−	0.998284i	\(-0.481351\pi\)
							−0.797512	+	0.603304i	\(0.793851\pi\)
\(17\)	0.706373	+	0.292589i	0.171321	+	0.0709633i	0.466695	−	0.884418i	\(-0.345444\pi\)
							−0.295375	+	0.955381i	\(0.595444\pi\)
\(19\)	0.347855	+	1.14672i	0.0798034	+	0.263077i	0.987758	−	0.155991i	\(-0.0498570\pi\)
							−0.907955	+	0.419068i	\(0.862357\pi\)
\(23\)	7.23454	−	4.83397i	1.50851	−	1.00795i	0.520444	−	0.853896i	\(-0.325766\pi\)
							0.988062	−	0.154056i	\(-0.0492337\pi\)
\(29\)	0.806075	−	0.661529i	0.149684	−	0.122843i	−0.556571	−	0.830800i	\(-0.687883\pi\)
							0.706255	+	0.707957i	\(0.250383\pi\)
\(31\)	4.36685	+	4.36685i	0.784310	+	0.784310i	0.980555	−	0.196245i	\(-0.0628748\pi\)
							−0.196245	+	0.980555i	\(0.562875\pi\)
\(37\)	3.22856	−	10.6431i	0.530772	−	1.74972i	−0.122368	−	0.992485i	\(-0.539049\pi\)
							0.653140	−	0.757237i	\(-0.273451\pi\)
\(41\)	0.972588	−	0.649862i	0.151893	−	0.101491i	−0.477301	−	0.878740i	\(-0.658385\pi\)
							0.629194	+	0.777248i	\(0.283385\pi\)
\(43\)	0.495666	+	5.03258i	0.0755883	+	0.767461i	0.956023	+	0.293293i	\(0.0947511\pi\)
							−0.880434	+	0.474168i	\(0.842749\pi\)
\(47\)	−6.98651	−	2.89391i	−1.01909	−	0.422120i	−0.190325	−	0.981721i	\(-0.560954\pi\)
							−0.828762	+	0.559602i	\(0.810954\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.35.17	yes	992
3.2	odd	2	inner	384.2.w.a.35.46	yes	992
128.11	odd	32	inner	384.2.w.a.11.46	yes	992
384.11	even	32	inner	384.2.w.a.11.17	✓	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.17	✓	992	384.11	even	32	inner
384.2.w.a.11.46	yes	992	128.11	odd	32	inner
384.2.w.a.35.17	yes	992	1.1	even	1	trivial
384.2.w.a.35.46	yes	992	3.2	odd	2	inner