Embedded newform 384.2.w.a.323.36

• Label: 384.2.w.a.323.36
• Level: $384$
• Weight: $2$
• Character: 384.323
• 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			323.36
Character	\(\chi\)	\(=\)	384.323
Dual form			384.2.w.a.107.36

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.348649 - 1.37056i) q^{2} +(0.545128 - 1.64403i) q^{3} +(-1.75689 - 0.955692i) q^{4} +(-2.82396 - 0.856639i) q^{5} +(-2.06319 - 1.32032i) q^{6} +(-0.0745750 - 0.374914i) q^{7} +(-1.92237 + 2.07472i) q^{8} +(-2.40567 - 1.79241i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{19}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.348649	−	1.37056i	0.246532	−	0.969135i
							\(3\)	0.545128	−	1.64403i	0.314730	−	0.949181i
\(5\)	−2.82396	−	0.856639i	−1.26291	−	0.383101i								−0.413343	−	0.910575i	\(-0.635639\pi\)
							−0.849571	+	0.527475i	\(0.823139\pi\)
\(7\)	−0.0745750	−	0.374914i	−0.0281867	−	0.141704i	0.964129	−	0.265433i	\(-0.0855148\pi\)
							−0.992316	+	0.123729i	\(0.960515\pi\)
\(11\)	0.231027	+	2.34566i	0.0696573	+	0.707242i	0.965068	+	0.262001i	\(0.0843822\pi\)
							−0.895410	+	0.445242i	\(0.853118\pi\)
\(13\)	3.12979	−	0.949411i	0.868048	−	0.263319i	0.175314	−	0.984513i	\(-0.443906\pi\)
							0.692734	+	0.721193i	\(0.256406\pi\)
\(17\)	−0.745322	−	0.308723i	−0.180767	−	0.0748762i	0.290464	−	0.956886i	\(-0.406190\pi\)
							−0.471231	+	0.882010i	\(0.656190\pi\)
\(19\)	0.936386	+	0.500509i	0.214822	+	0.114825i	0.575298	−	0.817944i	\(-0.304886\pi\)
							−0.360477	+	0.932768i	\(0.617386\pi\)
\(23\)	−5.22949	−	7.82648i	−1.09042	−	1.63193i	−0.705022	−	0.709185i	\(-0.749063\pi\)
							−0.385402	−	0.922749i	\(-0.625937\pi\)
\(29\)	0.407758	−	4.14004i	0.0757188	−	0.768786i	−0.880092	−	0.474803i	\(-0.842519\pi\)
							0.955811	−	0.293983i	\(-0.0949807\pi\)
\(31\)	−2.67551	−	2.67551i	−0.480536	−	0.480536i	0.424767	−	0.905303i	\(-0.360356\pi\)
							−0.905303	+	0.424767i	\(0.860356\pi\)
\(37\)	−5.34624	+	2.85762i	−0.878916	+	0.469791i	−0.848162	−	0.529736i	\(-0.822291\pi\)
							−0.0307540	+	0.999527i	\(0.509791\pi\)
\(41\)	−1.74994	−	2.61897i	−0.273294	−	0.409014i	0.669280	−	0.743010i	\(-0.266603\pi\)
							−0.942574	+	0.333996i	\(0.891603\pi\)
\(43\)	5.57238	−	6.78997i	0.849780	−	1.03546i	−0.149131	−	0.988817i	\(-0.547648\pi\)
							0.998912	−	0.0466430i	\(-0.0148523\pi\)
\(47\)	−10.9974	−	4.55527i	−1.60413	−	0.664454i	−0.612141	−	0.790749i	\(-0.709691\pi\)
							−0.991993	+	0.126295i	\(0.959691\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.323.36	yes	992
3.2	odd	2	inner	384.2.w.a.323.27	yes	992
128.107	odd	32	inner	384.2.w.a.107.27	✓	992
384.107	even	32	inner	384.2.w.a.107.36	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.107.27	✓	992	128.107	odd	32	inner
384.2.w.a.107.36	yes	992	384.107	even	32	inner
384.2.w.a.323.27	yes	992	3.2	odd	2	inner
384.2.w.a.323.36	yes	992	1.1	even	1	trivial