Embedded newform 384.2.w.a.155.61

• Label: 384.2.w.a.155.61
• Level: $384$
• Weight: $2$
• Character: 384.155
• 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			155.61
Character	\(\chi\)	\(=\)	384.155
Dual form			384.2.w.a.275.61

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41268 - 0.0658980i) q^{2} +(0.141669 - 1.72625i) q^{3} +(1.99131 - 0.186185i) q^{4} +(-0.635192 + 0.521289i) q^{5} +(0.0863763 - 2.44797i) q^{6} +(-2.61198 - 3.90910i) q^{7} +(2.80082 - 0.394243i) q^{8} +(-2.95986 - 0.489111i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41268	−	0.0658980i	0.998914	−	0.0465969i
							\(3\)	0.141669	−	1.72625i	0.0817926	−	0.996649i
\(5\)	−0.635192	+	0.521289i	−0.284067	+	0.233127i								−0.765626	−	0.643286i	\(-0.777571\pi\)
							0.481559	+	0.876413i	\(0.340071\pi\)
\(7\)	−2.61198	−	3.90910i	−0.987235	−	1.47750i	−0.875178	−	0.483800i	\(-0.839256\pi\)
							−0.112057	−	0.993702i	\(-0.535744\pi\)
\(11\)	1.17966	−	0.357847i	0.355682	−	0.107895i	−0.107387	−	0.994217i	\(-0.534248\pi\)
							0.463069	+	0.886322i	\(0.346748\pi\)
\(13\)	0.549357	+	0.450845i	0.152364	+	0.125042i	0.707487	−	0.706726i	\(-0.249829\pi\)
							−0.555123	+	0.831768i	\(0.687329\pi\)
\(17\)	0.884985	+	2.13654i	0.214640	+	0.518188i	0.994126	−	0.108233i	\(-0.0345193\pi\)
							−0.779485	+	0.626421i	\(0.784519\pi\)
\(19\)	3.14341	−	0.309599i	0.721148	−	0.0710269i	0.269217	−	0.963080i	\(-0.413235\pi\)
							0.451931	+	0.892053i	\(0.350735\pi\)
\(23\)	5.92437	−	1.17843i	1.23532	−	0.245720i	0.466115	−	0.884724i	\(-0.345653\pi\)
							0.769203	+	0.639004i	\(0.220653\pi\)
\(29\)	3.75930	+	1.14037i	0.698085	+	0.211762i	0.619328	−	0.785132i	\(-0.287405\pi\)
							0.0787564	+	0.996894i	\(0.474905\pi\)
\(31\)	−5.90175	+	5.90175i	−1.05999	+	1.05999i	−0.0619041	+	0.998082i	\(0.519717\pi\)
							−0.998082	+	0.0619041i	\(0.980283\pi\)
\(37\)	−4.73256	−	0.466116i	−0.778027	−	0.0766290i	−0.298795	−	0.954317i	\(-0.596585\pi\)
							−0.479232	+	0.877688i	\(0.659085\pi\)
\(41\)	8.25119	−	1.64126i	1.28862	−	0.256322i	0.497221	−	0.867624i	\(-0.334354\pi\)
							0.791398	+	0.611302i	\(0.209354\pi\)
\(43\)	0.894942	−	1.67432i	0.136477	−	0.255331i	−0.804393	−	0.594097i	\(-0.797509\pi\)
							0.940871	+	0.338766i	\(0.110009\pi\)
\(47\)	−2.26955	−	5.47919i	−0.331048	−	0.799221i	−0.998510	−	0.0545769i	\(-0.982619\pi\)
							0.667461	−	0.744645i	\(-0.267381\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.155.61	yes	992
3.2	odd	2	inner	384.2.w.a.155.2	✓	992
128.19	odd	32	inner	384.2.w.a.275.2	yes	992
384.275	even	32	inner	384.2.w.a.275.61	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.155.2	✓	992	3.2	odd	2	inner
384.2.w.a.155.61	yes	992	1.1	even	1	trivial
384.2.w.a.275.2	yes	992	128.19	odd	32	inner
384.2.w.a.275.61	yes	992	384.275	even	32	inner