Embedded newform 384.2.w.a.227.22

• Label: 384.2.w.a.227.22
• Level: $384$
• Weight: $2$
• Character: 384.227
• 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			227.22
Character	\(\chi\)	\(=\)	384.227
Dual form			384.2.w.a.203.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.617502 - 1.27228i) q^{2} +(-0.948084 + 1.44953i) q^{3} +(-1.23738 + 1.57127i) q^{4} +(0.857825 + 1.60488i) q^{5} +(2.42965 + 0.311138i) q^{6} +(-1.64546 + 0.327303i) q^{7} +(2.76318 + 0.604033i) q^{8} +(-1.20227 - 2.74855i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{27}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.617502	−	1.27228i	−0.436640	−	0.899636i
							\(3\)	−0.948084	+	1.44953i	−0.547376	+	0.836887i
\(5\)	0.857825	+	1.60488i	0.383631	+	0.717723i								0.997423	−	0.0717448i	\(-0.0228567\pi\)
							−0.613792	+	0.789468i	\(0.710357\pi\)
\(7\)	−1.64546	+	0.327303i	−0.621927	+	0.123709i	−0.495988	−	0.868329i	\(-0.665194\pi\)
							−0.125938	+	0.992038i	\(0.540194\pi\)
\(11\)	2.14203	−	2.61007i	0.645846	−	0.786965i	−0.342446	−	0.939537i	\(-0.611256\pi\)
							0.988292	+	0.152572i	\(0.0487555\pi\)
\(13\)	−2.50305	+	4.68288i	−0.694222	+	1.29880i	0.249602	+	0.968348i	\(0.419700\pi\)
							−0.943824	+	0.330449i	\(0.892800\pi\)
\(17\)	−1.09689	−	0.454347i	−0.266035	−	0.110195i	0.245679	−	0.969351i	\(-0.420989\pi\)
							−0.511714	+	0.859156i	\(0.670989\pi\)
\(19\)	−8.04397	+	2.44011i	−1.84541	+	0.559800i	−0.845759	+	0.533566i	\(0.820852\pi\)
							−0.999656	+	0.0262345i	\(0.991648\pi\)
\(23\)	−6.42338	+	4.29197i	−1.33937	+	0.894937i	−0.998971	−	0.0453445i	\(-0.985561\pi\)
							−0.340397	+	0.940282i	\(0.610561\pi\)
\(29\)	1.18960	+	1.44953i	0.220903	+	0.269171i	0.871656	−	0.490119i	\(-0.163047\pi\)
							−0.650753	+	0.759290i	\(0.725547\pi\)
\(31\)	−5.85015	−	5.85015i	−1.05072	−	1.05072i	−0.998643	−	0.0520753i	\(-0.983416\pi\)
							−0.0520753	−	0.998643i	\(-0.516584\pi\)
\(37\)	6.63463	+	2.01259i	1.09073	+	0.330868i	0.783894	−	0.620895i	\(-0.213231\pi\)
							0.306833	+	0.951763i	\(0.400731\pi\)
\(41\)	−3.28266	+	2.19340i	−0.512665	+	0.342552i	−0.784843	−	0.619695i	\(-0.787256\pi\)
							0.272177	+	0.962247i	\(0.412256\pi\)
\(43\)	−8.90461	+	0.877028i	−1.35794	+	0.133745i	−0.750632	−	0.660721i	\(-0.770251\pi\)
							−0.607308	+	0.794466i	\(0.707751\pi\)
\(47\)	7.69592	+	3.18775i	1.12256	+	0.464982i	0.865247	−	0.501345i	\(-0.167161\pi\)
							0.257317	+	0.966327i	\(0.417161\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.227.22	yes	992
3.2	odd	2	inner	384.2.w.a.227.41	yes	992
128.75	odd	32	inner	384.2.w.a.203.41	yes	992
384.203	even	32	inner	384.2.w.a.203.22	✓	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.203.22	✓	992	384.203	even	32	inner
384.2.w.a.203.41	yes	992	128.75	odd	32	inner
384.2.w.a.227.22	yes	992	1.1	even	1	trivial
384.2.w.a.227.41	yes	992	3.2	odd	2	inner