Embedded newform 384.2.w.a.155.27

• Label: 384.2.w.a.155.27
• 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			155.27
Character	\(\chi\)	\(=\)	384.155
Dual form			384.2.w.a.275.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.256580 + 1.39074i) q^{2} +(1.07322 - 1.35948i) q^{3} +(-1.86833 - 0.713673i) q^{4} +(-2.89270 + 2.37398i) q^{5} +(1.61532 + 1.84139i) q^{6} +(-0.529368 - 0.792255i) q^{7} +(1.47191 - 2.41526i) q^{8} +(-0.696387 - 2.91806i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.256580	+	1.39074i	−0.181429	+	0.983404i
							\(3\)	1.07322	−	1.35948i	0.619625	−	0.784898i
\(5\)	−2.89270	+	2.37398i	−1.29366	+	1.06168i								−0.299459	+	0.954109i	\(0.596806\pi\)
							−0.994198	+	0.107567i	\(0.965694\pi\)
\(7\)	−0.529368	−	0.792255i	−0.200082	−	0.299444i	0.717837	−	0.696211i	\(-0.245132\pi\)
							−0.917920	+	0.396767i	\(0.870132\pi\)
\(11\)	−2.28790	+	0.694028i	−0.689829	+	0.209257i	−0.615687	−	0.787990i	\(-0.711122\pi\)
							−0.0741414	+	0.997248i	\(0.523622\pi\)
\(13\)	−5.12067	−	4.20242i	−1.42022	−	1.16554i	−0.960773	−	0.277335i	\(-0.910549\pi\)
							−0.459444	−	0.888207i	\(-0.651951\pi\)
\(17\)	0.0824864	+	0.199140i	0.0200059	+	0.0482985i	0.933567	−	0.358403i	\(-0.116679\pi\)
							−0.913561	+	0.406702i	\(0.866679\pi\)
\(19\)	−1.72717	+	0.170112i	−0.396240	+	0.0390263i	−0.294175	−	0.955751i	\(-0.595045\pi\)
							−0.102065	+	0.994778i	\(0.532545\pi\)
\(23\)	2.10427	−	0.418565i	0.438770	−	0.0872768i	0.0292378	−	0.999572i	\(-0.490692\pi\)
							0.409532	+	0.912296i	\(0.365692\pi\)
\(29\)	−1.48234	−	0.449662i	−0.275263	−	0.0835001i	0.149635	−	0.988741i	\(-0.452190\pi\)
							−0.424898	+	0.905241i	\(0.639690\pi\)
\(31\)	−6.65319	+	6.65319i	−1.19495	+	1.19495i	−0.219287	+	0.975660i	\(0.570373\pi\)
							−0.975660	+	0.219287i	\(0.929627\pi\)
\(37\)	−1.57148	−	0.154777i	−0.258349	−	0.0254452i	−0.0319866	−	0.999488i	\(-0.510183\pi\)
							−0.226362	+	0.974043i	\(0.572683\pi\)
\(41\)	−0.579605	+	0.115291i	−0.0905190	+	0.0180054i	−0.240142	−	0.970738i	\(-0.577194\pi\)
							0.149623	+	0.988743i	\(0.452194\pi\)
\(43\)	4.89019	−	9.14891i	0.745748	−	1.39520i	−0.166702	−	0.986007i	\(-0.553312\pi\)
							0.912450	−	0.409188i	\(-0.134188\pi\)
\(47\)	1.41857	+	3.42473i	0.206920	+	0.499549i	0.992935	−	0.118658i	\(-0.0378593\pi\)
							−0.786015	+	0.618207i	\(0.787859\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.155.27	✓	992
3.2	odd	2	inner	384.2.w.a.155.36	yes	992
128.19	odd	32	inner	384.2.w.a.275.36	yes	992
384.275	even	32	inner	384.2.w.a.275.27	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.155.27	✓	992	1.1	even	1	trivial
384.2.w.a.155.36	yes	992	3.2	odd	2	inner
384.2.w.a.275.27	yes	992	384.275	even	32	inner
384.2.w.a.275.36	yes	992	128.19	odd	32	inner