Embedded newform 384.2.w.a.179.10

• Label: 384.2.w.a.179.10
• Level: $384$
• Weight: $2$
• Character: 384.179
• 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			179.10
Character	\(\chi\)	\(=\)	384.179
Dual form			384.2.w.a.251.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28535 + 0.589818i) q^{2} +(1.67971 + 0.422569i) q^{3} +(1.30423 - 1.51624i) q^{4} +(-0.286907 - 2.91302i) q^{5} +(-2.40825 + 0.447577i) q^{6} +(0.492319 + 0.328957i) q^{7} +(-0.782083 + 2.71815i) q^{8} +(2.64287 + 1.41959i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{31}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28535	+	0.589818i	−0.908877	+	0.417064i
							\(3\)	1.67971	+	0.422569i	0.969783	+	0.243970i
\(5\)	−0.286907	−	2.91302i	−0.128309	−	1.30274i								−0.816557	−	0.577265i	\(-0.804120\pi\)
							0.688248	−	0.725475i	\(-0.258380\pi\)
\(7\)	0.492319	+	0.328957i	0.186079	+	0.124334i	0.645121	−	0.764081i	\(-0.276807\pi\)
							−0.459042	+	0.888415i	\(0.651807\pi\)
\(11\)	−0.718313	−	1.34387i	−0.216579	−	0.405192i	0.750046	−	0.661386i	\(-0.230031\pi\)
							−0.966625	+	0.256194i	\(0.917531\pi\)
\(13\)	−0.558452	+	5.67006i	−0.154887	+	1.57259i	0.532583	+	0.846378i	\(0.321222\pi\)
							−0.687469	+	0.726213i	\(0.741278\pi\)
\(17\)	2.49299	−	6.01860i	0.604638	−	1.45972i	−0.264121	−	0.964489i	\(-0.585082\pi\)
							0.868759	−	0.495235i	\(-0.164918\pi\)
\(19\)	6.66165	−	5.46708i	1.52829	−	1.25423i	0.668334	−	0.743862i	\(-0.267008\pi\)
							0.859954	−	0.510372i	\(-0.170492\pi\)
\(23\)	0.296389	−	1.49005i	0.0618014	−	0.310697i	−0.937503	−	0.347976i	\(-0.886869\pi\)
							0.999305	+	0.0372793i	\(0.0118691\pi\)
\(29\)	−1.08636	+	2.03244i	−0.201733	+	0.377415i	−0.962362	−	0.271771i	\(-0.912391\pi\)
							0.760629	+	0.649187i	\(0.224891\pi\)
\(31\)	−4.65095	−	4.65095i	−0.835335	−	0.835335i	0.152906	−	0.988241i	\(-0.451137\pi\)
							−0.988241	+	0.152906i	\(0.951137\pi\)
\(37\)	2.11790	+	1.73811i	0.348180	+	0.285744i	0.792186	−	0.610280i	\(-0.208943\pi\)
							−0.444006	+	0.896024i	\(0.646443\pi\)
\(41\)	−1.21104	+	6.08832i	−0.189133	+	0.950835i	0.763290	+	0.646056i	\(0.223583\pi\)
							−0.952423	+	0.304779i	\(0.901417\pi\)
\(43\)	1.02142	−	3.36717i	0.155765	−	0.513489i	−0.843973	−	0.536386i	\(-0.819789\pi\)
							0.999738	+	0.0228974i	\(0.00728910\pi\)
\(47\)	−2.18867	+	5.28391i	−0.319250	+	0.770738i	0.680044	+	0.733171i	\(0.261961\pi\)
							−0.999294	+	0.0375665i	\(0.988039\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.179.10	✓	992
3.2	odd	2	inner	384.2.w.a.179.53	yes	992
128.123	odd	32	inner	384.2.w.a.251.53	yes	992
384.251	even	32	inner	384.2.w.a.251.10	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.179.10	✓	992	1.1	even	1	trivial
384.2.w.a.179.53	yes	992	3.2	odd	2	inner
384.2.w.a.251.10	yes	992	384.251	even	32	inner
384.2.w.a.251.53	yes	992	128.123	odd	32	inner