Embedded newform 384.2.w.a.35.19

• Label: 384.2.w.a.35.19
• Level: $384$
• Weight: $2$
• Character: 384.35
• 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			35.19
Character	\(\chi\)	\(=\)	384.35
Dual form			384.2.w.a.11.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.845116 + 1.13392i) q^{2} +(-0.765283 - 1.55382i) q^{3} +(-0.571557 - 1.91659i) q^{4} +(-0.974296 + 0.520772i) q^{5} +(2.40866 + 0.445384i) q^{6} +(-1.27617 + 0.253846i) q^{7} +(2.65630 + 0.971642i) q^{8} +(-1.82869 + 2.37822i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.845116	+	1.13392i	−0.597587	+	0.801804i
							\(3\)	−0.765283	−	1.55382i	−0.441836	−	0.897096i
\(5\)	−0.974296	+	0.520772i	−0.435719	+	0.232896i								−0.674647	−	0.738141i	\(-0.735704\pi\)
							0.238928	+	0.971037i	\(0.423204\pi\)
\(7\)	−1.27617	+	0.253846i	−0.482347	+	0.0959447i	−0.430274	−	0.902699i	\(-0.641583\pi\)
							−0.0520732	+	0.998643i	\(0.516583\pi\)
\(11\)	−2.45402	−	2.01397i	−0.739916	−	0.607233i	0.187062	−	0.982348i	\(-0.440103\pi\)
							−0.926978	+	0.375115i	\(0.877603\pi\)
\(13\)	2.24004	+	1.19733i	0.621276	+	0.332079i	0.751825	−	0.659363i	\(-0.229174\pi\)
							−0.130549	+	0.991442i	\(0.541674\pi\)
\(17\)	5.66801	+	2.34777i	1.37469	+	0.569417i	0.943057	−	0.332631i	\(-0.107936\pi\)
							0.431637	+	0.902048i	\(0.357936\pi\)
\(19\)	1.05544	+	3.47933i	0.242136	+	0.798214i	0.990808	+	0.135274i	\(0.0431915\pi\)
							−0.748673	+	0.662940i	\(0.769308\pi\)
\(23\)	−7.00650	+	4.68160i	−1.46096	+	0.976180i	−0.465099	+	0.885259i	\(0.653981\pi\)
							−0.995858	+	0.0909212i	\(0.971019\pi\)
\(29\)	−2.10632	+	1.72861i	−0.391134	+	0.320995i	−0.809322	−	0.587365i	\(-0.800165\pi\)
							0.418188	+	0.908360i	\(0.362665\pi\)
\(31\)	6.37098	+	6.37098i	1.14426	+	1.14426i	0.987662	+	0.156599i	\(0.0500530\pi\)
							0.156599	+	0.987662i	\(0.449947\pi\)
\(37\)	−1.88396	+	6.21057i	−0.309721	+	1.02101i	0.654534	+	0.756033i	\(0.272865\pi\)
							−0.964254	+	0.264979i	\(0.914635\pi\)
\(41\)	0.168224	−	0.112403i	0.0262721	−	0.0175545i	−0.542365	−	0.840143i	\(-0.682471\pi\)
							0.568637	+	0.822589i	\(0.307471\pi\)
\(43\)	−1.21655	−	12.3519i	−0.185523	−	1.88364i	−0.414264	−	0.910157i	\(-0.635961\pi\)
							0.228741	−	0.973487i	\(-0.426539\pi\)
\(47\)	−3.43982	−	1.42482i	−0.501750	−	0.207832i	0.117429	−	0.993081i	\(-0.462535\pi\)
							−0.619179	+	0.785250i	\(0.712535\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.35.19	yes	992
3.2	odd	2	inner	384.2.w.a.35.44	yes	992
128.11	odd	32	inner	384.2.w.a.11.44	yes	992
384.11	even	32	inner	384.2.w.a.11.19	✓	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.19	✓	992	384.11	even	32	inner
384.2.w.a.11.44	yes	992	128.11	odd	32	inner
384.2.w.a.35.19	yes	992	1.1	even	1	trivial
384.2.w.a.35.44	yes	992	3.2	odd	2	inner