Embedded newform 384.2.w.a.107.3

• Label: 384.2.w.a.107.3
• Level: $384$
• Weight: $2$
• Character: 384.107
• 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			107.3
Character	\(\chi\)	\(=\)	384.107
Dual form			384.2.w.a.323.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40130 + 0.190668i) q^{2} +(1.64387 + 0.545607i) q^{3} +(1.92729 - 0.534366i) q^{4} +(-0.951600 + 0.288665i) q^{5} +(-2.40759 - 0.451127i) q^{6} +(0.819407 - 4.11944i) q^{7} +(-2.59883 + 1.11628i) q^{8} +(2.40463 + 1.79382i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{13}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40130	+	0.190668i	−0.990870	+	0.134822i
							\(3\)	1.64387	+	0.545607i	0.949090	+	0.315006i
\(5\)	−0.951600	+	0.288665i	−0.425568	+	0.129095i								−0.495802	−	0.868436i	\(-0.665126\pi\)
							0.0702332	+	0.997531i	\(0.477626\pi\)
\(7\)	0.819407	−	4.11944i	0.309707	−	1.55700i	−0.441700	−	0.897163i	\(-0.645624\pi\)
							0.751407	−	0.659839i	\(-0.229376\pi\)
\(11\)	0.569368	−	5.78089i	0.171671	−	1.74300i	−0.394662	−	0.918827i	\(-0.629138\pi\)
							0.566333	−	0.824177i	\(-0.308362\pi\)
\(13\)	−3.84905	−	1.16760i	−1.06753	−	0.323833i	−0.292835	−	0.956163i	\(-0.594599\pi\)
							−0.774699	+	0.632330i	\(0.782099\pi\)
\(17\)	2.30930	−	0.956545i	0.560088	−	0.231996i	−0.0846352	−	0.996412i	\(-0.526972\pi\)
							0.644724	+	0.764416i	\(0.276972\pi\)
\(19\)	5.02117	−	2.68387i	1.15194	−	0.615722i	0.218985	−	0.975728i	\(-0.429725\pi\)
							0.932950	+	0.360006i	\(0.117225\pi\)
\(23\)	−3.44880	+	5.16149i	−0.719124	+	1.07624i	0.274288	+	0.961648i	\(0.411558\pi\)
							−0.993412	+	0.114597i	\(0.963442\pi\)
\(29\)	−0.184182	−	1.87003i	−0.0342017	−	0.347256i	−0.997022	−	0.0771140i	\(-0.975429\pi\)
							0.962821	−	0.270142i	\(-0.0870705\pi\)
\(31\)	−1.94970	+	1.94970i	−0.350177	+	0.350177i	−0.860175	−	0.509999i	\(-0.829646\pi\)
							0.509999	+	0.860175i	\(0.329646\pi\)
\(37\)	10.0177	+	5.35456i	1.64689	+	0.880284i	0.992237	+	0.124358i	\(0.0396871\pi\)
							0.654657	+	0.755926i	\(0.272813\pi\)
\(41\)	1.50639	−	2.25448i	0.235259	−	0.352090i	−0.694990	−	0.719020i	\(-0.744591\pi\)
							0.930249	+	0.366930i	\(0.119591\pi\)
\(43\)	2.53690	+	3.09123i	0.386874	+	0.471408i	0.929636	−	0.368479i	\(-0.120121\pi\)
							−0.542762	+	0.839887i	\(0.682621\pi\)
\(47\)	2.44237	−	1.01166i	0.356256	−	0.147566i	−0.197375	−	0.980328i	\(-0.563242\pi\)
							0.553631	+	0.832762i	\(0.313242\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.107.3	✓	992
3.2	odd	2	inner	384.2.w.a.107.60	yes	992
128.67	odd	32	inner	384.2.w.a.323.60	yes	992
384.323	even	32	inner	384.2.w.a.323.3	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.107.3	✓	992	1.1	even	1	trivial
384.2.w.a.107.60	yes	992	3.2	odd	2	inner
384.2.w.a.323.3	yes	992	384.323	even	32	inner
384.2.w.a.323.60	yes	992	128.67	odd	32	inner