Embedded newform 384.2.w.a.155.41

• Label: 384.2.w.a.155.41
• 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			155.41
Character	\(\chi\)	\(=\)	384.155
Dual form			384.2.w.a.275.41

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.601247 - 1.28004i) q^{2} +(1.69881 + 0.337726i) q^{3} +(-1.27700 - 1.53924i) q^{4} +(1.73972 - 1.42775i) q^{5} +(1.45370 - 1.97148i) q^{6} +(-2.21518 - 3.31526i) q^{7} +(-2.73808 + 0.709154i) q^{8} +(2.77188 + 1.14746i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.601247	−	1.28004i	0.425146	−	0.905125i
							\(3\)	1.69881	+	0.337726i	0.980806	+	0.194986i
\(5\)	1.73972	−	1.42775i	0.778026	−	0.638509i								−0.159195	−	0.987247i	\(-0.550890\pi\)
							0.937220	+	0.348738i	\(0.113390\pi\)
\(7\)	−2.21518	−	3.31526i	−0.837261	−	1.25305i	−0.965264	−	0.261275i	\(-0.915857\pi\)
							0.128004	−	0.991774i	\(-0.459143\pi\)
\(11\)	−2.99520	+	0.908583i	−0.903086	+	0.273948i	−0.707479	−	0.706734i	\(-0.750168\pi\)
							−0.195607	+	0.980682i	\(0.562668\pi\)
\(13\)	1.35365	+	1.11091i	0.375434	+	0.308111i	0.803121	−	0.595816i	\(-0.203171\pi\)
							−0.427687	+	0.903927i	\(0.640671\pi\)
\(17\)	1.39278	+	3.36247i	0.337799	+	0.815519i	0.997926	+	0.0643655i	\(0.0205023\pi\)
							−0.660127	+	0.751154i	\(0.729498\pi\)
\(19\)	0.238967	−	0.0235362i	0.0548229	−	0.00539958i	−0.0705685	−	0.997507i	\(-0.522481\pi\)
							0.125391	+	0.992107i	\(0.459981\pi\)
\(23\)	2.14589	−	0.426843i	0.447448	−	0.0890030i	0.0337771	−	0.999429i	\(-0.489246\pi\)
							0.413671	+	0.910426i	\(0.364246\pi\)
\(29\)	6.57751	+	1.99526i	1.22141	+	0.370511i	0.834191	−	0.551475i	\(-0.185935\pi\)
							0.387221	+	0.921987i	\(0.373435\pi\)
\(31\)	4.97558	−	4.97558i	0.893640	−	0.893640i	−0.101224	−	0.994864i	\(-0.532276\pi\)
							0.994864	+	0.101224i	\(0.0322758\pi\)
\(37\)	−7.54619	−	0.743235i	−1.24059	−	0.122187i	−0.543645	−	0.839315i	\(-0.682956\pi\)
							−0.696941	+	0.717128i	\(0.745456\pi\)
\(41\)	−0.161061	+	0.0320370i	−0.0251535	+	0.00500334i	−0.207651	−	0.978203i	\(-0.566582\pi\)
							0.182498	+	0.983206i	\(0.441582\pi\)
\(43\)	6.07652	−	11.3684i	0.926660	−	1.73366i	0.300876	−	0.953663i	\(-0.402721\pi\)
							0.625784	−	0.779996i	\(-0.284779\pi\)
\(47\)	3.66510	+	8.84835i	0.534610	+	1.29066i	0.928441	+	0.371481i	\(0.121150\pi\)
							−0.393830	+	0.919183i	\(0.628850\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.155.41	yes	992
3.2	odd	2	inner	384.2.w.a.155.22	✓	992
128.19	odd	32	inner	384.2.w.a.275.22	yes	992
384.275	even	32	inner	384.2.w.a.275.41	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.155.22	✓	992	3.2	odd	2	inner
384.2.w.a.155.41	yes	992	1.1	even	1	trivial
384.2.w.a.275.22	yes	992	128.19	odd	32	inner
384.2.w.a.275.41	yes	992	384.275	even	32	inner