Embedded newform 384.2.w.a.299.5

• Label: 384.2.w.a.299.5
• Level: $384$
• Weight: $2$
• Character: 384.299
• 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			299.5
Character	\(\chi\)	\(=\)	384.299
Dual form			384.2.w.a.131.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37910 + 0.313196i) q^{2} +(-1.73187 + 0.0251802i) q^{3} +(1.80382 - 0.863856i) q^{4} +(-0.847138 - 2.79264i) q^{5} +(2.38053 - 0.577140i) q^{6} +(0.874231 - 4.39506i) q^{7} +(-2.21708 + 1.75629i) q^{8} +(2.99873 - 0.0872176i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{29}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37910	+	0.313196i	−0.975169	+	0.221463i
							\(3\)	−1.73187	+	0.0251802i	−0.999894	+	0.0145378i
\(5\)	−0.847138	−	2.79264i	−0.378852	−	1.24891i								−0.914542	−	0.404492i	\(-0.867448\pi\)
							0.535690	−	0.844415i	\(-0.320052\pi\)
\(7\)	0.874231	−	4.39506i	0.330428	−	1.66118i	−0.356383	−	0.934340i	\(-0.615990\pi\)
							0.686811	−	0.726836i	\(-0.259010\pi\)
\(11\)	0.263965	+	0.0259983i	0.0795886	+	0.00783879i	0.137733	−	0.990469i	\(-0.456018\pi\)
							−0.0581445	+	0.998308i	\(0.518518\pi\)
\(13\)	−1.43291	+	4.72366i	−0.397417	+	1.31011i	0.498887	+	0.866667i	\(0.333742\pi\)
							−0.896304	+	0.443440i	\(0.853758\pi\)
\(17\)	−5.15360	+	2.13469i	−1.24993	+	0.517738i	−0.906805	−	0.421551i	\(-0.861486\pi\)
							−0.343126	+	0.939289i	\(0.611486\pi\)
\(19\)	−2.20182	−	4.11931i	−0.505131	−	0.945034i	−0.997222	−	0.0744898i	\(-0.976267\pi\)
							0.492091	−	0.870544i	\(-0.336233\pi\)
\(23\)	3.95476	−	5.91872i	0.824624	−	1.23414i	−0.144974	−	0.989435i	\(-0.546310\pi\)
							0.969598	−	0.244702i	\(-0.0786902\pi\)
\(29\)	0.738285	−	0.0727147i	0.137096	−	0.0135028i	−0.0292363	−	0.999573i	\(-0.509308\pi\)
							0.166332	+	0.986070i	\(0.446808\pi\)
\(31\)	−1.24357	+	1.24357i	−0.223352	+	0.223352i	−0.809908	−	0.586557i	\(-0.800483\pi\)
							0.586557	+	0.809908i	\(0.300483\pi\)
\(37\)	−0.796534	+	1.49021i	−0.130949	+	0.244989i	−0.938848	−	0.344333i	\(-0.888105\pi\)
							0.807898	+	0.589322i	\(0.200605\pi\)
\(41\)	0.0428556	−	0.0641379i	0.00669291	−	0.0100167i	−0.828108	−	0.560568i	\(-0.810583\pi\)
							0.834801	+	0.550552i	\(0.185583\pi\)
\(43\)	−8.82340	+	7.24118i	−1.34556	+	1.10427i	−0.360365	+	0.932811i	\(0.617348\pi\)
							−0.985191	+	0.171458i	\(0.945152\pi\)
\(47\)	−0.670700	+	0.277813i	−0.0978317	+	0.0405232i	−0.431062	−	0.902322i	\(-0.641861\pi\)
							0.333231	+	0.942845i	\(0.391861\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.299.5	yes	992
3.2	odd	2	inner	384.2.w.a.299.58	yes	992
128.3	odd	32	inner	384.2.w.a.131.58	yes	992
384.131	even	32	inner	384.2.w.a.131.5	✓	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.131.5	✓	992	384.131	even	32	inner
384.2.w.a.131.58	yes	992	128.3	odd	32	inner
384.2.w.a.299.5	yes	992	1.1	even	1	trivial
384.2.w.a.299.58	yes	992	3.2	odd	2	inner