Embedded newform 384.2.w.a.11.2

• Label: 384.2.w.a.11.2
• Level: $384$
• Weight: $2$
• Character: 384.11
• 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			11.2
Character	\(\chi\)	\(=\)	384.11
Dual form			384.2.w.a.35.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41347 + 0.0458784i) q^{2} +(-0.204040 - 1.71999i) q^{3} +(1.99579 - 0.129695i) q^{4} +(-0.561681 - 0.300225i) q^{5} +(0.367315 + 2.42179i) q^{6} +(-0.946225 - 0.188216i) q^{7} +(-2.81504 + 0.274884i) q^{8} +(-2.91674 + 0.701894i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{21}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41347	+	0.0458784i	−0.999474	+	0.0324409i
							\(3\)	−0.204040	−	1.71999i	−0.117803	−	0.993037i
\(5\)	−0.561681	−	0.300225i	−0.251191	−	0.134265i								0.340996	−	0.940065i	\(-0.389236\pi\)
							−0.592188	+	0.805800i	\(0.701736\pi\)
\(7\)	−0.946225	−	0.188216i	−0.357640	−	0.0711389i	0.0130006	−	0.999915i	\(-0.495862\pi\)
							−0.370640	+	0.928777i	\(0.620862\pi\)
\(11\)	−4.33792	+	3.56004i	−1.30793	+	1.07339i	−0.315719	+	0.948853i	\(0.602246\pi\)
							−0.992215	+	0.124540i	\(0.960254\pi\)
\(13\)	−1.08354	+	0.579164i	−0.300520	+	0.160631i	−0.614764	−	0.788711i	\(-0.710749\pi\)
							0.314244	+	0.949342i	\(0.398249\pi\)
\(17\)	2.91472	−	1.20732i	0.706924	−	0.292818i	−0.000106898	−	1.00000i	\(-0.500034\pi\)
							0.707031	+	0.707182i	\(0.250034\pi\)
\(19\)	−0.764121	+	2.51897i	−0.175301	+	0.577891i	0.824624	+	0.565681i	\(0.191387\pi\)
							−0.999925	+	0.0122099i	\(0.996113\pi\)
\(23\)	−3.58426	−	2.39493i	−0.747371	−	0.499377i	0.122615	−	0.992454i	\(-0.460872\pi\)
							−0.869986	+	0.493077i	\(0.835872\pi\)
\(29\)	4.42389	+	3.63059i	0.821495	+	0.674183i	0.948171	−	0.317761i	\(-0.102931\pi\)
							−0.126676	+	0.991944i	\(0.540431\pi\)
\(31\)	−5.09112	+	5.09112i	−0.914392	+	0.914392i	−0.996614	−	0.0822222i	\(-0.973798\pi\)
							0.0822222	+	0.996614i	\(0.473798\pi\)
\(37\)	2.98132	+	9.82808i	0.490125	+	1.61573i	0.757987	+	0.652270i	\(0.226183\pi\)
							−0.267862	+	0.963457i	\(0.586317\pi\)
\(41\)	−6.17366	−	4.12511i	−0.964163	−	0.644233i	−0.0294238	−	0.999567i	\(-0.509367\pi\)
							−0.934740	+	0.355334i	\(0.884367\pi\)
\(43\)	0.337787	−	3.42961i	0.0515121	−	0.523011i	−0.934456	−	0.356078i	\(-0.884114\pi\)
							0.985968	−	0.166933i	\(-0.0533863\pi\)
\(47\)	−5.09963	+	2.11234i	−0.743858	+	0.308116i	−0.722233	−	0.691650i	\(-0.756884\pi\)
							−0.0216253	+	0.999766i	\(0.506884\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.11.2	✓	992
3.2	odd	2	inner	384.2.w.a.11.61	yes	992
128.35	odd	32	inner	384.2.w.a.35.61	yes	992
384.35	even	32	inner	384.2.w.a.35.2	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.11.2	✓	992	1.1	even	1	trivial
384.2.w.a.11.61	yes	992	3.2	odd	2	inner
384.2.w.a.35.2	yes	992	384.35	even	32	inner
384.2.w.a.35.61	yes	992	128.35	odd	32	inner