Embedded newform 384.2.w.a.275.40

• Label: 384.2.w.a.275.40
• Level: $384$
• Weight: $2$
• Character: 384.275
• 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			275.40
Character	\(\chi\)	\(=\)	384.275
Dual form			384.2.w.a.155.40

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.503011 - 1.32173i) q^{2} +(-1.58078 - 0.707917i) q^{3} +(-1.49396 - 1.32969i) q^{4} +(-2.77004 - 2.27332i) q^{5} +(-1.73083 + 1.73327i) q^{6} +(1.08926 - 1.63019i) q^{7} +(-2.50898 + 1.30577i) q^{8} +(1.99771 + 2.23812i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{23}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.503011	−	1.32173i	0.355682	−	0.934607i
							\(3\)	−1.58078	−	0.707917i	−0.912661	−	0.408716i
\(5\)	−2.77004	−	2.27332i	−1.23880	−	1.01666i								−0.998948	−	0.0458488i	\(-0.985401\pi\)
							−0.239853	−	0.970809i	\(-0.577099\pi\)
\(7\)	1.08926	−	1.63019i	0.411702	−	0.616155i	−0.566438	−	0.824104i	\(-0.691679\pi\)
							0.978140	+	0.207949i	\(0.0666789\pi\)
\(11\)	0.773514	+	0.234643i	0.233223	+	0.0707475i	0.404733	−	0.914435i	\(-0.367365\pi\)
							−0.171510	+	0.985182i	\(0.554865\pi\)
\(13\)	1.37777	−	1.13071i	0.382125	−	0.313602i	−0.423647	−	0.905827i	\(-0.639250\pi\)
							0.805773	+	0.592225i	\(0.201750\pi\)
\(17\)	−1.91039	+	4.61208i	−0.463337	+	1.11859i	0.503682	+	0.863889i	\(0.331978\pi\)
							−0.967019	+	0.254705i	\(0.918022\pi\)
\(19\)	−4.68594	−	0.461525i	−1.07503	−	0.105881i	−0.455024	−	0.890479i	\(-0.650369\pi\)
							−0.620005	+	0.784598i	\(0.712869\pi\)
\(23\)	−1.09942	−	0.218689i	−0.229246	−	0.0455998i	0.0791306	−	0.996864i	\(-0.474786\pi\)
							−0.308376	+	0.951264i	\(0.599786\pi\)
\(29\)	8.16321	−	2.47628i	1.51587	−	0.459834i	0.580725	−	0.814100i	\(-0.302769\pi\)
							0.935145	+	0.354266i	\(0.115269\pi\)
\(31\)	−3.89061	−	3.89061i	−0.698775	−	0.698775i	0.265371	−	0.964146i	\(-0.414505\pi\)
							−0.964146	+	0.265371i	\(0.914505\pi\)
\(37\)	−6.27380	+	0.617915i	−1.03141	+	0.101585i	−0.599533	−	0.800350i	\(-0.704647\pi\)
							−0.431873	+	0.901934i	\(0.642147\pi\)
\(41\)	−7.63277	−	1.51825i	−1.19204	−	0.237111i	−0.441085	−	0.897465i	\(-0.645406\pi\)
							−0.750954	+	0.660354i	\(0.770406\pi\)
\(43\)	−5.80320	−	10.8570i	−0.884980	−	1.65568i	−0.746570	−	0.665307i	\(-0.768301\pi\)
							−0.138410	−	0.990375i	\(-0.544199\pi\)
\(47\)	1.62661	−	3.92699i	0.237266	−	0.572810i	−0.759732	−	0.650236i	\(-0.774670\pi\)
							0.996998	+	0.0774256i	\(0.0246700\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.275.40	yes	992
3.2	odd	2	inner	384.2.w.a.275.23	yes	992
128.27	odd	32	inner	384.2.w.a.155.23	✓	992
384.155	even	32	inner	384.2.w.a.155.40	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.155.23	✓	992	128.27	odd	32	inner
384.2.w.a.155.40	yes	992	384.155	even	32	inner
384.2.w.a.275.23	yes	992	3.2	odd	2	inner
384.2.w.a.275.40	yes	992	1.1	even	1	trivial