Embedded newform 384.2.w.a.155.40

• Label: 384.2.w.a.155.40
• 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			155.40
Character	\(\chi\)	\(=\)	384.155
Dual form			384.2.w.a.275.40

$q$-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 \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{32}\right)\)	\(-1\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.155.23	✓	992	3.2	odd	2	inner
384.2.w.a.155.40	yes	992	1.1	even	1	trivial
384.2.w.a.275.23	yes	992	128.19	odd	32	inner
384.2.w.a.275.40	yes	992	384.275	even	32	inner