Embedded newform 384.2.w.a.107.6

• Label: 384.2.w.a.107.6
• Level: $384$
• Weight: $2$
• Character: 384.107
• 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			107.6
Character	\(\chi\)	\(=\)	384.107
Dual form			384.2.w.a.323.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34824 - 0.426895i) q^{2} +(0.438452 - 1.67564i) q^{3} +(1.63552 + 1.15112i) q^{4} +(1.15753 - 0.351132i) q^{5} +(-1.30646 + 2.07199i) q^{6} +(0.316363 - 1.59046i) q^{7} +(-1.71367 - 2.25018i) q^{8} +(-2.61552 - 1.46937i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{13}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34824	−	0.426895i	−0.953352	−	0.301861i
							\(3\)	0.438452	−	1.67564i	0.253140	−	0.967430i
\(5\)	1.15753	−	0.351132i	0.517662	−	0.157031i								−0.0206339	−	0.999787i	\(-0.506568\pi\)
							0.538295	+	0.842756i	\(0.319068\pi\)
\(7\)	0.316363	−	1.59046i	0.119574	−	0.601139i	−0.873807	−	0.486273i	\(-0.838356\pi\)
							0.993381	−	0.114866i	\(-0.0366439\pi\)
\(11\)	0.0667361	−	0.677583i	0.0201217	−	0.204299i	−0.979867	−	0.199649i	\(-0.936020\pi\)
							0.999989	−	0.00465004i	\(-0.00148016\pi\)
\(13\)	−1.42420	−	0.432026i	−0.395002	−	0.119823i	0.0865339	−	0.996249i	\(-0.472421\pi\)
							−0.481536	+	0.876426i	\(0.659921\pi\)
\(17\)	1.97876	−	0.819630i	0.479920	−	0.198790i	−0.129590	−	0.991568i	\(-0.541366\pi\)
							0.609510	+	0.792778i	\(0.291366\pi\)
\(19\)	2.76964	−	1.48041i	0.635400	−	0.339628i	−0.122048	−	0.992524i	\(-0.538946\pi\)
							0.757448	+	0.652896i	\(0.226446\pi\)
\(23\)	3.71250	−	5.55615i	0.774110	−	1.15854i	−0.209426	−	0.977824i	\(-0.567160\pi\)
							0.983536	−	0.180713i	\(-0.0578404\pi\)
\(29\)	−0.000620160	−	0.00629659i	−0.000115161	−	0.00116925i	0.995127	−	0.0986017i	\(-0.0314370\pi\)
							−0.995242	+	0.0974325i	\(0.968937\pi\)
\(31\)	−0.512370	+	0.512370i	−0.0920244	+	0.0920244i	−0.751620	−	0.659596i	\(-0.770727\pi\)
							0.659596	+	0.751620i	\(0.270727\pi\)
\(37\)	−8.88448	−	4.74885i	−1.46060	−	0.780707i	−0.466029	−	0.884770i	\(-0.654316\pi\)
							−0.994571	+	0.104063i	\(0.966816\pi\)
\(41\)	−4.32021	+	6.46565i	−0.674704	+	1.00977i	0.323281	+	0.946303i	\(0.395214\pi\)
							−0.997984	+	0.0634622i	\(0.979786\pi\)
\(43\)	−3.35505	−	4.08814i	−0.511640	−	0.623435i	0.451553	−	0.892244i	\(-0.350870\pi\)
							−0.963194	+	0.268809i	\(0.913370\pi\)
\(47\)	4.53151	−	1.87701i	0.660989	−	0.273791i	−0.0268657	−	0.999639i	\(-0.508553\pi\)
							0.687855	+	0.725848i	\(0.258553\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.107.6	✓	992
3.2	odd	2	inner	384.2.w.a.107.57	yes	992
128.67	odd	32	inner	384.2.w.a.323.57	yes	992
384.323	even	32	inner	384.2.w.a.323.6	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.107.6	✓	992	1.1	even	1	trivial
384.2.w.a.107.57	yes	992	3.2	odd	2	inner
384.2.w.a.323.6	yes	992	384.323	even	32	inner
384.2.w.a.323.57	yes	992	128.67	odd	32	inner