Embedded newform 96.3.m.a.19.13

• Label: 96.3.m.a.19.13
• Level: $96$
• Weight: $3$
• Character: 96.19
• Analytic conductor: $2.616$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 96 = 2^{5} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	96.m (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.61581053786\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			19.13
Character	\(\chi\)	\(=\)	96.19
Dual form			96.3.m.a.91.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.32170 - 1.50104i) q^{2} +(1.60021 + 0.662827i) q^{3} +(-0.506212 - 3.96784i) q^{4} +(2.17599 - 0.901324i) q^{5} +(3.10992 - 1.52591i) q^{6} +(0.825541 + 0.825541i) q^{7} +(-6.62493 - 4.48446i) q^{8} +(2.12132 + 2.12132i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(37\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{8}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.32170	−	1.50104i	0.660851	−	0.750518i
							\(3\)	1.60021	+	0.662827i	0.533402	+	0.220942i
\(5\)	2.17599	−	0.901324i	0.435198	−	0.180265i								−0.154319	−	0.988021i	\(-0.549318\pi\)
							0.589516	+	0.807756i	\(0.299318\pi\)
\(7\)	0.825541	+	0.825541i	0.117934	+	0.117934i	0.763611	−	0.645677i	\(-0.223425\pi\)
							−0.645677	+	0.763611i	\(0.723425\pi\)
\(11\)	−2.70334	+	1.11976i	−0.245759	+	0.101797i	−0.502163	−	0.864773i	\(-0.667462\pi\)
							0.256404	+	0.966570i	\(0.417462\pi\)
\(13\)	4.09036	+	1.69428i	0.314643	+	0.130329i	0.534416	−	0.845222i	\(-0.320532\pi\)
							−0.219773	+	0.975551i	\(0.570532\pi\)
\(17\)		−	3.54559i		−	0.208564i	−0.994548	−	0.104282i	\(-0.966746\pi\)
							0.994548	−	0.104282i	\(-0.0332544\pi\)
\(19\)	−9.76692	+	23.5794i	−0.514048	+	1.24102i	0.427461	+	0.904034i	\(0.359408\pi\)
							−0.941509	+	0.336988i	\(0.890592\pi\)
\(23\)	−2.48098	+	2.48098i	−0.107869	+	0.107869i	−0.758981	−	0.651112i	\(-0.774303\pi\)
							0.651112	+	0.758981i	\(0.274303\pi\)
\(29\)	14.4543	−	34.8957i	0.498423	−	1.20330i	−0.451910	−	0.892064i	\(-0.649257\pi\)
							0.950333	−	0.311236i	\(-0.100743\pi\)
\(31\)			40.3729i			1.30235i	0.758928	+	0.651175i	\(0.225724\pi\)
							−0.758928	+	0.651175i	\(0.774276\pi\)
\(37\)	21.5761	−	8.93713i	0.583139	−	0.241544i	−0.0715569	−	0.997437i	\(-0.522797\pi\)
							0.654696	+	0.755892i	\(0.272797\pi\)
\(41\)	13.9896	+	13.9896i	0.341210	+	0.341210i	0.856822	−	0.515612i	\(-0.172435\pi\)
							−0.515612	+	0.856822i	\(0.672435\pi\)
\(43\)	−64.8172	+	26.8482i	−1.50738	+	0.624376i	−0.975015	−	0.222141i	\(-0.928695\pi\)
							−0.532362	+	0.846517i	\(0.678695\pi\)
\(47\)	44.6533			0.950071			0.475035	−	0.879967i	\(-0.342435\pi\)
							0.475035	+	0.879967i	\(0.342435\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.3.m.a.19.13	✓	64
3.2	odd	2		288.3.u.b.19.4		64
4.3	odd	2		384.3.m.a.367.5		64
32.5	even	8		384.3.m.a.271.5		64
32.27	odd	8	inner	96.3.m.a.91.13	yes	64
96.59	even	8		288.3.u.b.91.4		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.m.a.19.13	✓	64	1.1	even	1	trivial
96.3.m.a.91.13	yes	64	32.27	odd	8	inner
288.3.u.b.19.4		64	3.2	odd	2
288.3.u.b.91.4		64	96.59	even	8
384.3.m.a.271.5		64	32.5	even	8
384.3.m.a.367.5		64	4.3	odd	2