Embedded newform 96.3.m.a.19.14

• Label: 96.3.m.a.19.14
• 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			19.14
Character	\(\chi\)	\(=\)	96.19
Dual form			96.3.m.a.91.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.64362 + 1.13952i) q^{2} +(-1.60021 - 0.662827i) q^{3} +(1.40297 + 3.74589i) q^{4} +(0.838363 - 0.347262i) q^{5} +(-1.87482 - 2.91291i) q^{6} +(8.29065 + 8.29065i) q^{7} +(-1.96258 + 7.75553i) q^{8} +(2.12132 + 2.12132i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.64362	+	1.13952i	0.821810	+	0.569762i
							\(3\)	−1.60021	−	0.662827i	−0.533402	−	0.220942i
\(5\)	0.838363	−	0.347262i	0.167673	−	0.0694523i								−0.297268	−	0.954794i	\(-0.596076\pi\)
							0.464941	+	0.885342i	\(0.346076\pi\)
\(7\)	8.29065	+	8.29065i	1.18438	+	1.18438i	0.978598	+	0.205781i	\(0.0659734\pi\)
							0.205781	+	0.978598i	\(0.434027\pi\)
\(11\)	6.14387	−	2.54487i	0.558534	−	0.231352i	−0.0855148	−	0.996337i	\(-0.527254\pi\)
							0.644048	+	0.764985i	\(0.277254\pi\)
\(13\)	−17.7917	−	7.36955i	−1.36859	−	0.566889i	−0.427184	−	0.904165i	\(-0.640494\pi\)
							−0.941406	+	0.337276i	\(0.890494\pi\)
\(17\)		−	24.0244i		−	1.41320i	−0.707612	−	0.706601i	\(-0.750227\pi\)
							0.707612	−	0.706601i	\(-0.249773\pi\)
\(19\)	5.48690	−	13.2466i	0.288784	−	0.697187i	−0.711199	−	0.702991i	\(-0.751847\pi\)
							0.999983	+	0.00580389i	\(0.00184745\pi\)
\(23\)	10.1957	−	10.1957i	0.443293	−	0.443293i	−0.449824	−	0.893117i	\(-0.648513\pi\)
							0.893117	+	0.449824i	\(0.148513\pi\)
\(29\)	4.28415	−	10.3428i	0.147729	−	0.356650i	−0.832642	−	0.553812i	\(-0.813173\pi\)
							0.980371	+	0.197162i	\(0.0631726\pi\)
\(31\)			17.7042i			0.571104i	0.958363	+	0.285552i	\(0.0921769\pi\)
							−0.958363	+	0.285552i	\(0.907823\pi\)
\(37\)	53.2287	−	22.0480i	1.43861	−	0.595893i	0.479151	−	0.877732i	\(-0.340945\pi\)
							0.959462	+	0.281839i	\(0.0909445\pi\)
\(41\)	−23.6060	−	23.6060i	−0.575756	−	0.575756i	0.357975	−	0.933731i	\(-0.383467\pi\)
							−0.933731	+	0.357975i	\(0.883467\pi\)
\(43\)	−52.0091	+	21.5429i	−1.20951	+	0.500997i	−0.894061	−	0.447944i	\(-0.852156\pi\)
							−0.315452	+	0.948941i	\(0.602156\pi\)
\(47\)	23.8851			0.508194			0.254097	−	0.967179i	\(-0.418222\pi\)
							0.254097	+	0.967179i	\(0.418222\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.3.m.a.19.14	✓	64
3.2	odd	2		288.3.u.b.19.3		64
4.3	odd	2		384.3.m.a.367.13		64
32.5	even	8		384.3.m.a.271.13		64
32.27	odd	8	inner	96.3.m.a.91.14	yes	64
96.59	even	8		288.3.u.b.91.3		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.m.a.19.14	✓	64	1.1	even	1	trivial
96.3.m.a.91.14	yes	64	32.27	odd	8	inner
288.3.u.b.19.3		64	3.2	odd	2
288.3.u.b.91.3		64	96.59	even	8
384.3.m.a.271.13		64	32.5	even	8
384.3.m.a.367.13		64	4.3	odd	2