Embedded newform 96.3.m.a.19.15

• Label: 96.3.m.a.19.15
• 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			19.15
Character	\(\chi\)	\(=\)	96.19
Dual form			96.3.m.a.91.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.92938 - 0.526758i) q^{2} +(-1.60021 - 0.662827i) q^{3} +(3.44505 - 2.03264i) q^{4} +(3.51382 - 1.45547i) q^{5} +(-3.43656 - 0.435928i) q^{6} +(-1.77216 - 1.77216i) q^{7} +(5.57613 - 5.73645i) q^{8} +(2.12132 + 2.12132i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.92938	−	0.526758i	0.964692	−	0.263379i
							\(3\)	−1.60021	−	0.662827i	−0.533402	−	0.220942i
\(5\)	3.51382	−	1.45547i	0.702763	−	0.291094i								−0.00254304	−	0.999997i	\(-0.500809\pi\)
							0.705306	+	0.708903i	\(0.250809\pi\)
\(7\)	−1.77216	−	1.77216i	−0.253166	−	0.253166i	0.569101	−	0.822267i	\(-0.307291\pi\)
							−0.822267	+	0.569101i	\(0.807291\pi\)
\(11\)	−3.84851	+	1.59411i	−0.349865	+	0.144919i	−0.550693	−	0.834708i	\(-0.685637\pi\)
							0.200828	+	0.979626i	\(0.435637\pi\)
\(13\)	5.91643	+	2.45067i	0.455110	+	0.188513i	0.598449	−	0.801161i	\(-0.295784\pi\)
							−0.143339	+	0.989674i	\(0.545784\pi\)
\(17\)			10.3089i			0.606406i	0.952926	+	0.303203i	\(0.0980560\pi\)
							−0.952926	+	0.303203i	\(0.901944\pi\)
\(19\)	−2.86839	+	6.92491i	−0.150968	+	0.364469i	−0.981212	−	0.192931i	\(-0.938201\pi\)
							0.830244	+	0.557399i	\(0.188201\pi\)
\(23\)	−29.6688	+	29.6688i	−1.28995	+	1.28995i	−0.355131	+	0.934817i	\(0.615564\pi\)
							−0.934817	+	0.355131i	\(0.884436\pi\)
\(29\)	−10.0526	+	24.2691i	−0.346641	+	0.836865i	0.650371	+	0.759617i	\(0.274613\pi\)
							−0.997012	+	0.0772483i	\(0.975387\pi\)
\(31\)		−	7.83864i		−	0.252859i	−0.991976	−	0.126430i	\(-0.959648\pi\)
							0.991976	−	0.126430i	\(-0.0403518\pi\)
\(37\)	56.8298	−	23.5397i	1.53594	−	0.636208i	0.555235	−	0.831693i	\(-0.312628\pi\)
							0.980707	+	0.195485i	\(0.0626282\pi\)
\(41\)	−18.3495	−	18.3495i	−0.447550	−	0.447550i	0.446989	−	0.894539i	\(-0.352496\pi\)
							−0.894539	+	0.446989i	\(0.852496\pi\)
\(43\)	54.2671	−	22.4782i	1.26202	−	0.522748i	0.351495	−	0.936190i	\(-0.385674\pi\)
							0.910530	+	0.413442i	\(0.135674\pi\)
\(47\)	−55.9090			−1.18955			−0.594777	−	0.803891i	\(-0.702760\pi\)
							−0.594777	+	0.803891i	\(0.702760\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.3.m.a.19.15	✓	64
3.2	odd	2		288.3.u.b.19.2		64
4.3	odd	2		384.3.m.a.367.14		64
32.5	even	8		384.3.m.a.271.14		64
32.27	odd	8	inner	96.3.m.a.91.15	yes	64
96.59	even	8		288.3.u.b.91.2		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.m.a.19.15	✓	64	1.1	even	1	trivial
96.3.m.a.91.15	yes	64	32.27	odd	8	inner
288.3.u.b.19.2		64	3.2	odd	2
288.3.u.b.91.2		64	96.59	even	8
384.3.m.a.271.14		64	32.5	even	8
384.3.m.a.367.14		64	4.3	odd	2