Embedded newform 96.3.p.a.5.20

• Label: 96.3.p.a.5.20
• Level: $96$
• Weight: $3$
• Character: 96.5
• Analytic conductor: $2.616$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.61581053786\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(30\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			5.20
Character	\(\chi\)	\(=\)	96.5
Dual form			96.3.p.a.77.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.896196 + 1.78797i) q^{2} +(-0.286103 + 2.98633i) q^{3} +(-2.39366 + 3.20474i) q^{4} +(0.794982 - 1.91926i) q^{5} +(-5.59586 + 2.16479i) q^{6} +(4.69711 + 4.69711i) q^{7} +(-7.87517 - 1.40772i) q^{8} +(-8.83629 - 1.70880i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.896196	+	1.78797i	0.448098	+	0.893984i
							\(3\)	−0.286103	+	2.98633i	−0.0953678	+	0.995442i
\(5\)	0.794982	−	1.91926i	0.158996	−	0.383851i								−0.824226	−	0.566261i	\(-0.808390\pi\)
							0.983223	+	0.182409i	\(0.0583896\pi\)
\(7\)	4.69711	+	4.69711i	0.671016	+	0.671016i	0.957950	−	0.286934i	\(-0.0926361\pi\)
							−0.286934	+	0.957950i	\(0.592636\pi\)
\(11\)	−0.393643	−	0.163052i	−0.0357857	−	0.0148229i	0.364719	−	0.931118i	\(-0.381165\pi\)
							−0.400505	+	0.916295i	\(0.631165\pi\)
\(13\)	2.31489	−	0.958858i	0.178068	−	0.0737583i	−0.291868	−	0.956459i	\(-0.594277\pi\)
							0.469936	+	0.882700i	\(0.344277\pi\)
\(17\)	−0.432707			−0.0254534			−0.0127267	−	0.999919i	\(-0.504051\pi\)
							−0.0127267	+	0.999919i	\(0.504051\pi\)
\(19\)	23.6610	−	9.80070i	1.24531	−	0.515826i	0.339943	−	0.940446i	\(-0.389592\pi\)
							0.905372	+	0.424620i	\(0.139592\pi\)
\(23\)	31.0794	+	31.0794i	1.35128	+	1.35128i	0.884235	+	0.467042i	\(0.154680\pi\)
							0.467042	+	0.884235i	\(0.345320\pi\)
\(29\)	5.18802	−	2.14895i	0.178897	−	0.0741016i	−0.291437	−	0.956590i	\(-0.594133\pi\)
							0.470334	+	0.882488i	\(0.344133\pi\)
\(31\)	−40.7959			−1.31600			−0.657998	−	0.753020i	\(-0.728596\pi\)
							−0.657998	+	0.753020i	\(0.728596\pi\)
\(37\)	−5.92626	−	2.45474i	−0.160169	−	0.0663442i	0.301158	−	0.953574i	\(-0.402627\pi\)
							−0.461328	+	0.887230i	\(0.652627\pi\)
\(41\)	−25.7408	−	25.7408i	−0.627824	−	0.627824i	0.319696	−	0.947520i	\(-0.396419\pi\)
							−0.947520	+	0.319696i	\(0.896419\pi\)
\(43\)	29.2668	−	70.6562i	0.680623	−	1.64317i	−0.0822441	−	0.996612i	\(-0.526209\pi\)
							0.762867	−	0.646556i	\(-0.223791\pi\)
\(47\)	32.4483			0.690389			0.345194	−	0.938531i	\(-0.387813\pi\)
							0.345194	+	0.938531i	\(0.387813\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.3.p.a.5.20	yes	120
3.2	odd	2	inner	96.3.p.a.5.11	✓	120
4.3	odd	2		384.3.p.a.113.15		120
12.11	even	2		384.3.p.a.113.7		120
32.13	even	8	inner	96.3.p.a.77.11	yes	120
32.19	odd	8		384.3.p.a.17.7		120
96.77	odd	8	inner	96.3.p.a.77.20	yes	120
96.83	even	8		384.3.p.a.17.15		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.p.a.5.11	✓	120	3.2	odd	2	inner
96.3.p.a.5.20	yes	120	1.1	even	1	trivial
96.3.p.a.77.11	yes	120	32.13	even	8	inner
96.3.p.a.77.20	yes	120	96.77	odd	8	inner
384.3.p.a.17.7		120	32.19	odd	8
384.3.p.a.17.15		120	96.83	even	8
384.3.p.a.113.7		120	12.11	even	2
384.3.p.a.113.15		120	4.3	odd	2