Embedded newform 96.3.p.a.5.10

• Label: 96.3.p.a.5.10
• 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			5.10
Character	\(\chi\)	\(=\)	96.5
Dual form			96.3.p.a.77.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.05135 + 1.70137i) q^{2} +(-1.64571 - 2.50831i) q^{3} +(-1.78934 - 3.57746i) q^{4} +(-1.03584 + 2.50074i) q^{5} +(5.99779 - 0.162867i) q^{6} +(7.44249 + 7.44249i) q^{7} +(7.96782 + 0.716807i) q^{8} +(-3.58326 + 8.25592i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.05135	+	1.70137i	−0.525673	+	0.850687i
							\(3\)	−1.64571	−	2.50831i	−0.548571	−	0.836104i
\(5\)	−1.03584	+	2.50074i	−0.207168	+	0.500147i								−0.992975	−	0.118324i	\(-0.962248\pi\)
							0.785807	+	0.618471i	\(0.212248\pi\)
\(7\)	7.44249	+	7.44249i	1.06321	+	1.06321i	0.997862	+	0.0653511i	\(0.0208167\pi\)
							0.0653511	+	0.997862i	\(0.479183\pi\)
\(11\)	15.3219	+	6.34656i	1.39290	+	0.576960i	0.947900	−	0.318569i	\(-0.103202\pi\)
							0.445004	+	0.895529i	\(0.353202\pi\)
\(13\)	13.3849	−	5.54420i	1.02961	−	0.426477i	0.197037	−	0.980396i	\(-0.436868\pi\)
							0.832571	+	0.553919i	\(0.186868\pi\)
\(17\)	−32.6736			−1.92198			−0.960989	−	0.276587i	\(-0.910797\pi\)
							−0.960989	+	0.276587i	\(0.910797\pi\)
\(19\)	5.22697	−	2.16508i	0.275104	−	0.113952i	−0.240866	−	0.970558i	\(-0.577432\pi\)
							0.515970	+	0.856607i	\(0.327432\pi\)
\(23\)	9.22752	+	9.22752i	0.401197	+	0.401197i	0.878655	−	0.477458i	\(-0.158442\pi\)
							−0.477458	+	0.878655i	\(0.658442\pi\)
\(29\)	7.45235	−	3.08686i	0.256978	−	0.106444i	−0.250476	−	0.968123i	\(-0.580587\pi\)
							0.507453	+	0.861679i	\(0.330587\pi\)
\(31\)	−5.05668			−0.163119			−0.0815594	−	0.996668i	\(-0.525990\pi\)
							−0.0815594	+	0.996668i	\(0.525990\pi\)
\(37\)	−6.39573	−	2.64920i	−0.172858	−	0.0716000i	0.294576	−	0.955628i	\(-0.404821\pi\)
							−0.467434	+	0.884028i	\(0.654821\pi\)
\(41\)	11.1413	+	11.1413i	0.271738	+	0.271738i	0.829800	−	0.558061i	\(-0.188455\pi\)
							−0.558061	+	0.829800i	\(0.688455\pi\)
\(43\)	−9.29530	+	22.4408i	−0.216170	+	0.521880i	−0.994349	−	0.106162i	\(-0.966144\pi\)
							0.778179	+	0.628043i	\(0.216144\pi\)
\(47\)	1.74893			0.0372114			0.0186057	−	0.999827i	\(-0.494077\pi\)
							0.0186057	+	0.999827i	\(0.494077\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.3.p.a.5.10	✓	120
3.2	odd	2	inner	96.3.p.a.5.21	yes	120
4.3	odd	2		384.3.p.a.113.22		120
12.11	even	2		384.3.p.a.113.17		120
32.13	even	8	inner	96.3.p.a.77.21	yes	120
32.19	odd	8		384.3.p.a.17.17		120
96.77	odd	8	inner	96.3.p.a.77.10	yes	120
96.83	even	8		384.3.p.a.17.22		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.p.a.5.10	✓	120	1.1	even	1	trivial
96.3.p.a.5.21	yes	120	3.2	odd	2	inner
96.3.p.a.77.10	yes	120	96.77	odd	8	inner
96.3.p.a.77.21	yes	120	32.13	even	8	inner
384.3.p.a.17.17		120	32.19	odd	8
384.3.p.a.17.22		120	96.83	even	8
384.3.p.a.113.17		120	12.11	even	2
384.3.p.a.113.22		120	4.3	odd	2