Embedded newform 96.3.p.a.5.2

• Label: 96.3.p.a.5.2
• 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			5.2
Character	\(\chi\)	\(=\)	96.5
Dual form			96.3.p.a.77.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99160 - 0.183158i) q^{2} +(2.25564 - 1.97790i) q^{3} +(3.93291 + 0.729554i) q^{4} +(-3.05886 + 7.38474i) q^{5} +(-4.85460 + 3.52603i) q^{6} +(6.52353 + 6.52353i) q^{7} +(-7.69913 - 2.17332i) q^{8} +(1.17585 - 8.92286i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99160	−	0.183158i	−0.995798	−	0.0915791i
							\(3\)	2.25564	−	1.97790i	0.751881	−	0.659299i
\(5\)	−3.05886	+	7.38474i	−0.611772	+	1.47695i								0.249281	+	0.968431i	\(0.419806\pi\)
							−0.861053	+	0.508516i	\(0.830194\pi\)
\(7\)	6.52353	+	6.52353i	0.931933	+	0.931933i	0.997827	−	0.0658939i	\(-0.0209899\pi\)
							−0.0658939	+	0.997827i	\(0.520990\pi\)
\(11\)	7.86759	+	3.25886i	0.715235	+	0.296260i	0.710469	−	0.703728i	\(-0.248483\pi\)
							0.00476633	+	0.999989i	\(0.498483\pi\)
\(13\)	−4.33713	+	1.79650i	−0.333625	+	0.138192i	−0.543206	−	0.839599i	\(-0.682790\pi\)
							0.209581	+	0.977791i	\(0.432790\pi\)
\(17\)	23.8192			1.40113			0.700565	−	0.713589i	\(-0.252931\pi\)
							0.700565	+	0.713589i	\(0.252931\pi\)
\(19\)	−2.84170	+	1.17707i	−0.149563	+	0.0619511i	−0.456209	−	0.889872i	\(-0.650793\pi\)
							0.306646	+	0.951824i	\(0.400793\pi\)
\(23\)	−10.2113	−	10.2113i	−0.443969	−	0.443969i	0.449374	−	0.893343i	\(-0.351647\pi\)
							−0.893343	+	0.449374i	\(0.851647\pi\)
\(29\)	9.02559	−	3.73852i	0.311227	−	0.128915i	−0.221602	−	0.975137i	\(-0.571129\pi\)
							0.532830	+	0.846223i	\(0.321129\pi\)
\(31\)	−25.2517			−0.814573			−0.407286	−	0.913301i	\(-0.633525\pi\)
							−0.407286	+	0.913301i	\(0.633525\pi\)
\(37\)	14.5468	+	6.02549i	0.393157	+	0.162851i	0.570499	−	0.821298i	\(-0.306750\pi\)
							−0.177342	+	0.984149i	\(0.556750\pi\)
\(41\)	−46.5680	−	46.5680i	−1.13580	−	1.13580i	−0.989194	−	0.146609i	\(-0.953164\pi\)
							−0.146609	−	0.989194i	\(-0.546836\pi\)
\(43\)	−17.5199	+	42.2969i	−0.407441	+	0.983649i	0.578368	+	0.815776i	\(0.303690\pi\)
							−0.985809	+	0.167873i	\(0.946310\pi\)
\(47\)	69.8057			1.48523			0.742613	−	0.669720i	\(-0.233586\pi\)
							0.742613	+	0.669720i	\(0.233586\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.3.p.a.5.2	✓	120
3.2	odd	2	inner	96.3.p.a.5.29	yes	120
4.3	odd	2		384.3.p.a.113.8		120
12.11	even	2		384.3.p.a.113.29		120
32.13	even	8	inner	96.3.p.a.77.29	yes	120
32.19	odd	8		384.3.p.a.17.29		120
96.77	odd	8	inner	96.3.p.a.77.2	yes	120
96.83	even	8		384.3.p.a.17.8		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.p.a.5.2	✓	120	1.1	even	1	trivial
96.3.p.a.5.29	yes	120	3.2	odd	2	inner
96.3.p.a.77.2	yes	120	96.77	odd	8	inner
96.3.p.a.77.29	yes	120	32.13	even	8	inner
384.3.p.a.17.8		120	96.83	even	8
384.3.p.a.17.29		120	32.19	odd	8
384.3.p.a.113.8		120	4.3	odd	2
384.3.p.a.113.29		120	12.11	even	2