Embedded newform 96.3.p.a.5.1

• Label: 96.3.p.a.5.1
• 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			5.1
Character	\(\chi\)	\(=\)	96.5
Dual form			96.3.p.a.77.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99935 - 0.0509440i) q^{2} +(2.77972 + 1.12834i) q^{3} +(3.99481 + 0.203710i) q^{4} +(2.35810 - 5.69297i) q^{5} +(-5.50016 - 2.39756i) q^{6} +(-1.55454 - 1.55454i) q^{7} +(-7.97665 - 0.610799i) q^{8} +(6.45370 + 6.27294i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99935	−	0.0509440i	−0.999676	−	0.0254720i
							\(3\)	2.77972	+	1.12834i	0.926574	+	0.376113i
\(5\)	2.35810	−	5.69297i	0.471621	−	1.13859i								−0.491826	−	0.870694i	\(-0.663670\pi\)
							0.963447	−	0.267900i	\(-0.0863297\pi\)
\(7\)	−1.55454	−	1.55454i	−0.222077	−	0.222077i	0.587296	−	0.809372i	\(-0.300193\pi\)
							−0.809372	+	0.587296i	\(0.800193\pi\)
\(11\)	−1.06147	−	0.439675i	−0.0964972	−	0.0399705i	0.333913	−	0.942604i	\(-0.391631\pi\)
							−0.430410	+	0.902634i	\(0.641631\pi\)
\(13\)	21.0080	−	8.70178i	1.61600	−	0.669368i	0.622436	−	0.782670i	\(-0.286143\pi\)
							0.993561	+	0.113302i	\(0.0361429\pi\)
\(17\)	−12.3370			−0.725706			−0.362853	−	0.931846i	\(-0.618197\pi\)
							−0.362853	+	0.931846i	\(0.618197\pi\)
\(19\)	−7.49164	+	3.10314i	−0.394297	+	0.163323i	−0.571017	−	0.820938i	\(-0.693451\pi\)
							0.176720	+	0.984261i	\(0.443451\pi\)
\(23\)	27.3855	+	27.3855i	1.19068	+	1.19068i	0.976878	+	0.213797i	\(0.0685832\pi\)
							0.213797	+	0.976878i	\(0.431417\pi\)
\(29\)	−16.2919	+	6.74831i	−0.561789	+	0.232701i	−0.645462	−	0.763793i	\(-0.723335\pi\)
							0.0836728	+	0.996493i	\(0.473335\pi\)
\(31\)	−36.9734			−1.19269			−0.596346	−	0.802728i	\(-0.703381\pi\)
							−0.596346	+	0.802728i	\(0.703381\pi\)
\(37\)	−42.4386	−	17.5786i	−1.14699	−	0.475098i	−0.273466	−	0.961882i	\(-0.588170\pi\)
							−0.873522	+	0.486784i	\(0.838170\pi\)
\(41\)	−34.5226	−	34.5226i	−0.842016	−	0.842016i	0.147105	−	0.989121i	\(-0.453004\pi\)
							−0.989121	+	0.147105i	\(0.953004\pi\)
\(43\)	−17.0666	+	41.2024i	−0.396898	+	0.958196i	0.591500	+	0.806305i	\(0.298536\pi\)
							−0.988397	+	0.151891i	\(0.951464\pi\)
\(47\)	−9.71170			−0.206632			−0.103316	−	0.994649i	\(-0.532945\pi\)
							−0.103316	+	0.994649i	\(0.532945\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.3.p.a.5.1	✓	120
3.2	odd	2	inner	96.3.p.a.5.30	yes	120
4.3	odd	2		384.3.p.a.113.5		120
12.11	even	2		384.3.p.a.113.19		120
32.13	even	8	inner	96.3.p.a.77.30	yes	120
32.19	odd	8		384.3.p.a.17.19		120
96.77	odd	8	inner	96.3.p.a.77.1	yes	120
96.83	even	8		384.3.p.a.17.5		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.p.a.5.1	✓	120	1.1	even	1	trivial
96.3.p.a.5.30	yes	120	3.2	odd	2	inner
96.3.p.a.77.1	yes	120	96.77	odd	8	inner
96.3.p.a.77.30	yes	120	32.13	even	8	inner
384.3.p.a.17.5		120	96.83	even	8
384.3.p.a.17.19		120	32.19	odd	8
384.3.p.a.113.5		120	4.3	odd	2
384.3.p.a.113.19		120	12.11	even	2