Embedded newform 96.2.o.a.35.2

• Label: 96.2.o.a.35.2
• Level: $96$
• Weight: $2$
• Character: 96.35
• Analytic conductor: $0.767$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.766563859404\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			35.2
Character	\(\chi\)	\(=\)	96.35
Dual form			96.2.o.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.12377 - 0.858568i) q^{2} +(-0.0380372 - 1.73163i) q^{3} +(0.525721 + 1.92967i) q^{4} +(2.18808 - 0.906333i) q^{5} +(-1.44398 + 1.97862i) q^{6} +(-1.93241 - 1.93241i) q^{7} +(1.06596 - 2.61987i) q^{8} +(-2.99711 + 0.131733i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(37\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{8}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.12377	−	0.858568i	−0.794626	−	0.607100i
							\(3\)	−0.0380372	−	1.73163i	−0.0219608	−	0.999759i
\(5\)	2.18808	−	0.906333i	0.978540	−	0.405324i								0.164655	−	0.986351i	\(-0.447349\pi\)
							0.813884	+	0.581027i	\(0.197349\pi\)
\(7\)	−1.93241	−	1.93241i	−0.730381	−	0.730381i	0.240314	−	0.970695i	\(-0.422750\pi\)
							−0.970695	+	0.240314i	\(0.922750\pi\)
\(11\)	−1.42447	+	0.590036i	−0.429495	+	0.177903i	−0.586949	−	0.809624i	\(-0.699671\pi\)
							0.157454	+	0.987526i	\(0.449671\pi\)
\(13\)	−0.110405	+	0.266541i	−0.0306208	+	0.0739252i	−0.938450	−	0.345414i	\(-0.887738\pi\)
							0.907829	+	0.419340i	\(0.137738\pi\)
\(17\)	6.17031			1.49652			0.748260	−	0.663406i	\(-0.230890\pi\)
							0.748260	+	0.663406i	\(0.230890\pi\)
\(19\)	7.34269	+	3.04144i	1.68453	+	0.697755i	0.999526	−	0.0307845i	\(-0.00980057\pi\)
							0.685004	+	0.728540i	\(0.259801\pi\)
\(23\)	−1.85295	−	1.85295i	−0.386366	−	0.386366i	0.487023	−	0.873389i	\(-0.338083\pi\)
							−0.873389	+	0.487023i	\(0.838083\pi\)
\(29\)	−2.11574	+	5.10784i	−0.392882	+	0.948501i	0.596427	+	0.802667i	\(0.296587\pi\)
							−0.989309	+	0.145834i	\(0.953413\pi\)
\(31\)		−	3.42046i		−	0.614332i	−0.951656	−	0.307166i	\(-0.900619\pi\)
							0.951656	−	0.307166i	\(-0.0993807\pi\)
\(37\)	2.52377	+	6.09293i	0.414906	+	1.00167i	0.983801	+	0.179262i	\(0.0573709\pi\)
							−0.568895	+	0.822410i	\(0.692629\pi\)
\(41\)	0.753641	−	0.753641i	0.117699	−	0.117699i	−0.645804	−	0.763503i	\(-0.723478\pi\)
							0.763503	+	0.645804i	\(0.223478\pi\)
\(43\)	1.57129	+	3.79343i	0.239619	+	0.578492i	0.997243	−	0.0741989i	\(-0.0236400\pi\)
							−0.757624	+	0.652691i	\(0.773640\pi\)
\(47\)			1.54798i			0.225796i	0.993607	+	0.112898i	\(0.0360133\pi\)
							−0.993607	+	0.112898i	\(0.963987\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.2.o.a.35.2	yes	56
3.2	odd	2	inner	96.2.o.a.35.13	yes	56
4.3	odd	2		384.2.o.a.47.8		56
8.3	odd	2		768.2.o.a.95.7		56
8.5	even	2		768.2.o.b.95.8		56
12.11	even	2		384.2.o.a.47.3		56
24.5	odd	2		768.2.o.b.95.3		56
24.11	even	2		768.2.o.a.95.12		56
32.5	even	8		768.2.o.a.671.12		56
32.11	odd	8	inner	96.2.o.a.11.13	yes	56
32.21	even	8		384.2.o.a.335.3		56
32.27	odd	8		768.2.o.b.671.3		56
96.5	odd	8		768.2.o.a.671.7		56
96.11	even	8	inner	96.2.o.a.11.2	✓	56
96.53	odd	8		384.2.o.a.335.8		56
96.59	even	8		768.2.o.b.671.8		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.2.o.a.11.2	✓	56	96.11	even	8	inner
96.2.o.a.11.13	yes	56	32.11	odd	8	inner
96.2.o.a.35.2	yes	56	1.1	even	1	trivial
96.2.o.a.35.13	yes	56	3.2	odd	2	inner
384.2.o.a.47.3		56	12.11	even	2
384.2.o.a.47.8		56	4.3	odd	2
384.2.o.a.335.3		56	32.21	even	8
384.2.o.a.335.8		56	96.53	odd	8
768.2.o.a.95.7		56	8.3	odd	2
768.2.o.a.95.12		56	24.11	even	2
768.2.o.a.671.7		56	96.5	odd	8
768.2.o.a.671.12		56	32.5	even	8
768.2.o.b.95.3		56	24.5	odd	2
768.2.o.b.95.8		56	8.5	even	2
768.2.o.b.671.3		56	32.27	odd	8
768.2.o.b.671.8		56	96.59	even	8