Embedded newform 96.2.o.a.11.3

• Label: 96.2.o.a.11.3
• Level: $96$
• Weight: $2$
• Character: 96.11
• 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			11.3
Character	\(\chi\)	\(=\)	96.11
Dual form			96.2.o.a.35.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.09490 - 0.895090i) q^{2} +(-1.68370 + 0.406387i) q^{3} +(0.397627 + 1.96007i) q^{4} +(2.81491 + 1.16597i) q^{5} +(2.20724 + 1.06211i) q^{6} +(-0.543879 + 0.543879i) q^{7} +(1.31908 - 2.50200i) q^{8} +(2.66970 - 1.36847i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(37\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{8}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.09490	−	0.895090i	−0.774214	−	0.632924i
							\(3\)	−1.68370	+	0.406387i	−0.972085	+	0.234627i
\(5\)	2.81491	+	1.16597i	1.25886	+	0.521438i								0.909561	−	0.415571i	\(-0.136418\pi\)
							0.349303	+	0.937010i	\(0.386418\pi\)
\(7\)	−0.543879	+	0.543879i	−0.205567	+	0.205567i	−0.802380	−	0.596813i	\(-0.796433\pi\)
							0.596813	+	0.802380i	\(0.296433\pi\)
\(11\)	3.96678	+	1.64309i	1.19603	+	0.495411i	0.889713	−	0.456520i	\(-0.150904\pi\)
							0.306314	+	0.951930i	\(0.400904\pi\)
\(13\)	1.13904	+	2.74988i	0.315912	+	0.762680i	0.999463	+	0.0327767i	\(0.0104350\pi\)
							−0.683550	+	0.729903i	\(0.739565\pi\)
\(17\)	−5.73443			−1.39080			−0.695402	−	0.718621i	\(-0.744774\pi\)
							−0.695402	+	0.718621i	\(0.744774\pi\)
\(19\)	0.0793211	−	0.0328559i	0.0181975	−	0.00753765i	−0.373566	−	0.927604i	\(-0.621865\pi\)
							0.391764	+	0.920066i	\(0.371865\pi\)
\(23\)	1.46457	−	1.46457i	0.305385	−	0.305385i	−0.537731	−	0.843116i	\(-0.680719\pi\)
							0.843116	+	0.537731i	\(0.180719\pi\)
\(29\)	−0.520422	−	1.25641i	−0.0966400	−	0.233310i	0.868165	−	0.496275i	\(-0.165299\pi\)
							−0.964805	+	0.262966i	\(0.915299\pi\)
\(31\)		−	5.64072i		−	1.01310i	−0.862210	−	0.506552i	\(-0.830920\pi\)
							0.862210	−	0.506552i	\(-0.169080\pi\)
\(37\)	4.19628	−	10.1307i	0.689864	−	1.66548i	−0.0551907	−	0.998476i	\(-0.517577\pi\)
							0.745055	−	0.667003i	\(-0.232423\pi\)
\(41\)	−4.93573	−	4.93573i	−0.770831	−	0.770831i	0.207421	−	0.978252i	\(-0.433493\pi\)
							−0.978252	+	0.207421i	\(0.933493\pi\)
\(43\)	−3.50661	+	8.46571i	−0.534753	+	1.29101i	0.393591	+	0.919286i	\(0.371233\pi\)
							−0.928344	+	0.371722i	\(0.878767\pi\)
\(47\)		−	8.98904i		−	1.31119i	−0.755115	−	0.655593i	\(-0.772419\pi\)
							0.755115	−	0.655593i	\(-0.227581\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.2.o.a.11.3	✓	56
3.2	odd	2	inner	96.2.o.a.11.12	yes	56
4.3	odd	2		384.2.o.a.335.13		56
8.3	odd	2		768.2.o.a.671.2		56
8.5	even	2		768.2.o.b.671.13		56
12.11	even	2		384.2.o.a.335.2		56
24.5	odd	2		768.2.o.b.671.2		56
24.11	even	2		768.2.o.a.671.13		56
32.3	odd	8	inner	96.2.o.a.35.12	yes	56
32.13	even	8		768.2.o.a.95.13		56
32.19	odd	8		768.2.o.b.95.2		56
32.29	even	8		384.2.o.a.47.2		56
96.29	odd	8		384.2.o.a.47.13		56
96.35	even	8	inner	96.2.o.a.35.3	yes	56
96.77	odd	8		768.2.o.a.95.2		56
96.83	even	8		768.2.o.b.95.13		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.2.o.a.11.3	✓	56	1.1	even	1	trivial
96.2.o.a.11.12	yes	56	3.2	odd	2	inner
96.2.o.a.35.3	yes	56	96.35	even	8	inner
96.2.o.a.35.12	yes	56	32.3	odd	8	inner
384.2.o.a.47.2		56	32.29	even	8
384.2.o.a.47.13		56	96.29	odd	8
384.2.o.a.335.2		56	12.11	even	2
384.2.o.a.335.13		56	4.3	odd	2
768.2.o.a.95.2		56	96.77	odd	8
768.2.o.a.95.13		56	32.13	even	8
768.2.o.a.671.2		56	8.3	odd	2
768.2.o.a.671.13		56	24.11	even	2
768.2.o.b.95.2		56	32.19	odd	8
768.2.o.b.95.13		56	96.83	even	8
768.2.o.b.671.2		56	24.5	odd	2
768.2.o.b.671.13		56	8.5	even	2