Embedded newform 96.5.p.a.5.8

• Label: 96.5.p.a.5.8
• Level: $96$
• Weight: $5$
• Character: 96.5
• Analytic conductor: $9.924$
• Analytic rank: $0$
• Dimension: $248$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.92351645605\)
Analytic rank:	\(0\)
Dimension:	\(248\)
Relative dimension:	\(62\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			5.8
Character	\(\chi\)	\(=\)	96.5
Dual form			96.5.p.a.77.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.65302 - 1.62955i) q^{2} +(2.58774 - 8.61995i) q^{3} +(10.6892 + 11.9055i) q^{4} +(-2.37119 + 5.72455i) q^{5} +(-23.4997 + 27.2720i) q^{6} +(6.96147 + 6.96147i) q^{7} +(-19.6471 - 60.9097i) q^{8} +(-67.6072 - 44.6124i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 248 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.65302	−	1.62955i	−0.913256	−	0.407386i
							\(3\)	2.58774	−	8.61995i	0.287527	−	0.957773i
\(5\)	−2.37119	+	5.72455i	−0.0948475	+	0.228982i								−0.964182	−	0.265243i	\(-0.914548\pi\)
							0.869334	+	0.494225i	\(0.164548\pi\)
\(7\)	6.96147	+	6.96147i	0.142071	+	0.142071i	0.774565	−	0.632494i	\(-0.217969\pi\)
							−0.632494	+	0.774565i	\(0.717969\pi\)
\(11\)	−174.043	−	72.0910i	−1.43837	−	0.595793i	−0.478969	−	0.877832i	\(-0.658989\pi\)
							−0.959403	+	0.282039i	\(0.908989\pi\)
\(13\)	123.424	−	51.1238i	0.730318	−	0.302508i	0.0136356	−	0.999907i	\(-0.495660\pi\)
							0.716683	+	0.697399i	\(0.245660\pi\)
\(17\)	−211.676			−0.732442			−0.366221	−	0.930528i	\(-0.619349\pi\)
							−0.366221	+	0.930528i	\(0.619349\pi\)
\(19\)	−349.180	+	144.635i	−0.967258	+	0.400651i	−0.809691	−	0.586857i	\(-0.800365\pi\)
							−0.157567	+	0.987508i	\(0.550365\pi\)
\(23\)	−79.1918	−	79.1918i	−0.149701	−	0.149701i	0.628284	−	0.777984i	\(-0.283758\pi\)
							−0.777984	+	0.628284i	\(0.783758\pi\)
\(29\)	−802.893	+	332.569i	−0.954689	+	0.395445i	−0.804991	−	0.593287i	\(-0.797830\pi\)
							−0.149698	+	0.988732i	\(0.547830\pi\)
\(31\)	−1499.22			−1.56007			−0.780033	−	0.625739i	\(-0.784798\pi\)
							−0.780033	+	0.625739i	\(0.784798\pi\)
\(37\)	49.4119	+	20.4671i	0.0360934	+	0.0149504i	0.400657	−	0.916228i	\(-0.368782\pi\)
							−0.364564	+	0.931178i	\(0.618782\pi\)
\(41\)	1484.60	+	1484.60i	0.883165	+	0.883165i	0.993855	−	0.110690i	\(-0.0353061\pi\)
							−0.110690	+	0.993855i	\(0.535306\pi\)
\(43\)	502.754	−	1213.76i	0.271906	−	0.656439i	−0.727659	−	0.685939i	\(-0.759392\pi\)
							0.999565	+	0.0295002i	\(0.00939156\pi\)
\(47\)	1632.94			0.739220			0.369610	−	0.929187i	\(-0.379491\pi\)
							0.369610	+	0.929187i	\(0.379491\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.5.p.a.5.8	✓	248
3.2	odd	2	inner	96.5.p.a.5.55	yes	248
32.13	even	8	inner	96.5.p.a.77.55	yes	248
96.77	odd	8	inner	96.5.p.a.77.8	yes	248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.5.p.a.5.8	✓	248	1.1	even	1	trivial
96.5.p.a.5.55	yes	248	3.2	odd	2	inner
96.5.p.a.77.8	yes	248	96.77	odd	8	inner
96.5.p.a.77.55	yes	248	32.13	even	8	inner