Embedded newform 96.5.p.a.5.3

• Label: 96.5.p.a.5.3
• 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			5.3
Character	\(\chi\)	\(=\)	96.5
Dual form			96.5.p.a.77.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.95130 + 0.622306i) q^{2} +(8.32544 + 3.41864i) q^{3} +(15.2255 - 4.91783i) q^{4} +(-16.1290 + 38.9389i) q^{5} +(-35.0237 - 8.32709i) q^{6} +(-56.6723 - 56.6723i) q^{7} +(-57.0999 + 28.9067i) q^{8} +(57.6258 + 56.9233i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 248 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.95130	+	0.622306i	−0.987824	+	0.155576i
							\(3\)	8.32544	+	3.41864i	0.925049	+	0.379849i
\(5\)	−16.1290	+	38.9389i	−0.645161	+	1.55756i								0.174468	+	0.984663i	\(0.444179\pi\)
							−0.819629	+	0.572894i	\(0.805821\pi\)
\(7\)	−56.6723	−	56.6723i	−1.15658	−	1.15658i	−0.985206	−	0.171372i	\(-0.945180\pi\)
							−0.171372	−	0.985206i	\(-0.554820\pi\)
\(11\)	−142.220	−	58.9094i	−1.17537	−	0.486855i	−0.292407	−	0.956294i	\(-0.594456\pi\)
							−0.882965	+	0.469439i	\(0.844456\pi\)
\(13\)	37.2089	−	15.4124i	0.220171	−	0.0911977i	−0.269872	−	0.962896i	\(-0.586981\pi\)
							0.490042	+	0.871699i	\(0.336981\pi\)
\(17\)	143.952			0.498105			0.249053	−	0.968490i	\(-0.419881\pi\)
							0.249053	+	0.968490i	\(0.419881\pi\)
\(19\)	9.97734	−	4.13275i	0.0276381	−	0.0114481i	−0.368822	−	0.929500i	\(-0.620239\pi\)
							0.396460	+	0.918052i	\(0.370239\pi\)
\(23\)	−648.254	−	648.254i	−1.22543	−	1.22543i	−0.965673	−	0.259760i	\(-0.916357\pi\)
							−0.259760	−	0.965673i	\(-0.583643\pi\)
\(29\)	−241.511	+	100.037i	−0.287171	+	0.118950i	−0.521619	−	0.853178i	\(-0.674672\pi\)
							0.234448	+	0.972129i	\(0.424672\pi\)
\(31\)	−1278.70			−1.33059			−0.665296	−	0.746580i	\(-0.731695\pi\)
							−0.665296	+	0.746580i	\(0.731695\pi\)
\(37\)	612.442	+	253.682i	0.447364	+	0.185304i	0.594980	−	0.803740i	\(-0.297160\pi\)
							−0.147616	+	0.989045i	\(0.547160\pi\)
\(41\)	812.885	+	812.885i	0.483572	+	0.483572i	0.906270	−	0.422698i	\(-0.138917\pi\)
							−0.422698	+	0.906270i	\(0.638917\pi\)
\(43\)	−813.613	+	1964.24i	−0.440029	+	1.06232i	0.535909	+	0.844276i	\(0.319969\pi\)
							−0.975938	+	0.218048i	\(0.930031\pi\)
\(47\)	−3387.06			−1.53330			−0.766650	−	0.642065i	\(-0.778078\pi\)
							−0.766650	+	0.642065i	\(0.778078\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.5.p.a.5.3	✓	248
3.2	odd	2	inner	96.5.p.a.5.60	yes	248
32.13	even	8	inner	96.5.p.a.77.60	yes	248
96.77	odd	8	inner	96.5.p.a.77.3	yes	248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.5.p.a.5.3	✓	248	1.1	even	1	trivial
96.5.p.a.5.60	yes	248	3.2	odd	2	inner
96.5.p.a.77.3	yes	248	96.77	odd	8	inner
96.5.p.a.77.60	yes	248	32.13	even	8	inner