Embedded newform 96.5.p.a.5.2

• Label: 96.5.p.a.5.2
• 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			5.2
Character	\(\chi\)	\(=\)	96.5
Dual form			96.5.p.a.77.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.98973 - 0.286445i) q^{2} +(-7.27242 + 5.30207i) q^{3} +(15.8359 + 2.28568i) q^{4} +(1.03929 - 2.50907i) q^{5} +(30.5337 - 19.0707i) q^{6} +(-13.2001 - 13.2001i) q^{7} +(-62.5262 - 13.6553i) q^{8} +(24.7760 - 77.1178i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 248 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.98973	−	0.286445i	−0.997433	−	0.0716112i
							\(3\)	−7.27242	+	5.30207i	−0.808046	+	0.589119i
\(5\)	1.03929	−	2.50907i	0.0415716	−	0.100363i								−0.901730	−	0.432300i	\(-0.857702\pi\)
							0.943301	+	0.331937i	\(0.107702\pi\)
\(7\)	−13.2001	−	13.2001i	−0.269390	−	0.269390i	0.559464	−	0.828854i	\(-0.311007\pi\)
							−0.828854	+	0.559464i	\(0.811007\pi\)
\(11\)	−65.7914	−	27.2517i	−0.543731	−	0.225221i	0.0938743	−	0.995584i	\(-0.470075\pi\)
							−0.637605	+	0.770363i	\(0.720075\pi\)
\(13\)	−27.4112	+	11.3541i	−0.162196	+	0.0671839i	−0.462304	−	0.886721i	\(-0.652977\pi\)
							0.300108	+	0.953905i	\(0.402977\pi\)
\(17\)	325.003			1.12458			0.562290	−	0.826940i	\(-0.309921\pi\)
							0.562290	+	0.826940i	\(0.309921\pi\)
\(19\)	−27.7172	+	11.4808i	−0.0767790	+	0.0318029i	−0.420742	−	0.907180i	\(-0.638230\pi\)
							0.343963	+	0.938983i	\(0.388230\pi\)
\(23\)	472.307	+	472.307i	0.892830	+	0.892830i	0.994789	−	0.101959i	\(-0.0325110\pi\)
							−0.101959	+	0.994789i	\(0.532511\pi\)
\(29\)	834.619	−	345.710i	0.992412	−	0.411071i	0.173403	−	0.984851i	\(-0.444524\pi\)
							0.819009	+	0.573780i	\(0.194524\pi\)
\(31\)	−132.453			−0.137828			−0.0689141	−	0.997623i	\(-0.521953\pi\)
							−0.0689141	+	0.997623i	\(0.521953\pi\)
\(37\)	137.996	+	57.1600i	0.100801	+	0.0417531i	0.432514	−	0.901627i	\(-0.357627\pi\)
							−0.331713	+	0.943380i	\(0.607627\pi\)
\(41\)	1041.58	+	1041.58i	0.619618	+	0.619618i	0.945434	−	0.325815i	\(-0.105639\pi\)
							−0.325815	+	0.945434i	\(0.605639\pi\)
\(43\)	−883.601	+	2133.20i	−0.477880	+	1.15371i	0.482721	+	0.875774i	\(0.339649\pi\)
							−0.960601	+	0.277931i	\(0.910351\pi\)
\(47\)	638.565			0.289074			0.144537	−	0.989499i	\(-0.453831\pi\)
							0.144537	+	0.989499i	\(0.453831\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.5.p.a.5.2	✓	248
3.2	odd	2	inner	96.5.p.a.5.61	yes	248
32.13	even	8	inner	96.5.p.a.77.61	yes	248
96.77	odd	8	inner	96.5.p.a.77.2	yes	248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.5.p.a.5.2	✓	248	1.1	even	1	trivial
96.5.p.a.5.61	yes	248	3.2	odd	2	inner
96.5.p.a.77.2	yes	248	96.77	odd	8	inner
96.5.p.a.77.61	yes	248	32.13	even	8	inner