Embedded newform 96.5.p.a.77.54

• Label: 96.5.p.a.77.54
• Level: $96$
• Weight: $5$
• Character: 96.77
• Analytic conductor: $9.924$
• Analytic rank: $0$
• Dimension: $248$
• 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			77.54
Character	\(\chi\)	\(=\)	96.77
Dual form			96.5.p.a.5.54

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.65028 - 1.63569i) q^{2} +(-3.81668 - 8.15064i) q^{3} +(10.6490 - 11.9414i) q^{4} +(0.344704 + 0.832190i) q^{5} +(-27.2638 - 23.5092i) q^{6} +(38.0513 - 38.0513i) q^{7} +(19.3394 - 61.0081i) q^{8} +(-51.8660 + 62.2167i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 248 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(37\)	\(65\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.65028	−	1.63569i	0.912569	−	0.408923i
							\(3\)	−3.81668	−	8.15064i	−0.424075	−	0.905627i
\(5\)	0.344704	+	0.832190i	0.0137882	+	0.0332876i								0.930624	−	0.365978i	\(-0.119265\pi\)
							−0.916836	+	0.399265i	\(0.869265\pi\)
\(7\)	38.0513	−	38.0513i	0.776558	−	0.776558i	−0.202686	−	0.979244i	\(-0.564967\pi\)
							0.979244	+	0.202686i	\(0.0649671\pi\)
\(11\)	−2.53448	+	1.04982i	−0.0209462	+	0.00867618i	−0.393132	−	0.919482i	\(-0.628608\pi\)
							0.372186	+	0.928158i	\(0.378608\pi\)
\(13\)	−207.043	−	85.7601i	−1.22511	−	0.507456i	−0.326077	−	0.945343i	\(-0.605727\pi\)
							−0.899030	+	0.437887i	\(0.855727\pi\)
\(17\)	15.2288			0.0526949			0.0263474	−	0.999653i	\(-0.491612\pi\)
							0.0263474	+	0.999653i	\(0.491612\pi\)
\(19\)	71.9968	+	29.8220i	0.199437	+	0.0826095i	0.480166	−	0.877177i	\(-0.340576\pi\)
							−0.280729	+	0.959787i	\(0.590576\pi\)
\(23\)	−417.892	+	417.892i	−0.789966	+	0.789966i	−0.981488	−	0.191522i	\(-0.938658\pi\)
							0.191522	+	0.981488i	\(0.438658\pi\)
\(29\)	336.642	+	139.442i	0.400288	+	0.165805i	0.573740	−	0.819038i	\(-0.305492\pi\)
							−0.173452	+	0.984842i	\(0.555492\pi\)
\(31\)	1127.72			1.17349			0.586745	−	0.809772i	\(-0.300409\pi\)
							0.586745	+	0.809772i	\(0.300409\pi\)
\(37\)	1039.33	−	430.506i	0.759192	−	0.314468i	0.0307062	−	0.999528i	\(-0.490224\pi\)
							0.728486	+	0.685061i	\(0.240224\pi\)
\(41\)	800.472	−	800.472i	0.476188	−	0.476188i	−0.427722	−	0.903910i	\(-0.640684\pi\)
							0.903910	+	0.427722i	\(0.140684\pi\)
\(43\)	−21.1600	−	51.0848i	−0.0114440	−	0.0276283i	0.918054	−	0.396455i	\(-0.129760\pi\)
							−0.929498	+	0.368826i	\(0.879760\pi\)
\(47\)	3946.50			1.78656			0.893278	−	0.449504i	\(-0.148399\pi\)
							0.893278	+	0.449504i	\(0.148399\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.5.p.a.77.54	yes	248
3.2	odd	2	inner	96.5.p.a.77.9	yes	248
32.5	even	8	inner	96.5.p.a.5.9	✓	248
96.5	odd	8	inner	96.5.p.a.5.54	yes	248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.5.p.a.5.9	✓	248	32.5	even	8	inner
96.5.p.a.5.54	yes	248	96.5	odd	8	inner
96.5.p.a.77.9	yes	248	3.2	odd	2	inner
96.5.p.a.77.54	yes	248	1.1	even	1	trivial