Embedded newform 16.8.e.a.5.2

• Label: 16.8.e.a.5.2
• Level: $16$
• Weight: $8$
• Character: 16.5
• Analytic conductor: $4.998$
• Analytic rank: $0$
• Dimension: $26$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.99816040775\)
Analytic rank:	\(0\)
Dimension:	\(26\)
Relative dimension:	\(13\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.2
Character	\(\chi\)	\(=\)	16.5
Dual form			16.8.e.a.13.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.5089 - 4.19081i) q^{2} +(42.2145 - 42.2145i) q^{3} +(92.8743 + 88.0816i) q^{4} +(46.9348 + 46.9348i) q^{5} +(-620.541 + 266.716i) q^{6} -1384.27i q^{7} +(-606.875 - 1314.86i) q^{8} -1377.13i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 2 q^{2} - 2 q^{3} - 184 q^{4} - 2 q^{5} - 176 q^{6} - 1004 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.5089	−	4.19081i	−0.928865	−	0.370418i
							\(3\)	42.2145	−	42.2145i	0.902688	−	0.902688i	−0.0929803	−	0.995668i	\(-0.529639\pi\)
0.995668	+	0.0929803i	\(0.0296394\pi\)
\(5\)	46.9348	+	46.9348i	0.167919	+	0.167919i	0.786064	−	0.618145i	\(-0.212116\pi\)
							−0.618145	+	0.786064i	\(0.712116\pi\)
\(7\)		−	1384.27i		−	1.52537i	−0.646767	−	0.762687i	\(-0.723880\pi\)
							0.646767	−	0.762687i	\(-0.276120\pi\)
\(11\)	−3275.70	−	3275.70i	−0.742044	−	0.742044i	0.230927	−	0.972971i	\(-0.425824\pi\)
							−0.972971	+	0.230927i	\(0.925824\pi\)
\(13\)	−2195.79	+	2195.79i	−0.277197	+	0.277197i	−0.831989	−	0.554792i	\(-0.812798\pi\)
							0.554792	+	0.831989i	\(0.312798\pi\)
\(17\)	13239.8			0.653599			0.326800	−	0.945094i	\(-0.394030\pi\)
							0.326800	+	0.945094i	\(0.394030\pi\)
\(19\)	34559.1	−	34559.1i	1.15591	−	1.15591i	0.170564	−	0.985347i	\(-0.445441\pi\)
							0.985347	−	0.170564i	\(-0.0545589\pi\)
\(23\)			96231.5i			1.64919i	0.565726	+	0.824594i	\(0.308596\pi\)
							−0.565726	+	0.824594i	\(0.691404\pi\)
\(29\)	−80629.4	+	80629.4i	−0.613904	+	0.613904i	−0.943961	−	0.330057i	\(-0.892932\pi\)
							0.330057	+	0.943961i	\(0.392932\pi\)
\(31\)	121994.			0.735485			0.367743	−	0.929928i	\(-0.380131\pi\)
							0.367743	+	0.929928i	\(0.380131\pi\)
\(37\)	209240.	+	209240.i	0.679109	+	0.679109i	0.959799	−	0.280690i	\(-0.0905632\pi\)
							−0.280690	+	0.959799i	\(0.590563\pi\)
\(41\)			65043.0i			0.147386i	0.997281	+	0.0736931i	\(0.0234785\pi\)
							−0.997281	+	0.0736931i	\(0.976521\pi\)
\(43\)	467435.	+	467435.i	0.896565	+	0.896565i	0.995131	−	0.0985657i	\(-0.0314255\pi\)
							−0.0985657	+	0.995131i	\(0.531425\pi\)
\(47\)	−498778.			−0.700752			−0.350376	−	0.936609i	\(-0.613946\pi\)
							−0.350376	+	0.936609i	\(0.613946\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.8.e.a.5.2	✓	26
4.3	odd	2		64.8.e.a.49.3		26
8.3	odd	2		128.8.e.a.97.11		26
8.5	even	2		128.8.e.b.97.3		26
16.3	odd	4		64.8.e.a.17.3		26
16.5	even	4		128.8.e.b.33.3		26
16.11	odd	4		128.8.e.a.33.11		26
16.13	even	4	inner	16.8.e.a.13.2	yes	26

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.8.e.a.5.2	✓	26	1.1	even	1	trivial
16.8.e.a.13.2	yes	26	16.13	even	4	inner
64.8.e.a.17.3		26	16.3	odd	4
64.8.e.a.49.3		26	4.3	odd	2
128.8.e.a.33.11		26	16.11	odd	4
128.8.e.a.97.11		26	8.3	odd	2
128.8.e.b.33.3		26	16.5	even	4
128.8.e.b.97.3		26	8.5	even	2