Embedded newform 16.13.f.a.3.17

• Label: 16.13.f.a.3.17
• Level: $16$
• Weight: $13$
• Character: 16.3
• Analytic conductor: $14.624$
• Analytic rank: $0$
• Dimension: $46$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.6239010764\)
Analytic rank:	\(0\)
Dimension:	\(46\)
Relative dimension:	\(23\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.17
Character	\(\chi\)	\(=\)	16.3
Dual form			16.13.f.a.11.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(40.2856 + 49.7300i) q^{2} +(87.0633 - 87.0633i) q^{3} +(-850.147 + 4006.80i) q^{4} +(8119.76 - 8119.76i) q^{5} +(7837.05 + 822.264i) q^{6} +149756. q^{7} +(-233507. + 119138. i) q^{8} +516281. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 2 q^{2} - 2 q^{3} - 4232 q^{4} - 2 q^{5} + 72928 q^{6} - 4 q^{7} - 478172 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	40.2856	+	49.7300i	0.629462	+	0.777031i
							\(3\)	87.0633	−	87.0633i	0.119428	−	0.119428i	−0.644867	−	0.764295i	\(-0.723087\pi\)
0.764295	+	0.644867i	\(0.223087\pi\)
\(5\)	8119.76	−	8119.76i	0.519665	−	0.519665i	−0.397805	−	0.917470i	\(-0.630228\pi\)
							0.917470	+	0.397805i	\(0.130228\pi\)
\(7\)	149756.			1.27290			0.636451	−	0.771317i	\(-0.280402\pi\)
							0.636451	+	0.771317i	\(0.280402\pi\)
\(11\)	193498.	+	193498.i	0.109224	+	0.109224i	0.759607	−	0.650382i	\(-0.225391\pi\)
							−0.650382	+	0.759607i	\(0.725391\pi\)
\(13\)	2.19385e6	+	2.19385e6i	0.454514	+	0.454514i	0.896850	−	0.442335i	\(-0.145850\pi\)
							−0.442335	+	0.896850i	\(0.645850\pi\)
\(17\)	3.62781e6			0.150297			0.0751487	−	0.997172i	\(-0.476057\pi\)
							0.0751487	+	0.997172i	\(0.476057\pi\)
\(19\)	−5.40686e7	+	5.40686e7i	−1.14927	+	1.14927i	−0.162577	+	0.986696i	\(0.551981\pi\)
							−0.986696	+	0.162577i	\(0.948019\pi\)
\(23\)	1.88267e8			1.27176			0.635882	−	0.771786i	\(-0.280637\pi\)
							0.635882	+	0.771786i	\(0.280637\pi\)
\(29\)	−2.28285e7	−	2.28285e7i	−0.0383786	−	0.0383786i	0.687657	−	0.726036i	\(-0.258639\pi\)
							−0.726036	+	0.687657i	\(0.758639\pi\)
\(31\)		−	1.07059e9i		−	1.20630i	−0.797629	−	0.603149i	\(-0.793912\pi\)
							0.797629	−	0.603149i	\(-0.206088\pi\)
\(37\)	2.33298e9	−	2.33298e9i	0.909286	−	0.909286i	−0.0869285	−	0.996215i	\(-0.527705\pi\)
							0.996215	+	0.0869285i	\(0.0277052\pi\)
\(41\)		−	8.66967e9i		−	1.82515i	−0.408905	−	0.912577i	\(-0.634089\pi\)
							0.408905	−	0.912577i	\(-0.365911\pi\)
\(43\)	1.65062e9	+	1.65062e9i	0.261118	+	0.261118i	0.825508	−	0.564390i	\(-0.190889\pi\)
							−0.564390	+	0.825508i	\(0.690889\pi\)
\(47\)			1.96236e9i			0.182050i	0.995849	+	0.0910250i	\(0.0290143\pi\)
							−0.995849	+	0.0910250i	\(0.970986\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.13.f.a.3.17	✓	46
4.3	odd	2		64.13.f.a.47.11		46
8.3	odd	2		128.13.f.a.95.13		46
8.5	even	2		128.13.f.b.95.11		46
16.3	odd	4		128.13.f.b.31.11		46
16.5	even	4		64.13.f.a.15.11		46
16.11	odd	4	inner	16.13.f.a.11.17	yes	46
16.13	even	4		128.13.f.a.31.13		46

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.13.f.a.3.17	✓	46	1.1	even	1	trivial
16.13.f.a.11.17	yes	46	16.11	odd	4	inner
64.13.f.a.15.11		46	16.5	even	4
64.13.f.a.47.11		46	4.3	odd	2
128.13.f.a.31.13		46	16.13	even	4
128.13.f.a.95.13		46	8.3	odd	2
128.13.f.b.31.11		46	16.3	odd	4
128.13.f.b.95.11		46	8.5	even	2