Embedded newform 16.13.f.a.3.20

• Label: 16.13.f.a.3.20
• 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			3.20
Character	\(\chi\)	\(=\)	16.3
Dual form			16.13.f.a.11.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(58.9783 - 24.8508i) q^{2} +(-513.579 + 513.579i) q^{3} +(2860.88 - 2931.31i) q^{4} +(208.882 - 208.882i) q^{5} +(-17527.2 + 43052.8i) q^{6} +42912.4 q^{7} +(95884.1 - 243979. i) q^{8} +3914.51i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 2 q^{2} - 2 q^{3} - 4232 q^{4} - 2 q^{5} + 72928 q^{6} - 4 q^{7} - 478172 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	58.9783	−	24.8508i	0.921536	−	0.388294i
							\(3\)	−513.579	+	513.579i	−0.704498	+	0.704498i	−0.965373	−	0.260875i	\(-0.915989\pi\)
0.260875	+	0.965373i	\(0.415989\pi\)
\(5\)	208.882	−	208.882i	0.0133684	−	0.0133684i	−0.700391	−	0.713759i	\(-0.746991\pi\)
							0.713759	+	0.700391i	\(0.246991\pi\)
\(7\)	42912.4			0.364750			0.182375	−	0.983229i	\(-0.441622\pi\)
							0.182375	+	0.983229i	\(0.441622\pi\)
\(11\)	1.88113e6	+	1.88113e6i	1.06185	+	1.06185i	0.997957	+	0.0638897i	\(0.0203506\pi\)
							0.0638897	+	0.997957i	\(0.479649\pi\)
\(13\)	4.89666e6	+	4.89666e6i	1.01447	+	1.01447i	0.999894	+	0.0145777i	\(0.00464038\pi\)
							0.0145777	+	0.999894i	\(0.495360\pi\)
\(17\)	160536.			0.00665088			0.00332544	−	0.999994i	\(-0.498941\pi\)
							0.00332544	+	0.999994i	\(0.498941\pi\)
\(19\)	2.46125e7	−	2.46125e7i	0.523161	−	0.523161i	−0.395364	−	0.918525i	\(-0.629382\pi\)
							0.918525	+	0.395364i	\(0.129382\pi\)
\(23\)	7.27551e7			0.491469			0.245735	−	0.969337i	\(-0.420971\pi\)
							0.245735	+	0.969337i	\(0.420971\pi\)
\(29\)	−1.76501e7	−	1.76501e7i	−0.0296728	−	0.0296728i	0.692115	−	0.721788i	\(-0.256679\pi\)
							−0.721788	+	0.692115i	\(0.756679\pi\)
\(31\)		−	2.66281e8i		−	0.300034i	−0.988683	−	0.150017i	\(-0.952067\pi\)
							0.988683	−	0.150017i	\(-0.0479328\pi\)
\(37\)	−1.26642e9	+	1.26642e9i	−0.493591	+	0.493591i	−0.909436	−	0.415844i	\(-0.863486\pi\)
							0.415844	+	0.909436i	\(0.363486\pi\)
\(41\)		−	1.57572e9i		−	0.331724i	−0.986149	−	0.165862i	\(-0.946959\pi\)
							0.986149	−	0.165862i	\(-0.0530406\pi\)
\(43\)	−5.30330e9	−	5.30330e9i	−0.838949	−	0.838949i	0.149771	−	0.988721i	\(-0.452146\pi\)
							−0.988721	+	0.149771i	\(0.952146\pi\)
\(47\)			1.04580e10i			0.970197i	0.874460	+	0.485098i	\(0.161216\pi\)
							−0.874460	+	0.485098i	\(0.838784\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.13.f.a.3.20	✓	46
4.3	odd	2		64.13.f.a.47.18		46
8.3	odd	2		128.13.f.a.95.6		46
8.5	even	2		128.13.f.b.95.18		46
16.3	odd	4		128.13.f.b.31.18		46
16.5	even	4		64.13.f.a.15.18		46
16.11	odd	4	inner	16.13.f.a.11.20	yes	46
16.13	even	4		128.13.f.a.31.6		46

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.13.f.a.3.20	✓	46	1.1	even	1	trivial
16.13.f.a.11.20	yes	46	16.11	odd	4	inner
64.13.f.a.15.18		46	16.5	even	4
64.13.f.a.47.18		46	4.3	odd	2
128.13.f.a.31.6		46	16.13	even	4
128.13.f.a.95.6		46	8.3	odd	2
128.13.f.b.31.18		46	16.3	odd	4
128.13.f.b.95.18		46	8.5	even	2