Embedded newform 16.13.f.a.3.11

• Label: 16.13.f.a.3.11
• 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			3.11
Character	\(\chi\)	\(=\)	16.3
Dual form			16.13.f.a.11.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.53819 + 63.9496i) q^{2} +(-322.288 + 322.288i) q^{3} +(-4083.12 + 324.633i) q^{4} +(-16463.9 + 16463.9i) q^{5} +(-21428.3 - 19792.2i) q^{6} +21051.9 q^{7} +(-31123.9 - 260290. i) q^{8} +323701. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 2 q^{2} - 2 q^{3} - 4232 q^{4} - 2 q^{5} + 72928 q^{6} - 4 q^{7} - 478172 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.53819	+	63.9496i	0.0396593	+	0.999213i
							\(3\)	−322.288	+	322.288i	−0.442097	+	0.442097i	−0.892716	−	0.450620i	\(-0.851203\pi\)
0.450620	+	0.892716i	\(0.351203\pi\)
\(5\)	−16463.9	+	16463.9i	−1.05369	+	1.05369i	−0.0552178	+	0.998474i	\(0.517585\pi\)
							−0.998474	+	0.0552178i	\(0.982415\pi\)
\(7\)	21051.9			0.178938			0.0894689	−	0.995990i	\(-0.471483\pi\)
							0.0894689	+	0.995990i	\(0.471483\pi\)
\(11\)	−684046.	−	684046.i	−0.386126	−	0.386126i	0.487177	−	0.873303i	\(-0.338027\pi\)
							−0.873303	+	0.487177i	\(0.838027\pi\)
\(13\)	1.16664e6	+	1.16664e6i	0.241699	+	0.241699i	0.817553	−	0.575854i	\(-0.195330\pi\)
							−0.575854	+	0.817553i	\(0.695330\pi\)
\(17\)	1.51231e7			0.626537			0.313269	−	0.949665i	\(-0.398576\pi\)
							0.313269	+	0.949665i	\(0.398576\pi\)
\(19\)	1.09747e7	−	1.09747e7i	0.233278	−	0.233278i	−0.580782	−	0.814059i	\(-0.697253\pi\)
							0.814059	+	0.580782i	\(0.197253\pi\)
\(23\)	7.74575e7			0.523234			0.261617	−	0.965172i	\(-0.415744\pi\)
							0.261617	+	0.965172i	\(0.415744\pi\)
\(29\)	−8.16722e8	−	8.16722e8i	−1.37305	−	1.37305i	−0.855888	−	0.517161i	\(-0.826989\pi\)
							−0.517161	−	0.855888i	\(-0.673011\pi\)
\(31\)			1.34968e9i			1.52076i	0.649477	+	0.760382i	\(0.274988\pi\)
							−0.649477	+	0.760382i	\(0.725012\pi\)
\(37\)	−2.54864e9	+	2.54864e9i	−0.993340	+	0.993340i	−0.999978	−	0.00663752i	\(-0.997887\pi\)
							0.00663752	+	0.999978i	\(0.497887\pi\)
\(41\)			5.12620e8i			0.107918i	0.998543	+	0.0539588i	\(0.0171840\pi\)
							−0.998543	+	0.0539588i	\(0.982816\pi\)
\(43\)	−6.63624e9	−	6.63624e9i	−1.04981	−	1.04981i	−0.998693	−	0.0511192i	\(-0.983721\pi\)
							−0.0511192	−	0.998693i	\(-0.516279\pi\)
\(47\)		−	1.21981e10i		−	1.13163i	−0.824531	−	0.565817i	\(-0.808561\pi\)
							0.824531	−	0.565817i	\(-0.191439\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.13.f.a.3.11	✓	46
4.3	odd	2		64.13.f.a.47.15		46
8.3	odd	2		128.13.f.a.95.9		46
8.5	even	2		128.13.f.b.95.15		46
16.3	odd	4		128.13.f.b.31.15		46
16.5	even	4		64.13.f.a.15.15		46
16.11	odd	4	inner	16.13.f.a.11.11	yes	46
16.13	even	4		128.13.f.a.31.9		46

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.13.f.a.3.11	✓	46	1.1	even	1	trivial
16.13.f.a.11.11	yes	46	16.11	odd	4	inner
64.13.f.a.15.15		46	16.5	even	4
64.13.f.a.47.15		46	4.3	odd	2
128.13.f.a.31.9		46	16.13	even	4
128.13.f.a.95.9		46	8.3	odd	2
128.13.f.b.31.15		46	16.3	odd	4
128.13.f.b.95.15		46	8.5	even	2