Embedded newform 16.13.f.a.3.18

• Label: 16.13.f.a.3.18
• 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			3.18
Character	\(\chi\)	\(=\)	16.3
Dual form			16.13.f.a.11.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(41.7334 - 48.5213i) q^{2} +(193.300 - 193.300i) q^{3} +(-612.641 - 4049.92i) q^{4} +(-10355.8 + 10355.8i) q^{5} +(-1312.10 - 17446.2i) q^{6} -108466. q^{7} +(-222075. - 139291. i) q^{8} +456712. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 2 q^{2} - 2 q^{3} - 4232 q^{4} - 2 q^{5} + 72928 q^{6} - 4 q^{7} - 478172 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	41.7334	−	48.5213i	0.652085	−	0.758146i
							\(3\)	193.300	−	193.300i	0.265157	−	0.265157i	−0.561988	−	0.827145i	\(-0.689963\pi\)
0.827145	+	0.561988i	\(0.189963\pi\)
\(5\)	−10355.8	+	10355.8i	−0.662774	+	0.662774i	−0.956033	−	0.293259i	\(-0.905260\pi\)
							0.293259	+	0.956033i	\(0.405260\pi\)
\(7\)	−108466.			−0.921944			−0.460972	−	0.887415i	\(-0.652499\pi\)
							−0.460972	+	0.887415i	\(0.652499\pi\)
\(11\)	−1.04650e6	−	1.04650e6i	−0.590721	−	0.590721i	0.347105	−	0.937826i	\(-0.387165\pi\)
							−0.937826	+	0.347105i	\(0.887165\pi\)
\(13\)	−1.27642e6	−	1.27642e6i	−0.264445	−	0.264445i	0.562412	−	0.826857i	\(-0.309873\pi\)
							−0.826857	+	0.562412i	\(0.809873\pi\)
\(17\)	1.95112e7			0.808332			0.404166	−	0.914686i	\(-0.367562\pi\)
							0.404166	+	0.914686i	\(0.367562\pi\)
\(19\)	−2.72724e7	+	2.72724e7i	−0.579698	+	0.579698i	−0.934820	−	0.355122i	\(-0.884439\pi\)
							0.355122	+	0.934820i	\(0.384439\pi\)
\(23\)	−1.76773e8			−1.19412			−0.597060	−	0.802197i	\(-0.703665\pi\)
							−0.597060	+	0.802197i	\(0.703665\pi\)
\(29\)	−3.27663e8	−	3.27663e8i	−0.550858	−	0.550858i	0.375830	−	0.926688i	\(-0.377358\pi\)
							−0.926688	+	0.375830i	\(0.877358\pi\)
\(31\)		−	1.40311e9i		−	1.58096i	−0.612487	−	0.790480i	\(-0.709831\pi\)
							0.612487	−	0.790480i	\(-0.290169\pi\)
\(37\)	3.08303e9	−	3.08303e9i	1.20162	−	1.20162i	0.227949	−	0.973673i	\(-0.426798\pi\)
							0.973673	−	0.227949i	\(-0.0732019\pi\)
\(41\)		−	1.10296e9i		−	0.232197i	−0.993238	−	0.116098i	\(-0.962961\pi\)
							0.993238	−	0.116098i	\(-0.0370388\pi\)
\(43\)	−4.13826e9	−	4.13826e9i	−0.654646	−	0.654646i	0.299462	−	0.954108i	\(-0.403193\pi\)
							−0.954108	+	0.299462i	\(0.903193\pi\)
\(47\)		−	7.39074e9i		−	0.685647i	−0.939400	−	0.342824i	\(-0.888617\pi\)
							0.939400	−	0.342824i	\(-0.111383\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.13.f.a.3.18	✓	46
4.3	odd	2		64.13.f.a.47.10		46
8.3	odd	2		128.13.f.a.95.14		46
8.5	even	2		128.13.f.b.95.10		46
16.3	odd	4		128.13.f.b.31.10		46
16.5	even	4		64.13.f.a.15.10		46
16.11	odd	4	inner	16.13.f.a.11.18	yes	46
16.13	even	4		128.13.f.a.31.14		46

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.13.f.a.3.18	✓	46	1.1	even	1	trivial
16.13.f.a.11.18	yes	46	16.11	odd	4	inner
64.13.f.a.15.10		46	16.5	even	4
64.13.f.a.47.10		46	4.3	odd	2
128.13.f.a.31.14		46	16.13	even	4
128.13.f.a.95.14		46	8.3	odd	2
128.13.f.b.31.10		46	16.3	odd	4
128.13.f.b.95.10		46	8.5	even	2