Embedded newform 16.13.f.a.3.16

• Label: 16.13.f.a.3.16
• 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			3.16
Character	\(\chi\)	\(=\)	16.3
Dual form			16.13.f.a.11.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(26.6058 - 58.2077i) q^{2} +(-435.597 + 435.597i) q^{3} +(-2680.26 - 3097.32i) q^{4} +(17337.6 - 17337.6i) q^{5} +(13765.7 + 36944.5i) q^{6} +102408. q^{7} +(-251598. + 73605.0i) q^{8} +151952. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 2 q^{2} - 2 q^{3} - 4232 q^{4} - 2 q^{5} + 72928 q^{6} - 4 q^{7} - 478172 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	26.6058	−	58.2077i	0.415716	−	0.909495i
							\(3\)	−435.597	+	435.597i	−0.597526	+	0.597526i	−0.939654	−	0.342127i	\(-0.888853\pi\)
0.342127	+	0.939654i	\(0.388853\pi\)
\(5\)	17337.6	−	17337.6i	1.10961	−	1.10961i	0.116406	−	0.993202i	\(-0.462862\pi\)
							0.993202	−	0.116406i	\(-0.0371375\pi\)
\(7\)	102408.			0.870452			0.435226	−	0.900321i	\(-0.356669\pi\)
							0.435226	+	0.900321i	\(0.356669\pi\)
\(11\)	−1.38379e6	−	1.38379e6i	−0.781112	−	0.781112i	0.198907	−	0.980018i	\(-0.436261\pi\)
							−0.980018	+	0.198907i	\(0.936261\pi\)
\(13\)	−5.77232e6	−	5.77232e6i	−1.19589	−	1.19589i	−0.975387	−	0.220502i	\(-0.929231\pi\)
							−0.220502	−	0.975387i	\(-0.570769\pi\)
\(17\)	−3.78167e7			−1.56672			−0.783358	−	0.621571i	\(-0.786495\pi\)
							−0.783358	+	0.621571i	\(0.786495\pi\)
\(19\)	2.52129e7	−	2.52129e7i	0.535922	−	0.535922i	−0.386406	−	0.922329i	\(-0.626284\pi\)
							0.922329	+	0.386406i	\(0.126284\pi\)
\(23\)	1.57577e8			1.06445			0.532227	−	0.846602i	\(-0.321355\pi\)
							0.532227	+	0.846602i	\(0.321355\pi\)
\(29\)	5.46743e7	+	5.46743e7i	0.0919168	+	0.0919168i	0.751570	−	0.659653i	\(-0.229297\pi\)
							−0.659653	+	0.751570i	\(0.729297\pi\)
\(31\)			6.13795e8i			0.691597i	0.938309	+	0.345799i	\(0.112392\pi\)
							−0.938309	+	0.345799i	\(0.887608\pi\)
\(37\)	−5.91379e8	+	5.91379e8i	−0.230492	+	0.230492i	−0.812898	−	0.582406i	\(-0.802111\pi\)
							0.582406	+	0.812898i	\(0.302111\pi\)
\(41\)		−	6.47790e9i		−	1.36374i	−0.731474	−	0.681870i	\(-0.761167\pi\)
							0.731474	−	0.681870i	\(-0.238833\pi\)
\(43\)	−6.63091e8	−	6.63091e8i	−0.104897	−	0.104897i	0.652711	−	0.757607i	\(-0.273632\pi\)
							−0.757607	+	0.652711i	\(0.773632\pi\)
\(47\)		−	6.80100e9i		−	0.630937i	−0.948936	−	0.315468i	\(-0.897838\pi\)
							0.948936	−	0.315468i	\(-0.102162\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.13.f.a.3.16	✓	46
4.3	odd	2		64.13.f.a.47.17		46
8.3	odd	2		128.13.f.a.95.7		46
8.5	even	2		128.13.f.b.95.17		46
16.3	odd	4		128.13.f.b.31.17		46
16.5	even	4		64.13.f.a.15.17		46
16.11	odd	4	inner	16.13.f.a.11.16	yes	46
16.13	even	4		128.13.f.a.31.7		46

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.13.f.a.3.16	✓	46	1.1	even	1	trivial
16.13.f.a.11.16	yes	46	16.11	odd	4	inner
64.13.f.a.15.17		46	16.5	even	4
64.13.f.a.47.17		46	4.3	odd	2
128.13.f.a.31.7		46	16.13	even	4
128.13.f.a.95.7		46	8.3	odd	2
128.13.f.b.31.17		46	16.3	odd	4
128.13.f.b.95.17		46	8.5	even	2