Embedded newform 16.13.f.a.3.7

• Label: 16.13.f.a.3.7
• 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			3.7
Character	\(\chi\)	\(=\)	16.3
Dual form			16.13.f.a.11.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-48.1592 - 42.1508i) q^{2} +(-640.742 + 640.742i) q^{3} +(542.622 + 4059.90i) q^{4} +(10737.8 - 10737.8i) q^{5} +(57865.4 - 3849.85i) q^{6} +25233.9 q^{7} +(144996. - 218394. i) q^{8} -289658. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 2 q^{2} - 2 q^{3} - 4232 q^{4} - 2 q^{5} + 72928 q^{6} - 4 q^{7} - 478172 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−48.1592	−	42.1508i	−0.752488	−	0.658606i
							\(3\)	−640.742	+	640.742i	−0.878932	+	0.878932i	−0.993424	−	0.114492i	\(-0.963476\pi\)
0.114492	+	0.993424i	\(0.463476\pi\)
\(5\)	10737.8	−	10737.8i	0.687217	−	0.687217i	−0.274399	−	0.961616i	\(-0.588479\pi\)
							0.961616	+	0.274399i	\(0.0884788\pi\)
\(7\)	25233.9			0.214485			0.107242	−	0.994233i	\(-0.465798\pi\)
							0.107242	+	0.994233i	\(0.465798\pi\)
\(11\)	−899119.	−	899119.i	−0.507529	−	0.507529i	0.406238	−	0.913767i	\(-0.366840\pi\)
							−0.913767	+	0.406238i	\(0.866840\pi\)
\(13\)	4.57967e6	+	4.57967e6i	0.948798	+	0.948798i	0.998752	−	0.0499533i	\(-0.0159072\pi\)
							−0.0499533	+	0.998752i	\(0.515907\pi\)
\(17\)	1.36851e7			0.566964			0.283482	−	0.958978i	\(-0.408510\pi\)
							0.283482	+	0.958978i	\(0.408510\pi\)
\(19\)	−3.74694e7	+	3.74694e7i	−0.796443	+	0.796443i	−0.982533	−	0.186089i	\(-0.940419\pi\)
							0.186089	+	0.982533i	\(0.440419\pi\)
\(23\)	−2.83954e8			−1.91814			−0.959071	−	0.283166i	\(-0.908615\pi\)
							−0.959071	+	0.283166i	\(0.908615\pi\)
\(29\)	−2.95130e8	−	2.95130e8i	−0.496165	−	0.496165i	0.414077	−	0.910242i	\(-0.364104\pi\)
							−0.910242	+	0.414077i	\(0.864104\pi\)
\(31\)			6.90139e8i			0.777619i	0.921318	+	0.388809i	\(0.127113\pi\)
							−0.921318	+	0.388809i	\(0.872887\pi\)
\(37\)	−1.48153e9	+	1.48153e9i	−0.577433	+	0.577433i	−0.934195	−	0.356763i	\(-0.883881\pi\)
							0.356763	+	0.934195i	\(0.383881\pi\)
\(41\)		−	3.18480e9i		−	0.670470i	−0.942135	−	0.335235i	\(-0.891184\pi\)
							0.942135	−	0.335235i	\(-0.108816\pi\)
\(43\)	1.98699e9	+	1.98699e9i	0.314330	+	0.314330i	0.846584	−	0.532254i	\(-0.178655\pi\)
							−0.532254	+	0.846584i	\(0.678655\pi\)
\(47\)		−	3.62834e9i		−	0.336605i	−0.985735	−	0.168303i	\(-0.946171\pi\)
							0.985735	−	0.168303i	\(-0.0538286\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.13.f.a.3.7	✓	46
4.3	odd	2		64.13.f.a.47.20		46
8.3	odd	2		128.13.f.a.95.4		46
8.5	even	2		128.13.f.b.95.20		46
16.3	odd	4		128.13.f.b.31.20		46
16.5	even	4		64.13.f.a.15.20		46
16.11	odd	4	inner	16.13.f.a.11.7	yes	46
16.13	even	4		128.13.f.a.31.4		46

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.13.f.a.3.7	✓	46	1.1	even	1	trivial
16.13.f.a.11.7	yes	46	16.11	odd	4	inner
64.13.f.a.15.20		46	16.5	even	4
64.13.f.a.47.20		46	4.3	odd	2
128.13.f.a.31.4		46	16.13	even	4
128.13.f.a.95.4		46	8.3	odd	2
128.13.f.b.31.20		46	16.3	odd	4
128.13.f.b.95.20		46	8.5	even	2