Embedded newform 16.13.f.a.3.5

• Label: 16.13.f.a.3.5
• 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			3.5
Character	\(\chi\)	\(=\)	16.3
Dual form			16.13.f.a.11.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-50.3472 - 39.5115i) q^{2} +(739.301 - 739.301i) q^{3} +(973.684 + 3978.59i) q^{4} +(-805.082 + 805.082i) q^{5} +(-66432.6 + 8010.85i) q^{6} -97582.7 q^{7} +(108178. - 238782. i) q^{8} -561690. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 2 q^{2} - 2 q^{3} - 4232 q^{4} - 2 q^{5} + 72928 q^{6} - 4 q^{7} - 478172 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−50.3472	−	39.5115i	−0.786675	−	0.617367i
							\(3\)	739.301	−	739.301i	1.01413	−	1.01413i	0.0142311	−	0.999899i	\(-0.495470\pi\)
0.999899	−	0.0142311i	\(-0.00453006\pi\)
\(5\)	−805.082	+	805.082i	−0.0515252	+	0.0515252i	−0.732400	−	0.680875i	\(-0.761600\pi\)
							0.680875	+	0.732400i	\(0.261600\pi\)
\(7\)	−97582.7			−0.829439			−0.414719	−	0.909949i	\(-0.636120\pi\)
							−0.414719	+	0.909949i	\(0.636120\pi\)
\(11\)	−1.24686e6	−	1.24686e6i	−0.703821	−	0.703821i	0.261408	−	0.965228i	\(-0.415813\pi\)
							−0.965228	+	0.261408i	\(0.915813\pi\)
\(13\)	−1.01971e6	−	1.01971e6i	−0.211260	−	0.211260i	0.593543	−	0.804803i	\(-0.297729\pi\)
							−0.804803	+	0.593543i	\(0.797729\pi\)
\(17\)	−3.87419e7			−1.60505			−0.802524	−	0.596620i	\(-0.796510\pi\)
							−0.802524	+	0.596620i	\(0.796510\pi\)
\(19\)	−2.64090e7	+	2.64090e7i	−0.561346	+	0.561346i	−0.929690	−	0.368343i	\(-0.879925\pi\)
							0.368343	+	0.929690i	\(0.379925\pi\)
\(23\)	1.86324e8			1.25864			0.629320	−	0.777147i	\(-0.283334\pi\)
							0.629320	+	0.777147i	\(0.283334\pi\)
\(29\)	−5.99082e8	−	5.99082e8i	−1.00716	−	1.00716i	−0.999974	−	0.00718598i	\(-0.997713\pi\)
							−0.00718598	−	0.999974i	\(-0.502287\pi\)
\(31\)			1.96510e7i			0.0221419i	0.999939	+	0.0110709i	\(0.00352406\pi\)
							−0.999939	+	0.0110709i	\(0.996476\pi\)
\(37\)	7.61799e8	−	7.61799e8i	0.296914	−	0.296914i	−0.542890	−	0.839804i	\(-0.682670\pi\)
							0.839804	+	0.542890i	\(0.182670\pi\)
\(41\)			3.33992e8i			0.0703125i	0.999382	+	0.0351563i	\(0.0111929\pi\)
							−0.999382	+	0.0351563i	\(0.988807\pi\)
\(43\)	−7.52677e9	−	7.52677e9i	−1.19069	−	1.19069i	−0.976875	−	0.213813i	\(-0.931412\pi\)
							−0.213813	−	0.976875i	\(-0.568588\pi\)
\(47\)			1.17553e10i			1.09055i	0.838257	+	0.545275i	\(0.183575\pi\)
							−0.838257	+	0.545275i	\(0.816425\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.13.f.a.3.5	✓	46
4.3	odd	2		64.13.f.a.47.4		46
8.3	odd	2		128.13.f.a.95.20		46
8.5	even	2		128.13.f.b.95.4		46
16.3	odd	4		128.13.f.b.31.4		46
16.5	even	4		64.13.f.a.15.4		46
16.11	odd	4	inner	16.13.f.a.11.5	yes	46
16.13	even	4		128.13.f.a.31.20		46

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.13.f.a.3.5	✓	46	1.1	even	1	trivial
16.13.f.a.11.5	yes	46	16.11	odd	4	inner
64.13.f.a.15.4		46	16.5	even	4
64.13.f.a.47.4		46	4.3	odd	2
128.13.f.a.31.20		46	16.13	even	4
128.13.f.a.95.20		46	8.3	odd	2
128.13.f.b.31.4		46	16.3	odd	4
128.13.f.b.95.4		46	8.5	even	2