Embedded newform 16.11.f.a.3.16

• Label: 16.11.f.a.3.16
• Level: $16$
• Weight: $11$
• Character: 16.3
• Analytic conductor: $10.166$
• Analytic rank: $0$
• Dimension: $38$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1657160428\)
Analytic rank:	\(0\)
Dimension:	\(38\)
Relative dimension:	\(19\) 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			16.11.f.a.11.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(27.4041 + 16.5231i) q^{2} +(305.982 - 305.982i) q^{3} +(477.973 + 905.603i) q^{4} +(155.264 - 155.264i) q^{5} +(13440.9 - 3329.39i) q^{6} +8523.80 q^{7} +(-1864.95 + 32714.9i) q^{8} -128200. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 2 q^{3} - 1256 q^{4} - 2 q^{5} - 17216 q^{6} - 4 q^{7} + 69124 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	27.4041	+	16.5231i	0.856379	+	0.516347i
							\(3\)	305.982	−	305.982i	1.25918	−	1.25918i	0.307700	−	0.951484i	\(-0.400441\pi\)
0.951484	−	0.307700i	\(-0.0995592\pi\)
\(5\)	155.264	−	155.264i	0.0496845	−	0.0496845i	−0.681828	−	0.731513i	\(-0.738815\pi\)
							0.731513	+	0.681828i	\(0.238815\pi\)
\(7\)	8523.80			0.507158			0.253579	−	0.967315i	\(-0.418392\pi\)
							0.253579	+	0.967315i	\(0.418392\pi\)
\(11\)	−164652.	−	164652.i	−1.02236	−	1.02236i	−0.999744	−	0.0226151i	\(-0.992801\pi\)
							−0.0226151	−	0.999744i	\(-0.507199\pi\)
\(13\)	409060.	+	409060.i	1.10172	+	1.10172i	0.994204	+	0.107514i	\(0.0342890\pi\)
							0.107514	+	0.994204i	\(0.465711\pi\)
\(17\)	941406.			0.663029			0.331514	−	0.943450i	\(-0.392440\pi\)
							0.331514	+	0.943450i	\(0.392440\pi\)
\(19\)	−1.22085e6	+	1.22085e6i	−0.493053	+	0.493053i	−0.909267	−	0.416213i	\(-0.863357\pi\)
							0.416213	+	0.909267i	\(0.363357\pi\)
\(23\)	−7.27437e6			−1.13020			−0.565101	−	0.825022i	\(-0.691163\pi\)
							−0.565101	+	0.825022i	\(0.691163\pi\)
\(29\)	−1.86281e7	−	1.86281e7i	−0.908191	−	0.908191i	0.0879348	−	0.996126i	\(-0.471973\pi\)
							−0.996126	+	0.0879348i	\(0.971973\pi\)
\(31\)			1.16717e7i			0.407684i	0.979004	+	0.203842i	\(0.0653429\pi\)
							−0.979004	+	0.203842i	\(0.934657\pi\)
\(37\)	1.73137e7	−	1.73137e7i	0.249678	−	0.249678i	−0.571160	−	0.820838i	\(-0.693507\pi\)
							0.820838	+	0.571160i	\(0.193507\pi\)
\(41\)			1.55012e8i			1.33796i	0.743278	+	0.668982i	\(0.233270\pi\)
							−0.743278	+	0.668982i	\(0.766730\pi\)
\(43\)	−9.07108e7	−	9.07108e7i	−0.617045	−	0.617045i	0.327727	−	0.944772i	\(-0.393717\pi\)
							−0.944772	+	0.327727i	\(0.893717\pi\)
\(47\)			1.93016e7i			0.0841596i	0.999114	+	0.0420798i	\(0.0133984\pi\)
							−0.999114	+	0.0420798i	\(0.986602\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.11.f.a.3.16	✓	38
4.3	odd	2		64.11.f.a.47.1		38
8.3	odd	2		128.11.f.a.95.19		38
8.5	even	2		128.11.f.b.95.1		38
16.3	odd	4		128.11.f.b.31.1		38
16.5	even	4		64.11.f.a.15.1		38
16.11	odd	4	inner	16.11.f.a.11.16	yes	38
16.13	even	4		128.11.f.a.31.19		38

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.11.f.a.3.16	✓	38	1.1	even	1	trivial
16.11.f.a.11.16	yes	38	16.11	odd	4	inner
64.11.f.a.15.1		38	16.5	even	4
64.11.f.a.47.1		38	4.3	odd	2
128.11.f.a.31.19		38	16.13	even	4
128.11.f.a.95.19		38	8.3	odd	2
128.11.f.b.31.1		38	16.3	odd	4
128.11.f.b.95.1		38	8.5	even	2