Embedded newform 16.11.f.a.3.3

• Label: 16.11.f.a.3.3
• 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			3.3
Character	\(\chi\)	\(=\)	16.3
Dual form			16.11.f.a.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-29.1906 + 13.1115i) q^{2} +(32.7265 - 32.7265i) q^{3} +(680.178 - 765.463i) q^{4} +(-1645.29 + 1645.29i) q^{5} +(-526.211 + 1384.40i) q^{6} +16644.7 q^{7} +(-9818.42 + 31262.4i) q^{8} +56907.0i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 2 q^{3} - 1256 q^{4} - 2 q^{5} - 17216 q^{6} - 4 q^{7} + 69124 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−29.1906	+	13.1115i	−0.912205	+	0.409734i
							\(3\)	32.7265	−	32.7265i	0.134677	−	0.134677i	−0.636555	−	0.771232i	\(-0.719641\pi\)
0.771232	+	0.636555i	\(0.219641\pi\)
\(5\)	−1645.29	+	1645.29i	−0.526491	+	0.526491i	−0.919524	−	0.393033i	\(-0.871426\pi\)
							0.393033	+	0.919524i	\(0.371426\pi\)
\(7\)	16644.7			0.990345			0.495172	−	0.868795i	\(-0.335105\pi\)
							0.495172	+	0.868795i	\(0.335105\pi\)
\(11\)	−206920.	−	206920.i	−1.28481	−	1.28481i	−0.937897	−	0.346914i	\(-0.887229\pi\)
							−0.346914	−	0.937897i	\(-0.612771\pi\)
\(13\)	−260892.	−	260892.i	−0.702659	−	0.702659i	0.262322	−	0.964980i	\(-0.415512\pi\)
							−0.964980	+	0.262322i	\(0.915512\pi\)
\(17\)	−1.45423e6			−1.02421			−0.512106	−	0.858922i	\(-0.671134\pi\)
							−0.512106	+	0.858922i	\(0.671134\pi\)
\(19\)	38220.5	−	38220.5i	0.0154358	−	0.0154358i	−0.699347	−	0.714783i	\(-0.746526\pi\)
							0.714783	+	0.699347i	\(0.246526\pi\)
\(23\)	−7.64364e6			−1.18758			−0.593788	−	0.804622i	\(-0.702368\pi\)
							−0.593788	+	0.804622i	\(0.702368\pi\)
\(29\)	−1.59706e7	−	1.59706e7i	−0.778630	−	0.778630i	0.200968	−	0.979598i	\(-0.435591\pi\)
							−0.979598	+	0.200968i	\(0.935591\pi\)
\(31\)		−	1.35480e6i		−	0.0473223i	−0.999720	−	0.0236612i	\(-0.992468\pi\)
							0.999720	−	0.0236612i	\(-0.00753229\pi\)
\(37\)	−3.25003e7	+	3.25003e7i	−0.468683	+	0.468683i	−0.901488	−	0.432805i	\(-0.857524\pi\)
							0.432805	+	0.901488i	\(0.357524\pi\)
\(41\)		−	1.46638e8i		−	1.26569i	−0.774280	−	0.632843i	\(-0.781888\pi\)
							0.774280	−	0.632843i	\(-0.218112\pi\)
\(43\)	1.13208e7	+	1.13208e7i	0.0770077	+	0.0770077i	0.744562	−	0.667554i	\(-0.232659\pi\)
							−0.667554	+	0.744562i	\(0.732659\pi\)
\(47\)			2.26944e8i			0.989530i	0.869027	+	0.494765i	\(0.164746\pi\)
							−0.869027	+	0.494765i	\(0.835254\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.11.f.a.3.3	✓	38
4.3	odd	2		64.11.f.a.47.9		38
8.3	odd	2		128.11.f.a.95.11		38
8.5	even	2		128.11.f.b.95.9		38
16.3	odd	4		128.11.f.b.31.9		38
16.5	even	4		64.11.f.a.15.9		38
16.11	odd	4	inner	16.11.f.a.11.3	yes	38
16.13	even	4		128.11.f.a.31.11		38

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.11.f.a.3.3	✓	38	1.1	even	1	trivial
16.11.f.a.11.3	yes	38	16.11	odd	4	inner
64.11.f.a.15.9		38	16.5	even	4
64.11.f.a.47.9		38	4.3	odd	2
128.11.f.a.31.11		38	16.13	even	4
128.11.f.a.95.11		38	8.3	odd	2
128.11.f.b.31.9		38	16.3	odd	4
128.11.f.b.95.9		38	8.5	even	2