Embedded newform 16.11.f.a.3.13

• Label: 16.11.f.a.3.13
• 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			3.13
Character	\(\chi\)	\(=\)	16.3
Dual form			16.11.f.a.11.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(12.5336 + 29.4433i) q^{2} +(44.0645 - 44.0645i) q^{3} +(-709.815 + 738.064i) q^{4} +(-126.270 + 126.270i) q^{5} +(1849.69 + 745.115i) q^{6} -19594.2 q^{7} +(-30627.6 - 11648.7i) q^{8} +55165.6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 2 q^{3} - 1256 q^{4} - 2 q^{5} - 17216 q^{6} - 4 q^{7} + 69124 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	12.5336	+	29.4433i	0.391677	+	0.920103i
							\(3\)	44.0645	−	44.0645i	0.181335	−	0.181335i	−0.610602	−	0.791937i	\(-0.709073\pi\)
0.791937	+	0.610602i	\(0.209073\pi\)
\(5\)	−126.270	+	126.270i	−0.0404063	+	0.0404063i	−0.727021	−	0.686615i	\(-0.759096\pi\)
							0.686615	+	0.727021i	\(0.259096\pi\)
\(7\)	−19594.2			−1.16584			−0.582919	−	0.812530i	\(-0.698089\pi\)
							−0.582919	+	0.812530i	\(0.698089\pi\)
\(11\)	−35597.2	−	35597.2i	−0.221030	−	0.221030i	0.587902	−	0.808932i	\(-0.299954\pi\)
							−0.808932	+	0.587902i	\(0.799954\pi\)
\(13\)	−316993.	−	316993.i	−0.853754	−	0.853754i	0.136839	−	0.990593i	\(-0.456306\pi\)
							−0.990593	+	0.136839i	\(0.956306\pi\)
\(17\)	357825.			0.252015			0.126007	−	0.992029i	\(-0.459784\pi\)
							0.126007	+	0.992029i	\(0.459784\pi\)
\(19\)	−1.92949e6	+	1.92949e6i	−0.779247	+	0.779247i	−0.979703	−	0.200456i	\(-0.935758\pi\)
							0.200456	+	0.979703i	\(0.435758\pi\)
\(23\)	1.53144e6			0.237936			0.118968	−	0.992898i	\(-0.462041\pi\)
							0.118968	+	0.992898i	\(0.462041\pi\)
\(29\)	1.81553e7	+	1.81553e7i	0.885144	+	0.885144i	0.994052	−	0.108908i	\(-0.0347352\pi\)
							−0.108908	+	0.994052i	\(0.534735\pi\)
\(31\)			2.17972e7i			0.761363i	0.924706	+	0.380681i	\(0.124311\pi\)
							−0.924706	+	0.380681i	\(0.875689\pi\)
\(37\)	−2.50024e7	+	2.50024e7i	−0.360556	+	0.360556i	−0.864018	−	0.503462i	\(-0.832060\pi\)
							0.503462	+	0.864018i	\(0.332060\pi\)
\(41\)			1.41691e8i			1.22299i	0.791248	+	0.611496i	\(0.209432\pi\)
							−0.791248	+	0.611496i	\(0.790568\pi\)
\(43\)	−9.95826e7	−	9.95826e7i	−0.677394	−	0.677394i	0.282016	−	0.959410i	\(-0.408997\pi\)
							−0.959410	+	0.282016i	\(0.908997\pi\)
\(47\)		−	3.56813e8i		−	1.55579i	−0.628394	−	0.777895i	\(-0.716287\pi\)
							0.628394	−	0.777895i	\(-0.283713\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.11.f.a.3.13	✓	38
4.3	odd	2		64.11.f.a.47.8		38
8.3	odd	2		128.11.f.a.95.12		38
8.5	even	2		128.11.f.b.95.8		38
16.3	odd	4		128.11.f.b.31.8		38
16.5	even	4		64.11.f.a.15.8		38
16.11	odd	4	inner	16.11.f.a.11.13	yes	38
16.13	even	4		128.11.f.a.31.12		38

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.11.f.a.3.13	✓	38	1.1	even	1	trivial
16.11.f.a.11.13	yes	38	16.11	odd	4	inner
64.11.f.a.15.8		38	16.5	even	4
64.11.f.a.47.8		38	4.3	odd	2
128.11.f.a.31.12		38	16.13	even	4
128.11.f.a.95.12		38	8.3	odd	2
128.11.f.b.31.8		38	16.3	odd	4
128.11.f.b.95.8		38	8.5	even	2