Embedded newform 16.4.e.a.13.5

• Label: 16.4.e.a.13.5
• Level: $16$
• Weight: $4$
• Character: 16.13
• Analytic conductor: $0.944$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.944030560092\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Relative dimension:	\(5\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	\( x^{10} - 2x^{9} - x^{8} + 6x^{7} + 14x^{6} - 80x^{5} + 56x^{4} + 96x^{3} - 64x^{2} - 512x + 1024 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{10} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			13.5
Root			\(-1.56339 - 1.24732i\) of defining polynomial
Character	\(\chi\)	\(=\)	16.13
Dual form			16.4.e.a.5.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.81071 + 0.316066i) q^{2} +(-3.27139 - 3.27139i) q^{3} +(7.80020 + 1.77674i) q^{4} +(-12.6449 + 12.6449i) q^{5} +(-8.16095 - 10.2289i) q^{6} -13.8754i q^{7} +(21.3626 + 7.45928i) q^{8} -5.59607i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{2} - 2 q^{3} + 8 q^{4} - 2 q^{5} - 32 q^{6} - 44 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.81071	+	0.316066i	0.993737	+	0.111746i
							\(3\)	−3.27139	−	3.27139i	−0.629578	−	0.629578i	0.318384	−	0.947962i	\(-0.396860\pi\)
−0.947962	+	0.318384i	\(0.896860\pi\)
\(5\)	−12.6449	+	12.6449i	−1.13099	+	1.13099i	−0.140981	+	0.990012i	\(0.545026\pi\)
							−0.990012	+	0.140981i	\(0.954974\pi\)
\(7\)		−	13.8754i		−	0.749200i	−0.927187	−	0.374600i	\(-0.877780\pi\)
							0.927187	−	0.374600i	\(-0.122220\pi\)
\(11\)	1.54694	−	1.54694i	0.0424019	−	0.0424019i	−0.685588	−	0.727990i	\(-0.740455\pi\)
							0.727990	+	0.685588i	\(0.240455\pi\)
\(13\)	32.7875	+	32.7875i	0.699509	+	0.699509i	0.964304	−	0.264796i	\(-0.0853046\pi\)
							−0.264796	+	0.964304i	\(0.585305\pi\)
\(17\)	18.6531			0.266119			0.133060	−	0.991108i	\(-0.457520\pi\)
							0.133060	+	0.991108i	\(0.457520\pi\)
\(19\)	−86.4042	−	86.4042i	−1.04329	−	1.04329i	−0.999020	−	0.0442688i	\(-0.985904\pi\)
							−0.0442688	−	0.999020i	\(-0.514096\pi\)
\(23\)			134.006i			1.21488i	0.794367	+	0.607438i	\(0.207803\pi\)
							−0.794367	+	0.607438i	\(0.792197\pi\)
\(29\)	−59.7949	−	59.7949i	−0.382884	−	0.382884i	0.489256	−	0.872140i	\(-0.337268\pi\)
							−0.872140	+	0.489256i	\(0.837268\pi\)
\(31\)	−31.5391			−0.182729			−0.0913645	−	0.995818i	\(-0.529123\pi\)
							−0.0913645	+	0.995818i	\(0.529123\pi\)
\(37\)	89.1866	−	89.1866i	0.396275	−	0.396275i	−0.480642	−	0.876917i	\(-0.659596\pi\)
							0.876917	+	0.480642i	\(0.159596\pi\)
\(41\)			210.504i			0.801834i	0.916114	+	0.400917i	\(0.131308\pi\)
							−0.916114	+	0.400917i	\(0.868692\pi\)
\(43\)	119.402	−	119.402i	0.423456	−	0.423456i	−0.462936	−	0.886392i	\(-0.653204\pi\)
							0.886392	+	0.462936i	\(0.153204\pi\)
\(47\)	−182.902			−0.567638			−0.283819	−	0.958878i	\(-0.591602\pi\)
							−0.283819	+	0.958878i	\(0.591602\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.4.e.a.13.5	yes	10
3.2	odd	2		144.4.k.a.109.1		10
4.3	odd	2		64.4.e.a.17.4		10
8.3	odd	2		128.4.e.a.33.2		10
8.5	even	2		128.4.e.b.33.4		10
12.11	even	2		576.4.k.a.145.5		10
16.3	odd	4		128.4.e.a.97.2		10
16.5	even	4	inner	16.4.e.a.5.5	✓	10
16.11	odd	4		64.4.e.a.49.4		10
16.13	even	4		128.4.e.b.97.4		10
32.3	odd	8		1024.4.b.k.513.8		10
32.5	even	8		1024.4.a.n.1.8		10
32.11	odd	8		1024.4.a.m.1.8		10
32.13	even	8		1024.4.b.j.513.8		10
32.19	odd	8		1024.4.b.k.513.3		10
32.21	even	8		1024.4.a.n.1.3		10
32.27	odd	8		1024.4.a.m.1.3		10
32.29	even	8		1024.4.b.j.513.3		10
48.5	odd	4		144.4.k.a.37.1		10
48.11	even	4		576.4.k.a.433.5		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.5	✓	10	16.5	even	4	inner
16.4.e.a.13.5	yes	10	1.1	even	1	trivial
64.4.e.a.17.4		10	4.3	odd	2
64.4.e.a.49.4		10	16.11	odd	4
128.4.e.a.33.2		10	8.3	odd	2
128.4.e.a.97.2		10	16.3	odd	4
128.4.e.b.33.4		10	8.5	even	2
128.4.e.b.97.4		10	16.13	even	4
144.4.k.a.37.1		10	48.5	odd	4
144.4.k.a.109.1		10	3.2	odd	2
576.4.k.a.145.5		10	12.11	even	2
576.4.k.a.433.5		10	48.11	even	4
1024.4.a.m.1.3		10	32.27	odd	8
1024.4.a.m.1.8		10	32.11	odd	8
1024.4.a.n.1.3		10	32.21	even	8
1024.4.a.n.1.8		10	32.5	even	8
1024.4.b.j.513.3		10	32.29	even	8
1024.4.b.j.513.8		10	32.13	even	8
1024.4.b.k.513.3		10	32.19	odd	8
1024.4.b.k.513.8		10	32.3	odd	8