Embedded newform 16.4.e.a.13.1

• Label: 16.4.e.a.13.1
• 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			13.1
Root			\(1.28199 + 1.53509i\) of defining polynomial
Character	\(\chi\)	\(=\)	16.13
Dual form			16.4.e.a.5.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.81708 + 0.253099i) q^{2} +(5.49618 + 5.49618i) q^{3} +(7.87188 - 1.42600i) q^{4} +(-4.66372 + 4.66372i) q^{5} +(-16.8743 - 14.0921i) q^{6} -24.8965i q^{7} +(-21.8148 + 6.00953i) q^{8} +33.4160i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{2} - 2 q^{3} + 8 q^{4} - 2 q^{5} - 32 q^{6} - 44 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.81708	+	0.253099i	−0.995988	+	0.0894841i
							\(3\)	5.49618	+	5.49618i	1.05774	+	1.05774i	0.998228	+	0.0595129i	\(0.0189548\pi\)
0.0595129	+	0.998228i	\(0.481045\pi\)
\(5\)	−4.66372	+	4.66372i	−0.417136	+	0.417136i	−0.884215	−	0.467079i	\(-0.845306\pi\)
							0.467079	+	0.884215i	\(0.345306\pi\)
\(7\)		−	24.8965i		−	1.34429i	−0.740422	−	0.672143i	\(-0.765374\pi\)
							0.740422	−	0.672143i	\(-0.234626\pi\)
\(11\)	22.3431	−	22.3431i	0.612427	−	0.612427i	−0.331151	−	0.943578i	\(-0.607437\pi\)
							0.943578	+	0.331151i	\(0.107437\pi\)
\(13\)	−11.2714	−	11.2714i	−0.240471	−	0.240471i	0.576574	−	0.817045i	\(-0.304389\pi\)
							−0.817045	+	0.576574i	\(0.804389\pi\)
\(17\)	−88.4846			−1.26239			−0.631196	−	0.775623i	\(-0.717436\pi\)
							−0.631196	+	0.775623i	\(0.717436\pi\)
\(19\)	37.8187	+	37.8187i	0.456643	+	0.456643i	0.897552	−	0.440909i	\(-0.145344\pi\)
							−0.440909	+	0.897552i	\(0.645344\pi\)
\(23\)			48.1224i			0.436270i	0.975919	+	0.218135i	\(0.0699973\pi\)
							−0.975919	+	0.218135i	\(0.930003\pi\)
\(29\)	10.4432	+	10.4432i	0.0668705	+	0.0668705i	0.739751	−	0.672881i	\(-0.234943\pi\)
							−0.672881	+	0.739751i	\(0.734943\pi\)
\(31\)	−96.9578			−0.561746			−0.280873	−	0.959745i	\(-0.590624\pi\)
							−0.280873	+	0.959745i	\(0.590624\pi\)
\(37\)	−163.279	+	163.279i	−0.725484	+	0.725484i	−0.969717	−	0.244233i	\(-0.921464\pi\)
							0.244233	+	0.969717i	\(0.421464\pi\)
\(41\)		−	360.519i		−	1.37326i	−0.727008	−	0.686629i	\(-0.759090\pi\)
							0.727008	−	0.686629i	\(-0.240910\pi\)
\(43\)	−100.249	+	100.249i	−0.355531	+	0.355531i	−0.862163	−	0.506632i	\(-0.830890\pi\)
							0.506632	+	0.862163i	\(0.330890\pi\)
\(47\)	220.669			0.684849			0.342425	−	0.939545i	\(-0.388752\pi\)
							0.342425	+	0.939545i	\(0.388752\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.4.e.a.13.1	yes	10
3.2	odd	2		144.4.k.a.109.5		10
4.3	odd	2		64.4.e.a.17.1		10
8.3	odd	2		128.4.e.a.33.5		10
8.5	even	2		128.4.e.b.33.1		10
12.11	even	2		576.4.k.a.145.4		10
16.3	odd	4		128.4.e.a.97.5		10
16.5	even	4	inner	16.4.e.a.5.1	✓	10
16.11	odd	4		64.4.e.a.49.1		10
16.13	even	4		128.4.e.b.97.1		10
32.3	odd	8		1024.4.b.k.513.2		10
32.5	even	8		1024.4.a.n.1.2		10
32.11	odd	8		1024.4.a.m.1.2		10
32.13	even	8		1024.4.b.j.513.2		10
32.19	odd	8		1024.4.b.k.513.9		10
32.21	even	8		1024.4.a.n.1.9		10
32.27	odd	8		1024.4.a.m.1.9		10
32.29	even	8		1024.4.b.j.513.9		10
48.5	odd	4		144.4.k.a.37.5		10
48.11	even	4		576.4.k.a.433.4		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.1	✓	10	16.5	even	4	inner
16.4.e.a.13.1	yes	10	1.1	even	1	trivial
64.4.e.a.17.1		10	4.3	odd	2
64.4.e.a.49.1		10	16.11	odd	4
128.4.e.a.33.5		10	8.3	odd	2
128.4.e.a.97.5		10	16.3	odd	4
128.4.e.b.33.1		10	8.5	even	2
128.4.e.b.97.1		10	16.13	even	4
144.4.k.a.37.5		10	48.5	odd	4
144.4.k.a.109.5		10	3.2	odd	2
576.4.k.a.145.4		10	12.11	even	2
576.4.k.a.433.4		10	48.11	even	4
1024.4.a.m.1.2		10	32.11	odd	8
1024.4.a.m.1.9		10	32.27	odd	8
1024.4.a.n.1.2		10	32.5	even	8
1024.4.a.n.1.9		10	32.21	even	8
1024.4.b.j.513.2		10	32.13	even	8
1024.4.b.j.513.9		10	32.29	even	8
1024.4.b.k.513.2		10	32.3	odd	8
1024.4.b.k.513.9		10	32.19	odd	8