Embedded newform 16.4.e.a.13.4

• Label: 16.4.e.a.13.4
• 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			13.4
Root			\(0.932438 - 1.76934i\) of defining polynomial
Character	\(\chi\)	\(=\)	16.13
Dual form			16.4.e.a.5.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.836901 - 2.70178i) q^{2} +(1.98356 + 1.98356i) q^{3} +(-6.59919 - 4.52224i) q^{4} +(-0.596848 + 0.596848i) q^{5} +(7.01918 - 3.69910i) q^{6} +29.0828i q^{7} +(-17.7410 + 14.0449i) q^{8} -19.1310i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{2} - 2 q^{3} + 8 q^{4} - 2 q^{5} - 32 q^{6} - 44 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.836901	−	2.70178i	0.295889	−	0.955222i
							\(3\)	1.98356	+	1.98356i	0.381737	+	0.381737i	0.871728	−	0.489991i	\(-0.163000\pi\)
−0.489991	+	0.871728i	\(0.663000\pi\)
\(5\)	−0.596848	+	0.596848i	−0.0533837	+	0.0533837i	−0.733295	−	0.679911i	\(-0.762018\pi\)
							0.679911	+	0.733295i	\(0.262018\pi\)
\(7\)			29.0828i			1.57033i	0.619289	+	0.785163i	\(0.287421\pi\)
							−0.619289	+	0.785163i	\(0.712579\pi\)
\(11\)	12.1291	−	12.1291i	0.332461	−	0.332461i	−0.521059	−	0.853520i	\(-0.674463\pi\)
							0.853520	+	0.521059i	\(0.174463\pi\)
\(13\)	−48.5658	−	48.5658i	−1.03613	−	1.03613i	−0.999322	−	0.0368113i	\(-0.988280\pi\)
							−0.0368113	−	0.999322i	\(-0.511720\pi\)
\(17\)	86.7193			1.23721			0.618604	−	0.785703i	\(-0.287699\pi\)
							0.618604	+	0.785703i	\(0.287699\pi\)
\(19\)	−54.8442	−	54.8442i	−0.662217	−	0.662217i	0.293685	−	0.955902i	\(-0.405118\pi\)
							−0.955902	+	0.293685i	\(0.905118\pi\)
\(23\)			70.2145i			0.636554i	0.947998	+	0.318277i	\(0.103104\pi\)
							−0.947998	+	0.318277i	\(0.896896\pi\)
\(29\)	63.4021	+	63.4021i	0.405982	+	0.405982i	0.880335	−	0.474353i	\(-0.157318\pi\)
							−0.474353	+	0.880335i	\(0.657318\pi\)
\(31\)	−8.86868			−0.0513826			−0.0256913	−	0.999670i	\(-0.508179\pi\)
							−0.0256913	+	0.999670i	\(0.508179\pi\)
\(37\)	−21.7145	+	21.7145i	−0.0964820	+	0.0964820i	−0.753700	−	0.657218i	\(-0.771733\pi\)
							0.657218	+	0.753700i	\(0.271733\pi\)
\(41\)		−	153.274i		−	0.583840i	−0.956443	−	0.291920i	\(-0.905706\pi\)
							0.956443	−	0.291920i	\(-0.0942941\pi\)
\(43\)	−120.951	+	120.951i	−0.428949	+	0.428949i	−0.888270	−	0.459322i	\(-0.848093\pi\)
							0.459322	+	0.888270i	\(0.348093\pi\)
\(47\)	−99.9792			−0.310286			−0.155143	−	0.987892i	\(-0.549584\pi\)
							−0.155143	+	0.987892i	\(0.549584\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.4.e.a.13.4	yes	10
3.2	odd	2		144.4.k.a.109.2		10
4.3	odd	2		64.4.e.a.17.2		10
8.3	odd	2		128.4.e.a.33.4		10
8.5	even	2		128.4.e.b.33.2		10
12.11	even	2		576.4.k.a.145.3		10
16.3	odd	4		128.4.e.a.97.4		10
16.5	even	4	inner	16.4.e.a.5.4	✓	10
16.11	odd	4		64.4.e.a.49.2		10
16.13	even	4		128.4.e.b.97.2		10
32.3	odd	8		1024.4.b.k.513.4		10
32.5	even	8		1024.4.a.n.1.4		10
32.11	odd	8		1024.4.a.m.1.4		10
32.13	even	8		1024.4.b.j.513.4		10
32.19	odd	8		1024.4.b.k.513.7		10
32.21	even	8		1024.4.a.n.1.7		10
32.27	odd	8		1024.4.a.m.1.7		10
32.29	even	8		1024.4.b.j.513.7		10
48.5	odd	4		144.4.k.a.37.2		10
48.11	even	4		576.4.k.a.433.3		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.4	✓	10	16.5	even	4	inner
16.4.e.a.13.4	yes	10	1.1	even	1	trivial
64.4.e.a.17.2		10	4.3	odd	2
64.4.e.a.49.2		10	16.11	odd	4
128.4.e.a.33.4		10	8.3	odd	2
128.4.e.a.97.4		10	16.3	odd	4
128.4.e.b.33.2		10	8.5	even	2
128.4.e.b.97.2		10	16.13	even	4
144.4.k.a.37.2		10	48.5	odd	4
144.4.k.a.109.2		10	3.2	odd	2
576.4.k.a.145.3		10	12.11	even	2
576.4.k.a.433.3		10	48.11	even	4
1024.4.a.m.1.4		10	32.11	odd	8
1024.4.a.m.1.7		10	32.27	odd	8
1024.4.a.n.1.4		10	32.5	even	8
1024.4.a.n.1.7		10	32.21	even	8
1024.4.b.j.513.4		10	32.13	even	8
1024.4.b.j.513.7		10	32.29	even	8
1024.4.b.k.513.4		10	32.3	odd	8
1024.4.b.k.513.7		10	32.19	odd	8