Embedded newform 16.4.e.a.13.3

• Label: 16.4.e.a.13.3
• 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			13.3
Root			\(-1.62580 + 1.16481i\) of defining polynomial
Character	\(\chi\)	\(=\)	16.13
Dual form			16.4.e.a.5.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.460984 + 2.79061i) q^{2} +(0.756776 + 0.756776i) q^{3} +(-7.57499 + 2.57285i) q^{4} +(8.22587 - 8.22587i) q^{5} +(-1.76300 + 2.46073i) q^{6} -2.67171i q^{7} +(-10.6718 - 19.9528i) q^{8} -25.8546i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{2} - 2 q^{3} + 8 q^{4} - 2 q^{5} - 32 q^{6} - 44 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.460984	+	2.79061i	0.162982	+	0.986629i
							\(3\)	0.756776	+	0.756776i	0.145642	+	0.145642i	0.776168	−	0.630526i	\(-0.217161\pi\)
−0.630526	+	0.776168i	\(0.717161\pi\)
\(5\)	8.22587	−	8.22587i	0.735744	−	0.735744i	−0.236007	−	0.971751i	\(-0.575839\pi\)
							0.971751	+	0.236007i	\(0.0758389\pi\)
\(7\)		−	2.67171i		−	0.144259i	−0.997395	−	0.0721293i	\(-0.977021\pi\)
							0.997395	−	0.0721293i	\(-0.0229794\pi\)
\(11\)	−45.2213	+	45.2213i	−1.23952	+	1.23952i	−0.279323	+	0.960197i	\(0.590110\pi\)
							−0.960197	+	0.279323i	\(0.909890\pi\)
\(13\)	35.3968	+	35.3968i	0.755176	+	0.755176i	0.975440	−	0.220264i	\(-0.0706918\pi\)
							−0.220264	+	0.975440i	\(0.570692\pi\)
\(17\)	−72.4991			−1.03433			−0.517165	−	0.855886i	\(-0.673013\pi\)
							−0.517165	+	0.855886i	\(0.673013\pi\)
\(19\)	19.4427	+	19.4427i	0.234761	+	0.234761i	0.814676	−	0.579916i	\(-0.196915\pi\)
							−0.579916	+	0.814676i	\(0.696915\pi\)
\(23\)		−	139.462i		−	1.26434i	−0.774830	−	0.632170i	\(-0.782165\pi\)
							0.774830	−	0.632170i	\(-0.217835\pi\)
\(29\)	66.0434	+	66.0434i	0.422895	+	0.422895i	0.886199	−	0.463304i	\(-0.153336\pi\)
							−0.463304	+	0.886199i	\(0.653336\pi\)
\(31\)	188.682			1.09317			0.546584	−	0.837404i	\(-0.315928\pi\)
							0.546584	+	0.837404i	\(0.315928\pi\)
\(37\)	−84.0653	+	84.0653i	−0.373520	+	0.373520i	−0.868758	−	0.495237i	\(-0.835081\pi\)
							0.495237	+	0.868758i	\(0.335081\pi\)
\(41\)			104.629i			0.398545i	0.979944	+	0.199272i	\(0.0638578\pi\)
							−0.979944	+	0.199272i	\(0.936142\pi\)
\(43\)	−31.4857	+	31.4857i	−0.111663	+	0.111663i	−0.760731	−	0.649067i	\(-0.775159\pi\)
							0.649067	+	0.760731i	\(0.275159\pi\)
\(47\)	−488.151			−1.51498			−0.757491	−	0.652846i	\(-0.773575\pi\)
							−0.757491	+	0.652846i	\(0.773575\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.4.e.a.13.3	yes	10
3.2	odd	2		144.4.k.a.109.3		10
4.3	odd	2		64.4.e.a.17.3		10
8.3	odd	2		128.4.e.a.33.3		10
8.5	even	2		128.4.e.b.33.3		10
12.11	even	2		576.4.k.a.145.2		10
16.3	odd	4		128.4.e.a.97.3		10
16.5	even	4	inner	16.4.e.a.5.3	✓	10
16.11	odd	4		64.4.e.a.49.3		10
16.13	even	4		128.4.e.b.97.3		10
32.3	odd	8		1024.4.b.k.513.5		10
32.5	even	8		1024.4.a.n.1.5		10
32.11	odd	8		1024.4.a.m.1.5		10
32.13	even	8		1024.4.b.j.513.5		10
32.19	odd	8		1024.4.b.k.513.6		10
32.21	even	8		1024.4.a.n.1.6		10
32.27	odd	8		1024.4.a.m.1.6		10
32.29	even	8		1024.4.b.j.513.6		10
48.5	odd	4		144.4.k.a.37.3		10
48.11	even	4		576.4.k.a.433.2		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.3	✓	10	16.5	even	4	inner
16.4.e.a.13.3	yes	10	1.1	even	1	trivial
64.4.e.a.17.3		10	4.3	odd	2
64.4.e.a.49.3		10	16.11	odd	4
128.4.e.a.33.3		10	8.3	odd	2
128.4.e.a.97.3		10	16.3	odd	4
128.4.e.b.33.3		10	8.5	even	2
128.4.e.b.97.3		10	16.13	even	4
144.4.k.a.37.3		10	48.5	odd	4
144.4.k.a.109.3		10	3.2	odd	2
576.4.k.a.145.2		10	12.11	even	2
576.4.k.a.433.2		10	48.11	even	4
1024.4.a.m.1.5		10	32.11	odd	8
1024.4.a.m.1.6		10	32.27	odd	8
1024.4.a.n.1.5		10	32.5	even	8
1024.4.a.n.1.6		10	32.21	even	8
1024.4.b.j.513.5		10	32.13	even	8
1024.4.b.j.513.6		10	32.29	even	8
1024.4.b.k.513.5		10	32.3	odd	8
1024.4.b.k.513.6		10	32.19	odd	8