Embedded newform 16.4.e.a.13.2

• Label: 16.4.e.a.13.2
• 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			13.2
Root			\(1.97476 + 0.316760i\) of defining polynomial
Character	\(\chi\)	\(=\)	16.13
Dual form			16.4.e.a.5.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.29152 - 1.65800i) q^{2} +(-5.96513 - 5.96513i) q^{3} +(2.50210 + 7.59865i) q^{4} +(8.67959 - 8.67959i) q^{5} +(3.77903 + 23.5594i) q^{6} -1.63924i q^{7} +(6.86495 - 21.5609i) q^{8} +44.1656i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{2} - 2 q^{3} + 8 q^{4} - 2 q^{5} - 32 q^{6} - 44 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.29152	−	1.65800i	−0.810173	−	0.586190i
							\(3\)	−5.96513	−	5.96513i	−1.14799	−	1.14799i	−0.986947	−	0.161043i	\(-0.948514\pi\)
−0.161043	−	0.986947i	\(-0.551486\pi\)
\(5\)	8.67959	−	8.67959i	0.776326	−	0.776326i	−0.202878	−	0.979204i	\(-0.565029\pi\)
							0.979204	+	0.202878i	\(0.0650295\pi\)
\(7\)		−	1.63924i		−	0.0885109i	−0.999020	−	0.0442554i	\(-0.985908\pi\)
							0.999020	−	0.0442554i	\(-0.0140915\pi\)
\(11\)	18.2021	−	18.2021i	0.498921	−	0.498921i	−0.412181	−	0.911102i	\(-0.635233\pi\)
							0.911102	+	0.412181i	\(0.135233\pi\)
\(13\)	−9.34700	−	9.34700i	−0.199415	−	0.199415i	0.600334	−	0.799749i	\(-0.295034\pi\)
							−0.799749	+	0.600334i	\(0.795034\pi\)
\(17\)	53.6113			0.764862			0.382431	−	0.923984i	\(-0.375087\pi\)
							0.382431	+	0.923984i	\(0.375087\pi\)
\(19\)	70.9870	+	70.9870i	0.857133	+	0.857133i	0.990999	−	0.133866i	\(-0.0427392\pi\)
							−0.133866	+	0.990999i	\(0.542739\pi\)
\(23\)			25.1189i			0.227724i	0.993497	+	0.113862i	\(0.0363222\pi\)
							−0.993497	+	0.113862i	\(0.963678\pi\)
\(29\)	−181.094	−	181.094i	−1.15960	−	1.15960i	−0.984563	−	0.175033i	\(-0.943997\pi\)
							−0.175033	−	0.984563i	\(-0.556003\pi\)
\(31\)	132.684			0.768733			0.384367	−	0.923180i	\(-0.374420\pi\)
							0.384367	+	0.923180i	\(0.374420\pi\)
\(37\)	174.872	−	174.872i	0.776994	−	0.776994i	−0.202324	−	0.979319i	\(-0.564849\pi\)
							0.979319	+	0.202324i	\(0.0648495\pi\)
\(41\)			198.660i			0.756720i	0.925658	+	0.378360i	\(0.123512\pi\)
							−0.925658	+	0.378360i	\(0.876488\pi\)
\(43\)	−285.717	+	285.717i	−1.01329	+	1.01329i	−0.0133770	+	0.999911i	\(0.504258\pi\)
							−0.999911	+	0.0133770i	\(0.995742\pi\)
\(47\)	78.3629			0.243200			0.121600	−	0.992579i	\(-0.461197\pi\)
							0.121600	+	0.992579i	\(0.461197\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.4.e.a.13.2	yes	10
3.2	odd	2		144.4.k.a.109.4		10
4.3	odd	2		64.4.e.a.17.5		10
8.3	odd	2		128.4.e.a.33.1		10
8.5	even	2		128.4.e.b.33.5		10
12.11	even	2		576.4.k.a.145.1		10
16.3	odd	4		128.4.e.a.97.1		10
16.5	even	4	inner	16.4.e.a.5.2	✓	10
16.11	odd	4		64.4.e.a.49.5		10
16.13	even	4		128.4.e.b.97.5		10
32.3	odd	8		1024.4.b.k.513.10		10
32.5	even	8		1024.4.a.n.1.10		10
32.11	odd	8		1024.4.a.m.1.10		10
32.13	even	8		1024.4.b.j.513.10		10
32.19	odd	8		1024.4.b.k.513.1		10
32.21	even	8		1024.4.a.n.1.1		10
32.27	odd	8		1024.4.a.m.1.1		10
32.29	even	8		1024.4.b.j.513.1		10
48.5	odd	4		144.4.k.a.37.4		10
48.11	even	4		576.4.k.a.433.1		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.2	✓	10	16.5	even	4	inner
16.4.e.a.13.2	yes	10	1.1	even	1	trivial
64.4.e.a.17.5		10	4.3	odd	2
64.4.e.a.49.5		10	16.11	odd	4
128.4.e.a.33.1		10	8.3	odd	2
128.4.e.a.97.1		10	16.3	odd	4
128.4.e.b.33.5		10	8.5	even	2
128.4.e.b.97.5		10	16.13	even	4
144.4.k.a.37.4		10	48.5	odd	4
144.4.k.a.109.4		10	3.2	odd	2
576.4.k.a.145.1		10	12.11	even	2
576.4.k.a.433.1		10	48.11	even	4
1024.4.a.m.1.1		10	32.27	odd	8
1024.4.a.m.1.10		10	32.11	odd	8
1024.4.a.n.1.1		10	32.21	even	8
1024.4.a.n.1.10		10	32.5	even	8
1024.4.b.j.513.1		10	32.29	even	8
1024.4.b.j.513.10		10	32.13	even	8
1024.4.b.k.513.1		10	32.19	odd	8
1024.4.b.k.513.10		10	32.3	odd	8