Embedded newform 16.11.f.a.3.17

• Label: 16.11.f.a.3.17
• Level: $16$
• Weight: $11$
• Character: 16.3
• Analytic conductor: $10.166$
• Analytic rank: $0$
• Dimension: $38$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1657160428\)
Analytic rank:	\(0\)
Dimension:	\(38\)
Relative dimension:	\(19\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.17
Character	\(\chi\)	\(=\)	16.3
Dual form			16.11.f.a.11.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(28.0563 - 15.3897i) q^{2} +(-317.338 + 317.338i) q^{3} +(550.315 - 863.556i) q^{4} +(1564.48 - 1564.48i) q^{5} +(-4019.61 + 13787.1i) q^{6} -15496.4 q^{7} +(2149.97 - 32697.4i) q^{8} -142358. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 2 q^{3} - 1256 q^{4} - 2 q^{5} - 17216 q^{6} - 4 q^{7} + 69124 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	28.0563	−	15.3897i	0.876760	−	0.480928i
							\(3\)	−317.338	+	317.338i	−1.30592	+	1.30592i	−0.381587	+	0.924333i	\(0.624622\pi\)
−0.924333	+	0.381587i	\(0.875378\pi\)
\(5\)	1564.48	−	1564.48i	0.500635	−	0.500635i	−0.411000	−	0.911635i	\(-0.634820\pi\)
							0.911635	+	0.411000i	\(0.134820\pi\)
\(7\)	−15496.4			−0.922021			−0.461011	−	0.887395i	\(-0.652513\pi\)
							−0.461011	+	0.887395i	\(0.652513\pi\)
\(11\)	−141839.	−	141839.i	−0.880711	−	0.880711i	0.112896	−	0.993607i	\(-0.463987\pi\)
							−0.993607	+	0.112896i	\(0.963987\pi\)
\(13\)	−215697.	−	215697.i	−0.580935	−	0.580935i	0.354225	−	0.935160i	\(-0.384745\pi\)
							−0.935160	+	0.354225i	\(0.884745\pi\)
\(17\)	1.64558e6			1.15898			0.579489	−	0.814980i	\(-0.303252\pi\)
							0.579489	+	0.814980i	\(0.303252\pi\)
\(19\)	−1.70512e6	+	1.70512e6i	−0.688630	+	0.688630i	−0.961929	−	0.273299i	\(-0.911885\pi\)
							0.273299	+	0.961929i	\(0.411885\pi\)
\(23\)	−1.04483e6			−0.162333			−0.0811665	−	0.996701i	\(-0.525865\pi\)
							−0.0811665	+	0.996701i	\(0.525865\pi\)
\(29\)	−7.08635e6	−	7.08635e6i	−0.345488	−	0.345488i	0.512938	−	0.858426i	\(-0.328557\pi\)
							−0.858426	+	0.512938i	\(0.828557\pi\)
\(31\)		−	3.47811e7i		−	1.21488i	−0.794364	−	0.607442i	\(-0.792196\pi\)
							0.794364	−	0.607442i	\(-0.207804\pi\)
\(37\)	−9.40015e7	+	9.40015e7i	−1.35558	+	1.35558i	−0.476302	+	0.879282i	\(0.658023\pi\)
							−0.879282	+	0.476302i	\(0.841977\pi\)
\(41\)		−	4.61571e7i		−	0.398400i	−0.979959	−	0.199200i	\(-0.936166\pi\)
							0.979959	−	0.199200i	\(-0.0638344\pi\)
\(43\)	6.50907e7	+	6.50907e7i	0.442768	+	0.442768i	0.892941	−	0.450173i	\(-0.148638\pi\)
							−0.450173	+	0.892941i	\(0.648638\pi\)
\(47\)		−	1.86795e8i		−	0.814473i	−0.913323	−	0.407236i	\(-0.866493\pi\)
							0.913323	−	0.407236i	\(-0.133507\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.11.f.a.3.17	✓	38
4.3	odd	2		64.11.f.a.47.19		38
8.3	odd	2		128.11.f.a.95.1		38
8.5	even	2		128.11.f.b.95.19		38
16.3	odd	4		128.11.f.b.31.19		38
16.5	even	4		64.11.f.a.15.19		38
16.11	odd	4	inner	16.11.f.a.11.17	yes	38
16.13	even	4		128.11.f.a.31.1		38

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.11.f.a.3.17	✓	38	1.1	even	1	trivial
16.11.f.a.11.17	yes	38	16.11	odd	4	inner
64.11.f.a.15.19		38	16.5	even	4
64.11.f.a.47.19		38	4.3	odd	2
128.11.f.a.31.1		38	16.13	even	4
128.11.f.a.95.1		38	8.3	odd	2
128.11.f.b.31.19		38	16.3	odd	4
128.11.f.b.95.19		38	8.5	even	2