Embedded newform 16.11.f.a.3.6

• Label: 16.11.f.a.3.6
• 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			3.6
Character	\(\chi\)	\(=\)	16.3
Dual form			16.11.f.a.11.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-19.3314 - 25.5009i) q^{2} +(-196.215 + 196.215i) q^{3} +(-276.596 + 985.936i) q^{4} +(-1311.50 + 1311.50i) q^{5} +(8796.79 + 1210.56i) q^{6} -1120.35 q^{7} +(30489.3 - 12006.1i) q^{8} -17952.0i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 2 q^{3} - 1256 q^{4} - 2 q^{5} - 17216 q^{6} - 4 q^{7} + 69124 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−19.3314	−	25.5009i	−0.604105	−	0.796904i
							\(3\)	−196.215	+	196.215i	−0.807471	+	0.807471i	−0.984251	−	0.176779i	\(-0.943432\pi\)
0.176779	+	0.984251i	\(0.443432\pi\)
\(5\)	−1311.50	+	1311.50i	−0.419680	+	0.419680i	−0.885094	−	0.465413i	\(-0.845906\pi\)
							0.465413	+	0.885094i	\(0.345906\pi\)
\(7\)	−1120.35			−0.0666594			−0.0333297	−	0.999444i	\(-0.510611\pi\)
							−0.0333297	+	0.999444i	\(0.510611\pi\)
\(11\)	−70269.0	−	70269.0i	−0.436315	−	0.436315i	0.454455	−	0.890770i	\(-0.349834\pi\)
							−0.890770	+	0.454455i	\(0.849834\pi\)
\(13\)	−229159.	−	229159.i	−0.617193	−	0.617193i	0.327618	−	0.944810i	\(-0.393754\pi\)
							−0.944810	+	0.327618i	\(0.893754\pi\)
\(17\)	38414.0			0.0270549			0.0135274	−	0.999909i	\(-0.495694\pi\)
							0.0135274	+	0.999909i	\(0.495694\pi\)
\(19\)	2.95322e6	−	2.95322e6i	1.19269	−	1.19269i	0.216380	−	0.976309i	\(-0.430575\pi\)
							0.976309	−	0.216380i	\(-0.0694249\pi\)
\(23\)	8.89185e6			1.38151			0.690753	−	0.723091i	\(-0.257279\pi\)
							0.690753	+	0.723091i	\(0.257279\pi\)
\(29\)	−5.33687e6	−	5.33687e6i	−0.260194	−	0.260194i	0.564939	−	0.825133i	\(-0.308900\pi\)
							−0.825133	+	0.564939i	\(0.808900\pi\)
\(31\)		−	5.79285e6i		−	0.202341i	−0.994869	−	0.101170i	\(-0.967741\pi\)
							0.994869	−	0.101170i	\(-0.0322587\pi\)
\(37\)	8.06230e7	−	8.06230e7i	1.16265	−	1.16265i	0.178761	−	0.983892i	\(-0.442791\pi\)
							0.983892	−	0.178761i	\(-0.0572090\pi\)
\(41\)			1.06427e8i			0.918616i	0.888277	+	0.459308i	\(0.151903\pi\)
							−0.888277	+	0.459308i	\(0.848097\pi\)
\(43\)	−1.52989e8	−	1.52989e8i	−1.04068	−	1.04068i	−0.999137	−	0.0415445i	\(-0.986772\pi\)
							−0.0415445	−	0.999137i	\(-0.513228\pi\)
\(47\)		−	1.83042e8i		−	0.798108i	−0.916928	−	0.399054i	\(-0.869339\pi\)
							0.916928	−	0.399054i	\(-0.130661\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.11.f.a.3.6	✓	38
4.3	odd	2		64.11.f.a.47.15		38
8.3	odd	2		128.11.f.a.95.5		38
8.5	even	2		128.11.f.b.95.15		38
16.3	odd	4		128.11.f.b.31.15		38
16.5	even	4		64.11.f.a.15.15		38
16.11	odd	4	inner	16.11.f.a.11.6	yes	38
16.13	even	4		128.11.f.a.31.5		38

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.11.f.a.3.6	✓	38	1.1	even	1	trivial
16.11.f.a.11.6	yes	38	16.11	odd	4	inner
64.11.f.a.15.15		38	16.5	even	4
64.11.f.a.47.15		38	4.3	odd	2
128.11.f.a.31.5		38	16.13	even	4
128.11.f.a.95.5		38	8.3	odd	2
128.11.f.b.31.15		38	16.3	odd	4
128.11.f.b.95.15		38	8.5	even	2