Embedded newform 16.22.e.a.5.1

• Label: 16.22.e.a.5.1
• Level: $16$
• Weight: $22$
• Character: 16.5
• Analytic conductor: $44.716$
• Analytic rank: $0$
• Dimension: $82$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(44.7163750859\)
Analytic rank:	\(0\)
Dimension:	\(82\)
Relative dimension:	\(41\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.1
Character	\(\chi\)	\(=\)	16.5
Dual form			16.22.e.a.13.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1439.53 + 157.777i) q^{2} +(-112866. + 112866. i) q^{3} +(2.04736e6 - 454251. i) q^{4} +(-2.81638e6 - 2.81638e6i) q^{5} +(1.44667e8 - 1.80282e8i) q^{6} -1.39005e9i q^{7} +(-2.87558e9 + 9.76936e8i) q^{8} -1.50171e10i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 2 q^{2} - 2 q^{3} + 1604328 q^{4} - 2 q^{5} - 142760912 q^{6} - 5437750796 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1439.53	+	157.777i	−0.994047	+	0.108950i
							\(3\)	−112866.	+	112866.i	−1.10355	+	1.10355i	−0.109566	+	0.993980i	\(0.534946\pi\)
−0.993980	+	0.109566i	\(0.965054\pi\)
\(5\)	−2.81638e6	−	2.81638e6i	−0.128975	−	0.128975i	0.639672	−	0.768648i	\(-0.279070\pi\)
							−0.768648	+	0.639672i	\(0.779070\pi\)
\(7\)		−	1.39005e9i		−	1.85994i	−0.367630	−	0.929972i	\(-0.619831\pi\)
							0.367630	−	0.929972i	\(-0.380169\pi\)
\(11\)	−7.64619e10	−	7.64619e10i	−0.888837	−	0.888837i	0.105575	−	0.994411i	\(-0.466332\pi\)
							−0.994411	+	0.105575i	\(0.966332\pi\)
\(13\)	3.49069e11	−	3.49069e11i	0.702274	−	0.702274i	−0.262624	−	0.964898i	\(-0.584588\pi\)
							0.964898	+	0.262624i	\(0.0845880\pi\)
\(17\)	−8.22099e12			−0.989032			−0.494516	−	0.869168i	\(-0.664655\pi\)
							−0.494516	+	0.869168i	\(0.664655\pi\)
\(19\)	2.48513e13	−	2.48513e13i	0.929901	−	0.929901i	−0.0677978	−	0.997699i	\(-0.521597\pi\)
							0.997699	+	0.0677978i	\(0.0215973\pi\)
\(23\)			1.30981e14i			0.659272i	0.944108	+	0.329636i	\(0.106926\pi\)
							−0.944108	+	0.329636i	\(0.893074\pi\)
\(29\)	−2.61387e14	+	2.61387e14i	−0.115373	+	0.115373i	−0.762436	−	0.647063i	\(-0.775997\pi\)
							0.647063	+	0.762436i	\(0.275997\pi\)
\(31\)	−1.44578e14			−0.0316814			−0.0158407	−	0.999875i	\(-0.505042\pi\)
							−0.0158407	+	0.999875i	\(0.505042\pi\)
\(37\)	2.28944e16	+	2.28944e16i	0.782729	+	0.782729i	0.980290	−	0.197562i	\(-0.0633023\pi\)
							−0.197562	+	0.980290i	\(0.563302\pi\)
\(41\)			2.63753e16i			0.306879i	0.988158	+	0.153439i	\(0.0490350\pi\)
							−0.988158	+	0.153439i	\(0.950965\pi\)
\(43\)	−9.63524e16	−	9.63524e16i	−0.679898	−	0.679898i	0.280079	−	0.959977i	\(-0.409639\pi\)
							−0.959977	+	0.280079i	\(0.909639\pi\)
\(47\)	1.10699e17			0.306985			0.153492	−	0.988150i	\(-0.450948\pi\)
							0.153492	+	0.988150i	\(0.450948\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.22.e.a.5.1	✓	82
4.3	odd	2		64.22.e.a.49.37		82
16.3	odd	4		64.22.e.a.17.37		82
16.13	even	4	inner	16.22.e.a.13.1	yes	82

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.22.e.a.5.1	✓	82	1.1	even	1	trivial
16.22.e.a.13.1	yes	82	16.13	even	4	inner
64.22.e.a.17.37		82	16.3	odd	4
64.22.e.a.49.37		82	4.3	odd	2