Embedded newform 16.22.e.a.5.4

• Label: 16.22.e.a.5.4
• 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			5.4
Character	\(\chi\)	\(=\)	16.5
Dual form			16.22.e.a.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1393.10 - 395.509i) q^{2} +(-81293.3 + 81293.3i) q^{3} +(1.78430e6 + 1.10197e6i) q^{4} +(-1.56158e7 - 1.56158e7i) q^{5} +(1.45402e8 - 8.10974e7i) q^{6} +1.06061e9i q^{7} +(-2.04986e9 - 2.24085e9i) q^{8} -2.75685e9i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 2 q^{2} - 2 q^{3} + 1604328 q^{4} - 2 q^{5} - 142760912 q^{6} - 5437750796 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1393.10	−	395.509i	−0.961982	−	0.273113i
							\(3\)	−81293.3	+	81293.3i	−0.794843	+	0.794843i	−0.982277	−	0.187434i	\(-0.939983\pi\)
0.187434	+	0.982277i	\(0.439983\pi\)
\(5\)	−1.56158e7	−	1.56158e7i	−0.715122	−	0.715122i	0.252480	−	0.967602i	\(-0.418754\pi\)
							−0.967602	+	0.252480i	\(0.918754\pi\)
\(7\)			1.06061e9i			1.41915i	0.704631	+	0.709574i	\(0.251113\pi\)
							−0.704631	+	0.709574i	\(0.748887\pi\)
\(11\)	4.35301e9	+	4.35301e9i	0.0506018	+	0.0506018i	0.731955	−	0.681353i	\(-0.238608\pi\)
							−0.681353	+	0.731955i	\(0.738608\pi\)
\(13\)	−1.03737e10	+	1.03737e10i	−0.0208702	+	0.0208702i	−0.717465	−	0.696595i	\(-0.754698\pi\)
							0.696595	+	0.717465i	\(0.254698\pi\)
\(17\)	1.03434e13			1.24437			0.622183	−	0.782872i	\(-0.286246\pi\)
							0.622183	+	0.782872i	\(0.286246\pi\)
\(19\)	1.01407e13	−	1.01407e13i	0.379449	−	0.379449i	−0.491454	−	0.870903i	\(-0.663535\pi\)
							0.870903	+	0.491454i	\(0.163535\pi\)
\(23\)		−	3.46353e14i		−	1.74332i	−0.490111	−	0.871660i	\(-0.663044\pi\)
							0.490111	−	0.871660i	\(-0.336956\pi\)
\(29\)	−1.20079e15	+	1.20079e15i	−0.530014	+	0.530014i	−0.920576	−	0.390562i	\(-0.872281\pi\)
							0.390562	+	0.920576i	\(0.372281\pi\)
\(31\)	8.00285e15			1.75367			0.876833	−	0.480795i	\(-0.159652\pi\)
							0.876833	+	0.480795i	\(0.159652\pi\)
\(37\)	9.92427e15	+	9.92427e15i	0.339297	+	0.339297i	0.856103	−	0.516806i	\(-0.172879\pi\)
							−0.516806	+	0.856103i	\(0.672879\pi\)
\(41\)			3.42326e16i			0.398299i	0.979969	+	0.199149i	\(0.0638179\pi\)
							−0.979969	+	0.199149i	\(0.936182\pi\)
\(43\)	9.71270e16	+	9.71270e16i	0.685364	+	0.685364i	0.961204	−	0.275840i	\(-0.0889560\pi\)
							−0.275840	+	0.961204i	\(0.588956\pi\)
\(47\)	2.72466e17			0.755588			0.377794	−	0.925890i	\(-0.376683\pi\)
							0.377794	+	0.925890i	\(0.376683\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.22.e.a.5.4	✓	82
4.3	odd	2		64.22.e.a.49.32		82
16.3	odd	4		64.22.e.a.17.32		82
16.13	even	4	inner	16.22.e.a.13.4	yes	82

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.22.e.a.5.4	✓	82	1.1	even	1	trivial
16.22.e.a.13.4	yes	82	16.13	even	4	inner
64.22.e.a.17.32		82	16.3	odd	4
64.22.e.a.49.32		82	4.3	odd	2