Embedded newform 16.22.e.a.5.20

• Label: 16.22.e.a.5.20
• 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			5.20
Character	\(\chi\)	\(=\)	16.5
Dual form			16.22.e.a.13.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-26.6184 + 1447.91i) q^{2} +(-103140. + 103140. i) q^{3} +(-2.09573e6 - 77082.0i) q^{4} +(8.68960e6 + 8.68960e6i) q^{5} +(-1.46592e8 - 1.52083e8i) q^{6} -1.15126e9i q^{7} +(1.67393e8 - 3.03238e9i) q^{8} -1.08153e10i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 2 q^{2} - 2 q^{3} + 1604328 q^{4} - 2 q^{5} - 142760912 q^{6} - 5437750796 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−26.6184	+	1447.91i	−0.0183809	+	0.999831i
							\(3\)	−103140.	+	103140.i	−1.00845	+	1.00845i	−0.00848298	+	0.999964i	\(0.502700\pi\)
−0.999964	+	0.00848298i	\(0.997300\pi\)
\(5\)	8.68960e6	+	8.68960e6i	0.397937	+	0.397937i	0.877505	−	0.479568i	\(-0.159206\pi\)
							−0.479568	+	0.877505i	\(0.659206\pi\)
\(7\)		−	1.15126e9i		−	1.54043i	−0.637783	−	0.770216i	\(-0.720149\pi\)
							0.637783	−	0.770216i	\(-0.279851\pi\)
\(11\)	1.08903e11	+	1.08903e11i	1.26595	+	1.26595i	0.948163	+	0.317785i	\(0.102939\pi\)
							0.317785	+	0.948163i	\(0.397061\pi\)
\(13\)	−3.65898e11	+	3.65898e11i	−0.736130	+	0.736130i	−0.971827	−	0.235697i	\(-0.924263\pi\)
							0.235697	+	0.971827i	\(0.424263\pi\)
\(17\)	8.97017e12			1.07916			0.539581	−	0.841933i	\(-0.318582\pi\)
							0.539581	+	0.841933i	\(0.318582\pi\)
\(19\)	7.41836e12	−	7.41836e12i	0.277584	−	0.277584i	−0.554560	−	0.832144i	\(-0.687113\pi\)
							0.832144	+	0.554560i	\(0.187113\pi\)
\(23\)			5.37414e13i			0.270500i	0.990812	+	0.135250i	\(0.0431837\pi\)
							−0.990812	+	0.135250i	\(0.956816\pi\)
\(29\)	5.93288e14	−	5.93288e14i	0.261870	−	0.261870i	−0.563943	−	0.825814i	\(-0.690716\pi\)
							0.825814	+	0.563943i	\(0.190716\pi\)
\(31\)	6.66774e15			1.46110			0.730552	−	0.682857i	\(-0.239263\pi\)
							0.730552	+	0.682857i	\(0.239263\pi\)
\(37\)	1.84534e16	+	1.84534e16i	0.630897	+	0.630897i	0.948293	−	0.317396i	\(-0.102808\pi\)
							−0.317396	+	0.948293i	\(0.602808\pi\)
\(41\)		−	1.40340e17i		−	1.63287i	−0.577437	−	0.816435i	\(-0.695947\pi\)
							0.577437	−	0.816435i	\(-0.304053\pi\)
\(43\)	1.03361e17	+	1.03361e17i	0.729355	+	0.729355i	0.970491	−	0.241136i	\(-0.0775201\pi\)
							−0.241136	+	0.970491i	\(0.577520\pi\)
\(47\)	−5.25241e17			−1.45657			−0.728284	−	0.685275i	\(-0.759682\pi\)
							−0.728284	+	0.685275i	\(0.759682\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.22.e.a.5.20	✓	82
4.3	odd	2		64.22.e.a.49.36		82
16.3	odd	4		64.22.e.a.17.36		82
16.13	even	4	inner	16.22.e.a.13.20	yes	82

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.22.e.a.5.20	✓	82	1.1	even	1	trivial
16.22.e.a.13.20	yes	82	16.13	even	4	inner
64.22.e.a.17.36		82	16.3	odd	4
64.22.e.a.49.36		82	4.3	odd	2