Embedded newform 16.14.e.a.13.6

• Label: 16.14.e.a.13.6
• Level: $16$
• Weight: $14$
• Character: 16.13
• Analytic conductor: $17.157$
• Analytic rank: $0$
• Dimension: $50$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.1569486323\)
Analytic rank:	\(0\)
Dimension:	\(50\)
Relative dimension:	\(25\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			13.6
Character	\(\chi\)	\(=\)	16.13
Dual form			16.14.e.a.5.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-71.4691 - 55.5353i) q^{2} +(293.446 + 293.446i) q^{3} +(2023.65 + 7938.12i) q^{4} +(22716.6 - 22716.6i) q^{5} +(-4675.68 - 37268.9i) q^{6} +343566. i q^{7} +(296217. - 679714. i) q^{8} -1.42210e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 2 q^{2} - 2 q^{3} + 360 q^{4} - 2 q^{5} - 255056 q^{6} - 1076876 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−71.4691	−	55.5353i	−0.789629	−	0.613585i
							\(3\)	293.446	+	293.446i	0.232402	+	0.232402i	0.813695	−	0.581293i	\(-0.197453\pi\)
−0.581293	+	0.813695i	\(0.697453\pi\)
\(5\)	22716.6	−	22716.6i	0.650187	−	0.650187i	−0.302851	−	0.953038i	\(-0.597938\pi\)
							0.953038	+	0.302851i	\(0.0979384\pi\)
\(7\)			343566.i			1.10375i	0.833925	+	0.551877i	\(0.186088\pi\)
							−0.833925	+	0.551877i	\(0.813912\pi\)
\(11\)	−7.55337e6	+	7.55337e6i	−1.28555	+	1.28555i	−0.348085	+	0.937463i	\(0.613168\pi\)
							−0.937463	+	0.348085i	\(0.886832\pi\)
\(13\)	−6.02369e6	−	6.02369e6i	−0.346123	−	0.346123i	0.512540	−	0.858663i	\(-0.328705\pi\)
							−0.858663	+	0.512540i	\(0.828705\pi\)
\(17\)	1.22712e8			1.23302			0.616508	−	0.787349i	\(-0.288547\pi\)
							0.616508	+	0.787349i	\(0.288547\pi\)
\(19\)	1.66266e8	+	1.66266e8i	0.810783	+	0.810783i	0.984751	−	0.173968i	\(-0.0556589\pi\)
							−0.173968	+	0.984751i	\(0.555659\pi\)
\(23\)			7.77189e8i			1.09470i	0.836904	+	0.547350i	\(0.184363\pi\)
							−0.836904	+	0.547350i	\(0.815637\pi\)
\(29\)	3.04458e9	+	3.04458e9i	0.950476	+	0.950476i	0.998830	−	0.0483544i	\(-0.0153977\pi\)
							−0.0483544	+	0.998830i	\(0.515398\pi\)
\(31\)	−6.68256e9			−1.35236			−0.676180	−	0.736737i	\(-0.736366\pi\)
							−0.676180	+	0.736737i	\(0.736366\pi\)
\(37\)	7.06457e9	−	7.06457e9i	0.452663	−	0.452663i	−0.443575	−	0.896237i	\(-0.646290\pi\)
							0.896237	+	0.443575i	\(0.146290\pi\)
\(41\)			4.77022e10i			1.56835i	0.620538	+	0.784176i	\(0.286914\pi\)
							−0.620538	+	0.784176i	\(0.713086\pi\)
\(43\)	−3.20320e10	+	3.20320e10i	−0.772749	+	0.772749i	−0.978586	−	0.205837i	\(-0.934008\pi\)
							0.205837	+	0.978586i	\(0.434008\pi\)
\(47\)	4.56238e10			0.617384			0.308692	−	0.951162i	\(-0.400109\pi\)
							0.308692	+	0.951162i	\(0.400109\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.14.e.a.13.6	yes	50
4.3	odd	2		64.14.e.a.17.11		50
8.3	odd	2		128.14.e.a.33.15		50
8.5	even	2		128.14.e.b.33.11		50
16.3	odd	4		128.14.e.a.97.15		50
16.5	even	4	inner	16.14.e.a.5.6	✓	50
16.11	odd	4		64.14.e.a.49.11		50
16.13	even	4		128.14.e.b.97.11		50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.14.e.a.5.6	✓	50	16.5	even	4	inner
16.14.e.a.13.6	yes	50	1.1	even	1	trivial
64.14.e.a.17.11		50	4.3	odd	2
64.14.e.a.49.11		50	16.11	odd	4
128.14.e.a.33.15		50	8.3	odd	2
128.14.e.a.97.15		50	16.3	odd	4
128.14.e.b.33.11		50	8.5	even	2
128.14.e.b.97.11		50	16.13	even	4