Embedded newform 16.14.e.a.5.20

• Label: 16.14.e.a.5.20
• Level: $16$
• Weight: $14$
• Character: 16.5
• 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			5.20
Character	\(\chi\)	\(=\)	16.5
Dual form			16.14.e.a.13.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(64.2863 - 63.7124i) q^{2} +(-1618.47 + 1618.47i) q^{3} +(73.4667 - 8191.67i) q^{4} +(-23191.8 - 23191.8i) q^{5} +(-928.945 + 207162. i) q^{6} +280152. i q^{7} +(-517188. - 531293. i) q^{8} -3.64459e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 2 q^{2} - 2 q^{3} + 360 q^{4} - 2 q^{5} - 255056 q^{6} - 1076876 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	64.2863	−	63.7124i	0.710270	−	0.703929i
							\(3\)	−1618.47	+	1618.47i	−1.28179	+	1.28179i	−0.342143	+	0.939648i	\(0.611153\pi\)
−0.939648	+	0.342143i	\(0.888847\pi\)
\(5\)	−23191.8	−	23191.8i	−0.663789	−	0.663789i	0.292482	−	0.956271i	\(-0.405519\pi\)
							−0.956271	+	0.292482i	\(0.905519\pi\)
\(7\)			280152.i			0.900029i	0.893021	+	0.450015i	\(0.148581\pi\)
							−0.893021	+	0.450015i	\(0.851419\pi\)
\(11\)	6.11068e6	+	6.11068e6i	1.04001	+	1.04001i	0.999166	+	0.0408442i	\(0.0130047\pi\)
							0.0408442	+	0.999166i	\(0.486995\pi\)
\(13\)	1.15263e7	−	1.15263e7i	0.662303	−	0.662303i	−0.293619	−	0.955923i	\(-0.594860\pi\)
							0.955923	+	0.293619i	\(0.0948597\pi\)
\(17\)	1.68871e8			1.69683			0.848414	−	0.529334i	\(-0.177558\pi\)
							0.848414	+	0.529334i	\(0.177558\pi\)
\(19\)	5.82970e6	−	5.82970e6i	0.0284281	−	0.0284281i	−0.692750	−	0.721178i	\(-0.743601\pi\)
							0.721178	+	0.692750i	\(0.243601\pi\)
\(23\)		−	2.84306e8i		−	0.400456i	−0.979749	−	0.200228i	\(-0.935832\pi\)
							0.979749	−	0.200228i	\(-0.0641683\pi\)
\(29\)	−1.26029e9	+	1.26029e9i	−0.393444	+	0.393444i	−0.875913	−	0.482469i	\(-0.839740\pi\)
							0.482469	+	0.875913i	\(0.339740\pi\)
\(31\)	1.42208e9			0.287788			0.143894	−	0.989593i	\(-0.454038\pi\)
							0.143894	+	0.989593i	\(0.454038\pi\)
\(37\)	1.30091e10	+	1.30091e10i	0.833561	+	0.833561i	0.988002	−	0.154441i	\(-0.0493578\pi\)
							−0.154441	+	0.988002i	\(0.549358\pi\)
\(41\)		−	3.32305e10i		−	1.09255i	−0.837606	−	0.546275i	\(-0.816045\pi\)
							0.837606	−	0.546275i	\(-0.183955\pi\)
\(43\)	−8.67795e9	−	8.67795e9i	−0.209350	−	0.209350i	0.594641	−	0.803991i	\(-0.297294\pi\)
							−0.803991	+	0.594641i	\(0.797294\pi\)
\(47\)	2.33273e10			0.315667			0.157833	−	0.987466i	\(-0.449549\pi\)
							0.157833	+	0.987466i	\(0.449549\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.14.e.a.5.20	✓	50
4.3	odd	2		64.14.e.a.49.24		50
8.3	odd	2		128.14.e.a.97.2		50
8.5	even	2		128.14.e.b.97.24		50
16.3	odd	4		64.14.e.a.17.24		50
16.5	even	4		128.14.e.b.33.24		50
16.11	odd	4		128.14.e.a.33.2		50
16.13	even	4	inner	16.14.e.a.13.20	yes	50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.14.e.a.5.20	✓	50	1.1	even	1	trivial
16.14.e.a.13.20	yes	50	16.13	even	4	inner
64.14.e.a.17.24		50	16.3	odd	4
64.14.e.a.49.24		50	4.3	odd	2
128.14.e.a.33.2		50	16.11	odd	4
128.14.e.a.97.2		50	8.3	odd	2
128.14.e.b.33.24		50	16.5	even	4
128.14.e.b.97.24		50	8.5	even	2