Embedded newform 16.14.e.a.5.11

• Label: 16.14.e.a.5.11
• 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			5.11
Character	\(\chi\)	\(=\)	16.5
Dual form			16.14.e.a.13.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-21.6798 - 87.8748i) q^{2} +(-1056.56 + 1056.56i) q^{3} +(-7251.97 + 3810.23i) q^{4} +(11923.3 + 11923.3i) q^{5} +(115752. + 69939.2i) q^{6} +170872. i q^{7} +(492044. + 554660. i) q^{8} -638332. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 2 q^{2} - 2 q^{3} + 360 q^{4} - 2 q^{5} - 255056 q^{6} - 1076876 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−21.6798	−	87.8748i	−0.239531	−	0.970889i
							\(3\)	−1056.56	+	1056.56i	−0.836773	+	0.836773i	−0.988433	−	0.151660i	\(-0.951538\pi\)
0.151660	+	0.988433i	\(0.451538\pi\)
\(5\)	11923.3	+	11923.3i	0.341266	+	0.341266i	0.856843	−	0.515577i	\(-0.172422\pi\)
							−0.515577	+	0.856843i	\(0.672422\pi\)
\(7\)			170872.i			0.548952i	0.961594	+	0.274476i	\(0.0885044\pi\)
							−0.961594	+	0.274476i	\(0.911496\pi\)
\(11\)	−2.37753e6	−	2.37753e6i	−0.404645	−	0.404645i	0.475222	−	0.879866i	\(-0.342368\pi\)
							−0.879866	+	0.475222i	\(0.842368\pi\)
\(13\)	4.28234e6	−	4.28234e6i	0.246065	−	0.246065i	−0.573289	−	0.819354i	\(-0.694333\pi\)
							0.819354	+	0.573289i	\(0.194333\pi\)
\(17\)	−9.43907e7			−0.948443			−0.474222	−	0.880406i	\(-0.657270\pi\)
							−0.474222	+	0.880406i	\(0.657270\pi\)
\(19\)	−1.74041e8	+	1.74041e8i	−0.848697	+	0.848697i	−0.989971	−	0.141274i	\(-0.954880\pi\)
							0.141274	+	0.989971i	\(0.454880\pi\)
\(23\)		−	1.73142e8i		−	0.243877i	−0.992538	−	0.121938i	\(-0.961089\pi\)
							0.992538	−	0.121938i	\(-0.0389110\pi\)
\(29\)	3.73585e9	−	3.73585e9i	1.16628	−	1.16628i	0.183203	−	0.983075i	\(-0.441353\pi\)
							0.983075	−	0.183203i	\(-0.0586467\pi\)
\(31\)	−1.65346e9			−0.334613			−0.167306	−	0.985905i	\(-0.553507\pi\)
							−0.167306	+	0.985905i	\(0.553507\pi\)
\(37\)	1.63242e9	+	1.63242e9i	0.104598	+	0.104598i	0.757469	−	0.652871i	\(-0.226436\pi\)
							−0.652871	+	0.757469i	\(0.726436\pi\)
\(41\)		−	3.24143e10i		−	1.06572i	−0.846205	−	0.532858i	\(-0.821118\pi\)
							0.846205	−	0.532858i	\(-0.178882\pi\)
\(43\)	−2.71755e10	−	2.71755e10i	−0.655591	−	0.655591i	0.298743	−	0.954334i	\(-0.403433\pi\)
							−0.954334	+	0.298743i	\(0.903433\pi\)
\(47\)	1.24361e11			1.68285			0.841427	−	0.540370i	\(-0.181716\pi\)
							0.841427	+	0.540370i	\(0.181716\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.14.e.a.5.11	✓	50
4.3	odd	2		64.14.e.a.49.21		50
8.3	odd	2		128.14.e.a.97.5		50
8.5	even	2		128.14.e.b.97.21		50
16.3	odd	4		64.14.e.a.17.21		50
16.5	even	4		128.14.e.b.33.21		50
16.11	odd	4		128.14.e.a.33.5		50
16.13	even	4	inner	16.14.e.a.13.11	yes	50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.14.e.a.5.11	✓	50	1.1	even	1	trivial
16.14.e.a.13.11	yes	50	16.13	even	4	inner
64.14.e.a.17.21		50	16.3	odd	4
64.14.e.a.49.21		50	4.3	odd	2
128.14.e.a.33.5		50	16.11	odd	4
128.14.e.a.97.5		50	8.3	odd	2
128.14.e.b.33.21		50	16.5	even	4
128.14.e.b.97.21		50	8.5	even	2