Embedded newform 16.14.e.a.5.16

• Label: 16.14.e.a.5.16
• 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			5.16
Character	\(\chi\)	\(=\)	16.5
Dual form			16.14.e.a.13.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(34.4647 + 83.6910i) q^{2} +(-787.355 + 787.355i) q^{3} +(-5816.37 + 5768.77i) q^{4} +(13376.5 + 13376.5i) q^{5} +(-93030.5 - 38758.6i) q^{6} +547213. i q^{7} +(-683254. - 287958. i) q^{8} +354466. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 2 q^{2} - 2 q^{3} + 360 q^{4} - 2 q^{5} - 255056 q^{6} - 1076876 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	34.4647	+	83.6910i	0.380785	+	0.924664i
							\(3\)	−787.355	+	787.355i	−0.623566	+	0.623566i	−0.946442	−	0.322875i	\(-0.895351\pi\)
0.322875	+	0.946442i	\(0.395351\pi\)
\(5\)	13376.5	+	13376.5i	0.382857	+	0.382857i	0.872130	−	0.489274i	\(-0.162738\pi\)
							−0.489274	+	0.872130i	\(0.662738\pi\)
\(7\)			547213.i			1.75800i	0.476820	+	0.879001i	\(0.341789\pi\)
							−0.476820	+	0.879001i	\(0.658211\pi\)
\(11\)	−4.63585e6	−	4.63585e6i	−0.789000	−	0.789000i	0.192330	−	0.981330i	\(-0.438395\pi\)
							−0.981330	+	0.192330i	\(0.938395\pi\)
\(13\)	4.58853e6	−	4.58853e6i	0.263659	−	0.263659i	−0.562880	−	0.826539i	\(-0.690307\pi\)
							0.826539	+	0.562880i	\(0.190307\pi\)
\(17\)	1.39554e8			1.40225			0.701125	−	0.713039i	\(-0.252682\pi\)
							0.701125	+	0.713039i	\(0.252682\pi\)
\(19\)	9.33014e7	−	9.33014e7i	0.454977	−	0.454977i	−0.442025	−	0.897003i	\(-0.645740\pi\)
							0.897003	+	0.442025i	\(0.145740\pi\)
\(23\)			6.57682e8i			0.926371i	0.886261	+	0.463185i	\(0.153294\pi\)
							−0.886261	+	0.463185i	\(0.846706\pi\)
\(29\)	−1.55661e9	+	1.55661e9i	−0.485952	+	0.485952i	−0.907026	−	0.421074i	\(-0.861653\pi\)
							0.421074	+	0.907026i	\(0.361653\pi\)
\(31\)	−6.80557e9			−1.37725			−0.688627	−	0.725116i	\(-0.741786\pi\)
							−0.688627	+	0.725116i	\(0.741786\pi\)
\(37\)	5.41980e9	+	5.41980e9i	0.347274	+	0.347274i	0.859093	−	0.511819i	\(-0.171028\pi\)
							−0.511819	+	0.859093i	\(0.671028\pi\)
\(41\)			2.37231e10i			0.779967i	0.920822	+	0.389983i	\(0.127519\pi\)
							−0.920822	+	0.389983i	\(0.872481\pi\)
\(43\)	5.22826e10	+	5.22826e10i	1.26128	+	1.26128i	0.950472	+	0.310810i	\(0.100600\pi\)
							0.310810	+	0.950472i	\(0.399400\pi\)
\(47\)	−6.79598e10			−0.919636			−0.459818	−	0.888013i	\(-0.652085\pi\)
							−0.459818	+	0.888013i	\(0.652085\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.14.e.a.5.16	✓	50
4.3	odd	2		64.14.e.a.49.18		50
8.3	odd	2		128.14.e.a.97.8		50
8.5	even	2		128.14.e.b.97.18		50
16.3	odd	4		64.14.e.a.17.18		50
16.5	even	4		128.14.e.b.33.18		50
16.11	odd	4		128.14.e.a.33.8		50
16.13	even	4	inner	16.14.e.a.13.16	yes	50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.14.e.a.5.16	✓	50	1.1	even	1	trivial
16.14.e.a.13.16	yes	50	16.13	even	4	inner
64.14.e.a.17.18		50	16.3	odd	4
64.14.e.a.49.18		50	4.3	odd	2
128.14.e.a.33.8		50	16.11	odd	4
128.14.e.a.97.8		50	8.3	odd	2
128.14.e.b.33.18		50	16.5	even	4
128.14.e.b.97.18		50	8.5	even	2