Embedded newform 16.28.e.a.5.18

• Label: 16.28.e.a.5.18
• Level: $16$
• Weight: $28$
• Character: 16.5
• Analytic conductor: $73.897$
• Analytic rank: $0$
• Dimension: $106$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 28 \)
Character orbit:	\([\chi]\)	\(=\)	16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(73.8968919741\)
Analytic rank:	\(0\)
Dimension:	\(106\)
Relative dimension:	\(53\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.18
Character	\(\chi\)	\(=\)	16.5
Dual form			16.28.e.a.13.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5041.21 - 10430.9i) q^{2} +(-329571. + 329571. i) q^{3} +(-8.33902e7 + 1.05169e8i) q^{4} +(-2.69387e8 - 2.69387e8i) q^{5} +(5.09916e9 + 1.77629e9i) q^{6} -2.71695e11i q^{7} +(1.51739e12 + 3.39659e11i) q^{8} +7.40836e12i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 106 q - 2 q^{2} - 2 q^{3} + 125095688 q^{4} - 2 q^{5} - 59499919472 q^{6} - 287180933804 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5041.21	−	10430.9i	−0.435140	−	0.900363i
							\(3\)	−329571.	+	329571.i	−0.119347	+	0.119347i	−0.764258	−	0.644911i	\(-0.776894\pi\)
0.644911	+	0.764258i	\(0.276894\pi\)
\(5\)	−2.69387e8	−	2.69387e8i	−0.0986920	−	0.0986920i	0.656037	−	0.754729i	\(-0.272232\pi\)
							−0.754729	+	0.656037i	\(0.772232\pi\)
\(7\)		−	2.71695e11i		−	1.05988i	−0.848035	−	0.529941i	\(-0.822214\pi\)
							0.848035	−	0.529941i	\(-0.177786\pi\)
\(11\)	2.66489e13	+	2.66489e13i	0.232744	+	0.232744i	0.813837	−	0.581093i	\(-0.197375\pi\)
							−0.581093	+	0.813837i	\(0.697375\pi\)
\(13\)	−4.16447e14	+	4.16447e14i	−0.381351	+	0.381351i	−0.871589	−	0.490238i	\(-0.836910\pi\)
							0.490238	+	0.871589i	\(0.336910\pi\)
\(17\)	2.48650e16			0.608875			0.304437	−	0.952532i	\(-0.401532\pi\)
							0.304437	+	0.952532i	\(0.401532\pi\)
\(19\)	3.21749e16	−	3.21749e16i	0.175527	−	0.175527i	−0.613876	−	0.789403i	\(-0.710390\pi\)
							0.789403	+	0.613876i	\(0.210390\pi\)
\(23\)		−	1.49823e18i		−	0.619800i	−0.950769	−	0.309900i	\(-0.899704\pi\)
							0.950769	−	0.309900i	\(-0.100296\pi\)
\(29\)	8.47986e18	−	8.47986e18i	0.153467	−	0.153467i	−0.626197	−	0.779665i	\(-0.715390\pi\)
							0.779665	+	0.626197i	\(0.215390\pi\)
\(31\)	−1.32940e20			−0.977851			−0.488925	−	0.872326i	\(-0.662611\pi\)
							−0.488925	+	0.872326i	\(0.662611\pi\)
\(37\)	7.33132e20	+	7.33132e20i	0.494833	+	0.494833i	0.909825	−	0.414992i	\(-0.136216\pi\)
							−0.414992	+	0.909825i	\(0.636216\pi\)
\(41\)			9.82451e20i			0.165855i	0.996556	+	0.0829276i	\(0.0264270\pi\)
							−0.996556	+	0.0829276i	\(0.973573\pi\)
\(43\)	−1.30008e22	−	1.30008e22i	−1.15385	−	1.15385i	−0.985775	−	0.168070i	\(-0.946247\pi\)
							−0.168070	−	0.985775i	\(-0.553753\pi\)
\(47\)	−4.26559e22			−1.13935			−0.569676	−	0.821869i	\(-0.692931\pi\)
							−0.569676	+	0.821869i	\(0.692931\pi\)