Embedded newform 16.28.e.a.5.13

• Label: 16.28.e.a.5.13
• 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			5.13
Character	\(\chi\)	\(=\)	16.5
Dual form			16.28.e.a.13.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8692.02 - 7659.41i) q^{2} +(-956364. + 956364. i) q^{3} +(1.68847e7 + 1.33151e8i) q^{4} +(-3.72252e9 - 3.72252e9i) q^{5} +(1.56379e10 - 9.87553e8i) q^{6} +2.96868e11i q^{7} +(8.73099e11 - 1.28668e12i) q^{8} +5.79634e12i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 106 q - 2 q^{2} - 2 q^{3} + 125095688 q^{4} - 2 q^{5} - 59499919472 q^{6} - 287180933804 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8692.02	−	7659.41i	−0.750267	−	0.661135i
							\(3\)	−956364.	+	956364.i	−0.346327	+	0.346327i	−0.858739	−	0.512413i	\(-0.828752\pi\)
0.512413	+	0.858739i	\(0.328752\pi\)
\(5\)	−3.72252e9	−	3.72252e9i	−1.36377	−	1.36377i	−0.869053	−	0.494718i	\(-0.835271\pi\)
							−0.494718	−	0.869053i	\(-0.664729\pi\)
\(7\)			2.96868e11i			1.15808i	0.815298	+	0.579042i	\(0.196573\pi\)
							−0.815298	+	0.579042i	\(0.803427\pi\)
\(11\)	−7.72808e13	−	7.72808e13i	−0.674948	−	0.674948i	0.283904	−	0.958853i	\(-0.408370\pi\)
							−0.958853	+	0.283904i	\(0.908370\pi\)
\(13\)	−3.14495e14	+	3.14495e14i	−0.287991	+	0.287991i	−0.836285	−	0.548295i	\(-0.815277\pi\)
							0.548295	+	0.836285i	\(0.315277\pi\)
\(17\)	3.52910e16			0.864179			0.432090	−	0.901831i	\(-0.357776\pi\)
							0.432090	+	0.901831i	\(0.357776\pi\)
\(19\)	−2.33440e16	+	2.33440e16i	−0.127351	+	0.127351i	−0.767909	−	0.640559i	\(-0.778703\pi\)
							0.640559	+	0.767909i	\(0.278703\pi\)
\(23\)			2.78338e18i			1.15146i	0.817641	+	0.575728i	\(0.195281\pi\)
							−0.817641	+	0.575728i	\(0.804719\pi\)
\(29\)	6.25137e19	−	6.25137e19i	1.13136	−	1.13136i	0.141413	−	0.989951i	\(-0.454835\pi\)
							0.989951	−	0.141413i	\(-0.0451646\pi\)
\(31\)	1.67223e20			1.23002			0.615012	−	0.788517i	\(-0.289151\pi\)
							0.615012	+	0.788517i	\(0.289151\pi\)
\(37\)	9.41768e20	+	9.41768e20i	0.635654	+	0.635654i	0.949480	−	0.313826i	\(-0.101611\pi\)
							−0.313826	+	0.949480i	\(0.601611\pi\)
\(41\)		−	4.31811e21i		−	0.728974i	−0.931208	−	0.364487i	\(-0.881244\pi\)
							0.931208	−	0.364487i	\(-0.118756\pi\)
\(43\)	9.34576e21	+	9.34576e21i	0.829451	+	0.829451i	0.987441	−	0.157990i	\(-0.0505013\pi\)
							−0.157990	+	0.987441i	\(0.550501\pi\)
\(47\)	−3.69526e22			−0.987016			−0.493508	−	0.869741i	\(-0.664286\pi\)
							−0.493508	+	0.869741i	\(0.664286\pi\)