Embedded newform 16.28.e.a.5.9

• Label: 16.28.e.a.5.9
• 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			5.9
Character	\(\chi\)	\(=\)	16.5
Dual form			16.28.e.a.13.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9759.73 + 6242.23i) q^{2} +(-2.84375e6 + 2.84375e6i) q^{3} +(5.62868e7 - 1.21845e8i) q^{4} +(1.41523e9 + 1.41523e9i) q^{5} +(1.00029e10 - 4.55056e10i) q^{6} -4.92833e10i q^{7} +(2.11241e11 + 1.54053e12i) q^{8} -8.54826e12i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 106 q - 2 q^{2} - 2 q^{3} + 125095688 q^{4} - 2 q^{5} - 59499919472 q^{6} - 287180933804 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9759.73	+	6242.23i	−0.842428	+	0.538809i
							\(3\)	−2.84375e6	+	2.84375e6i	−1.02980	+	1.02980i	−0.0302626	+	0.999542i	\(0.509634\pi\)
−0.999542	+	0.0302626i	\(0.990366\pi\)
\(5\)	1.41523e9	+	1.41523e9i	0.518480	+	0.518480i	0.917111	−	0.398631i	\(-0.130515\pi\)
							−0.398631	+	0.917111i	\(0.630515\pi\)
\(7\)		−	4.92833e10i		−	0.192255i	−0.995369	−	0.0961273i	\(-0.969354\pi\)
							0.995369	−	0.0961273i	\(-0.0306456\pi\)
\(11\)	−7.50786e13	−	7.50786e13i	−0.655715	−	0.655715i	0.298648	−	0.954363i	\(-0.403464\pi\)
							−0.954363	+	0.298648i	\(0.903464\pi\)
\(13\)	−2.11138e14	+	2.11138e14i	−0.193344	+	0.193344i	−0.797139	−	0.603795i	\(-0.793654\pi\)
							0.603795	+	0.797139i	\(0.293654\pi\)
\(17\)	−3.49310e16			−0.855364			−0.427682	−	0.903929i	\(-0.640670\pi\)
							−0.427682	+	0.903929i	\(0.640670\pi\)
\(19\)	−9.88946e16	+	9.88946e16i	−0.539509	+	0.539509i	−0.923385	−	0.383875i	\(-0.874589\pi\)
							0.383875	+	0.923385i	\(0.374589\pi\)
\(23\)		−	1.36338e17i		−	0.0564018i	−0.999602	−	0.0282009i	\(-0.991022\pi\)
							0.999602	−	0.0282009i	\(-0.00897781\pi\)
\(29\)	4.33571e19	−	4.33571e19i	0.784670	−	0.784670i	−0.195945	−	0.980615i	\(-0.562777\pi\)
							0.980615	+	0.195945i	\(0.0627775\pi\)
\(31\)	1.79832e20			1.32277			0.661386	−	0.750046i	\(-0.269969\pi\)
							0.661386	+	0.750046i	\(0.269969\pi\)
\(37\)	3.56328e20	+	3.56328e20i	0.240507	+	0.240507i	0.817060	−	0.576553i	\(-0.195603\pi\)
							−0.576553	+	0.817060i	\(0.695603\pi\)
\(41\)			8.20369e20i			0.138493i	0.997600	+	0.0692464i	\(0.0220595\pi\)
							−0.997600	+	0.0692464i	\(0.977941\pi\)
\(43\)	−6.39705e21	−	6.39705e21i	−0.567748	−	0.567748i	0.363749	−	0.931497i	\(-0.381497\pi\)
							−0.931497	+	0.363749i	\(0.881497\pi\)
\(47\)	1.41378e22			0.377625			0.188813	−	0.982013i	\(-0.439536\pi\)
							0.188813	+	0.982013i	\(0.439536\pi\)