Embedded newform 16.28.e.a.5.12

• Label: 16.28.e.a.5.12
• 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			5.12
Character	\(\chi\)	\(=\)	16.5
Dual form			16.28.e.a.13.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9065.18 - 7213.90i) q^{2} +(-3.30251e6 + 3.30251e6i) q^{3} +(3.01372e7 + 1.30790e8i) q^{4} +(3.04061e8 + 3.04061e8i) q^{5} +(5.37618e10 - 6.11388e9i) q^{6} -2.05985e11i q^{7} +(6.70310e11 - 1.40305e12i) q^{8} -1.41875e13i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 106 q - 2 q^{2} - 2 q^{3} + 125095688 q^{4} - 2 q^{5} - 59499919472 q^{6} - 287180933804 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9065.18	−	7213.90i	−0.782477	−	0.622680i
							\(3\)	−3.30251e6	+	3.30251e6i	−1.19593	+	1.19593i	−0.220560	+	0.975373i	\(0.570789\pi\)
−0.975373	+	0.220560i	\(0.929211\pi\)
\(5\)	3.04061e8	+	3.04061e8i	0.111395	+	0.111395i	0.760607	−	0.649212i	\(-0.224901\pi\)
							−0.649212	+	0.760607i	\(0.724901\pi\)
\(7\)		−	2.05985e11i		−	0.803548i	−0.915739	−	0.401774i	\(-0.868394\pi\)
							0.915739	−	0.401774i	\(-0.131606\pi\)
\(11\)	−7.46488e13	−	7.46488e13i	−0.651961	−	0.651961i	0.301504	−	0.953465i	\(-0.402511\pi\)
							−0.953465	+	0.301504i	\(0.902511\pi\)
\(13\)	4.03416e14	−	4.03416e14i	0.369418	−	0.369418i	−0.497847	−	0.867265i	\(-0.665876\pi\)
							0.867265	+	0.497847i	\(0.165876\pi\)
\(17\)	−2.29769e14			−0.00562642			−0.00281321	−	0.999996i	\(-0.500895\pi\)
							−0.00281321	+	0.999996i	\(0.500895\pi\)
\(19\)	1.92943e17	−	1.92943e17i	1.05258	−	1.05258i	0.0540436	−	0.998539i	\(-0.482789\pi\)
							0.998539	−	0.0540436i	\(-0.0172110\pi\)
\(23\)			9.65370e17i			0.399363i	0.979861	+	0.199682i	\(0.0639908\pi\)
							−0.979861	+	0.199682i	\(0.936009\pi\)
\(29\)	−5.76285e19	+	5.76285e19i	−1.04295	+	1.04295i	−0.0439173	+	0.999035i	\(0.513984\pi\)
							−0.999035	+	0.0439173i	\(0.986016\pi\)
\(31\)	1.11797e20			0.822329			0.411165	−	0.911561i	\(-0.365122\pi\)
							0.411165	+	0.911561i	\(0.365122\pi\)
\(37\)	−1.14792e21	−	1.14792e21i	−0.774794	−	0.774794i	0.204146	−	0.978940i	\(-0.434558\pi\)
							−0.978940	+	0.204146i	\(0.934558\pi\)
\(41\)		−	2.52651e21i		−	0.426520i	−0.976995	−	0.213260i	\(-0.931592\pi\)
							0.976995	−	0.213260i	\(-0.0684081\pi\)
\(43\)	3.23875e19	+	3.23875e19i	0.00287444	+	0.00287444i	0.708543	−	0.705668i	\(-0.249353\pi\)
							−0.705668	+	0.708543i	\(0.749353\pi\)
\(47\)	1.55896e22			0.416402			0.208201	−	0.978086i	\(-0.433239\pi\)
							0.208201	+	0.978086i	\(0.433239\pi\)