Embedded newform 16.28.e.a.5.19

• Label: 16.28.e.a.5.19
• 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			5.19
Character	\(\chi\)	\(=\)	16.5
Dual form			16.28.e.a.13.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4635.48 + 10617.4i) q^{2} +(631535. - 631535. i) q^{3} +(-9.12424e7 - 9.84339e7i) q^{4} +(2.52651e9 + 2.52651e9i) q^{5} +(3.77782e9 + 9.63275e9i) q^{6} +3.51372e11i q^{7} +(1.46807e12 - 5.12473e11i) q^{8} +6.82793e12i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 106 q - 2 q^{2} - 2 q^{3} + 125095688 q^{4} - 2 q^{5} - 59499919472 q^{6} - 287180933804 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4635.48	+	10617.4i	−0.400119	+	0.916463i
							\(3\)	631535.	−	631535.i	0.228697	−	0.228697i	−0.583451	−	0.812148i	\(-0.698298\pi\)
0.812148	+	0.583451i	\(0.198298\pi\)
\(5\)	2.52651e9	+	2.52651e9i	0.925606	+	0.925606i	0.997418	−	0.0718122i	\(-0.0228782\pi\)
							−0.0718122	+	0.997418i	\(0.522878\pi\)
\(7\)			3.51372e11i			1.37070i	0.728213	+	0.685351i	\(0.240351\pi\)
							−0.728213	+	0.685351i	\(0.759649\pi\)
\(11\)	−1.25184e14	−	1.25184e14i	−1.09332	−	1.09332i	−0.995171	−	0.0981517i	\(-0.968707\pi\)
							−0.0981517	−	0.995171i	\(-0.531293\pi\)
\(13\)	−3.05437e14	+	3.05437e14i	−0.279696	+	0.279696i	−0.832988	−	0.553292i	\(-0.813372\pi\)
							0.553292	+	0.832988i	\(0.313372\pi\)
\(17\)	1.23233e15			0.0301762			0.0150881	−	0.999886i	\(-0.495197\pi\)
							0.0150881	+	0.999886i	\(0.495197\pi\)
\(19\)	7.03680e15	−	7.03680e15i	0.0383885	−	0.0383885i	−0.687652	−	0.726040i	\(-0.741358\pi\)
							0.726040	+	0.687652i	\(0.241358\pi\)
\(23\)			4.13969e18i			1.71255i	0.516524	+	0.856273i	\(0.327226\pi\)
							−0.516524	+	0.856273i	\(0.672774\pi\)
\(29\)	3.43194e19	−	3.43194e19i	0.621108	−	0.621108i	−0.324707	−	0.945815i	\(-0.605266\pi\)
							0.945815	+	0.324707i	\(0.105266\pi\)
\(31\)	−2.05027e20			−1.50809			−0.754047	−	0.656820i	\(-0.771901\pi\)
							−0.754047	+	0.656820i	\(0.771901\pi\)
\(37\)	1.64863e20	+	1.64863e20i	0.111275	+	0.111275i	0.760552	−	0.649277i	\(-0.224928\pi\)
							−0.649277	+	0.760552i	\(0.724928\pi\)
\(41\)			3.35911e21i			0.567078i	0.958961	+	0.283539i	\(0.0915085\pi\)
							−0.958961	+	0.283539i	\(0.908491\pi\)
\(43\)	−2.49774e20	−	2.49774e20i	−0.0221678	−	0.0221678i	0.695936	−	0.718104i	\(-0.254990\pi\)
							−0.718104	+	0.695936i	\(0.754990\pi\)
\(47\)	4.20881e21			0.112419			0.0562093	−	0.998419i	\(-0.482099\pi\)
							0.0562093	+	0.998419i	\(0.482099\pi\)