Embedded newform 16.15.f.a.3.17

• Label: 16.15.f.a.3.17
• Level: $16$
• Weight: $15$
• Character: 16.3
• Analytic conductor: $19.893$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.8926349043\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(27\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.17
Character	\(\chi\)	\(=\)	16.3
Dual form			16.15.f.a.11.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(47.5205 + 118.852i) q^{2} +(2009.03 - 2009.03i) q^{3} +(-11867.6 + 11295.8i) q^{4} +(-90795.6 + 90795.6i) q^{5} +(334247. + 143307. i) q^{6} +656977. q^{7} +(-1.90649e6 - 873704. i) q^{8} -3.28943e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 2 q^{2} - 2 q^{3} + 15832 q^{4} - 2 q^{5} - 182528 q^{6} - 4 q^{7} + 2748484 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	47.5205	+	118.852i	0.371254	+	0.928531i
							\(3\)	2009.03	−	2009.03i	0.918623	−	0.918623i	−0.0783061	−	0.996929i	\(-0.524951\pi\)
0.996929	+	0.0783061i	\(0.0249511\pi\)
\(5\)	−90795.6	+	90795.6i	−1.16218	+	1.16218i	−0.178187	+	0.983997i	\(0.557023\pi\)
							−0.983997	+	0.178187i	\(0.942977\pi\)
\(7\)	656977.			0.797745			0.398873	−	0.917006i	\(-0.369402\pi\)
							0.398873	+	0.917006i	\(0.369402\pi\)
\(11\)	−1.21770e7	−	1.21770e7i	−0.624871	−	0.624871i	0.321902	−	0.946773i	\(-0.395678\pi\)
							−0.946773	+	0.321902i	\(0.895678\pi\)
\(13\)	−3.40778e6	−	3.40778e6i	−0.0543085	−	0.0543085i	0.679431	−	0.733739i	\(-0.262227\pi\)
							−0.733739	+	0.679431i	\(0.762227\pi\)
\(17\)	−6.45627e8			−1.57340			−0.786700	−	0.617335i	\(-0.788212\pi\)
							−0.786700	+	0.617335i	\(0.788212\pi\)
\(19\)	−1.07210e9	+	1.07210e9i	−1.19938	+	1.19938i	−0.225032	+	0.974351i	\(0.572249\pi\)
							−0.974351	+	0.225032i	\(0.927751\pi\)
\(23\)	−4.00748e9			−1.17700			−0.588500	−	0.808497i	\(-0.700281\pi\)
							−0.588500	+	0.808497i	\(0.700281\pi\)
\(29\)	5.92781e9	+	5.92781e9i	0.343644	+	0.343644i	0.857735	−	0.514092i	\(-0.171871\pi\)
							−0.514092	+	0.857735i	\(0.671871\pi\)
\(31\)			1.22435e9i			0.0445014i	0.999752	+	0.0222507i	\(0.00708320\pi\)
							−0.999752	+	0.0222507i	\(0.992917\pi\)
\(37\)	7.92855e10	−	7.92855e10i	0.835184	−	0.835184i	−0.153037	−	0.988220i	\(-0.548905\pi\)
							0.988220	+	0.153037i	\(0.0489053\pi\)
\(41\)			1.55849e11i			0.800235i	0.916464	+	0.400118i	\(0.131031\pi\)
							−0.916464	+	0.400118i	\(0.868969\pi\)
\(43\)	3.18163e11	+	3.18163e11i	1.17050	+	1.17050i	0.982091	+	0.188406i	\(0.0603320\pi\)
							0.188406	+	0.982091i	\(0.439668\pi\)
\(47\)			4.08632e10i			0.0806579i	0.999186	+	0.0403290i	\(0.0128406\pi\)
							−0.999186	+	0.0403290i	\(0.987159\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.15.f.a.3.17	✓	54
4.3	odd	2		64.15.f.a.47.5		54
16.5	even	4		64.15.f.a.15.5		54
16.11	odd	4	inner	16.15.f.a.11.17	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.15.f.a.3.17	✓	54	1.1	even	1	trivial
16.15.f.a.11.17	yes	54	16.11	odd	4	inner
64.15.f.a.15.5		54	16.5	even	4
64.15.f.a.47.5		54	4.3	odd	2