Embedded newform 16.13.f.a.3.13

• Label: 16.13.f.a.3.13
• Level: $16$
• Weight: $13$
• Character: 16.3
• Analytic conductor: $14.624$
• Analytic rank: $0$
• Dimension: $46$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.6239010764\)
Analytic rank:	\(0\)
Dimension:	\(46\)
Relative dimension:	\(23\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.13
Character	\(\chi\)	\(=\)	16.3
Dual form			16.13.f.a.11.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.15344 - 63.8651i) q^{2} +(-980.484 + 980.484i) q^{3} +(-4061.50 - 530.520i) q^{4} +(-15901.7 + 15901.7i) q^{5} +(58546.3 + 66691.1i) q^{6} -94838.7 q^{7} +(-50750.9 + 257184. i) q^{8} -1.39126e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 2 q^{2} - 2 q^{3} - 4232 q^{4} - 2 q^{5} + 72928 q^{6} - 4 q^{7} - 478172 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.15344	−	63.8651i	0.0648976	−	0.997892i
							\(3\)	−980.484	+	980.484i	−1.34497	+	1.34497i	−0.453938	+	0.891033i	\(0.649981\pi\)
−0.891033	+	0.453938i	\(0.850019\pi\)
\(5\)	−15901.7	+	15901.7i	−1.01771	+	1.01771i	−0.0178661	+	0.999840i	\(0.505687\pi\)
							−0.999840	+	0.0178661i	\(0.994313\pi\)
\(7\)	−94838.7			−0.806116			−0.403058	−	0.915174i	\(-0.632053\pi\)
							−0.403058	+	0.915174i	\(0.632053\pi\)
\(11\)	464056.	+	464056.i	0.261948	+	0.261948i	0.825845	−	0.563897i	\(-0.190699\pi\)
							−0.563897	+	0.825845i	\(0.690699\pi\)
\(13\)	1.28356e6	+	1.28356e6i	0.265923	+	0.265923i	0.827455	−	0.561532i	\(-0.189788\pi\)
							−0.561532	+	0.827455i	\(0.689788\pi\)
\(17\)	−2.93346e7			−1.21531			−0.607654	−	0.794202i	\(-0.707889\pi\)
							−0.607654	+	0.794202i	\(0.707889\pi\)
\(19\)	−3.66596e7	+	3.66596e7i	−0.779231	+	0.779231i	−0.979700	−	0.200469i	\(-0.935753\pi\)
							0.200469	+	0.979700i	\(0.435753\pi\)
\(23\)	1.23199e8			0.832227			0.416113	−	0.909313i	\(-0.363392\pi\)
							0.416113	+	0.909313i	\(0.363392\pi\)
\(29\)	−2.94878e8	−	2.94878e8i	−0.495740	−	0.495740i	0.414369	−	0.910109i	\(-0.364002\pi\)
							−0.910109	+	0.414369i	\(0.864002\pi\)
\(31\)			9.22849e8i			1.03983i	0.854219	+	0.519913i	\(0.174036\pi\)
							−0.854219	+	0.519913i	\(0.825964\pi\)
\(37\)	1.29360e9	−	1.29360e9i	0.504184	−	0.504184i	−0.408551	−	0.912735i	\(-0.633966\pi\)
							0.912735	+	0.408551i	\(0.133966\pi\)
\(41\)		−	8.03976e8i		−	0.169254i	−0.996413	−	0.0846272i	\(-0.973030\pi\)
							0.996413	−	0.0846272i	\(-0.0269699\pi\)
\(43\)	4.51882e9	+	4.51882e9i	0.714849	+	0.714849i	0.967546	−	0.252696i	\(-0.0813173\pi\)
							−0.252696	+	0.967546i	\(0.581317\pi\)
\(47\)		−	3.52976e9i		−	0.327460i	−0.986505	−	0.163730i	\(-0.947647\pi\)
							0.986505	−	0.163730i	\(-0.0523525\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.13.f.a.3.13	✓	46
4.3	odd	2		64.13.f.a.47.23		46
8.3	odd	2		128.13.f.a.95.1		46
8.5	even	2		128.13.f.b.95.23		46
16.3	odd	4		128.13.f.b.31.23		46
16.5	even	4		64.13.f.a.15.23		46
16.11	odd	4	inner	16.13.f.a.11.13	yes	46
16.13	even	4		128.13.f.a.31.1		46

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.13.f.a.3.13	✓	46	1.1	even	1	trivial
16.13.f.a.11.13	yes	46	16.11	odd	4	inner
64.13.f.a.15.23		46	16.5	even	4
64.13.f.a.47.23		46	4.3	odd	2
128.13.f.a.31.1		46	16.13	even	4
128.13.f.a.95.1		46	8.3	odd	2
128.13.f.b.31.23		46	16.3	odd	4
128.13.f.b.95.23		46	8.5	even	2