Embedded newform 16.13.f.a.3.10

• Label: 16.13.f.a.3.10
• 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			3.10
Character	\(\chi\)	\(=\)	16.3
Dual form			16.13.f.a.11.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.7829 - 62.9060i) q^{2} +(252.451 - 252.451i) q^{3} +(-3818.33 + 1482.43i) q^{4} +(-9569.73 + 9569.73i) q^{5} +(-18855.3 - 12906.1i) q^{6} +168743. q^{7} +(138244. + 222729. i) q^{8} +403978. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 2 q^{2} - 2 q^{3} - 4232 q^{4} - 2 q^{5} + 72928 q^{6} - 4 q^{7} - 478172 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.7829	−	62.9060i	−0.184107	−	0.982906i
							\(3\)	252.451	−	252.451i	0.346298	−	0.346298i	−0.512431	−	0.858729i	\(-0.671255\pi\)
0.858729	+	0.512431i	\(0.171255\pi\)
\(5\)	−9569.73	+	9569.73i	−0.612463	+	0.612463i	−0.943587	−	0.331125i	\(-0.892572\pi\)
							0.331125	+	0.943587i	\(0.392572\pi\)
\(7\)	168743.			1.43429			0.717144	−	0.696925i	\(-0.245449\pi\)
							0.717144	+	0.696925i	\(0.245449\pi\)
\(11\)	−1.62851e6	−	1.62851e6i	−0.919253	−	0.919253i	0.0777219	−	0.996975i	\(-0.475235\pi\)
							−0.996975	+	0.0777219i	\(0.975235\pi\)
\(13\)	6.06856e6	+	6.06856e6i	1.25726	+	1.25726i	0.952395	+	0.304867i	\(0.0986122\pi\)
							0.304867	+	0.952395i	\(0.401388\pi\)
\(17\)	−7.47202e6			−0.309560			−0.154780	−	0.987949i	\(-0.549467\pi\)
							−0.154780	+	0.987949i	\(0.549467\pi\)
\(19\)	6.07939e7	−	6.07939e7i	1.29223	−	1.29223i	0.358820	−	0.933407i	\(-0.383179\pi\)
							0.933407	−	0.358820i	\(-0.116821\pi\)
\(23\)	−1.47815e7			−0.0998507			−0.0499254	−	0.998753i	\(-0.515898\pi\)
							−0.0499254	+	0.998753i	\(0.515898\pi\)
\(29\)	6.03037e8	+	6.03037e8i	1.01381	+	1.01381i	0.999903	+	0.0139048i	\(0.00442619\pi\)
							0.0139048	+	0.999903i	\(0.495574\pi\)
\(31\)		−	1.95732e8i		−	0.220542i	−0.993902	−	0.110271i	\(-0.964828\pi\)
							0.993902	−	0.110271i	\(-0.0351719\pi\)
\(37\)	2.13066e8	−	2.13066e8i	0.0830430	−	0.0830430i	−0.664365	−	0.747408i	\(-0.731298\pi\)
							0.747408	+	0.664365i	\(0.231298\pi\)
\(41\)			7.06106e9i			1.48651i	0.669011	+	0.743253i	\(0.266718\pi\)
							−0.669011	+	0.743253i	\(0.733282\pi\)
\(43\)	5.07382e9	+	5.07382e9i	0.802647	+	0.802647i	0.983508	−	0.180862i	\(-0.0578887\pi\)
							−0.180862	+	0.983508i	\(0.557889\pi\)
\(47\)			6.27503e9i			0.582141i	0.956702	+	0.291071i	\(0.0940115\pi\)
							−0.956702	+	0.291071i	\(0.905989\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.13.f.a.3.10	✓	46
4.3	odd	2		64.13.f.a.47.9		46
8.3	odd	2		128.13.f.a.95.15		46
8.5	even	2		128.13.f.b.95.9		46
16.3	odd	4		128.13.f.b.31.9		46
16.5	even	4		64.13.f.a.15.9		46
16.11	odd	4	inner	16.13.f.a.11.10	yes	46
16.13	even	4		128.13.f.a.31.15		46

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.13.f.a.3.10	✓	46	1.1	even	1	trivial
16.13.f.a.11.10	yes	46	16.11	odd	4	inner
64.13.f.a.15.9		46	16.5	even	4
64.13.f.a.47.9		46	4.3	odd	2
128.13.f.a.31.15		46	16.13	even	4
128.13.f.a.95.15		46	8.3	odd	2
128.13.f.b.31.9		46	16.3	odd	4
128.13.f.b.95.9		46	8.5	even	2