Embedded newform 16.13.f.a.3.14

• Label: 16.13.f.a.3.14
• 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			3.14
Character	\(\chi\)	\(=\)	16.3
Dual form			16.13.f.a.11.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(14.0860 + 62.4306i) q^{2} +(-832.058 + 832.058i) q^{3} +(-3699.17 + 1758.79i) q^{4} +(17612.4 - 17612.4i) q^{5} +(-63666.3 - 40225.6i) q^{6} -164049. q^{7} +(-161909. - 206167. i) q^{8} -853201. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 2 q^{2} - 2 q^{3} - 4232 q^{4} - 2 q^{5} + 72928 q^{6} - 4 q^{7} - 478172 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	14.0860	+	62.4306i	0.220093	+	0.975479i
							\(3\)	−832.058	+	832.058i	−1.14137	+	1.14137i	−0.153169	+	0.988200i	\(0.548948\pi\)
−0.988200	+	0.153169i	\(0.951052\pi\)
\(5\)	17612.4	−	17612.4i	1.12719	−	1.12719i	0.136563	−	0.990631i	\(-0.456394\pi\)
							0.990631	−	0.136563i	\(-0.0436055\pi\)
\(7\)	−164049.			−1.39439			−0.697197	−	0.716879i	\(-0.745570\pi\)
							−0.697197	+	0.716879i	\(0.745570\pi\)
\(11\)	1.61310e6	+	1.61310e6i	0.910553	+	0.910553i	0.996316	−	0.0857627i	\(-0.0273327\pi\)
							−0.0857627	+	0.996316i	\(0.527333\pi\)
\(13\)	−1.20902e6	−	1.20902e6i	−0.250481	−	0.250481i	0.570687	−	0.821168i	\(-0.306677\pi\)
							−0.821168	+	0.570687i	\(0.806677\pi\)
\(17\)	−6.99072e6			−0.289620			−0.144810	−	0.989459i	\(-0.546257\pi\)
							−0.144810	+	0.989459i	\(0.546257\pi\)
\(19\)	2.17498e7	−	2.17498e7i	0.462310	−	0.462310i	−0.437102	−	0.899412i	\(-0.643995\pi\)
							0.899412	+	0.437102i	\(0.143995\pi\)
\(23\)	−6.39300e7			−0.431855			−0.215927	−	0.976409i	\(-0.569277\pi\)
							−0.215927	+	0.976409i	\(0.569277\pi\)
\(29\)	−7.89361e7	−	7.89361e7i	−0.132705	−	0.132705i	0.637634	−	0.770339i	\(-0.279913\pi\)
							−0.770339	+	0.637634i	\(0.779913\pi\)
\(31\)		−	1.11358e9i		−	1.25473i	−0.778726	−	0.627364i	\(-0.784134\pi\)
							0.778726	−	0.627364i	\(-0.215866\pi\)
\(37\)	2.71644e9	−	2.71644e9i	1.05874	−	1.05874i	0.0605783	−	0.998163i	\(-0.480706\pi\)
							0.998163	−	0.0605783i	\(-0.0192945\pi\)
\(41\)			4.18655e9i			0.881359i	0.897665	+	0.440679i	\(0.145262\pi\)
							−0.897665	+	0.440679i	\(0.854738\pi\)
\(43\)	−5.33511e9	−	5.33511e9i	−0.843981	−	0.843981i	0.145393	−	0.989374i	\(-0.453555\pi\)
							−0.989374	+	0.145393i	\(0.953555\pi\)
\(47\)		−	1.47143e10i		−	1.36506i	−0.730856	−	0.682531i	\(-0.760879\pi\)
							0.730856	−	0.682531i	\(-0.239121\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.13.f.a.3.14	✓	46
4.3	odd	2		64.13.f.a.47.21		46
8.3	odd	2		128.13.f.a.95.3		46
8.5	even	2		128.13.f.b.95.21		46
16.3	odd	4		128.13.f.b.31.21		46
16.5	even	4		64.13.f.a.15.21		46
16.11	odd	4	inner	16.13.f.a.11.14	yes	46
16.13	even	4		128.13.f.a.31.3		46

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.13.f.a.3.14	✓	46	1.1	even	1	trivial
16.13.f.a.11.14	yes	46	16.11	odd	4	inner
64.13.f.a.15.21		46	16.5	even	4
64.13.f.a.47.21		46	4.3	odd	2
128.13.f.a.31.3		46	16.13	even	4
128.13.f.a.95.3		46	8.3	odd	2
128.13.f.b.31.21		46	16.3	odd	4
128.13.f.b.95.21		46	8.5	even	2