Embedded newform 16.10.e.a.5.11

• Label: 16.10.e.a.5.11
• Level: $16$
• Weight: $10$
• Character: 16.5
• Analytic conductor: $8.241$
• Analytic rank: $0$
• Dimension: $34$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.24057337862\)
Analytic rank:	\(0\)
Dimension:	\(34\)
Relative dimension:	\(17\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.11
Character	\(\chi\)	\(=\)	16.5
Dual form			16.10.e.a.13.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.52251 - 20.9611i) q^{2} +(-77.8288 + 77.8288i) q^{3} +(-366.734 - 357.282i) q^{4} +(206.519 + 206.519i) q^{5} +(968.080 + 2294.67i) q^{6} +6914.62i q^{7} +(-10614.5 + 4642.20i) q^{8} +7568.35i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q - 2 q^{2} - 2 q^{3} + 168 q^{4} - 2 q^{5} - 2192 q^{6} - 716 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.52251	−	20.9611i	0.376645	−	0.926358i
							\(3\)	−77.8288	+	77.8288i	−0.554747	+	0.554747i	−0.927807	−	0.373060i	\(-0.878308\pi\)
0.373060	+	0.927807i	\(0.378308\pi\)
\(5\)	206.519	+	206.519i	0.147773	+	0.147773i	0.777123	−	0.629349i	\(-0.216679\pi\)
							−0.629349	+	0.777123i	\(0.716679\pi\)
\(7\)			6914.62i			1.08850i	0.838924	+	0.544248i	\(0.183185\pi\)
							−0.838924	+	0.544248i	\(0.816815\pi\)
\(11\)	31462.1	+	31462.1i	0.647919	+	0.647919i	0.952490	−	0.304570i	\(-0.0985129\pi\)
							−0.304570	+	0.952490i	\(0.598513\pi\)
\(13\)	9963.82	−	9963.82i	0.0967567	−	0.0967567i	−0.657072	−	0.753828i	\(-0.728205\pi\)
							0.753828	+	0.657072i	\(0.228205\pi\)
\(17\)	−378055.			−1.09783			−0.548914	−	0.835879i	\(-0.684959\pi\)
							−0.548914	+	0.835879i	\(0.684959\pi\)
\(19\)	−390247.	+	390247.i	−0.686988	+	0.686988i	−0.961565	−	0.274577i	\(-0.911462\pi\)
							0.274577	+	0.961565i	\(0.411462\pi\)
\(23\)			157609.i			0.117437i	0.998275	+	0.0587187i	\(0.0187015\pi\)
							−0.998275	+	0.0587187i	\(0.981298\pi\)
\(29\)	−4.13607e6	+	4.13607e6i	−1.08592	+	1.08592i	−0.0899724	+	0.995944i	\(0.528678\pi\)
							−0.995944	+	0.0899724i	\(0.971322\pi\)
\(31\)	3.53705e6			0.687882			0.343941	−	0.938991i	\(-0.388238\pi\)
							0.343941	+	0.938991i	\(0.388238\pi\)
\(37\)	1.43393e6	+	1.43393e6i	0.125783	+	0.125783i	0.767196	−	0.641413i	\(-0.221651\pi\)
							−0.641413	+	0.767196i	\(0.721651\pi\)
\(41\)		−	1.28079e7i		−	0.707863i	−0.935271	−	0.353932i	\(-0.884845\pi\)
							0.935271	−	0.353932i	\(-0.115155\pi\)
\(43\)	2.78572e7	+	2.78572e7i	1.24260	+	1.24260i	0.958920	+	0.283676i	\(0.0915541\pi\)
							0.283676	+	0.958920i	\(0.408446\pi\)
\(47\)	4.54507e7			1.35863			0.679313	−	0.733848i	\(-0.262278\pi\)
							0.679313	+	0.733848i	\(0.262278\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.10.e.a.5.11	✓	34
4.3	odd	2		64.10.e.a.49.12		34
8.3	odd	2		128.10.e.a.97.6		34
8.5	even	2		128.10.e.b.97.12		34
16.3	odd	4		64.10.e.a.17.12		34
16.5	even	4		128.10.e.b.33.12		34
16.11	odd	4		128.10.e.a.33.6		34
16.13	even	4	inner	16.10.e.a.13.11	yes	34

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.10.e.a.5.11	✓	34	1.1	even	1	trivial
16.10.e.a.13.11	yes	34	16.13	even	4	inner
64.10.e.a.17.12		34	16.3	odd	4
64.10.e.a.49.12		34	4.3	odd	2
128.10.e.a.33.6		34	16.11	odd	4
128.10.e.a.97.6		34	8.3	odd	2
128.10.e.b.33.12		34	16.5	even	4
128.10.e.b.97.12		34	8.5	even	2