Embedded newform 16.10.e.a.5.4

• Label: 16.10.e.a.5.4
• 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			5.4
Character	\(\chi\)	\(=\)	16.5
Dual form			16.10.e.a.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-16.2245 - 15.7723i) q^{2} +(111.916 - 111.916i) q^{3} +(14.4713 + 511.795i) q^{4} +(1353.12 + 1353.12i) q^{5} +(-3580.96 + 50.6167i) q^{6} +6223.61i q^{7} +(7837.38 - 8531.89i) q^{8} -5367.52i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q - 2 q^{2} - 2 q^{3} + 168 q^{4} - 2 q^{5} - 2192 q^{6} - 716 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−16.2245	−	15.7723i	−0.717030	−	0.697042i
							\(3\)	111.916	−	111.916i	0.797715	−	0.797715i	−0.185020	−	0.982735i	\(-0.559235\pi\)
0.982735	+	0.185020i	\(0.0592349\pi\)
\(5\)	1353.12	+	1353.12i	0.968216	+	0.968216i	0.999510	−	0.0312942i	\(-0.00996289\pi\)
							−0.0312942	+	0.999510i	\(0.509963\pi\)
\(7\)			6223.61i			0.979718i	0.871802	+	0.489859i	\(0.162952\pi\)
							−0.871802	+	0.489859i	\(0.837048\pi\)
\(11\)	30508.6	+	30508.6i	0.628282	+	0.628282i	0.947636	−	0.319354i	\(-0.103466\pi\)
							−0.319354	+	0.947636i	\(0.603466\pi\)
\(13\)	112559.	−	112559.i	1.09304	−	1.09304i	0.0978373	−	0.995202i	\(-0.468808\pi\)
							0.995202	−	0.0978373i	\(-0.0311925\pi\)
\(17\)	−441500.			−1.28207			−0.641033	−	0.767514i	\(-0.721494\pi\)
							−0.641033	+	0.767514i	\(0.721494\pi\)
\(19\)	115513.	−	115513.i	0.203348	−	0.203348i	−0.598085	−	0.801433i	\(-0.704071\pi\)
							0.801433	+	0.598085i	\(0.204071\pi\)
\(23\)			744234.i			0.554542i	0.960792	+	0.277271i	\(0.0894300\pi\)
							−0.960792	+	0.277271i	\(0.910570\pi\)
\(29\)	4.63228e6	−	4.63228e6i	1.21620	−	1.21620i	0.247243	−	0.968954i	\(-0.420475\pi\)
							0.968954	−	0.247243i	\(-0.0795246\pi\)
\(31\)	−4.65587e6			−0.905468			−0.452734	−	0.891646i	\(-0.649551\pi\)
							−0.452734	+	0.891646i	\(0.649551\pi\)
\(37\)	6.50581e6	+	6.50581e6i	0.570681	+	0.570681i	0.932319	−	0.361638i	\(-0.117782\pi\)
							−0.361638	+	0.932319i	\(0.617782\pi\)
\(41\)		−	8.16858e6i		−	0.451460i	−0.974190	−	0.225730i	\(-0.927523\pi\)
							0.974190	−	0.225730i	\(-0.0724767\pi\)
\(43\)	9.42532e6	+	9.42532e6i	0.420424	+	0.420424i	0.885350	−	0.464925i	\(-0.153919\pi\)
							−0.464925	+	0.885350i	\(0.653919\pi\)
\(47\)	−4.14989e7			−1.24050			−0.620249	−	0.784405i	\(-0.712968\pi\)
							−0.620249	+	0.784405i	\(0.712968\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.10.e.a.5.4	✓	34
4.3	odd	2		64.10.e.a.49.4		34
8.3	odd	2		128.10.e.a.97.14		34
8.5	even	2		128.10.e.b.97.4		34
16.3	odd	4		64.10.e.a.17.4		34
16.5	even	4		128.10.e.b.33.4		34
16.11	odd	4		128.10.e.a.33.14		34
16.13	even	4	inner	16.10.e.a.13.4	yes	34

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.10.e.a.5.4	✓	34	1.1	even	1	trivial
16.10.e.a.13.4	yes	34	16.13	even	4	inner
64.10.e.a.17.4		34	16.3	odd	4
64.10.e.a.49.4		34	4.3	odd	2
128.10.e.a.33.14		34	16.11	odd	4
128.10.e.a.97.14		34	8.3	odd	2
128.10.e.b.33.4		34	16.5	even	4
128.10.e.b.97.4		34	8.5	even	2