Embedded newform 16.10.e.a.5.14

• Label: 16.10.e.a.5.14
• 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			5.14
Character	\(\chi\)	\(=\)	16.5
Dual form			16.10.e.a.13.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(18.5834 - 12.9096i) q^{2} +(62.9496 - 62.9496i) q^{3} +(178.687 - 479.807i) q^{4} +(1829.14 + 1829.14i) q^{5} +(357.167 - 1982.47i) q^{6} -7941.79i q^{7} +(-2873.49 - 11223.2i) q^{8} +11757.7i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q - 2 q^{2} - 2 q^{3} + 168 q^{4} - 2 q^{5} - 2192 q^{6} - 716 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	18.5834	−	12.9096i	0.821279	−	0.570527i
							\(3\)	62.9496	−	62.9496i	0.448691	−	0.448691i	−0.446228	−	0.894919i	\(-0.647233\pi\)
0.894919	+	0.446228i	\(0.147233\pi\)
\(5\)	1829.14	+	1829.14i	1.30883	+	1.30883i	0.922262	+	0.386566i	\(0.126339\pi\)
							0.386566	+	0.922262i	\(0.373661\pi\)
\(7\)		−	7941.79i		−	1.25019i	−0.780547	−	0.625097i	\(-0.785059\pi\)
							0.780547	−	0.625097i	\(-0.214941\pi\)
\(11\)	−34503.8	−	34503.8i	−0.710558	−	0.710558i	0.256094	−	0.966652i	\(-0.417564\pi\)
							−0.966652	+	0.256094i	\(0.917564\pi\)
\(13\)	−3725.15	+	3725.15i	−0.0361741	+	0.0361741i	−0.724962	−	0.688788i	\(-0.758143\pi\)
							0.688788	+	0.724962i	\(0.258143\pi\)
\(17\)	438940.			1.27463			0.637316	−	0.770603i	\(-0.280045\pi\)
							0.637316	+	0.770603i	\(0.280045\pi\)
\(19\)	−617270.	+	617270.i	−1.08664	+	1.08664i	−0.0907642	+	0.995872i	\(0.528931\pi\)
							−0.995872	+	0.0907642i	\(0.971069\pi\)
\(23\)			508098.i			0.378593i	0.981920	+	0.189296i	\(0.0606207\pi\)
							−0.981920	+	0.189296i	\(0.939379\pi\)
\(29\)	−2.04068e6	+	2.04068e6i	−0.535777	+	0.535777i	−0.922286	−	0.386508i	\(-0.873681\pi\)
							0.386508	+	0.922286i	\(0.373681\pi\)
\(31\)	−1.29666e6			−0.252173			−0.126087	−	0.992019i	\(-0.540242\pi\)
							−0.126087	+	0.992019i	\(0.540242\pi\)
\(37\)	2.27886e6	+	2.27886e6i	0.199899	+	0.199899i	0.799957	−	0.600058i	\(-0.204856\pi\)
							−0.600058	+	0.799957i	\(0.704856\pi\)
\(41\)		−	1.77743e7i		−	0.982345i	−0.871062	−	0.491173i	\(-0.836568\pi\)
							0.871062	−	0.491173i	\(-0.163432\pi\)
\(43\)	−2.37777e6	−	2.37777e6i	−0.106063	−	0.106063i	0.652084	−	0.758147i	\(-0.273895\pi\)
							−0.758147	+	0.652084i	\(0.773895\pi\)
\(47\)	−1.63778e6			−0.0489570			−0.0244785	−	0.999700i	\(-0.507793\pi\)
							−0.0244785	+	0.999700i	\(0.507793\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.10.e.a.5.14	✓	34
4.3	odd	2		64.10.e.a.49.7		34
8.3	odd	2		128.10.e.a.97.11		34
8.5	even	2		128.10.e.b.97.7		34
16.3	odd	4		64.10.e.a.17.7		34
16.5	even	4		128.10.e.b.33.7		34
16.11	odd	4		128.10.e.a.33.11		34
16.13	even	4	inner	16.10.e.a.13.14	yes	34

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.10.e.a.5.14	✓	34	1.1	even	1	trivial
16.10.e.a.13.14	yes	34	16.13	even	4	inner
64.10.e.a.17.7		34	16.3	odd	4
64.10.e.a.49.7		34	4.3	odd	2
128.10.e.a.33.11		34	16.11	odd	4
128.10.e.a.97.11		34	8.3	odd	2
128.10.e.b.33.7		34	16.5	even	4
128.10.e.b.97.7		34	8.5	even	2