Embedded newform 16.10.e.a.5.15

• Label: 16.10.e.a.5.15
• 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			5.15
Character	\(\chi\)	\(=\)	16.5
Dual form			16.10.e.a.13.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(20.5458 - 9.47996i) q^{2} +(117.096 - 117.096i) q^{3} +(332.261 - 389.547i) q^{4} +(-1418.63 - 1418.63i) q^{5} +(1295.77 - 3515.91i) q^{6} +7313.79i q^{7} +(3133.68 - 11153.4i) q^{8} -7740.11i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q - 2 q^{2} - 2 q^{3} + 168 q^{4} - 2 q^{5} - 2192 q^{6} - 716 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	20.5458	−	9.47996i	0.908005	−	0.418959i
							\(3\)	117.096	−	117.096i	0.834637	−	0.834637i	−0.153510	−	0.988147i	\(-0.549058\pi\)
0.988147	+	0.153510i	\(0.0490577\pi\)
\(5\)	−1418.63	−	1418.63i	−1.01509	−	1.01509i	−0.999884	−	0.0152030i	\(-0.995161\pi\)
							−0.0152030	−	0.999884i	\(-0.504839\pi\)
\(7\)			7313.79i			1.15133i	0.817685	+	0.575667i	\(0.195257\pi\)
							−0.817685	+	0.575667i	\(0.804743\pi\)
\(11\)	14792.3	+	14792.3i	0.304626	+	0.304626i	0.842821	−	0.538194i	\(-0.180893\pi\)
							−0.538194	+	0.842821i	\(0.680893\pi\)
\(13\)	72858.5	−	72858.5i	0.707515	−	0.707515i	−0.258497	−	0.966012i	\(-0.583227\pi\)
							0.966012	+	0.258497i	\(0.0832274\pi\)
\(17\)	187631.			0.544858			0.272429	−	0.962176i	\(-0.412173\pi\)
							0.272429	+	0.962176i	\(0.412173\pi\)
\(19\)	542888.	−	542888.i	0.955695	−	0.955695i	−0.0433647	−	0.999059i	\(-0.513808\pi\)
							0.999059	+	0.0433647i	\(0.0138077\pi\)
\(23\)			2.20605e6i			1.64377i	0.569657	+	0.821883i	\(0.307076\pi\)
							−0.569657	+	0.821883i	\(0.692924\pi\)
\(29\)	−2.92526e6	+	2.92526e6i	−0.768022	+	0.768022i	−0.977758	−	0.209736i	\(-0.932740\pi\)
							0.209736	+	0.977758i	\(0.432740\pi\)
\(31\)	2.41397e6			0.469466			0.234733	−	0.972060i	\(-0.424578\pi\)
							0.234733	+	0.972060i	\(0.424578\pi\)
\(37\)	−1.33521e7	−	1.33521e7i	−1.17123	−	1.17123i	−0.981917	−	0.189311i	\(-0.939375\pi\)
							−0.189311	−	0.981917i	\(-0.560625\pi\)
\(41\)			3.14802e7i			1.73984i	0.493189	+	0.869922i	\(0.335831\pi\)
							−0.493189	+	0.869922i	\(0.664169\pi\)
\(43\)	−9.51806e6	−	9.51806e6i	−0.424561	−	0.424561i	0.462209	−	0.886771i	\(-0.347057\pi\)
							−0.886771	+	0.462209i	\(0.847057\pi\)
\(47\)	−2.77704e7			−0.830120			−0.415060	−	0.909794i	\(-0.636239\pi\)
							−0.415060	+	0.909794i	\(0.636239\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.10.e.a.5.15	✓	34
4.3	odd	2		64.10.e.a.49.3		34
8.3	odd	2		128.10.e.a.97.15		34
8.5	even	2		128.10.e.b.97.3		34
16.3	odd	4		64.10.e.a.17.3		34
16.5	even	4		128.10.e.b.33.3		34
16.11	odd	4		128.10.e.a.33.15		34
16.13	even	4	inner	16.10.e.a.13.15	yes	34

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.10.e.a.5.15	✓	34	1.1	even	1	trivial
16.10.e.a.13.15	yes	34	16.13	even	4	inner
64.10.e.a.17.3		34	16.3	odd	4
64.10.e.a.49.3		34	4.3	odd	2
128.10.e.a.33.15		34	16.11	odd	4
128.10.e.a.97.15		34	8.3	odd	2
128.10.e.b.33.3		34	16.5	even	4
128.10.e.b.97.3		34	8.5	even	2