Embedded newform 16.11.f.a.3.4

• Label: 16.11.f.a.3.4
• Level: $16$
• Weight: $11$
• Character: 16.3
• Analytic conductor: $10.166$
• Analytic rank: $0$
• Dimension: $38$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1657160428\)
Analytic rank:	\(0\)
Dimension:	\(38\)
Relative dimension:	\(19\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.4
Character	\(\chi\)	\(=\)	16.3
Dual form			16.11.f.a.11.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-25.9865 + 18.6735i) q^{2} +(280.546 - 280.546i) q^{3} +(326.598 - 970.520i) q^{4} +(3102.01 - 3102.01i) q^{5} +(-2051.63 + 12529.2i) q^{6} -27408.1 q^{7} +(9635.89 + 31319.2i) q^{8} -98363.4i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 2 q^{3} - 1256 q^{4} - 2 q^{5} - 17216 q^{6} - 4 q^{7} + 69124 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−25.9865	+	18.6735i	−0.812079	+	0.583548i
							\(3\)	280.546	−	280.546i	1.15451	−	1.15451i	0.168874	−	0.985638i	\(-0.445987\pi\)
0.985638	−	0.168874i	\(-0.0540129\pi\)
\(5\)	3102.01	−	3102.01i	0.992644	−	0.992644i	−0.00732890	−	0.999973i	\(-0.502333\pi\)
							0.999973	+	0.00732890i	\(0.00233288\pi\)
\(7\)	−27408.1			−1.63075			−0.815377	−	0.578931i	\(-0.803470\pi\)
							−0.815377	+	0.578931i	\(0.803470\pi\)
\(11\)	56353.4	+	56353.4i	0.349910	+	0.349910i	0.860076	−	0.510166i	\(-0.170416\pi\)
							−0.510166	+	0.860076i	\(0.670416\pi\)
\(13\)	−181622.	−	181622.i	−0.489161	−	0.489161i	0.418880	−	0.908041i	\(-0.362423\pi\)
							−0.908041	+	0.418880i	\(0.862423\pi\)
\(17\)	−180430.			−0.127076			−0.0635382	−	0.997979i	\(-0.520238\pi\)
							−0.0635382	+	0.997979i	\(0.520238\pi\)
\(19\)	−121261.	+	121261.i	−0.0489727	+	0.0489727i	−0.731169	−	0.682196i	\(-0.761025\pi\)
							0.682196	+	0.731169i	\(0.261025\pi\)
\(23\)	7.96819e6			1.23800			0.619000	−	0.785391i	\(-0.287538\pi\)
							0.619000	+	0.785391i	\(0.287538\pi\)
\(29\)	−2.05620e7	−	2.05620e7i	−1.00248	−	1.00248i	−0.999997	−	0.00248392i	\(-0.999209\pi\)
							−0.00248392	−	0.999997i	\(-0.500791\pi\)
\(31\)		−	1.51066e7i		−	0.527664i	−0.964569	−	0.263832i	\(-0.915014\pi\)
							0.964569	−	0.263832i	\(-0.0849865\pi\)
\(37\)	5.65742e7	−	5.65742e7i	0.815850	−	0.815850i	−0.169654	−	0.985504i	\(-0.554265\pi\)
							0.985504	+	0.169654i	\(0.0542651\pi\)
\(41\)		−	2.70501e7i		−	0.233480i	−0.993163	−	0.116740i	\(-0.962756\pi\)
							0.993163	−	0.116740i	\(-0.0372444\pi\)
\(43\)	2.01095e8	+	2.01095e8i	1.36791	+	1.36791i	0.863408	+	0.504507i	\(0.168326\pi\)
							0.504507	+	0.863408i	\(0.331674\pi\)
\(47\)		−	1.85496e8i		−	0.808809i	−0.914580	−	0.404404i	\(-0.867479\pi\)
							0.914580	−	0.404404i	\(-0.132521\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.11.f.a.3.4	✓	38
4.3	odd	2		64.11.f.a.47.3		38
8.3	odd	2		128.11.f.a.95.17		38
8.5	even	2		128.11.f.b.95.3		38
16.3	odd	4		128.11.f.b.31.3		38
16.5	even	4		64.11.f.a.15.3		38
16.11	odd	4	inner	16.11.f.a.11.4	yes	38
16.13	even	4		128.11.f.a.31.17		38

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.11.f.a.3.4	✓	38	1.1	even	1	trivial
16.11.f.a.11.4	yes	38	16.11	odd	4	inner
64.11.f.a.15.3		38	16.5	even	4
64.11.f.a.47.3		38	4.3	odd	2
128.11.f.a.31.17		38	16.13	even	4
128.11.f.a.95.17		38	8.3	odd	2
128.11.f.b.31.3		38	16.3	odd	4
128.11.f.b.95.3		38	8.5	even	2