Embedded newform 16.10.e.a.5.13

• Label: 16.10.e.a.5.13
• 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			5.13
Character	\(\chi\)	\(=\)	16.5
Dual form			16.10.e.a.13.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(17.2877 + 14.5992i) q^{2} +(185.041 - 185.041i) q^{3} +(85.7268 + 504.772i) q^{4} +(356.050 + 356.050i) q^{5} +(5900.38 - 497.477i) q^{6} -6406.17i q^{7} +(-5887.25 + 9977.87i) q^{8} -48797.5i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q - 2 q^{2} - 2 q^{3} + 168 q^{4} - 2 q^{5} - 2192 q^{6} - 716 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	17.2877	+	14.5992i	0.764014	+	0.645200i
							\(3\)	185.041	−	185.041i	1.31893	−	1.31893i	0.404312	−	0.914621i	\(-0.367511\pi\)
0.914621	−	0.404312i	\(-0.132489\pi\)
\(5\)	356.050	+	356.050i	0.254768	+	0.254768i	0.822922	−	0.568154i	\(-0.192342\pi\)
							−0.568154	+	0.822922i	\(0.692342\pi\)
\(7\)		−	6406.17i		−	1.00846i	−0.863570	−	0.504228i	\(-0.831777\pi\)
							0.863570	−	0.504228i	\(-0.168223\pi\)
\(11\)	50050.0	+	50050.0i	1.03071	+	1.03071i	0.999513	+	0.0311985i	\(0.00993239\pi\)
							0.0311985	+	0.999513i	\(0.490068\pi\)
\(13\)	−60858.7	+	60858.7i	−0.590986	+	0.590986i	−0.937898	−	0.346912i	\(-0.887230\pi\)
							0.346912	+	0.937898i	\(0.387230\pi\)
\(17\)	−143797.			−0.417570			−0.208785	−	0.977962i	\(-0.566951\pi\)
							−0.208785	+	0.977962i	\(0.566951\pi\)
\(19\)	34366.1	−	34366.1i	0.0604977	−	0.0604977i	−0.676211	−	0.736708i	\(-0.736379\pi\)
							0.736708	+	0.676211i	\(0.236379\pi\)
\(23\)			1.27145e6i			0.947379i	0.880692	+	0.473689i	\(0.157078\pi\)
							−0.880692	+	0.473689i	\(0.842922\pi\)
\(29\)	−3.36620e6	+	3.36620e6i	−0.883790	+	0.883790i	−0.993917	−	0.110128i	\(-0.964874\pi\)
							0.110128	+	0.993917i	\(0.464874\pi\)
\(31\)	−6.77127e6			−1.31687			−0.658434	−	0.752638i	\(-0.728781\pi\)
							−0.658434	+	0.752638i	\(0.728781\pi\)
\(37\)	5.33733e6	+	5.33733e6i	0.468184	+	0.468184i	0.901326	−	0.433142i	\(-0.142595\pi\)
							−0.433142	+	0.901326i	\(0.642595\pi\)
\(41\)		−	6.67342e6i		−	0.368826i	−0.982849	−	0.184413i	\(-0.940962\pi\)
							0.982849	−	0.184413i	\(-0.0590384\pi\)
\(43\)	−1.55358e7	−	1.55358e7i	−0.692989	−	0.692989i	0.269900	−	0.962888i	\(-0.413009\pi\)
							−0.962888	+	0.269900i	\(0.913009\pi\)
\(47\)	1.73631e7			0.519023			0.259512	−	0.965740i	\(-0.416438\pi\)
							0.259512	+	0.965740i	\(0.416438\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.10.e.a.5.13	✓	34
4.3	odd	2		64.10.e.a.49.1		34
8.3	odd	2		128.10.e.a.97.17		34
8.5	even	2		128.10.e.b.97.1		34
16.3	odd	4		64.10.e.a.17.1		34
16.5	even	4		128.10.e.b.33.1		34
16.11	odd	4		128.10.e.a.33.17		34
16.13	even	4	inner	16.10.e.a.13.13	yes	34

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.10.e.a.5.13	✓	34	1.1	even	1	trivial
16.10.e.a.13.13	yes	34	16.13	even	4	inner
64.10.e.a.17.1		34	16.3	odd	4
64.10.e.a.49.1		34	4.3	odd	2
128.10.e.a.33.17		34	16.11	odd	4
128.10.e.a.97.17		34	8.3	odd	2
128.10.e.b.33.1		34	16.5	even	4
128.10.e.b.97.1		34	8.5	even	2