Embedded newform 16.10.e.a.5.9

• Label: 16.10.e.a.5.9
• 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			5.9
Character	\(\chi\)	\(=\)	16.5
Dual form			16.10.e.a.13.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.735037 - 22.6155i) q^{2} +(100.059 - 100.059i) q^{3} +(-510.919 + 33.2464i) q^{4} +(-601.526 - 601.526i) q^{5} +(-2336.42 - 2189.32i) q^{6} -5438.71i q^{7} +(1127.43 + 11530.2i) q^{8} -340.415i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q - 2 q^{2} - 2 q^{3} + 168 q^{4} - 2 q^{5} - 2192 q^{6} - 716 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.735037	−	22.6155i	−0.0324844	−	0.999472i
							\(3\)	100.059	−	100.059i	0.713195	−	0.713195i	−0.254007	−	0.967202i	\(-0.581749\pi\)
0.967202	+	0.254007i	\(0.0817487\pi\)
\(5\)	−601.526	−	601.526i	−0.430417	−	0.430417i	0.458353	−	0.888770i	\(-0.348440\pi\)
							−0.888770	+	0.458353i	\(0.848440\pi\)
\(7\)		−	5438.71i		−	0.856160i	−0.903741	−	0.428080i	\(-0.859190\pi\)
							0.903741	−	0.428080i	\(-0.140810\pi\)
\(11\)	−22864.0	−	22864.0i	−0.470853	−	0.470853i	0.431337	−	0.902191i	\(-0.358042\pi\)
							−0.902191	+	0.431337i	\(0.858042\pi\)
\(13\)	−73965.3	+	73965.3i	−0.718262	+	0.718262i	−0.968249	−	0.249987i	\(-0.919574\pi\)
							0.249987	+	0.968249i	\(0.419574\pi\)
\(17\)	−109146.			−0.316948			−0.158474	−	0.987363i	\(-0.550657\pi\)
							−0.158474	+	0.987363i	\(0.550657\pi\)
\(19\)	422754.	−	422754.i	0.744211	−	0.744211i	−0.229174	−	0.973385i	\(-0.573603\pi\)
							0.973385	+	0.229174i	\(0.0736026\pi\)
\(23\)		−	1.59496e6i		−	1.18844i	−0.804304	−	0.594218i	\(-0.797462\pi\)
							0.804304	−	0.594218i	\(-0.202538\pi\)
\(29\)	4.82465e6	−	4.82465e6i	1.26670	−	1.26670i	0.318924	−	0.947780i	\(-0.396679\pi\)
							0.947780	−	0.318924i	\(-0.103321\pi\)
\(31\)	−5.33161e6			−1.03689			−0.518443	−	0.855112i	\(-0.673488\pi\)
							−0.518443	+	0.855112i	\(0.673488\pi\)
\(37\)	−1.16948e7	−	1.16948e7i	−1.02585	−	1.02585i	−0.999657	−	0.0261970i	\(-0.991660\pi\)
							−0.0261970	−	0.999657i	\(-0.508340\pi\)
\(41\)			2.31435e6i			0.127909i	0.997953	+	0.0639545i	\(0.0203712\pi\)
							−0.997953	+	0.0639545i	\(0.979629\pi\)
\(43\)	2.23553e7	+	2.23553e7i	0.997179	+	0.997179i	0.999996	−	0.00281693i	\(-0.000896657\pi\)
							−0.00281693	+	0.999996i	\(0.500897\pi\)
\(47\)	2.41225e7			0.721077			0.360539	−	0.932744i	\(-0.382593\pi\)
							0.360539	+	0.932744i	\(0.382593\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.10.e.a.5.9	✓	34
4.3	odd	2		64.10.e.a.49.5		34
8.3	odd	2		128.10.e.a.97.13		34
8.5	even	2		128.10.e.b.97.5		34
16.3	odd	4		64.10.e.a.17.5		34
16.5	even	4		128.10.e.b.33.5		34
16.11	odd	4		128.10.e.a.33.13		34
16.13	even	4	inner	16.10.e.a.13.9	yes	34

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.10.e.a.5.9	✓	34	1.1	even	1	trivial
16.10.e.a.13.9	yes	34	16.13	even	4	inner
64.10.e.a.17.5		34	16.3	odd	4
64.10.e.a.49.5		34	4.3	odd	2
128.10.e.a.33.13		34	16.11	odd	4
128.10.e.a.97.13		34	8.3	odd	2
128.10.e.b.33.5		34	16.5	even	4
128.10.e.b.97.5		34	8.5	even	2