Embedded newform 16.9.f.a.3.2

• Label: 16.9.f.a.3.2
• Level: $16$
• Weight: $9$
• Character: 16.3
• Analytic conductor: $6.518$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.51805776098\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(15\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.2
Character	\(\chi\)	\(=\)	16.3
Dual form			16.9.f.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-15.9118 - 1.67767i) q^{2} +(-93.3844 + 93.3844i) q^{3} +(250.371 + 53.3896i) q^{4} +(-695.406 + 695.406i) q^{5} +(1642.58 - 1329.25i) q^{6} +805.482 q^{7} +(-3894.28 - 1269.56i) q^{8} -10880.3i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 2 q^{2} - 2 q^{3} + 184 q^{4} - 2 q^{5} + 3232 q^{6} - 4 q^{7} - 8732 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−15.9118	−	1.67767i	−0.994488	−	0.104854i
							\(3\)	−93.3844	+	93.3844i	−1.15289	+	1.15289i	−0.166925	+	0.985970i	\(0.553384\pi\)
−0.985970	+	0.166925i	\(0.946616\pi\)
\(5\)	−695.406	+	695.406i	−1.11265	+	1.11265i	−0.119858	+	0.992791i	\(0.538244\pi\)
							−0.992791	+	0.119858i	\(0.961756\pi\)
\(7\)	805.482			0.335478			0.167739	−	0.985831i	\(-0.446353\pi\)
							0.167739	+	0.985831i	\(0.446353\pi\)
\(11\)	−11976.3	−	11976.3i	−0.818000	−	0.818000i	0.167818	−	0.985818i	\(-0.446328\pi\)
							−0.985818	+	0.167818i	\(0.946328\pi\)
\(13\)	24598.3	+	24598.3i	0.861254	+	0.861254i	0.991484	−	0.130229i	\(-0.0415714\pi\)
							−0.130229	+	0.991484i	\(0.541571\pi\)
\(17\)	86193.0			1.03199			0.515996	−	0.856591i	\(-0.327422\pi\)
							0.515996	+	0.856591i	\(0.327422\pi\)
\(19\)	−64254.1	+	64254.1i	−0.493045	+	0.493045i	−0.909264	−	0.416219i	\(-0.863355\pi\)
							0.416219	+	0.909264i	\(0.363355\pi\)
\(23\)	−193598.			−0.691814			−0.345907	−	0.938269i	\(-0.612429\pi\)
							−0.345907	+	0.938269i	\(0.612429\pi\)
\(29\)	−352710.	−	352710.i	−0.498684	−	0.498684i	0.412344	−	0.911028i	\(-0.364710\pi\)
							−0.911028	+	0.412344i	\(0.864710\pi\)
\(31\)		−	910874.i		−	0.986305i	−0.869943	−	0.493153i	\(-0.835844\pi\)
							0.869943	−	0.493153i	\(-0.164156\pi\)
\(37\)	−51233.6	+	51233.6i	−0.0273368	+	0.0273368i	−0.720643	−	0.693306i	\(-0.756153\pi\)
							0.693306	+	0.720643i	\(0.256153\pi\)
\(41\)			4.16291e6i			1.47320i	0.676328	+	0.736601i	\(0.263570\pi\)
							−0.676328	+	0.736601i	\(0.736430\pi\)
\(43\)	−1.53143e6	−	1.53143e6i	−0.447943	−	0.447943i	0.446727	−	0.894670i	\(-0.352589\pi\)
							−0.894670	+	0.446727i	\(0.852589\pi\)
\(47\)			186063.i			0.0381303i	0.999818	+	0.0190651i	\(0.00606899\pi\)
							−0.999818	+	0.0190651i	\(0.993931\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.9.f.a.3.2	✓	30
4.3	odd	2		64.9.f.a.47.14		30
8.3	odd	2		128.9.f.a.95.2		30
8.5	even	2		128.9.f.b.95.14		30
16.3	odd	4		128.9.f.b.31.14		30
16.5	even	4		64.9.f.a.15.14		30
16.11	odd	4	inner	16.9.f.a.11.2	yes	30
16.13	even	4		128.9.f.a.31.2		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.9.f.a.3.2	✓	30	1.1	even	1	trivial
16.9.f.a.11.2	yes	30	16.11	odd	4	inner
64.9.f.a.15.14		30	16.5	even	4
64.9.f.a.47.14		30	4.3	odd	2
128.9.f.a.31.2		30	16.13	even	4
128.9.f.a.95.2		30	8.3	odd	2
128.9.f.b.31.14		30	16.3	odd	4
128.9.f.b.95.14		30	8.5	even	2