Embedded newform 16.9.f.a.3.8

• Label: 16.9.f.a.3.8
• 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			3.8
Character	\(\chi\)	\(=\)	16.3
Dual form			16.9.f.a.11.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.81651 + 15.8965i) q^{2} +(-71.2555 + 71.2555i) q^{3} +(-249.401 - 57.7524i) q^{4} +(-8.38358 + 8.38358i) q^{5} +(-1003.28 - 1262.15i) q^{6} +1087.59 q^{7} +(1371.10 - 3859.70i) q^{8} -3593.69i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 2 q^{2} - 2 q^{3} + 184 q^{4} - 2 q^{5} + 3232 q^{6} - 4 q^{7} - 8732 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.81651	+	15.8965i	−0.113532	+	0.993534i
							\(3\)	−71.2555	+	71.2555i	−0.879697	+	0.879697i	−0.993503	−	0.113806i	\(-0.963696\pi\)
0.113806	+	0.993503i	\(0.463696\pi\)
\(5\)	−8.38358	+	8.38358i	−0.0134137	+	0.0134137i	−0.713782	−	0.700368i	\(-0.753019\pi\)
							0.700368	+	0.713782i	\(0.253019\pi\)
\(7\)	1087.59			0.452972			0.226486	−	0.974014i	\(-0.427276\pi\)
							0.226486	+	0.974014i	\(0.427276\pi\)
\(11\)	983.291	+	983.291i	0.0671601	+	0.0671601i	0.739889	−	0.672729i	\(-0.234878\pi\)
							−0.672729	+	0.739889i	\(0.734878\pi\)
\(13\)	−38218.6	−	38218.6i	−1.33814	−	1.33814i	−0.897860	−	0.440281i	\(-0.854879\pi\)
							−0.440281	−	0.897860i	\(-0.645121\pi\)
\(17\)	−68193.9			−0.816488			−0.408244	−	0.912873i	\(-0.633859\pi\)
							−0.408244	+	0.912873i	\(0.633859\pi\)
\(19\)	−116287.	+	116287.i	−0.892313	+	0.892313i	−0.994740	−	0.102427i	\(-0.967339\pi\)
							0.102427	+	0.994740i	\(0.467339\pi\)
\(23\)	217798.			0.778290			0.389145	−	0.921176i	\(-0.372770\pi\)
							0.389145	+	0.921176i	\(0.372770\pi\)
\(29\)	−727014.	−	727014.i	−1.02790	−	1.02790i	−0.999599	−	0.0283007i	\(-0.990990\pi\)
							−0.0283007	−	0.999599i	\(-0.509010\pi\)
\(31\)		−	597856.i		−	0.647366i	−0.946166	−	0.323683i	\(-0.895079\pi\)
							0.946166	−	0.323683i	\(-0.104921\pi\)
\(37\)	−530978.	+	530978.i	−0.283315	+	0.283315i	−0.834430	−	0.551115i	\(-0.814203\pi\)
							0.551115	+	0.834430i	\(0.314203\pi\)
\(41\)		−	2.04250e6i		−	0.722816i	−0.932408	−	0.361408i	\(-0.882296\pi\)
							0.932408	−	0.361408i	\(-0.117704\pi\)
\(43\)	3.58242e6	+	3.58242e6i	1.04786	+	1.04786i	0.998796	+	0.0490641i	\(0.0156239\pi\)
							0.0490641	+	0.998796i	\(0.484376\pi\)
\(47\)			5.76838e6i			1.18212i	0.806627	+	0.591061i	\(0.201291\pi\)
							−0.806627	+	0.591061i	\(0.798709\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.9.f.a.3.8	✓	30
4.3	odd	2		64.9.f.a.47.13		30
8.3	odd	2		128.9.f.a.95.3		30
8.5	even	2		128.9.f.b.95.13		30
16.3	odd	4		128.9.f.b.31.13		30
16.5	even	4		64.9.f.a.15.13		30
16.11	odd	4	inner	16.9.f.a.11.8	yes	30
16.13	even	4		128.9.f.a.31.3		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.9.f.a.3.8	✓	30	1.1	even	1	trivial
16.9.f.a.11.8	yes	30	16.11	odd	4	inner
64.9.f.a.15.13		30	16.5	even	4
64.9.f.a.47.13		30	4.3	odd	2
128.9.f.a.31.3		30	16.13	even	4
128.9.f.a.95.3		30	8.3	odd	2
128.9.f.b.31.13		30	16.3	odd	4
128.9.f.b.95.13		30	8.5	even	2