Embedded newform 16.9.f.a.3.1

• Label: 16.9.f.a.3.1
• 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			3.1
Character	\(\chi\)	\(=\)	16.3
Dual form			16.9.f.a.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-15.9265 - 1.53176i) q^{2} +(110.050 - 110.050i) q^{3} +(251.307 + 48.7911i) q^{4} +(-128.245 + 128.245i) q^{5} +(-1921.29 + 1584.14i) q^{6} +3057.02 q^{7} +(-3927.71 - 1162.01i) q^{8} -17661.1i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 2 q^{2} - 2 q^{3} + 184 q^{4} - 2 q^{5} + 3232 q^{6} - 4 q^{7} - 8732 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−15.9265	−	1.53176i	−0.995407	−	0.0957349i
							\(3\)	110.050	−	110.050i	1.35864	−	1.35864i	0.483052	−	0.875591i	\(-0.339528\pi\)
0.875591	−	0.483052i	\(-0.160472\pi\)
\(5\)	−128.245	+	128.245i	−0.205192	+	0.205192i	−0.802220	−	0.597028i	\(-0.796348\pi\)
							0.597028	+	0.802220i	\(0.296348\pi\)
\(7\)	3057.02			1.27323			0.636614	−	0.771182i	\(-0.280334\pi\)
							0.636614	+	0.771182i	\(0.280334\pi\)
\(11\)	−987.205	−	987.205i	−0.0674274	−	0.0674274i	0.672589	−	0.740016i	\(-0.265182\pi\)
							−0.740016	+	0.672589i	\(0.765182\pi\)
\(13\)	−11883.0	−	11883.0i	−0.416057	−	0.416057i	0.467785	−	0.883842i	\(-0.345052\pi\)
							−0.883842	+	0.467785i	\(0.845052\pi\)
\(17\)	−92854.9			−1.11176			−0.555878	−	0.831264i	\(-0.687618\pi\)
							−0.555878	+	0.831264i	\(0.687618\pi\)
\(19\)	49427.7	−	49427.7i	0.379277	−	0.379277i	−0.491564	−	0.870841i	\(-0.663575\pi\)
							0.870841	+	0.491564i	\(0.163575\pi\)
\(23\)	126923.			0.453555			0.226778	−	0.973947i	\(-0.427181\pi\)
							0.226778	+	0.973947i	\(0.427181\pi\)
\(29\)	813796.	+	813796.i	1.15060	+	1.15060i	0.986433	+	0.164165i	\(0.0524929\pi\)
							0.164165	+	0.986433i	\(0.447507\pi\)
\(31\)			458469.i			0.496436i	0.968704	+	0.248218i	\(0.0798449\pi\)
							−0.968704	+	0.248218i	\(0.920155\pi\)
\(37\)	1.42527e6	−	1.42527e6i	0.760482	−	0.760482i	−0.215928	−	0.976409i	\(-0.569278\pi\)
							0.976409	+	0.215928i	\(0.0692775\pi\)
\(41\)			1.04918e6i			0.371293i	0.982617	+	0.185646i	\(0.0594379\pi\)
							−0.982617	+	0.185646i	\(0.940562\pi\)
\(43\)	3.03720e6	+	3.03720e6i	0.888381	+	0.888381i	0.994368	−	0.105987i	\(-0.0338001\pi\)
							−0.105987	+	0.994368i	\(0.533800\pi\)
\(47\)		−	1.32060e6i		−	0.270633i	−0.990802	−	0.135316i	\(-0.956795\pi\)
							0.990802	−	0.135316i	\(-0.0432051\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.9.f.a.3.1	✓	30
4.3	odd	2		64.9.f.a.47.1		30
8.3	odd	2		128.9.f.a.95.15		30
8.5	even	2		128.9.f.b.95.1		30
16.3	odd	4		128.9.f.b.31.1		30
16.5	even	4		64.9.f.a.15.1		30
16.11	odd	4	inner	16.9.f.a.11.1	yes	30
16.13	even	4		128.9.f.a.31.15		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.9.f.a.3.1	✓	30	1.1	even	1	trivial
16.9.f.a.11.1	yes	30	16.11	odd	4	inner
64.9.f.a.15.1		30	16.5	even	4
64.9.f.a.47.1		30	4.3	odd	2
128.9.f.a.31.15		30	16.13	even	4
128.9.f.a.95.15		30	8.3	odd	2
128.9.f.b.31.1		30	16.3	odd	4
128.9.f.b.95.1		30	8.5	even	2