Embedded newform 16.11.f.a.3.9

• Label: 16.11.f.a.3.9
• Level: $16$
• Weight: $11$
• Character: 16.3
• Analytic conductor: $10.166$
• Analytic rank: $0$
• Dimension: $38$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1657160428\)
Analytic rank:	\(0\)
Dimension:	\(38\)
Relative dimension:	\(19\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.9
Character	\(\chi\)	\(=\)	16.3
Dual form			16.11.f.a.11.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.25659 + 31.3824i) q^{2} +(196.669 - 196.669i) q^{3} +(-945.710 - 392.694i) q^{4} +(-1812.92 + 1812.92i) q^{5} +(4941.46 + 7402.41i) q^{6} +22339.6 q^{7} +(18240.6 - 27221.7i) q^{8} -18308.2i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 2 q^{3} - 1256 q^{4} - 2 q^{5} - 17216 q^{6} - 4 q^{7} + 69124 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.25659	+	31.3824i	−0.195518	+	0.980700i
							\(3\)	196.669	−	196.669i	0.809336	−	0.809336i	−0.175197	−	0.984533i	\(-0.556056\pi\)
0.984533	+	0.175197i	\(0.0560562\pi\)
\(5\)	−1812.92	+	1812.92i	−0.580134	+	0.580134i	−0.934940	−	0.354806i	\(-0.884547\pi\)
							0.354806	+	0.934940i	\(0.384547\pi\)
\(7\)	22339.6			1.32918			0.664591	−	0.747207i	\(-0.268606\pi\)
							0.664591	+	0.747207i	\(0.268606\pi\)
\(11\)	211685.	+	211685.i	1.31440	+	1.31440i	0.918139	+	0.396259i	\(0.129692\pi\)
							0.396259	+	0.918139i	\(0.370308\pi\)
\(13\)	117432.	+	117432.i	0.316279	+	0.316279i	0.847336	−	0.531057i	\(-0.178205\pi\)
							−0.531057	+	0.847336i	\(0.678205\pi\)
\(17\)	−1.27320e6			−0.896711			−0.448355	−	0.893855i	\(-0.647990\pi\)
							−0.448355	+	0.893855i	\(0.647990\pi\)
\(19\)	621143.	−	621143.i	0.250856	−	0.250856i	−0.570466	−	0.821321i	\(-0.693237\pi\)
							0.821321	+	0.570466i	\(0.193237\pi\)
\(23\)	6.08199e6			0.944945			0.472473	−	0.881345i	\(-0.343362\pi\)
							0.472473	+	0.881345i	\(0.343362\pi\)
\(29\)	−8.09088e6	−	8.09088e6i	−0.394463	−	0.394463i	0.481812	−	0.876275i	\(-0.339979\pi\)
							−0.876275	+	0.481812i	\(0.839979\pi\)
\(31\)		−	4.52952e7i		−	1.58213i	−0.611729	−	0.791067i	\(-0.709526\pi\)
							0.611729	−	0.791067i	\(-0.290474\pi\)
\(37\)	3.83236e7	−	3.83236e7i	0.552660	−	0.552660i	−0.374548	−	0.927208i	\(-0.622202\pi\)
							0.927208	+	0.374548i	\(0.122202\pi\)
\(41\)			1.45266e8i			1.25385i	0.779081	+	0.626923i	\(0.215686\pi\)
							−0.779081	+	0.626923i	\(0.784314\pi\)
\(43\)	−1.49199e8	−	1.49199e8i	−1.01490	−	1.01490i	−0.999887	−	0.0150168i	\(-0.995220\pi\)
							−0.0150168	−	0.999887i	\(-0.504780\pi\)
\(47\)			1.28417e8i			0.559928i	0.960011	+	0.279964i	\(0.0903225\pi\)
							−0.960011	+	0.279964i	\(0.909678\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.11.f.a.3.9	✓	38
4.3	odd	2		64.11.f.a.47.4		38
8.3	odd	2		128.11.f.a.95.16		38
8.5	even	2		128.11.f.b.95.4		38
16.3	odd	4		128.11.f.b.31.4		38
16.5	even	4		64.11.f.a.15.4		38
16.11	odd	4	inner	16.11.f.a.11.9	yes	38
16.13	even	4		128.11.f.a.31.16		38

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.11.f.a.3.9	✓	38	1.1	even	1	trivial
16.11.f.a.11.9	yes	38	16.11	odd	4	inner
64.11.f.a.15.4		38	16.5	even	4
64.11.f.a.47.4		38	4.3	odd	2
128.11.f.a.31.16		38	16.13	even	4
128.11.f.a.95.16		38	8.3	odd	2
128.11.f.b.31.4		38	16.3	odd	4
128.11.f.b.95.4		38	8.5	even	2