Embedded newform 16.11.f.a.3.11

• Label: 16.11.f.a.3.11
• 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			3.11
Character	\(\chi\)	\(=\)	16.3
Dual form			16.11.f.a.11.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.46002 - 31.9053i) q^{2} +(-46.0373 + 46.0373i) q^{3} +(-1011.90 - 156.975i) q^{4} +(1933.37 - 1933.37i) q^{5} +(1355.58 + 1582.09i) q^{6} -25927.9 q^{7} +(-7497.63 + 31898.7i) q^{8} +54810.1i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 2 q^{3} - 1256 q^{4} - 2 q^{5} - 17216 q^{6} - 4 q^{7} + 69124 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.46002	−	31.9053i	0.0768756	−	0.997041i
							\(3\)	−46.0373	+	46.0373i	−0.189454	+	0.189454i	−0.795460	−	0.606006i	\(-0.792771\pi\)
0.606006	+	0.795460i	\(0.292771\pi\)
\(5\)	1933.37	−	1933.37i	0.618677	−	0.618677i	−0.326515	−	0.945192i	\(-0.605874\pi\)
							0.945192	+	0.326515i	\(0.105874\pi\)
\(7\)	−25927.9			−1.54268			−0.771341	−	0.636422i	\(-0.780414\pi\)
							−0.771341	+	0.636422i	\(0.780414\pi\)
\(11\)	−12347.2	−	12347.2i	−0.0766667	−	0.0766667i	0.667734	−	0.744400i	\(-0.267265\pi\)
							−0.744400	+	0.667734i	\(0.767265\pi\)
\(13\)	398527.	+	398527.i	1.07335	+	1.07335i	0.997088	+	0.0762617i	\(0.0242984\pi\)
							0.0762617	+	0.997088i	\(0.475702\pi\)
\(17\)	−2.30085e6			−1.62048			−0.810240	−	0.586098i	\(-0.800663\pi\)
							−0.810240	+	0.586098i	\(0.800663\pi\)
\(19\)	−356963.	+	356963.i	−0.144163	+	0.144163i	−0.775505	−	0.631342i	\(-0.782505\pi\)
							0.631342	+	0.775505i	\(0.282505\pi\)
\(23\)	−7.31152e6			−1.13597			−0.567987	−	0.823037i	\(-0.692278\pi\)
							−0.567987	+	0.823037i	\(0.692278\pi\)
\(29\)	−1.42804e7	−	1.42804e7i	−0.696225	−	0.696225i	0.267369	−	0.963594i	\(-0.413846\pi\)
							−0.963594	+	0.267369i	\(0.913846\pi\)
\(31\)		−	3.35533e7i		−	1.17200i	−0.810311	−	0.586000i	\(-0.800702\pi\)
							0.810311	−	0.586000i	\(-0.199298\pi\)
\(37\)	−1.40774e7	+	1.40774e7i	−0.203008	+	0.203008i	−0.801287	−	0.598280i	\(-0.795851\pi\)
							0.598280	+	0.801287i	\(0.295851\pi\)
\(41\)			1.71717e7i			0.148216i	0.997250	+	0.0741080i	\(0.0236110\pi\)
							−0.997250	+	0.0741080i	\(0.976389\pi\)
\(43\)	−1.08540e8	−	1.08540e8i	−0.738326	−	0.738326i	0.233928	−	0.972254i	\(-0.424842\pi\)
							−0.972254	+	0.233928i	\(0.924842\pi\)
\(47\)			2.53066e8i			1.10343i	0.834033	+	0.551714i	\(0.186026\pi\)
							−0.834033	+	0.551714i	\(0.813974\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.11.f.a.3.11	✓	38
4.3	odd	2		64.11.f.a.47.11		38
8.3	odd	2		128.11.f.a.95.9		38
8.5	even	2		128.11.f.b.95.11		38
16.3	odd	4		128.11.f.b.31.11		38
16.5	even	4		64.11.f.a.15.11		38
16.11	odd	4	inner	16.11.f.a.11.11	yes	38
16.13	even	4		128.11.f.a.31.9		38

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.11.f.a.3.11	✓	38	1.1	even	1	trivial
16.11.f.a.11.11	yes	38	16.11	odd	4	inner
64.11.f.a.15.11		38	16.5	even	4
64.11.f.a.47.11		38	4.3	odd	2
128.11.f.a.31.9		38	16.13	even	4
128.11.f.a.95.9		38	8.3	odd	2
128.11.f.b.31.11		38	16.3	odd	4
128.11.f.b.95.11		38	8.5	even	2