Embedded newform 16.11.f.a.3.14

• Label: 16.11.f.a.3.14
• 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			3.14
Character	\(\chi\)	\(=\)	16.3
Dual form			16.11.f.a.11.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(17.5699 + 26.7451i) q^{2} +(-286.288 + 286.288i) q^{3} +(-406.599 + 939.816i) q^{4} +(-1338.28 + 1338.28i) q^{5} +(-12686.8 - 2626.75i) q^{6} +19657.4 q^{7} +(-32279.3 + 5637.94i) q^{8} -104873. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 2 q^{3} - 1256 q^{4} - 2 q^{5} - 17216 q^{6} - 4 q^{7} + 69124 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	17.5699	+	26.7451i	0.549059	+	0.835784i
							\(3\)	−286.288	+	286.288i	−1.17814	+	1.17814i	−0.197923	+	0.980218i	\(0.563420\pi\)
−0.980218	+	0.197923i	\(0.936580\pi\)
\(5\)	−1338.28	+	1338.28i	−0.428250	+	0.428250i	−0.888032	−	0.459782i	\(-0.847928\pi\)
							0.459782	+	0.888032i	\(0.347928\pi\)
\(7\)	19657.4			1.16959			0.584797	−	0.811180i	\(-0.301174\pi\)
							0.584797	+	0.811180i	\(0.301174\pi\)
\(11\)	−19287.1	−	19287.1i	−0.119758	−	0.119758i	0.644688	−	0.764446i	\(-0.276987\pi\)
							−0.764446	+	0.644688i	\(0.776987\pi\)
\(13\)	334654.	+	334654.i	0.901320	+	0.901320i	0.995550	−	0.0942301i	\(-0.0300389\pi\)
							−0.0942301	+	0.995550i	\(0.530039\pi\)
\(17\)	−2.63201e6			−1.85372			−0.926858	−	0.375413i	\(-0.877501\pi\)
							−0.926858	+	0.375413i	\(0.877501\pi\)
\(19\)	1.60841e6	−	1.60841e6i	0.649576	−	0.649576i	−0.303315	−	0.952891i	\(-0.598093\pi\)
							0.952891	+	0.303315i	\(0.0980933\pi\)
\(23\)	−4.82072e6			−0.748984			−0.374492	−	0.927230i	\(-0.622183\pi\)
							−0.374492	+	0.927230i	\(0.622183\pi\)
\(29\)	1.60002e7	+	1.60002e7i	0.780071	+	0.780071i	0.979843	−	0.199771i	\(-0.0640199\pi\)
							−0.199771	+	0.979843i	\(0.564020\pi\)
\(31\)			1.59760e6i			0.0558033i	0.999611	+	0.0279016i	\(0.00888252\pi\)
							−0.999611	+	0.0279016i	\(0.991117\pi\)
\(37\)	−6.81154e7	+	6.81154e7i	−0.982283	+	0.982283i	−0.999846	−	0.0175625i	\(-0.994409\pi\)
							0.0175625	+	0.999846i	\(0.494409\pi\)
\(41\)		−	1.62530e8i		−	1.40286i	−0.712740	−	0.701428i	\(-0.752546\pi\)
							0.712740	−	0.701428i	\(-0.247454\pi\)
\(43\)	3.72035e7	+	3.72035e7i	0.253070	+	0.253070i	0.822228	−	0.569158i	\(-0.192731\pi\)
							−0.569158	+	0.822228i	\(0.692731\pi\)
\(47\)		−	2.22161e8i		−	0.968678i	−0.874880	−	0.484339i	\(-0.839060\pi\)
							0.874880	−	0.484339i	\(-0.160940\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.11.f.a.3.14	✓	38
4.3	odd	2		64.11.f.a.47.18		38
8.3	odd	2		128.11.f.a.95.2		38
8.5	even	2		128.11.f.b.95.18		38
16.3	odd	4		128.11.f.b.31.18		38
16.5	even	4		64.11.f.a.15.18		38
16.11	odd	4	inner	16.11.f.a.11.14	yes	38
16.13	even	4		128.11.f.a.31.2		38

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.11.f.a.3.14	✓	38	1.1	even	1	trivial
16.11.f.a.11.14	yes	38	16.11	odd	4	inner
64.11.f.a.15.18		38	16.5	even	4
64.11.f.a.47.18		38	4.3	odd	2
128.11.f.a.31.2		38	16.13	even	4
128.11.f.a.95.2		38	8.3	odd	2
128.11.f.b.31.18		38	16.3	odd	4
128.11.f.b.95.18		38	8.5	even	2