Embedded newform 16.11.f.a.3.8

• Label: 16.11.f.a.3.8
• 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			3.8
Character	\(\chi\)	\(=\)	16.3
Dual form			16.11.f.a.11.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.61264 + 30.5221i) q^{2} +(-107.144 + 107.144i) q^{3} +(-839.194 - 586.796i) q^{4} +(3434.17 - 3434.17i) q^{5} +(-2240.31 - 4300.18i) q^{6} +8771.09 q^{7} +(25977.1 - 19973.3i) q^{8} +36089.4i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 2 q^{3} - 1256 q^{4} - 2 q^{5} - 17216 q^{6} - 4 q^{7} + 69124 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.61264	+	30.5221i	−0.300395	+	0.953815i
							\(3\)	−107.144	+	107.144i	−0.440921	+	0.440921i	−0.892321	−	0.451401i	\(-0.850925\pi\)
0.451401	+	0.892321i	\(0.350925\pi\)
\(5\)	3434.17	−	3434.17i	1.09893	−	1.09893i	0.104399	−	0.994535i	\(-0.466708\pi\)
							0.994535	−	0.104399i	\(-0.0332919\pi\)
\(7\)	8771.09			0.521871			0.260936	−	0.965356i	\(-0.415969\pi\)
							0.260936	+	0.965356i	\(0.415969\pi\)
\(11\)	−170937.	−	170937.i	−1.06138	−	1.06138i	−0.997988	−	0.0633958i	\(-0.979807\pi\)
							−0.0633958	−	0.997988i	\(-0.520193\pi\)
\(13\)	335619.	+	335619.i	0.903918	+	0.903918i	0.995772	−	0.0918541i	\(-0.0292793\pi\)
							−0.0918541	+	0.995772i	\(0.529279\pi\)
\(17\)	2.18235e6			1.53702			0.768512	−	0.639836i	\(-0.220998\pi\)
							0.768512	+	0.639836i	\(0.220998\pi\)
\(19\)	1.50658e6	−	1.50658e6i	0.608449	−	0.608449i	−0.334092	−	0.942541i	\(-0.608430\pi\)
							0.942541	+	0.334092i	\(0.108430\pi\)
\(23\)	5.06073e6			0.786274			0.393137	−	0.919480i	\(-0.371390\pi\)
							0.393137	+	0.919480i	\(0.371390\pi\)
\(29\)	7.02130e6	+	7.02130e6i	0.342316	+	0.342316i	0.857237	−	0.514921i	\(-0.172179\pi\)
							−0.514921	+	0.857237i	\(0.672179\pi\)
\(31\)			4.44115e6i			0.155127i	0.996987	+	0.0775634i	\(0.0247140\pi\)
							−0.996987	+	0.0775634i	\(0.975286\pi\)
\(37\)	5.14101e7	−	5.14101e7i	0.741378	−	0.741378i	−0.231465	−	0.972843i	\(-0.574352\pi\)
							0.972843	+	0.231465i	\(0.0743520\pi\)
\(41\)		−	2.23859e7i		−	0.193221i	−0.995322	−	0.0966105i	\(-0.969200\pi\)
							0.995322	−	0.0966105i	\(-0.0308001\pi\)
\(43\)	1.18868e8	+	1.18868e8i	0.808576	+	0.808576i	0.984418	−	0.175842i	\(-0.0562648\pi\)
							−0.175842	+	0.984418i	\(0.556265\pi\)
\(47\)			4.89607e7i			0.213480i	0.994287	+	0.106740i	\(0.0340413\pi\)
							−0.994287	+	0.106740i	\(0.965959\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.11.f.a.3.8	✓	38
4.3	odd	2		64.11.f.a.47.13		38
8.3	odd	2		128.11.f.a.95.7		38
8.5	even	2		128.11.f.b.95.13		38
16.3	odd	4		128.11.f.b.31.13		38
16.5	even	4		64.11.f.a.15.13		38
16.11	odd	4	inner	16.11.f.a.11.8	yes	38
16.13	even	4		128.11.f.a.31.7		38

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.11.f.a.3.8	✓	38	1.1	even	1	trivial
16.11.f.a.11.8	yes	38	16.11	odd	4	inner
64.11.f.a.15.13		38	16.5	even	4
64.11.f.a.47.13		38	4.3	odd	2
128.11.f.a.31.7		38	16.13	even	4
128.11.f.a.95.7		38	8.3	odd	2
128.11.f.b.31.13		38	16.3	odd	4
128.11.f.b.95.13		38	8.5	even	2