Embedded newform 192.2.s.a.11.3

• Label: 192.2.s.a.11.3
• Level: $192$
• Weight: $2$
• Character: 192.11
• Analytic conductor: $1.533$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	192.s (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.53312771881\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(30\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			11.3
Character	\(\chi\)	\(=\)	192.11
Dual form			192.2.s.a.35.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35305 + 0.411422i) q^{2} +(-0.145818 - 1.72590i) q^{3} +(1.66146 - 1.11334i) q^{4} +(0.731423 - 0.488721i) q^{5} +(0.907373 + 2.27523i) q^{6} +(0.683144 - 1.64925i) q^{7} +(-1.78998 + 2.18997i) q^{8} +(-2.95747 + 0.503337i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(133\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35305	+	0.411422i	−0.956748	+	0.290919i
							\(3\)	−0.145818	−	1.72590i	−0.0841883	−	0.996450i
\(5\)	0.731423	−	0.488721i	0.327102	−	0.218563i								−0.381165	−	0.924507i	\(-0.624477\pi\)
							0.708267	+	0.705944i	\(0.249477\pi\)
\(7\)	0.683144	−	1.64925i	0.258204	−	0.623360i	−0.740616	−	0.671929i	\(-0.765466\pi\)
							0.998820	+	0.0485690i	\(0.0154661\pi\)
\(11\)	0.385603	−	1.93856i	0.116264	−	0.584497i	−0.878100	−	0.478477i	\(-0.841189\pi\)
							0.994364	−	0.106021i	\(-0.0338109\pi\)
\(13\)	−0.659632	+	0.987209i	−0.182949	+	0.273803i	−0.911596	−	0.411087i	\(-0.865149\pi\)
							0.728647	+	0.684889i	\(0.240149\pi\)
\(17\)	−2.96995	−	2.96995i	−0.720319	−	0.720319i	0.248351	−	0.968670i	\(-0.420111\pi\)
							−0.968670	+	0.248351i	\(0.920111\pi\)
\(19\)	2.88241	−	4.31384i	0.661271	−	0.989662i	−0.337559	−	0.941304i	\(-0.609601\pi\)
							0.998830	−	0.0483579i	\(-0.0153988\pi\)
\(23\)	−1.53714	−	3.71099i	−0.320516	−	0.773794i	−0.999224	−	0.0393859i	\(-0.987460\pi\)
							0.678708	−	0.734408i	\(-0.262540\pi\)
\(29\)	8.74967	−	1.74042i	1.62477	−	0.323188i	0.703081	−	0.711109i	\(-0.251807\pi\)
							0.921692	+	0.387922i	\(0.126807\pi\)
\(31\)	2.66183			0.478079			0.239039	−	0.971010i	\(-0.423167\pi\)
							0.239039	+	0.971010i	\(0.423167\pi\)
\(37\)	−2.69661	+	1.80182i	−0.443320	+	0.296217i	−0.757124	−	0.653271i	\(-0.773396\pi\)
							0.313805	+	0.949488i	\(0.398396\pi\)
\(41\)	−9.64627	+	3.99562i	−1.50649	+	0.624011i	−0.974831	−	0.222945i	\(-0.928433\pi\)
							−0.531664	+	0.846955i	\(0.678433\pi\)
\(43\)	−0.964098	+	4.84685i	−0.147024	+	0.739138i	0.834981	+	0.550279i	\(0.185479\pi\)
							−0.982004	+	0.188858i	\(0.939521\pi\)
\(47\)	2.39391	−	2.39391i	0.349187	−	0.349187i	−0.510620	−	0.859807i	\(-0.670584\pi\)
							0.859807	+	0.510620i	\(0.170584\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.2.s.a.11.3	✓	240
3.2	odd	2	inner	192.2.s.a.11.28	yes	240
4.3	odd	2		768.2.s.a.719.16		240
12.11	even	2		768.2.s.a.719.19		240
64.29	even	16		768.2.s.a.47.19		240
64.35	odd	16	inner	192.2.s.a.35.28	yes	240
192.29	odd	16		768.2.s.a.47.16		240
192.35	even	16	inner	192.2.s.a.35.3	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.2.s.a.11.3	✓	240	1.1	even	1	trivial
192.2.s.a.11.28	yes	240	3.2	odd	2	inner
192.2.s.a.35.3	yes	240	192.35	even	16	inner
192.2.s.a.35.28	yes	240	64.35	odd	16	inner
768.2.s.a.47.16		240	192.29	odd	16
768.2.s.a.47.19		240	64.29	even	16
768.2.s.a.719.16		240	4.3	odd	2
768.2.s.a.719.19		240	12.11	even	2