Embedded newform 192.3.t.a.19.19

• Label: 192.3.t.a.19.19
• Level: $192$
• Weight: $3$
• Character: 192.19
• Analytic conductor: $5.232$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.23162107572\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(32\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			19.19
Character	\(\chi\)	\(=\)	192.19
Dual form			192.3.t.a.91.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.377294 - 1.96409i) q^{2} +(1.44015 - 0.962276i) q^{3} +(-3.71530 - 1.48208i) q^{4} +(1.80347 + 9.06666i) q^{5} +(-1.34664 - 3.19164i) q^{6} +(9.72216 - 4.02705i) q^{7} +(-4.31270 + 6.73800i) q^{8} +(1.14805 - 2.77164i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(133\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{7}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.377294	−	1.96409i	0.188647	−	0.982045i
							\(3\)	1.44015	−	0.962276i	0.480049	−	0.320759i
\(5\)	1.80347	+	9.06666i	0.360694	+	1.81333i								0.554374	+	0.832267i	\(0.312958\pi\)
							−0.193680	+	0.981065i	\(0.562042\pi\)
\(7\)	9.72216	−	4.02705i	1.38888	−	0.575293i	0.442039	−	0.896996i	\(-0.354255\pi\)
							0.946841	+	0.321703i	\(0.104255\pi\)
\(11\)	8.64183	−	12.9334i	0.785621	−	1.17576i	−0.195183	−	0.980767i	\(-0.562530\pi\)
							0.980804	−	0.194997i	\(-0.0624699\pi\)
\(13\)	−0.571515	+	2.87320i	−0.0439627	+	0.221015i	−0.996521	−	0.0833403i	\(-0.973441\pi\)
							0.952558	+	0.304356i	\(0.0984411\pi\)
\(17\)	2.53096	+	2.53096i	0.148880	+	0.148880i	0.777618	−	0.628737i	\(-0.216428\pi\)
							−0.628737	+	0.777618i	\(0.716428\pi\)
\(19\)	4.06879	−	20.4552i	0.214147	−	1.07659i	−0.712792	−	0.701376i	\(-0.752570\pi\)
							0.926939	−	0.375213i	\(-0.122430\pi\)
\(23\)	−14.1204	+	34.0896i	−0.613929	+	1.48216i	0.244721	+	0.969594i	\(0.421304\pi\)
							−0.858650	+	0.512562i	\(0.828696\pi\)
\(29\)	9.43514	−	6.30436i	0.325350	−	0.217392i	−0.382159	−	0.924097i	\(-0.624819\pi\)
							0.707508	+	0.706705i	\(0.249819\pi\)
\(31\)	−11.0208			−0.355510			−0.177755	−	0.984075i	\(-0.556883\pi\)
							−0.177755	+	0.984075i	\(0.556883\pi\)
\(37\)	6.41981	−	1.27698i	0.173508	−	0.0345129i	−0.107571	−	0.994197i	\(-0.534307\pi\)
							0.281080	+	0.959684i	\(0.409307\pi\)
\(41\)	24.0990	+	9.98213i	0.587780	+	0.243466i	0.656695	−	0.754156i	\(-0.271954\pi\)
							−0.0689152	+	0.997623i	\(0.521954\pi\)
\(43\)	−44.3689	−	29.6464i	−1.03184	−	0.689450i	−0.0802311	−	0.996776i	\(-0.525566\pi\)
							−0.951604	+	0.307326i	\(0.900566\pi\)
\(47\)	1.79886	−	1.79886i	0.0382736	−	0.0382736i	−0.687711	−	0.725985i	\(-0.741384\pi\)
							0.725985	+	0.687711i	\(0.241384\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.t.a.19.19	✓	256
64.27	odd	16	inner	192.3.t.a.91.19	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.t.a.19.19	✓	256	1.1	even	1	trivial
192.3.t.a.91.19	yes	256	64.27	odd	16	inner