Embedded newform 192.3.t.a.19.2

• Label: 192.3.t.a.19.2
• 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			19.2
Character	\(\chi\)	\(=\)	192.19
Dual form			192.3.t.a.91.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.97540 - 0.312752i) q^{2} +(-1.44015 + 0.962276i) q^{3} +(3.80437 + 1.23562i) q^{4} +(-0.279220 - 1.40373i) q^{5} +(3.14581 - 1.45047i) q^{6} +(-3.62990 + 1.50355i) q^{7} +(-7.12870 - 3.63066i) q^{8} +(1.14805 - 2.77164i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.97540	−	0.312752i	−0.987698	−	0.156376i
							\(3\)	−1.44015	+	0.962276i	−0.480049	+	0.320759i
\(5\)	−0.279220	−	1.40373i	−0.0558440	−	0.280747i								0.942766	−	0.333455i	\(-0.108215\pi\)
							−0.998610	+	0.0527084i	\(0.983215\pi\)
\(7\)	−3.62990	+	1.50355i	−0.518558	+	0.214794i	−0.626583	−	0.779355i	\(-0.715547\pi\)
							0.108026	+	0.994148i	\(0.465547\pi\)
\(11\)	8.51083	−	12.7374i	0.773711	−	1.15794i	−0.209915	−	0.977720i	\(-0.567319\pi\)
							0.983626	−	0.180221i	\(-0.0576814\pi\)
\(13\)	−3.60171	+	18.1070i	−0.277055	+	1.39285i	0.552077	+	0.833793i	\(0.313836\pi\)
							−0.829131	+	0.559054i	\(0.811164\pi\)
\(17\)	1.82043	+	1.82043i	0.107084	+	0.107084i	0.758619	−	0.651535i	\(-0.225875\pi\)
							−0.651535	+	0.758619i	\(0.725875\pi\)
\(19\)	5.22193	−	26.2524i	0.274838	−	1.38171i	−0.558758	−	0.829331i	\(-0.688722\pi\)
							0.833596	−	0.552375i	\(-0.186278\pi\)
\(23\)	13.8451	−	33.4250i	0.601961	−	1.45326i	−0.269599	−	0.962973i	\(-0.586891\pi\)
							0.871560	−	0.490289i	\(-0.163109\pi\)
\(29\)	27.9872	−	18.7004i	0.965074	−	0.644842i	0.0300959	−	0.999547i	\(-0.490419\pi\)
							0.934978	+	0.354705i	\(0.115419\pi\)
\(31\)	18.2240			0.587872			0.293936	−	0.955825i	\(-0.405035\pi\)
							0.293936	+	0.955825i	\(0.405035\pi\)
\(37\)	63.1584	−	12.5630i	1.70698	−	0.339540i	0.757376	−	0.652980i	\(-0.226481\pi\)
							0.949608	+	0.313439i	\(0.101481\pi\)
\(41\)	−51.0426	−	21.1426i	−1.24494	−	0.515672i	−0.339686	−	0.940539i	\(-0.610321\pi\)
							−0.905256	+	0.424867i	\(0.860321\pi\)
\(43\)	−26.6638	−	17.8162i	−0.620089	−	0.414330i	0.205456	−	0.978666i	\(-0.434132\pi\)
							−0.825545	+	0.564336i	\(0.809132\pi\)
\(47\)	−31.6086	+	31.6086i	−0.672524	+	0.672524i	−0.958297	−	0.285773i	\(-0.907750\pi\)
							0.285773	+	0.958297i	\(0.407750\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.t.a.19.2	✓	256
64.27	odd	16	inner	192.3.t.a.91.2	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.t.a.19.2	✓	256	1.1	even	1	trivial
192.3.t.a.91.2	yes	256	64.27	odd	16	inner