Embedded newform 192.3.t.a.19.18

• Label: 192.3.t.a.19.18
• 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			19.18
Character	\(\chi\)	\(=\)	192.19
Dual form			192.3.t.a.91.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.195531 + 1.99042i) q^{2} +(-1.44015 + 0.962276i) q^{3} +(-3.92354 + 0.778375i) q^{4} +(1.68587 + 8.47543i) q^{5} +(-2.19693 - 2.67834i) q^{6} +(-3.67509 + 1.52227i) q^{7} +(-2.31646 - 7.65728i) q^{8} +(1.14805 - 2.77164i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.195531	+	1.99042i	0.0977653	+	0.995210i
							\(3\)	−1.44015	+	0.962276i	−0.480049	+	0.320759i
\(5\)	1.68587	+	8.47543i	0.337173	+	1.69509i								0.662156	+	0.749366i	\(0.269641\pi\)
							−0.324983	+	0.945720i	\(0.605359\pi\)
\(7\)	−3.67509	+	1.52227i	−0.525013	+	0.217468i	−0.629417	−	0.777067i	\(-0.716706\pi\)
							0.104404	+	0.994535i	\(0.466706\pi\)
\(11\)	6.39297	−	9.56776i	0.581179	−	0.869796i	−0.418075	−	0.908413i	\(-0.637295\pi\)
							0.999254	+	0.0386164i	\(0.0122951\pi\)
\(13\)	−3.92400	+	19.7273i	−0.301846	+	1.51748i	0.470568	+	0.882364i	\(0.344049\pi\)
							−0.772414	+	0.635119i	\(0.780951\pi\)
\(17\)	−14.8311	−	14.8311i	−0.872418	−	0.872418i	0.120318	−	0.992735i	\(-0.461609\pi\)
							−0.992735	+	0.120318i	\(0.961609\pi\)
\(19\)	3.11044	−	15.6372i	0.163707	−	0.823012i	−0.808429	−	0.588594i	\(-0.799682\pi\)
							0.972136	−	0.234418i	\(-0.0753183\pi\)
\(23\)	−7.64256	+	18.4508i	−0.332285	+	0.802208i	0.666125	+	0.745840i	\(0.267952\pi\)
							−0.998410	+	0.0563673i	\(0.982048\pi\)
\(29\)	−1.72226	+	1.15078i	−0.0593883	+	0.0396820i	−0.584910	−	0.811098i	\(-0.698870\pi\)
							0.525522	+	0.850780i	\(0.323870\pi\)
\(31\)	3.09295			0.0997725			0.0498862	−	0.998755i	\(-0.484114\pi\)
							0.0498862	+	0.998755i	\(0.484114\pi\)
\(37\)	−26.1963	+	5.21077i	−0.708009	+	0.140832i	−0.535944	−	0.844254i	\(-0.680044\pi\)
							−0.172065	+	0.985086i	\(0.555044\pi\)
\(41\)	58.4524	+	24.2118i	1.42567	+	0.590531i	0.956278	−	0.292459i	\(-0.0944736\pi\)
							0.469391	+	0.882991i	\(0.344474\pi\)
\(43\)	63.9772	+	42.7482i	1.48784	+	0.994143i	0.992070	+	0.125690i	\(0.0401146\pi\)
							0.495771	+	0.868453i	\(0.334885\pi\)
\(47\)	−32.1820	+	32.1820i	−0.684724	+	0.684724i	−0.961061	−	0.276337i	\(-0.910879\pi\)
							0.276337	+	0.961061i	\(0.410879\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.t.a.19.18	✓	256
64.27	odd	16	inner	192.3.t.a.91.18	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.t.a.19.18	✓	256	1.1	even	1	trivial
192.3.t.a.91.18	yes	256	64.27	odd	16	inner