Embedded newform 192.3.t.a.19.15

• Label: 192.3.t.a.19.15
• 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			19.15
Character	\(\chi\)	\(=\)	192.19
Dual form			192.3.t.a.91.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.399279 - 1.95974i) q^{2} +(1.44015 - 0.962276i) q^{3} +(-3.68115 + 1.56497i) q^{4} +(0.758038 + 3.81092i) q^{5} +(-2.46083 - 2.43810i) q^{6} +(-10.7874 + 4.46830i) q^{7} +(4.53673 + 6.58924i) q^{8} +(1.14805 - 2.77164i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.399279	−	1.95974i	−0.199640	−	0.979869i
							\(3\)	1.44015	−	0.962276i	0.480049	−	0.320759i
\(5\)	0.758038	+	3.81092i	0.151608	+	0.762183i								0.979523	+	0.201331i	\(0.0645266\pi\)
							−0.827916	+	0.560853i	\(0.810473\pi\)
\(7\)	−10.7874	+	4.46830i	−1.54106	+	0.638329i	−0.981672	−	0.190577i	\(-0.938964\pi\)
							−0.559389	+	0.828905i	\(0.688964\pi\)
\(11\)	−9.48063	+	14.1888i	−0.861876	+	1.28989i	0.0938370	+	0.995588i	\(0.470087\pi\)
							−0.955713	+	0.294301i	\(0.904913\pi\)
\(13\)	−1.36849	+	6.87985i	−0.105268	+	0.529220i	0.891782	+	0.452466i	\(0.149456\pi\)
							−0.997050	+	0.0767538i	\(0.975544\pi\)
\(17\)	−20.9763	−	20.9763i	−1.23390	−	1.23390i	−0.962453	−	0.271447i	\(-0.912498\pi\)
							−0.271447	−	0.962453i	\(-0.587502\pi\)
\(19\)	−0.378141	+	1.90104i	−0.0199021	+	0.100055i	−0.989460	−	0.144809i	\(-0.953743\pi\)
							0.969557	+	0.244864i	\(0.0787432\pi\)
\(23\)	−7.93619	+	19.1597i	−0.345052	+	0.833029i	0.652137	+	0.758101i	\(0.273873\pi\)
							−0.997189	+	0.0749279i	\(0.976127\pi\)
\(29\)	16.3100	−	10.8980i	0.562415	−	0.375793i	−0.241624	−	0.970370i	\(-0.577680\pi\)
							0.804039	+	0.594576i	\(0.202680\pi\)
\(31\)	55.8755			1.80243			0.901217	−	0.433368i	\(-0.142675\pi\)
							0.901217	+	0.433368i	\(0.142675\pi\)
\(37\)	−33.6204	+	6.68752i	−0.908661	+	0.180744i	−0.627227	−	0.778837i	\(-0.715810\pi\)
							−0.281434	+	0.959581i	\(0.590810\pi\)
\(41\)	−24.7355	−	10.2458i	−0.603305	−	0.249897i	0.0600579	−	0.998195i	\(-0.480871\pi\)
							−0.663363	+	0.748298i	\(0.730871\pi\)
\(43\)	18.8694	+	12.6081i	0.438822	+	0.293212i	0.755289	−	0.655392i	\(-0.227497\pi\)
							−0.316467	+	0.948604i	\(0.602497\pi\)
\(47\)	−55.0072	+	55.0072i	−1.17037	+	1.17037i	−0.188245	+	0.982122i	\(0.560280\pi\)
							−0.982122	+	0.188245i	\(0.939720\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.t.a.19.15	✓	256
64.27	odd	16	inner	192.3.t.a.91.15	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.t.a.19.15	✓	256	1.1	even	1	trivial
192.3.t.a.91.15	yes	256	64.27	odd	16	inner