Embedded newform 192.3.t.a.19.16

• Label: 192.3.t.a.19.16
• 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			19.16
Character	\(\chi\)	\(=\)	192.19
Dual form			192.3.t.a.91.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.358174 + 1.96767i) q^{2} +(1.44015 - 0.962276i) q^{3} +(-3.74342 - 1.40954i) q^{4} +(-0.809137 - 4.06781i) q^{5} +(1.37761 + 3.17839i) q^{6} +(-6.21464 + 2.57419i) q^{7} +(4.11429 - 6.86095i) q^{8} +(1.14805 - 2.77164i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.358174	+	1.96767i	−0.179087	+	0.983833i
							\(3\)	1.44015	−	0.962276i	0.480049	−	0.320759i
\(5\)	−0.809137	−	4.06781i	−0.161827	−	0.813562i								−0.973365	−	0.229261i	\(-0.926369\pi\)
							0.811537	−	0.584300i	\(-0.198631\pi\)
\(7\)	−6.21464	+	2.57419i	−0.887805	+	0.367741i	−0.779519	−	0.626379i	\(-0.784536\pi\)
							−0.108287	+	0.994120i	\(0.534536\pi\)
\(11\)	2.01559	−	3.01654i	0.183235	−	0.274231i	−0.728469	−	0.685079i	\(-0.759768\pi\)
							0.911704	+	0.410849i	\(0.134768\pi\)
\(13\)	3.63041	−	18.2513i	0.279262	−	1.40394i	−0.545332	−	0.838220i	\(-0.683596\pi\)
							0.824594	−	0.565725i	\(-0.191404\pi\)
\(17\)	−11.3413	−	11.3413i	−0.667136	−	0.667136i	0.289916	−	0.957052i	\(-0.406372\pi\)
							−0.957052	+	0.289916i	\(0.906372\pi\)
\(19\)	6.12084	−	30.7715i	0.322150	−	1.61956i	−0.392279	−	0.919846i	\(-0.628313\pi\)
							0.714428	−	0.699709i	\(-0.246687\pi\)
\(23\)	−6.56526	+	15.8499i	−0.285446	+	0.689127i	−0.999945	−	0.0105093i	\(-0.996655\pi\)
							0.714499	+	0.699637i	\(0.246655\pi\)
\(29\)	−31.3238	+	20.9299i	−1.08013	+	0.721721i	−0.962485	−	0.271334i	\(-0.912535\pi\)
							−0.117647	+	0.993055i	\(0.537535\pi\)
\(31\)	3.04888			0.0983511			0.0491756	−	0.998790i	\(-0.484341\pi\)
							0.0491756	+	0.998790i	\(0.484341\pi\)
\(37\)	22.0194	−	4.37994i	0.595120	−	0.118377i	0.111665	−	0.993746i	\(-0.464382\pi\)
							0.483455	+	0.875369i	\(0.339382\pi\)
\(41\)	−2.85967	−	1.18452i	−0.0697481	−	0.0288906i	0.347537	−	0.937666i	\(-0.387018\pi\)
							−0.417285	+	0.908776i	\(0.637018\pi\)
\(43\)	52.6472	+	35.1777i	1.22435	+	0.818087i	0.988131	−	0.153612i	\(-0.0490906\pi\)
							0.236222	+	0.971699i	\(0.424091\pi\)
\(47\)	−28.8969	+	28.8969i	−0.614827	+	0.614827i	−0.944200	−	0.329373i	\(-0.893163\pi\)
							0.329373	+	0.944200i	\(0.393163\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.t.a.19.16	✓	256
64.27	odd	16	inner	192.3.t.a.91.16	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.t.a.19.16	✓	256	1.1	even	1	trivial
192.3.t.a.91.16	yes	256	64.27	odd	16	inner