Embedded newform 192.3.q.a.5.19

• Label: 192.3.q.a.5.19
• Level: $192$
• Weight: $3$
• Character: 192.5
• Analytic conductor: $5.232$
• Analytic rank: $0$
• Dimension: $496$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.23162107572\)
Analytic rank:	\(0\)
Dimension:	\(496\)
Relative dimension:	\(62\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			5.19
Character	\(\chi\)	\(=\)	192.5
Dual form			192.3.q.a.77.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15682 + 1.63149i) q^{2} +(1.21225 + 2.74417i) q^{3} +(-1.32354 - 3.77468i) q^{4} +(-0.581021 + 2.92099i) q^{5} +(-5.87944 - 1.19674i) q^{6} +(-0.312379 + 0.754149i) q^{7} +(7.68947 + 2.20728i) q^{8} +(-6.06092 + 6.65321i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15682	+	1.63149i	−0.578410	+	0.815746i
							\(3\)	1.21225	+	2.74417i	0.404082	+	0.914723i
\(5\)	−0.581021	+	2.92099i	−0.116204	+	0.584198i								0.878177	+	0.478337i	\(0.158760\pi\)
							−0.994381	+	0.105862i	\(0.966240\pi\)
\(7\)	−0.312379	+	0.754149i	−0.0446255	+	0.107736i	−0.944621	−	0.328165i	\(-0.893570\pi\)
							0.899995	+	0.435900i	\(0.143570\pi\)
\(11\)	−7.29924	+	4.87719i	−0.663567	+	0.443381i	−0.841206	−	0.540714i	\(-0.818154\pi\)
							0.177639	+	0.984096i	\(0.443154\pi\)
\(13\)	−10.1937	+	2.02765i	−0.784128	+	0.155973i	−0.570889	−	0.821027i	\(-0.693401\pi\)
							−0.213239	+	0.977000i	\(0.568401\pi\)
\(17\)	−0.785945	−	0.785945i	−0.0462321	−	0.0462321i	0.683613	−	0.729845i	\(-0.260408\pi\)
							−0.729845	+	0.683613i	\(0.760408\pi\)
\(19\)	0.543827	+	2.73400i	0.0286225	+	0.143895i	0.992454	−	0.122617i	\(-0.0391287\pi\)
							−0.963832	+	0.266512i	\(0.914129\pi\)
\(23\)	−7.07840	−	17.0888i	−0.307756	−	0.742990i	−0.999777	−	0.0211120i	\(-0.993279\pi\)
							0.692021	−	0.721878i	\(-0.256721\pi\)
\(29\)	2.80825	+	1.87641i	0.0968362	+	0.0647039i	0.603046	−	0.797707i	\(-0.293954\pi\)
							−0.506209	+	0.862411i	\(0.668954\pi\)
\(31\)		−	16.5141i		−	0.532714i	−0.963874	−	0.266357i	\(-0.914180\pi\)
							0.963874	−	0.266357i	\(-0.0858200\pi\)
\(37\)	−6.32165	+	31.7811i	−0.170855	+	0.858948i	0.796327	+	0.604866i	\(0.206773\pi\)
							−0.967183	+	0.254082i	\(0.918227\pi\)
\(41\)	3.75215	+	9.05848i	0.0915158	+	0.220939i	0.963009	−	0.269469i	\(-0.0868480\pi\)
							−0.871493	+	0.490407i	\(0.836848\pi\)
\(43\)	−48.7097	+	32.5468i	−1.13278	+	0.756902i	−0.973127	−	0.230271i	\(-0.926039\pi\)
							−0.159657	+	0.987173i	\(0.551039\pi\)
\(47\)	−1.94171	−	1.94171i	−0.0413130	−	0.0413130i	0.686149	−	0.727461i	\(-0.259300\pi\)
							−0.727461	+	0.686149i	\(0.759300\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.q.a.5.19	✓	496
3.2	odd	2	inner	192.3.q.a.5.44	yes	496
64.13	even	16	inner	192.3.q.a.77.44	yes	496
192.77	odd	16	inner	192.3.q.a.77.19	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.q.a.5.19	✓	496	1.1	even	1	trivial
192.3.q.a.5.44	yes	496	3.2	odd	2	inner
192.3.q.a.77.19	yes	496	192.77	odd	16	inner
192.3.q.a.77.44	yes	496	64.13	even	16	inner