Embedded newform 192.3.q.a.5.1

• Label: 192.3.q.a.5.1
• 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			5.1
Character	\(\chi\)	\(=\)	192.5
Dual form			192.3.q.a.77.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99952 - 0.0439169i) q^{2} +(-0.469761 + 2.96299i) q^{3} +(3.99614 + 0.175625i) q^{4} +(-1.47453 + 7.41296i) q^{5} +(1.06942 - 5.90393i) q^{6} +(-4.15461 + 10.0301i) q^{7} +(-7.98265 - 0.526664i) q^{8} +(-8.55865 - 2.78380i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99952	−	0.0439169i	−0.999759	−	0.0219584i
							\(3\)	−0.469761	+	2.96299i	−0.156587	+	0.987664i
\(5\)	−1.47453	+	7.41296i	−0.294906	+	1.48259i								0.494746	+	0.869037i	\(0.335261\pi\)
							−0.789652	+	0.613555i	\(0.789739\pi\)
\(7\)	−4.15461	+	10.0301i	−0.593516	+	1.43287i	0.286570	+	0.958059i	\(0.407485\pi\)
							−0.880086	+	0.474814i	\(0.842515\pi\)
\(11\)	13.4252	−	8.97040i	1.22047	−	0.815491i	0.232872	−	0.972507i	\(-0.425188\pi\)
							0.987596	+	0.157016i	\(0.0501875\pi\)
\(13\)	−0.609524	+	0.121242i	−0.0468865	+	0.00932630i	−0.218478	−	0.975842i	\(-0.570109\pi\)
							0.171591	+	0.985168i	\(0.445109\pi\)
\(17\)	8.17940	+	8.17940i	0.481141	+	0.481141i	0.905496	−	0.424355i	\(-0.139499\pi\)
							−0.424355	+	0.905496i	\(0.639499\pi\)
\(19\)	0.681909	+	3.42819i	0.0358900	+	0.180431i	0.994572	−	0.104048i	\(-0.0331795\pi\)
							−0.958682	+	0.284479i	\(0.908179\pi\)
\(23\)	−7.73804	−	18.6813i	−0.336437	−	0.812230i	−0.998052	−	0.0623865i	\(-0.980129\pi\)
							0.661615	−	0.749843i	\(-0.269871\pi\)
\(29\)	−33.1821	−	22.1716i	−1.14421	−	0.764537i	−0.168956	−	0.985624i	\(-0.554040\pi\)
							−0.975254	+	0.221087i	\(0.929040\pi\)
\(31\)		−	5.90307i		−	0.190422i	−0.995457	−	0.0952109i	\(-0.969647\pi\)
							0.995457	−	0.0952109i	\(-0.0303525\pi\)
\(37\)	−12.2632	+	61.6513i	−0.331438	+	1.66625i	0.351816	+	0.936069i	\(0.385564\pi\)
							−0.683253	+	0.730181i	\(0.739436\pi\)
\(41\)	23.1646	+	55.9243i	0.564990	+	1.36401i	0.905733	+	0.423850i	\(0.139322\pi\)
							−0.340743	+	0.940157i	\(0.610678\pi\)
\(43\)	25.7199	−	17.1855i	0.598136	−	0.399662i	−0.219320	−	0.975653i	\(-0.570384\pi\)
							0.817456	+	0.575991i	\(0.195384\pi\)
\(47\)	50.2594	+	50.2594i	1.06935	+	1.06935i	0.997409	+	0.0719400i	\(0.0229190\pi\)
							0.0719400	+	0.997409i	\(0.477081\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.q.a.5.1	✓	496
3.2	odd	2	inner	192.3.q.a.5.62	yes	496
64.13	even	16	inner	192.3.q.a.77.62	yes	496
192.77	odd	16	inner	192.3.q.a.77.1	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.q.a.5.1	✓	496	1.1	even	1	trivial
192.3.q.a.5.62	yes	496	3.2	odd	2	inner
192.3.q.a.77.1	yes	496	192.77	odd	16	inner
192.3.q.a.77.62	yes	496	64.13	even	16	inner