Embedded newform 192.3.q.a.5.3

• Label: 192.3.q.a.5.3
• 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			5.3
Character	\(\chi\)	\(=\)	192.5
Dual form			192.3.q.a.77.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.98950 + 0.204660i) q^{2} +(-0.243775 + 2.99008i) q^{3} +(3.91623 - 0.814343i) q^{4} +(1.66868 - 8.38903i) q^{5} +(-0.126959 - 5.99866i) q^{6} +(0.401965 - 0.970430i) q^{7} +(-7.62468 + 2.42163i) q^{8} +(-8.88115 - 1.45781i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.98950	+	0.204660i	−0.994750	+	0.102330i
							\(3\)	−0.243775	+	2.99008i	−0.0812583	+	0.996693i
\(5\)	1.66868	−	8.38903i	0.333736	−	1.67781i								−0.341247	−	0.939974i	\(-0.610849\pi\)
							0.674984	−	0.737833i	\(-0.264151\pi\)
\(7\)	0.401965	−	0.970430i	0.0574236	−	0.138633i	−0.892564	−	0.450921i	\(-0.851096\pi\)
							0.949987	+	0.312288i	\(0.101096\pi\)
\(11\)	−5.32917	+	3.56084i	−0.484470	+	0.323712i	−0.773705	−	0.633546i	\(-0.781599\pi\)
							0.289236	+	0.957258i	\(0.406599\pi\)
\(13\)	−24.7079	+	4.91471i	−1.90061	+	0.378054i	−0.998563	−	0.0535873i	\(-0.982934\pi\)
							−0.902045	+	0.431642i	\(0.857934\pi\)
\(17\)	−9.29664	−	9.29664i	−0.546861	−	0.546861i	0.378671	−	0.925532i	\(-0.376381\pi\)
							−0.925532	+	0.378671i	\(0.876381\pi\)
\(19\)	−5.23469	−	26.3166i	−0.275510	−	1.38508i	−0.832252	−	0.554397i	\(-0.812949\pi\)
							0.556742	−	0.830685i	\(-0.312051\pi\)
\(23\)	2.53808	+	6.12746i	0.110351	+	0.266411i	0.969402	−	0.245479i	\(-0.0789453\pi\)
							−0.859051	+	0.511891i	\(0.828945\pi\)
\(29\)	22.6892	+	15.1604i	0.782387	+	0.522774i	0.881433	−	0.472309i	\(-0.156579\pi\)
							−0.0990465	+	0.995083i	\(0.531579\pi\)
\(31\)			10.4730i			0.337837i	0.985630	+	0.168919i	\(0.0540275\pi\)
							−0.985630	+	0.168919i	\(0.945973\pi\)
\(37\)	5.50053	−	27.6531i	0.148663	−	0.747380i	−0.832474	−	0.554065i	\(-0.813076\pi\)
							0.981137	−	0.193315i	\(-0.0619240\pi\)
\(41\)	−19.2957	−	46.5839i	−0.470627	−	1.13619i	−0.963887	−	0.266312i	\(-0.914195\pi\)
							0.493260	−	0.869882i	\(-0.335805\pi\)
\(43\)	−34.1961	+	22.8491i	−0.795259	+	0.531375i	−0.885552	−	0.464540i	\(-0.846220\pi\)
							0.0902933	+	0.995915i	\(0.471220\pi\)
\(47\)	7.03823	+	7.03823i	0.149750	+	0.149750i	0.778006	−	0.628257i	\(-0.216231\pi\)
							−0.628257	+	0.778006i	\(0.716231\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.q.a.5.3	✓	496
3.2	odd	2	inner	192.3.q.a.5.60	yes	496
64.13	even	16	inner	192.3.q.a.77.60	yes	496
192.77	odd	16	inner	192.3.q.a.77.3	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.q.a.5.3	✓	496	1.1	even	1	trivial
192.3.q.a.5.60	yes	496	3.2	odd	2	inner
192.3.q.a.77.3	yes	496	192.77	odd	16	inner
192.3.q.a.77.60	yes	496	64.13	even	16	inner