Embedded newform 192.3.q.a.5.15

• Label: 192.3.q.a.5.15
• 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			5.15
Character	\(\chi\)	\(=\)	192.5
Dual form			192.3.q.a.77.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40959 - 1.41882i) q^{2} +(-2.45976 + 1.71744i) q^{3} +(-0.0261177 + 3.99991i) q^{4} +(1.04139 - 5.23542i) q^{5} +(5.90399 + 1.06907i) q^{6} +(-4.45319 + 10.7510i) q^{7} +(5.71199 - 5.60118i) q^{8} +(3.10080 - 8.44897i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40959	−	1.41882i	−0.704794	−	0.709412i
							\(3\)	−2.45976	+	1.71744i	−0.819919	+	0.572480i
\(5\)	1.04139	−	5.23542i	0.208278	−	1.04708i								−0.725224	−	0.688513i	\(-0.758264\pi\)
							0.933502	−	0.358572i	\(-0.116736\pi\)
\(7\)	−4.45319	+	10.7510i	−0.636170	+	1.53585i	0.195573	+	0.980689i	\(0.437343\pi\)
							−0.831743	+	0.555161i	\(0.812657\pi\)
\(11\)	−1.91896	+	1.28221i	−0.174451	+	0.116564i	−0.639730	−	0.768600i	\(-0.720954\pi\)
							0.465279	+	0.885164i	\(0.345954\pi\)
\(13\)	14.8661	−	2.95704i	1.14354	−	0.227465i	0.413262	−	0.910612i	\(-0.364389\pi\)
							0.730281	+	0.683147i	\(0.239389\pi\)
\(17\)	−15.2405	−	15.2405i	−0.896501	−	0.896501i	0.0986236	−	0.995125i	\(-0.468556\pi\)
							−0.995125	+	0.0986236i	\(0.968556\pi\)
\(19\)	−5.76485	−	28.9819i	−0.303413	−	1.52536i	−0.768357	−	0.640022i	\(-0.778925\pi\)
							0.464944	−	0.885340i	\(-0.346075\pi\)
\(23\)	−8.46204	−	20.4292i	−0.367915	−	0.888225i	−0.994092	−	0.108544i	\(-0.965381\pi\)
							0.626177	−	0.779681i	\(-0.284619\pi\)
\(29\)	−15.0040	−	10.0254i	−0.517381	−	0.345703i	0.269306	−	0.963055i	\(-0.413206\pi\)
							−0.786687	+	0.617352i	\(0.788206\pi\)
\(31\)		−	49.5103i		−	1.59711i	−0.601924	−	0.798553i	\(-0.705599\pi\)
							0.601924	−	0.798553i	\(-0.294401\pi\)
\(37\)	−2.08384	+	10.4762i	−0.0563199	+	0.283139i	−0.998674	−	0.0514777i	\(-0.983607\pi\)
							0.942354	+	0.334617i	\(0.108607\pi\)
\(41\)	5.40653	+	13.0525i	0.131867	+	0.318354i	0.975997	−	0.217784i	\(-0.0698827\pi\)
							−0.844130	+	0.536138i	\(0.819883\pi\)
\(43\)	26.8175	−	17.9189i	0.623662	−	0.416718i	−0.203188	−	0.979140i	\(-0.565130\pi\)
							0.826850	+	0.562422i	\(0.190130\pi\)
\(47\)	−14.6977	−	14.6977i	−0.312718	−	0.312718i	0.533244	−	0.845962i	\(-0.320973\pi\)
							−0.845962	+	0.533244i	\(0.820973\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.q.a.5.15	✓	496
3.2	odd	2	inner	192.3.q.a.5.48	yes	496
64.13	even	16	inner	192.3.q.a.77.48	yes	496
192.77	odd	16	inner	192.3.q.a.77.15	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.q.a.5.15	✓	496	1.1	even	1	trivial
192.3.q.a.5.48	yes	496	3.2	odd	2	inner
192.3.q.a.77.15	yes	496	192.77	odd	16	inner
192.3.q.a.77.48	yes	496	64.13	even	16	inner