Embedded newform 192.3.q.a.5.12

• Label: 192.3.q.a.5.12
• 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			5.12
Character	\(\chi\)	\(=\)	192.5
Dual form			192.3.q.a.77.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.73173 - 1.00056i) q^{2} +(2.76570 + 1.16228i) q^{3} +(1.99775 + 3.46540i) q^{4} +(-0.965897 + 4.85589i) q^{5} +(-3.62650 - 4.78001i) q^{6} +(-1.32825 + 3.20667i) q^{7} +(0.00779347 - 8.00000i) q^{8} +(6.29820 + 6.42905i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.73173	−	1.00056i	−0.865863	−	0.500281i
							\(3\)	2.76570	+	1.16228i	0.921900	+	0.387427i
\(5\)	−0.965897	+	4.85589i	−0.193179	+	0.971179i								0.755550	+	0.655091i	\(0.227370\pi\)
							−0.948730	+	0.316088i	\(0.897630\pi\)
\(7\)	−1.32825	+	3.20667i	−0.189750	+	0.458096i	−0.989911	−	0.141688i	\(-0.954747\pi\)
							0.800162	+	0.599784i	\(0.204747\pi\)
\(11\)	−10.4577	+	6.98760i	−0.950699	+	0.635236i	−0.931175	−	0.364573i	\(-0.881215\pi\)
							−0.0195237	+	0.999809i	\(0.506215\pi\)
\(13\)	−11.2195	+	2.23171i	−0.863042	+	0.171670i	−0.606714	−	0.794920i	\(-0.707513\pi\)
							−0.256328	+	0.966590i	\(0.582513\pi\)
\(17\)	−9.17851	−	9.17851i	−0.539912	−	0.539912i	0.383591	−	0.923503i	\(-0.374687\pi\)
							−0.923503	+	0.383591i	\(0.874687\pi\)
\(19\)	−4.95827	−	24.9269i	−0.260962	−	1.31194i	−0.859618	−	0.510937i	\(-0.829299\pi\)
							0.598657	−	0.801006i	\(-0.295701\pi\)
\(23\)	11.6127	+	28.0357i	0.504902	+	1.21894i	0.946784	+	0.321868i	\(0.104311\pi\)
							−0.441882	+	0.897073i	\(0.645689\pi\)
\(29\)	26.0679	+	17.4180i	0.898894	+	0.600621i	0.916856	−	0.399217i	\(-0.130718\pi\)
							−0.0179629	+	0.999839i	\(0.505718\pi\)
\(31\)			18.2569i			0.588931i	0.955662	+	0.294466i	\(0.0951416\pi\)
							−0.955662	+	0.294466i	\(0.904858\pi\)
\(37\)	0.545909	−	2.74447i	0.0147543	−	0.0741748i	−0.972707	−	0.232038i	\(-0.925461\pi\)
							0.987461	+	0.157863i	\(0.0504606\pi\)
\(41\)	18.5098	+	44.6866i	0.451458	+	1.08992i	0.971768	+	0.235939i	\(0.0758165\pi\)
							−0.520310	+	0.853978i	\(0.674184\pi\)
\(43\)	62.2076	−	41.5658i	1.44669	−	0.966646i	0.449378	−	0.893342i	\(-0.351646\pi\)
							0.997310	−	0.0733038i	\(-0.0233543\pi\)
\(47\)	−0.0616761	−	0.0616761i	−0.00131226	−	0.00131226i	0.706450	−	0.707763i	\(-0.250295\pi\)
							−0.707763	+	0.706450i	\(0.750295\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.q.a.5.12	✓	496
3.2	odd	2	inner	192.3.q.a.5.51	yes	496
64.13	even	16	inner	192.3.q.a.77.51	yes	496
192.77	odd	16	inner	192.3.q.a.77.12	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.q.a.5.12	✓	496	1.1	even	1	trivial
192.3.q.a.5.51	yes	496	3.2	odd	2	inner
192.3.q.a.77.12	yes	496	192.77	odd	16	inner
192.3.q.a.77.51	yes	496	64.13	even	16	inner