Embedded newform 192.3.q.a.5.4

• Label: 192.3.q.a.5.4
• 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			5.4
Character	\(\chi\)	\(=\)	192.5
Dual form			192.3.q.a.77.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.94960 - 0.446149i) q^{2} +(-2.81493 + 1.03739i) q^{3} +(3.60190 + 1.73963i) q^{4} +(-0.199626 + 1.00359i) q^{5} +(5.95082 - 0.766618i) q^{6} +(2.15070 - 5.19226i) q^{7} +(-6.24615 - 4.99856i) q^{8} +(6.84765 - 5.84035i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.94960	−	0.446149i	−0.974801	−	0.223074i
							\(3\)	−2.81493	+	1.03739i	−0.938310	+	0.345796i
\(5\)	−0.199626	+	1.00359i	−0.0399251	+	0.200717i								−0.995600	−	0.0937078i	\(-0.970128\pi\)
							0.955675	+	0.294425i	\(0.0951281\pi\)
\(7\)	2.15070	−	5.19226i	0.307243	−	0.741751i	−0.692549	−	0.721371i	\(-0.743512\pi\)
							0.999792	−	0.0203801i	\(-0.00648765\pi\)
\(11\)	−7.54518	+	5.04153i	−0.685925	+	0.458321i	−0.849069	−	0.528282i	\(-0.822837\pi\)
							0.163144	+	0.986602i	\(0.447837\pi\)
\(13\)	0.537697	−	0.106955i	0.0413613	−	0.00822727i	−0.174366	−	0.984681i	\(-0.555788\pi\)
							0.215728	+	0.976454i	\(0.430788\pi\)
\(17\)	10.5394	+	10.5394i	0.619965	+	0.619965i	0.945522	−	0.325557i	\(-0.105552\pi\)
							−0.325557	+	0.945522i	\(0.605552\pi\)
\(19\)	3.02700	+	15.2178i	0.159316	+	0.800934i	0.974960	+	0.222381i	\(0.0713827\pi\)
							−0.815644	+	0.578554i	\(0.803617\pi\)
\(23\)	1.39978	+	3.37936i	0.0608599	+	0.146929i	0.951384	−	0.308007i	\(-0.0996621\pi\)
							−0.890524	+	0.454936i	\(0.849662\pi\)
\(29\)	19.0631	+	12.7376i	0.657349	+	0.439227i	0.838996	−	0.544137i	\(-0.183143\pi\)
							−0.181647	+	0.983364i	\(0.558143\pi\)
\(31\)			26.0950i			0.841773i	0.907113	+	0.420886i	\(0.138281\pi\)
							−0.907113	+	0.420886i	\(0.861719\pi\)
\(37\)	−9.73470	+	48.9397i	−0.263100	+	1.32269i	0.592715	+	0.805412i	\(0.298056\pi\)
							−0.855815	+	0.517282i	\(0.826944\pi\)
\(41\)	14.6731	+	35.4240i	0.357881	+	0.864001i	0.995598	+	0.0937301i	\(0.0298791\pi\)
							−0.637717	+	0.770271i	\(0.720121\pi\)
\(43\)	−26.8942	+	17.9701i	−0.625446	+	0.417910i	−0.827501	−	0.561464i	\(-0.810238\pi\)
							0.202055	+	0.979374i	\(0.435238\pi\)
\(47\)	−54.1001	−	54.1001i	−1.15107	−	1.15107i	−0.986339	−	0.164727i	\(-0.947326\pi\)
							−0.164727	−	0.986339i	\(-0.552674\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.q.a.5.4	✓	496
3.2	odd	2	inner	192.3.q.a.5.59	yes	496
64.13	even	16	inner	192.3.q.a.77.59	yes	496
192.77	odd	16	inner	192.3.q.a.77.4	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.q.a.5.4	✓	496	1.1	even	1	trivial
192.3.q.a.5.59	yes	496	3.2	odd	2	inner
192.3.q.a.77.4	yes	496	192.77	odd	16	inner
192.3.q.a.77.59	yes	496	64.13	even	16	inner