Embedded newform 192.3.q.a.5.2

• Label: 192.3.q.a.5.2
• 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			5.2
Character	\(\chi\)	\(=\)	192.5
Dual form			192.3.q.a.77.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99227 - 0.175619i) q^{2} +(-1.83061 - 2.37674i) q^{3} +(3.93832 + 0.699764i) q^{4} +(0.541397 - 2.72179i) q^{5} +(3.22967 + 5.05660i) q^{6} +(-4.07794 + 9.84503i) q^{7} +(-7.72331 - 2.08577i) q^{8} +(-2.29776 + 8.70174i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99227	−	0.175619i	−0.996137	−	0.0878096i
							\(3\)	−1.83061	−	2.37674i	−0.610202	−	0.792246i
\(5\)	0.541397	−	2.72179i	0.108279	−	0.544357i								−0.888123	−	0.459606i	\(-0.847991\pi\)
							0.996402	−	0.0847509i	\(-0.0270094\pi\)
\(7\)	−4.07794	+	9.84503i	−0.582563	+	1.40643i	0.307918	+	0.951413i	\(0.400368\pi\)
							−0.890482	+	0.455019i	\(0.849632\pi\)
\(11\)	1.95081	−	1.30349i	0.177346	−	0.118499i	−0.463729	−	0.885977i	\(-0.653489\pi\)
							0.641075	+	0.767478i	\(0.278489\pi\)
\(13\)	5.01019	−	0.996589i	0.385399	−	0.0766607i	0.00141188	−	0.999999i	\(-0.499551\pi\)
							0.383987	+	0.923338i	\(0.374551\pi\)
\(17\)	−6.89047	−	6.89047i	−0.405321	−	0.405321i	0.474782	−	0.880103i	\(-0.342527\pi\)
							−0.880103	+	0.474782i	\(0.842527\pi\)
\(19\)	3.83145	+	19.2620i	0.201655	+	1.01379i	0.940469	+	0.339880i	\(0.110386\pi\)
							−0.738814	+	0.673910i	\(0.764614\pi\)
\(23\)	8.14125	+	19.6547i	0.353967	+	0.854553i	0.996122	+	0.0879782i	\(0.0280406\pi\)
							−0.642155	+	0.766575i	\(0.721959\pi\)
\(29\)	40.7659	+	27.2389i	1.40572	+	0.939272i	0.999677	+	0.0254176i	\(0.00809153\pi\)
							0.406043	+	0.913854i	\(0.366908\pi\)
\(31\)			23.9040i			0.771095i	0.922688	+	0.385548i	\(0.125987\pi\)
							−0.922688	+	0.385548i	\(0.874013\pi\)
\(37\)	10.9659	−	55.1291i	0.296375	−	1.48998i	−0.489726	−	0.871877i	\(-0.662903\pi\)
							0.786100	−	0.618099i	\(-0.212097\pi\)
\(41\)	−11.0037	−	26.5652i	−0.268383	−	0.647933i	0.731025	−	0.682351i	\(-0.239042\pi\)
							−0.999408	+	0.0344179i	\(0.989042\pi\)
\(43\)	−8.95595	+	5.98417i	−0.208278	+	0.139167i	−0.655334	−	0.755339i	\(-0.727472\pi\)
							0.447057	+	0.894506i	\(0.352472\pi\)
\(47\)	30.0579	+	30.0579i	0.639529	+	0.639529i	0.950439	−	0.310910i	\(-0.100634\pi\)
							−0.310910	+	0.950439i	\(0.600634\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.q.a.5.2	✓	496
3.2	odd	2	inner	192.3.q.a.5.61	yes	496
64.13	even	16	inner	192.3.q.a.77.61	yes	496
192.77	odd	16	inner	192.3.q.a.77.2	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.q.a.5.2	✓	496	1.1	even	1	trivial
192.3.q.a.5.61	yes	496	3.2	odd	2	inner
192.3.q.a.77.2	yes	496	192.77	odd	16	inner
192.3.q.a.77.61	yes	496	64.13	even	16	inner