Embedded newform 192.3.q.a.5.13

• Label: 192.3.q.a.5.13
• 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			5.13
Character	\(\chi\)	\(=\)	192.5
Dual form			192.3.q.a.77.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.55855 - 1.25336i) q^{2} +(1.01740 + 2.82222i) q^{3} +(0.858181 + 3.90686i) q^{4} +(0.512850 - 2.57827i) q^{5} +(1.95158 - 5.67374i) q^{6} +(1.92886 - 4.65669i) q^{7} +(3.55917 - 7.16466i) q^{8} +(-6.92980 + 5.74263i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.55855	−	1.25336i	−0.779277	−	0.626680i
							\(3\)	1.01740	+	2.82222i	0.339133	+	0.940739i
\(5\)	0.512850	−	2.57827i	0.102570	−	0.515654i								−0.895005	−	0.446056i	\(-0.852828\pi\)
							0.997575	−	0.0695981i	\(-0.0221717\pi\)
\(7\)	1.92886	−	4.65669i	0.275552	−	0.665241i	−0.724150	−	0.689642i	\(-0.757768\pi\)
							0.999702	+	0.0244011i	\(0.00776788\pi\)
\(11\)	7.43373	−	4.96706i	0.675794	−	0.451551i	−0.169729	−	0.985491i	\(-0.554289\pi\)
							0.845522	+	0.533940i	\(0.179289\pi\)
\(13\)	11.4761	−	2.28274i	0.882778	−	0.175595i	0.267171	−	0.963649i	\(-0.413911\pi\)
							0.615607	+	0.788054i	\(0.288911\pi\)
\(17\)	18.9774	+	18.9774i	1.11632	+	1.11632i	0.992277	+	0.124041i	\(0.0395856\pi\)
							0.124041	+	0.992277i	\(0.460414\pi\)
\(19\)	2.42039	+	12.1681i	0.127389	+	0.640428i	0.990734	+	0.135814i	\(0.0433651\pi\)
							−0.863345	+	0.504614i	\(0.831635\pi\)
\(23\)	3.66553	+	8.84937i	0.159371	+	0.384755i	0.983314	−	0.181918i	\(-0.0582304\pi\)
							−0.823943	+	0.566673i	\(0.808230\pi\)
\(29\)	15.1546	+	10.1260i	0.522573	+	0.349172i	0.788712	−	0.614762i	\(-0.210748\pi\)
							−0.266139	+	0.963935i	\(0.585748\pi\)
\(31\)		−	51.6698i		−	1.66677i	−0.552695	−	0.833384i	\(-0.686400\pi\)
							0.552695	−	0.833384i	\(-0.313600\pi\)
\(37\)	4.44234	−	22.3331i	0.120063	−	0.603598i	−0.873166	−	0.487424i	\(-0.837937\pi\)
							0.993229	−	0.116175i	\(-0.0370632\pi\)
\(41\)	−18.0443	−	43.5629i	−0.440106	−	1.06251i	−0.975911	−	0.218168i	\(-0.929992\pi\)
							0.535805	−	0.844342i	\(-0.320008\pi\)
\(43\)	−41.8815	+	27.9843i	−0.973988	+	0.650798i	−0.937298	−	0.348529i	\(-0.886681\pi\)
							−0.0366901	+	0.999327i	\(0.511681\pi\)
\(47\)	33.3328	+	33.3328i	0.709209	+	0.709209i	0.966369	−	0.257160i	\(-0.0827866\pi\)
							−0.257160	+	0.966369i	\(0.582787\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.q.a.5.13	✓	496
3.2	odd	2	inner	192.3.q.a.5.50	yes	496
64.13	even	16	inner	192.3.q.a.77.50	yes	496
192.77	odd	16	inner	192.3.q.a.77.13	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.q.a.5.13	✓	496	1.1	even	1	trivial
192.3.q.a.5.50	yes	496	3.2	odd	2	inner
192.3.q.a.77.13	yes	496	192.77	odd	16	inner
192.3.q.a.77.50	yes	496	64.13	even	16	inner