Embedded newform 192.3.t.a.19.1

• Label: 192.3.t.a.19.1
• Level: $192$
• Weight: $3$
• Character: 192.19
• Analytic conductor: $5.232$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	192.t (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.23162107572\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(32\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			19.1
Character	\(\chi\)	\(=\)	192.19
Dual form			192.3.t.a.91.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99032 - 0.196561i) q^{2} +(1.44015 - 0.962276i) q^{3} +(3.92273 + 0.782437i) q^{4} +(-1.54570 - 7.77077i) q^{5} +(-3.05550 + 1.63216i) q^{6} +(-11.5402 + 4.78010i) q^{7} +(-7.65368 - 2.32835i) q^{8} +(1.14805 - 2.77164i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(133\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{7}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99032	−	0.196561i	−0.995159	−	0.0982804i
							\(3\)	1.44015	−	0.962276i	0.480049	−	0.320759i
\(5\)	−1.54570	−	7.77077i	−0.309141	−	1.55415i								−0.752977	−	0.658047i	\(-0.771383\pi\)
							0.443836	−	0.896108i	\(-0.353617\pi\)
\(7\)	−11.5402	+	4.78010i	−1.64860	+	0.682871i	−0.997123	−	0.0758007i	\(-0.975849\pi\)
							−0.651473	+	0.758672i	\(0.725849\pi\)
\(11\)	−6.85993	+	10.2666i	−0.623630	+	0.933329i	0.376346	+	0.926479i	\(0.377180\pi\)
							−0.999977	+	0.00684949i	\(0.997820\pi\)
\(13\)	0.0452983	−	0.227730i	0.00348449	−	0.0175177i	−0.979004	−	0.203841i	\(-0.934657\pi\)
							0.982488	+	0.186324i	\(0.0596573\pi\)
\(17\)	2.67644	+	2.67644i	0.157438	+	0.157438i	0.781430	−	0.623993i	\(-0.214490\pi\)
							−0.623993	+	0.781430i	\(0.714490\pi\)
\(19\)	−4.65885	+	23.4216i	−0.245203	+	1.23272i	0.640316	+	0.768112i	\(0.278804\pi\)
							−0.885518	+	0.464605i	\(0.846196\pi\)
\(23\)	9.60992	−	23.2004i	0.417823	−	1.00871i	−0.565154	−	0.824985i	\(-0.691183\pi\)
							0.982977	−	0.183728i	\(-0.0588166\pi\)
\(29\)	−13.2325	+	8.84165i	−0.456292	+	0.304884i	−0.762393	−	0.647115i	\(-0.775975\pi\)
							0.306101	+	0.951999i	\(0.400975\pi\)
\(31\)	−51.3325			−1.65589			−0.827943	−	0.560812i	\(-0.810489\pi\)
							−0.827943	+	0.560812i	\(0.810489\pi\)
\(37\)	16.1481	−	3.21205i	0.436434	−	0.0868121i	0.0280168	−	0.999607i	\(-0.491081\pi\)
							0.408417	+	0.912795i	\(0.366081\pi\)
\(41\)	−51.8490	−	21.4765i	−1.26461	−	0.523818i	−0.353288	−	0.935515i	\(-0.614936\pi\)
							−0.911321	+	0.411696i	\(0.864936\pi\)
\(43\)	−27.1505	−	18.1414i	−0.631406	−	0.421892i	0.198262	−	0.980149i	\(-0.436470\pi\)
							−0.829668	+	0.558257i	\(0.811470\pi\)
\(47\)	52.6279	−	52.6279i	1.11974	−	1.11974i	0.127964	−	0.991779i	\(-0.459156\pi\)
							0.991779	−	0.127964i	\(-0.0408443\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.t.a.19.1	✓	256
64.27	odd	16	inner	192.3.t.a.91.1	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.t.a.19.1	✓	256	1.1	even	1	trivial
192.3.t.a.91.1	yes	256	64.27	odd	16	inner