Embedded newform 192.3.t.a.19.13

• Label: 192.3.t.a.19.13
• 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			19.13
Character	\(\chi\)	\(=\)	192.19
Dual form			192.3.t.a.91.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.760241 + 1.84987i) q^{2} +(-1.44015 + 0.962276i) q^{3} +(-2.84407 - 2.81270i) q^{4} +(-0.194779 - 0.979220i) q^{5} +(-0.685230 - 3.39565i) q^{6} +(6.99962 - 2.89934i) q^{7} +(7.36532 - 3.12284i) q^{8} +(1.14805 - 2.77164i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.760241	+	1.84987i	−0.380121	+	0.924937i
							\(3\)	−1.44015	+	0.962276i	−0.480049	+	0.320759i
\(5\)	−0.194779	−	0.979220i	−0.0389558	−	0.195844i								0.956407	−	0.292038i	\(-0.0943332\pi\)
							−0.995363	+	0.0961935i	\(0.969333\pi\)
\(7\)	6.99962	−	2.89934i	0.999946	−	0.414191i	0.178169	−	0.984000i	\(-0.442983\pi\)
							0.821777	+	0.569809i	\(0.192983\pi\)
\(11\)	0.896298	−	1.34141i	0.0814817	−	0.121946i	−0.788490	−	0.615047i	\(-0.789137\pi\)
							0.869972	+	0.493101i	\(0.164137\pi\)
\(13\)	0.378678	−	1.90374i	0.0291291	−	0.146442i	−0.963484	−	0.267767i	\(-0.913714\pi\)
							0.992613	+	0.121325i	\(0.0387143\pi\)
\(17\)	22.9178	+	22.9178i	1.34810	+	1.34810i	0.887719	+	0.460386i	\(0.152289\pi\)
							0.460386	+	0.887719i	\(0.347711\pi\)
\(19\)	3.73814	−	18.7929i	0.196744	−	0.989100i	−0.748599	−	0.663023i	\(-0.769273\pi\)
							0.945343	−	0.326077i	\(-0.105727\pi\)
\(23\)	−10.8425	+	26.1761i	−0.471413	+	1.13809i	0.492126	+	0.870524i	\(0.336220\pi\)
							−0.963539	+	0.267568i	\(0.913780\pi\)
\(29\)	17.4603	−	11.6666i	0.602078	−	0.402296i	−0.216839	−	0.976207i	\(-0.569575\pi\)
							0.818917	+	0.573911i	\(0.194575\pi\)
\(31\)	−20.8120			−0.671355			−0.335678	−	0.941977i	\(-0.608965\pi\)
							−0.335678	+	0.941977i	\(0.608965\pi\)
\(37\)	45.3735	−	9.02536i	1.22631	−	0.243929i	0.460890	−	0.887457i	\(-0.347530\pi\)
							0.765422	+	0.643529i	\(0.222530\pi\)
\(41\)	50.1086	+	20.7557i	1.22216	+	0.506236i	0.898096	−	0.439799i	\(-0.144950\pi\)
							0.324065	+	0.946035i	\(0.394950\pi\)
\(43\)	−9.24491	−	6.17725i	−0.214998	−	0.143657i	0.443408	−	0.896320i	\(-0.353769\pi\)
							−0.658406	+	0.752663i	\(0.728769\pi\)
\(47\)	54.4743	−	54.4743i	1.15903	−	1.15903i	0.174343	−	0.984685i	\(-0.444220\pi\)
							0.984685	−	0.174343i	\(-0.0557802\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.t.a.19.13	✓	256
64.27	odd	16	inner	192.3.t.a.91.13	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.t.a.19.13	✓	256	1.1	even	1	trivial
192.3.t.a.91.13	yes	256	64.27	odd	16	inner