Embedded newform 192.3.t.a.19.20

• Label: 192.3.t.a.19.20
• 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			19.20
Character	\(\chi\)	\(=\)	192.19
Dual form			192.3.t.a.91.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.679497 - 1.88103i) q^{2} +(-1.44015 + 0.962276i) q^{3} +(-3.07657 - 2.55631i) q^{4} +(0.892221 + 4.48550i) q^{5} +(0.831497 + 3.36283i) q^{6} +(-2.40197 + 0.994929i) q^{7} +(-6.89902 + 4.05012i) q^{8} +(1.14805 - 2.77164i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.679497	−	1.88103i	0.339748	−	0.940516i
							\(3\)	−1.44015	+	0.962276i	−0.480049	+	0.320759i
\(5\)	0.892221	+	4.48550i	0.178444	+	0.897100i								0.961428	+	0.275057i	\(0.0886969\pi\)
							−0.782984	+	0.622042i	\(0.786303\pi\)
\(7\)	−2.40197	+	0.994929i	−0.343139	+	0.142133i	−0.547596	−	0.836743i	\(-0.684457\pi\)
							0.204457	+	0.978875i	\(0.434457\pi\)
\(11\)	−6.75055	+	10.1029i	−0.613687	+	0.918447i	−0.999992	−	0.00392368i	\(-0.998751\pi\)
							0.386305	+	0.922371i	\(0.373751\pi\)
\(13\)	−2.52962	+	12.7172i	−0.194586	+	0.978249i	0.752823	+	0.658223i	\(0.228692\pi\)
							−0.947409	+	0.320026i	\(0.896308\pi\)
\(17\)	8.84539	+	8.84539i	0.520317	+	0.520317i	0.917667	−	0.397350i	\(-0.130070\pi\)
							−0.397350	+	0.917667i	\(0.630070\pi\)
\(19\)	−1.65803	+	8.33547i	−0.0872646	+	0.438709i	0.912309	+	0.409503i	\(0.134298\pi\)
							−0.999573	+	0.0292058i	\(0.990702\pi\)
\(23\)	3.43704	−	8.29776i	0.149437	−	0.360772i	−0.831380	−	0.555704i	\(-0.812449\pi\)
							0.980817	+	0.194932i	\(0.0624487\pi\)
\(29\)	8.27977	−	5.53237i	0.285509	−	0.190771i	−0.404566	−	0.914509i	\(-0.632577\pi\)
							0.690075	+	0.723738i	\(0.257577\pi\)
\(31\)	−46.4392			−1.49804			−0.749019	−	0.662548i	\(-0.769475\pi\)
							−0.749019	+	0.662548i	\(0.769475\pi\)
\(37\)	−27.9892	+	5.56739i	−0.756464	+	0.150470i	−0.558232	−	0.829685i	\(-0.688520\pi\)
							−0.198232	+	0.980155i	\(0.563520\pi\)
\(41\)	6.11833	+	2.53429i	0.149227	+	0.0618120i	0.456047	−	0.889956i	\(-0.349265\pi\)
							−0.306820	+	0.951768i	\(0.599265\pi\)
\(43\)	−44.5743	−	29.7836i	−1.03661	−	0.692642i	−0.0838868	−	0.996475i	\(-0.526733\pi\)
							−0.952725	+	0.303833i	\(0.901733\pi\)
\(47\)	0.593669	−	0.593669i	0.0126312	−	0.0126312i	−0.700763	−	0.713394i	\(-0.747157\pi\)
							0.713394	+	0.700763i	\(0.247157\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.t.a.19.20	✓	256
64.27	odd	16	inner	192.3.t.a.91.20	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.t.a.19.20	✓	256	1.1	even	1	trivial
192.3.t.a.91.20	yes	256	64.27	odd	16	inner