Embedded newform 192.3.t.a.91.9

• Label: 192.3.t.a.91.9
• Level: $192$
• Weight: $3$
• Character: 192.91
• 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			91.9
Character	\(\chi\)	\(=\)	192.91
Dual form			192.3.t.a.19.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27767 - 1.53868i) q^{2} +(-1.44015 - 0.962276i) q^{3} +(-0.735101 + 3.93187i) q^{4} +(0.251501 - 1.26438i) q^{5} +(0.359399 + 3.44541i) q^{6} +(-9.71768 - 4.02520i) q^{7} +(6.98913 - 3.89256i) q^{8} +(1.14805 + 2.77164i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(133\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{9}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27767	−	1.53868i	−0.638837	−	0.769342i
							\(3\)	−1.44015	−	0.962276i	−0.480049	−	0.320759i
\(5\)	0.251501	−	1.26438i	0.0503003	−	0.252877i								−0.947451	−	0.319901i	\(-0.896350\pi\)
							0.997751	+	0.0670241i	\(0.0213504\pi\)
\(7\)	−9.71768	−	4.02520i	−1.38824	−	0.575028i	−0.441569	−	0.897228i	\(-0.645578\pi\)
							−0.946672	+	0.322200i	\(0.895578\pi\)
\(11\)	3.29494	+	4.93123i	0.299540	+	0.448294i	0.950457	−	0.310855i	\(-0.100615\pi\)
							−0.650917	+	0.759149i	\(0.725615\pi\)
\(13\)	3.70442	+	18.6234i	0.284955	+	1.43257i	0.812461	+	0.583016i	\(0.198128\pi\)
							−0.527505	+	0.849552i	\(0.676872\pi\)
\(17\)	−10.9269	+	10.9269i	−0.642757	+	0.642757i	−0.951232	−	0.308475i	\(-0.900181\pi\)
							0.308475	+	0.951232i	\(0.400181\pi\)
\(19\)	−3.80562	−	19.1321i	−0.200296	−	1.00695i	−0.941843	−	0.336054i	\(-0.890908\pi\)
							0.741547	−	0.670901i	\(-0.234092\pi\)
\(23\)	16.8958	+	40.7901i	0.734600	+	1.77348i	0.626614	+	0.779330i	\(0.284440\pi\)
							0.107987	+	0.994152i	\(0.465560\pi\)
\(29\)	30.6210	+	20.4603i	1.05590	+	0.705527i	0.957152	−	0.289586i	\(-0.0935176\pi\)
							0.0987438	+	0.995113i	\(0.468518\pi\)
\(31\)	0.0562603			0.00181485			0.000907424	−	1.00000i	\(-0.499711\pi\)
							0.000907424		1.00000i	\(0.499711\pi\)
\(37\)	−63.9224	−	12.7150i	−1.72763	−	0.343648i	−0.771422	−	0.636324i	\(-0.780454\pi\)
							−0.956211	+	0.292677i	\(0.905454\pi\)
\(41\)	21.9614	−	9.09672i	0.535645	−	0.221871i	−0.0984288	−	0.995144i	\(-0.531382\pi\)
							0.634073	+	0.773273i	\(0.281382\pi\)
\(43\)	−47.3088	+	31.6108i	−1.10021	+	0.735134i	−0.966700	−	0.255914i	\(-0.917624\pi\)
							−0.133506	+	0.991048i	\(0.542624\pi\)
\(47\)	−30.0083	−	30.0083i	−0.638475	−	0.638475i	0.311704	−	0.950179i	\(-0.399100\pi\)
							−0.950179	+	0.311704i	\(0.899100\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.t.a.91.9	yes	256
64.19	odd	16	inner	192.3.t.a.19.9	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.t.a.19.9	✓	256	64.19	odd	16	inner
192.3.t.a.91.9	yes	256	1.1	even	1	trivial