Embedded newform 192.3.t.a.19.6

• Label: 192.3.t.a.19.6
• 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			19.6
Character	\(\chi\)	\(=\)	192.19
Dual form			192.3.t.a.91.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.75449 + 0.960092i) q^{2} +(-1.44015 + 0.962276i) q^{3} +(2.15645 - 3.36894i) q^{4} +(-0.982422 - 4.93897i) q^{5} +(1.60285 - 3.07097i) q^{6} +(2.14513 - 0.888543i) q^{7} +(-0.548972 + 7.98114i) q^{8} +(1.14805 - 2.77164i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.75449	+	0.960092i	−0.877243	+	0.480046i
							\(3\)	−1.44015	+	0.962276i	−0.480049	+	0.320759i
\(5\)	−0.982422	−	4.93897i	−0.196484	−	0.987794i								−0.945594	−	0.325348i	\(-0.894518\pi\)
							0.749110	−	0.662446i	\(-0.230482\pi\)
\(7\)	2.14513	−	0.888543i	0.306448	−	0.126935i	−0.224160	−	0.974552i	\(-0.571964\pi\)
							0.530608	+	0.847618i	\(0.321964\pi\)
\(11\)	−8.86512	+	13.2676i	−0.805920	+	1.20614i	0.169444	+	0.985540i	\(0.445803\pi\)
							−0.975363	+	0.220604i	\(0.929197\pi\)
\(13\)	−2.92209	+	14.6904i	−0.224776	+	1.13003i	0.689297	+	0.724479i	\(0.257919\pi\)
							−0.914074	+	0.405549i	\(0.867081\pi\)
\(17\)	−19.2565	−	19.2565i	−1.13274	−	1.13274i	−0.989721	−	0.143014i	\(-0.954320\pi\)
							−0.143014	−	0.989721i	\(-0.545680\pi\)
\(19\)	0.890946	−	4.47909i	0.0468919	−	0.235741i	−0.950231	−	0.311547i	\(-0.899153\pi\)
							0.997123	+	0.0758056i	\(0.0241529\pi\)
\(23\)	−8.49699	+	20.5136i	−0.369434	+	0.891894i	0.624409	+	0.781098i	\(0.285340\pi\)
							−0.993843	+	0.110796i	\(0.964660\pi\)
\(29\)	−31.3323	+	20.9356i	−1.08042	+	0.721916i	−0.962548	−	0.271112i	\(-0.912609\pi\)
							−0.117876	+	0.993028i	\(0.537609\pi\)
\(31\)	−32.8326			−1.05912			−0.529558	−	0.848274i	\(-0.677642\pi\)
							−0.529558	+	0.848274i	\(0.677642\pi\)
\(37\)	−23.8453	+	4.74312i	−0.644467	+	0.128193i	−0.506494	−	0.862243i	\(-0.669059\pi\)
							−0.137973	+	0.990436i	\(0.544059\pi\)
\(41\)	−57.6280	−	23.8703i	−1.40556	−	0.582203i	−0.454373	−	0.890812i	\(-0.650137\pi\)
							−0.951189	+	0.308609i	\(0.900137\pi\)
\(43\)	59.5001	+	39.7567i	1.38372	+	0.924575i	0.999999	+	0.00112690i	\(0.000358703\pi\)
							0.383724	+	0.923448i	\(0.374641\pi\)
\(47\)	−16.4416	+	16.4416i	−0.349822	+	0.349822i	−0.860043	−	0.510221i	\(-0.829564\pi\)
							0.510221	+	0.860043i	\(0.329564\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.t.a.19.6	✓	256
64.27	odd	16	inner	192.3.t.a.91.6	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.t.a.19.6	✓	256	1.1	even	1	trivial
192.3.t.a.91.6	yes	256	64.27	odd	16	inner