Embedded newform 722.2.k.a.139.6

• Label: 722.2.k.a.139.6
• Level: $722$
• Weight: $2$
• Character: 722.139
• Analytic conductor: $5.765$
• Analytic rank: $0$
• Dimension: $1620$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 722 = 2 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	722.k (of order \(171\), degree \(108\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.76519902594\)
Analytic rank:	\(0\)
Dimension:	\(1620\)
Relative dimension:	\(15\) over \(\Q(\zeta_{171})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{171}]$

Embedding invariants

Embedding label			139.6
Character	\(\chi\)	\(=\)	722.139
Dual form			722.2.k.a.187.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.991742 + 0.128249i) q^{2} +(-0.380847 + 0.560178i) q^{3} +(0.967104 + 0.254380i) q^{4} +(0.0402235 + 4.37868i) q^{5} +(-0.449544 + 0.506708i) q^{6} +(-3.89231 - 1.83682i) q^{7} +(0.926494 + 0.376309i) q^{8} +(0.934594 + 2.36305i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1620 q - 3 q^{7} - 45 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/722\mathbb{Z}\right)^\times\).

\(n\)	\(363\)
\(\chi(n)\)	\(e\left(\frac{7}{171}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.991742	+	0.128249i	0.701267	+	0.0906857i
							\(3\)	−0.380847	+	0.560178i	−0.219882	+	0.323419i	−0.920954	−	0.389672i	\(-0.872588\pi\)
0.701072	+	0.713091i	\(0.252705\pi\)
\(5\)	0.0402235	+	4.37868i	0.0179885	+	1.95821i	0.212152	+	0.977237i	\(0.431953\pi\)
							−0.194163	+	0.980969i	\(0.562199\pi\)
\(7\)	−3.89231	−	1.83682i	−1.47115	−	0.694254i	−0.487301	−	0.873234i	\(-0.662018\pi\)
							−0.983853	+	0.178980i	\(0.942720\pi\)
\(11\)	1.71740	+	1.05568i	0.517816	+	0.318300i	0.757811	−	0.652474i	\(-0.226269\pi\)
							−0.239996	+	0.970774i	\(0.577146\pi\)
\(13\)	−0.281085	−	2.77229i	−0.0779588	−	0.768895i	−0.956598	−	0.291410i	\(-0.905876\pi\)
							0.878640	−	0.477486i	\(-0.158452\pi\)
\(17\)	−2.57608	−	2.59985i	−0.624790	−	0.630556i	0.324153	−	0.946005i	\(-0.394921\pi\)
							−0.948943	+	0.315449i	\(0.897845\pi\)
\(19\)	−3.25868	+	2.89500i	−0.747592	+	0.664159i
							\(23\)	5.87833	−	0.432763i	1.22572	−	0.0902374i	0.554936	−	0.831893i	\(-0.312743\pi\)
0.670781	+	0.741656i	\(0.265959\pi\)
\(29\)	−5.12387	+	7.24657i	−0.951478	+	1.34565i	−0.0131482	+	0.999914i	\(0.504185\pi\)
							−0.938330	+	0.345741i	\(0.887628\pi\)
\(31\)	0.576764	+	1.84402i	0.103590	+	0.331195i	0.991724	−	0.128389i	\(-0.0409808\pi\)
							−0.888134	+	0.459585i	\(0.847998\pi\)
\(37\)	−0.437306	+	0.236658i	−0.0718926	+	0.0389063i	−0.511496	−	0.859286i	\(-0.670908\pi\)
							0.439603	+	0.898192i	\(0.355119\pi\)
\(41\)	0.395842	−	0.414453i	0.0618201	−	0.0647267i	−0.691523	−	0.722355i	\(-0.743060\pi\)
							0.753343	+	0.657628i	\(0.228440\pi\)
\(43\)	4.79987	+	2.05305i	0.731973	+	0.313087i	0.726753	−	0.686899i	\(-0.241029\pi\)
							0.00521990	+	0.999986i	\(0.498338\pi\)
\(47\)	9.27975	+	1.90131i	1.35359	+	0.277334i	0.821913	−	0.569613i	\(-0.192907\pi\)
							0.531677	+	0.846947i	\(0.321562\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	722.2.k.a.139.6	✓	1620
361.187	even	171	inner	722.2.k.a.187.6	yes	1620

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
722.2.k.a.139.6	✓	1620	1.1	even	1	trivial
722.2.k.a.187.6	yes	1620	361.187	even	171	inner