Embedded newform 722.2.k.b.139.11

• Label: 722.2.k.b.139.11
• Level: $722$
• Weight: $2$
• Character: 722.139
• Analytic conductor: $5.765$
• Analytic rank: $0$
• Dimension: $1728$
• 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:	\(1728\)
Relative dimension:	\(16\) over \(\Q(\zeta_{171})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{171}]$

Embedding invariants

Embedding label			139.11
Character	\(\chi\)	\(=\)	722.139
Dual form			722.2.k.b.187.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.991742 - 0.128249i) q^{2} +(0.859405 - 1.26408i) q^{3} +(0.967104 + 0.254380i) q^{4} +(0.0119159 + 1.29715i) q^{5} +(-1.01442 + 1.14342i) q^{6} +(-1.19276 - 0.562876i) q^{7} +(-0.926494 - 0.376309i) q^{8} +(0.244037 + 0.617030i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1728 q + 3 q^{3} + 3 q^{6} + 3 q^{7} + 48 q^{8} + 3 q^{9}+O(q^{10}) \)

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
							\(3\)	0.859405	−	1.26408i	0.496178	−	0.729815i	−0.494026	−	0.869447i	\(-0.664475\pi\)
0.990203	+	0.139633i	\(0.0445922\pi\)
\(5\)	0.0119159	+	1.29715i	0.00532894	+	0.580103i	0.959631	+	0.281260i	\(0.0907525\pi\)
							−0.954303	+	0.298842i	\(0.903400\pi\)
\(7\)	−1.19276	−	0.562876i	−0.450820	−	0.212747i	0.187897	−	0.982189i	\(-0.439833\pi\)
							−0.638717	+	0.769441i	\(0.720535\pi\)
\(11\)	0.00993440	+	0.00610665i	0.00299533	+	0.00184122i	0.525170	−	0.850997i	\(-0.324002\pi\)
							−0.522174	+	0.852839i	\(0.674879\pi\)
\(13\)	0.289152	+	2.85186i	0.0801962	+	0.790962i	0.953016	+	0.302920i	\(0.0979615\pi\)
							−0.872820	+	0.488043i	\(0.837711\pi\)
\(17\)	2.50412	+	2.52723i	0.607337	+	0.612942i	0.944503	−	0.328503i	\(-0.106544\pi\)
							−0.337165	+	0.941445i	\(0.609468\pi\)
\(19\)	2.05634	−	3.84337i	0.471757	−	0.881728i
							\(23\)	5.06103	−	0.372593i	1.05530	−	0.0776911i	0.465345	−	0.885129i	\(-0.345930\pi\)
0.589952	+	0.807438i	\(0.299147\pi\)
\(29\)	−2.63424	+	3.72554i	−0.489165	+	0.691816i	−0.984201	−	0.177057i	\(-0.943342\pi\)
							0.495035	+	0.868873i	\(0.335155\pi\)
\(31\)	0.797394	+	2.54941i	0.143216	+	0.457887i	0.998147	−	0.0608551i	\(-0.0193828\pi\)
							−0.854930	+	0.518743i	\(0.826400\pi\)
\(37\)	−0.539290	+	0.291849i	−0.0886588	+	0.0479797i	−0.519672	−	0.854366i	\(-0.673946\pi\)
							0.431013	+	0.902346i	\(0.358156\pi\)
\(41\)	−0.288030	+	0.301572i	−0.0449827	+	0.0470977i	−0.745265	−	0.666768i	\(-0.767677\pi\)
							0.700283	+	0.713866i	\(0.253057\pi\)
\(43\)	0.403737	+	0.172691i	0.0615694	+	0.0263351i	0.423831	−	0.905742i	\(-0.360685\pi\)
							−0.362261	+	0.932077i	\(0.617995\pi\)
\(47\)	2.13538	+	0.437513i	0.311477	+	0.0638178i	0.353904	−	0.935282i	\(-0.384854\pi\)
							−0.0424269	+	0.999100i	\(0.513509\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	722.2.k.b.139.11	✓	1728
361.187	even	171	inner	722.2.k.b.187.11	yes	1728

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
722.2.k.b.139.11	✓	1728	1.1	even	1	trivial
722.2.k.b.187.11	yes	1728	361.187	even	171	inner