Embedded newform 722.2.k.b.5.16

• Label: 722.2.k.b.5.16
• Level: $722$
• Weight: $2$
• Character: 722.5
• 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			5.16
Character	\(\chi\)	\(=\)	722.5
Dual form			722.2.k.b.289.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.531476 + 0.847073i) q^{2} +(1.96711 + 2.57582i) q^{3} +(-0.435066 - 0.900399i) q^{4} +(-0.113996 - 0.288231i) q^{5} +(-3.22738 + 0.297302i) q^{6} +(0.284348 - 0.276618i) q^{7} +(0.993931 + 0.110008i) q^{8} +(-1.97554 + 7.23965i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1728 q + 3 q^{3} + 3 q^{6} + 3 q^{7} + 48 q^{8} + 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(363\)
\(\chi(n)\)	\(e\left(\frac{116}{171}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.531476	+	0.847073i	−0.375811	+	0.598971i
							\(3\)	1.96711	+	2.57582i	1.13571	+	1.48715i	0.848434	+	0.529301i	\(0.177546\pi\)
0.287278	+	0.957847i	\(0.407250\pi\)
\(5\)	−0.113996	−	0.288231i	−0.0509807	−	0.128901i	0.902185	−	0.431349i	\(-0.141962\pi\)
							−0.953166	+	0.302448i	\(0.902196\pi\)
\(7\)	0.284348	−	0.276618i	0.107473	−	0.104552i	−0.641598	−	0.767041i	\(-0.721728\pi\)
							0.749071	+	0.662489i	\(0.230500\pi\)
\(11\)	4.45947	+	3.09213i	1.34458	+	0.932312i	0.999979	−	0.00643636i	\(-0.00204877\pi\)
							0.344603	+	0.938749i	\(0.388014\pi\)
\(13\)	3.93398	+	2.51894i	1.09109	+	0.698628i	0.956341	−	0.292252i	\(-0.0944046\pi\)
							0.134747	+	0.990880i	\(0.456978\pi\)
\(17\)	−2.84741	−	4.18818i	−0.690599	−	1.01578i	−0.997938	−	0.0641839i	\(-0.979556\pi\)
							0.307339	−	0.951600i	\(-0.400561\pi\)
\(19\)	−2.03556	−	3.85441i	−0.466990	−	0.884263i
							\(23\)	−1.87570	−	0.242559i	−0.391110	−	0.0505771i	−0.0698238	−	0.997559i	\(-0.522244\pi\)
−0.321286	+	0.946982i	\(0.604115\pi\)
\(29\)	−0.531112	+	5.23827i	−0.0986251	+	0.972723i	0.818603	+	0.574360i	\(0.194749\pi\)
							−0.917228	+	0.398363i	\(0.869578\pi\)
\(31\)	−1.32227	+	9.53555i	−0.237487	+	1.71264i	0.379105	+	0.925353i	\(0.376232\pi\)
							−0.616593	+	0.787282i	\(0.711488\pi\)
\(37\)	0.755420	−	9.11656i	0.124190	−	1.49875i	−0.594879	−	0.803815i	\(-0.702800\pi\)
							0.719069	−	0.694938i	\(-0.244568\pi\)
\(41\)	−1.34511	−	8.54345i	−0.210070	−	1.33426i	−0.833559	−	0.552430i	\(-0.813701\pi\)
							0.623489	−	0.781832i	\(-0.285715\pi\)
\(43\)	−5.11475	−	5.98303i	−0.779991	−	0.912404i	0.217987	−	0.975952i	\(-0.430051\pi\)
							−0.997979	+	0.0635478i	\(0.979758\pi\)
\(47\)	−2.78757	+	6.05037i	−0.406609	+	0.882537i	0.590535	+	0.807012i	\(0.298917\pi\)
							−0.997144	+	0.0755244i	\(0.975937\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	722.2.k.b.5.16	✓	1728
361.289	even	171	inner	722.2.k.b.289.16	yes	1728

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
722.2.k.b.5.16	✓	1728	1.1	even	1	trivial
722.2.k.b.289.16	yes	1728	361.289	even	171	inner