Embedded newform 441.2.z.a.142.49

• Label: 441.2.z.a.142.49
• Level: $441$
• Weight: $2$
• Character: 441.142
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.z (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			142.49
Character	\(\chi\)	\(=\)	441.142
Dual form			441.2.z.a.382.49

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.837831 - 2.13476i) q^{2} +(-0.00927680 + 1.73203i) q^{3} +(-2.38913 - 2.21679i) q^{4} +(-0.0993933 + 0.0478653i) q^{5} +(3.68969 + 1.47095i) q^{6} +(2.41306 - 1.08495i) q^{7} +(-2.60165 + 1.25289i) q^{8} +(-2.99983 - 0.0321353i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.837831	−	2.13476i	0.592436	−	1.50950i	−0.248645	−	0.968595i	\(-0.579985\pi\)
							0.841082	−	0.540908i	\(-0.181919\pi\)
\(3\)	−0.00927680	+	1.73203i	−0.00535596	+	0.999986i
							\(5\)	−0.0993933	+	0.0478653i	−0.0444501	+	0.0214060i	−0.455977	−	0.889992i	\(-0.650710\pi\)
0.411527	+	0.911398i	\(0.364996\pi\)
\(7\)	2.41306	−	1.08495i	0.912053	−	0.410073i
							\(11\)	1.61037	+	2.01934i	0.485545	+	0.608854i	0.962901	−	0.269856i	\(-0.0869761\pi\)
−0.477356	+	0.878710i	\(0.658405\pi\)
\(13\)	1.40412	−	3.57765i	0.389434	−	0.992260i	−0.592926	−	0.805257i	\(-0.702027\pi\)
							0.982359	−	0.187003i	\(-0.0598775\pi\)
\(17\)	5.78712	−	5.36966i	1.40358	−	1.30233i	0.505280	−	0.862956i	\(-0.331389\pi\)
							0.898302	−	0.439378i	\(-0.144801\pi\)
\(19\)	0.178675	−	0.309474i	0.0409908	−	0.0709982i	−0.844802	−	0.535079i	\(-0.820282\pi\)
							0.885793	+	0.464081i	\(0.153615\pi\)
\(23\)	−0.551333	+	2.41555i	−0.114961	+	0.503677i	0.884359	+	0.466807i	\(0.154596\pi\)
							−0.999320	+	0.0368695i	\(0.988261\pi\)
\(29\)	9.38459	+	2.89476i	1.74267	+	0.537544i	0.992342	−	0.123517i	\(-0.0394175\pi\)
							0.750332	+	0.661061i	\(0.229894\pi\)
\(31\)	−2.98663	+	5.17299i	−0.536414	+	0.929096i	0.462680	+	0.886526i	\(0.346888\pi\)
							−0.999093	+	0.0425704i	\(0.986445\pi\)
\(37\)	−8.76752	−	2.70442i	−1.44137	−	0.444604i	−0.527133	−	0.849783i	\(-0.676733\pi\)
							−0.914238	+	0.405179i	\(0.867209\pi\)
\(41\)	−0.0868448	−	0.0592098i	−0.0135629	−	0.00924701i	0.556520	−	0.830834i	\(-0.312136\pi\)
							−0.570083	+	0.821587i	\(0.693089\pi\)
\(43\)	−10.3103	+	7.02943i	−1.57230	+	1.07198i	−0.612817	+	0.790224i	\(0.709964\pi\)
							−0.959487	+	0.281754i	\(0.909084\pi\)
\(47\)	1.47379	−	3.75515i	0.214974	−	0.547745i	−0.782187	−	0.623044i	\(-0.785896\pi\)
							0.997161	+	0.0752991i	\(0.0239911\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.142.49	yes	648
9.4	even	3		441.2.y.a.436.49	yes	648
49.39	even	21		441.2.y.a.88.49	✓	648
441.382	even	21	inner	441.2.z.a.382.49	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.88.49	✓	648	49.39	even	21
441.2.y.a.436.49	yes	648	9.4	even	3
441.2.z.a.142.49	yes	648	1.1	even	1	trivial
441.2.z.a.382.49	yes	648	441.382	even	21	inner