Embedded newform 441.2.y.a.151.46

• Label: 441.2.y.a.151.46
• Level: $441$
• Weight: $2$
• Character: 441.151
• 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.y (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			151.46
Character	\(\chi\)	\(=\)	441.151
Dual form			441.2.y.a.184.46

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.95416 - 0.941073i) q^{2} +(-1.63364 + 0.575513i) q^{3} +(1.68614 - 2.11435i) q^{4} +(1.30801 + 1.21366i) q^{5} +(-2.65079 + 2.66202i) q^{6} +(1.60591 - 2.10263i) q^{7} +(0.339946 - 1.48940i) q^{8} +(2.33757 - 1.88036i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.95416	−	0.941073i	1.38180	−	0.665439i	0.412416	−	0.910996i	\(-0.364685\pi\)
							0.969382	+	0.245557i	\(0.0789707\pi\)
\(3\)	−1.63364	+	0.575513i	−0.943183	+	0.332273i
							\(5\)	1.30801	+	1.21366i	0.584960	+	0.542764i	0.916168	−	0.400794i	\(-0.131266\pi\)
−0.331208	+	0.943558i	\(0.607456\pi\)
\(7\)	1.60591	−	2.10263i	0.606978	−	0.794718i
							\(11\)	−3.10858	−	2.11939i	−0.937271	−	0.639021i	−0.00469299	−	0.999989i	\(-0.501494\pi\)
−0.932578	+	0.360968i	\(0.882446\pi\)
\(13\)	4.16752	+	2.84136i	1.15586	+	0.788053i	0.980513	−	0.196454i	\(-0.0629426\pi\)
							0.175348	+	0.984507i	\(0.443895\pi\)
\(17\)	6.99567	−	1.05443i	1.69670	−	0.255736i	0.771777	−	0.635893i	\(-0.219368\pi\)
							0.924922	+	0.380156i	\(0.124130\pi\)
\(19\)	0.0857190	+	0.148470i	0.0196653	+	0.0340613i	0.875691	−	0.482873i	\(-0.160407\pi\)
							−0.856025	+	0.516934i	\(0.827073\pi\)
\(23\)	−1.23761	−	3.15339i	−0.258060	−	0.657527i	0.741855	−	0.670561i	\(-0.233946\pi\)
							−0.999915	+	0.0130338i	\(0.995851\pi\)
\(29\)	−5.76436	+	0.868838i	−1.07042	+	0.161339i	−0.660531	−	0.750799i	\(-0.729669\pi\)
							−0.409884	+	0.912138i	\(0.634431\pi\)
\(31\)	−7.76067			−1.39386			−0.696929	−	0.717141i	\(-0.745451\pi\)
							−0.696929	+	0.717141i	\(0.745451\pi\)
\(37\)	−1.41656	+	3.60935i	−0.232882	+	0.593373i	−0.998769	−	0.0496018i	\(-0.984205\pi\)
							0.765887	+	0.642975i	\(0.222300\pi\)
\(41\)	−5.38763	+	1.66186i	−0.841406	+	0.259539i	−0.685355	−	0.728209i	\(-0.740353\pi\)
							−0.156052	+	0.987749i	\(0.549877\pi\)
\(43\)	6.18503	+	1.90783i	0.943208	+	0.290941i	0.727961	−	0.685619i	\(-0.240468\pi\)
							0.215247	+	0.976560i	\(0.430944\pi\)
\(47\)	−9.92852	+	4.78132i	−1.44822	+	0.697428i	−0.982286	−	0.187390i	\(-0.939997\pi\)
							−0.465938	+	0.884818i	\(0.654283\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.151.46	✓	648
9.4	even	3		441.2.z.a.4.9	yes	648
49.37	even	21		441.2.z.a.331.9	yes	648
441.184	even	21	inner	441.2.y.a.184.46	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.46	✓	648	1.1	even	1	trivial
441.2.y.a.184.46	yes	648	441.184	even	21	inner
441.2.z.a.4.9	yes	648	9.4	even	3
441.2.z.a.331.9	yes	648	49.37	even	21