Embedded newform 441.2.z.a.4.9

• Label: 441.2.z.a.4.9
• Level: $441$
• Weight: $2$
• Character: 441.4
• 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			4.9
Character	\(\chi\)	\(=\)	441.4
Dual form			441.2.z.a.331.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.162086 + 2.16289i) q^{2} +(0.451822 - 1.67208i) q^{3} +(-2.67415 - 0.403063i) q^{4} +(0.397052 - 1.73960i) q^{5} +(3.54329 + 1.24826i) q^{6} +(1.01797 + 2.44208i) q^{7} +(0.339946 - 1.48940i) q^{8} +(-2.59171 - 1.51097i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.162086	+	2.16289i	−0.114612	+	1.52939i	0.582737	+	0.812661i	\(0.301982\pi\)
							−0.697349	+	0.716731i	\(0.745637\pi\)
\(3\)	0.451822	−	1.67208i	0.260859	−	0.965377i
							\(5\)	0.397052	−	1.73960i	0.177567	−	0.777972i	−0.805182	−	0.593028i	\(-0.797932\pi\)
0.982749	−	0.184944i	\(-0.0592105\pi\)
\(7\)	1.01797	+	2.44208i	0.384757	+	0.923018i
							\(11\)	3.38974	−	1.63241i	1.02204	−	0.492190i	0.153681	−	0.988120i	\(-0.450887\pi\)
0.868362	+	0.495930i	\(0.165173\pi\)
\(13\)	0.376936	−	5.02986i	0.104543	−	1.39503i	−0.660391	−	0.750922i	\(-0.729609\pi\)
							0.764934	−	0.644109i	\(-0.222772\pi\)
\(17\)	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	\(0.827073\pi\)
\(23\)	−2.11211	+	2.64850i	−0.440405	+	0.552250i	−0.951650	−	0.307185i	\(-0.900613\pi\)
							0.511245	+	0.859435i	\(0.329184\pi\)
\(29\)	2.12975	−	5.42650i	0.395484	−	1.00768i	−0.584993	−	0.811039i	\(-0.698903\pi\)
							0.980476	−	0.196637i	\(-0.0630022\pi\)
\(31\)	3.88033	+	6.72094i	0.696929	+	1.20712i	0.969526	+	0.244988i	\(0.0787839\pi\)
							−0.272598	+	0.962128i	\(0.587883\pi\)
\(37\)	−1.41656	+	3.60935i	−0.232882	+	0.593373i	−0.998769	−	0.0496018i	\(-0.984205\pi\)
							0.765887	+	0.642975i	\(0.222300\pi\)
\(41\)	4.13303	+	3.83489i	0.645471	+	0.598909i	0.933441	−	0.358730i	\(-0.116790\pi\)
							−0.287971	+	0.957639i	\(0.592981\pi\)
\(43\)	−4.74474	+	4.40248i	−0.723566	+	0.671371i	−0.953349	−	0.301871i	\(-0.902389\pi\)
							0.229783	+	0.973242i	\(0.426198\pi\)
\(47\)	0.823512	−	10.9890i	0.120122	−	1.60291i	−0.533306	−	0.845923i	\(-0.679050\pi\)
							0.653427	−	0.756989i	\(-0.273331\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.46	✓	648	9.7	even	3
441.2.y.a.184.46	yes	648	49.37	even	21
441.2.z.a.4.9	yes	648	1.1	even	1	trivial
441.2.z.a.331.9	yes	648	441.331	even	21	inner