Embedded newform 441.2.z.a.331.9

• Label: 441.2.z.a.331.9
• Level: $441$
• Weight: $2$
• Character: 441.331
• 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			331.9
Character	\(\chi\)	\(=\)	441.331
Dual form			441.2.z.a.4.9

$q$-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 \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.162086	−	2.16289i	−0.114612	−	1.52939i	−0.697349	−	0.716731i	\(-0.745637\pi\)
							0.582737	−	0.812661i	\(-0.301982\pi\)
\(3\)	0.451822	+	1.67208i	0.260859	+	0.965377i
							\(5\)	0.397052	+	1.73960i	0.177567	+	0.777972i	0.982749	+	0.184944i	\(0.0592105\pi\)
−0.805182	+	0.593028i	\(0.797932\pi\)
\(7\)	1.01797	−	2.44208i	0.384757	−	0.923018i
							\(11\)	3.38974	+	1.63241i	1.02204	+	0.492190i	0.868362	−	0.495930i	\(-0.165173\pi\)
0.153681	+	0.988120i	\(0.450887\pi\)
\(13\)	0.376936	+	5.02986i	0.104543	+	1.39503i	0.764934	+	0.644109i	\(0.222772\pi\)
							−0.660391	+	0.750922i	\(0.729609\pi\)
\(17\)	6.99567	+	1.05443i	1.69670	+	0.255736i	0.924922	−	0.380156i	\(-0.124130\pi\)
							0.771777	+	0.635893i	\(0.219368\pi\)
\(19\)	0.0857190	−	0.148470i	0.0196653	−	0.0340613i	−0.856025	−	0.516934i	\(-0.827073\pi\)
							0.875691	+	0.482873i	\(0.160407\pi\)
\(23\)	−2.11211	−	2.64850i	−0.440405	−	0.552250i	0.511245	−	0.859435i	\(-0.329184\pi\)
							−0.951650	+	0.307185i	\(0.900613\pi\)
\(29\)	2.12975	+	5.42650i	0.395484	+	1.00768i	0.980476	+	0.196637i	\(0.0630022\pi\)
							−0.584993	+	0.811039i	\(0.698903\pi\)
\(31\)	3.88033	−	6.72094i	0.696929	−	1.20712i	−0.272598	−	0.962128i	\(-0.587883\pi\)
							0.969526	−	0.244988i	\(-0.0787839\pi\)
\(37\)	−1.41656	−	3.60935i	−0.232882	−	0.593373i	0.765887	−	0.642975i	\(-0.222300\pi\)
							−0.998769	+	0.0496018i	\(0.984205\pi\)
\(41\)	4.13303	−	3.83489i	0.645471	−	0.598909i	−0.287971	−	0.957639i	\(-0.592981\pi\)
							0.933441	+	0.358730i	\(0.116790\pi\)
\(43\)	−4.74474	−	4.40248i	−0.723566	−	0.671371i	0.229783	−	0.973242i	\(-0.426198\pi\)
							−0.953349	+	0.301871i	\(0.902389\pi\)
\(47\)	0.823512	+	10.9890i	0.120122	+	1.60291i	0.653427	+	0.756989i	\(0.273331\pi\)
							−0.533306	+	0.845923i	\(0.679050\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.46	✓	648	49.4	even	21
441.2.y.a.184.46	yes	648	9.4	even	3
441.2.z.a.4.9	yes	648	441.4	even	21	inner
441.2.z.a.331.9	yes	648	1.1	even	1	trivial