Embedded newform 441.2.z.a.331.13

• Label: 441.2.z.a.331.13
• 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			331.13
Character	\(\chi\)	\(=\)	441.331
Dual form			441.2.z.a.4.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.122748 - 1.63795i) q^{2} +(0.324451 - 1.70139i) q^{3} +(-0.690165 + 0.104026i) q^{4} +(0.354390 + 1.55268i) q^{5} +(-2.82663 - 0.322594i) q^{6} +(2.24116 - 1.40613i) q^{7} +(-0.475897 - 2.08504i) q^{8} +(-2.78946 - 1.10404i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.122748	−	1.63795i	−0.0867957	−	1.15821i	−0.855219	−	0.518267i	\(-0.826578\pi\)
							0.768423	−	0.639942i	\(-0.221042\pi\)
\(3\)	0.324451	−	1.70139i	0.187322	−	0.982299i
							\(5\)	0.354390	+	1.55268i	0.158488	+	0.694382i	0.990256	+	0.139258i	\(0.0444719\pi\)
−0.831768	+	0.555123i	\(0.812671\pi\)
\(7\)	2.24116	−	1.40613i	0.847079	−	0.531467i
							\(11\)	1.86872	+	0.899929i	0.563441	+	0.271339i	0.693848	−	0.720122i	\(-0.255914\pi\)
−0.130407	+	0.991461i	\(0.541628\pi\)
\(13\)	−0.129127	−	1.72308i	−0.0358134	−	0.477896i	−0.986059	−	0.166395i	\(-0.946787\pi\)
							0.950246	−	0.311501i	\(-0.100832\pi\)
\(17\)	−3.41500	−	0.514728i	−0.828259	−	0.124840i	−0.278788	−	0.960353i	\(-0.589933\pi\)
							−0.549471	+	0.835513i	\(0.685171\pi\)
\(19\)	1.13809	−	1.97123i	0.261095	−	0.452231i	−0.705438	−	0.708772i	\(-0.749250\pi\)
							0.966533	+	0.256541i	\(0.0825829\pi\)
\(23\)	−0.552405	−	0.692694i	−0.115184	−	0.144437i	0.720897	−	0.693043i	\(-0.243730\pi\)
							−0.836081	+	0.548606i	\(0.815159\pi\)
\(29\)	0.740080	+	1.88569i	0.137429	+	0.350164i	0.982975	−	0.183739i	\(-0.0588200\pi\)
							−0.845546	+	0.533903i	\(0.820725\pi\)
\(31\)	0.776379	−	1.34473i	0.139442	−	0.241520i	−0.787844	−	0.615875i	\(-0.788803\pi\)
							0.927285	+	0.374355i	\(0.122136\pi\)
\(37\)	0.883157	+	2.25025i	0.145190	+	0.369938i	0.984907	−	0.173087i	\(-0.0553741\pi\)
							−0.839717	+	0.543025i	\(0.817279\pi\)
\(41\)	−7.62226	+	7.07243i	−1.19040	+	1.10453i	−0.198160	+	0.980170i	\(0.563497\pi\)
							−0.992237	+	0.124358i	\(0.960313\pi\)
\(43\)	6.98388	+	6.48009i	1.06503	+	0.988205i	0.999953	−	0.00967557i	\(-0.00307988\pi\)
							0.0650781	+	0.997880i	\(0.479270\pi\)
\(47\)	0.861749	+	11.4992i	0.125699	+	1.67734i	0.602341	+	0.798239i	\(0.294235\pi\)
							−0.476641	+	0.879098i	\(0.658146\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.331.13	yes	648
9.4	even	3		441.2.y.a.184.42	yes	648
49.4	even	21		441.2.y.a.151.42	✓	648
441.4	even	21	inner	441.2.z.a.4.13	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.42	✓	648	49.4	even	21
441.2.y.a.184.42	yes	648	9.4	even	3
441.2.z.a.4.13	yes	648	441.4	even	21	inner
441.2.z.a.331.13	yes	648	1.1	even	1	trivial