Embedded newform 441.2.ba.a.22.2

• Label: 441.2.ba.a.22.2
• Level: $441$
• Weight: $2$
• Character: 441.22
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.ba (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			22.2
Character	\(\chi\)	\(=\)	441.22
Dual form			441.2.ba.a.421.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.20796 - 1.50536i) q^{2} +(-1.57686 + 0.716598i) q^{3} +(1.87830 + 4.78582i) q^{4} +(2.79913 + 2.59721i) q^{5} +(4.56039 + 0.791525i) q^{6} +(-2.53111 - 0.770385i) q^{7} +(1.86790 - 8.18382i) q^{8} +(1.97297 - 2.25995i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 10 q^{3} + 47 q^{4} - 9 q^{5} - 34 q^{6} - 7 q^{7} - 28 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.20796	−	1.50536i	−1.56126	−	1.06445i	−0.965199	−	0.261517i	\(-0.915777\pi\)
							−0.596066	−	0.802935i	\(-0.703270\pi\)
\(3\)	−1.57686	+	0.716598i	−0.910400	+	0.413728i
							\(5\)	2.79913	+	2.59721i	1.25181	+	1.16151i	0.979965	+	0.199168i	\(0.0638240\pi\)
0.271844	+	0.962341i	\(0.412366\pi\)
\(7\)	−2.53111	−	0.770385i	−0.956669	−	0.291178i
							\(11\)	3.20609	+	2.18587i	0.966672	+	0.659066i	0.940214	−	0.340584i	\(-0.110625\pi\)
0.0264579	+	0.999650i	\(0.491577\pi\)
\(13\)	0.284458	−	3.79582i	0.0788943	−	1.05277i	−0.807508	−	0.589856i	\(-0.799184\pi\)
							0.886402	−	0.462915i	\(-0.153197\pi\)
\(17\)	2.29667	+	2.87993i	0.557023	+	0.698485i	0.978004	−	0.208584i	\(-0.0668856\pi\)
							−0.420981	+	0.907069i	\(0.638314\pi\)
\(19\)	0.292141			0.0670218			0.0335109	−	0.999438i	\(-0.489331\pi\)
							0.0335109	+	0.999438i	\(0.489331\pi\)
\(23\)	0.889906	+	2.26744i	0.185558	+	0.472795i	0.993136	−	0.116967i	\(-0.0373170\pi\)
							−0.807578	+	0.589761i	\(0.799222\pi\)
\(29\)	2.21095	−	5.63341i	0.410563	−	1.04610i	−0.564748	−	0.825263i	\(-0.691027\pi\)
							0.975311	−	0.220835i	\(-0.0708781\pi\)
\(31\)	−2.26723	+	3.92696i	−0.407207	+	0.705303i	−0.994576	−	0.104016i	\(-0.966831\pi\)
							0.587369	+	0.809320i	\(0.300164\pi\)
\(37\)	0.194278	+	0.243617i	0.0319391	+	0.0400504i	0.797545	−	0.603260i	\(-0.206132\pi\)
							−0.765605	+	0.643310i	\(0.777560\pi\)
\(41\)	4.51576	+	4.19001i	0.705243	+	0.654369i	0.948931	−	0.315484i	\(-0.102167\pi\)
							−0.243688	+	0.969854i	\(0.578357\pi\)
\(43\)	0.240665	−	0.223305i	0.0367011	−	0.0340537i	−0.661609	−	0.749849i	\(-0.730126\pi\)
							0.698310	+	0.715795i	\(0.253936\pi\)
\(47\)	−1.50532	−	1.02631i	−0.219573	−	0.149702i	0.448539	−	0.893763i	\(-0.351945\pi\)
							−0.668111	+	0.744061i	\(0.732897\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.ba.a.22.2	✓	648
9.7	even	3	inner	441.2.ba.a.169.2	yes	648
49.29	even	7	inner	441.2.ba.a.274.2	yes	648
441.421	even	21	inner	441.2.ba.a.421.2	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.ba.a.22.2	✓	648	1.1	even	1	trivial
441.2.ba.a.169.2	yes	648	9.7	even	3	inner
441.2.ba.a.274.2	yes	648	49.29	even	7	inner
441.2.ba.a.421.2	yes	648	441.421	even	21	inner