Embedded newform 441.2.ba.a.22.13

• Label: 441.2.ba.a.22.13
• 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			22.13
Character	\(\chi\)	\(=\)	441.22
Dual form			441.2.ba.a.421.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.48693 - 1.01377i) q^{2} +(1.13536 - 1.30804i) q^{3} +(0.452539 + 1.15305i) q^{4} +(1.39969 + 1.29873i) q^{5} +(-3.01424 + 0.793970i) q^{6} +(-0.229184 + 2.63581i) q^{7} +(-0.304877 + 1.33575i) q^{8} +(-0.421935 - 2.97018i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 10 q^{3} + 47 q^{4} - 9 q^{5} - 34 q^{6} - 7 q^{7} - 28 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.48693	−	1.01377i	−1.05142	−	0.716843i	−0.0911234	−	0.995840i	\(-0.529046\pi\)
							−0.960292	+	0.278997i	\(0.909998\pi\)
\(3\)	1.13536	−	1.30804i	0.655498	−	0.755197i
							\(5\)	1.39969	+	1.29873i	0.625962	+	0.580808i	0.928042	−	0.372475i	\(-0.121491\pi\)
−0.302080	+	0.953282i	\(0.597681\pi\)
\(7\)	−0.229184	+	2.63581i	−0.0866235	+	0.996241i
							\(11\)	1.34280	+	0.915504i	0.404869	+	0.276035i	0.748585	−	0.663039i	\(-0.230733\pi\)
−0.343716	+	0.939074i	\(0.611686\pi\)
\(13\)	0.424753	−	5.66794i	0.117805	−	1.57200i	−0.554788	−	0.831991i	\(-0.687201\pi\)
							0.672594	−	0.740012i	\(-0.265180\pi\)
\(17\)	2.49941	+	3.13416i	0.606195	+	0.760145i	0.986329	−	0.164787i	\(-0.0526935\pi\)
							−0.380134	+	0.924931i	\(0.624122\pi\)
\(19\)	6.69536			1.53602			0.768011	−	0.640437i	\(-0.221247\pi\)
							0.768011	+	0.640437i	\(0.221247\pi\)
\(23\)	−0.805071	−	2.05129i	−0.167869	−	0.427723i	0.822040	−	0.569430i	\(-0.192836\pi\)
							−0.989909	+	0.141707i	\(0.954741\pi\)
\(29\)	−0.208798	+	0.532007i	−0.0387727	+	0.0987913i	−0.948949	−	0.315431i	\(-0.897851\pi\)
							0.910176	+	0.414222i	\(0.135946\pi\)
\(31\)	1.42129	−	2.46174i	0.255271	−	0.442142i	−0.709698	−	0.704506i	\(-0.751169\pi\)
							0.964969	+	0.262364i	\(0.0845020\pi\)
\(37\)	2.81579	+	3.53089i	0.462913	+	0.580474i	0.957420	−	0.288700i	\(-0.0932229\pi\)
							−0.494507	+	0.869174i	\(0.664651\pi\)
\(41\)	4.02576	+	3.73536i	0.628718	+	0.583365i	0.928815	−	0.370543i	\(-0.120829\pi\)
							−0.300097	+	0.953909i	\(0.597019\pi\)
\(43\)	0.706423	−	0.655464i	0.107728	−	0.0999574i	−0.624470	−	0.781049i	\(-0.714685\pi\)
							0.732198	+	0.681092i	\(0.238494\pi\)
\(47\)	2.35826	+	1.60783i	0.343987	+	0.234527i	0.722974	−	0.690876i	\(-0.242775\pi\)
							−0.378986	+	0.925402i	\(0.623727\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.ba.a.22.13	✓	648
9.7	even	3	inner	441.2.ba.a.169.13	yes	648
49.29	even	7	inner	441.2.ba.a.274.13	yes	648
441.421	even	21	inner	441.2.ba.a.421.13	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.ba.a.22.13	✓	648	1.1	even	1	trivial
441.2.ba.a.169.13	yes	648	9.7	even	3	inner
441.2.ba.a.274.13	yes	648	49.29	even	7	inner
441.2.ba.a.421.13	yes	648	441.421	even	21	inner