Embedded newform 441.2.ba.a.106.53

• Label: 441.2.ba.a.106.53
• Level: $441$
• Weight: $2$
• Character: 441.106
• 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			106.53
Character	\(\chi\)	\(=\)	441.106
Dual form			441.2.ba.a.337.53

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.66904 - 0.402293i) q^{2} +(-1.66495 - 0.477417i) q^{3} +(5.05080 - 1.55797i) q^{4} +(1.57905 + 1.07658i) q^{5} +(-4.63590 - 0.604446i) q^{6} +(-0.157382 - 2.64107i) q^{7} +(7.99028 - 3.84791i) q^{8} +(2.54415 + 1.58975i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 10 q^{3} + 47 q^{4} - 9 q^{5} - 34 q^{6} - 7 q^{7} - 28 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.66904	−	0.402293i	1.88730	−	0.284464i	0.899112	−	0.437720i	\(-0.144214\pi\)
							0.988187	+	0.153255i	\(0.0489756\pi\)
\(3\)	−1.66495	−	0.477417i	−0.961262	−	0.275637i
							\(5\)	1.57905	+	1.07658i	0.706173	+	0.481461i	0.862379	−	0.506264i	\(-0.168974\pi\)
−0.156206	+	0.987725i	\(0.549926\pi\)
\(7\)	−0.157382	−	2.64107i	−0.0594848	−	0.998229i
							\(11\)	−3.91838	+	0.590601i	−1.18144	+	0.178073i	−0.710243	−	0.703957i	\(-0.751415\pi\)
−0.471194	+	0.882030i	\(0.656177\pi\)
\(13\)	−1.69856	+	4.32787i	−0.471096	+	1.20033i	0.476017	+	0.879436i	\(0.342080\pi\)
							−0.947114	+	0.320898i	\(0.896015\pi\)
\(17\)	−0.299635	−	1.31279i	−0.0726723	−	0.318398i	0.925506	−	0.378733i	\(-0.123640\pi\)
							−0.998178	+	0.0603352i	\(0.980783\pi\)
\(19\)	−2.09983			−0.481733			−0.240866	−	0.970558i	\(-0.577432\pi\)
							−0.240866	+	0.970558i	\(0.577432\pi\)
\(23\)	6.65484	−	2.05275i	1.38763	−	0.428027i	0.491136	−	0.871083i	\(-0.336582\pi\)
							0.896494	+	0.443056i	\(0.146106\pi\)
\(29\)	−1.95313	−	0.602459i	−0.362686	−	0.111874i	0.108056	−	0.994145i	\(-0.465538\pi\)
							−0.470742	+	0.882271i	\(0.656014\pi\)
\(31\)	2.15723	+	3.73643i	0.387450	+	0.671082i	0.992106	−	0.125404i	\(-0.0400228\pi\)
							−0.604656	+	0.796487i	\(0.706689\pi\)
\(37\)	1.70026	+	7.44935i	0.279522	+	1.22466i	0.898400	+	0.439178i	\(0.144730\pi\)
							−0.618878	+	0.785487i	\(0.712413\pi\)
\(41\)	−1.39472	−	0.950901i	−0.217818	−	0.148506i	0.449496	−	0.893282i	\(-0.351604\pi\)
							−0.667314	+	0.744777i	\(0.732556\pi\)
\(43\)	−9.84687	+	6.71348i	−1.50163	+	1.02380i	−0.515680	+	0.856782i	\(0.672461\pi\)
							−0.985954	+	0.167015i	\(0.946587\pi\)
\(47\)	−7.00144	+	1.05530i	−1.02126	+	0.153931i	−0.638264	−	0.769817i	\(-0.720347\pi\)
							−0.383000	+	0.923748i	\(0.625109\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.ba.a.106.53	yes	648
9.4	even	3	inner	441.2.ba.a.400.2	yes	648
49.43	even	7	inner	441.2.ba.a.43.2	✓	648
441.337	even	21	inner	441.2.ba.a.337.53	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.ba.a.43.2	✓	648	49.43	even	7	inner
441.2.ba.a.106.53	yes	648	1.1	even	1	trivial
441.2.ba.a.337.53	yes	648	441.337	even	21	inner
441.2.ba.a.400.2	yes	648	9.4	even	3	inner