Embedded newform 441.2.ba.a.106.27

• Label: 441.2.ba.a.106.27
• 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			106.27
Character	\(\chi\)	\(=\)	441.106
Dual form			441.2.ba.a.337.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.119443 + 0.0180031i) q^{2} +(1.72845 - 0.111557i) q^{3} +(-1.89720 + 0.585210i) q^{4} +(-0.771365 - 0.525908i) q^{5} +(-0.204443 + 0.0444422i) q^{6} +(0.740744 - 2.53994i) q^{7} +(0.433731 - 0.208874i) q^{8} +(2.97511 - 0.385644i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 10 q^{3} + 47 q^{4} - 9 q^{5} - 34 q^{6} - 7 q^{7} - 28 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.119443	+	0.0180031i	−0.0844587	+	0.0127301i	−0.191136	−	0.981564i	\(-0.561217\pi\)
							0.106677	+	0.994294i	\(0.465979\pi\)
\(3\)	1.72845	−	0.111557i	0.997924	−	0.0644077i
							\(5\)	−0.771365	−	0.525908i	−0.344965	−	0.235193i	0.378426	−	0.925631i	\(-0.376465\pi\)
−0.723391	+	0.690438i	\(0.757418\pi\)
\(7\)	0.740744	−	2.53994i	0.279975	−	0.960007i
							\(11\)	2.43629	−	0.367211i	0.734568	−	0.110718i	0.228904	−	0.973449i	\(-0.426486\pi\)
0.505664	+	0.862731i	\(0.331248\pi\)
\(13\)	0.128164	−	0.326556i	0.0355462	−	0.0905703i	−0.911998	−	0.410194i	\(-0.865461\pi\)
							0.947545	+	0.319623i	\(0.103556\pi\)
\(17\)	−0.442848	−	1.94024i	−0.107406	−	0.470578i	−0.999813	−	0.0193457i	\(-0.993842\pi\)
							0.892406	−	0.451232i	\(-0.149015\pi\)
\(19\)	−0.530612			−0.121731			−0.0608654	−	0.998146i	\(-0.519386\pi\)
							−0.0608654	+	0.998146i	\(0.519386\pi\)
\(23\)	4.16686	−	1.28531i	0.868851	−	0.268005i	0.171892	−	0.985116i	\(-0.445012\pi\)
							0.696959	+	0.717111i	\(0.254536\pi\)
\(29\)	3.07904	+	0.949759i	0.571764	+	0.176366i	0.567137	−	0.823624i	\(-0.308051\pi\)
							0.00462685	+	0.999989i	\(0.498527\pi\)
\(31\)	0.489424	+	0.847707i	0.0879031	+	0.152253i	0.906625	−	0.421938i	\(-0.138650\pi\)
							−0.818722	+	0.574191i	\(0.805317\pi\)
\(37\)	−0.916175	−	4.01402i	−0.150618	−	0.659902i	−0.992706	−	0.120561i	\(-0.961531\pi\)
							0.842088	−	0.539341i	\(-0.181326\pi\)
\(41\)	9.58101	+	6.53222i	1.49630	+	1.02016i	0.987206	+	0.159451i	\(0.0509725\pi\)
							0.509096	+	0.860710i	\(0.329980\pi\)
\(43\)	−7.74917	+	5.28330i	−1.18174	+	0.805695i	−0.984624	−	0.174685i	\(-0.944109\pi\)
							−0.197114	+	0.980381i	\(0.563157\pi\)
\(47\)	−7.75783	+	1.16930i	−1.13160	+	0.170561i	−0.688031	−	0.725682i	\(-0.741525\pi\)
							−0.443565	+	0.896242i	\(0.646286\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.ba.a.106.27	yes	648
9.4	even	3	inner	441.2.ba.a.400.28	yes	648
49.43	even	7	inner	441.2.ba.a.43.28	✓	648
441.337	even	21	inner	441.2.ba.a.337.27	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.ba.a.43.28	✓	648	49.43	even	7	inner
441.2.ba.a.106.27	yes	648	1.1	even	1	trivial
441.2.ba.a.337.27	yes	648	441.337	even	21	inner
441.2.ba.a.400.28	yes	648	9.4	even	3	inner