Embedded newform 441.2.ba.a.43.10

• Label: 441.2.ba.a.43.10
• Level: $441$
• Weight: $2$
• Character: 441.43
• 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			43.10
Character	\(\chi\)	\(=\)	441.43
Dual form			441.2.ba.a.400.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.698389 - 1.77947i) q^{2} +(1.73166 - 0.0367326i) q^{3} +(-1.21265 + 1.12518i) q^{4} +(-0.0460817 - 0.614918i) q^{5} +(-1.27474 - 3.05578i) q^{6} +(-0.671251 - 2.55918i) q^{7} +(-0.595487 - 0.286771i) q^{8} +(2.99730 - 0.127217i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 10 q^{3} + 47 q^{4} - 9 q^{5} - 34 q^{6} - 7 q^{7} - 28 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.698389	−	1.77947i	−0.493836	−	1.25827i	−0.933003	−	0.359868i	\(-0.882822\pi\)
							0.439168	−	0.898405i	\(-0.355273\pi\)
\(3\)	1.73166	−	0.0367326i	0.999775	−	0.0212076i
							\(5\)	−0.0460817	−	0.614918i	−0.0206084	−	0.275000i	−0.997983	−	0.0634835i	\(-0.979779\pi\)
0.977375	−	0.211516i	\(-0.0678401\pi\)
\(7\)	−0.671251	−	2.55918i	−0.253709	−	0.967281i
							\(11\)	−0.679503	−	1.73135i	−0.204878	−	0.522020i	0.791093	−	0.611696i	\(-0.209512\pi\)
−0.995971	+	0.0896753i	\(0.971417\pi\)
\(13\)	3.66184	+	0.551934i	1.01561	+	0.153079i	0.635694	−	0.771941i	\(-0.280714\pi\)
							0.379918	+	0.925020i	\(0.375952\pi\)
\(17\)	0.108842	−	0.476870i	0.0263982	−	0.115658i	−0.960012	−	0.279958i	\(-0.909679\pi\)
							0.986410	+	0.164300i	\(0.0525366\pi\)
\(19\)	−4.81600			−1.10487			−0.552433	−	0.833557i	\(-0.686300\pi\)
							−0.552433	+	0.833557i	\(0.686300\pi\)
\(23\)	−1.91742	+	1.77911i	−0.399810	+	0.370969i	−0.854298	−	0.519784i	\(-0.826013\pi\)
							0.454488	+	0.890753i	\(0.349822\pi\)
\(29\)	−7.12935	−	6.61507i	−1.32389	−	1.22839i	−0.954177	−	0.299241i	\(-0.903266\pi\)
							−0.369711	−	0.929147i	\(-0.620543\pi\)
\(31\)	3.05299	+	5.28793i	0.548333	+	0.949741i	0.998389	+	0.0567403i	\(0.0180707\pi\)
							−0.450056	+	0.893000i	\(0.648596\pi\)
\(37\)	1.66442	−	7.29229i	0.273629	−	1.19885i	−0.632066	−	0.774915i	\(-0.717793\pi\)
							0.905695	−	0.423931i	\(-0.139350\pi\)
\(41\)	−0.647285	−	8.63742i	−0.101089	−	1.34894i	−0.785056	−	0.619425i	\(-0.787366\pi\)
							0.683967	−	0.729513i	\(-0.260253\pi\)
\(43\)	−0.504390	+	6.73061i	−0.0769187	+	1.02641i	0.816547	+	0.577278i	\(0.195885\pi\)
							−0.893466	+	0.449130i	\(0.851734\pi\)
\(47\)	0.213968	+	0.545183i	0.0312105	+	0.0795230i	0.945629	−	0.325246i	\(-0.105447\pi\)
							−0.914419	+	0.404769i	\(0.867352\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.ba.a.43.10	✓	648
9.4	even	3	inner	441.2.ba.a.337.45	yes	648
49.8	even	7	inner	441.2.ba.a.106.45	yes	648
441.400	even	21	inner	441.2.ba.a.400.10	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.ba.a.43.10	✓	648	1.1	even	1	trivial
441.2.ba.a.106.45	yes	648	49.8	even	7	inner
441.2.ba.a.337.45	yes	648	9.4	even	3	inner
441.2.ba.a.400.10	yes	648	441.400	even	21	inner