Embedded newform 441.2.bd.a.110.18

• Label: 441.2.bd.a.110.18
• Level: $441$
• Weight: $2$
• Character: 441.110
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.bd (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			110.18
Character	\(\chi\)	\(=\)	441.110
Dual form			441.2.bd.a.437.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.11818 + 0.0837964i) q^{2} +(1.49366 + 0.876908i) q^{3} +(-0.734346 + 0.110685i) q^{4} +(-0.402620 - 1.76399i) q^{5} +(-1.74367 - 0.855381i) q^{6} +(-2.31474 - 1.28141i) q^{7} +(2.99827 - 0.684336i) q^{8} +(1.46207 + 2.61961i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{11}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.11818	+	0.0837964i	−0.790676	+	0.0592530i	−0.463945	−	0.885864i	\(-0.653566\pi\)
							−0.326731	+	0.945117i	\(0.605947\pi\)
\(3\)	1.49366	+	0.876908i	0.862367	+	0.506283i
							\(5\)	−0.402620	−	1.76399i	−0.180057	−	0.788882i	−0.981601	−	0.190945i	\(-0.938845\pi\)
0.801544	−	0.597936i	\(-0.204012\pi\)
\(7\)	−2.31474	−	1.28141i	−0.874888	−	0.484326i
							\(11\)	−2.65497	+	5.51310i	−0.800503	+	1.66226i	−0.0524503	+	0.998624i	\(0.516703\pi\)
−0.748053	+	0.663639i	\(0.769011\pi\)
\(13\)	5.61722	−	0.420953i	1.55794	−	0.116751i	0.732312	−	0.680970i	\(-0.238441\pi\)
							0.825626	+	0.564218i	\(0.190822\pi\)
\(17\)	2.95262	+	0.445036i	0.716117	+	0.107937i	0.496986	−	0.867758i	\(-0.334440\pi\)
							0.219130	+	0.975696i	\(0.429678\pi\)
\(19\)	5.50949	+	3.18091i	1.26396	+	0.729750i	0.973839	−	0.227238i	\(-0.0729696\pi\)
							0.290126	+	0.956989i	\(0.406303\pi\)
\(23\)	2.91399	−	2.32383i	0.607609	−	0.484552i	−0.270688	−	0.962667i	\(-0.587251\pi\)
							0.878297	+	0.478115i	\(0.158680\pi\)
\(29\)	−2.10773	+	0.827224i	−0.391396	+	0.153612i	−0.552874	−	0.833265i	\(-0.686469\pi\)
							0.161478	+	0.986876i	\(0.448374\pi\)
\(31\)	6.04555	+	3.49040i	1.08581	+	0.626894i	0.932459	−	0.361277i	\(-0.117659\pi\)
							0.153354	+	0.988171i	\(0.450992\pi\)
\(37\)	−2.79770	−	7.12842i	−0.459939	−	1.17190i	−0.953285	−	0.302072i	\(-0.902322\pi\)
							0.493346	−	0.869833i	\(-0.335773\pi\)
\(41\)	−6.65025	+	6.17053i	−1.03859	+	0.963674i	−0.999368	−	0.0355560i	\(-0.988680\pi\)
							−0.0392263	+	0.999230i	\(0.512489\pi\)
\(43\)	1.99603	+	1.85205i	0.304392	+	0.282434i	0.817547	−	0.575861i	\(-0.195333\pi\)
							−0.513156	+	0.858296i	\(0.671524\pi\)
\(47\)	−0.355283	−	4.74092i	−0.0518233	−	0.691534i	−0.961251	−	0.275673i	\(-0.911099\pi\)
							0.909428	−	0.415861i	\(-0.136520\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.110.18	✓	648
9.5	odd	6		441.2.bn.a.257.18	yes	648
49.45	odd	42		441.2.bn.a.290.18	yes	648
441.437	even	42	inner	441.2.bd.a.437.18	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.110.18	✓	648	1.1	even	1	trivial
441.2.bd.a.437.18	yes	648	441.437	even	42	inner
441.2.bn.a.257.18	yes	648	9.5	odd	6
441.2.bn.a.290.18	yes	648	49.45	odd	42