Embedded newform 441.2.bn.a.353.6

• Label: 441.2.bn.a.353.6
• Level: $441$
• Weight: $2$
• Character: 441.353
• 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.bn (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			353.6
Character	\(\chi\)	\(=\)	441.353
Dual form			441.2.bn.a.5.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.86842 - 1.49001i) q^{2} +(-1.41768 - 0.995078i) q^{3} +(0.825803 + 3.61808i) q^{4} +(-1.88505 + 1.28520i) q^{5} +(1.16614 + 3.97158i) q^{6} +(-0.922050 - 2.47988i) q^{7} +(1.77425 - 3.68427i) q^{8} +(1.01964 + 2.82141i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.86842	−	1.49001i	−1.32117	−	1.05360i	−0.994085	−	0.108604i	\(-0.965362\pi\)
							−0.327086	−	0.944995i	\(-0.606067\pi\)
\(3\)	−1.41768	−	0.995078i	−0.818499	−	0.574509i
							\(5\)	−1.88505	+	1.28520i	−0.843018	+	0.574760i	−0.906011	−	0.423254i	\(-0.860888\pi\)
0.0629931	+	0.998014i	\(0.479935\pi\)
\(7\)	−0.922050	−	2.47988i	−0.348502	−	0.937308i
							\(11\)	−0.775241	+	5.14339i	−0.233744	+	1.55079i	0.496763	+	0.867886i	\(0.334522\pi\)
−0.730507	+	0.682905i	\(0.760716\pi\)
\(13\)	0.0919355	−	0.609952i	0.0254983	−	0.169170i	−0.972457	−	0.233081i	\(-0.925119\pi\)
							0.997956	+	0.0639108i	\(0.0203573\pi\)
\(17\)	−1.39998	−	1.29899i	−0.339544	−	0.315051i	0.491908	−	0.870647i	\(-0.336300\pi\)
							−0.831452	+	0.555596i	\(0.812490\pi\)
\(19\)	4.54187	−	2.62225i	1.04198	−	0.601586i	0.121584	−	0.992581i	\(-0.461203\pi\)
							0.920393	+	0.390996i	\(0.127869\pi\)
\(23\)	−0.184426	+	0.597895i	−0.0384555	+	0.124670i	−0.972765	−	0.231793i	\(-0.925541\pi\)
							0.934310	+	0.356462i	\(0.116017\pi\)
\(29\)	5.12493	−	5.52337i	0.951676	−	1.02566i	−0.0479640	−	0.998849i	\(-0.515273\pi\)
							0.999640	−	0.0268142i	\(-0.00853625\pi\)
\(31\)			4.69739i			0.843677i	0.906671	+	0.421838i	\(0.138615\pi\)
							−0.906671	+	0.421838i	\(0.861385\pi\)
\(37\)	−5.42578	+	1.67363i	−0.891993	+	0.275143i	−0.706678	−	0.707536i	\(-0.749807\pi\)
							−0.185315	+	0.982679i	\(0.559331\pi\)
\(41\)	−0.543928	−	7.25822i	−0.0849473	−	1.13354i	−0.862952	−	0.505285i	\(-0.831387\pi\)
							0.778005	−	0.628258i	\(-0.216232\pi\)
\(43\)	0.446996	−	5.96474i	0.0681662	−	0.909615i	−0.853366	−	0.521312i	\(-0.825443\pi\)
							0.921532	−	0.388302i	\(-0.126938\pi\)
\(47\)	1.56059	−	1.95692i	0.227635	−	0.285446i	−0.654877	−	0.755736i	\(-0.727279\pi\)
							0.882512	+	0.470290i	\(0.155851\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.353.6	yes	648
9.5	odd	6		441.2.bd.a.59.6	✓	648
49.5	odd	42		441.2.bd.a.299.6	yes	648
441.5	even	42	inner	441.2.bn.a.5.6	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.6	✓	648	9.5	odd	6
441.2.bd.a.299.6	yes	648	49.5	odd	42
441.2.bn.a.5.6	yes	648	441.5	even	42	inner
441.2.bn.a.353.6	yes	648	1.1	even	1	trivial