Embedded newform 441.2.bn.a.101.53

• Label: 441.2.bn.a.101.53
• Level: $441$
• Weight: $2$
• Character: 441.101
• 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			101.53
Character	\(\chi\)	\(=\)	441.101
Dual form			441.2.bn.a.131.53

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.10146 - 1.67586i) q^{2} +(-0.574256 - 1.63408i) q^{3} +(1.16259 - 5.09363i) q^{4} +(-0.190264 + 2.53889i) q^{5} +(-3.94527 - 2.47159i) q^{6} +(-1.20290 - 2.35649i) q^{7} +(-3.76062 - 7.80901i) q^{8} +(-2.34046 + 1.87677i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.10146	−	1.67586i	1.48596	−	1.18501i	0.548871	−	0.835907i	\(-0.315058\pi\)
							0.937085	−	0.349103i	\(-0.113513\pi\)
\(3\)	−0.574256	−	1.63408i	−0.331547	−	0.943439i
							\(5\)	−0.190264	+	2.53889i	−0.0850886	+	1.13543i	0.777283	+	0.629151i	\(0.216597\pi\)
−0.862371	+	0.506276i	\(0.831022\pi\)
\(7\)	−1.20290	−	2.35649i	−0.454654	−	0.890668i
							\(11\)	−1.07254	−	0.420941i	−0.323383	−	0.126919i	0.198096	−	0.980183i	\(-0.436524\pi\)
−0.521479	+	0.853264i	\(0.674620\pi\)
\(13\)	5.20615	+	2.04326i	1.44393	+	0.566699i	0.952586	−	0.304269i	\(-0.0984120\pi\)
							0.491340	+	0.870968i	\(0.336507\pi\)
\(17\)	−2.91640	−	0.899589i	−0.707330	−	0.218182i	−0.0798409	−	0.996808i	\(-0.525441\pi\)
							−0.627489	+	0.778625i	\(0.715917\pi\)
\(19\)	5.90833	−	3.41117i	1.35546	−	0.782577i	0.366454	−	0.930436i	\(-0.380572\pi\)
							0.989008	+	0.147859i	\(0.0472383\pi\)
\(23\)	−0.156274	+	0.168423i	−0.0325854	+	0.0351187i	−0.749134	−	0.662418i	\(-0.769530\pi\)
							0.716549	+	0.697537i	\(0.245721\pi\)
\(29\)	1.38618	−	4.49390i	0.257408	−	0.834496i	−0.730961	−	0.682419i	\(-0.760928\pi\)
							0.988369	−	0.152076i	\(-0.0485960\pi\)
\(31\)			8.38508i			1.50600i	0.658018	+	0.753002i	\(0.271395\pi\)
							−0.658018	+	0.753002i	\(0.728605\pi\)
\(37\)	0.586868	−	0.544534i	0.0964805	−	0.0895208i	−0.630458	−	0.776223i	\(-0.717133\pi\)
							0.726939	+	0.686702i	\(0.240942\pi\)
\(41\)	4.95221	+	3.37636i	0.773404	+	0.527298i	0.884516	−	0.466510i	\(-0.154489\pi\)
							−0.111112	+	0.993808i	\(0.535441\pi\)
\(43\)	5.71931	−	3.89936i	0.872187	−	0.594647i	−0.0423710	−	0.999102i	\(-0.513491\pi\)
							0.914558	+	0.404455i	\(0.132539\pi\)
\(47\)	1.77594	+	2.22696i	0.259048	+	0.324836i	0.894299	−	0.447470i	\(-0.147675\pi\)
							−0.635251	+	0.772306i	\(0.719103\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.101.53	yes	648
9.5	odd	6		441.2.bd.a.248.2	✓	648
49.33	odd	42		441.2.bd.a.425.2	yes	648
441.131	even	42	inner	441.2.bn.a.131.53	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.248.2	✓	648	9.5	odd	6
441.2.bd.a.425.2	yes	648	49.33	odd	42
441.2.bn.a.101.53	yes	648	1.1	even	1	trivial
441.2.bn.a.131.53	yes	648	441.131	even	42	inner