Embedded newform 441.2.bd.a.59.1

• Label: 441.2.bd.a.59.1
• Level: $441$
• Weight: $2$
• Character: 441.59
• 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			59.1
Character	\(\chi\)	\(=\)	441.59
Dual form			441.2.bd.a.299.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.57597 + 1.01099i) q^{2} +(1.30663 + 1.13698i) q^{3} +(4.14739 - 3.84822i) q^{4} +(3.17550 + 1.52924i) q^{5} +(-4.51531 - 1.60783i) q^{6} +(2.46063 - 0.972261i) q^{7} +(-4.39168 + 9.11943i) q^{8} +(0.414560 + 2.97122i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.57597	+	1.01099i	−1.82148	+	0.714879i	−0.835170	+	0.549991i	\(0.814631\pi\)
							−0.986312	+	0.164888i	\(0.947274\pi\)
\(3\)	1.30663	+	1.13698i	0.754383	+	0.656435i
							\(5\)	3.17550	+	1.52924i	1.42013	+	0.683897i	0.977133	−	0.212628i	\(-0.0682021\pi\)
0.442994	+	0.896525i	\(0.353916\pi\)
\(7\)	2.46063	−	0.972261i	0.930031	−	0.367480i
							\(11\)	−0.673629	−	0.537201i	−0.203107	−	0.161972i	0.516654	−	0.856194i	\(-0.327177\pi\)
−0.719761	+	0.694222i	\(0.755749\pi\)
\(13\)	−3.72778	+	1.46305i	−1.03390	+	0.405776i	−0.820777	−	0.571249i	\(-0.806459\pi\)
							−0.213124	+	0.977025i	\(0.568364\pi\)
\(17\)	1.51858	+	1.40904i	0.368311	+	0.341742i	0.842515	−	0.538672i	\(-0.181074\pi\)
							−0.474205	+	0.880415i	\(0.657264\pi\)
\(19\)	1.72443	−	0.995602i	0.395612	−	0.228407i	−0.288977	−	0.957336i	\(-0.593315\pi\)
							0.684589	+	0.728929i	\(0.259982\pi\)
\(23\)	1.25652	−	0.286793i	0.262003	−	0.0598004i	−0.0895015	−	0.995987i	\(-0.528527\pi\)
							0.351504	+	0.936186i	\(0.385670\pi\)
\(29\)	−2.61703	−	8.48421i	−0.485970	−	1.57548i	−0.783898	−	0.620889i	\(-0.786772\pi\)
							0.297928	−	0.954588i	\(-0.403705\pi\)
\(31\)	2.40588	−	1.38903i	0.432108	−	0.249478i	−0.268136	−	0.963381i	\(-0.586408\pi\)
							0.700244	+	0.713903i	\(0.253074\pi\)
\(37\)	1.25748	−	0.387883i	0.206729	−	0.0637675i	−0.189661	−	0.981850i	\(-0.560739\pi\)
							0.396390	+	0.918082i	\(0.370263\pi\)
\(41\)	−4.21667	+	2.87488i	−0.658533	+	0.448980i	−0.845923	−	0.533305i	\(-0.820950\pi\)
							0.187390	+	0.982286i	\(0.439997\pi\)
\(43\)	−3.57231	−	2.43556i	−0.544773	−	0.371420i	0.259442	−	0.965759i	\(-0.416461\pi\)
							−0.804215	+	0.594339i	\(0.797414\pi\)
\(47\)	−1.71760	−	4.37636i	−0.250537	−	0.638358i	0.749175	−	0.662373i	\(-0.230450\pi\)
							−0.999712	+	0.0240146i	\(0.992355\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.59.1	✓	648
9.2	odd	6		441.2.bn.a.353.1	yes	648
49.5	odd	42		441.2.bn.a.5.1	yes	648
441.299	even	42	inner	441.2.bd.a.299.1	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.1	✓	648	1.1	even	1	trivial
441.2.bd.a.299.1	yes	648	441.299	even	42	inner
441.2.bn.a.5.1	yes	648	49.5	odd	42
441.2.bn.a.353.1	yes	648	9.2	odd	6