Embedded newform 441.2.bd.a.299.1

• Label: 441.2.bd.a.299.1
• Level: $441$
• Weight: $2$
• Character: 441.299
• 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			299.1
Character	\(\chi\)	\(=\)	441.299
Dual form			441.2.bd.a.59.1

$q$-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 \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.1	✓	648	441.59	even	42	inner
441.2.bd.a.299.1	yes	648	1.1	even	1	trivial
441.2.bn.a.5.1	yes	648	9.5	odd	6
441.2.bn.a.353.1	yes	648	49.10	odd	42