Embedded newform 441.2.bn.a.353.3

• Label: 441.2.bn.a.353.3
• 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			353.3
Character	\(\chi\)	\(=\)	441.353
Dual form			441.2.bn.a.5.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.01930 - 1.61034i) q^{2} +(0.161619 - 1.72449i) q^{3} +(1.03935 + 4.55368i) q^{4} +(0.846045 - 0.576824i) q^{5} +(-3.10338 + 3.22201i) q^{6} +(-1.90950 + 1.83134i) q^{7} +(2.99296 - 6.21494i) q^{8} +(-2.94776 - 0.557422i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.01930	−	1.61034i	−1.42786	−	1.13868i	−0.968077	−	0.250653i	\(-0.919355\pi\)
							−0.459786	−	0.888030i	\(-0.652074\pi\)
\(3\)	0.161619	−	1.72449i	0.0933107	−	0.995637i
							\(5\)	0.846045	−	0.576824i	0.378363	−	0.257963i	−0.359174	−	0.933271i	\(-0.616941\pi\)
0.737536	+	0.675307i	\(0.235989\pi\)
\(7\)	−1.90950	+	1.83134i	−0.721724	+	0.692181i
							\(11\)	−0.313064	+	2.07704i	−0.0943924	+	0.626253i	0.890747	+	0.454499i	\(0.150182\pi\)
−0.985140	+	0.171754i	\(0.945057\pi\)
\(13\)	−0.369197	+	2.44946i	−0.102397	+	0.679359i	0.877459	+	0.479652i	\(0.159237\pi\)
							−0.979856	+	0.199707i	\(0.936001\pi\)
\(17\)	2.20085	+	2.04209i	0.533785	+	0.495280i	0.900382	−	0.435100i	\(-0.143287\pi\)
							−0.366597	+	0.930380i	\(0.619477\pi\)
\(19\)	−4.42780	+	2.55639i	−1.01581	+	0.586476i	−0.912887	−	0.408213i	\(-0.866152\pi\)
							−0.102920	+	0.994690i	\(0.532819\pi\)
\(23\)	−0.423980	+	1.37451i	−0.0884059	+	0.286605i	−0.988969	−	0.148120i	\(-0.952678\pi\)
							0.900564	+	0.434724i	\(0.143154\pi\)
\(29\)	−3.00560	+	3.23927i	−0.558126	+	0.601517i	−0.947576	−	0.319531i	\(-0.896475\pi\)
							0.389450	+	0.921048i	\(0.372665\pi\)
\(31\)			6.92810i			1.24432i	0.782889	+	0.622162i	\(0.213745\pi\)
							−0.782889	+	0.622162i	\(0.786255\pi\)
\(37\)	8.66701	−	2.67342i	1.42485	−	0.439507i	0.515983	−	0.856599i	\(-0.327427\pi\)
							0.908865	+	0.417091i	\(0.136950\pi\)
\(41\)	−0.864467	−	11.5355i	−0.135007	−	1.80154i	−0.495319	−	0.868711i	\(-0.664949\pi\)
							0.360312	−	0.932832i	\(-0.382670\pi\)
\(43\)	0.178340	−	2.37978i	0.0271966	−	0.362913i	−0.966957	−	0.254939i	\(-0.917945\pi\)
							0.994154	−	0.107974i	\(-0.0344364\pi\)
\(47\)	3.35857	−	4.21151i	0.489898	−	0.614312i	−0.474019	−	0.880514i	\(-0.657197\pi\)
							0.963917	+	0.266202i	\(0.0857688\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.353.3	yes	648
9.5	odd	6		441.2.bd.a.59.3	✓	648
49.5	odd	42		441.2.bd.a.299.3	yes	648
441.5	even	42	inner	441.2.bn.a.5.3	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.3	✓	648	9.5	odd	6
441.2.bd.a.299.3	yes	648	49.5	odd	42
441.2.bn.a.5.3	yes	648	441.5	even	42	inner
441.2.bn.a.353.3	yes	648	1.1	even	1	trivial