Embedded newform 441.2.bn.a.5.3

• Label: 441.2.bn.a.5.3
• Level: $441$
• Weight: $2$
• Character: 441.5
• 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			5.3
Character	\(\chi\)	\(=\)	441.5
Dual form			441.2.bn.a.353.3

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.3	✓	648	49.10	odd	42
441.2.bd.a.299.3	yes	648	9.2	odd	6
441.2.bn.a.5.3	yes	648	1.1	even	1	trivial
441.2.bn.a.353.3	yes	648	441.353	even	42	inner