Embedded newform 441.2.bd.a.59.6

• Label: 441.2.bd.a.59.6
• 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			59.6
Character	\(\chi\)	\(=\)	441.59
Dual form			441.2.bd.a.299.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.22460 + 0.873090i) q^{2} +(-0.408355 - 1.68322i) q^{3} +(2.72045 - 2.52420i) q^{4} +(-2.05554 - 0.989896i) q^{5} +(2.37803 + 3.38797i) q^{6} +(2.60867 + 0.441423i) q^{7} +(-1.77425 + 3.68427i) q^{8} +(-2.66649 + 1.37471i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.22460	+	0.873090i	−1.57303	+	0.617368i	−0.981932	−	0.189233i	\(-0.939400\pi\)
							−0.591096	+	0.806601i	\(0.701305\pi\)
\(3\)	−0.408355	−	1.68322i	−0.235764	−	0.971810i
							\(5\)	−2.05554	−	0.989896i	−0.919266	−	0.442695i	−0.0864568	−	0.996256i	\(-0.527554\pi\)
−0.832809	+	0.553561i	\(0.813269\pi\)
\(7\)	2.60867	+	0.441423i	0.985984	+	0.166842i
							\(11\)	4.06669	+	3.24307i	1.22615	+	0.977824i	0.999993	+	0.00373334i	\(0.00118836\pi\)
0.226159	+	0.974090i	\(0.427383\pi\)
\(13\)	−0.574202	+	0.225358i	−0.159255	+	0.0625030i	−0.443629	−	0.896210i	\(-0.646309\pi\)
							0.284374	+	0.958713i	\(0.408214\pi\)
\(17\)	1.39998	+	1.29899i	0.339544	+	0.315051i	0.831452	−	0.555596i	\(-0.187510\pi\)
							−0.491908	+	0.870647i	\(0.663700\pi\)
\(19\)	4.54187	−	2.62225i	1.04198	−	0.601586i	0.121584	−	0.992581i	\(-0.461203\pi\)
							0.920393	+	0.390996i	\(0.127869\pi\)
\(23\)	−0.610005	+	0.139230i	−0.127195	+	0.0290314i	−0.285644	−	0.958336i	\(-0.592208\pi\)
							0.158450	+	0.987367i	\(0.449350\pi\)
\(29\)	−2.22091	−	7.20001i	−0.412412	−	1.33701i	−0.889011	−	0.457887i	\(-0.848607\pi\)
							0.476598	−	0.879121i	\(-0.341870\pi\)
\(31\)	4.06806	−	2.34870i	0.730645	−	0.421838i	−0.0880129	−	0.996119i	\(-0.528052\pi\)
							0.818658	+	0.574281i	\(0.194718\pi\)
\(37\)	−5.42578	+	1.67363i	−0.891993	+	0.275143i	−0.706678	−	0.707536i	\(-0.749807\pi\)
							−0.185315	+	0.982679i	\(0.559331\pi\)
\(41\)	6.01384	−	4.10017i	0.939204	−	0.640338i	0.00611385	−	0.999981i	\(-0.498054\pi\)
							0.933090	+	0.359643i	\(0.117102\pi\)
\(43\)	4.94212	+	3.36948i	0.753666	+	0.513841i	0.878153	−	0.478380i	\(-0.158776\pi\)
							−0.124486	+	0.992221i	\(0.539728\pi\)
\(47\)	−0.914445	−	2.32997i	−0.133385	−	0.339861i	0.848539	−	0.529132i	\(-0.177483\pi\)
							−0.981925	+	0.189272i	\(0.939387\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.59.6	✓	648
9.2	odd	6		441.2.bn.a.353.6	yes	648
49.5	odd	42		441.2.bn.a.5.6	yes	648
441.299	even	42	inner	441.2.bd.a.299.6	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.6	✓	648	1.1	even	1	trivial
441.2.bd.a.299.6	yes	648	441.299	even	42	inner
441.2.bn.a.5.6	yes	648	49.5	odd	42
441.2.bn.a.353.6	yes	648	9.2	odd	6