Embedded newform 441.2.bh.a.20.12

• Label: 441.2.bh.a.20.12
• Level: $441$
• Weight: $2$
• Character: 441.20
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.bh (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			20.12
Character	\(\chi\)	\(=\)	441.20
Dual form			441.2.bh.a.419.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.01131 - 1.48331i) q^{2} +(1.68226 - 0.412308i) q^{3} +(-0.446798 + 1.13842i) q^{4} +(-1.62492 + 1.50771i) q^{5} +(-2.31286 - 2.07835i) q^{6} +(-2.44442 - 1.01232i) q^{7} +(-1.36001 + 0.310414i) q^{8} +(2.66000 - 1.38722i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 15 q^{2} - 14 q^{3} - 57 q^{4} - 21 q^{5} + 14 q^{6} - 5 q^{7} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.01131	−	1.48331i	−0.715102	−	1.04886i	−0.996020	−	0.0891297i	\(-0.971591\pi\)
							0.280919	−	0.959732i	\(-0.409361\pi\)
\(3\)	1.68226	−	0.412308i	0.971254	−	0.238046i
							\(5\)	−1.62492	+	1.50771i	−0.726687	+	0.674267i	−0.954085	−	0.299535i	\(-0.903168\pi\)
0.227399	+	0.973802i	\(0.426978\pi\)
\(7\)	−2.44442	−	1.01232i	−0.923905	−	0.382623i
							\(11\)	−2.32313	−	3.40741i	−0.700451	−	1.02737i	−0.997371	−	0.0724705i	\(-0.976912\pi\)
0.296920	−	0.954903i	\(-0.404041\pi\)
\(13\)	−2.63562	+	0.197513i	−0.730990	+	0.0547802i	−0.435025	−	0.900418i	\(-0.643260\pi\)
							−0.295966	+	0.955199i	\(0.595641\pi\)
\(17\)	−2.06268	+	2.58652i	−0.500274	+	0.627324i	−0.966291	−	0.257452i	\(-0.917117\pi\)
							0.466017	+	0.884776i	\(0.345689\pi\)
\(19\)		−	4.64750i		−	1.06621i	−0.846049	−	0.533104i	\(-0.821025\pi\)
							0.846049	−	0.533104i	\(-0.178975\pi\)
\(23\)	−5.97432	−	2.34475i	−1.24573	−	0.488914i	−0.351361	−	0.936240i	\(-0.614281\pi\)
							−0.894371	+	0.447326i	\(0.852376\pi\)
\(29\)	−5.01597	+	1.96862i	−0.931443	+	0.365564i	−0.782051	−	0.623214i	\(-0.785827\pi\)
							−0.149391	+	0.988778i	\(0.547731\pi\)
\(31\)	1.15408	−	0.666311i	0.207280	−	0.119673i	−0.392767	−	0.919638i	\(-0.628482\pi\)
							0.600046	+	0.799965i	\(0.295149\pi\)
\(37\)	5.77118	−	7.23683i	0.948777	−	1.18973i	−0.0329550	−	0.999457i	\(-0.510492\pi\)
							0.981732	−	0.190271i	\(-0.0609368\pi\)
\(41\)	−0.585215	+	0.543000i	−0.0913952	+	0.0848024i	−0.724547	−	0.689225i	\(-0.757951\pi\)
							0.633152	+	0.774027i	\(0.281761\pi\)
\(43\)	0.0385621	+	0.0357804i	0.00588067	+	0.00545647i	0.683108	−	0.730318i	\(-0.260628\pi\)
							−0.677227	+	0.735774i	\(0.736818\pi\)
\(47\)	5.37806	−	3.66670i	0.784471	−	0.534843i	−0.103562	−	0.994623i	\(-0.533024\pi\)
							0.888033	+	0.459780i	\(0.152072\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bh.a.20.12	✓	648
9.5	odd	6	inner	441.2.bh.a.167.12	yes	648
49.27	odd	14	inner	441.2.bh.a.272.12	yes	648
441.419	even	42	inner	441.2.bh.a.419.12	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bh.a.20.12	✓	648	1.1	even	1	trivial
441.2.bh.a.167.12	yes	648	9.5	odd	6	inner
441.2.bh.a.272.12	yes	648	49.27	odd	14	inner
441.2.bh.a.419.12	yes	648	441.419	even	42	inner