Embedded newform 441.2.bh.a.20.20

• Label: 441.2.bh.a.20.20
• 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			20.20
Character	\(\chi\)	\(=\)	441.20
Dual form			441.2.bh.a.419.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.558540 - 0.819228i) q^{2} +(0.799969 - 1.53625i) q^{3} +(0.371515 - 0.946605i) q^{4} +(0.352468 - 0.327042i) q^{5} +(-1.70535 + 0.202698i) q^{6} +(2.11315 + 1.59204i) q^{7} +(-2.91630 + 0.665627i) q^{8} +(-1.72010 - 2.45790i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 15 q^{2} - 14 q^{3} - 57 q^{4} - 21 q^{5} + 14 q^{6} - 5 q^{7} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.558540	−	0.819228i	−0.394947	−	0.579281i	0.576334	−	0.817214i	\(-0.304483\pi\)
							−0.971282	+	0.237933i	\(0.923530\pi\)
\(3\)	0.799969	−	1.53625i	0.461862	−	0.886952i
							\(5\)	0.352468	−	0.327042i	0.157628	−	0.146258i	−0.597416	−	0.801932i	\(-0.703806\pi\)
0.755044	+	0.655674i	\(0.227615\pi\)
\(7\)	2.11315	+	1.59204i	0.798695	+	0.601736i
							\(11\)	−2.06122	−	3.02325i	−0.621480	−	0.911544i	0.378420	−	0.925634i	\(-0.376467\pi\)
−0.999901	+	0.0140896i	\(0.995515\pi\)
\(13\)	−0.295014	+	0.0221083i	−0.0818223	+	0.00613173i	−0.115578	−	0.993298i	\(-0.536872\pi\)
							0.0337560	+	0.999430i	\(0.489253\pi\)
\(17\)	3.30019	−	4.13830i	0.800413	−	1.00369i	−0.199305	−	0.979937i	\(-0.563869\pi\)
							0.999718	−	0.0237483i	\(-0.00756004\pi\)
\(19\)		−	0.874844i		−	0.200703i	−0.994952	−	0.100351i	\(-0.968003\pi\)
							0.994952	−	0.100351i	\(-0.0319967\pi\)
\(23\)	1.84490	+	0.724069i	0.384688	+	0.150979i	0.549801	−	0.835296i	\(-0.314704\pi\)
							−0.165113	+	0.986275i	\(0.552799\pi\)
\(29\)	−5.24955	+	2.06030i	−0.974816	+	0.382587i	−0.798665	−	0.601776i	\(-0.794460\pi\)
							−0.176151	+	0.984363i	\(0.556365\pi\)
\(31\)	1.34937	−	0.779061i	0.242355	−	0.139924i	−0.373904	−	0.927467i	\(-0.621981\pi\)
							0.616258	+	0.787544i	\(0.288648\pi\)
\(37\)	5.31442	−	6.66408i	0.873686	−	1.09557i	−0.121004	−	0.992652i	\(-0.538612\pi\)
							0.994690	−	0.102915i	\(-0.0328170\pi\)
\(41\)	0.0916130	−	0.0850045i	0.0143075	−	0.0132755i	−0.672987	−	0.739655i	\(-0.734989\pi\)
							0.687294	+	0.726379i	\(0.258798\pi\)
\(43\)	4.95192	+	4.59471i	0.755161	+	0.700687i	0.960585	−	0.277986i	\(-0.0896669\pi\)
							−0.205424	+	0.978673i	\(0.565857\pi\)
\(47\)	−4.59512	+	3.13290i	−0.670267	+	0.456980i	−0.850033	−	0.526729i	\(-0.823418\pi\)
							0.179766	+	0.983709i	\(0.442466\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bh.a.20.20	✓	648
9.5	odd	6	inner	441.2.bh.a.167.20	yes	648
49.27	odd	14	inner	441.2.bh.a.272.20	yes	648
441.419	even	42	inner	441.2.bh.a.419.20	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bh.a.20.20	✓	648	1.1	even	1	trivial
441.2.bh.a.167.20	yes	648	9.5	odd	6	inner
441.2.bh.a.272.20	yes	648	49.27	odd	14	inner
441.2.bh.a.419.20	yes	648	441.419	even	42	inner