Embedded newform 441.2.bd.a.110.3

• Label: 441.2.bd.a.110.3
• Level: $441$
• Weight: $2$
• Character: 441.110
• 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			110.3
Character	\(\chi\)	\(=\)	441.110
Dual form			441.2.bd.a.437.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.63305 + 0.197320i) q^{2} +(1.19240 + 1.25625i) q^{3} +(4.91638 - 0.741025i) q^{4} +(-0.535855 - 2.34773i) q^{5} +(-3.38755 - 3.07250i) q^{6} +(2.62454 + 0.334340i) q^{7} +(-7.65041 + 1.74616i) q^{8} +(-0.156350 + 2.99592i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{11}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.63305	+	0.197320i	−1.86185	+	0.139526i	−0.957715	−	0.287718i	\(-0.907103\pi\)
							−0.904136	+	0.427245i	\(0.859484\pi\)
\(3\)	1.19240	+	1.25625i	0.688434	+	0.725299i
							\(5\)	−0.535855	−	2.34773i	−0.239641	−	1.04994i	−0.941339	−	0.337463i	\(-0.890431\pi\)
0.701697	−	0.712475i	\(-0.252426\pi\)
\(7\)	2.62454	+	0.334340i	0.991983	+	0.126368i
							\(11\)	−1.46282	+	3.03758i	−0.441058	+	0.915866i	0.555385	+	0.831593i	\(0.312571\pi\)
−0.996443	+	0.0842727i	\(0.973143\pi\)
\(13\)	−3.74127	+	0.280370i	−1.03764	+	0.0777605i	−0.582640	−	0.812730i	\(-0.697980\pi\)
							−0.455002	+	0.890491i	\(0.650361\pi\)
\(17\)	7.66852	+	1.15584i	1.85989	+	0.280333i	0.980609	−	0.195972i	\(-0.0627862\pi\)
							0.879280	+	0.476305i	\(0.158024\pi\)
\(19\)	1.55966	+	0.900472i	0.357811	+	0.206583i	0.668120	−	0.744053i	\(-0.267099\pi\)
							−0.310309	+	0.950636i	\(0.600432\pi\)
\(23\)	−0.221470	+	0.176616i	−0.0461797	+	0.0368271i	−0.646308	−	0.763077i	\(-0.723688\pi\)
							0.600128	+	0.799904i	\(0.295116\pi\)
\(29\)	4.27736	−	1.67874i	0.794286	−	0.311734i	0.0667195	−	0.997772i	\(-0.478747\pi\)
							0.727566	+	0.686038i	\(0.240651\pi\)
\(31\)	−0.817145	−	0.471779i	−0.146764	−	0.0847340i	0.424820	−	0.905278i	\(-0.360337\pi\)
							−0.571584	+	0.820544i	\(0.693671\pi\)
\(37\)	1.22251	+	3.11491i	0.200980	+	0.512088i	0.995458	−	0.0952050i	\(-0.0303507\pi\)
							−0.794478	+	0.607293i	\(0.792255\pi\)
\(41\)	2.85718	−	2.65107i	0.446216	−	0.414028i	−0.424796	−	0.905289i	\(-0.639654\pi\)
							0.871013	+	0.491261i	\(0.163464\pi\)
\(43\)	3.13453	+	2.90842i	0.478012	+	0.443530i	0.882011	−	0.471229i	\(-0.156190\pi\)
							−0.403999	+	0.914760i	\(0.632380\pi\)
\(47\)	0.727989	+	9.71433i	0.106188	+	1.41698i	0.754846	+	0.655902i	\(0.227711\pi\)
							−0.648658	+	0.761080i	\(0.724670\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.110.3	✓	648
9.5	odd	6		441.2.bn.a.257.3	yes	648
49.45	odd	42		441.2.bn.a.290.3	yes	648
441.437	even	42	inner	441.2.bd.a.437.3	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.110.3	✓	648	1.1	even	1	trivial
441.2.bd.a.437.3	yes	648	441.437	even	42	inner
441.2.bn.a.257.3	yes	648	9.5	odd	6
441.2.bn.a.290.3	yes	648	49.45	odd	42