Embedded newform 441.2.y.a.151.42

• Label: 441.2.y.a.151.42
• Level: $441$
• Weight: $2$
• Character: 441.151
• 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.y (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			151.42
Character	\(\chi\)	\(=\)	441.151
Dual form			441.2.y.a.184.42

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.47988 - 0.712674i) q^{2} +(1.72088 + 0.196399i) q^{3} +(0.435171 - 0.545687i) q^{4} +(1.16747 + 1.08325i) q^{5} +(2.68667 - 0.935780i) q^{6} +(0.0971625 - 2.64397i) q^{7} +(-0.475897 + 2.08504i) q^{8} +(2.92285 + 0.675958i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.47988	−	0.712674i	1.04644	−	0.503937i	0.169994	−	0.985445i	\(-0.445625\pi\)
							0.876442	+	0.481508i	\(0.159911\pi\)
\(3\)	1.72088	+	0.196399i	0.993550	+	0.113391i
							\(5\)	1.16747	+	1.08325i	0.522108	+	0.484446i	0.896635	−	0.442770i	\(-0.146004\pi\)
−0.374527	+	0.927216i	\(0.622195\pi\)
\(7\)	0.0971625	−	2.64397i	0.0367240	−	0.999325i
							\(11\)	−1.71372	−	1.16840i	−0.516707	−	0.352285i	0.276720	−	0.960951i	\(-0.410753\pi\)
−0.793427	+	0.608666i	\(0.791705\pi\)
\(13\)	−1.42767	−	0.973367i	−0.395964	−	0.269963i	0.348927	−	0.937150i	\(-0.386546\pi\)
							−0.744891	+	0.667186i	\(0.767498\pi\)
\(17\)	−3.41500	+	0.514728i	−0.828259	+	0.124840i	−0.549471	−	0.835513i	\(-0.685171\pi\)
							−0.278788	+	0.960353i	\(0.589933\pi\)
\(19\)	1.13809	+	1.97123i	0.261095	+	0.452231i	0.966533	−	0.256541i	\(-0.0825829\pi\)
							−0.705438	+	0.708772i	\(0.749250\pi\)
\(23\)	−0.323688	−	0.824744i	−0.0674936	−	0.171971i	0.893147	−	0.449765i	\(-0.148492\pi\)
							−0.960641	+	0.277794i	\(0.910397\pi\)
\(29\)	−2.00310	+	0.301918i	−0.371966	+	0.0560648i	−0.332365	−	0.943151i	\(-0.607847\pi\)
							−0.0396006	+	0.999216i	\(0.512609\pi\)
\(31\)	−1.55276			−0.278884			−0.139442	−	0.990230i	\(-0.544531\pi\)
							−0.139442	+	0.990230i	\(0.544531\pi\)
\(37\)	0.883157	−	2.25025i	0.145190	−	0.369938i	−0.839717	−	0.543025i	\(-0.817279\pi\)
							0.984907	+	0.173087i	\(0.0553741\pi\)
\(41\)	9.93603	−	3.06486i	1.55175	−	0.478651i	0.603816	−	0.797124i	\(-0.293646\pi\)
							0.947932	+	0.318473i	\(0.103170\pi\)
\(43\)	−9.10386	−	2.80817i	−1.38833	−	0.428242i	−0.491597	−	0.870823i	\(-0.663587\pi\)
							−0.896729	+	0.442581i	\(0.854063\pi\)
\(47\)	−10.3895	+	5.00332i	−1.51547	+	0.729810i	−0.992466	−	0.122523i	\(-0.960902\pi\)
							−0.523000	+	0.852333i	\(0.675187\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.151.42	✓	648
9.4	even	3		441.2.z.a.4.13	yes	648
49.37	even	21		441.2.z.a.331.13	yes	648
441.184	even	21	inner	441.2.y.a.184.42	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.42	✓	648	1.1	even	1	trivial
441.2.y.a.184.42	yes	648	441.184	even	21	inner
441.2.z.a.4.13	yes	648	9.4	even	3
441.2.z.a.331.13	yes	648	49.37	even	21