Embedded newform 441.2.y.a.25.8

• Label: 441.2.y.a.25.8
• Level: $441$
• Weight: $2$
• Character: 441.25
• 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			25.8
Character	\(\chi\)	\(=\)	441.25
Dual form			441.2.y.a.247.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.502908 + 2.20339i) q^{2} +(-0.976404 - 1.43061i) q^{3} +(-2.80006 - 1.34844i) q^{4} +(-0.712094 - 1.81439i) q^{5} +(3.64322 - 1.43193i) q^{6} +(-2.27396 + 1.35244i) q^{7} +(1.56106 - 1.95751i) q^{8} +(-1.09327 + 2.79370i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.502908	+	2.20339i	−0.355610	+	1.55803i	0.408387	+	0.912809i	\(0.366091\pi\)
							−0.763997	+	0.645220i	\(0.776766\pi\)
\(3\)	−0.976404	−	1.43061i	−0.563727	−	0.825961i
							\(5\)	−0.712094	−	1.81439i	−0.318458	−	0.811418i	−0.997039	−	0.0768925i	\(-0.975500\pi\)
0.678581	−	0.734525i	\(-0.262595\pi\)
\(7\)	−2.27396	+	1.35244i	−0.859477	+	0.511174i
							\(11\)	2.81030	−	0.866864i	0.847338	−	0.261369i	0.159468	−	0.987203i	\(-0.449022\pi\)
0.687870	+	0.725834i	\(0.258546\pi\)
\(13\)	5.66736	−	1.74815i	1.57184	−	0.484849i	0.618259	−	0.785975i	\(-0.287838\pi\)
							0.953585	+	0.301125i	\(0.0973623\pi\)
\(17\)	−0.339436	+	4.52946i	−0.0823253	+	1.09855i	0.791129	+	0.611649i	\(0.209494\pi\)
							−0.873455	+	0.486906i	\(0.838126\pi\)
\(19\)	3.26637	+	5.65752i	0.749356	+	1.29792i	0.948132	+	0.317878i	\(0.102970\pi\)
							−0.198775	+	0.980045i	\(0.563696\pi\)
\(23\)	−4.92188	+	3.35568i	−1.02628	+	0.699707i	−0.954652	−	0.297724i	\(-0.903773\pi\)
							−0.0716303	+	0.997431i	\(0.522820\pi\)
\(29\)	−0.371450	+	4.95666i	−0.0689766	+	0.920429i	0.850169	+	0.526510i	\(0.176500\pi\)
							−0.919145	+	0.393918i	\(0.871119\pi\)
\(31\)	5.38510			0.967192			0.483596	−	0.875291i	\(-0.339330\pi\)
							0.483596	+	0.875291i	\(0.339330\pi\)
\(37\)	3.85221	+	2.62639i	0.633299	+	0.431776i	0.836959	−	0.547266i	\(-0.184331\pi\)
							−0.203660	+	0.979042i	\(0.565284\pi\)
\(41\)	−2.95932	−	0.446046i	−0.462168	−	0.0696607i	−0.0861689	−	0.996281i	\(-0.527462\pi\)
							−0.376000	+	0.926620i	\(0.622701\pi\)
\(43\)	−0.543360	+	0.0818984i	−0.0828617	+	0.0124894i	−0.190342	−	0.981718i	\(-0.560960\pi\)
							0.107481	+	0.994207i	\(0.465722\pi\)
\(47\)	−1.53213	+	6.71270i	−0.223484	+	0.979147i	0.731349	+	0.682004i	\(0.238891\pi\)
							−0.954833	+	0.297144i	\(0.903966\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.25.8	✓	648
9.4	even	3		441.2.z.a.319.8	yes	648
49.2	even	21		441.2.z.a.394.8	yes	648
441.247	even	21	inner	441.2.y.a.247.8	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.25.8	✓	648	1.1	even	1	trivial
441.2.y.a.247.8	yes	648	441.247	even	21	inner
441.2.z.a.319.8	yes	648	9.4	even	3
441.2.z.a.394.8	yes	648	49.2	even	21