Embedded newform 441.2.y.a.247.19

• Label: 441.2.y.a.247.19
• Level: $441$
• Weight: $2$
• Character: 441.247
• 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			247.19
Character	\(\chi\)	\(=\)	441.247
Dual form			441.2.y.a.25.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.234242 - 1.02628i) q^{2} +(-0.449041 - 1.67283i) q^{3} +(0.803556 - 0.386972i) q^{4} +(0.973235 - 2.47976i) q^{5} +(-1.61161 + 0.852689i) q^{6} +(-1.43872 + 2.22038i) q^{7} +(-1.89803 - 2.38005i) q^{8} +(-2.59672 + 1.50234i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.234242	−	1.02628i	−0.165634	−	0.725690i	−0.987708	−	0.156309i	\(-0.950040\pi\)
							0.822074	−	0.569380i	\(-0.192817\pi\)
\(3\)	−0.449041	−	1.67283i	−0.259254	−	0.965809i
							\(5\)	0.973235	−	2.47976i	0.435244	−	1.10898i	−0.530084	−	0.847945i	\(-0.677840\pi\)
0.965328	−	0.261039i	\(-0.0840651\pi\)
\(7\)	−1.43872	+	2.22038i	−0.543786	+	0.839224i
							\(11\)	−2.81587	−	0.868582i	−0.849018	−	0.261887i	−0.160436	−	0.987046i	\(-0.551290\pi\)
−0.688582	+	0.725159i	\(0.741766\pi\)
\(13\)	0.821387	+	0.253364i	0.227812	+	0.0702706i	0.406559	−	0.913624i	\(-0.366728\pi\)
							−0.178748	+	0.983895i	\(0.557205\pi\)
\(17\)	−0.176020	−	2.34882i	−0.0426911	−	0.569673i	−0.977004	−	0.213220i	\(-0.931605\pi\)
							0.934313	−	0.356453i	\(-0.116014\pi\)
\(19\)	1.20770	−	2.09179i	0.277065	−	0.479891i	−0.693589	−	0.720371i	\(-0.743972\pi\)
							0.970654	+	0.240480i	\(0.0773049\pi\)
\(23\)	4.84253	+	3.30158i	1.00974	+	0.688427i	0.950791	−	0.309834i	\(-0.100273\pi\)
							0.0589466	+	0.998261i	\(0.481226\pi\)
\(29\)	−0.461086	−	6.15276i	−0.0856214	−	1.14254i	−0.860164	−	0.510017i	\(-0.829639\pi\)
							0.774543	−	0.632522i	\(-0.217980\pi\)
\(31\)	−2.40618			−0.432163			−0.216082	−	0.976375i	\(-0.569328\pi\)
							−0.216082	+	0.976375i	\(0.569328\pi\)
\(37\)	3.02263	−	2.06079i	0.496917	−	0.338793i	−0.288788	−	0.957393i	\(-0.593252\pi\)
							0.785706	+	0.618600i	\(0.212300\pi\)
\(41\)	6.15025	−	0.927001i	0.960508	−	0.144773i	0.349971	−	0.936761i	\(-0.386192\pi\)
							0.610537	+	0.791987i	\(0.290954\pi\)
\(43\)	−6.82726	−	1.02904i	−1.04115	−	0.156928i	−0.393857	−	0.919172i	\(-0.628860\pi\)
							−0.647290	+	0.762244i	\(0.724098\pi\)
\(47\)	1.70109	+	7.45295i	0.248129	+	1.08712i	0.933401	+	0.358836i	\(0.116826\pi\)
							−0.685272	+	0.728288i	\(0.740317\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.247.19	yes	648
9.7	even	3		441.2.z.a.394.19	yes	648
49.25	even	21		441.2.z.a.319.19	yes	648
441.25	even	21	inner	441.2.y.a.25.19	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.25.19	✓	648	441.25	even	21	inner
441.2.y.a.247.19	yes	648	1.1	even	1	trivial
441.2.z.a.319.19	yes	648	49.25	even	21
441.2.z.a.394.19	yes	648	9.7	even	3