Embedded newform 441.2.y.a.121.33

• Label: 441.2.y.a.121.33
• Level: $441$
• Weight: $2$
• Character: 441.121
• 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			121.33
Character	\(\chi\)	\(=\)	441.121
Dual form			441.2.y.a.277.33

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.672232 - 0.323730i) q^{2} +(0.453039 - 1.67175i) q^{3} +(-0.899885 + 1.12842i) q^{4} +(-3.38637 + 1.04456i) q^{5} +(-0.236648 - 1.27047i) q^{6} +(1.93065 + 1.80903i) q^{7} +(-0.571683 + 2.50471i) q^{8} +(-2.58951 - 1.51474i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.672232	−	0.323730i	0.475339	−	0.228911i	−0.180845	−	0.983512i	\(-0.557883\pi\)
							0.656185	+	0.754600i	\(0.272169\pi\)
\(3\)	0.453039	−	1.67175i	0.261562	−	0.965187i
							\(5\)	−3.38637	+	1.04456i	−1.51443	+	0.467139i	−0.937008	−	0.349307i	\(-0.886417\pi\)
−0.577421	+	0.816447i	\(0.695941\pi\)
\(7\)	1.93065	+	1.80903i	0.729719	+	0.683747i
							\(11\)	−0.0201462	+	0.268833i	−0.00607432	+	0.0810561i	−0.999417	−	0.0341403i	\(-0.989131\pi\)
0.993343	+	0.115196i	\(0.0367497\pi\)
\(13\)	−0.418942	+	5.59039i	−0.116194	+	1.55049i	0.569125	+	0.822251i	\(0.307282\pi\)
							−0.685318	+	0.728244i	\(0.740337\pi\)
\(17\)	1.31622	−	3.35367i	0.319230	−	0.813385i	−0.677718	−	0.735322i	\(-0.737031\pi\)
							0.996948	−	0.0780636i	\(-0.0248737\pi\)
\(19\)	−3.46534	+	6.00215i	−0.795005	+	1.37699i	0.127832	+	0.991796i	\(0.459198\pi\)
							−0.922836	+	0.385192i	\(0.874135\pi\)
\(23\)	3.26784	+	0.492548i	0.681393	+	0.102703i	0.480613	−	0.876933i	\(-0.340414\pi\)
							0.200780	+	0.979636i	\(0.435652\pi\)
\(29\)	−2.71271	+	6.91186i	−0.503737	+	1.28350i	0.422433	+	0.906394i	\(0.361176\pi\)
							−0.926170	+	0.377107i	\(0.876919\pi\)
\(31\)	−5.55353			−0.997443			−0.498722	−	0.866762i	\(-0.666197\pi\)
							−0.498722	+	0.866762i	\(0.666197\pi\)
\(37\)	6.61905	−	0.997661i	1.08817	−	0.164014i	0.419630	−	0.907695i	\(-0.362160\pi\)
							0.668535	+	0.743681i	\(0.266922\pi\)
\(41\)	−8.53017	−	7.91485i	−1.33219	−	1.23609i	−0.950092	−	0.311969i	\(-0.899012\pi\)
							−0.382097	−	0.924122i	\(-0.624798\pi\)
\(43\)	4.24597	−	3.93969i	0.647505	−	0.600796i	−0.286490	−	0.958083i	\(-0.592489\pi\)
							0.933995	+	0.357287i	\(0.116298\pi\)
\(47\)	2.55841	−	1.23206i	0.373182	−	0.179715i	−0.237886	−	0.971293i	\(-0.576455\pi\)
							0.611068	+	0.791578i	\(0.290740\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.121.33	✓	648
9.7	even	3		441.2.z.a.268.22	yes	648
49.32	even	21		441.2.z.a.130.22	yes	648
441.277	even	21	inner	441.2.y.a.277.33	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.121.33	✓	648	1.1	even	1	trivial
441.2.y.a.277.33	yes	648	441.277	even	21	inner
441.2.z.a.130.22	yes	648	49.32	even	21
441.2.z.a.268.22	yes	648	9.7	even	3