Embedded newform 441.2.y.a.25.17

• Label: 441.2.y.a.25.17
• 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			25.17
Character	\(\chi\)	\(=\)	441.25
Dual form			441.2.y.a.247.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.279438 + 1.22430i) q^{2} +(-0.359714 - 1.69429i) q^{3} +(0.381117 + 0.183536i) q^{4} +(0.462772 + 1.17912i) q^{5} +(2.17483 + 0.0330508i) q^{6} +(2.42137 - 1.06628i) q^{7} +(-1.89714 + 2.37894i) q^{8} +(-2.74121 + 1.21892i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.279438	+	1.22430i	−0.197593	+	0.865710i	0.774772	+	0.632241i	\(0.217865\pi\)
							−0.972364	+	0.233469i	\(0.924992\pi\)
\(3\)	−0.359714	−	1.69429i	−0.207681	−	0.978197i
							\(5\)	0.462772	+	1.17912i	0.206958	+	0.527321i	0.996233	−	0.0867197i	\(-0.0276385\pi\)
−0.789275	+	0.614040i	\(0.789543\pi\)
\(7\)	2.42137	−	1.06628i	0.915192	−	0.403017i
							\(11\)	4.75556	−	1.46690i	1.43385	−	0.442286i	0.522052	−	0.852914i	\(-0.325167\pi\)
0.911803	+	0.410628i	\(0.134691\pi\)
\(13\)	−1.23794	+	0.381853i	−0.343342	+	0.105907i	−0.461631	−	0.887072i	\(-0.652735\pi\)
							0.118288	+	0.992979i	\(0.462259\pi\)
\(17\)	−0.447947	+	5.97743i	−0.108643	+	1.44974i	0.630499	+	0.776190i	\(0.282850\pi\)
							−0.739142	+	0.673550i	\(0.764769\pi\)
\(19\)	−1.89422	−	3.28088i	−0.434564	−	0.752686i	0.562696	−	0.826664i	\(-0.309764\pi\)
							−0.997260	+	0.0739774i	\(0.976431\pi\)
\(23\)	7.22775	−	4.92780i	1.50709	−	1.02752i	0.522501	−	0.852639i	\(-0.324999\pi\)
							0.984590	−	0.174879i	\(-0.0559533\pi\)
\(29\)	−0.665662	+	8.88264i	−0.123610	+	1.64946i	0.498759	+	0.866741i	\(0.333789\pi\)
							−0.622369	+	0.782724i	\(0.713830\pi\)
\(31\)	1.52967			0.274737			0.137369	−	0.990520i	\(-0.456136\pi\)
							0.137369	+	0.990520i	\(0.456136\pi\)
\(37\)	4.00466	+	2.73033i	0.658361	+	0.448863i	0.845862	−	0.533401i	\(-0.179086\pi\)
							−0.187501	+	0.982264i	\(0.560039\pi\)
\(41\)	−5.10400	−	0.769304i	−0.797111	−	0.120145i	−0.262156	−	0.965026i	\(-0.584433\pi\)
							−0.534955	+	0.844880i	\(0.679672\pi\)
\(43\)	3.15177	−	0.475052i	0.480640	−	0.0724448i	0.0957463	−	0.995406i	\(-0.469476\pi\)
							0.384894	+	0.922961i	\(0.374238\pi\)
\(47\)	−0.869950	+	3.81150i	−0.126895	+	0.555965i	0.871010	+	0.491266i	\(0.163466\pi\)
							−0.997905	+	0.0646986i	\(0.979391\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.25.17	✓	648
9.4	even	3		441.2.z.a.319.17	yes	648
49.2	even	21		441.2.z.a.394.17	yes	648
441.247	even	21	inner	441.2.y.a.247.17	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.25.17	✓	648	1.1	even	1	trivial
441.2.y.a.247.17	yes	648	441.247	even	21	inner
441.2.z.a.319.17	yes	648	9.4	even	3
441.2.z.a.394.17	yes	648	49.2	even	21