Embedded newform 441.2.y.a.25.15

• Label: 441.2.y.a.25.15
• 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			25.15
Character	\(\chi\)	\(=\)	441.25
Dual form			441.2.y.a.247.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.316624 + 1.38722i) q^{2} +(-1.35055 + 1.08445i) q^{3} +(-0.0221929 - 0.0106875i) q^{4} +(-1.32101 - 3.36589i) q^{5} +(-1.07675 - 2.21687i) q^{6} +(-1.86586 + 1.87578i) q^{7} +(-1.75247 + 2.19753i) q^{8} +(0.647946 - 2.92919i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.316624	+	1.38722i	−0.223887	+	0.980913i	0.730634	+	0.682769i	\(0.239225\pi\)
							−0.954521	+	0.298144i	\(0.903633\pi\)
\(3\)	−1.35055	+	1.08445i	−0.779738	+	0.626106i
							\(5\)	−1.32101	−	3.36589i	−0.590776	−	1.50527i	−0.843170	−	0.537647i	\(-0.819313\pi\)
0.252394	−	0.967624i	\(-0.418782\pi\)
\(7\)	−1.86586	+	1.87578i	−0.705229	+	0.708980i
							\(11\)	−2.70852	+	0.835468i	−0.816650	+	0.251903i	−0.674815	−	0.737987i	\(-0.735776\pi\)
−0.141835	+	0.989890i	\(0.545300\pi\)
\(13\)	4.22678	−	1.30379i	1.17230	−	0.361606i	0.353360	−	0.935487i	\(-0.385039\pi\)
							0.818939	+	0.573881i	\(0.194563\pi\)
\(17\)	0.466690	−	6.22754i	0.113189	−	1.51040i	−0.594626	−	0.804002i	\(-0.702700\pi\)
							0.707815	−	0.706398i	\(-0.249681\pi\)
\(19\)	−1.82287	−	3.15731i	−0.418196	−	0.724336i	0.577562	−	0.816347i	\(-0.304004\pi\)
							−0.995758	+	0.0920105i	\(0.970671\pi\)
\(23\)	5.81866	−	3.96709i	1.21327	−	0.827196i	0.224190	−	0.974546i	\(-0.428027\pi\)
							0.989085	+	0.147349i	\(0.0470742\pi\)
\(29\)	0.304515	−	4.06347i	0.0565470	−	0.754568i	−0.894804	−	0.446459i	\(-0.852685\pi\)
							0.951351	−	0.308109i	\(-0.0996960\pi\)
\(31\)	−1.27338			−0.228706			−0.114353	−	0.993440i	\(-0.536480\pi\)
							−0.114353	+	0.993440i	\(0.536480\pi\)
\(37\)	2.38600	+	1.62675i	0.392256	+	0.267435i	0.743347	−	0.668906i	\(-0.233237\pi\)
							−0.351092	+	0.936341i	\(0.614189\pi\)
\(41\)	−10.0061	−	1.50817i	−1.56269	−	0.235537i	−0.689998	−	0.723812i	\(-0.742388\pi\)
							−0.872690	+	0.488274i	\(0.837627\pi\)
\(43\)	−6.22727	+	0.938610i	−0.949650	+	0.143137i	−0.605560	−	0.795800i	\(-0.707051\pi\)
							−0.344090	+	0.938937i	\(0.611813\pi\)
\(47\)	0.0122529	−	0.0536836i	0.00178728	−	0.00783056i	−0.974026	−	0.226434i	\(-0.927293\pi\)
							0.975814	+	0.218604i	\(0.0701502\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.25.15	✓	648
9.4	even	3		441.2.z.a.319.15	yes	648
49.2	even	21		441.2.z.a.394.15	yes	648
441.247	even	21	inner	441.2.y.a.247.15	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.25.15	✓	648	1.1	even	1	trivial
441.2.y.a.247.15	yes	648	441.247	even	21	inner
441.2.z.a.319.15	yes	648	9.4	even	3
441.2.z.a.394.15	yes	648	49.2	even	21