Embedded newform 441.2.y.a.25.1

• Label: 441.2.y.a.25.1
• 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			25.1
Character	\(\chi\)	\(=\)	441.25
Dual form			441.2.y.a.247.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.620557 + 2.71884i) q^{2} +(-1.73004 + 0.0835206i) q^{3} +(-5.20506 - 2.50662i) q^{4} +(-0.446610 - 1.13794i) q^{5} +(0.846508 - 4.75552i) q^{6} +(2.43233 + 1.04104i) q^{7} +(6.56762 - 8.23553i) q^{8} +(2.98605 - 0.288987i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.620557	+	2.71884i	−0.438800	+	1.92251i	−0.0566933	+	0.998392i	\(0.518056\pi\)
							−0.382107	+	0.924118i	\(0.624801\pi\)
\(3\)	−1.73004	+	0.0835206i	−0.998837	+	0.0482206i
							\(5\)	−0.446610	−	1.13794i	−0.199730	−	0.508904i	0.795557	−	0.605879i	\(-0.207179\pi\)
−0.995287	+	0.0969750i	\(0.969083\pi\)
\(7\)	2.43233	+	1.04104i	0.919335	+	0.393475i
							\(11\)	−2.80334	+	0.864717i	−0.845239	+	0.260722i	−0.686981	−	0.726676i	\(-0.741064\pi\)
−0.158259	+	0.987398i	\(0.550588\pi\)
\(13\)	2.70262	−	0.833647i	0.749571	−	0.231212i	0.103647	−	0.994614i	\(-0.466949\pi\)
							0.645923	+	0.763402i	\(0.276473\pi\)
\(17\)	−0.0273221	+	0.364589i	−0.00662659	+	0.0884257i	−0.999536	−	0.0304437i	\(-0.990308\pi\)
							0.992910	+	0.118869i	\(0.0379270\pi\)
\(19\)	−2.60649	−	4.51458i	−0.597970	−	1.03571i	−0.993120	−	0.117098i	\(-0.962641\pi\)
							0.395150	−	0.918617i	\(-0.370693\pi\)
\(23\)	2.49226	−	1.69920i	0.519673	−	0.354307i	−0.274903	−	0.961472i	\(-0.588646\pi\)
							0.794576	+	0.607165i	\(0.207693\pi\)
\(29\)	−0.755074	+	10.0758i	−0.140214	+	1.87102i	0.275198	+	0.961388i	\(0.411257\pi\)
							−0.415412	+	0.909634i	\(0.636362\pi\)
\(31\)	2.84390			0.510780			0.255390	−	0.966838i	\(-0.417796\pi\)
							0.255390	+	0.966838i	\(0.417796\pi\)
\(37\)	5.41836	+	3.69418i	0.890773	+	0.607319i	0.919855	−	0.392258i	\(-0.128306\pi\)
							−0.0290819	+	0.999577i	\(0.509258\pi\)
\(41\)	8.64517	+	1.30305i	1.35015	+	0.203502i	0.783979	−	0.620787i	\(-0.213187\pi\)
							0.566169	+	0.824289i	\(0.308425\pi\)
\(43\)	4.30561	−	0.648967i	0.656600	−	0.0989665i	0.187712	−	0.982224i	\(-0.439893\pi\)
							0.468888	+	0.883258i	\(0.344655\pi\)
\(47\)	0.0379583	−	0.166306i	0.00553678	−	0.0242582i	−0.972084	−	0.234633i	\(-0.924611\pi\)
							0.977621	+	0.210375i	\(0.0674684\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.25.1	✓	648
9.4	even	3		441.2.z.a.319.1	yes	648
49.2	even	21		441.2.z.a.394.1	yes	648
441.247	even	21	inner	441.2.y.a.247.1	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.25.1	✓	648	1.1	even	1	trivial
441.2.y.a.247.1	yes	648	441.247	even	21	inner
441.2.z.a.319.1	yes	648	9.4	even	3
441.2.z.a.394.1	yes	648	49.2	even	21