Embedded newform 441.2.y.a.121.23

• Label: 441.2.y.a.121.23
• 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			121.23
Character	\(\chi\)	\(=\)	441.121
Dual form			441.2.y.a.277.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.482381 + 0.232302i) q^{2} +(-1.59820 - 0.667645i) q^{3} +(-1.06825 + 1.33955i) q^{4} +(3.58963 - 1.10725i) q^{5} +(0.926038 - 0.0492068i) q^{6} +(-2.64249 + 0.131335i) q^{7} +(0.442401 - 1.93829i) q^{8} +(2.10850 + 2.13406i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.482381	+	0.232302i	−0.341095	+	0.164263i	−0.596587	−	0.802549i	\(-0.703477\pi\)
							0.255492	+	0.966811i	\(0.417763\pi\)
\(3\)	−1.59820	−	0.667645i	−0.922722	−	0.385465i
							\(5\)	3.58963	−	1.10725i	1.60533	−	0.495179i	0.642729	−	0.766094i	\(-0.277802\pi\)
0.962603	+	0.270914i	\(0.0873260\pi\)
\(7\)	−2.64249	+	0.131335i	−0.998767	+	0.0496401i
							\(11\)	0.209473	−	2.79522i	0.0631585	−	0.842792i	−0.872213	−	0.489126i	\(-0.837316\pi\)
0.935372	−	0.353666i	\(-0.115065\pi\)
\(13\)	−0.248953	+	3.32205i	−0.0690473	+	0.921372i	0.849888	+	0.526963i	\(0.176669\pi\)
							−0.918935	+	0.394408i	\(0.870950\pi\)
\(17\)	−0.492578	+	1.25507i	−0.119468	+	0.304399i	−0.978081	−	0.208225i	\(-0.933231\pi\)
							0.858613	+	0.512624i	\(0.171326\pi\)
\(19\)	2.71357	−	4.70005i	0.622537	−	1.07827i	−0.366475	−	0.930428i	\(-0.619436\pi\)
							0.989012	−	0.147837i	\(-0.0472312\pi\)
\(23\)	7.03171	+	1.05986i	1.46621	+	0.220996i	0.833125	−	0.553085i	\(-0.186549\pi\)
							0.633087	+	0.774081i	\(0.281788\pi\)
\(29\)	1.53670	−	3.91546i	0.285359	−	0.727083i	−0.714271	−	0.699870i	\(-0.753241\pi\)
							0.999630	−	0.0272132i	\(-0.00866330\pi\)
\(31\)	6.16905			1.10799			0.553997	−	0.832518i	\(-0.313102\pi\)
							0.553997	+	0.832518i	\(0.313102\pi\)
\(37\)	4.08330	−	0.615459i	0.671290	−	0.101181i	0.195452	−	0.980713i	\(-0.437383\pi\)
							0.475839	+	0.879533i	\(0.342145\pi\)
\(41\)	−0.513342	−	0.476312i	−0.0801706	−	0.0743874i	0.639070	−	0.769149i	\(-0.279320\pi\)
							−0.719240	+	0.694761i	\(0.755510\pi\)
\(43\)	3.36917	−	3.12614i	0.513794	−	0.476731i	−0.380140	−	0.924929i	\(-0.624124\pi\)
							0.893935	+	0.448198i	\(0.147934\pi\)
\(47\)	−2.92918	+	1.41062i	−0.427265	+	0.205760i	−0.635143	−	0.772395i	\(-0.719059\pi\)
							0.207878	+	0.978155i	\(0.433344\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.121.23	✓	648
9.7	even	3		441.2.z.a.268.32	yes	648
49.32	even	21		441.2.z.a.130.32	yes	648
441.277	even	21	inner	441.2.y.a.277.23	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.121.23	✓	648	1.1	even	1	trivial
441.2.y.a.277.23	yes	648	441.277	even	21	inner
441.2.z.a.130.32	yes	648	49.32	even	21
441.2.z.a.268.32	yes	648	9.7	even	3