Embedded newform 441.2.z.a.4.14

• Label: 441.2.z.a.4.14
• Level: $441$
• Weight: $2$
• Character: 441.4
• 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.z (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			4.14
Character	\(\chi\)	\(=\)	441.4
Dual form			441.2.z.a.331.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.109094 + 1.45576i) q^{2} +(1.61060 + 0.637154i) q^{3} +(-0.129671 - 0.0195447i) q^{4} +(-0.0506126 + 0.221748i) q^{5} +(-1.10325 + 2.27514i) q^{6} +(1.03301 - 2.43575i) q^{7} +(-0.607092 + 2.65984i) q^{8} +(2.18807 + 2.05240i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.109094	+	1.45576i	−0.0771411	+	1.02938i	0.815544	+	0.578696i	\(0.196438\pi\)
							−0.892685	+	0.450681i	\(0.851181\pi\)
\(3\)	1.61060	+	0.637154i	0.929881	+	0.367861i
							\(5\)	−0.0506126	+	0.221748i	−0.0226346	+	0.0991688i	−0.984983	−	0.172649i	\(-0.944768\pi\)
0.962349	+	0.271817i	\(0.0876247\pi\)
\(7\)	1.03301	−	2.43575i	0.390441	−	0.920628i
							\(11\)	−4.68175	+	2.25461i	−1.41160	+	0.679791i	−0.975478	−	0.220096i	\(-0.929363\pi\)
−0.436122	+	0.899887i	\(0.643648\pi\)
\(13\)	−0.0162605	+	0.216981i	−0.00450985	+	0.0601797i	−0.999005	−	0.0446062i	\(-0.985797\pi\)
							0.994495	+	0.104786i	\(0.0334157\pi\)
\(17\)	6.00548	−	0.905180i	1.45654	−	0.219538i	0.627453	−	0.778655i	\(-0.284098\pi\)
							0.829089	+	0.559116i	\(0.188859\pi\)
\(19\)	1.38539	+	2.39957i	0.317831	+	0.550500i	0.980035	−	0.198824i	\(-0.0637121\pi\)
							−0.662204	+	0.749324i	\(0.730379\pi\)
\(23\)	−3.70233	+	4.64258i	−0.771990	+	0.968045i	−0.999984	−	0.00561750i	\(-0.998212\pi\)
							0.227994	+	0.973663i	\(0.426783\pi\)
\(29\)	3.32444	−	8.47053i	0.617333	−	1.57294i	−0.189322	−	0.981915i	\(-0.560629\pi\)
							0.806655	−	0.591023i	\(-0.201276\pi\)
\(31\)	−3.33019	−	5.76806i	−0.598120	−	1.03597i	−0.993098	−	0.117283i	\(-0.962581\pi\)
							0.394979	−	0.918690i	\(-0.370752\pi\)
\(37\)	1.39309	−	3.54954i	0.229023	−	0.583542i	−0.769454	−	0.638702i	\(-0.779472\pi\)
							0.998478	+	0.0551603i	\(0.0175670\pi\)
\(41\)	−3.64611	−	3.38309i	−0.569426	−	0.528350i	0.342062	−	0.939678i	\(-0.388875\pi\)
							−0.911488	+	0.411327i	\(0.865065\pi\)
\(43\)	2.69874	−	2.50406i	0.411553	−	0.381866i	−0.447048	−	0.894510i	\(-0.647525\pi\)
							0.858601	+	0.512644i	\(0.171334\pi\)
\(47\)	0.0503245	−	0.671534i	0.00734058	−	0.0979533i	−0.992330	−	0.123617i	\(-0.960551\pi\)
							0.999671	+	0.0256633i	\(0.00816977\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.4.14	yes	648
9.7	even	3		441.2.y.a.151.41	✓	648
49.37	even	21		441.2.y.a.184.41	yes	648
441.331	even	21	inner	441.2.z.a.331.14	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.41	✓	648	9.7	even	3
441.2.y.a.184.41	yes	648	49.37	even	21
441.2.z.a.4.14	yes	648	1.1	even	1	trivial
441.2.z.a.331.14	yes	648	441.331	even	21	inner