Embedded newform 441.2.z.a.142.42

• Label: 441.2.z.a.142.42
• Level: $441$
• Weight: $2$
• Character: 441.142
• 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			142.42
Character	\(\chi\)	\(=\)	441.142
Dual form			441.2.z.a.382.42

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.617138 - 1.57244i) q^{2} +(-0.0875646 - 1.72984i) q^{3} +(-0.625612 - 0.580484i) q^{4} +(2.02014 - 0.972848i) q^{5} +(-2.77411 - 0.929858i) q^{6} +(-2.54277 - 0.730966i) q^{7} +(1.74499 - 0.840342i) q^{8} +(-2.98466 + 0.302945i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{4}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.617138	−	1.57244i	0.436383	−	1.11188i	−0.528437	−	0.848972i	\(-0.677222\pi\)
							0.964820	−	0.262912i	\(-0.0846830\pi\)
\(3\)	−0.0875646	−	1.72984i	−0.0505554	−	0.998721i
							\(5\)	2.02014	−	0.972848i	0.903434	−	0.435071i	0.0763061	−	0.997084i	\(-0.475687\pi\)
0.827128	+	0.562013i	\(0.189973\pi\)
\(7\)	−2.54277	−	0.730966i	−0.961077	−	0.276279i
							\(11\)	0.293345	+	0.367843i	0.0884468	+	0.110909i	0.824086	−	0.566465i	\(-0.191689\pi\)
−0.735639	+	0.677374i	\(0.763118\pi\)
\(13\)	−0.290222	+	0.739473i	−0.0804930	+	0.205093i	−0.965455	−	0.260571i	\(-0.916089\pi\)
							0.884962	+	0.465664i	\(0.154184\pi\)
\(17\)	−0.101566	+	0.0942390i	−0.0246333	+	0.0228563i	−0.692393	−	0.721520i	\(-0.743444\pi\)
							0.667760	+	0.744377i	\(0.267253\pi\)
\(19\)	2.14455	−	3.71448i	0.491994	−	0.852159i	−0.507963	−	0.861379i	\(-0.669601\pi\)
							0.999957	+	0.00921960i	\(0.00293473\pi\)
\(23\)	−0.596871	+	2.61506i	−0.124456	+	0.545278i	0.873802	+	0.486282i	\(0.161647\pi\)
							−0.998258	+	0.0589963i	\(0.981210\pi\)
\(29\)	1.30872	+	0.403687i	0.243023	+	0.0749627i	0.413874	−	0.910334i	\(-0.364175\pi\)
							−0.170851	+	0.985297i	\(0.554652\pi\)
\(31\)	0.681939	−	1.18115i	0.122480	−	0.212141i	−0.798265	−	0.602306i	\(-0.794249\pi\)
							0.920745	+	0.390165i	\(0.127582\pi\)
\(37\)	9.82663	+	3.03111i	1.61549	+	0.498312i	0.965230	−	0.261402i	\(-0.0841847\pi\)
							0.650258	+	0.759714i	\(0.274661\pi\)
\(41\)	4.87878	+	3.32630i	0.761938	+	0.519481i	0.880834	−	0.473426i	\(-0.156983\pi\)
							−0.118895	+	0.992907i	\(0.537935\pi\)
\(43\)	−5.40132	+	3.68255i	−0.823693	+	0.561584i	−0.900196	−	0.435485i	\(-0.856577\pi\)
							0.0765030	+	0.997069i	\(0.475625\pi\)
\(47\)	1.41103	−	3.59524i	0.205820	−	0.524420i	−0.790271	−	0.612758i	\(-0.790060\pi\)
							0.996091	+	0.0883376i	\(0.0281554\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.142.42	yes	648
9.4	even	3		441.2.y.a.436.42	yes	648
49.39	even	21		441.2.y.a.88.42	✓	648
441.382	even	21	inner	441.2.z.a.382.42	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.88.42	✓	648	49.39	even	21
441.2.y.a.436.42	yes	648	9.4	even	3
441.2.z.a.142.42	yes	648	1.1	even	1	trivial
441.2.z.a.382.42	yes	648	441.382	even	21	inner