Embedded newform 441.2.y.a.277.14

• Label: 441.2.y.a.277.14
• Level: $441$
• Weight: $2$
• Character: 441.277
• 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			277.14
Character	\(\chi\)	\(=\)	441.277
Dual form			441.2.y.a.121.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33704 - 0.643883i) q^{2} +(1.30409 - 1.13989i) q^{3} +(0.126103 + 0.158129i) q^{4} +(3.60505 + 1.11201i) q^{5} +(-2.47757 + 0.684393i) q^{6} +(-0.518907 - 2.59437i) q^{7} +(0.593653 + 2.60096i) q^{8} +(0.401303 - 2.97304i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33704	−	0.643883i	−0.945428	−	0.455294i	−0.103347	−	0.994645i	\(-0.532955\pi\)
							−0.842081	+	0.539351i	\(0.818670\pi\)
\(3\)	1.30409	−	1.13989i	0.752917	−	0.658116i
							\(5\)	3.60505	+	1.11201i	1.61223	+	0.497307i	0.964393	−	0.264473i	\(-0.0851979\pi\)
0.647836	+	0.761780i	\(0.275674\pi\)
\(7\)	−0.518907	−	2.59437i	−0.196129	−	0.980578i
							\(11\)	−0.253822	−	3.38702i	−0.0765303	−	1.02123i	−0.894822	−	0.446423i	\(-0.852698\pi\)
0.818292	−	0.574803i	\(-0.194921\pi\)
\(13\)	0.343112	+	4.57851i	0.0951621	+	1.26985i	0.816532	+	0.577300i	\(0.195894\pi\)
							−0.721370	+	0.692550i	\(0.756487\pi\)
\(17\)	−1.28155	−	3.26534i	−0.310822	−	0.791961i	−0.997863	−	0.0653426i	\(-0.979186\pi\)
							0.687041	−	0.726619i	\(-0.258909\pi\)
\(19\)	2.34967	+	4.06975i	0.539052	+	0.933665i	0.998955	+	0.0456960i	\(0.0145506\pi\)
							−0.459904	+	0.887969i	\(0.652116\pi\)
\(23\)	4.92073	−	0.741681i	1.02604	−	0.154651i	0.385608	−	0.922663i	\(-0.373992\pi\)
							0.640436	+	0.768012i	\(0.278754\pi\)
\(29\)	0.819201	+	2.08729i	0.152122	+	0.387600i	0.986537	−	0.163536i	\(-0.0522899\pi\)
							−0.834416	+	0.551136i	\(0.814195\pi\)
\(31\)	−5.72451			−1.02815			−0.514076	−	0.857745i	\(-0.671865\pi\)
							−0.514076	+	0.857745i	\(0.671865\pi\)
\(37\)	−8.60873	−	1.29756i	−1.41527	−	0.213317i	−0.603530	−	0.797340i	\(-0.706240\pi\)
							−0.811737	+	0.584023i	\(0.801478\pi\)
\(41\)	2.46576	−	2.28789i	0.385086	−	0.357308i	−0.463748	−	0.885967i	\(-0.653496\pi\)
							0.848834	+	0.528659i	\(0.177305\pi\)
\(43\)	1.90507	+	1.76765i	0.290521	+	0.269564i	0.811945	−	0.583734i	\(-0.198409\pi\)
							−0.521425	+	0.853297i	\(0.674599\pi\)
\(47\)	−5.93255	−	2.85696i	−0.865351	−	0.416731i	−0.0520984	−	0.998642i	\(-0.516591\pi\)
							−0.813253	+	0.581911i	\(0.802305\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.277.14	yes	648
9.4	even	3		441.2.z.a.130.41	yes	648
49.23	even	21		441.2.z.a.268.41	yes	648
441.121	even	21	inner	441.2.y.a.121.14	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.121.14	✓	648	441.121	even	21	inner
441.2.y.a.277.14	yes	648	1.1	even	1	trivial
441.2.z.a.130.41	yes	648	9.4	even	3
441.2.z.a.268.41	yes	648	49.23	even	21