Embedded newform 441.2.y.a.247.20

• Label: 441.2.y.a.247.20
• Level: $441$
• Weight: $2$
• Character: 441.247
• 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			247.20
Character	\(\chi\)	\(=\)	441.247
Dual form			441.2.y.a.25.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.222631 - 0.975409i) q^{2} +(-0.211834 + 1.71905i) q^{3} +(0.900079 - 0.433455i) q^{4} +(0.692294 - 1.76394i) q^{5} +(1.72394 - 0.176088i) q^{6} +(-0.942297 - 2.47226i) q^{7} +(-1.87078 - 2.34588i) q^{8} +(-2.91025 - 0.728307i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.222631	−	0.975409i	−0.157424	−	0.689718i	−0.990609	−	0.136725i	\(-0.956342\pi\)
							0.833185	−	0.552994i	\(-0.186515\pi\)
\(3\)	−0.211834	+	1.71905i	−0.122303	+	0.992493i
							\(5\)	0.692294	−	1.76394i	0.309603	−	0.788856i	−0.688378	−	0.725352i	\(-0.741677\pi\)
0.997982	−	0.0635044i	\(-0.0202277\pi\)
\(7\)	−0.942297	−	2.47226i	−0.356155	−	0.934427i
							\(11\)	−5.99512	−	1.84925i	−1.80760	−	0.557570i	−0.808093	−	0.589055i	\(-0.799500\pi\)
−0.999504	+	0.0314855i	\(0.989976\pi\)
\(13\)	−0.194644	−	0.0600396i	−0.0539844	−	0.0166520i	0.267645	−	0.963517i	\(-0.413754\pi\)
							−0.321630	+	0.946866i	\(0.604231\pi\)
\(17\)	0.0911454	+	1.21625i	0.0221060	+	0.294984i	0.997289	+	0.0735839i	\(0.0234437\pi\)
							−0.975183	+	0.221400i	\(0.928937\pi\)
\(19\)	1.58184	−	2.73983i	0.362899	−	0.628560i	−0.625538	−	0.780194i	\(-0.715120\pi\)
							0.988437	+	0.151634i	\(0.0484536\pi\)
\(23\)	1.64394	+	1.12082i	0.342786	+	0.233708i	0.722460	−	0.691413i	\(-0.243011\pi\)
							−0.379674	+	0.925120i	\(0.623964\pi\)
\(29\)	0.226244	+	3.01901i	0.0420124	+	0.560617i	0.978003	+	0.208591i	\(0.0668878\pi\)
							−0.935991	+	0.352025i	\(0.885493\pi\)
\(31\)	−1.41429			−0.254015			−0.127007	−	0.991902i	\(-0.540537\pi\)
							−0.127007	+	0.991902i	\(0.540537\pi\)
\(37\)	1.94111	−	1.32343i	0.319116	−	0.217570i	−0.393158	−	0.919471i	\(-0.628618\pi\)
							0.712275	+	0.701901i	\(0.247665\pi\)
\(41\)	6.93989	−	1.04602i	1.08383	−	0.163361i	0.417247	−	0.908793i	\(-0.362995\pi\)
							0.666582	+	0.745432i	\(0.267757\pi\)
\(43\)	9.05595	+	1.36496i	1.38102	+	0.208155i	0.797193	−	0.603724i	\(-0.206317\pi\)
							0.583826	+	0.811879i	\(0.301555\pi\)
\(47\)	−2.36394	−	10.3571i	−0.344816	−	1.51074i	−0.788770	−	0.614688i	\(-0.789282\pi\)
							0.443954	−	0.896050i	\(-0.353575\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.247.20	yes	648
9.7	even	3		441.2.z.a.394.20	yes	648
49.25	even	21		441.2.z.a.319.20	yes	648
441.25	even	21	inner	441.2.y.a.25.20	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.25.20	✓	648	441.25	even	21	inner
441.2.y.a.247.20	yes	648	1.1	even	1	trivial
441.2.z.a.319.20	yes	648	49.25	even	21
441.2.z.a.394.20	yes	648	9.7	even	3