Embedded newform 441.2.y.a.88.47

• Label: 441.2.y.a.88.47
• Level: $441$
• Weight: $2$
• Character: 441.88
• 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			88.47
Character	\(\chi\)	\(=\)	441.88
Dual form			441.2.y.a.436.47

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.35928 - 1.70448i) q^{2} +(1.01160 - 1.40594i) q^{3} +(-0.612572 - 2.68385i) q^{4} +(-0.0312270 + 0.0212902i) q^{5} +(-1.02136 - 3.63531i) q^{6} +(-2.62974 - 0.290623i) q^{7} +(-1.47880 - 0.712152i) q^{8} +(-0.953346 - 2.84449i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{17}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.35928	−	1.70448i	0.961153	−	1.20525i	−0.0175252	−	0.999846i	\(-0.505579\pi\)
							0.978678	−	0.205401i	\(-0.0658499\pi\)
\(3\)	1.01160	−	1.40594i	0.584045	−	0.811721i
							\(5\)	−0.0312270	+	0.0212902i	−0.0139651	+	0.00952126i	−0.570283	−	0.821449i	\(-0.693166\pi\)
0.556317	+	0.830970i	\(0.312214\pi\)
\(7\)	−2.62974	−	0.290623i	−0.993949	−	0.109845i
							\(11\)	3.07397	+	0.463326i	0.926835	+	0.139698i	0.595079	−	0.803667i	\(-0.297121\pi\)
0.331756	+	0.943365i	\(0.392359\pi\)
\(13\)	−1.40141	−	0.211229i	−0.388682	−	0.0585844i	−0.0482055	−	0.998837i	\(-0.515350\pi\)
							−0.340476	+	0.940253i	\(0.610588\pi\)
\(17\)	−0.288207	−	0.267417i	−0.0699004	−	0.0648581i	0.644449	−	0.764647i	\(-0.277087\pi\)
							−0.714349	+	0.699789i	\(0.753277\pi\)
\(19\)	0.0784099	+	0.135810i	0.0179885	+	0.0311569i	0.874880	−	0.484341i	\(-0.160940\pi\)
							−0.856891	+	0.515498i	\(0.827607\pi\)
\(23\)	−1.40981	−	0.434867i	−0.293965	−	0.0906761i	0.144265	−	0.989539i	\(-0.453918\pi\)
							−0.438230	+	0.898863i	\(0.644394\pi\)
\(29\)	5.40918	+	5.01899i	1.00446	+	0.932003i	0.997687	−	0.0679743i	\(-0.0216536\pi\)
							0.00677298	+	0.999977i	\(0.497844\pi\)
\(31\)	7.08221			1.27200			0.636001	−	0.771688i	\(-0.280587\pi\)
							0.636001	+	0.771688i	\(0.280587\pi\)
\(37\)	6.77609	−	2.09015i	1.11398	−	0.343618i	0.317487	−	0.948263i	\(-0.397161\pi\)
							0.796495	+	0.604645i	\(0.206685\pi\)
\(41\)	0.232341	+	3.10038i	0.0362856	+	0.484197i	0.985512	+	0.169604i	\(0.0542490\pi\)
							−0.949227	+	0.314593i	\(0.898132\pi\)
\(43\)	−0.372534	+	4.97112i	−0.0568109	+	0.758089i	0.893951	+	0.448165i	\(0.147922\pi\)
							−0.950762	+	0.309923i	\(0.899697\pi\)
\(47\)	0.831220	−	1.04232i	0.121246	−	0.152038i	−0.717504	−	0.696554i	\(-0.754716\pi\)
							0.838750	+	0.544517i	\(0.183287\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.88.47	✓	648
9.4	even	3		441.2.z.a.382.47	yes	648
49.44	even	21		441.2.z.a.142.47	yes	648
441.436	even	21	inner	441.2.y.a.436.47	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.88.47	✓	648	1.1	even	1	trivial
441.2.y.a.436.47	yes	648	441.436	even	21	inner
441.2.z.a.142.47	yes	648	49.44	even	21
441.2.z.a.382.47	yes	648	9.4	even	3