Embedded newform 441.2.z.a.142.1

• Label: 441.2.z.a.142.1
• 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			142.1
Character	\(\chi\)	\(=\)	441.142
Dual form			441.2.z.a.382.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.02455 + 2.61051i) q^{2} +(1.47366 + 0.910122i) q^{3} +(-4.29896 - 3.98885i) q^{4} +(1.33021 - 0.640594i) q^{5} +(-3.88572 + 2.91454i) q^{6} +(0.948892 - 2.46974i) q^{7} +(9.76416 - 4.70217i) q^{8} +(1.34336 + 2.68242i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.02455	+	2.61051i	−0.724466	+	1.84591i	−0.241073	+	0.970507i	\(0.577499\pi\)
							−0.483393	+	0.875403i	\(0.660596\pi\)
\(3\)	1.47366	+	0.910122i	0.850819	+	0.525459i
							\(5\)	1.33021	−	0.640594i	0.594887	−	0.286483i	−0.112114	−	0.993695i	\(-0.535762\pi\)
0.707001	+	0.707213i	\(0.250048\pi\)
\(7\)	0.948892	−	2.46974i	0.358647	−	0.933473i
							\(11\)	2.77950	+	3.48539i	0.838052	+	1.05088i	0.997966	+	0.0637528i	\(0.0203069\pi\)
−0.159914	+	0.987131i	\(0.551122\pi\)
\(13\)	1.57920	−	4.02375i	0.437992	−	1.11599i	−0.526101	−	0.850422i	\(-0.676346\pi\)
							0.964093	−	0.265564i	\(-0.0855583\pi\)
\(17\)	−1.37534	+	1.27613i	−0.333570	+	0.309508i	−0.829119	−	0.559072i	\(-0.811157\pi\)
							0.495549	+	0.868580i	\(0.334967\pi\)
\(19\)	−0.601339	+	1.04155i	−0.137957	+	0.238948i	−0.926723	−	0.375745i	\(-0.877387\pi\)
							0.788766	+	0.614693i	\(0.210720\pi\)
\(23\)	1.20924	−	5.29801i	0.252143	−	1.10471i	−0.677288	−	0.735718i	\(-0.736845\pi\)
							0.929432	−	0.368994i	\(-0.120298\pi\)
\(29\)	1.85137	+	0.571071i	0.343790	+	0.106045i	0.461842	−	0.886962i	\(-0.347189\pi\)
							−0.118052	+	0.993007i	\(0.537665\pi\)
\(31\)	−0.861777	+	1.49264i	−0.154780	+	0.268086i	−0.932979	−	0.359931i	\(-0.882800\pi\)
							0.778199	+	0.628018i	\(0.216133\pi\)
\(37\)	4.63633	+	1.43012i	0.762208	+	0.235110i	0.651403	−	0.758732i	\(-0.274181\pi\)
							0.110805	+	0.993842i	\(0.464657\pi\)
\(41\)	0.243537	+	0.166041i	0.0380341	+	0.0259312i	0.582188	−	0.813054i	\(-0.302197\pi\)
							−0.544154	+	0.838986i	\(0.683149\pi\)
\(43\)	3.85182	−	2.62612i	0.587396	−	0.400480i	−0.232831	−	0.972517i	\(-0.574799\pi\)
							0.820228	+	0.572037i	\(0.193847\pi\)
\(47\)	1.75082	−	4.46101i	0.255383	−	0.650705i	−0.744473	−	0.667652i	\(-0.767299\pi\)
							0.999856	+	0.0169468i	\(0.00539460\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.142.1	yes	648
9.4	even	3		441.2.y.a.436.1	yes	648
49.39	even	21		441.2.y.a.88.1	✓	648
441.382	even	21	inner	441.2.z.a.382.1	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.88.1	✓	648	49.39	even	21
441.2.y.a.436.1	yes	648	9.4	even	3
441.2.z.a.142.1	yes	648	1.1	even	1	trivial
441.2.z.a.382.1	yes	648	441.382	even	21	inner