Embedded newform 441.2.z.a.142.11

• Label: 441.2.z.a.142.11
• 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			142.11
Character	\(\chi\)	\(=\)	441.142
Dual form			441.2.z.a.382.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.701186 + 1.78659i) q^{2} +(0.322748 - 1.70172i) q^{3} +(-1.23415 - 1.14512i) q^{4} +(0.740914 - 0.356805i) q^{5} +(2.81396 + 1.76984i) q^{6} +(0.756123 - 2.53540i) q^{7} +(-0.547165 + 0.263501i) q^{8} +(-2.79167 - 1.09845i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.701186	+	1.78659i	−0.495813	+	1.26331i	0.435860	+	0.900014i	\(0.356444\pi\)
							−0.931673	+	0.363297i	\(0.881651\pi\)
\(3\)	0.322748	−	1.70172i	0.186338	−	0.982486i
							\(5\)	0.740914	−	0.356805i	0.331347	−	0.159568i	−0.260812	−	0.965390i	\(-0.583990\pi\)
0.592159	+	0.805821i	\(0.298276\pi\)
\(7\)	0.756123	−	2.53540i	0.285788	−	0.958293i
							\(11\)	1.56961	+	1.96823i	0.473255	+	0.593443i	0.959965	−	0.280121i	\(-0.0903746\pi\)
−0.486710	+	0.873564i	\(0.661803\pi\)
\(13\)	−0.227504	+	0.579670i	−0.0630982	+	0.160772i	−0.958945	−	0.283590i	\(-0.908474\pi\)
							0.895847	+	0.444362i	\(0.146570\pi\)
\(17\)	2.66716	−	2.47476i	0.646881	−	0.600218i	−0.286945	−	0.957947i	\(-0.592640\pi\)
							0.933825	+	0.357730i	\(0.116449\pi\)
\(19\)	3.83557	−	6.64341i	0.879941	−	1.52410i	0.0285357	−	0.999593i	\(-0.490916\pi\)
							0.851405	−	0.524509i	\(-0.175751\pi\)
\(23\)	0.990752	−	4.34077i	0.206586	−	0.905113i	−0.760233	−	0.649651i	\(-0.774915\pi\)
							0.966819	−	0.255462i	\(-0.0822277\pi\)
\(29\)	−1.23858	−	0.382052i	−0.229999	−	0.0709454i	0.177613	−	0.984100i	\(-0.443162\pi\)
							−0.407613	+	0.913155i	\(0.633639\pi\)
\(31\)	5.56302	−	9.63544i	0.999149	−	1.73058i	0.463849	−	0.885914i	\(-0.346468\pi\)
							0.535300	−	0.844662i	\(-0.320199\pi\)
\(37\)	−1.48751	−	0.458835i	−0.244544	−	0.0754319i	0.170060	−	0.985434i	\(-0.445604\pi\)
							−0.414604	+	0.910002i	\(0.636080\pi\)
\(41\)	8.03860	+	5.48062i	1.25542	+	0.855929i	0.994009	−	0.109299i	\(-0.0348605\pi\)
							0.261409	+	0.965228i	\(0.415813\pi\)
\(43\)	−0.484759	+	0.330503i	−0.0739250	+	0.0504012i	−0.599719	−	0.800211i	\(-0.704721\pi\)
							0.525794	+	0.850612i	\(0.323768\pi\)
\(47\)	−4.22503	+	10.7652i	−0.616283	+	1.57026i	0.191952	+	0.981404i	\(0.438518\pi\)
							−0.808235	+	0.588860i	\(0.799577\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.142.11	yes	648
9.4	even	3		441.2.y.a.436.11	yes	648
49.39	even	21		441.2.y.a.88.11	✓	648
441.382	even	21	inner	441.2.z.a.382.11	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.88.11	✓	648	49.39	even	21
441.2.y.a.436.11	yes	648	9.4	even	3
441.2.z.a.142.11	yes	648	1.1	even	1	trivial
441.2.z.a.382.11	yes	648	441.382	even	21	inner