Embedded newform 441.2.y.a.121.46

• Label: 441.2.y.a.121.46
• Level: $441$
• Weight: $2$
• Character: 441.121
• 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			121.46
Character	\(\chi\)	\(=\)	441.121
Dual form			441.2.y.a.277.46

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.87576 - 0.903317i) q^{2} +(-1.02234 + 1.39815i) q^{3} +(1.45550 - 1.82514i) q^{4} +(3.54626 - 1.09388i) q^{5} +(-0.654701 + 3.54608i) q^{6} +(-0.389198 - 2.61697i) q^{7} +(0.154941 - 0.678840i) q^{8} +(-0.909624 - 2.85877i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.87576	−	0.903317i	1.32636	−	0.638742i	0.369484	−	0.929237i	\(-0.379534\pi\)
							0.956876	+	0.290495i	\(0.0938201\pi\)
\(3\)	−1.02234	+	1.39815i	−0.590251	+	0.807220i
							\(5\)	3.54626	−	1.09388i	1.58594	−	0.489197i	0.628492	−	0.777816i	\(-0.283672\pi\)
0.957444	+	0.288619i	\(0.0931962\pi\)
\(7\)	−0.389198	−	2.61697i	−0.147103	−	0.989121i
							\(11\)	0.165900	−	2.21379i	0.0500208	−	0.667482i	−0.914688	−	0.404161i	\(-0.867563\pi\)
0.964709	−	0.263320i	\(-0.0848175\pi\)
\(13\)	−0.363777	+	4.85427i	−0.100894	+	1.34633i	0.685259	+	0.728299i	\(0.259689\pi\)
							−0.786153	+	0.618032i	\(0.787930\pi\)
\(17\)	−1.40024	+	3.56776i	−0.339609	+	0.865310i	0.654414	+	0.756137i	\(0.272916\pi\)
							−0.994023	+	0.109173i	\(0.965180\pi\)
\(19\)	3.03118	−	5.25016i	0.695400	−	1.20447i	−0.274645	−	0.961546i	\(-0.588560\pi\)
							0.970046	−	0.242923i	\(-0.0781063\pi\)
\(23\)	−4.24838	−	0.640340i	−0.885849	−	0.133520i	−0.309670	−	0.950844i	\(-0.600219\pi\)
							−0.576179	+	0.817324i	\(0.695457\pi\)
\(29\)	−0.721699	+	1.83886i	−0.134016	+	0.341468i	−0.982091	−	0.188410i	\(-0.939667\pi\)
							0.848074	+	0.529877i	\(0.177762\pi\)
\(31\)	4.12498			0.740869			0.370434	−	0.928859i	\(-0.379209\pi\)
							0.370434	+	0.928859i	\(0.379209\pi\)
\(37\)	−3.95404	+	0.595975i	−0.650040	+	0.0979777i	−0.465781	−	0.884900i	\(-0.654227\pi\)
							−0.184259	+	0.982878i	\(0.558988\pi\)
\(41\)	−0.404886	−	0.375679i	−0.0632325	−	0.0586712i	0.647924	−	0.761705i	\(-0.275638\pi\)
							−0.711156	+	0.703034i	\(0.751828\pi\)
\(43\)	−9.53448	+	8.84671i	−1.45399	+	1.34911i	−0.639759	+	0.768575i	\(0.720966\pi\)
							−0.814236	+	0.580534i	\(0.802844\pi\)
\(47\)	−0.548744	+	0.264261i	−0.0800425	+	0.0385465i	−0.473477	−	0.880806i	\(-0.657001\pi\)
							0.393434	+	0.919353i	\(0.371287\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.121.46	✓	648
9.7	even	3		441.2.z.a.268.9	yes	648
49.32	even	21		441.2.z.a.130.9	yes	648
441.277	even	21	inner	441.2.y.a.277.46	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.121.46	✓	648	1.1	even	1	trivial
441.2.y.a.277.46	yes	648	441.277	even	21	inner
441.2.z.a.130.9	yes	648	49.32	even	21
441.2.z.a.268.9	yes	648	9.7	even	3