Embedded newform 441.2.y.a.121.8

• Label: 441.2.y.a.121.8
• 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			121.8
Character	\(\chi\)	\(=\)	441.121
Dual form			441.2.y.a.277.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.96950 + 0.948460i) q^{2} +(-0.329565 - 1.70041i) q^{3} +(1.73236 - 2.17232i) q^{4} +(1.99129 - 0.614233i) q^{5} +(2.26185 + 3.03637i) q^{6} +(0.361067 + 2.62100i) q^{7} +(-0.378680 + 1.65911i) q^{8} +(-2.78277 + 1.12079i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.96950	+	0.948460i	−1.39264	+	0.670662i	−0.971655	−	0.236403i	\(-0.924031\pi\)
							−0.420990	+	0.907065i	\(0.638317\pi\)
\(3\)	−0.329565	−	1.70041i	−0.190275	−	0.981731i
							\(5\)	1.99129	−	0.614233i	0.890534	−	0.274693i	0.184467	−	0.982839i	\(-0.440944\pi\)
0.706067	+	0.708145i	\(0.250468\pi\)
\(7\)	0.361067	+	2.62100i	0.136471	+	0.990644i
							\(11\)	−0.126123	+	1.68300i	−0.0380276	+	0.507443i	0.945372	+	0.325992i	\(0.105698\pi\)
−0.983400	+	0.181451i	\(0.941921\pi\)
\(13\)	−0.452632	+	6.03995i	−0.125538	+	1.67518i	0.478394	+	0.878145i	\(0.341219\pi\)
							−0.603931	+	0.797036i	\(0.706400\pi\)
\(17\)	1.86452	−	4.75071i	0.452212	−	1.15222i	−0.505075	−	0.863076i	\(-0.668535\pi\)
							0.957286	−	0.289141i	\(-0.0933697\pi\)
\(19\)	−0.222533	+	0.385439i	−0.0510526	+	0.0884257i	−0.890422	−	0.455135i	\(-0.849591\pi\)
							0.839370	+	0.543561i	\(0.182924\pi\)
\(23\)	1.43700	+	0.216593i	0.299635	+	0.0451627i	0.297139	−	0.954834i	\(-0.403968\pi\)
							0.00249565	+	0.999997i	\(0.499206\pi\)
\(29\)	−1.89818	+	4.83649i	−0.352484	+	0.898114i	0.639156	+	0.769077i	\(0.279284\pi\)
							−0.991640	+	0.129037i	\(0.958811\pi\)
\(31\)	9.11273			1.63669			0.818347	−	0.574725i	\(-0.194891\pi\)
							0.818347	+	0.574725i	\(0.194891\pi\)
\(37\)	−3.67673	+	0.554178i	−0.604451	+	0.0911063i	−0.444136	−	0.895960i	\(-0.646489\pi\)
							−0.160315	+	0.987066i	\(0.551251\pi\)
\(41\)	7.07827	+	6.56767i	1.10544	+	1.02570i	0.999509	+	0.0313242i	\(0.00997243\pi\)
							0.105930	+	0.994374i	\(0.466218\pi\)
\(43\)	6.04162	−	5.60581i	0.921339	−	0.854877i	−0.0684158	−	0.997657i	\(-0.521794\pi\)
							0.989755	+	0.142779i	\(0.0456040\pi\)
\(47\)	−7.24565	+	3.48932i	−1.05689	+	0.508970i	−0.879859	−	0.475236i	\(-0.842363\pi\)
							−0.177029	+	0.984206i	\(0.556649\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.121.8	✓	648
9.7	even	3		441.2.z.a.268.47	yes	648
49.32	even	21		441.2.z.a.130.47	yes	648
441.277	even	21	inner	441.2.y.a.277.8	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.121.8	✓	648	1.1	even	1	trivial
441.2.y.a.277.8	yes	648	441.277	even	21	inner
441.2.z.a.130.47	yes	648	49.32	even	21
441.2.z.a.268.47	yes	648	9.7	even	3