Embedded newform 441.2.y.a.151.37

• Label: 441.2.y.a.151.37
• Level: $441$
• Weight: $2$
• Character: 441.151
• 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			151.37
Character	\(\chi\)	\(=\)	441.151
Dual form			441.2.y.a.184.37

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00423 - 0.483613i) q^{2} +(1.36655 - 1.06420i) q^{3} +(-0.472379 + 0.592345i) q^{4} +(2.49565 + 2.31562i) q^{5} +(0.857674 - 1.72959i) q^{6} +(-1.45589 + 2.20916i) q^{7} +(-0.683962 + 2.99663i) q^{8} +(0.734939 - 2.90858i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00423	−	0.483613i	0.710099	−	0.341966i	−0.0437235	−	0.999044i	\(-0.513922\pi\)
							0.753823	+	0.657078i	\(0.228208\pi\)
\(3\)	1.36655	−	1.06420i	0.788980	−	0.614419i
							\(5\)	2.49565	+	2.31562i	1.11609	+	1.03558i	0.999099	+	0.0424449i	\(0.0135147\pi\)
0.116989	+	0.993133i	\(0.462676\pi\)
\(7\)	−1.45589	+	2.20916i	−0.550275	+	0.834983i
							\(11\)	−2.86127	−	1.95078i	−0.862704	−	0.588182i	0.0491060	−	0.998794i	\(-0.484363\pi\)
−0.911810	+	0.410612i	\(0.865315\pi\)
\(13\)	3.95424	+	2.69596i	1.09671	+	0.747724i	0.969730	−	0.244181i	\(-0.0785191\pi\)
							0.126980	+	0.991905i	\(0.459472\pi\)
\(17\)	3.97020	−	0.598412i	0.962915	−	0.145136i	0.351275	−	0.936272i	\(-0.385748\pi\)
							0.611640	+	0.791136i	\(0.290510\pi\)
\(19\)	−2.15962	−	3.74057i	−0.495450	−	0.858145i	0.504536	−	0.863391i	\(-0.331664\pi\)
							−0.999986	+	0.00524540i	\(0.998330\pi\)
\(23\)	−2.70698	−	6.89727i	−0.564444	−	1.43818i	−0.873353	−	0.487088i	\(-0.838059\pi\)
							0.308909	−	0.951092i	\(-0.400036\pi\)
\(29\)	−1.61173	+	0.242930i	−0.299291	+	0.0451109i	−0.296971	−	0.954886i	\(-0.595977\pi\)
							−0.00232004	+	0.999997i	\(0.500738\pi\)
\(31\)	−4.51740			−0.811348			−0.405674	−	0.914018i	\(-0.632963\pi\)
							−0.405674	+	0.914018i	\(0.632963\pi\)
\(37\)	2.43477	−	6.20370i	0.400274	−	1.01988i	−0.578635	−	0.815587i	\(-0.696414\pi\)
							0.978909	−	0.204296i	\(-0.0654904\pi\)
\(41\)	1.45406	−	0.448518i	0.227086	−	0.0700467i	−0.179124	−	0.983826i	\(-0.557326\pi\)
							0.406210	+	0.913780i	\(0.366850\pi\)
\(43\)	1.38388	+	0.426869i	0.211039	+	0.0650969i	0.398472	−	0.917180i	\(-0.369541\pi\)
							−0.187433	+	0.982277i	\(0.560017\pi\)
\(47\)	9.86397	−	4.75024i	1.43881	−	0.692894i	0.458196	−	0.888851i	\(-0.348496\pi\)
							0.980612	+	0.195957i	\(0.0627814\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.151.37	✓	648
9.4	even	3		441.2.z.a.4.18	yes	648
49.37	even	21		441.2.z.a.331.18	yes	648
441.184	even	21	inner	441.2.y.a.184.37	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.37	✓	648	1.1	even	1	trivial
441.2.y.a.184.37	yes	648	441.184	even	21	inner
441.2.z.a.4.18	yes	648	9.4	even	3
441.2.z.a.331.18	yes	648	49.37	even	21