Embedded newform 441.2.y.a.184.36

• Label: 441.2.y.a.184.36
• Level: $441$
• Weight: $2$
• Character: 441.184
• 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			184.36
Character	\(\chi\)	\(=\)	441.184
Dual form			441.2.y.a.151.36

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.971896 + 0.468040i) q^{2} +(-1.40353 + 1.01494i) q^{3} +(-0.521460 - 0.653890i) q^{4} +(1.57571 - 1.46205i) q^{5} +(-1.83912 + 0.329504i) q^{6} +(-2.14883 + 1.54355i) q^{7} +(-0.680835 - 2.98293i) q^{8} +(0.939804 - 2.84899i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.971896	+	0.468040i	0.687234	+	0.330955i	0.744706	−	0.667393i	\(-0.232590\pi\)
							−0.0574718	+	0.998347i	\(0.518304\pi\)
\(3\)	−1.40353	+	1.01494i	−0.810330	+	0.585974i
							\(5\)	1.57571	−	1.46205i	0.704680	−	0.653847i	−0.244113	−	0.969747i	\(-0.578497\pi\)
0.948793	+	0.315900i	\(0.102306\pi\)
\(7\)	−2.14883	+	1.54355i	−0.812180	+	0.583406i
							\(11\)	1.27139	−	0.866821i	0.383339	−	0.261356i	−0.356284	−	0.934378i	\(-0.615956\pi\)
0.739623	+	0.673022i	\(0.235004\pi\)
\(13\)	5.49838	−	3.74873i	1.52498	−	1.03971i	0.545509	−	0.838105i	\(-0.316336\pi\)
							0.979467	−	0.201606i	\(-0.0646159\pi\)
\(17\)	−1.49795	−	0.225779i	−0.363305	−	0.0547595i	−0.0351469	−	0.999382i	\(-0.511190\pi\)
							−0.328158	+	0.944623i	\(0.606428\pi\)
\(19\)	0.0797763	−	0.138177i	0.0183019	−	0.0316999i	−0.856729	−	0.515766i	\(-0.827507\pi\)
							0.875031	+	0.484066i	\(0.160841\pi\)
\(23\)	3.01050	−	7.67062i	0.627732	−	1.59943i	−0.162556	−	0.986699i	\(-0.551974\pi\)
							0.790288	−	0.612735i	\(-0.209931\pi\)
\(29\)	−0.550634	−	0.0829947i	−0.102250	−	0.0154117i	0.0977178	−	0.995214i	\(-0.468846\pi\)
							−0.199968	+	0.979802i	\(0.564084\pi\)
\(31\)	2.50671			0.450218			0.225109	−	0.974334i	\(-0.427726\pi\)
							0.225109	+	0.974334i	\(0.427726\pi\)
\(37\)	−2.78152	−	7.08721i	−0.457280	−	1.16513i	−0.954687	−	0.297612i	\(-0.903810\pi\)
							0.497407	−	0.867517i	\(-0.334285\pi\)
\(41\)	1.12125	+	0.345861i	0.175111	+	0.0540145i	0.381071	−	0.924546i	\(-0.375555\pi\)
							−0.205960	+	0.978560i	\(0.566032\pi\)
\(43\)	−10.4898	+	3.23568i	−1.59968	+	0.493436i	−0.961119	−	0.276134i	\(-0.910947\pi\)
							−0.638562	+	0.769570i	\(0.720470\pi\)
\(47\)	10.6179	+	5.11333i	1.54878	+	0.745855i	0.996158	−	0.0875740i	\(-0.0279114\pi\)
							0.552626	+	0.833429i	\(0.313626\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.184.36	yes	648
9.7	even	3		441.2.z.a.331.19	yes	648
49.4	even	21		441.2.z.a.4.19	yes	648
441.151	even	21	inner	441.2.y.a.151.36	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.36	✓	648	441.151	even	21	inner
441.2.y.a.184.36	yes	648	1.1	even	1	trivial
441.2.z.a.4.19	yes	648	49.4	even	21
441.2.z.a.331.19	yes	648	9.7	even	3