Embedded newform 441.2.y.a.184.16

• Label: 441.2.y.a.184.16
• 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			184.16
Character	\(\chi\)	\(=\)	441.184
Dual form			441.2.y.a.151.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.14087 - 0.549413i) q^{2} +(-1.62798 - 0.591337i) q^{3} +(-0.247253 - 0.310046i) q^{4} +(1.11762 - 1.03700i) q^{5} +(1.53242 + 1.56907i) q^{6} +(1.67910 - 2.04466i) q^{7} +(0.675283 + 2.95861i) q^{8} +(2.30064 + 1.92537i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.14087	−	0.549413i	−0.806716	−	0.388494i	−0.0153847	−	0.999882i	\(-0.504897\pi\)
							−0.791331	+	0.611388i	\(0.790612\pi\)
\(3\)	−1.62798	−	0.591337i	−0.939915	−	0.341409i
							\(5\)	1.11762	−	1.03700i	0.499814	−	0.463760i	−0.389520	−	0.921018i	\(-0.627359\pi\)
0.889334	+	0.457259i	\(0.151169\pi\)
\(7\)	1.67910	−	2.04466i	0.634639	−	0.772809i
							\(11\)	4.51953	−	3.08136i	1.36269	−	0.929066i	0.362693	−	0.931909i	\(-0.381857\pi\)
0.999996	+	0.00284314i	\(0.000905000\pi\)
\(13\)	−0.984498	+	0.671219i	−0.273051	+	0.186163i	−0.692102	−	0.721800i	\(-0.743315\pi\)
							0.419051	+	0.907963i	\(0.362363\pi\)
\(17\)	6.40805	+	0.965859i	1.55418	+	0.234255i	0.869257	−	0.494360i	\(-0.164597\pi\)
							0.684924	+	0.728615i	\(0.259836\pi\)
\(19\)	−1.48970	+	2.58024i	−0.341761	+	0.591947i	−0.984760	−	0.173920i	\(-0.944357\pi\)
							0.642999	+	0.765867i	\(0.277690\pi\)
\(23\)	3.07522	−	7.83552i	0.641227	−	1.63382i	−0.125687	−	0.992070i	\(-0.540114\pi\)
							0.766914	−	0.641750i	\(-0.221791\pi\)
\(29\)	−7.37605	−	1.11176i	−1.36970	−	0.206449i	−0.577339	−	0.816504i	\(-0.695909\pi\)
							−0.792359	+	0.610056i	\(0.791147\pi\)
\(31\)	−0.379165			−0.0681000			−0.0340500	−	0.999420i	\(-0.510841\pi\)
							−0.0340500	+	0.999420i	\(0.510841\pi\)
\(37\)	−1.36219	−	3.47079i	−0.223942	−	0.570595i	0.774105	−	0.633057i	\(-0.218200\pi\)
							−0.998047	+	0.0624619i	\(0.980105\pi\)
\(41\)	−4.25784	−	1.31337i	−0.664963	−	0.205114i	−0.0561434	−	0.998423i	\(-0.517880\pi\)
							−0.608819	+	0.793309i	\(0.708357\pi\)
\(43\)	−1.67550	+	0.516822i	−0.255511	+	0.0788146i	−0.419864	−	0.907587i	\(-0.637922\pi\)
							0.164354	+	0.986401i	\(0.447446\pi\)
\(47\)	7.36605	+	3.54730i	1.07445	+	0.517427i	0.885538	−	0.464567i	\(-0.153790\pi\)
							0.188910	+	0.981994i	\(0.439504\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.184.16	yes	648
9.7	even	3		441.2.z.a.331.39	yes	648
49.4	even	21		441.2.z.a.4.39	yes	648
441.151	even	21	inner	441.2.y.a.151.16	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.16	✓	648	441.151	even	21	inner
441.2.y.a.184.16	yes	648	1.1	even	1	trivial
441.2.z.a.4.39	yes	648	49.4	even	21
441.2.z.a.331.39	yes	648	9.7	even	3