Embedded newform 441.2.z.a.4.1

• Label: 441.2.z.a.4.1
• Level: $441$
• Weight: $2$
• Character: 441.4
• 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.z (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			4.1
Character	\(\chi\)	\(=\)	441.4
Dual form			441.2.z.a.331.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.209537 + 2.79608i) q^{2} +(-0.318381 - 1.70254i) q^{3} +(-5.79651 - 0.873684i) q^{4} +(-0.429093 + 1.87998i) q^{5} +(4.82715 - 0.533474i) q^{6} +(0.156954 - 2.64109i) q^{7} +(2.40961 - 10.5572i) q^{8} +(-2.79727 + 1.08411i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.209537	+	2.79608i	−0.148165	+	1.97713i	0.0568965	+	0.998380i	\(0.481879\pi\)
							−0.205062	+	0.978749i	\(0.565740\pi\)
\(3\)	−0.318381	−	1.70254i	−0.183817	−	0.982960i
							\(5\)	−0.429093	+	1.87998i	−0.191896	+	0.840751i	0.783693	+	0.621148i	\(0.213333\pi\)
−0.975589	+	0.219603i	\(0.929524\pi\)
\(7\)	0.156954	−	2.64109i	0.0593231	−	0.998239i
							\(11\)	1.10421	−	0.531759i	0.332932	−	0.160331i	−0.259948	−	0.965623i	\(-0.583705\pi\)
0.592879	+	0.805291i	\(0.297991\pi\)
\(13\)	−0.0375385	+	0.500917i	−0.0104113	+	0.138929i	−0.999987	−	0.00508394i	\(-0.998382\pi\)
							0.989576	+	0.144013i	\(0.0460008\pi\)
\(17\)	5.37911	−	0.810771i	1.30463	−	0.196641i	0.540301	−	0.841472i	\(-0.318310\pi\)
							0.764325	+	0.644831i	\(0.223072\pi\)
\(19\)	−1.07046	−	1.85408i	−0.245579	−	0.425356i	0.716715	−	0.697366i	\(-0.245645\pi\)
							−0.962294	+	0.272010i	\(0.912312\pi\)
\(23\)	5.42246	−	6.79955i	1.13066	−	1.41780i	0.235605	−	0.971849i	\(-0.424293\pi\)
							0.895056	−	0.445954i	\(-0.147136\pi\)
\(29\)	−0.0305359	+	0.0778041i	−0.00567037	+	0.0144479i	−0.933681	−	0.358106i	\(-0.883423\pi\)
							0.928011	+	0.372554i	\(0.121518\pi\)
\(31\)	−4.39666	−	7.61524i	−0.789664	−	1.36774i	−0.926173	−	0.377099i	\(-0.876922\pi\)
							0.136509	−	0.990639i	\(-0.456412\pi\)
\(37\)	−0.726269	+	1.85050i	−0.119398	+	0.304221i	−0.978061	−	0.208320i	\(-0.933200\pi\)
							0.858663	+	0.512541i	\(0.171296\pi\)
\(41\)	−4.96825	−	4.60987i	−0.775911	−	0.719940i	0.189155	−	0.981947i	\(-0.439425\pi\)
							−0.965066	+	0.262007i	\(0.915616\pi\)
\(43\)	0.745904	−	0.692098i	0.113749	−	0.105544i	−0.621248	−	0.783614i	\(-0.713374\pi\)
							0.734997	+	0.678070i	\(0.237183\pi\)
\(47\)	0.296136	−	3.95166i	0.0431959	−	0.576409i	−0.933051	−	0.359745i	\(-0.882864\pi\)
							0.976246	−	0.216663i	\(-0.0695174\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.4.1	yes	648
9.7	even	3		441.2.y.a.151.54	✓	648
49.37	even	21		441.2.y.a.184.54	yes	648
441.331	even	21	inner	441.2.z.a.331.1	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.54	✓	648	9.7	even	3
441.2.y.a.184.54	yes	648	49.37	even	21
441.2.z.a.4.1	yes	648	1.1	even	1	trivial
441.2.z.a.331.1	yes	648	441.331	even	21	inner