Embedded newform 441.2.z.a.4.19

• Label: 441.2.z.a.4.19
• 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			4.19
Character	\(\chi\)	\(=\)	441.4
Dual form			441.2.z.a.331.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0806131 + 1.07571i) q^{2} +(-1.11699 - 1.32376i) q^{3} +(0.827015 + 0.124652i) q^{4} +(0.478314 - 2.09563i) q^{5} +(1.51402 - 1.09484i) q^{6} +(2.41117 - 1.08917i) q^{7} +(-0.680835 + 2.98293i) q^{8} +(-0.504687 + 2.95724i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0806131	+	1.07571i	−0.0570021	+	0.760640i	0.893330	+	0.449401i	\(0.148363\pi\)
							−0.950332	+	0.311238i	\(0.899256\pi\)
\(3\)	−1.11699	−	1.32376i	−0.644892	−	0.764274i
							\(5\)	0.478314	−	2.09563i	0.213908	−	0.937194i	−0.747973	−	0.663729i	\(-0.768973\pi\)
0.961882	−	0.273465i	\(-0.0881699\pi\)
\(7\)	2.41117	−	1.08917i	0.911335	−	0.411666i
							\(11\)	−1.38638	+	0.667648i	−0.418011	+	0.201303i	−0.631053	−	0.775740i	\(-0.717377\pi\)
0.213042	+	0.977043i	\(0.431663\pi\)
\(13\)	0.497307	−	6.63610i	0.137928	−	1.84052i	−0.315138	−	0.949046i	\(-0.602051\pi\)
							0.453066	−	0.891477i	\(-0.350330\pi\)
\(17\)	−1.49795	+	0.225779i	−0.363305	+	0.0547595i	−0.328158	−	0.944623i	\(-0.606428\pi\)
							−0.0351469	+	0.999382i	\(0.511190\pi\)
\(19\)	0.0797763	+	0.138177i	0.0183019	+	0.0316999i	0.875031	−	0.484066i	\(-0.160841\pi\)
							−0.856729	+	0.515766i	\(0.827507\pi\)
\(23\)	5.13770	−	6.44248i	1.07129	−	1.34335i	0.135500	−	0.990777i	\(-0.456736\pi\)
							0.935785	−	0.352572i	\(-0.114693\pi\)
\(29\)	0.203442	−	0.518361i	0.0377781	−	0.0962571i	−0.910740	−	0.412981i	\(-0.864488\pi\)
							0.948518	+	0.316724i	\(0.102583\pi\)
\(31\)	−1.25335	−	2.17087i	−0.225109	−	0.389900i	0.731243	−	0.682117i	\(-0.238940\pi\)
							−0.956352	+	0.292217i	\(0.905607\pi\)
\(37\)	−2.78152	+	7.08721i	−0.457280	+	1.16513i	0.497407	+	0.867517i	\(0.334285\pi\)
							−0.954687	+	0.297612i	\(0.903810\pi\)
\(41\)	−0.860152	−	0.798104i	−0.134333	−	0.124643i	0.610145	−	0.792290i	\(-0.291111\pi\)
							−0.744478	+	0.667647i	\(0.767302\pi\)
\(43\)	8.04708	−	7.46660i	1.22717	−	1.13865i	0.241420	−	0.970421i	\(-0.422387\pi\)
							0.985749	−	0.168226i	\(-0.0538037\pi\)
\(47\)	−0.880695	+	11.7521i	−0.128463	+	1.71421i	0.445458	+	0.895303i	\(0.353041\pi\)
							−0.573920	+	0.818911i	\(0.694578\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.36	✓	648	9.7	even	3
441.2.y.a.184.36	yes	648	49.37	even	21
441.2.z.a.4.19	yes	648	1.1	even	1	trivial
441.2.z.a.331.19	yes	648	441.331	even	21	inner