Embedded newform 441.2.z.a.331.19

• Label: 441.2.z.a.331.19
• Level: $441$
• Weight: $2$
• Character: 441.331
• 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			331.19
Character	\(\chi\)	\(=\)	441.331
Dual form			441.2.z.a.4.19

$q$-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 \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.36	✓	648	49.4	even	21
441.2.y.a.184.36	yes	648	9.4	even	3
441.2.z.a.4.19	yes	648	441.4	even	21	inner
441.2.z.a.331.19	yes	648	1.1	even	1	trivial