Embedded newform 441.2.z.a.130.32

• Label: 441.2.z.a.130.32
• Level: $441$
• Weight: $2$
• Character: 441.130
• 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			130.32
Character	\(\chi\)	\(=\)	441.130
Dual form			441.2.z.a.268.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.442370 - 0.301603i) q^{2} +(-0.944398 - 1.45193i) q^{3} +(-0.625955 + 1.59491i) q^{4} +(-0.835905 - 3.66234i) q^{5} +(-0.855681 - 0.357459i) q^{6} +(1.43498 - 2.22280i) q^{7} +(0.442401 + 1.93829i) q^{8} +(-1.21622 + 2.74241i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{2}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.442370	−	0.301603i	0.312803	−	0.213265i	−0.396734	−	0.917934i	\(-0.629856\pi\)
							0.709537	+	0.704668i	\(0.248904\pi\)
\(3\)	−0.944398	−	1.45193i	−0.545249	−	0.838274i
							\(5\)	−0.835905	−	3.66234i	−0.373828	−	1.63785i	−0.715920	−	0.698182i	\(-0.753993\pi\)
0.342092	−	0.939666i	\(-0.388865\pi\)
\(7\)	1.43498	−	2.22280i	0.542373	−	0.840138i
							\(11\)	−2.52547	−	1.21620i	−0.761458	−	0.366699i	0.0125112	−	0.999922i	\(-0.496017\pi\)
−0.773970	+	0.633223i	\(0.781732\pi\)
\(13\)	−2.75251	+	1.87663i	−0.763408	+	0.520483i	−0.881308	−	0.472543i	\(-0.843336\pi\)
							0.117900	+	0.993025i	\(0.462384\pi\)
\(17\)	−0.492578	−	1.25507i	−0.119468	−	0.304399i	0.858613	−	0.512624i	\(-0.171326\pi\)
							−0.978081	+	0.208225i	\(0.933231\pi\)
\(19\)	2.71357	+	4.70005i	0.622537	+	1.07827i	0.989012	+	0.147837i	\(0.0472312\pi\)
							−0.366475	+	0.930428i	\(0.619436\pi\)
\(23\)	−4.43372	−	5.55971i	−0.924494	−	1.15928i	−0.986917	−	0.161229i	\(-0.948454\pi\)
							0.0624230	−	0.998050i	\(-0.480117\pi\)
\(29\)	−4.15924	−	0.626905i	−0.772352	−	0.116413i	−0.248969	−	0.968511i	\(-0.580092\pi\)
							−0.523382	+	0.852098i	\(0.675330\pi\)
\(31\)	−3.08453	−	5.34256i	−0.553997	−	0.959552i	−0.997981	−	0.0635166i	\(-0.979768\pi\)
							0.443983	−	0.896035i	\(-0.353565\pi\)
\(37\)	4.08330	+	0.615459i	0.671290	+	0.101181i	0.475839	−	0.879533i	\(-0.342145\pi\)
							0.195452	+	0.980713i	\(0.437383\pi\)
\(41\)	0.669169	+	0.206411i	0.104507	+	0.0322360i	0.346567	−	0.938025i	\(-0.387347\pi\)
							−0.242061	+	0.970261i	\(0.577823\pi\)
\(43\)	−4.39190	+	1.35472i	−0.669758	+	0.206593i	−0.610942	−	0.791676i	\(-0.709209\pi\)
							−0.0588169	+	0.998269i	\(0.518733\pi\)
\(47\)	2.68622	−	1.83144i	0.391826	−	0.267142i	−0.351342	−	0.936247i	\(-0.614275\pi\)
							0.743168	+	0.669105i	\(0.233322\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.130.32	yes	648
9.7	even	3		441.2.y.a.277.23	yes	648
49.23	even	21		441.2.y.a.121.23	✓	648
441.268	even	21	inner	441.2.z.a.268.32	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.121.23	✓	648	49.23	even	21
441.2.y.a.277.23	yes	648	9.7	even	3
441.2.z.a.130.32	yes	648	1.1	even	1	trivial
441.2.z.a.268.32	yes	648	441.268	even	21	inner