Embedded newform 441.2.z.a.4.15

• Label: 441.2.z.a.4.15
• 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			4.15
Character	\(\chi\)	\(=\)	441.4
Dual form			441.2.z.a.331.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.107721 + 1.43743i) q^{2} +(0.0199394 - 1.73194i) q^{3} +(-0.0769418 - 0.0115971i) q^{4} +(-0.762536 + 3.34089i) q^{5} +(2.48739 + 0.215227i) q^{6} +(-2.60405 - 0.467908i) q^{7} +(-0.616552 + 2.70129i) q^{8} +(-2.99920 - 0.0690675i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.107721	+	1.43743i	−0.0761699	+	1.01642i	0.819901	+	0.572506i	\(0.194029\pi\)
							−0.896071	+	0.443911i	\(0.853591\pi\)
\(3\)	0.0199394	−	1.73194i	0.0115120	−	0.999934i
							\(5\)	−0.762536	+	3.34089i	−0.341016	+	1.49409i	0.455916	+	0.890023i	\(0.349312\pi\)
−0.796933	+	0.604068i	\(0.793546\pi\)
\(7\)	−2.60405	−	0.467908i	−0.984237	−	0.176852i
							\(11\)	−3.12026	+	1.50264i	−0.940795	+	0.453063i	−0.840449	−	0.541890i	\(-0.817709\pi\)
−0.100345	+	0.994953i	\(0.531995\pi\)
\(13\)	0.294409	−	3.92862i	0.0816545	−	1.08960i	−0.794401	−	0.607393i	\(-0.792215\pi\)
							0.876056	−	0.482210i	\(-0.160166\pi\)
\(17\)	−2.59959	+	0.391825i	−0.630493	+	0.0950316i	−0.456512	−	0.889717i	\(-0.650901\pi\)
							−0.173982	+	0.984749i	\(0.555663\pi\)
\(19\)	2.30654	+	3.99505i	0.529157	+	0.916527i	0.999422	+	0.0340014i	\(0.0108251\pi\)
							−0.470265	+	0.882525i	\(0.655842\pi\)
\(23\)	−0.689067	+	0.864062i	−0.143680	+	0.180169i	−0.848464	−	0.529252i	\(-0.822472\pi\)
							0.704784	+	0.709422i	\(0.251044\pi\)
\(29\)	1.32908	−	3.38644i	0.246804	−	0.628847i	−0.752762	−	0.658293i	\(-0.771279\pi\)
							0.999566	+	0.0294457i	\(0.00937421\pi\)
\(31\)	2.31322	+	4.00661i	0.415466	+	0.719608i	0.995477	−	0.0950004i	\(-0.0302853\pi\)
							−0.580011	+	0.814608i	\(0.696952\pi\)
\(37\)	−0.631627	+	1.60936i	−0.103839	+	0.264577i	−0.973345	−	0.229344i	\(-0.926342\pi\)
							0.869506	+	0.493922i	\(0.164437\pi\)
\(41\)	2.52395	+	2.34188i	0.394175	+	0.365741i	0.852216	−	0.523190i	\(-0.175258\pi\)
							−0.458041	+	0.888931i	\(0.651449\pi\)
\(43\)	−7.84707	+	7.28102i	−1.19667	+	1.11034i	−0.205373	+	0.978684i	\(0.565841\pi\)
							−0.991295	+	0.131661i	\(0.957969\pi\)
\(47\)	−0.382631	+	5.10585i	−0.0558124	+	0.744765i	0.897160	+	0.441707i	\(0.145627\pi\)
							−0.952972	+	0.303058i	\(0.901992\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.4.15	yes	648
9.7	even	3		441.2.y.a.151.40	✓	648
49.37	even	21		441.2.y.a.184.40	yes	648
441.331	even	21	inner	441.2.z.a.331.15	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.40	✓	648	9.7	even	3
441.2.y.a.184.40	yes	648	49.37	even	21
441.2.z.a.4.15	yes	648	1.1	even	1	trivial
441.2.z.a.331.15	yes	648	441.331	even	21	inner