Embedded newform 441.2.y.a.184.40

• Label: 441.2.y.a.184.40
• Level: $441$
• Weight: $2$
• Character: 441.184
• 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.y (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			184.40
Character	\(\chi\)	\(=\)	441.184
Dual form			441.2.y.a.151.40

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.29871 + 0.625427i) q^{2} +(-1.72560 - 0.149311i) q^{3} +(0.0485143 + 0.0608350i) q^{4} +(-2.51203 + 2.33082i) q^{5} +(-2.14768 - 1.27315i) q^{6} +(0.896804 - 2.48912i) q^{7} +(-0.616552 - 2.70129i) q^{8} +(2.95541 + 0.515304i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{16}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.29871	+	0.625427i	0.918328	+	0.442243i	0.832474	−	0.554064i	\(-0.186924\pi\)
							0.0858541	+	0.996308i	\(0.472638\pi\)
\(3\)	−1.72560	−	0.149311i	−0.996277	−	0.0862050i
							\(5\)	−2.51203	+	2.33082i	−1.12341	+	1.04237i	−0.124672	+	0.992198i	\(0.539788\pi\)
−0.998740	+	0.0501762i	\(0.984022\pi\)
\(7\)	0.896804	−	2.48912i	0.338960	−	0.940801i
							\(11\)	2.86145	−	1.95091i	0.862761	−	0.588221i	−0.0490657	−	0.998796i	\(-0.515624\pi\)
0.911827	+	0.410575i	\(0.134672\pi\)
\(13\)	3.25508	−	2.21928i	0.902797	−	0.615517i	−0.0204218	−	0.999791i	\(-0.506501\pi\)
							0.923219	+	0.384275i	\(0.125549\pi\)
\(17\)	−2.59959	−	0.391825i	−0.630493	−	0.0950316i	−0.173982	−	0.984749i	\(-0.555663\pi\)
							−0.456512	+	0.889717i	\(0.650901\pi\)
\(19\)	2.30654	−	3.99505i	0.529157	−	0.916527i	−0.470265	−	0.882525i	\(-0.655842\pi\)
							0.999422	−	0.0340014i	\(-0.0108251\pi\)
\(23\)	−0.403767	+	1.02878i	−0.0841911	+	0.214516i	−0.966769	−	0.255650i	\(-0.917711\pi\)
							0.882578	+	0.470166i	\(0.155806\pi\)
\(29\)	−3.59729	−	0.542204i	−0.667999	−	0.100685i	−0.193717	−	0.981057i	\(-0.562054\pi\)
							−0.474282	+	0.880373i	\(0.657292\pi\)
\(31\)	−4.62643			−0.830932			−0.415466	−	0.909609i	\(-0.636381\pi\)
							−0.415466	+	0.909609i	\(0.636381\pi\)
\(37\)	−0.631627	−	1.60936i	−0.103839	−	0.264577i	0.869506	−	0.493922i	\(-0.164437\pi\)
							−0.973345	+	0.229344i	\(0.926342\pi\)
\(41\)	−3.29010	−	1.01486i	−0.513828	−	0.158495i	0.0269882	−	0.999636i	\(-0.491408\pi\)
							−0.540816	+	0.841141i	\(0.681885\pi\)
\(43\)	10.2291	−	3.15525i	1.55992	−	0.481172i	0.609669	−	0.792656i	\(-0.291302\pi\)
							0.950251	+	0.311484i	\(0.100826\pi\)
\(47\)	4.61311	+	2.22156i	0.672892	+	0.324048i	0.738942	−	0.673769i	\(-0.235326\pi\)
							−0.0660506	+	0.997816i	\(0.521040\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.40	✓	648	441.151	even	21	inner
441.2.y.a.184.40	yes	648	1.1	even	1	trivial
441.2.z.a.4.15	yes	648	49.4	even	21
441.2.z.a.331.15	yes	648	9.7	even	3