Embedded newform 441.2.y.a.151.39

• Label: 441.2.y.a.151.39
• Level: $441$
• Weight: $2$
• Character: 441.151
• 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			151.39
Character	\(\chi\)	\(=\)	441.151
Dual form			441.2.y.a.184.39

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.27522 - 0.614113i) q^{2} +(-0.870611 + 1.49734i) q^{3} +(0.00206930 - 0.00259482i) q^{4} +(1.20496 + 1.11804i) q^{5} +(-0.190681 + 2.44409i) q^{6} +(-0.327178 + 2.62544i) q^{7} +(-0.628861 + 2.75522i) q^{8} +(-1.48407 - 2.60721i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.27522	−	0.614113i	0.901716	−	0.434244i	0.0752077	−	0.997168i	\(-0.476038\pi\)
							0.826508	+	0.562924i	\(0.190324\pi\)
\(3\)	−0.870611	+	1.49734i	−0.502647	+	0.864492i
							\(5\)	1.20496	+	1.11804i	0.538874	+	0.500002i	0.901999	−	0.431739i	\(-0.142100\pi\)
−0.363125	+	0.931740i	\(0.618290\pi\)
\(7\)	−0.327178	+	2.62544i	−0.123662	+	0.992324i
							\(11\)	−3.81349	−	2.60000i	−1.14981	−	0.783928i	−0.170315	−	0.985390i	\(-0.554479\pi\)
−0.979496	+	0.201462i	\(0.935431\pi\)
\(13\)	−2.22648	−	1.51799i	−0.617515	−	0.421015i	0.213752	−	0.976888i	\(-0.431432\pi\)
							−0.831267	+	0.555873i	\(0.812384\pi\)
\(17\)	0.301359	−	0.0454225i	0.0730902	−	0.0110166i	−0.112395	−	0.993664i	\(-0.535852\pi\)
							0.185486	+	0.982647i	\(0.440614\pi\)
\(19\)	2.48942	+	4.31181i	0.571113	+	0.989196i	0.996452	+	0.0841622i	\(0.0268214\pi\)
							−0.425339	+	0.905034i	\(0.639845\pi\)
\(23\)	2.50610	+	6.38544i	0.522558	+	1.33146i	0.911919	+	0.410370i	\(0.134601\pi\)
							−0.389361	+	0.921085i	\(0.627304\pi\)
\(29\)	−0.703391	+	0.106019i	−0.130617	+	0.0196873i	−0.214025	−	0.976828i	\(-0.568657\pi\)
							0.0834086	+	0.996515i	\(0.473419\pi\)
\(31\)	9.65537			1.73415			0.867077	−	0.498173i	\(-0.165996\pi\)
							0.867077	+	0.498173i	\(0.165996\pi\)
\(37\)	−2.64834	+	6.74785i	−0.435384	+	1.10934i	0.529882	+	0.848071i	\(0.322236\pi\)
							−0.965266	+	0.261269i	\(0.915859\pi\)
\(41\)	4.79178	−	1.47807i	0.748351	−	0.230836i	0.102957	−	0.994686i	\(-0.467170\pi\)
							0.645394	+	0.763850i	\(0.276693\pi\)
\(43\)	8.48787	+	2.61816i	1.29439	+	0.399266i	0.864052	−	0.503403i	\(-0.167919\pi\)
							0.430336	+	0.902669i	\(0.358395\pi\)
\(47\)	4.04943	−	1.95010i	0.590669	−	0.284451i	−0.114576	−	0.993414i	\(-0.536551\pi\)
							0.705246	+	0.708963i	\(0.250837\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.151.39	✓	648
9.4	even	3		441.2.z.a.4.16	yes	648
49.37	even	21		441.2.z.a.331.16	yes	648
441.184	even	21	inner	441.2.y.a.184.39	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.39	✓	648	1.1	even	1	trivial
441.2.y.a.184.39	yes	648	441.184	even	21	inner
441.2.z.a.4.16	yes	648	9.4	even	3
441.2.z.a.331.16	yes	648	49.37	even	21