Embedded newform 441.2.z.a.4.16

• Label: 441.2.z.a.4.16
• 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			4.16
Character	\(\chi\)	\(=\)	441.4
Dual form			441.2.z.a.331.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.105772 + 1.41143i) q^{2} +(1.42810 - 0.980073i) q^{3} +(-0.00328183 - 0.000494657i) q^{4} +(0.365770 - 1.60254i) q^{5} +(1.23225 + 2.11932i) q^{6} +(-2.11011 - 1.59607i) q^{7} +(-0.628861 + 2.75522i) q^{8} +(1.07891 - 2.79928i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.105772	+	1.41143i	−0.0747921	+	0.998031i	0.825969	+	0.563716i	\(0.190629\pi\)
							−0.900761	+	0.434315i	\(0.856990\pi\)
\(3\)	1.42810	−	0.980073i	0.824511	−	0.565845i
							\(5\)	0.365770	−	1.60254i	0.163577	−	0.716679i	−0.824896	−	0.565284i	\(-0.808766\pi\)
0.988473	−	0.151395i	\(-0.0483764\pi\)
\(7\)	−2.11011	−	1.59607i	−0.797547	−	0.603256i
							\(11\)	4.15841	−	2.00258i	1.25381	−	0.603802i	0.315277	−	0.949000i	\(-0.397902\pi\)
0.938530	+	0.345198i	\(0.112188\pi\)
\(13\)	−0.201377	+	2.68719i	−0.0558518	+	0.745291i	0.897034	+	0.441962i	\(0.145717\pi\)
							−0.952886	+	0.303329i	\(0.901902\pi\)
\(17\)	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	\(0.639845\pi\)
\(23\)	4.27690	−	5.36306i	0.891795	−	1.11828i	−0.100568	−	0.994930i	\(-0.532066\pi\)
							0.992364	−	0.123346i	\(-0.0393624\pi\)
\(29\)	0.259880	−	0.662164i	0.0482586	−	0.122961i	−0.904712	−	0.426024i	\(-0.859914\pi\)
							0.952971	+	0.303063i	\(0.0980092\pi\)
\(31\)	−4.82768	−	8.36179i	−0.867077	−	1.50182i	−0.864969	−	0.501824i	\(-0.832662\pi\)
							−0.00210797	−	0.999998i	\(-0.500671\pi\)
\(37\)	−2.64834	+	6.74785i	−0.435384	+	1.10934i	0.529882	+	0.848071i	\(0.322236\pi\)
							−0.965266	+	0.261269i	\(0.915859\pi\)
\(41\)	−3.67594	−	3.41077i	−0.574085	−	0.532673i	0.338817	−	0.940852i	\(-0.389973\pi\)
							−0.912902	+	0.408179i	\(0.866164\pi\)
\(43\)	−6.51133	+	6.04163i	−0.992968	+	0.921340i	−0.996902	−	0.0786522i	\(-0.974938\pi\)
							0.00393371	+	0.999992i	\(0.498748\pi\)
\(47\)	−0.335876	+	4.48196i	−0.0489926	+	0.653760i	0.917610	+	0.397481i	\(0.130116\pi\)
							−0.966603	+	0.256279i	\(0.917503\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.39	✓	648	9.7	even	3
441.2.y.a.184.39	yes	648	49.37	even	21
441.2.z.a.4.16	yes	648	1.1	even	1	trivial
441.2.z.a.331.16	yes	648	441.331	even	21	inner