Embedded newform 441.2.ba.a.43.18

• Label: 441.2.ba.a.43.18
• Level: $441$
• Weight: $2$
• Character: 441.43
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.ba (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			43.18
Character	\(\chi\)	\(=\)	441.43
Dual form			441.2.ba.a.400.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.422021 - 1.07529i) q^{2} +(-1.72764 - 0.123594i) q^{3} +(0.487951 - 0.452753i) q^{4} +(0.0478602 + 0.638649i) q^{5} +(0.596199 + 1.90987i) q^{6} +(2.41486 + 1.08094i) q^{7} +(-2.77426 - 1.33602i) q^{8} +(2.96945 + 0.427051i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 10 q^{3} + 47 q^{4} - 9 q^{5} - 34 q^{6} - 7 q^{7} - 28 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.422021	−	1.07529i	−0.298414	−	0.760347i	−0.998909	−	0.0466898i	\(-0.985133\pi\)
							0.700495	−	0.713657i	\(-0.252962\pi\)
\(3\)	−1.72764	−	0.123594i	−0.997451	−	0.0713571i
							\(5\)	0.0478602	+	0.638649i	0.0214037	+	0.285613i	0.997627	+	0.0688465i	\(0.0219319\pi\)
−0.976224	+	0.216766i	\(0.930449\pi\)
\(7\)	2.41486	+	1.08094i	0.912733	+	0.408557i
							\(11\)	0.777151	+	1.98015i	0.234320	+	0.597037i	0.998870	−	0.0475271i	\(-0.0151340\pi\)
−0.764550	+	0.644564i	\(0.777039\pi\)
\(13\)	3.31851	+	0.500184i	0.920388	+	0.138726i	0.592112	−	0.805856i	\(-0.298294\pi\)
							0.328276	+	0.944582i	\(0.393532\pi\)
\(17\)	−0.602864	+	2.64132i	−0.146216	+	0.640614i	0.847700	+	0.530475i	\(0.177987\pi\)
							−0.993916	+	0.110138i	\(0.964871\pi\)
\(19\)	−0.0842543			−0.0193293			−0.00966463	−	0.999953i	\(-0.503076\pi\)
							−0.00966463	+	0.999953i	\(0.503076\pi\)
\(23\)	6.23747	−	5.78752i	1.30060	−	1.20678i	0.336367	−	0.941731i	\(-0.390802\pi\)
							0.964234	−	0.265051i	\(-0.0853888\pi\)
\(29\)	−0.520736	−	0.483172i	−0.0966982	−	0.0897228i	0.630343	−	0.776317i	\(-0.282914\pi\)
							−0.727041	+	0.686594i	\(0.759105\pi\)
\(31\)	−2.81897	−	4.88260i	−0.506302	−	0.876941i	−0.999973	−	0.00729242i	\(-0.997679\pi\)
							0.493671	−	0.869649i	\(-0.335655\pi\)
\(37\)	2.30149	−	10.0835i	0.378363	−	1.65772i	−0.324121	−	0.946016i	\(-0.605068\pi\)
							0.702484	−	0.711700i	\(-0.252074\pi\)
\(41\)	0.460894	+	6.15021i	0.0719796	+	0.960501i	0.909932	+	0.414758i	\(0.136134\pi\)
							−0.837952	+	0.545743i	\(0.816247\pi\)
\(43\)	−0.540664	+	7.21466i	−0.0824504	+	1.10022i	0.790515	+	0.612443i	\(0.209813\pi\)
							−0.872965	+	0.487782i	\(0.837806\pi\)
\(47\)	−2.46782	−	6.28789i	−0.359968	−	0.917184i	−0.990058	−	0.140661i	\(-0.955077\pi\)
							0.630090	−	0.776522i	\(-0.283018\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.ba.a.43.18	✓	648
9.4	even	3	inner	441.2.ba.a.337.37	yes	648
49.8	even	7	inner	441.2.ba.a.106.37	yes	648
441.400	even	21	inner	441.2.ba.a.400.18	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.ba.a.43.18	✓	648	1.1	even	1	trivial
441.2.ba.a.106.37	yes	648	49.8	even	7	inner
441.2.ba.a.337.37	yes	648	9.4	even	3	inner
441.2.ba.a.400.18	yes	648	441.400	even	21	inner