Embedded newform 441.2.bh.a.20.11

• Label: 441.2.bh.a.20.11
• Level: $441$
• Weight: $2$
• Character: 441.20
• 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.bh (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			20.11
Character	\(\chi\)	\(=\)	441.20
Dual form			441.2.bh.a.419.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07603 - 1.57825i) q^{2} +(-1.16503 - 1.28168i) q^{3} +(-0.602346 + 1.53475i) q^{4} +(0.0991137 - 0.0919641i) q^{5} +(-0.769190 + 3.21784i) q^{6} +(1.56054 + 2.13652i) q^{7} +(-0.654172 + 0.149311i) q^{8} +(-0.285388 + 2.98639i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 15 q^{2} - 14 q^{3} - 57 q^{4} - 21 q^{5} + 14 q^{6} - 5 q^{7} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.07603	−	1.57825i	−0.760870	−	1.11599i	−0.989788	−	0.142546i	\(-0.954471\pi\)
							0.228918	−	0.973446i	\(-0.426481\pi\)
\(3\)	−1.16503	−	1.28168i	−0.672633	−	0.739976i
							\(5\)	0.0991137	−	0.0919641i	0.0443250	−	0.0411276i	−0.657699	−	0.753281i	\(-0.728470\pi\)
0.702024	+	0.712153i	\(0.252280\pi\)
\(7\)	1.56054	+	2.13652i	0.589829	+	0.807528i
							\(11\)	1.85976	+	2.72776i	0.560738	+	0.822452i	0.997003	−	0.0773666i	\(-0.0246512\pi\)
−0.436265	+	0.899818i	\(0.643699\pi\)
\(13\)	5.17626	−	0.387907i	1.43564	−	0.107586i	0.665944	−	0.746002i	\(-0.268029\pi\)
							0.769692	+	0.638416i	\(0.220410\pi\)
\(17\)	−2.61938	+	3.28460i	−0.635293	+	0.796633i	−0.990406	−	0.138191i	\(-0.955871\pi\)
							0.355112	+	0.934824i	\(0.384443\pi\)
\(19\)			1.64591i			0.377598i	0.982016	+	0.188799i	\(0.0604595\pi\)
							−0.982016	+	0.188799i	\(0.939541\pi\)
\(23\)	−0.559470	−	0.219576i	−0.116658	−	0.0457847i	0.306294	−	0.951937i	\(-0.400911\pi\)
							−0.422952	+	0.906152i	\(0.639006\pi\)
\(29\)	−2.45169	+	0.962216i	−0.455267	+	0.178679i	−0.581883	−	0.813273i	\(-0.697684\pi\)
							0.126616	+	0.991952i	\(0.459588\pi\)
\(31\)	−6.76447	+	3.90547i	−1.21493	+	0.701443i	−0.963830	−	0.266517i	\(-0.914127\pi\)
							−0.251104	+	0.967960i	\(0.580794\pi\)
\(37\)	4.71273	−	5.90958i	0.774768	−	0.971528i	−0.225228	−	0.974306i	\(-0.572313\pi\)
							0.999996	+	0.00277785i	\(0.000884219\pi\)
\(41\)	3.68571	−	3.41984i	0.575611	−	0.534089i	−0.337752	−	0.941235i	\(-0.609666\pi\)
							0.913363	+	0.407146i	\(0.133476\pi\)
\(43\)	−1.59405	−	1.47906i	−0.243090	−	0.225555i	0.549213	−	0.835683i	\(-0.314928\pi\)
							−0.792303	+	0.610128i	\(0.791118\pi\)
\(47\)	4.96154	−	3.38272i	0.723715	−	0.493421i	−0.144565	−	0.989495i	\(-0.546178\pi\)
							0.868280	+	0.496075i	\(0.165226\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bh.a.20.11	✓	648
9.5	odd	6	inner	441.2.bh.a.167.11	yes	648
49.27	odd	14	inner	441.2.bh.a.272.11	yes	648
441.419	even	42	inner	441.2.bh.a.419.11	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bh.a.20.11	✓	648	1.1	even	1	trivial
441.2.bh.a.167.11	yes	648	9.5	odd	6	inner
441.2.bh.a.272.11	yes	648	49.27	odd	14	inner
441.2.bh.a.419.11	yes	648	441.419	even	42	inner