Embedded newform 441.2.bh.a.20.19

• Label: 441.2.bh.a.20.19
• 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			20.19
Character	\(\chi\)	\(=\)	441.20
Dual form			441.2.bh.a.419.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.598227 - 0.877437i) q^{2} +(-1.65613 + 0.507172i) q^{3} +(0.318661 - 0.811934i) q^{4} +(-2.90237 + 2.69301i) q^{5} +(1.43575 + 1.14975i) q^{6} +(-1.67397 + 2.04886i) q^{7} +(-2.97373 + 0.678735i) q^{8} +(2.48555 - 1.67989i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 15 q^{2} - 14 q^{3} - 57 q^{4} - 21 q^{5} + 14 q^{6} - 5 q^{7} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.598227	−	0.877437i	−0.423010	−	0.620442i	0.554284	−	0.832327i	\(-0.312992\pi\)
							−0.977295	+	0.211885i	\(0.932040\pi\)
\(3\)	−1.65613	+	0.507172i	−0.956169	+	0.292816i
							\(5\)	−2.90237	+	2.69301i	−1.29798	+	1.20435i	−0.332729	+	0.943022i	\(0.607969\pi\)
−0.965250	+	0.261326i	\(0.915840\pi\)
\(7\)	−1.67397	+	2.04886i	−0.632702	+	0.774396i
							\(11\)	−2.48814	−	3.64943i	−0.750202	−	1.10034i	−0.991527	−	0.129900i	\(-0.958534\pi\)
0.241325	−	0.970444i	\(-0.422418\pi\)
\(13\)	6.56026	−	0.491624i	1.81949	−	0.136352i	0.879140	−	0.476563i	\(-0.158118\pi\)
							0.940349	+	0.340212i	\(0.110499\pi\)
\(17\)	2.93299	−	3.67786i	0.711355	−	0.892012i	−0.286459	−	0.958093i	\(-0.592478\pi\)
							0.997814	+	0.0660811i	\(0.0210496\pi\)
\(19\)			0.576838i			0.132336i	0.997809	+	0.0661678i	\(0.0210773\pi\)
							−0.997809	+	0.0661678i	\(0.978923\pi\)
\(23\)	3.47649	+	1.36442i	0.724899	+	0.284502i	0.698959	−	0.715161i	\(-0.253647\pi\)
							0.0259401	+	0.999664i	\(0.491742\pi\)
\(29\)	1.95009	−	0.765352i	0.362122	−	0.142122i	−0.177303	−	0.984156i	\(-0.556737\pi\)
							0.539424	+	0.842034i	\(0.318642\pi\)
\(31\)	−4.00040	+	2.30963i	−0.718492	+	0.414822i	−0.814198	−	0.580588i	\(-0.802823\pi\)
							0.0957052	+	0.995410i	\(0.469489\pi\)
\(37\)	3.56791	−	4.47402i	0.586561	−	0.735525i	−0.396655	−	0.917968i	\(-0.629829\pi\)
							0.983216	+	0.182443i	\(0.0584005\pi\)
\(41\)	−0.823274	+	0.763887i	−0.128574	+	0.119299i	−0.741839	−	0.670578i	\(-0.766046\pi\)
							0.613265	+	0.789877i	\(0.289856\pi\)
\(43\)	1.66515	+	1.54504i	0.253934	+	0.235616i	0.796860	−	0.604165i	\(-0.206493\pi\)
							−0.542926	+	0.839781i	\(0.682684\pi\)
\(47\)	9.12767	−	6.22314i	1.33141	−	0.907738i	0.332039	−	0.943266i	\(-0.392264\pi\)
							0.999369	+	0.0355275i	\(0.0113111\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bh.a.20.19	✓	648
9.5	odd	6	inner	441.2.bh.a.167.19	yes	648
49.27	odd	14	inner	441.2.bh.a.272.19	yes	648
441.419	even	42	inner	441.2.bh.a.419.19	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bh.a.20.19	✓	648	1.1	even	1	trivial
441.2.bh.a.167.19	yes	648	9.5	odd	6	inner
441.2.bh.a.272.19	yes	648	49.27	odd	14	inner
441.2.bh.a.419.19	yes	648	441.419	even	42	inner