Embedded newform 441.2.bh.a.20.16

• Label: 441.2.bh.a.20.16
• 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			20.16
Character	\(\chi\)	\(=\)	441.20
Dual form			441.2.bh.a.419.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.813844 - 1.19369i) q^{2} +(0.511765 + 1.65472i) q^{3} +(-0.0318715 + 0.0812073i) q^{4} +(0.0488140 - 0.0452928i) q^{5} +(1.55873 - 1.95757i) q^{6} +(-2.34525 + 1.22466i) q^{7} +(-2.69414 + 0.614919i) q^{8} +(-2.47619 + 1.69365i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 15 q^{2} - 14 q^{3} - 57 q^{4} - 21 q^{5} + 14 q^{6} - 5 q^{7} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.813844	−	1.19369i	−0.575475	−	0.844066i	0.422586	−	0.906323i	\(-0.361123\pi\)
							−0.998060	+	0.0622568i	\(0.980170\pi\)
\(3\)	0.511765	+	1.65472i	0.295468	+	0.955353i
							\(5\)	0.0488140	−	0.0452928i	0.0218303	−	0.0202555i	−0.669182	−	0.743098i	\(-0.733355\pi\)
0.691012	+	0.722843i	\(0.257165\pi\)
\(7\)	−2.34525	+	1.22466i	−0.886422	+	0.462878i
							\(11\)	1.31476	+	1.92839i	0.396414	+	0.581432i	0.971611	−	0.236582i	\(-0.0760273\pi\)
−0.575198	+	0.818014i	\(0.695075\pi\)
\(13\)	3.84634	−	0.288243i	1.06678	−	0.0799443i	0.470250	−	0.882533i	\(-0.344164\pi\)
							0.596533	+	0.802589i	\(0.296545\pi\)
\(17\)	−2.52782	+	3.16979i	−0.613086	+	0.768786i	−0.987353	−	0.158535i	\(-0.949323\pi\)
							0.374267	+	0.927321i	\(0.377894\pi\)
\(19\)			5.05566i			1.15985i	0.814670	+	0.579924i	\(0.196918\pi\)
							−0.814670	+	0.579924i	\(0.803082\pi\)
\(23\)	−3.33631	−	1.30941i	−0.695670	−	0.273030i	−0.00895767	−	0.999960i	\(-0.502851\pi\)
							−0.686712	+	0.726930i	\(0.740947\pi\)
\(29\)	−4.04680	+	1.58825i	−0.751472	+	0.294931i	−0.709994	−	0.704208i	\(-0.751302\pi\)
							−0.0414788	+	0.999139i	\(0.513207\pi\)
\(31\)	7.26116	−	4.19223i	1.30414	−	0.752947i	0.323031	−	0.946388i	\(-0.395298\pi\)
							0.981112	+	0.193441i	\(0.0619650\pi\)
\(37\)	−2.39868	+	3.00785i	−0.394340	+	0.494487i	−0.938878	−	0.344249i	\(-0.888133\pi\)
							0.544538	+	0.838736i	\(0.316705\pi\)
\(41\)	−2.58750	+	2.40085i	−0.404100	+	0.374950i	−0.855876	−	0.517182i	\(-0.826981\pi\)
							0.451776	+	0.892131i	\(0.350791\pi\)
\(43\)	3.96510	+	3.67907i	0.604672	+	0.561054i	0.921965	−	0.387274i	\(-0.126583\pi\)
							−0.317293	+	0.948328i	\(0.602774\pi\)
\(47\)	−0.253354	+	0.172734i	−0.0369555	+	0.0251958i	−0.581657	−	0.813434i	\(-0.697595\pi\)
							0.544701	+	0.838630i	\(0.316643\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bh.a.20.16	✓	648
9.5	odd	6	inner	441.2.bh.a.167.16	yes	648
49.27	odd	14	inner	441.2.bh.a.272.16	yes	648
441.419	even	42	inner	441.2.bh.a.419.16	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bh.a.20.16	✓	648	1.1	even	1	trivial
441.2.bh.a.167.16	yes	648	9.5	odd	6	inner
441.2.bh.a.272.16	yes	648	49.27	odd	14	inner
441.2.bh.a.419.16	yes	648	441.419	even	42	inner