Embedded newform 441.2.bd.a.173.42

• Label: 441.2.bd.a.173.42
• Level: $441$
• Weight: $2$
• Character: 441.173
• 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.bd (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			173.42
Character	\(\chi\)	\(=\)	441.173
Dual form			441.2.bd.a.311.42

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.928240 - 1.36148i) q^{2} +(-1.34896 - 1.08642i) q^{3} +(-0.261312 - 0.665811i) q^{4} +(0.872976 - 3.82476i) q^{5} +(-2.73130 + 0.828121i) q^{6} +(1.64298 - 2.07379i) q^{7} +(2.06393 + 0.471079i) q^{8} +(0.639385 + 2.93107i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{17}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.928240	−	1.36148i	0.656365	−	0.962711i	−0.343359	−	0.939204i	\(-0.611565\pi\)
							0.999724	−	0.0235063i	\(-0.00748298\pi\)
\(3\)	−1.34896	−	1.08642i	−0.778822	−	0.627245i
							\(5\)	0.872976	−	3.82476i	0.390407	−	1.71048i	−0.272819	−	0.962065i	\(-0.587956\pi\)
0.663226	−	0.748419i	\(-0.269187\pi\)
\(7\)	1.64298	−	2.07379i	0.620990	−	0.783819i
							\(11\)	0.266754	+	0.553920i	0.0804293	+	0.167013i	0.937305	−	0.348509i	\(-0.113312\pi\)
−0.856876	+	0.515523i	\(0.827598\pi\)
\(13\)	0.141311	−	0.207266i	0.0391927	−	0.0574852i	−0.806142	−	0.591722i	\(-0.798448\pi\)
							0.845335	+	0.534237i	\(0.179401\pi\)
\(17\)	−2.15951	+	5.50235i	−0.523759	+	1.33452i	0.387191	+	0.921999i	\(0.373445\pi\)
							−0.910950	+	0.412516i	\(0.864650\pi\)
\(19\)	−4.00625	−	2.31301i	−0.919097	−	0.530641i	−0.0357503	−	0.999361i	\(-0.511382\pi\)
							−0.883347	+	0.468720i	\(0.844715\pi\)
\(23\)	3.92237	+	3.12799i	0.817871	+	0.652230i	0.940338	−	0.340242i	\(-0.110509\pi\)
							−0.122467	+	0.992473i	\(0.539081\pi\)
\(29\)	0.790738	+	5.24620i	0.146836	+	0.974195i	0.933954	+	0.357393i	\(0.116334\pi\)
							−0.787118	+	0.616803i	\(0.788428\pi\)
\(31\)	−1.38459	−	0.799392i	−0.248679	−	0.143575i	0.370480	−	0.928840i	\(-0.379193\pi\)
							−0.619159	+	0.785265i	\(0.712527\pi\)
\(37\)	−5.30653	+	0.799831i	−0.872388	+	0.131491i	−0.569952	−	0.821678i	\(-0.693038\pi\)
							−0.302436	+	0.953170i	\(0.597800\pi\)
\(41\)	−5.05227	+	1.55842i	−0.789032	+	0.243384i	−0.662978	−	0.748639i	\(-0.730708\pi\)
							−0.126055	+	0.992023i	\(0.540231\pi\)
\(43\)	5.50607	+	1.69840i	0.839667	+	0.259003i	0.684617	−	0.728903i	\(-0.259970\pi\)
							0.155051	+	0.987906i	\(0.450446\pi\)
\(47\)	9.02292	+	6.15172i	1.31613	+	0.897321i	0.998701	−	0.0509620i	\(-0.0162287\pi\)
							0.317427	+	0.948283i	\(0.397181\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.173.42	✓	648
9.5	odd	6		441.2.bn.a.320.13	yes	648
49.17	odd	42		441.2.bn.a.164.13	yes	648
441.311	even	42	inner	441.2.bd.a.311.42	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.173.42	✓	648	1.1	even	1	trivial
441.2.bd.a.311.42	yes	648	441.311	even	42	inner
441.2.bn.a.164.13	yes	648	49.17	odd	42
441.2.bn.a.320.13	yes	648	9.5	odd	6