Embedded newform 441.2.bd.a.47.5

• Label: 441.2.bd.a.47.5
• Level: $441$
• Weight: $2$
• Character: 441.47
• 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			47.5
Character	\(\chi\)	\(=\)	441.47
Dual form			441.2.bd.a.122.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.67227 - 1.80228i) q^{2} +(-0.998913 + 1.41498i) q^{3} +(-0.302261 + 4.03339i) q^{4} +(2.61954 + 3.28480i) q^{5} +(4.22064 - 0.565913i) q^{6} +(2.36845 + 1.17916i) q^{7} +(3.93034 - 3.13434i) q^{8} +(-1.00435 - 2.82689i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{5}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.67227	−	1.80228i	−1.18247	−	1.27440i	−0.953644	−	0.300937i	\(-0.902700\pi\)
							−0.228830	−	0.973466i	\(-0.573490\pi\)
\(3\)	−0.998913	+	1.41498i	−0.576722	+	0.816940i
							\(5\)	2.61954	+	3.28480i	1.17149	+	1.46901i	0.853639	+	0.520866i	\(0.174391\pi\)
0.317854	+	0.948140i	\(0.397038\pi\)
\(7\)	2.36845	+	1.17916i	0.895192	+	0.445682i
							\(11\)	−1.89528	+	0.432586i	−0.571449	+	0.130429i	−0.498475	−	0.866904i	\(-0.666106\pi\)
−0.0729744	+	0.997334i	\(0.523249\pi\)
\(13\)	2.37751	+	2.56234i	0.659402	+	0.710666i	0.970874	−	0.239589i	\(-0.0770126\pi\)
							−0.311472	+	0.950255i	\(0.600822\pi\)
\(17\)	0.265862	+	3.54768i	0.0644810	+	0.860439i	0.931866	+	0.362802i	\(0.118180\pi\)
							−0.867385	+	0.497637i	\(0.834201\pi\)
\(19\)	1.25082	+	0.722163i	0.286958	+	0.165676i	0.636569	−	0.771219i	\(-0.280353\pi\)
							−0.349611	+	0.936895i	\(0.613686\pi\)
\(23\)	−3.73692	−	7.75979i	−0.779202	−	1.61803i	−0.786137	−	0.618053i	\(-0.787922\pi\)
							0.00693500	−	0.999976i	\(-0.497793\pi\)
\(29\)	−2.51898	−	3.69466i	−0.467762	−	0.686081i	0.517780	−	0.855514i	\(-0.326759\pi\)
							−0.985542	+	0.169433i	\(0.945806\pi\)
\(31\)	1.75246	+	1.01178i	0.314751	+	0.181722i	0.649051	−	0.760745i	\(-0.275166\pi\)
							−0.334299	+	0.942467i	\(0.608500\pi\)
\(37\)	2.14950	−	1.46550i	0.353376	−	0.240927i	−0.373601	−	0.927589i	\(-0.621877\pi\)
							0.726977	+	0.686662i	\(0.240925\pi\)
\(41\)	−2.28884	+	5.83188i	−0.357457	+	0.910786i	0.633148	+	0.774031i	\(0.281763\pi\)
							−0.990605	+	0.136755i	\(0.956333\pi\)
\(43\)	−2.16186	−	5.50833i	−0.329681	−	0.840012i	−0.995575	−	0.0939667i	\(-0.970045\pi\)
							0.665895	−	0.746046i	\(-0.268050\pi\)
\(47\)	3.04188	−	2.82245i	0.443704	−	0.411697i	−0.426424	−	0.904524i	\(-0.640227\pi\)
							0.870128	+	0.492826i	\(0.164036\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.47.5	✓	648
9.5	odd	6		441.2.bn.a.194.5	yes	648
49.24	odd	42		441.2.bn.a.416.5	yes	648
441.122	even	42	inner	441.2.bd.a.122.5	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.47.5	✓	648	1.1	even	1	trivial
441.2.bd.a.122.5	yes	648	441.122	even	42	inner
441.2.bn.a.194.5	yes	648	9.5	odd	6
441.2.bn.a.416.5	yes	648	49.24	odd	42