Embedded newform 441.2.bd.a.47.4

• Label: 441.2.bd.a.47.4
• 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			47.4
Character	\(\chi\)	\(=\)	441.47
Dual form			441.2.bd.a.122.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.75148 - 1.88764i) q^{2} +(0.0116260 + 1.73201i) q^{3} +(-0.346064 + 4.61791i) q^{4} +(-0.695350 - 0.871941i) q^{5} +(3.24905 - 3.05552i) q^{6} +(-2.60184 + 0.480046i) q^{7} +(5.29657 - 4.22388i) q^{8} +(-2.99973 + 0.0402729i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.75148	−	1.88764i	−1.23848	−	1.33476i	−0.922499	−	0.385998i	\(-0.873857\pi\)
							−0.315981	−	0.948766i	\(-0.602333\pi\)
\(3\)	0.0116260	+	1.73201i	0.00671230	+	0.999977i
							\(5\)	−0.695350	−	0.871941i	−0.310970	−	0.389944i	0.601645	−	0.798763i	\(-0.294512\pi\)
−0.912615	+	0.408819i	\(0.865941\pi\)
\(7\)	−2.60184	+	0.480046i	−0.983402	+	0.181440i
							\(11\)	1.68240	−	0.383996i	0.507261	−	0.115779i	0.0387717	−	0.999248i	\(-0.487655\pi\)
0.468490	+	0.883469i	\(0.344798\pi\)
\(13\)	2.33944	+	2.52131i	0.648843	+	0.699287i	0.968715	−	0.248174i	\(-0.0798305\pi\)
							−0.319872	+	0.947461i	\(0.603640\pi\)
\(17\)	−0.251022	−	3.34966i	−0.0608818	−	0.812411i	−0.941162	−	0.337956i	\(-0.890264\pi\)
							0.880280	−	0.474455i	\(-0.157355\pi\)
\(19\)	−2.18600	−	1.26209i	−0.501502	−	0.289542i	0.227831	−	0.973701i	\(-0.426836\pi\)
							−0.729334	+	0.684158i	\(0.760170\pi\)
\(23\)	0.848365	+	1.76165i	0.176896	+	0.367329i	0.970498	−	0.241108i	\(-0.0775109\pi\)
							−0.793602	+	0.608437i	\(0.791797\pi\)
\(29\)	−4.77828	−	7.00845i	−0.887304	−	1.30144i	−0.952805	−	0.303583i	\(-0.901817\pi\)
							0.0655009	−	0.997853i	\(-0.479135\pi\)
\(31\)	−9.00720	−	5.20031i	−1.61774	−	0.934004i	−0.987503	−	0.157602i	\(-0.949624\pi\)
							−0.630239	−	0.776401i	\(-0.717043\pi\)
\(37\)	2.28236	−	1.55609i	0.375218	−	0.255819i	−0.360997	−	0.932567i	\(-0.617564\pi\)
							0.736215	+	0.676748i	\(0.236611\pi\)
\(41\)	2.19413	−	5.59055i	0.342665	−	0.873097i	−0.650830	−	0.759224i	\(-0.725579\pi\)
							0.993495	−	0.113873i	\(-0.0363258\pi\)
\(43\)	0.249725	+	0.636288i	0.0380826	+	0.0970330i	0.948650	−	0.316328i	\(-0.102450\pi\)
							−0.910567	+	0.413361i	\(0.864355\pi\)
\(47\)	4.14511	−	3.84610i	0.604627	−	0.561012i	−0.317325	−	0.948317i	\(-0.602784\pi\)
							0.921952	+	0.387305i	\(0.126594\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.47.4	✓	648
9.5	odd	6		441.2.bn.a.194.4	yes	648
49.24	odd	42		441.2.bn.a.416.4	yes	648
441.122	even	42	inner	441.2.bd.a.122.4	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.47.4	✓	648	1.1	even	1	trivial
441.2.bd.a.122.4	yes	648	441.122	even	42	inner
441.2.bn.a.194.4	yes	648	9.5	odd	6
441.2.bn.a.416.4	yes	648	49.24	odd	42