Embedded newform 441.2.bd.a.122.4

• Label: 441.2.bd.a.122.4
• Level: $441$
• Weight: $2$
• Character: 441.122
• 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			122.4
Character	\(\chi\)	\(=\)	441.122
Dual form			441.2.bd.a.47.4

$q$-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 \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{37}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.47.4	✓	648	441.47	even	42	inner
441.2.bd.a.122.4	yes	648	1.1	even	1	trivial
441.2.bn.a.194.4	yes	648	49.47	odd	42
441.2.bn.a.416.4	yes	648	9.2	odd	6