Embedded newform 441.2.bd.a.59.3

• Label: 441.2.bd.a.59.3
• Level: $441$
• Weight: $2$
• Character: 441.59
• 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			59.3
Character	\(\chi\)	\(=\)	441.59
Dual form			441.2.bd.a.299.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.40425 + 0.943598i) q^{2} +(-1.66433 - 0.479582i) q^{3} +(3.42393 - 3.17694i) q^{4} +(0.922566 + 0.444285i) q^{5} +(4.45400 - 0.417427i) q^{6} +(-0.631236 - 2.56935i) q^{7} +(-2.99296 + 6.21494i) q^{8} +(2.54000 + 1.59637i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.40425	+	0.943598i	−1.70006	+	0.667224i	−0.998949	−	0.0458284i	\(-0.985407\pi\)
							−0.701110	+	0.713053i	\(0.747312\pi\)
\(3\)	−1.66433	−	0.479582i	−0.960903	−	0.276887i
							\(5\)	0.922566	+	0.444285i	0.412584	+	0.198690i	0.628649	−	0.777689i	\(-0.283608\pi\)
−0.216065	+	0.976379i	\(0.569322\pi\)
\(7\)	−0.631236	−	2.56935i	−0.238585	−	0.971122i
							\(11\)	1.64224	+	1.30964i	0.495154	+	0.394873i	0.838981	−	0.544161i	\(-0.183152\pi\)
−0.343827	+	0.939033i	\(0.611723\pi\)
\(13\)	2.30590	−	0.904998i	0.639541	−	0.251001i	−0.0233385	−	0.999728i	\(-0.507430\pi\)
							0.662879	+	0.748726i	\(0.269334\pi\)
\(17\)	−2.20085	−	2.04209i	−0.533785	−	0.495280i	0.366597	−	0.930380i	\(-0.380523\pi\)
							−0.900382	+	0.435100i	\(0.856713\pi\)
\(19\)	−4.42780	+	2.55639i	−1.01581	+	0.586476i	−0.912887	−	0.408213i	\(-0.866152\pi\)
							−0.102920	+	0.994690i	\(0.532819\pi\)
\(23\)	−1.40235	+	0.320077i	−0.292410	+	0.0667407i	−0.366209	−	0.930532i	\(-0.619345\pi\)
							0.0737996	+	0.997273i	\(0.476487\pi\)
\(29\)	1.30249	+	4.22256i	0.241866	+	0.784110i	0.992376	+	0.123250i	\(0.0393318\pi\)
							−0.750510	+	0.660859i	\(0.770192\pi\)
\(31\)	5.99991	−	3.46405i	1.07762	−	0.622162i	0.147364	−	0.989082i	\(-0.452921\pi\)
							0.930252	+	0.366921i	\(0.119588\pi\)
\(37\)	8.66701	−	2.67342i	1.42485	−	0.439507i	0.515983	−	0.856599i	\(-0.327427\pi\)
							0.908865	+	0.417091i	\(0.136950\pi\)
\(41\)	9.55781	−	6.51640i	1.49268	−	1.01769i	0.504666	−	0.863314i	\(-0.331615\pi\)
							0.988012	−	0.154377i	\(-0.0493369\pi\)
\(43\)	1.97178	+	1.34434i	0.300694	+	0.205010i	0.704262	−	0.709940i	\(-0.251278\pi\)
							−0.403568	+	0.914949i	\(0.632230\pi\)
\(47\)	−1.96799	−	5.01436i	−0.287061	−	0.731420i	−0.999558	−	0.0297442i	\(-0.990531\pi\)
							0.712496	−	0.701676i	\(-0.247565\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.59.3	✓	648
9.2	odd	6		441.2.bn.a.353.3	yes	648
49.5	odd	42		441.2.bn.a.5.3	yes	648
441.299	even	42	inner	441.2.bd.a.299.3	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.3	✓	648	1.1	even	1	trivial
441.2.bd.a.299.3	yes	648	441.299	even	42	inner
441.2.bn.a.5.3	yes	648	49.5	odd	42
441.2.bn.a.353.3	yes	648	9.2	odd	6