Embedded newform 441.2.bn.a.5.16

• Label: 441.2.bn.a.5.16
• Level: $441$
• Weight: $2$
• Character: 441.5
• 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.bn (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			5.16
Character	\(\chi\)	\(=\)	441.5
Dual form			441.2.bn.a.353.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.14829 + 0.915734i) q^{2} +(-0.441504 - 1.67484i) q^{3} +(0.0349690 - 0.153209i) q^{4} +(-1.40572 - 0.958401i) q^{5} +(2.04068 + 1.51890i) q^{6} +(-2.52036 - 0.804862i) q^{7} +(-1.17437 - 2.43860i) q^{8} +(-2.61015 + 1.47889i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.14829	+	0.915734i	−0.811967	+	0.647522i	−0.938823	−	0.344399i	\(-0.888083\pi\)
							0.126857	+	0.991921i	\(0.459511\pi\)
\(3\)	−0.441504	−	1.67484i	−0.254903	−	0.966967i
							\(5\)	−1.40572	−	0.958401i	−0.628655	−	0.428610i	0.206636	−	0.978418i	\(-0.433749\pi\)
−0.835291	+	0.549808i	\(0.814701\pi\)
\(7\)	−2.52036	−	0.804862i	−0.952605	−	0.304209i
							\(11\)	0.455826	+	3.02421i	0.137437	+	0.911832i	0.946173	+	0.323661i	\(0.104914\pi\)
−0.808736	+	0.588171i	\(0.799848\pi\)
\(13\)	0.885653	+	5.87592i	0.245636	+	1.62969i	0.683011	+	0.730408i	\(0.260670\pi\)
							−0.437375	+	0.899279i	\(0.644092\pi\)
\(17\)	5.35504	−	4.96875i	1.29879	−	1.20510i	0.333848	−	0.942627i	\(-0.391653\pi\)
							0.964940	−	0.262472i	\(-0.0845377\pi\)
\(19\)	6.22182	+	3.59217i	1.42738	+	0.824100i	0.996914	−	0.0785048i	\(-0.0250146\pi\)
							0.430470	+	0.902605i	\(0.358348\pi\)
\(23\)	−1.18239	−	3.83322i	−0.246546	−	0.799281i	−0.991263	−	0.131901i	\(-0.957892\pi\)
							0.744717	−	0.667380i	\(-0.232584\pi\)
\(29\)	2.62628	+	2.83046i	0.487688	+	0.525603i	0.928169	−	0.372159i	\(-0.121383\pi\)
							−0.440481	+	0.897762i	\(0.645192\pi\)
\(31\)		−	1.14292i		−	0.205275i	−0.994719	−	0.102637i	\(-0.967272\pi\)
							0.994719	−	0.102637i	\(-0.0327281\pi\)
\(37\)	2.20072	+	0.678834i	0.361797	+	0.111600i	0.470324	−	0.882494i	\(-0.344137\pi\)
							−0.108527	+	0.994094i	\(0.534613\pi\)
\(41\)	−0.696680	+	9.29655i	−0.108803	+	1.45188i	0.629286	+	0.777174i	\(0.283347\pi\)
							−0.738089	+	0.674703i	\(0.764272\pi\)
\(43\)	0.720078	+	9.60878i	0.109811	+	1.46533i	0.731382	+	0.681968i	\(0.238875\pi\)
							−0.621571	+	0.783357i	\(0.713505\pi\)
\(47\)	−1.97112	−	2.47171i	−0.287517	−	0.360535i	0.617007	−	0.786958i	\(-0.288345\pi\)
							−0.904524	+	0.426423i	\(0.859774\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.5.16	yes	648
9.2	odd	6		441.2.bd.a.299.16	yes	648
49.10	odd	42		441.2.bd.a.59.16	✓	648
441.353	even	42	inner	441.2.bn.a.353.16	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.16	✓	648	49.10	odd	42
441.2.bd.a.299.16	yes	648	9.2	odd	6
441.2.bn.a.5.16	yes	648	1.1	even	1	trivial
441.2.bn.a.353.16	yes	648	441.353	even	42	inner