Embedded newform 441.2.bd.a.437.51

• Label: 441.2.bd.a.437.51
• Level: $441$
• Weight: $2$
• Character: 441.437
• 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			437.51
Character	\(\chi\)	\(=\)	441.437
Dual form			441.2.bd.a.110.51

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.52537 + 0.189250i) q^{2} +(-0.882430 + 1.49041i) q^{3} +(4.36400 + 0.657768i) q^{4} +(-0.486593 + 2.13190i) q^{5} +(-2.51052 + 3.59683i) q^{6} +(1.52157 - 2.16445i) q^{7} +(5.95832 + 1.35995i) q^{8} +(-1.44263 - 2.63036i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{31}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.52537	+	0.189250i	1.78570	+	0.133820i	0.926133	−	0.377196i	\(-0.123112\pi\)
							0.859571	+	0.511016i	\(0.170731\pi\)
\(3\)	−0.882430	+	1.49041i	−0.509471	+	0.860488i
							\(5\)	−0.486593	+	2.13190i	−0.217611	+	0.953416i	0.741626	+	0.670813i	\(0.234055\pi\)
−0.959237	+	0.282602i	\(0.908802\pi\)
\(7\)	1.52157	−	2.16445i	0.575099	−	0.818084i
							\(11\)	0.444304	+	0.922606i	0.133963	+	0.278176i	0.957149	−	0.289594i	\(-0.0935204\pi\)
−0.823187	+	0.567771i	\(0.807806\pi\)
\(13\)	1.89601	+	0.142086i	0.525858	+	0.0394076i	0.335015	−	0.942213i	\(-0.391259\pi\)
							0.190843	+	0.981621i	\(0.438878\pi\)
\(17\)	−6.21921	+	0.937394i	−1.50838	+	0.227352i	−0.850581	−	0.525843i	\(-0.823750\pi\)
							−0.657797	+	0.753195i	\(0.728512\pi\)
\(19\)	−1.54472	+	0.891846i	−0.354384	+	0.204604i	−0.666614	−	0.745403i	\(-0.732257\pi\)
							0.312231	+	0.950006i	\(0.398924\pi\)
\(23\)	−2.10802	−	1.68109i	−0.439553	−	0.350532i	0.378572	−	0.925572i	\(-0.376415\pi\)
							−0.818125	+	0.575040i	\(0.804987\pi\)
\(29\)	−1.02994	−	0.404221i	−0.191255	−	0.0750619i	0.267781	−	0.963480i	\(-0.413710\pi\)
							−0.459035	+	0.888418i	\(0.651805\pi\)
\(31\)	9.15570	−	5.28605i	1.64441	−	0.949402i	0.665174	−	0.746688i	\(-0.268357\pi\)
							0.979238	−	0.202714i	\(-0.0649762\pi\)
\(37\)	2.78911	−	7.10654i	0.458527	−	1.16831i	−0.495506	−	0.868605i	\(-0.665017\pi\)
							0.954033	−	0.299703i	\(-0.0968875\pi\)
\(41\)	−3.65992	−	3.39591i	−0.571584	−	0.530352i	0.340560	−	0.940223i	\(-0.389383\pi\)
							−0.912144	+	0.409871i	\(0.865574\pi\)
\(43\)	−3.77572	+	3.50336i	−0.575792	+	0.534257i	−0.913418	−	0.407024i	\(-0.866567\pi\)
							0.337626	+	0.941280i	\(0.390376\pi\)
\(47\)	0.598936	−	7.99225i	0.0873638	−	1.16579i	−0.765422	−	0.643528i	\(-0.777470\pi\)
							0.852786	−	0.522260i	\(-0.174911\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.437.51	yes	648
9.2	odd	6		441.2.bn.a.290.51	yes	648
49.12	odd	42		441.2.bn.a.257.51	yes	648
441.110	even	42	inner	441.2.bd.a.110.51	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.110.51	✓	648	441.110	even	42	inner
441.2.bd.a.437.51	yes	648	1.1	even	1	trivial
441.2.bn.a.257.51	yes	648	49.12	odd	42
441.2.bn.a.290.51	yes	648	9.2	odd	6