Embedded newform 441.2.bd.a.47.18

• Label: 441.2.bd.a.47.18
• 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			47.18
Character	\(\chi\)	\(=\)	441.47
Dual form			441.2.bd.a.122.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.742532 - 0.800259i) q^{2} +(1.56641 + 0.739164i) q^{3} +(0.0603993 - 0.805973i) q^{4} +(1.83554 + 2.30169i) q^{5} +(-0.571585 - 1.80238i) q^{6} +(2.47412 - 0.937401i) q^{7} +(-2.39686 + 1.91143i) q^{8} +(1.90727 + 2.31567i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.742532	−	0.800259i	−0.525049	−	0.565868i	0.413725	−	0.910402i	\(-0.364228\pi\)
							−0.938774	+	0.344534i	\(0.888037\pi\)
\(3\)	1.56641	+	0.739164i	0.904366	+	0.426757i
							\(5\)	1.83554	+	2.30169i	0.820878	+	1.02935i	0.998972	+	0.0453350i	\(0.0144355\pi\)
−0.178094	+	0.984014i	\(0.556993\pi\)
\(7\)	2.47412	−	0.937401i	0.935130	−	0.354304i
							\(11\)	−1.97624	+	0.451064i	−0.595859	+	0.136001i	−0.509806	−	0.860290i	\(-0.670283\pi\)
−0.0860534	+	0.996291i	\(0.527426\pi\)
\(13\)	3.07356	+	3.31251i	0.852452	+	0.918725i	0.997498	−	0.0707000i	\(-0.0225233\pi\)
							−0.145045	+	0.989425i	\(0.546333\pi\)
\(17\)	−0.0740493	−	0.988119i	−0.0179596	−	0.239654i	−0.998958	−	0.0456385i	\(-0.985468\pi\)
							0.980998	−	0.194016i	\(-0.0621513\pi\)
\(19\)	−7.17949	−	4.14508i	−1.64709	−	0.950947i	−0.978221	−	0.207565i	\(-0.933446\pi\)
							−0.668867	−	0.743382i	\(-0.733221\pi\)
\(23\)	2.62271	+	5.44611i	0.546873	+	1.13559i	0.972973	+	0.230917i	\(0.0741727\pi\)
							−0.426101	+	0.904676i	\(0.640113\pi\)
\(29\)	−2.92863	−	4.29552i	−0.543834	−	0.797658i	0.451689	−	0.892175i	\(-0.350821\pi\)
							−0.995523	+	0.0945177i	\(0.969869\pi\)
\(31\)	−2.59751	−	1.49967i	−0.466527	−	0.269349i	0.248258	−	0.968694i	\(-0.420142\pi\)
							−0.714785	+	0.699345i	\(0.753475\pi\)
\(37\)	6.17119	−	4.20745i	1.01454	−	0.691700i	0.0626145	−	0.998038i	\(-0.480056\pi\)
							0.951923	+	0.306338i	\(0.0991037\pi\)
\(41\)	3.97249	−	10.1217i	0.620398	−	1.58075i	−0.181568	−	0.983378i	\(-0.558117\pi\)
							0.801966	−	0.597370i	\(-0.203788\pi\)
\(43\)	−1.69982	−	4.33106i	−0.259219	−	0.660480i	0.740716	−	0.671818i	\(-0.234486\pi\)
							−0.999936	+	0.0113377i	\(0.996391\pi\)
\(47\)	−6.94842	+	6.44719i	−1.01353	+	0.940419i	−0.998230	−	0.0594636i	\(-0.981061\pi\)
							−0.0153006	+	0.999883i	\(0.504871\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.47.18	✓	648
9.5	odd	6		441.2.bn.a.194.18	yes	648
49.24	odd	42		441.2.bn.a.416.18	yes	648
441.122	even	42	inner	441.2.bd.a.122.18	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.47.18	✓	648	1.1	even	1	trivial
441.2.bd.a.122.18	yes	648	441.122	even	42	inner
441.2.bn.a.194.18	yes	648	9.5	odd	6
441.2.bn.a.416.18	yes	648	49.24	odd	42