Embedded newform 441.2.bd.a.47.13

• Label: 441.2.bd.a.47.13
• 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			47.13
Character	\(\chi\)	\(=\)	441.47
Dual form			441.2.bd.a.122.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22764 - 1.32308i) q^{2} +(-1.33805 - 1.09983i) q^{3} +(-0.0939836 + 1.25412i) q^{4} +(-2.40505 - 3.01584i) q^{5} +(0.187488 + 3.12054i) q^{6} +(-1.92992 + 1.80981i) q^{7} +(-1.04756 + 0.835402i) q^{8} +(0.580768 + 2.94325i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22764	−	1.32308i	−0.868072	−	0.935559i	0.130364	−	0.991466i	\(-0.458385\pi\)
							−0.998436	+	0.0559070i	\(0.982195\pi\)
\(3\)	−1.33805	−	1.09983i	−0.772525	−	0.634985i
							\(5\)	−2.40505	−	3.01584i	−1.07557	−	1.34872i	−0.933382	−	0.358886i	\(-0.883157\pi\)
−0.142191	−	0.989839i	\(-0.545415\pi\)
\(7\)	−1.92992	+	1.80981i	−0.729443	+	0.684042i
							\(11\)	1.05209	−	0.240133i	0.317218	−	0.0724030i	−0.0609474	−	0.998141i	\(-0.519412\pi\)
0.378165	+	0.925738i	\(0.376555\pi\)
\(13\)	−0.184106	−	0.198419i	−0.0510617	−	0.0550315i	0.707004	−	0.707209i	\(-0.250046\pi\)
							−0.758066	+	0.652178i	\(0.773856\pi\)
\(17\)	−0.133789	−	1.78530i	−0.0324487	−	0.432998i	−0.989643	−	0.143551i	\(-0.954148\pi\)
							0.957194	−	0.289447i	\(-0.0934713\pi\)
\(19\)	−2.03265	−	1.17355i	−0.466321	−	0.269231i	0.248377	−	0.968663i	\(-0.420103\pi\)
							−0.714699	+	0.699433i	\(0.753436\pi\)
\(23\)	−3.64358	−	7.56596i	−0.759738	−	1.57761i	−0.815219	−	0.579153i	\(-0.803383\pi\)
							0.0554806	−	0.998460i	\(-0.482331\pi\)
\(29\)	3.45552	+	5.06831i	0.641673	+	0.941162i	0.999971	+	0.00755451i	\(0.00240470\pi\)
							−0.358298	+	0.933607i	\(0.616643\pi\)
\(31\)	−4.36320	−	2.51910i	−0.783654	−	0.452443i	0.0540694	−	0.998537i	\(-0.482781\pi\)
							−0.837724	+	0.546094i	\(0.816114\pi\)
\(37\)	3.96255	−	2.70162i	0.651439	−	0.444144i	−0.191981	−	0.981399i	\(-0.561491\pi\)
							0.843420	+	0.537255i	\(0.180539\pi\)
\(41\)	−2.32357	+	5.92037i	−0.362881	+	0.924606i	0.626521	+	0.779405i	\(0.284478\pi\)
							−0.989402	+	0.145202i	\(0.953617\pi\)
\(43\)	3.52153	+	8.97270i	0.537028	+	1.36832i	0.899741	+	0.436423i	\(0.143755\pi\)
							−0.362714	+	0.931901i	\(0.618150\pi\)
\(47\)	3.33891	−	3.09806i	0.487031	−	0.451898i	−0.398031	−	0.917372i	\(-0.630306\pi\)
							0.885062	+	0.465474i	\(0.154116\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.47.13	✓	648
9.5	odd	6		441.2.bn.a.194.13	yes	648
49.24	odd	42		441.2.bn.a.416.13	yes	648
441.122	even	42	inner	441.2.bd.a.122.13	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.47.13	✓	648	1.1	even	1	trivial
441.2.bd.a.122.13	yes	648	441.122	even	42	inner
441.2.bn.a.194.13	yes	648	9.5	odd	6
441.2.bn.a.416.13	yes	648	49.24	odd	42