Embedded newform 441.2.bd.a.173.51

• Label: 441.2.bd.a.173.51
• Level: $441$
• Weight: $2$
• Character: 441.173
• 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			173.51
Character	\(\chi\)	\(=\)	441.173
Dual form			441.2.bd.a.311.51

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.37100 - 2.01089i) q^{2} +(-0.537043 + 1.64669i) q^{3} +(-1.43336 - 3.65214i) q^{4} +(-0.751003 + 3.29036i) q^{5} +(2.57503 + 3.33755i) q^{6} +(-1.97891 + 1.75611i) q^{7} +(-4.56367 - 1.04163i) q^{8} +(-2.42317 - 1.76868i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{17}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.37100	−	2.01089i	0.969447	−	1.42192i	0.0637620	−	0.997965i	\(-0.479690\pi\)
							0.905685	−	0.423952i	\(-0.139357\pi\)
\(3\)	−0.537043	+	1.64669i	−0.310062	+	0.950716i
							\(5\)	−0.751003	+	3.29036i	−0.335859	+	1.47149i	0.471728	+	0.881744i	\(0.343630\pi\)
−0.807587	+	0.589748i	\(0.799227\pi\)
\(7\)	−1.97891	+	1.75611i	−0.747957	+	0.663747i
							\(11\)	1.03923	+	2.15798i	0.313339	+	0.650656i	0.996852	−	0.0792825i	\(-0.0252629\pi\)
−0.683513	+	0.729939i	\(0.739549\pi\)
\(13\)	−3.28623	+	4.82001i	−0.911436	+	1.33683i	0.0299679	+	0.999551i	\(0.490459\pi\)
							−0.941404	+	0.337281i	\(0.890493\pi\)
\(17\)	1.47076	−	3.74743i	0.356712	−	0.908886i	−0.634053	−	0.773290i	\(-0.718610\pi\)
							0.990764	−	0.135596i	\(-0.0432950\pi\)
\(19\)	3.32577	+	1.92013i	0.762984	+	0.440509i	0.830366	−	0.557218i	\(-0.188131\pi\)
							−0.0673821	+	0.997727i	\(0.521465\pi\)
\(23\)	−0.819296	−	0.653367i	−0.170835	−	0.136236i	0.534340	−	0.845270i	\(-0.320560\pi\)
							−0.705175	+	0.709033i	\(0.749132\pi\)
\(29\)	0.827953	+	5.49311i	0.153747	+	1.02005i	0.924024	+	0.382335i	\(0.124880\pi\)
							−0.770277	+	0.637710i	\(0.779882\pi\)
\(31\)	−2.55317	−	1.47408i	−0.458564	−	0.264752i	0.252876	−	0.967499i	\(-0.418623\pi\)
							−0.711440	+	0.702747i	\(0.751957\pi\)
\(37\)	11.0077	−	1.65914i	1.80965	−	0.272761i	0.844709	−	0.535225i	\(-0.179773\pi\)
							0.964942	+	0.262464i	\(0.0845352\pi\)
\(41\)	−4.92522	+	1.51923i	−0.769189	+	0.237264i	−0.654423	−	0.756129i	\(-0.727088\pi\)
							−0.114767	+	0.993392i	\(0.536612\pi\)
\(43\)	3.84316	+	1.18546i	0.586076	+	0.180781i	0.573592	−	0.819141i	\(-0.305550\pi\)
							0.0124848	+	0.999922i	\(0.496026\pi\)
\(47\)	−2.46593	−	1.68124i	−0.359693	−	0.245235i	0.369967	−	0.929045i	\(-0.379369\pi\)
							−0.729660	+	0.683810i	\(0.760322\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.173.51	✓	648
9.5	odd	6		441.2.bn.a.320.4	yes	648
49.17	odd	42		441.2.bn.a.164.4	yes	648
441.311	even	42	inner	441.2.bd.a.311.51	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.173.51	✓	648	1.1	even	1	trivial
441.2.bd.a.311.51	yes	648	441.311	even	42	inner
441.2.bn.a.164.4	yes	648	49.17	odd	42
441.2.bn.a.320.4	yes	648	9.5	odd	6