Embedded newform 441.2.bd.a.47.12

• Label: 441.2.bd.a.47.12
• 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			47.12
Character	\(\chi\)	\(=\)	441.47
Dual form			441.2.bd.a.122.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22870 - 1.32422i) q^{2} +(0.168894 + 1.72380i) q^{3} +(-0.0944028 + 1.25972i) q^{4} +(0.684410 + 0.858223i) q^{5} +(2.07517 - 2.34167i) q^{6} +(1.06533 - 2.42179i) q^{7} +(-1.04054 + 0.829802i) q^{8} +(-2.94295 + 0.582277i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22870	−	1.32422i	−0.868819	−	0.936364i	0.129656	−	0.991559i	\(-0.458613\pi\)
							−0.998476	+	0.0551946i	\(0.982422\pi\)
\(3\)	0.168894	+	1.72380i	0.0975109	+	0.995234i
							\(5\)	0.684410	+	0.858223i	0.306077	+	0.383809i	0.910952	−	0.412511i	\(-0.135348\pi\)
−0.604875	+	0.796320i	\(0.706777\pi\)
\(7\)	1.06533	−	2.42179i	0.402656	−	0.915351i
							\(11\)	4.90547	−	1.11964i	1.47906	−	0.337585i	0.594529	−	0.804074i	\(-0.297338\pi\)
0.884527	+	0.466489i	\(0.154481\pi\)
\(13\)	−2.31554	−	2.49556i	−0.642216	−	0.692144i	0.325111	−	0.945676i	\(-0.394598\pi\)
							−0.967327	+	0.253532i	\(0.918408\pi\)
\(17\)	0.0541590	+	0.722701i	0.0131355	+	0.175281i	0.999913	+	0.0131994i	\(0.00420164\pi\)
							−0.986777	+	0.162081i	\(0.948179\pi\)
\(19\)	3.19553	+	1.84494i	0.733105	+	0.423258i	0.819557	−	0.572998i	\(-0.194220\pi\)
							−0.0864523	+	0.996256i	\(0.527553\pi\)
\(23\)	1.87785	+	3.89939i	0.391558	+	0.813079i	0.999813	+	0.0193370i	\(0.00615554\pi\)
							−0.608255	+	0.793742i	\(0.708130\pi\)
\(29\)	4.96111	+	7.27661i	0.921255	+	1.35123i	0.936246	+	0.351345i	\(0.114275\pi\)
							−0.0149916	+	0.999888i	\(0.504772\pi\)
\(31\)	5.05494	+	2.91847i	0.907894	+	0.524173i	0.879753	−	0.475431i	\(-0.157708\pi\)
							0.0281408	+	0.999604i	\(0.491041\pi\)
\(37\)	8.62235	−	5.87862i	1.41750	−	0.966438i	0.419221	−	0.907884i	\(-0.362303\pi\)
							0.998284	−	0.0585541i	\(-0.0186490\pi\)
\(41\)	−0.00356075	+	0.00907264i	−0.000556096	+	0.00141691i	−0.931152	−	0.364632i	\(-0.881195\pi\)
							0.930595	+	0.366049i	\(0.119290\pi\)
\(43\)	−1.60697	−	4.09449i	−0.245061	−	0.624404i	0.754428	−	0.656383i	\(-0.227914\pi\)
							−0.999489	+	0.0319785i	\(0.989819\pi\)
\(47\)	0.570535	−	0.529379i	0.0832211	−	0.0772179i	−0.637466	−	0.770479i	\(-0.720017\pi\)
							0.720687	+	0.693261i	\(0.243827\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.47.12	✓	648
9.5	odd	6		441.2.bn.a.194.12	yes	648
49.24	odd	42		441.2.bn.a.416.12	yes	648
441.122	even	42	inner	441.2.bd.a.122.12	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.47.12	✓	648	1.1	even	1	trivial
441.2.bd.a.122.12	yes	648	441.122	even	42	inner
441.2.bn.a.194.12	yes	648	9.5	odd	6
441.2.bn.a.416.12	yes	648	49.24	odd	42