Embedded newform 441.2.bd.a.122.47

• Label: 441.2.bd.a.122.47
• Level: $441$
• Weight: $2$
• Character: 441.122
• 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			122.47
Character	\(\chi\)	\(=\)	441.122
Dual form			441.2.bd.a.47.47

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41528 - 1.52531i) q^{2} +(1.35556 - 1.07818i) q^{3} +(-0.174090 - 2.32308i) q^{4} +(-0.916315 + 1.14902i) q^{5} +(0.273934 - 3.59357i) q^{6} +(0.192616 - 2.63873i) q^{7} +(-0.536180 - 0.427589i) q^{8} +(0.675059 - 2.92306i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{37}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41528	−	1.52531i	1.00075	−	1.07856i	0.00384559	−	0.999993i	\(-0.498776\pi\)
							0.996909	−	0.0785644i	\(-0.0250336\pi\)
\(3\)	1.35556	−	1.07818i	0.782630	−	0.622487i
							\(5\)	−0.916315	+	1.14902i	−0.409789	+	0.513859i	−0.943303	−	0.331932i	\(-0.892300\pi\)
0.533515	+	0.845791i	\(0.320871\pi\)
\(7\)	0.192616	−	2.63873i	0.0728019	−	0.997346i
							\(11\)	−0.209086	−	0.0477225i	−0.0630418	−	0.0143889i	0.190884	−	0.981613i	\(-0.438865\pi\)
−0.253925	+	0.967224i	\(0.581722\pi\)
\(13\)	−0.777880	+	0.838355i	−0.215745	+	0.232518i	−0.831647	−	0.555305i	\(-0.812601\pi\)
							0.615902	+	0.787823i	\(0.288792\pi\)
\(17\)	−0.231084	+	3.08361i	−0.0560462	+	0.747884i	0.896413	+	0.443219i	\(0.146164\pi\)
							−0.952459	+	0.304665i	\(0.901455\pi\)
\(19\)	−2.52298	+	1.45664i	−0.578811	+	0.334177i	−0.760661	−	0.649150i	\(-0.775125\pi\)
							0.181850	+	0.983326i	\(0.441792\pi\)
\(23\)	0.271037	−	0.562813i	0.0565151	−	0.117355i	−0.870799	−	0.491640i	\(-0.836398\pi\)
							0.927314	+	0.374285i	\(0.122112\pi\)
\(29\)	−1.23835	+	1.81632i	−0.229955	+	0.337282i	−0.923823	−	0.382820i	\(-0.874953\pi\)
							0.693868	+	0.720102i	\(0.255905\pi\)
\(31\)	1.55633	−	0.898547i	0.279525	−	0.161384i	−0.353683	−	0.935365i	\(-0.615071\pi\)
							0.633208	+	0.773981i	\(0.281738\pi\)
\(37\)	−0.0841549	−	0.0573758i	−0.0138350	−	0.00943253i	0.556383	−	0.830926i	\(-0.312189\pi\)
							−0.570218	+	0.821494i	\(0.693141\pi\)
\(41\)	0.0371309	+	0.0946080i	0.00579887	+	0.0147753i	0.933744	−	0.357942i	\(-0.116521\pi\)
							−0.927945	+	0.372717i	\(0.878426\pi\)
\(43\)	0.816884	−	2.08139i	0.124574	−	0.317408i	−0.854958	−	0.518696i	\(-0.826417\pi\)
							0.979532	+	0.201288i	\(0.0645127\pi\)
\(47\)	7.30126	+	6.77458i	1.06500	+	0.988175i	0.999953	−	0.00970774i	\(-0.00309012\pi\)
							0.0650460	+	0.997882i	\(0.479281\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.122.47	yes	648
9.2	odd	6		441.2.bn.a.416.47	yes	648
49.47	odd	42		441.2.bn.a.194.47	yes	648
441.47	even	42	inner	441.2.bd.a.47.47	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.47.47	✓	648	441.47	even	42	inner
441.2.bd.a.122.47	yes	648	1.1	even	1	trivial
441.2.bn.a.194.47	yes	648	49.47	odd	42
441.2.bn.a.416.47	yes	648	9.2	odd	6