Embedded newform 441.2.bd.a.47.53

• Label: 441.2.bd.a.47.53
• 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			47.53
Character	\(\chi\)	\(=\)	441.47
Dual form			441.2.bd.a.122.53

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.76148 + 1.89842i) q^{2} +(-1.24560 - 1.20353i) q^{3} +(-0.351739 + 4.69364i) q^{4} +(2.19211 + 2.74882i) q^{5} +(0.0907032 - 4.48466i) q^{6} +(-0.860337 - 2.50196i) q^{7} +(-5.48058 + 4.37061i) q^{8} +(0.103044 + 2.99823i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.76148	+	1.89842i	1.24555	+	1.34239i	0.917505	+	0.397724i	\(0.130200\pi\)
							0.328047	+	0.944662i	\(0.393610\pi\)
\(3\)	−1.24560	−	1.20353i	−0.719148	−	0.694857i
							\(5\)	2.19211	+	2.74882i	0.980343	+	1.22931i	0.973347	+	0.229336i	\(0.0736556\pi\)
0.00699622	+	0.999976i	\(0.497773\pi\)
\(7\)	−0.860337	−	2.50196i	−0.325177	−	0.945653i
							\(11\)	−2.88472	+	0.658418i	−0.869775	+	0.198521i	−0.634043	−	0.773298i	\(-0.718606\pi\)
−0.235732	+	0.971818i	\(0.575749\pi\)
\(13\)	3.62264	+	3.90427i	1.00474	+	1.08285i	0.996559	+	0.0828838i	\(0.0264130\pi\)
							0.00817904	+	0.999967i	\(0.497397\pi\)
\(17\)	−0.169920	−	2.26742i	−0.0412116	−	0.549931i	−0.979151	−	0.203132i	\(-0.934888\pi\)
							0.937940	−	0.346798i	\(-0.112731\pi\)
\(19\)	0.908718	+	0.524648i	0.208474	+	0.120363i	0.600602	−	0.799548i	\(-0.294928\pi\)
							−0.392128	+	0.919911i	\(0.628261\pi\)
\(23\)	−0.819252	−	1.70119i	−0.170826	−	0.354723i	0.797926	−	0.602755i	\(-0.205931\pi\)
							−0.968752	+	0.248032i	\(0.920216\pi\)
\(29\)	−4.64473	−	6.81257i	−0.862505	−	1.26506i	−0.962830	−	0.270108i	\(-0.912941\pi\)
							0.100325	−	0.994955i	\(-0.468012\pi\)
\(31\)	4.78493	+	2.76258i	0.859399	+	0.496174i	0.863811	−	0.503816i	\(-0.168071\pi\)
							−0.00441186	+	0.999990i	\(0.501404\pi\)
\(37\)	2.52859	−	1.72396i	0.415697	−	0.283418i	−0.337356	−	0.941377i	\(-0.609533\pi\)
							0.753053	+	0.657960i	\(0.228580\pi\)
\(41\)	−0.380478	+	0.969441i	−0.0594206	+	0.151401i	−0.957501	−	0.288430i	\(-0.906867\pi\)
							0.898080	+	0.439832i	\(0.144962\pi\)
\(43\)	1.79260	+	4.56748i	0.273369	+	0.696534i	0.999955	+	0.00945833i	\(0.00301072\pi\)
							−0.726586	+	0.687076i	\(0.758894\pi\)
\(47\)	6.63338	−	6.15488i	0.967578	−	0.897781i	−0.0272221	−	0.999629i	\(-0.508666\pi\)
							0.994800	+	0.101848i	\(0.0324756\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.47.53	✓	648
9.5	odd	6		441.2.bn.a.194.53	yes	648
49.24	odd	42		441.2.bn.a.416.53	yes	648
441.122	even	42	inner	441.2.bd.a.122.53	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.47.53	✓	648	1.1	even	1	trivial
441.2.bd.a.122.53	yes	648	441.122	even	42	inner
441.2.bn.a.194.53	yes	648	9.5	odd	6
441.2.bn.a.416.53	yes	648	49.24	odd	42