Embedded newform 441.2.bd.a.110.36

• Label: 441.2.bd.a.110.36
• Level: $441$
• Weight: $2$
• Character: 441.110
• 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			110.36
Character	\(\chi\)	\(=\)	441.110
Dual form			441.2.bd.a.437.36

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.875192 - 0.0655866i) q^{2} +(1.63768 + 0.563919i) q^{3} +(-1.21600 + 0.183283i) q^{4} +(0.193683 + 0.848582i) q^{5} +(1.47027 + 0.386128i) q^{6} +(2.18345 + 1.49417i) q^{7} +(-2.76350 + 0.630751i) q^{8} +(2.36399 + 1.84704i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{11}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.875192	−	0.0655866i	0.618855	−	0.0463767i	0.238386	−	0.971171i	\(-0.423382\pi\)
							0.380469	+	0.924794i	\(0.375763\pi\)
\(3\)	1.63768	+	0.563919i	0.945515	+	0.325579i
							\(5\)	0.193683	+	0.848582i	0.0866178	+	0.379497i	0.999593	−	0.0285171i	\(-0.00907850\pi\)
−0.912976	+	0.408014i	\(0.866221\pi\)
\(7\)	2.18345	+	1.49417i	0.825267	+	0.564742i
							\(11\)	−0.798725	+	1.65857i	−0.240825	+	0.500077i	−0.985990	−	0.166802i	\(-0.946656\pi\)
0.745166	+	0.666879i	\(0.232370\pi\)
\(13\)	0.536240	−	0.0401857i	0.148726	−	0.0111455i	−0.000158874	−	1.00000i	\(-0.500051\pi\)
							0.148885	+	0.988854i	\(0.452432\pi\)
\(17\)	−0.957634	−	0.144340i	−0.232260	−	0.0350076i	0.0318807	−	0.999492i	\(-0.489850\pi\)
							−0.264141	+	0.964484i	\(0.585088\pi\)
\(19\)	−2.82319	−	1.62997i	−0.647684	−	0.373940i	0.139884	−	0.990168i	\(-0.455327\pi\)
							−0.787568	+	0.616227i	\(0.788660\pi\)
\(23\)	3.99347	−	3.18469i	0.832696	−	0.664053i	−0.111381	−	0.993778i	\(-0.535527\pi\)
							0.944077	+	0.329725i	\(0.106956\pi\)
\(29\)	−2.42698	+	0.952520i	−0.450679	+	0.176879i	−0.579815	−	0.814748i	\(-0.696875\pi\)
							0.129135	+	0.991627i	\(0.458780\pi\)
\(31\)	3.04210	+	1.75636i	0.546377	+	0.315451i	0.747660	−	0.664082i	\(-0.231177\pi\)
							−0.201282	+	0.979533i	\(0.564511\pi\)
\(37\)	−1.43960	−	3.66805i	−0.236669	−	0.603023i	0.762356	−	0.647157i	\(-0.224042\pi\)
							−0.999026	+	0.0441342i	\(0.985947\pi\)
\(41\)	2.25187	−	2.08943i	0.351683	−	0.326314i	−0.484473	−	0.874806i	\(-0.660989\pi\)
							0.836156	+	0.548492i	\(0.184798\pi\)
\(43\)	−3.25314	−	3.01848i	−0.496100	−	0.460313i	0.391999	−	0.919966i	\(-0.371784\pi\)
							−0.888099	+	0.459652i	\(0.847974\pi\)
\(47\)	0.0375101	+	0.500538i	0.00547141	+	0.0730109i	0.999271	−	0.0381746i	\(-0.0121543\pi\)
							−0.993800	+	0.111185i	\(0.964535\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.110.36	✓	648
9.5	odd	6		441.2.bn.a.257.36	yes	648
49.45	odd	42		441.2.bn.a.290.36	yes	648
441.437	even	42	inner	441.2.bd.a.437.36	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.110.36	✓	648	1.1	even	1	trivial
441.2.bd.a.437.36	yes	648	441.437	even	42	inner
441.2.bn.a.257.36	yes	648	9.5	odd	6
441.2.bn.a.290.36	yes	648	49.45	odd	42