Embedded newform 441.2.bn.a.164.24

• Label: 441.2.bn.a.164.24
• Level: $441$
• Weight: $2$
• Character: 441.164
• 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.bn (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			164.24
Character	\(\chi\)	\(=\)	441.164
Dual form			441.2.bn.a.320.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.123034 + 0.255482i) q^{2} +(1.44307 + 0.957892i) q^{3} +(1.19685 + 1.50080i) q^{4} +(1.22576 + 0.378095i) q^{5} +(-0.422270 + 0.250825i) q^{6} +(-2.64499 - 0.0636538i) q^{7} +(-1.08359 + 0.247322i) q^{8} +(1.16489 + 2.76461i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{25}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.123034	+	0.255482i	−0.0869980	+	0.180653i	−0.939929	−	0.341371i	\(-0.889109\pi\)
							0.852931	+	0.522024i	\(0.174823\pi\)
\(3\)	1.44307	+	0.957892i	0.833155	+	0.553039i
							\(5\)	1.22576	+	0.378095i	0.548174	+	0.169089i	0.556457	−	0.830876i	\(-0.312160\pi\)
−0.00828278	+	0.999966i	\(0.502637\pi\)
\(7\)	−2.64499	−	0.0636538i	−0.999711	−	0.0240589i
							\(11\)	−3.82613	+	0.286729i	−1.15362	+	0.0864520i	−0.637771	−	0.770226i	\(-0.720143\pi\)
−0.515851	+	0.856678i	\(0.672524\pi\)
\(13\)	4.22610	−	0.316703i	1.17211	−	0.0878375i	0.525592	−	0.850737i	\(-0.323844\pi\)
							0.646517	+	0.762899i	\(0.276225\pi\)
\(17\)	0.900045	+	2.29328i	0.218293	+	0.556201i	0.997508	−	0.0705556i	\(-0.0224772\pi\)
							−0.779215	+	0.626757i	\(0.784382\pi\)
\(19\)	2.20151	−	1.27104i	0.505061	−	0.291597i	−0.225740	−	0.974188i	\(-0.572480\pi\)
							0.730801	+	0.682591i	\(0.239147\pi\)
\(23\)	−1.21551	−	8.06436i	−0.253451	−	1.68153i	−0.647203	−	0.762318i	\(-0.724061\pi\)
							0.393752	−	0.919217i	\(-0.371177\pi\)
\(29\)	5.30891	−	2.08359i	0.985840	−	0.386914i	0.183003	−	0.983112i	\(-0.441418\pi\)
							0.802837	+	0.596199i	\(0.203323\pi\)
\(31\)		−	0.428403i		−	0.0769434i	−0.999260	−	0.0384717i	\(-0.987751\pi\)
							0.999260	−	0.0384717i	\(-0.0122489\pi\)
\(37\)	8.80969	+	1.32785i	1.44830	+	0.218297i	0.825642	−	0.564194i	\(-0.190813\pi\)
							0.622662	+	0.782491i	\(0.286051\pi\)
\(41\)	0.135859	−	0.126058i	0.0212175	−	0.0196870i	−0.669493	−	0.742819i	\(-0.733488\pi\)
							0.690710	+	0.723132i	\(0.257298\pi\)
\(43\)	5.21500	+	4.83881i	0.795280	+	0.737912i	0.969047	−	0.246875i	\(-0.0794036\pi\)
							−0.173767	+	0.984787i	\(0.555594\pi\)
\(47\)	−10.4905	−	5.05198i	−1.53020	−	0.736907i	−0.535980	−	0.844231i	\(-0.680058\pi\)
							−0.994224	+	0.107323i	\(0.965772\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.164.24	yes	648
9.5	odd	6		441.2.bd.a.311.31	yes	648
49.26	odd	42		441.2.bd.a.173.31	✓	648
441.320	even	42	inner	441.2.bn.a.320.24	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.173.31	✓	648	49.26	odd	42
441.2.bd.a.311.31	yes	648	9.5	odd	6
441.2.bn.a.164.24	yes	648	1.1	even	1	trivial
441.2.bn.a.320.24	yes	648	441.320	even	42	inner