Embedded newform 441.2.bd.a.173.31

• Label: 441.2.bd.a.173.31
• Level: $441$
• Weight: $2$
• Character: 441.173
• 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			173.31
Character	\(\chi\)	\(=\)	441.173
Dual form			441.2.bd.a.311.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.159737 - 0.234291i) q^{2} +(-1.73192 + 0.0214611i) q^{3} +(0.701306 + 1.78690i) q^{4} +(0.285437 - 1.25058i) q^{5} +(-0.271624 + 0.409202i) q^{6} +(1.37762 + 2.25880i) q^{7} +(1.08359 + 0.247322i) q^{8} +(2.99908 - 0.0743376i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{17}{42}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.159737	−	0.234291i	0.112951	−	0.165669i	−0.765600	−	0.643316i	\(-0.777558\pi\)
							0.878552	+	0.477647i	\(0.158510\pi\)
\(3\)	−1.73192	+	0.0214611i	−0.999923	+	0.0123905i
							\(5\)	0.285437	−	1.25058i	0.127651	−	0.559278i	−0.870137	−	0.492810i	\(-0.835970\pi\)
0.997789	−	0.0664679i	\(-0.0211730\pi\)
\(7\)	1.37762	+	2.25880i	0.520691	+	0.853745i
							\(11\)	−1.66475	−	3.45689i	−0.501941	−	1.04229i	−0.985921	−	0.167212i	\(-0.946523\pi\)
0.483980	−	0.875079i	\(-0.339191\pi\)
\(13\)	−2.38732	+	3.50156i	−0.662124	+	0.971158i	0.337431	+	0.941350i	\(0.390442\pi\)
							−0.999556	+	0.0298079i	\(0.990510\pi\)
\(17\)	−0.900045	+	2.29328i	−0.218293	+	0.556201i	−0.997508	−	0.0705556i	\(-0.977523\pi\)
							0.779215	+	0.626757i	\(0.215618\pi\)
\(19\)	2.20151	+	1.27104i	0.505061	+	0.291597i	0.730801	−	0.682591i	\(-0.239147\pi\)
							−0.225740	+	0.974188i	\(0.572480\pi\)
\(23\)	6.37618	+	5.08484i	1.32953	+	1.06026i	0.992941	+	0.118609i	\(0.0378435\pi\)
							0.336585	+	0.941653i	\(0.390728\pi\)
\(29\)	0.850010	+	5.63945i	0.157843	+	1.04722i	0.917742	+	0.397178i	\(0.130010\pi\)
							−0.759899	+	0.650041i	\(0.774752\pi\)
\(31\)	−0.371008	−	0.214201i	−0.0666350	−	0.0384717i	0.466312	−	0.884620i	\(-0.345582\pi\)
							−0.532947	+	0.846148i	\(0.678916\pi\)
\(37\)	8.80969	−	1.32785i	1.44830	−	0.218297i	0.622662	−	0.782491i	\(-0.286051\pi\)
							0.825642	+	0.564194i	\(0.190813\pi\)
\(41\)	0.177099	−	0.0546278i	0.0276582	−	0.00853143i	−0.280895	−	0.959738i	\(-0.590631\pi\)
							0.308553	+	0.951207i	\(0.400155\pi\)
\(43\)	−6.79803	−	2.09692i	−1.03669	−	0.319777i	−0.270724	−	0.962657i	\(-0.587263\pi\)
							−0.765967	+	0.642880i	\(0.777739\pi\)
\(47\)	−9.62042	−	6.55909i	−1.40328	−	0.956742i	−0.999115	−	0.0420567i	\(-0.986609\pi\)
							−0.404167	−	0.914685i	\(-0.632439\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.173.31	✓	648	1.1	even	1	trivial
441.2.bd.a.311.31	yes	648	441.311	even	42	inner
441.2.bn.a.164.24	yes	648	49.17	odd	42
441.2.bn.a.320.24	yes	648	9.5	odd	6