Embedded newform 441.2.bh.a.20.17

• Label: 441.2.bh.a.20.17
• Level: $441$
• Weight: $2$
• Character: 441.20
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.bh (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			20.17
Character	\(\chi\)	\(=\)	441.20
Dual form			441.2.bh.a.419.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.808964 - 1.18653i) q^{2} +(1.30536 + 1.13844i) q^{3} +(-0.0227537 + 0.0579756i) q^{4} +(-2.55074 + 2.36674i) q^{5} +(0.294803 - 2.46980i) q^{6} +(2.61161 + 0.423656i) q^{7} +(-2.71292 + 0.619207i) q^{8} +(0.407924 + 2.97214i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 15 q^{2} - 14 q^{3} - 57 q^{4} - 21 q^{5} + 14 q^{6} - 5 q^{7} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.808964	−	1.18653i	−0.572024	−	0.839005i	0.425808	−	0.904813i	\(-0.359990\pi\)
							−0.997832	+	0.0658085i	\(0.979037\pi\)
\(3\)	1.30536	+	1.13844i	0.753649	+	0.657277i
							\(5\)	−2.55074	+	2.36674i	−1.14073	+	1.05844i	−0.143091	+	0.989710i	\(0.545704\pi\)
−0.997635	+	0.0687296i	\(0.978105\pi\)
\(7\)	2.61161	+	0.423656i	0.987096	+	0.160127i
							\(11\)	−0.389146	−	0.570773i	−0.117332	−	0.172094i	0.763080	−	0.646304i	\(-0.223686\pi\)
−0.880412	+	0.474210i	\(0.842734\pi\)
\(13\)	−2.23977	+	0.167847i	−0.621199	+	0.0465525i	−0.381612	−	0.924322i	\(-0.624631\pi\)
							−0.239587	+	0.970875i	\(0.577012\pi\)
\(17\)	−2.73011	+	3.42345i	−0.662150	+	0.830310i	−0.993575	−	0.113172i	\(-0.963899\pi\)
							0.331426	+	0.943481i	\(0.392470\pi\)
\(19\)			4.55462i			1.04490i	0.852670	+	0.522450i	\(0.174982\pi\)
							−0.852670	+	0.522450i	\(0.825018\pi\)
\(23\)	5.56899	+	2.18567i	1.16121	+	0.455743i	0.866175	−	0.499742i	\(-0.166572\pi\)
							0.295040	+	0.955485i	\(0.404667\pi\)
\(29\)	3.82811	−	1.50242i	0.710862	−	0.278993i	0.0177702	−	0.999842i	\(-0.494343\pi\)
							0.693092	+	0.720849i	\(0.256248\pi\)
\(31\)	−6.48838	+	3.74607i	−1.16535	+	0.672814i	−0.952580	−	0.304289i	\(-0.901581\pi\)
							−0.212768	+	0.977103i	\(0.568248\pi\)
\(37\)	0.776918	−	0.974224i	0.127725	−	0.160162i	−0.713857	−	0.700291i	\(-0.753053\pi\)
							0.841582	+	0.540130i	\(0.181625\pi\)
\(41\)	1.17310	−	1.08848i	0.183208	−	0.169992i	−0.583237	−	0.812302i	\(-0.698214\pi\)
							0.766445	+	0.642310i	\(0.222024\pi\)
\(43\)	−0.343653	−	0.318863i	−0.0524065	−	0.0486262i	0.653535	−	0.756896i	\(-0.273285\pi\)
							−0.705941	+	0.708270i	\(0.749476\pi\)
\(47\)	−4.48811	+	3.05994i	−0.654658	+	0.446338i	−0.844557	−	0.535465i	\(-0.820136\pi\)
							0.189899	+	0.981804i	\(0.439184\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bh.a.20.17	✓	648
9.5	odd	6	inner	441.2.bh.a.167.17	yes	648
49.27	odd	14	inner	441.2.bh.a.272.17	yes	648
441.419	even	42	inner	441.2.bh.a.419.17	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bh.a.20.17	✓	648	1.1	even	1	trivial
441.2.bh.a.167.17	yes	648	9.5	odd	6	inner
441.2.bh.a.272.17	yes	648	49.27	odd	14	inner
441.2.bh.a.419.17	yes	648	441.419	even	42	inner