Embedded newform 441.2.bd.a.122.52

• Label: 441.2.bd.a.122.52
• Level: $441$
• Weight: $2$
• Character: 441.122
• 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			122.52
Character	\(\chi\)	\(=\)	441.122
Dual form			441.2.bd.a.47.52

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.73309 - 1.86783i) q^{2} +(1.41963 + 0.992289i) q^{3} +(-0.335716 - 4.47983i) q^{4} +(1.23495 - 1.54857i) q^{5} +(4.31378 - 0.931906i) q^{6} +(-2.58103 - 0.581642i) q^{7} +(-4.96513 - 3.95956i) q^{8} +(1.03072 + 2.81738i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{37}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.73309	−	1.86783i	1.22548	−	1.32075i	0.294515	−	0.955647i	\(-0.404842\pi\)
							0.930966	−	0.365107i	\(-0.118968\pi\)
\(3\)	1.41963	+	0.992289i	0.819626	+	0.572898i
							\(5\)	1.23495	−	1.54857i	0.552285	−	0.692543i	−0.424826	−	0.905275i	\(-0.639665\pi\)
0.977111	+	0.212732i	\(0.0682360\pi\)
\(7\)	−2.58103	−	0.581642i	−0.975536	−	0.219840i
							\(11\)	3.41384	+	0.779186i	1.02931	+	0.234933i	0.703645	−	0.710552i	\(-0.251554\pi\)
0.325666	+	0.945485i	\(0.394412\pi\)
\(13\)	−3.09441	+	3.33498i	−0.858235	+	0.924958i	−0.997869	−	0.0652455i	\(-0.979217\pi\)
							0.139634	+	0.990203i	\(0.455407\pi\)
\(17\)	−0.0470632	+	0.628015i	−0.0114145	+	0.152316i	0.988584	+	0.150670i	\(0.0481430\pi\)
							−0.999999	+	0.00164620i	\(0.999476\pi\)
\(19\)	−5.31120	+	3.06642i	−1.21847	+	0.703486i	−0.964591	−	0.263750i	\(-0.915041\pi\)
							−0.253882	+	0.967235i	\(0.581707\pi\)
\(23\)	0.622492	−	1.29262i	0.129799	−	0.269530i	−0.825934	−	0.563767i	\(-0.809351\pi\)
							0.955732	+	0.294238i	\(0.0950657\pi\)
\(29\)	1.94993	−	2.86002i	0.362093	−	0.531093i	−0.601357	−	0.798980i	\(-0.705373\pi\)
							0.963450	+	0.267887i	\(0.0863255\pi\)
\(31\)	7.04024	−	4.06469i	1.26446	−	0.730039i	0.290530	−	0.956866i	\(-0.406168\pi\)
							0.973935	+	0.226827i	\(0.0728352\pi\)
\(37\)	−3.85939	−	2.63129i	−0.634480	−	0.432581i	0.202902	−	0.979199i	\(-0.434963\pi\)
							−0.837382	+	0.546618i	\(0.815915\pi\)
\(41\)	0.825001	+	2.10207i	0.128844	+	0.328288i	0.980709	−	0.195472i	\(-0.0626240\pi\)
							−0.851866	+	0.523760i	\(0.824529\pi\)
\(43\)	1.99059	−	5.07195i	0.303563	−	0.773466i	−0.694956	−	0.719053i	\(-0.744576\pi\)
							0.998519	−	0.0544129i	\(-0.0173287\pi\)
\(47\)	−1.17080	−	1.08634i	−0.170779	−	0.158459i	0.590153	−	0.807291i	\(-0.299067\pi\)
							−0.760932	+	0.648832i	\(0.775258\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.122.52	yes	648
9.2	odd	6		441.2.bn.a.416.52	yes	648
49.47	odd	42		441.2.bn.a.194.52	yes	648
441.47	even	42	inner	441.2.bd.a.47.52	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.47.52	✓	648	441.47	even	42	inner
441.2.bd.a.122.52	yes	648	1.1	even	1	trivial
441.2.bn.a.194.52	yes	648	49.47	odd	42
441.2.bn.a.416.52	yes	648	9.2	odd	6