Embedded newform 441.2.bh.a.20.14

• Label: 441.2.bh.a.20.14
• 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			20.14
Character	\(\chi\)	\(=\)	441.20
Dual form			441.2.bh.a.419.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.961845 - 1.41077i) q^{2} +(0.474232 - 1.66586i) q^{3} +(-0.334437 + 0.852132i) q^{4} +(1.96011 - 1.81872i) q^{5} +(-2.80629 + 0.933272i) q^{6} +(-0.724718 - 2.54456i) q^{7} +(-1.80546 + 0.412084i) q^{8} +(-2.55021 - 1.58001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 15 q^{2} - 14 q^{3} - 57 q^{4} - 21 q^{5} + 14 q^{6} - 5 q^{7} - 20 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.961845	−	1.41077i	−0.680127	−	0.997563i	−0.998765	−	0.0496868i	\(-0.984178\pi\)
							0.318638	−	0.947877i	\(-0.396775\pi\)
\(3\)	0.474232	−	1.66586i	0.273798	−	0.961787i
							\(5\)	1.96011	−	1.81872i	0.876590	−	0.813357i	−0.106909	−	0.994269i	\(-0.534095\pi\)
0.983499	+	0.180912i	\(0.0579049\pi\)
\(7\)	−0.724718	−	2.54456i	−0.273918	−	0.961753i
							\(11\)	1.66533	+	2.44259i	0.502117	+	0.736470i	0.990703	−	0.136046i	\(-0.0434395\pi\)
−0.488586	+	0.872516i	\(0.662487\pi\)
\(13\)	1.94268	−	0.145583i	0.538801	−	0.0403776i	0.197449	−	0.980313i	\(-0.436734\pi\)
							0.341352	+	0.939936i	\(0.389115\pi\)
\(17\)	−2.89013	+	3.62411i	−0.700960	+	0.878976i	−0.997095	−	0.0761699i	\(-0.975731\pi\)
							0.296135	+	0.955146i	\(0.404302\pi\)
\(19\)		−	8.36039i		−	1.91800i	−0.283398	−	0.959002i	\(-0.591462\pi\)
							0.283398	−	0.959002i	\(-0.408538\pi\)
\(23\)	4.58149	+	1.79810i	0.955306	+	0.374930i	0.791231	−	0.611518i	\(-0.209441\pi\)
							0.164076	+	0.986448i	\(0.447536\pi\)
\(29\)	2.51129	−	0.985607i	0.466334	−	0.183023i	−0.120527	−	0.992710i	\(-0.538459\pi\)
							0.586861	+	0.809687i	\(0.300363\pi\)
\(31\)	4.45813	−	2.57390i	0.800703	−	0.462286i	−0.0430136	−	0.999074i	\(-0.513696\pi\)
							0.843717	+	0.536788i	\(0.180363\pi\)
\(37\)	−3.91356	+	4.90745i	−0.643386	+	0.806780i	−0.991422	−	0.130701i	\(-0.958277\pi\)
							0.348036	+	0.937481i	\(0.386849\pi\)
\(41\)	−0.147808	+	0.137146i	−0.0230838	+	0.0214186i	−0.691630	−	0.722252i	\(-0.743107\pi\)
							0.668547	+	0.743670i	\(0.266917\pi\)
\(43\)	6.95842	+	6.45647i	1.06115	+	0.984602i	0.999909	−	0.0135206i	\(-0.00430388\pi\)
							0.0612404	+	0.998123i	\(0.480494\pi\)
\(47\)	−5.97377	+	4.07285i	−0.871364	+	0.594086i	−0.914321	−	0.404991i	\(-0.867275\pi\)
							0.0429570	+	0.999077i	\(0.486322\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bh.a.20.14	✓	648
9.5	odd	6	inner	441.2.bh.a.167.14	yes	648
49.27	odd	14	inner	441.2.bh.a.272.14	yes	648
441.419	even	42	inner	441.2.bh.a.419.14	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bh.a.20.14	✓	648	1.1	even	1	trivial
441.2.bh.a.167.14	yes	648	9.5	odd	6	inner
441.2.bh.a.272.14	yes	648	49.27	odd	14	inner
441.2.bh.a.419.14	yes	648	441.419	even	42	inner