Embedded newform 864.2.bt.a.491.63

• Label: 864.2.bt.a.491.63
• Level: $864$
• Weight: $2$
• Character: 864.491
• Analytic conductor: $6.899$
• Analytic rank: $0$
• Dimension: $3408$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.bt (of order \(72\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89907473464\)
Analytic rank:	\(0\)
Dimension:	\(3408\)
Relative dimension:	\(142\) over \(\Q(\zeta_{72})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{72}]$

Embedding invariants

Embedding label			491.63
Character	\(\chi\)	\(=\)	864.491
Dual form			864.2.bt.a.227.63

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.256940 - 1.39068i) q^{2} +(-0.809653 + 1.53116i) q^{3} +(-1.86796 + 0.714641i) q^{4} +(2.40551 + 0.105027i) q^{5} +(2.33739 + 0.732548i) q^{6} +(3.60592 + 2.52489i) q^{7} +(1.47379 + 2.41411i) q^{8} +(-1.68892 - 2.47942i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3408 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 36 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(353\)	\(703\)
\(\chi(n)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.256940	−	1.39068i	−0.181684	−	0.983357i
							\(3\)	−0.809653	+	1.53116i	−0.467453	+	0.884018i
\(5\)	2.40551	+	0.105027i	1.07578	+	0.0469694i								0.574646	−	0.818402i	\(-0.305140\pi\)
							0.501131	+	0.865371i	\(0.332917\pi\)
\(7\)	3.60592	+	2.52489i	1.36291	+	0.954320i	0.999745	+	0.0225950i	\(0.00719283\pi\)
							0.363165	+	0.931725i	\(0.381696\pi\)
\(11\)	3.60108	+	3.29978i	1.08577	+	0.994921i	0.999999	−	0.00139422i	\(-0.000443794\pi\)
							0.0857667	+	0.996315i	\(0.472666\pi\)
\(13\)	−0.393261	+	1.24726i	−0.109071	+	0.345929i	−0.992437	−	0.122755i	\(-0.960827\pi\)
							0.883366	+	0.468684i	\(0.155272\pi\)
\(17\)	−2.98248	+	5.16581i	−0.723357	+	1.25289i	0.236289	+	0.971683i	\(0.424069\pi\)
							−0.959647	+	0.281209i	\(0.909265\pi\)
\(19\)	−3.52354	−	2.70370i	−0.808355	−	0.620272i	0.119660	−	0.992815i	\(-0.461819\pi\)
							−0.928015	+	0.372542i	\(0.878486\pi\)
\(23\)	−3.38429	−	4.83326i	−0.705673	−	1.00781i	−0.998663	−	0.0516899i	\(-0.983539\pi\)
							0.292990	−	0.956115i	\(-0.405350\pi\)
\(29\)	−5.10454	−	2.65726i	−0.947890	−	0.493440i	−0.0836362	−	0.996496i	\(-0.526653\pi\)
							−0.864254	+	0.503056i	\(0.832209\pi\)
\(31\)	7.05551	+	1.24408i	1.26721	+	0.223443i	0.766539	−	0.642197i	\(-0.221977\pi\)
							0.500667	+	0.865640i	\(0.333088\pi\)
\(37\)	0.665849	+	0.867751i	0.109465	+	0.142657i	0.844873	−	0.534967i	\(-0.179676\pi\)
							−0.735408	+	0.677625i	\(0.763009\pi\)
\(41\)	0.766475	+	8.76085i	0.119703	+	1.36821i	0.784007	+	0.620752i	\(0.213173\pi\)
							−0.664304	+	0.747463i	\(0.731272\pi\)
\(43\)	2.29368	−	2.50311i	0.349783	−	0.381721i	−0.537253	−	0.843421i	\(-0.680538\pi\)
							0.887036	+	0.461700i	\(0.152760\pi\)
\(47\)	−0.994210	+	0.175306i	−0.145020	+	0.0255710i	−0.245687	−	0.969349i	\(-0.579014\pi\)
							0.100667	+	0.994920i	\(0.467902\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bt.a.491.63	yes	3408
27.11	odd	18	inner	864.2.bt.a.11.17	✓	3408
32.3	odd	8	inner	864.2.bt.a.707.17	yes	3408
864.227	even	72	inner	864.2.bt.a.227.63	yes	3408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.bt.a.11.17	✓	3408	27.11	odd	18	inner
864.2.bt.a.227.63	yes	3408	864.227	even	72	inner
864.2.bt.a.491.63	yes	3408	1.1	even	1	trivial
864.2.bt.a.707.17	yes	3408	32.3	odd	8	inner