Embedded newform 864.2.bt.a.11.8

• Label: 864.2.bt.a.11.8
• Level: $864$
• Weight: $2$
• Character: 864.11
• 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			11.8
Character	\(\chi\)	\(=\)	864.11
Dual form			864.2.bt.a.707.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39800 + 0.213530i) q^{2} +(-1.67838 - 0.427824i) q^{3} +(1.90881 - 0.597031i) q^{4} +(-0.187368 - 0.171692i) q^{5} +(2.43773 + 0.239713i) q^{6} +(-0.568924 - 0.812508i) q^{7} +(-2.54103 + 1.24224i) q^{8} +(2.63393 + 1.43611i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3408 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 36 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(353\)	\(703\)
\(\chi(n)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39800	+	0.213530i	−0.988535	+	0.150989i
							\(3\)	−1.67838	−	0.427824i	−0.969014	−	0.247004i
\(5\)	−0.187368	−	0.171692i	−0.0837937	−	0.0767828i								0.632602	−	0.774477i	\(-0.281987\pi\)
							−0.716395	+	0.697695i	\(0.754209\pi\)
\(7\)	−0.568924	−	0.812508i	−0.215033	−	0.307099i	0.697115	−	0.716959i	\(-0.254467\pi\)
							−0.912149	+	0.409860i	\(0.865578\pi\)
\(11\)	4.30741	+	0.188066i	1.29873	+	0.0567039i	0.682515	−	0.730871i	\(-0.260886\pi\)
							0.616219	+	0.787575i	\(0.288664\pi\)
\(13\)	−4.26114	−	2.21821i	−1.18183	−	0.615221i	−0.246549	−	0.969130i	\(-0.579297\pi\)
							−0.935280	+	0.353909i	\(0.884852\pi\)
\(17\)	0.145145	+	0.251398i	0.0352027	+	0.0609729i	0.883090	−	0.469204i	\(-0.155459\pi\)
							−0.847887	+	0.530177i	\(0.822126\pi\)
\(19\)	−0.363517	+	2.76118i	−0.0833965	+	0.633459i	0.897794	+	0.440416i	\(0.145169\pi\)
							−0.981190	+	0.193043i	\(0.938164\pi\)
\(23\)	5.82048	+	4.07555i	1.21365	+	0.849810i	0.992114	−	0.125341i	\(-0.0400025\pi\)
							0.221541	+	0.975151i	\(0.428891\pi\)
\(29\)	−2.45581	+	7.78885i	−0.456033	+	1.44635i	0.393749	+	0.919218i	\(0.371178\pi\)
							−0.849782	+	0.527135i	\(0.823266\pi\)
\(31\)	3.58855	−	0.632758i	0.644522	−	0.113647i	0.158173	−	0.987411i	\(-0.449440\pi\)
							0.486349	+	0.873765i	\(0.338328\pi\)
\(37\)	−2.30772	+	0.303816i	−0.379386	+	0.0499471i	−0.317808	−	0.948155i	\(-0.602947\pi\)
							−0.0615782	+	0.998102i	\(0.519613\pi\)
\(41\)	−4.56870	−	0.399710i	−0.713511	−	0.0624242i	−0.275380	−	0.961335i	\(-0.588804\pi\)
							−0.438131	+	0.898911i	\(0.644359\pi\)
\(43\)	0.551934	−	12.6414i	0.0841691	−	1.92779i	−0.220602	−	0.975364i	\(-0.570802\pi\)
							0.304771	−	0.952426i	\(-0.401420\pi\)
\(47\)	5.23053	+	0.922283i	0.762951	+	0.134529i	0.541567	−	0.840658i	\(-0.317831\pi\)
							0.221384	+	0.975187i	\(0.428942\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bt.a.11.8	✓	3408
27.5	odd	18	inner	864.2.bt.a.491.71	yes	3408
32.3	odd	8	inner	864.2.bt.a.227.71	yes	3408
864.707	even	72	inner	864.2.bt.a.707.8	yes	3408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.bt.a.11.8	✓	3408	1.1	even	1	trivial
864.2.bt.a.227.71	yes	3408	32.3	odd	8	inner
864.2.bt.a.491.71	yes	3408	27.5	odd	18	inner
864.2.bt.a.707.8	yes	3408	864.707	even	72	inner