Embedded newform 864.2.bt.a.11.1

• Label: 864.2.bt.a.11.1
• 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			11.1
Character	\(\chi\)	\(=\)	864.11
Dual form			864.2.bt.a.707.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41414 - 0.0139384i) q^{2} +(0.185106 - 1.72213i) q^{3} +(1.99961 + 0.0394217i) q^{4} +(3.08870 + 2.83028i) q^{5} +(-0.285770 + 2.43276i) q^{6} +(1.76195 + 2.51633i) q^{7} +(-2.82719 - 0.0836194i) q^{8} +(-2.93147 - 0.637552i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3408 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 36 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41414	−	0.0139384i	−0.999951	−	0.00985591i
							\(3\)	0.185106	−	1.72213i	0.106871	−	0.994273i
\(5\)	3.08870	+	2.83028i	1.38131	+	1.26574i								0.927085	+	0.374851i	\(0.122306\pi\)
							0.454226	+	0.890887i	\(0.349916\pi\)
\(7\)	1.76195	+	2.51633i	0.665955	+	0.951083i	0.999964	+	0.00851266i	\(0.00270970\pi\)
							−0.334008	+	0.942570i	\(0.608401\pi\)
\(11\)	4.34827	+	0.189850i	1.31105	+	0.0572418i	0.688443	−	0.725291i	\(-0.258295\pi\)
							0.622610	+	0.782532i	\(0.286072\pi\)
\(13\)	1.59443	+	0.830008i	0.442215	+	0.230203i	0.668280	−	0.743909i	\(-0.267031\pi\)
							−0.226065	+	0.974112i	\(0.572586\pi\)
\(17\)	−3.93139	−	6.80937i	−0.953503	−	1.65151i	−0.737758	−	0.675065i	\(-0.764116\pi\)
							−0.215744	−	0.976450i	\(-0.569218\pi\)
\(19\)	−0.384127	+	2.91773i	−0.0881248	+	0.669374i	0.889183	+	0.457551i	\(0.151273\pi\)
							−0.977308	+	0.211823i	\(0.932060\pi\)
\(23\)	−1.22271	−	0.856152i	−0.254953	−	0.178520i	0.439112	−	0.898432i	\(-0.355293\pi\)
							−0.694065	+	0.719912i	\(0.744182\pi\)
\(29\)	−1.55459	+	4.93055i	−0.288681	+	0.915579i	0.692322	+	0.721588i	\(0.256588\pi\)
							−0.981003	+	0.193991i	\(0.937857\pi\)
\(31\)	4.77096	−	0.841248i	0.856889	−	0.151093i	0.272091	−	0.962272i	\(-0.412285\pi\)
							0.584798	+	0.811179i	\(0.301174\pi\)
\(37\)	1.10929	−	0.146041i	0.182366	−	0.0240090i	−0.0387899	−	0.999247i	\(-0.512350\pi\)
							0.221156	+	0.975238i	\(0.429017\pi\)
\(41\)	−3.52447	−	0.308351i	−0.550430	−	0.0481563i	−0.191451	−	0.981502i	\(-0.561319\pi\)
							−0.358979	+	0.933346i	\(0.616875\pi\)
\(43\)	−0.0519083	+	1.18890i	−0.00791594	+	0.181305i	0.990969	+	0.134092i	\(0.0428117\pi\)
							−0.998885	+	0.0472130i	\(0.984966\pi\)
\(47\)	−1.99034	−	0.350951i	−0.290321	−	0.0511915i	0.0265908	−	0.999646i	\(-0.491535\pi\)
							−0.316912	+	0.948455i	\(0.602646\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bt.a.11.1	✓	3408
27.5	odd	18	inner	864.2.bt.a.491.81	yes	3408
32.3	odd	8	inner	864.2.bt.a.227.81	yes	3408
864.707	even	72	inner	864.2.bt.a.707.1	yes	3408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.bt.a.11.1	✓	3408	1.1	even	1	trivial
864.2.bt.a.227.81	yes	3408	32.3	odd	8	inner
864.2.bt.a.491.81	yes	3408	27.5	odd	18	inner
864.2.bt.a.707.1	yes	3408	864.707	even	72	inner