Embedded newform 864.2.bu.a.13.10

• Label: 864.2.bu.a.13.10
• Level: $864$
• Weight: $2$
• Character: 864.13
• 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.bu (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			13.10
Character	\(\chi\)	\(=\)	864.13
Dual form			864.2.bu.a.133.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38811 - 0.270444i) q^{2} +(0.417493 - 1.68098i) q^{3} +(1.85372 + 0.750815i) q^{4} +(0.0983780 + 0.0310185i) q^{5} +(-1.03414 + 2.22049i) q^{6} +(2.27483 - 1.06077i) q^{7} +(-2.37012 - 1.54354i) q^{8} +(-2.65140 - 1.40360i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3408 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 12 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(353\)	\(703\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38811	−	0.270444i	−0.981545	−	0.191233i
							\(3\)	0.417493	−	1.68098i	0.241040	−	0.970515i
\(5\)	0.0983780	+	0.0310185i	0.0439960	+	0.0138719i								0.322624	−	0.946527i	\(-0.395435\pi\)
							−0.278628	+	0.960399i	\(0.589880\pi\)
\(7\)	2.27483	−	1.06077i	0.859805	−	0.400934i	0.0578551	−	0.998325i	\(-0.481574\pi\)
							0.801950	+	0.597391i	\(0.203796\pi\)
\(11\)	−1.64401	−	0.855817i	−0.495688	−	0.258039i	0.195514	−	0.980701i	\(-0.437363\pi\)
							−0.691201	+	0.722662i	\(0.742918\pi\)
\(13\)	−1.08623	+	1.70504i	−0.301266	+	0.472892i	−0.960202	−	0.279306i	\(-0.909896\pi\)
							0.658936	+	0.752199i	\(0.271007\pi\)
\(17\)	−3.61138	−	2.08503i	−0.875889	−	0.505695i	−0.00658807	−	0.999978i	\(-0.502097\pi\)
							−0.869301	+	0.494284i	\(0.835430\pi\)
\(19\)	−0.329265	−	2.50101i	−0.0755385	−	0.573772i	−0.986800	−	0.161944i	\(-0.948224\pi\)
							0.911262	−	0.411828i	\(-0.135110\pi\)
\(23\)	−1.47150	−	0.686173i	−0.306829	−	0.143077i	0.263105	−	0.964767i	\(-0.415254\pi\)
							−0.569934	+	0.821690i	\(0.693031\pi\)
\(29\)	−1.11477	−	0.247139i	−0.207008	−	0.0458926i	0.111716	−	0.993740i	\(-0.464365\pi\)
							−0.318724	+	0.947848i	\(0.603254\pi\)
\(31\)	1.79813	−	0.654467i	0.322954	−	0.117546i	−0.175455	−	0.984487i	\(-0.556140\pi\)
							0.498410	+	0.866942i	\(0.333918\pi\)
\(37\)	0.905031	−	6.87439i	0.148786	−	1.13014i	−0.740259	−	0.672322i	\(-0.765297\pi\)
							0.889045	−	0.457820i	\(-0.151370\pi\)
\(41\)	4.53183	−	3.17322i	0.707753	−	0.495574i	−0.163416	−	0.986557i	\(-0.552251\pi\)
							0.871169	+	0.490983i	\(0.163362\pi\)
\(43\)	3.44125	+	1.79140i	0.524786	+	0.273186i	0.703476	−	0.710719i	\(-0.251630\pi\)
							−0.178690	+	0.983905i	\(0.557186\pi\)
\(47\)	2.76909	−	7.60802i	0.403914	−	1.10974i	−0.556423	−	0.830899i	\(-0.687826\pi\)
							0.960336	−	0.278844i	\(-0.0899513\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bu.a.13.10	✓	3408
27.25	even	9	inner	864.2.bu.a.781.118	yes	3408
32.5	even	8	inner	864.2.bu.a.229.118	yes	3408
864.133	even	72	inner	864.2.bu.a.133.10	yes	3408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.bu.a.13.10	✓	3408	1.1	even	1	trivial
864.2.bu.a.133.10	yes	3408	864.133	even	72	inner
864.2.bu.a.229.118	yes	3408	32.5	even	8	inner
864.2.bu.a.781.118	yes	3408	27.25	even	9	inner