Embedded newform 864.2.bu.a.13.7

• Label: 864.2.bu.a.13.7
• 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			13.7
Character	\(\chi\)	\(=\)	864.13
Dual form			864.2.bu.a.133.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40459 + 0.164681i) q^{2} +(0.625973 + 1.61498i) q^{3} +(1.94576 - 0.462618i) q^{4} +(0.0622953 + 0.0196416i) q^{5} +(-1.14519 - 2.16530i) q^{6} +(-3.55538 + 1.65790i) q^{7} +(-2.65682 + 0.970219i) q^{8} +(-2.21632 + 2.02187i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3408 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 12 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40459	+	0.164681i	−0.993197	+	0.116447i
							\(3\)	0.625973	+	1.61498i	0.361406	+	0.932409i
\(5\)	0.0622953	+	0.0196416i	0.0278593	+	0.00878400i								0.314603	−	0.949223i	\(-0.398128\pi\)
							−0.286744	+	0.958007i	\(0.592573\pi\)
\(7\)	−3.55538	+	1.65790i	−1.34381	+	0.626628i	−0.955525	−	0.294911i	\(-0.904710\pi\)
							−0.388284	+	0.921540i	\(0.626932\pi\)
\(11\)	2.44137	+	1.27090i	0.736101	+	0.383190i	0.788173	−	0.615453i	\(-0.211027\pi\)
							−0.0520723	+	0.998643i	\(0.516583\pi\)
\(13\)	−1.96179	+	3.07939i	−0.544102	+	0.854069i	−0.999341	−	0.0363003i	\(-0.988443\pi\)
							0.455239	+	0.890369i	\(0.349554\pi\)
\(17\)	2.75415	+	1.59011i	0.667978	+	0.385658i	0.795310	−	0.606203i	\(-0.207308\pi\)
							−0.127332	+	0.991860i	\(0.540641\pi\)
\(19\)	−0.542797	−	4.12295i	−0.124526	−	0.945869i	−0.933639	−	0.358216i	\(-0.883385\pi\)
							0.809113	−	0.587654i	\(-0.199948\pi\)
\(23\)	−2.37115	−	1.10569i	−0.494420	−	0.230552i	0.159383	−	0.987217i	\(-0.449050\pi\)
							−0.653803	+	0.756665i	\(0.726827\pi\)
\(29\)	2.78213	+	0.616783i	0.516628	+	0.114534i	0.467040	−	0.884236i	\(-0.345320\pi\)
							0.0495878	+	0.998770i	\(0.484209\pi\)
\(31\)	−5.06936	+	1.84510i	−0.910484	+	0.331389i	−0.754446	−	0.656362i	\(-0.772094\pi\)
							−0.156038	+	0.987751i	\(0.549872\pi\)
\(37\)	0.623039	−	4.73245i	0.102427	−	0.778010i	−0.860729	−	0.509064i	\(-0.829992\pi\)
							0.963155	−	0.268945i	\(-0.0866751\pi\)
\(41\)	6.63338	−	4.64475i	1.03596	−	0.725387i	0.0736344	−	0.997285i	\(-0.476540\pi\)
							0.962326	+	0.271898i	\(0.0876513\pi\)
\(43\)	−5.79030	−	3.01424i	−0.883013	−	0.459667i	−0.0410213	−	0.999158i	\(-0.513061\pi\)
							−0.841991	+	0.539491i	\(0.818617\pi\)
\(47\)	−4.47030	+	12.2821i	−0.652061	+	1.79152i	−0.0420694	+	0.999115i	\(0.513395\pi\)
							−0.609992	+	0.792408i	\(0.708827\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bu.a.13.7	✓	3408
27.25	even	9	inner	864.2.bu.a.781.133	yes	3408
32.5	even	8	inner	864.2.bu.a.229.133	yes	3408
864.133	even	72	inner	864.2.bu.a.133.7	yes	3408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.bu.a.13.7	✓	3408	1.1	even	1	trivial
864.2.bu.a.133.7	yes	3408	864.133	even	72	inner
864.2.bu.a.229.133	yes	3408	32.5	even	8	inner
864.2.bu.a.781.133	yes	3408	27.25	even	9	inner