Embedded newform 864.2.bu.a.13.17

• Label: 864.2.bu.a.13.17
• 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			13.17
Character	\(\chi\)	\(=\)	864.13
Dual form			864.2.bu.a.133.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33912 - 0.454707i) q^{2} +(1.64935 - 0.528821i) q^{3} +(1.58648 + 1.21781i) q^{4} +(0.966263 + 0.304661i) q^{5} +(-2.44913 - 0.0418154i) q^{6} +(-2.27200 + 1.05945i) q^{7} +(-1.57074 - 2.35218i) q^{8} +(2.44070 - 1.74442i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3408 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 12 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33912	−	0.454707i	−0.946901	−	0.321526i
							\(3\)	1.64935	−	0.528821i	0.952251	−	0.305315i
\(5\)	0.966263	+	0.304661i	0.432126	+	0.136249i								0.508950	−	0.860796i	\(-0.330034\pi\)
							−0.0768245	+	0.997045i	\(0.524478\pi\)
\(7\)	−2.27200	+	1.05945i	−0.858735	+	0.400435i	−0.801549	−	0.597929i	\(-0.795990\pi\)
							−0.0571856	+	0.998364i	\(0.518213\pi\)
\(11\)	2.35276	+	1.22477i	0.709383	+	0.369281i	0.777917	−	0.628367i	\(-0.216277\pi\)
							−0.0685338	+	0.997649i	\(0.521832\pi\)
\(13\)	−0.479414	+	0.752530i	−0.132966	+	0.208714i	−0.903393	−	0.428813i	\(-0.858932\pi\)
							0.770427	+	0.637528i	\(0.220043\pi\)
\(17\)	0.0657605	+	0.0379668i	0.0159493	+	0.00920831i	0.507953	−	0.861385i	\(-0.330402\pi\)
							−0.492004	+	0.870593i	\(0.663736\pi\)
\(19\)	−0.576294	−	4.37739i	−0.132211	−	1.00424i	−0.920996	−	0.389572i	\(-0.872623\pi\)
							0.788785	−	0.614669i	\(-0.210710\pi\)
\(23\)	6.31864	+	2.94643i	1.31753	+	0.614373i	0.949009	−	0.315248i	\(-0.102088\pi\)
							0.368517	+	0.929621i	\(0.379865\pi\)
\(29\)	10.2329	+	2.26857i	1.90020	+	0.421263i	0.999902	−	0.0140178i	\(-0.00446214\pi\)
							0.900295	+	0.435281i	\(0.143351\pi\)
\(31\)	9.54705	−	3.47484i	1.71470	−	0.624100i	0.717341	−	0.696722i	\(-0.245359\pi\)
							0.997360	+	0.0726218i	\(0.0231366\pi\)
\(37\)	−0.584481	+	4.43958i	−0.0960881	+	0.729862i	0.873796	+	0.486293i	\(0.161651\pi\)
							−0.969884	+	0.243568i	\(0.921682\pi\)
\(41\)	−3.10035	+	2.17089i	−0.484193	+	0.339035i	−0.790050	−	0.613042i	\(-0.789946\pi\)
							0.305858	+	0.952077i	\(0.401057\pi\)
\(43\)	−0.282458	−	0.147038i	−0.0430744	−	0.0224231i	0.440075	−	0.897961i	\(-0.354952\pi\)
							−0.483150	+	0.875538i	\(0.660507\pi\)
\(47\)	0.222606	−	0.611604i	0.0324704	−	0.0892117i	−0.922398	−	0.386240i	\(-0.873774\pi\)
							0.954869	+	0.297029i	\(0.0959957\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bu.a.13.17	✓	3408
27.25	even	9	inner	864.2.bu.a.781.113	yes	3408
32.5	even	8	inner	864.2.bu.a.229.113	yes	3408
864.133	even	72	inner	864.2.bu.a.133.17	yes	3408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.bu.a.13.17	✓	3408	1.1	even	1	trivial
864.2.bu.a.133.17	yes	3408	864.133	even	72	inner
864.2.bu.a.229.113	yes	3408	32.5	even	8	inner
864.2.bu.a.781.113	yes	3408	27.25	even	9	inner