Embedded newform 864.2.bu.a.13.9

• Label: 864.2.bu.a.13.9
• 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			13.9
Character	\(\chi\)	\(=\)	864.13
Dual form			864.2.bu.a.133.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38819 + 0.270068i) q^{2} +(-1.71796 + 0.220516i) q^{3} +(1.85413 - 0.749809i) q^{4} +(-3.40435 - 1.07339i) q^{5} +(2.32529 - 0.770083i) q^{6} +(0.509100 - 0.237397i) q^{7} +(-2.37138 + 1.54162i) q^{8} +(2.90275 - 0.757675i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3408 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 12 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38819	+	0.270068i	−0.981596	+	0.190967i
							\(3\)	−1.71796	+	0.220516i	−0.991862	+	0.127315i
\(5\)	−3.40435	−	1.07339i	−1.52247	−	0.480034i								−0.580084	−	0.814556i	\(-0.696980\pi\)
							−0.942388	+	0.334523i	\(0.891425\pi\)
\(7\)	0.509100	−	0.237397i	0.192422	−	0.0897277i	−0.324019	−	0.946050i	\(-0.605034\pi\)
							0.516441	+	0.856323i	\(0.327257\pi\)
\(11\)	1.18960	+	0.619268i	0.358678	+	0.186716i	0.631552	−	0.775334i	\(-0.282418\pi\)
							−0.272873	+	0.962050i	\(0.587974\pi\)
\(13\)	−2.56901	+	4.03254i	−0.712515	+	1.11842i	0.275083	+	0.961421i	\(0.411295\pi\)
							−0.987598	+	0.157004i	\(0.949816\pi\)
\(17\)	3.60508	+	2.08140i	0.874361	+	0.504813i	0.868795	−	0.495172i	\(-0.164895\pi\)
							0.00556618	+	0.999985i	\(0.498228\pi\)
\(19\)	0.557436	+	4.23415i	0.127885	+	0.971380i	0.928273	+	0.371899i	\(0.121293\pi\)
							−0.800389	+	0.599481i	\(0.795374\pi\)
\(23\)	−7.60868	−	3.54799i	−1.58652	−	0.739806i	−0.588864	−	0.808232i	\(-0.700424\pi\)
							−0.997656	+	0.0684261i	\(0.978202\pi\)
\(29\)	−1.89793	−	0.420760i	−0.352436	−	0.0781332i	0.0366667	−	0.999328i	\(-0.488326\pi\)
							−0.389103	+	0.921194i	\(0.627215\pi\)
\(31\)	5.45580	−	1.98575i	0.979890	−	0.356651i	0.198093	−	0.980183i	\(-0.436525\pi\)
							0.781797	+	0.623533i	\(0.214303\pi\)
\(37\)	1.06461	−	8.08651i	0.175021	−	1.32941i	−0.648103	−	0.761553i	\(-0.724437\pi\)
							0.823124	−	0.567862i	\(-0.192229\pi\)
\(41\)	2.21808	−	1.55312i	0.346406	−	0.242556i	−0.387405	−	0.921910i	\(-0.626629\pi\)
							0.733811	+	0.679353i	\(0.237740\pi\)
\(43\)	−1.04730	−	0.545192i	−0.159712	−	0.0831410i	0.380017	−	0.924979i	\(-0.375918\pi\)
							−0.539730	+	0.841838i	\(0.681474\pi\)
\(47\)	0.599122	−	1.64607i	0.0873909	−	0.240105i	−0.888297	−	0.459270i	\(-0.848111\pi\)
							0.975688	+	0.219165i	\(0.0703334\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bu.a.13.9	✓	3408
27.25	even	9	inner	864.2.bu.a.781.138	yes	3408
32.5	even	8	inner	864.2.bu.a.229.138	yes	3408
864.133	even	72	inner	864.2.bu.a.133.9	yes	3408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.bu.a.13.9	✓	3408	1.1	even	1	trivial
864.2.bu.a.133.9	yes	3408	864.133	even	72	inner
864.2.bu.a.229.138	yes	3408	32.5	even	8	inner
864.2.bu.a.781.138	yes	3408	27.25	even	9	inner