Embedded newform 864.2.bu.a.13.3

• Label: 864.2.bu.a.13.3
• 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			13.3
Character	\(\chi\)	\(=\)	864.13
Dual form			864.2.bu.a.133.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41215 - 0.0762974i) q^{2} +(1.09754 - 1.33992i) q^{3} +(1.98836 + 0.215487i) q^{4} +(-2.03835 - 0.642689i) q^{5} +(-1.65213 + 1.80844i) q^{6} +(-2.07741 + 0.968715i) q^{7} +(-2.79143 - 0.456008i) q^{8} +(-0.590794 - 2.94125i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3408 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 12 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41215	−	0.0762974i	−0.998544	−	0.0539504i
							\(3\)	1.09754	−	1.33992i	0.633667	−	0.773606i
\(5\)	−2.03835	−	0.642689i	−0.911578	−	0.287419i								−0.191646	−	0.981464i	\(-0.561383\pi\)
							−0.719932	+	0.694045i	\(0.755827\pi\)
\(7\)	−2.07741	+	0.968715i	−0.785189	+	0.366140i	−0.773503	−	0.633792i	\(-0.781497\pi\)
							−0.0116856	+	0.999932i	\(0.503720\pi\)
\(11\)	−0.594029	−	0.309232i	−0.179107	−	0.0932370i	0.369836	−	0.929097i	\(-0.379414\pi\)
							−0.548943	+	0.835860i	\(0.684969\pi\)
\(13\)	3.48607	−	5.47203i	0.966861	−	1.51767i	0.115355	−	0.993324i	\(-0.463199\pi\)
							0.851507	−	0.524344i	\(-0.175689\pi\)
\(17\)	2.34663	+	1.35483i	0.569142	+	0.328595i	0.756807	−	0.653639i	\(-0.226758\pi\)
							−0.187664	+	0.982233i	\(0.560092\pi\)
\(19\)	0.516285	+	3.92157i	0.118444	+	0.899671i	0.942744	+	0.333518i	\(0.108236\pi\)
							−0.824300	+	0.566153i	\(0.808431\pi\)
\(23\)	−3.52538	−	1.64391i	−0.735093	−	0.342780i	0.0187585	−	0.999824i	\(-0.494029\pi\)
							−0.753852	+	0.657044i	\(0.771806\pi\)
\(29\)	−7.97514	−	1.76805i	−1.48095	−	0.328318i	−0.599413	−	0.800440i	\(-0.704599\pi\)
							−0.881533	+	0.472122i	\(0.843488\pi\)
\(31\)	−1.85183	+	0.674011i	−0.332598	+	0.121056i	−0.502921	−	0.864332i	\(-0.667741\pi\)
							0.170322	+	0.985388i	\(0.445519\pi\)
\(37\)	−0.609782	+	4.63175i	−0.100248	+	0.761456i	0.865305	+	0.501246i	\(0.167125\pi\)
							−0.965552	+	0.260209i	\(0.916208\pi\)
\(41\)	−8.23152	+	5.76377i	−1.28555	+	0.900149i	−0.998333	−	0.0577170i	\(-0.981618\pi\)
							−0.287214	+	0.957867i	\(0.592729\pi\)
\(43\)	−3.64019	−	1.89496i	−0.555124	−	0.288979i	0.160998	−	0.986955i	\(-0.448529\pi\)
							−0.716121	+	0.697976i	\(0.754084\pi\)
\(47\)	−3.98875	+	10.9590i	−0.581819	+	1.59853i	0.203251	+	0.979127i	\(0.434849\pi\)
							−0.785070	+	0.619408i	\(0.787373\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bu.a.13.3	✓	3408
27.25	even	9	inner	864.2.bu.a.781.125	yes	3408
32.5	even	8	inner	864.2.bu.a.229.125	yes	3408
864.133	even	72	inner	864.2.bu.a.133.3	yes	3408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.bu.a.13.3	✓	3408	1.1	even	1	trivial
864.2.bu.a.133.3	yes	3408	864.133	even	72	inner
864.2.bu.a.229.125	yes	3408	32.5	even	8	inner
864.2.bu.a.781.125	yes	3408	27.25	even	9	inner