Embedded newform 864.2.bu.a.13.14

• Label: 864.2.bu.a.13.14
• 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			13.14
Character	\(\chi\)	\(=\)	864.13
Dual form			864.2.bu.a.133.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35215 + 0.414342i) q^{2} +(1.46175 + 0.929134i) q^{3} +(1.65664 - 1.12051i) q^{4} +(-2.56205 - 0.807812i) q^{5} +(-2.36149 - 0.650669i) q^{6} +(0.176479 - 0.0822935i) q^{7} +(-1.77576 + 2.20152i) q^{8} +(1.27342 + 2.71632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3408 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 12 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35215	+	0.414342i	−0.956117	+	0.292984i
							\(3\)	1.46175	+	0.929134i	0.843941	+	0.536436i
\(5\)	−2.56205	−	0.807812i	−1.14578	−	0.361264i								−0.332484	−	0.943109i	\(-0.607887\pi\)
							−0.813300	+	0.581845i	\(0.802331\pi\)
\(7\)	0.176479	−	0.0822935i	0.0667028	−	0.0311040i	−0.388981	−	0.921246i	\(-0.627173\pi\)
							0.455683	+	0.890142i	\(0.349395\pi\)
\(11\)	−3.38539	−	1.76232i	−1.02073	−	0.531360i	−0.132708	−	0.991155i	\(-0.542367\pi\)
							−0.888025	+	0.459795i	\(0.847923\pi\)
\(13\)	2.20118	−	3.45516i	0.610497	−	0.958289i	−0.388819	−	0.921314i	\(-0.627117\pi\)
							0.999316	−	0.0369746i	\(-0.0117721\pi\)
\(17\)	4.80603	+	2.77476i	1.16563	+	0.672978i	0.952647	−	0.304077i	\(-0.0983481\pi\)
							0.212985	+	0.977055i	\(0.431681\pi\)
\(19\)	−0.362728	−	2.75519i	−0.0832156	−	0.632085i	−0.981331	−	0.192326i	\(-0.938397\pi\)
							0.898116	−	0.439759i	\(-0.144936\pi\)
\(23\)	6.85972	+	3.19874i	1.43035	+	0.666983i	0.974770	−	0.223211i	\(-0.0716539\pi\)
							0.455581	+	0.890195i	\(0.349432\pi\)
\(29\)	−0.0198281	−	0.00439578i	−0.00368199	−	0.000816277i	0.214598	−	0.976702i	\(-0.431156\pi\)
							−0.218280	+	0.975886i	\(0.570045\pi\)
\(31\)	7.68580	−	2.79740i	1.38041	−	0.502428i	0.458108	−	0.888897i	\(-0.348527\pi\)
							0.922303	+	0.386468i	\(0.126305\pi\)
\(37\)	−0.0577988	+	0.439025i	−0.00950206	+	0.0721753i	−0.995539	−	0.0943521i	\(-0.969922\pi\)
							0.986037	+	0.166527i	\(0.0532554\pi\)
\(41\)	4.98279	−	3.48899i	0.778181	−	0.544888i	−0.115651	−	0.993290i	\(-0.536895\pi\)
							0.893832	+	0.448402i	\(0.148007\pi\)
\(43\)	8.86198	+	4.61325i	1.35144	+	0.703514i	0.974849	−	0.222866i	\(-0.0715411\pi\)
							0.376590	+	0.926380i	\(0.377097\pi\)
\(47\)	3.24141	−	8.90570i	0.472808	−	1.29903i	−0.442679	−	0.896680i	\(-0.645972\pi\)
							0.915486	−	0.402349i	\(-0.131806\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bu.a.13.14	✓	3408
27.25	even	9	inner	864.2.bu.a.781.140	yes	3408
32.5	even	8	inner	864.2.bu.a.229.140	yes	3408
864.133	even	72	inner	864.2.bu.a.133.14	yes	3408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.bu.a.13.14	✓	3408	1.1	even	1	trivial
864.2.bu.a.133.14	yes	3408	864.133	even	72	inner
864.2.bu.a.229.140	yes	3408	32.5	even	8	inner
864.2.bu.a.781.140	yes	3408	27.25	even	9	inner