Embedded newform 864.2.bu.a.13.13

• Label: 864.2.bu.a.13.13
• 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			13.13
Character	\(\chi\)	\(=\)	864.13
Dual form			864.2.bu.a.133.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36276 - 0.378009i) q^{2} +(-1.60615 - 0.648285i) q^{3} +(1.71422 + 1.03027i) q^{4} +(1.57723 + 0.497299i) q^{5} +(1.94374 + 1.49060i) q^{6} +(1.30846 - 0.610147i) q^{7} +(-1.94661 - 2.05200i) q^{8} +(2.15945 + 2.08249i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3408 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 12 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36276	−	0.378009i	−0.963615	−	0.267293i
							\(3\)	−1.60615	−	0.648285i	−0.927313	−	0.374287i
\(5\)	1.57723	+	0.497299i	0.705359	+	0.222399i								0.632069	−	0.774912i	\(-0.282206\pi\)
							0.0732896	+	0.997311i	\(0.476650\pi\)
\(7\)	1.30846	−	0.610147i	0.494553	−	0.230614i	−0.159307	−	0.987229i	\(-0.550926\pi\)
							0.653861	+	0.756615i	\(0.273148\pi\)
\(11\)	5.52460	+	2.87593i	1.66573	+	0.867124i	0.991723	+	0.128399i	\(0.0409840\pi\)
							0.674007	+	0.738725i	\(0.264572\pi\)
\(13\)	−3.85325	+	6.04839i	−1.06870	+	1.67752i	−0.446093	+	0.894986i	\(0.647185\pi\)
							−0.622606	+	0.782536i	\(0.713926\pi\)
\(17\)	−1.23816	−	0.714852i	−0.300298	−	0.173377i	0.342279	−	0.939598i	\(-0.388801\pi\)
							−0.642577	+	0.766221i	\(0.722135\pi\)
\(19\)	0.0190112	+	0.144405i	0.00436148	+	0.0331287i	0.993487	−	0.113944i	\(-0.0363483\pi\)
							−0.989126	+	0.147072i	\(0.953015\pi\)
\(23\)	2.69224	+	1.25541i	0.561370	+	0.261771i	0.682525	−	0.730862i	\(-0.260882\pi\)
							−0.121155	+	0.992634i	\(0.538660\pi\)
\(29\)	−0.787903	−	0.174674i	−0.146310	−	0.0324361i	0.142676	−	0.989769i	\(-0.454429\pi\)
							−0.288986	+	0.957333i	\(0.593318\pi\)
\(31\)	−9.54195	+	3.47299i	−1.71379	+	0.623767i	−0.997273	−	0.0738075i	\(-0.976485\pi\)
							−0.716513	+	0.697574i	\(0.754263\pi\)
\(37\)	−0.663886	+	5.04272i	−0.109142	+	0.829017i	0.846065	+	0.533080i	\(0.178965\pi\)
							−0.955207	+	0.295938i	\(0.904368\pi\)
\(41\)	−5.40702	+	3.78604i	−0.844435	+	0.591280i	−0.913733	−	0.406316i	\(-0.866813\pi\)
							0.0692974	+	0.997596i	\(0.477924\pi\)
\(43\)	−0.797055	−	0.414921i	−0.121550	−	0.0632748i	0.399889	−	0.916564i	\(-0.369049\pi\)
							−0.521438	+	0.853289i	\(0.674604\pi\)
\(47\)	−0.276626	+	0.760024i	−0.0403500	+	0.110861i	−0.958231	−	0.285994i	\(-0.907676\pi\)
							0.917881	+	0.396855i	\(0.129898\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bu.a.13.13	✓	3408
27.25	even	9	inner	864.2.bu.a.781.115	yes	3408
32.5	even	8	inner	864.2.bu.a.229.115	yes	3408
864.133	even	72	inner	864.2.bu.a.133.13	yes	3408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.bu.a.13.13	✓	3408	1.1	even	1	trivial
864.2.bu.a.133.13	yes	3408	864.133	even	72	inner
864.2.bu.a.229.115	yes	3408	32.5	even	8	inner
864.2.bu.a.781.115	yes	3408	27.25	even	9	inner