Embedded newform 864.2.bu.a.13.15

• Label: 864.2.bu.a.13.15
• 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			13.15
Character	\(\chi\)	\(=\)	864.13
Dual form			864.2.bu.a.133.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34354 - 0.441482i) q^{2} +(-1.19271 - 1.25596i) q^{3} +(1.61019 + 1.18630i) q^{4} +(-2.66785 - 0.841171i) q^{5} +(1.04797 + 2.21399i) q^{6} +(-3.54003 + 1.65074i) q^{7} +(-1.63962 - 2.30470i) q^{8} +(-0.154877 + 2.99600i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3408 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 12 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34354	−	0.441482i	−0.950025	−	0.312175i
							\(3\)	−1.19271	−	1.25596i	−0.688613	−	0.725129i
\(5\)	−2.66785	−	0.841171i	−1.19310	−	0.376183i								−0.361933	−	0.932204i	\(-0.617883\pi\)
							−0.831168	+	0.556021i	\(0.812327\pi\)
\(7\)	−3.54003	+	1.65074i	−1.33801	+	0.623923i	−0.954111	−	0.299454i	\(-0.903196\pi\)
							−0.383896	+	0.923376i	\(0.625418\pi\)
\(11\)	−4.61654	−	2.40322i	−1.39194	−	0.724598i	−0.409698	−	0.912221i	\(-0.634366\pi\)
							−0.982241	+	0.187624i	\(0.939921\pi\)
\(13\)	−1.96123	+	3.07852i	−0.543948	+	0.853828i	−0.999335	−	0.0364658i	\(-0.988390\pi\)
							0.455386	+	0.890294i	\(0.349501\pi\)
\(17\)	−0.520602	−	0.300570i	−0.126265	−	0.0728988i	0.435537	−	0.900171i	\(-0.356558\pi\)
							−0.561802	+	0.827272i	\(0.689892\pi\)
\(19\)	−0.573703	−	4.35770i	−0.131616	−	0.999726i	−0.922021	−	0.387140i	\(-0.873463\pi\)
							0.790404	−	0.612585i	\(-0.209870\pi\)
\(23\)	2.93136	+	1.36691i	0.611230	+	0.285021i	0.703480	−	0.710715i	\(-0.251628\pi\)
							−0.0922505	+	0.995736i	\(0.529406\pi\)
\(29\)	−3.83929	−	0.851149i	−0.712938	−	0.158054i	−0.154972	−	0.987919i	\(-0.549529\pi\)
							−0.557965	+	0.829864i	\(0.688418\pi\)
\(31\)	1.78691	−	0.650383i	0.320939	−	0.116812i	−0.176527	−	0.984296i	\(-0.556486\pi\)
							0.497466	+	0.867484i	\(0.334264\pi\)
\(37\)	−0.736094	+	5.59119i	−0.121013	+	0.919186i	0.817979	+	0.575248i	\(0.195094\pi\)
							−0.938992	+	0.343938i	\(0.888239\pi\)
\(41\)	3.58828	−	2.51254i	0.560396	−	0.392393i	−0.258780	−	0.965936i	\(-0.583320\pi\)
							0.819176	+	0.573543i	\(0.194431\pi\)
\(43\)	−6.04035	−	3.14441i	−0.921145	−	0.479518i	−0.0659493	−	0.997823i	\(-0.521008\pi\)
							−0.855196	+	0.518305i	\(0.826563\pi\)
\(47\)	−1.86895	+	5.13490i	−0.272615	+	0.749002i	0.725534	+	0.688186i	\(0.241593\pi\)
							−0.998149	+	0.0608164i	\(0.980630\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bu.a.13.15	✓	3408
27.25	even	9	inner	864.2.bu.a.781.114	yes	3408
32.5	even	8	inner	864.2.bu.a.229.114	yes	3408
864.133	even	72	inner	864.2.bu.a.133.15	yes	3408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.bu.a.13.15	✓	3408	1.1	even	1	trivial
864.2.bu.a.133.15	yes	3408	864.133	even	72	inner
864.2.bu.a.229.114	yes	3408	32.5	even	8	inner
864.2.bu.a.781.114	yes	3408	27.25	even	9	inner