Embedded newform 864.2.bu.a.13.12

• Label: 864.2.bu.a.13.12
• 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			13.12
Character	\(\chi\)	\(=\)	864.13
Dual form			864.2.bu.a.133.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36689 - 0.362787i) q^{2} +(-0.738111 + 1.56691i) q^{3} +(1.73677 + 0.991780i) q^{4} +(0.0389079 + 0.0122676i) q^{5} +(1.57737 - 1.87401i) q^{6} +(2.13798 - 0.996959i) q^{7} +(-2.01417 - 1.98573i) q^{8} +(-1.91039 - 2.31310i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3408 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 12 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36689	−	0.362787i	−0.966536	−	0.256529i
							\(3\)	−0.738111	+	1.56691i	−0.426148	+	0.904653i
\(5\)	0.0389079	+	0.0122676i	0.0174002	+	0.00548625i								0.309393	−	0.950934i	\(-0.399874\pi\)
							−0.291993	+	0.956420i	\(0.594318\pi\)
\(7\)	2.13798	−	0.996959i	0.808082	−	0.376815i	0.0257438	−	0.999669i	\(-0.491805\pi\)
							0.782338	+	0.622854i	\(0.214027\pi\)
\(11\)	3.04495	+	1.58510i	0.918088	+	0.477926i	0.854149	−	0.520028i	\(-0.174078\pi\)
							0.0639384	+	0.997954i	\(0.479634\pi\)
\(13\)	1.73308	−	2.72039i	0.480669	−	0.754500i	−0.513981	−	0.857802i	\(-0.671830\pi\)
							0.994650	+	0.103302i	\(0.0329408\pi\)
\(17\)	1.62552	+	0.938496i	0.394247	+	0.227619i	0.683999	−	0.729483i	\(-0.260239\pi\)
							−0.289751	+	0.957102i	\(0.593573\pi\)
\(19\)	−1.05982	−	8.05013i	−0.243139	−	1.84683i	−0.482500	−	0.875896i	\(-0.660271\pi\)
							0.239360	−	0.970931i	\(-0.423062\pi\)
\(23\)	−7.22473	−	3.36895i	−1.50646	−	0.702474i	−0.518201	−	0.855259i	\(-0.673398\pi\)
							−0.988259	+	0.152785i	\(0.951176\pi\)
\(29\)	−5.65082	−	1.25276i	−1.04933	−	0.232631i	−0.342137	−	0.939650i	\(-0.611151\pi\)
							−0.707195	+	0.707019i	\(0.750040\pi\)
\(31\)	5.94957	−	2.16547i	1.06857	−	0.388929i	0.252931	−	0.967484i	\(-0.418606\pi\)
							0.815643	+	0.578555i	\(0.196383\pi\)
\(37\)	0.227316	−	1.72663i	0.0373705	−	0.283857i	−0.962548	−	0.271110i	\(-0.912609\pi\)
							0.999919	−	0.0127466i	\(-0.00405749\pi\)
\(41\)	−1.08560	+	0.760146i	−0.169542	+	0.118715i	−0.655268	−	0.755396i	\(-0.727444\pi\)
							0.485726	+	0.874111i	\(0.338555\pi\)
\(43\)	4.19461	+	2.18358i	0.639672	+	0.332992i	0.750523	−	0.660845i	\(-0.229802\pi\)
							−0.110850	+	0.993837i	\(0.535357\pi\)
\(47\)	1.64857	−	4.52940i	0.240468	−	0.660680i	−0.759481	−	0.650530i	\(-0.774547\pi\)
							0.999948	−	0.0101502i	\(-0.00323096\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bu.a.13.12	✓	3408
27.25	even	9	inner	864.2.bu.a.781.116	yes	3408
32.5	even	8	inner	864.2.bu.a.229.116	yes	3408
864.133	even	72	inner	864.2.bu.a.133.12	yes	3408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.bu.a.13.12	✓	3408	1.1	even	1	trivial
864.2.bu.a.133.12	yes	3408	864.133	even	72	inner
864.2.bu.a.229.116	yes	3408	32.5	even	8	inner
864.2.bu.a.781.116	yes	3408	27.25	even	9	inner