Embedded newform 912.2.bu.a.259.48

• Label: 912.2.bu.a.259.48
• Level: $912$
• Weight: $2$
• Character: 912.259
• Analytic conductor: $7.282$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.bu (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.28235666434\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(80\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			259.48
Character	\(\chi\)	\(=\)	912.259
Dual form			912.2.bu.a.331.48

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.374959 - 1.36360i) q^{2} +(0.258819 - 0.965926i) q^{3} +(-1.71881 - 1.02259i) q^{4} +(-1.86831 - 0.500613i) q^{5} +(-1.22009 - 0.715109i) q^{6} -1.82689 q^{7} +(-2.03889 + 1.96034i) q^{8} +(-0.866025 - 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(229\)	\(305\)	\(799\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.374959	−	1.36360i	0.265136	−	0.964211i
							\(3\)	0.258819	−	0.965926i	0.149429	−	0.557678i
\(5\)	−1.86831	−	0.500613i	−0.835536	−	0.223881i								−0.184408	−	0.982850i	\(-0.559037\pi\)
							−0.651127	+	0.758969i	\(0.725704\pi\)
\(7\)	−1.82689			−0.690498			−0.345249	−	0.938511i	\(-0.612206\pi\)
							−0.345249	+	0.938511i	\(0.612206\pi\)
\(11\)	1.39525	+	1.39525i	0.420682	+	0.420682i	0.885439	−	0.464756i	\(-0.153858\pi\)
							−0.464756	+	0.885439i	\(0.653858\pi\)
\(13\)	0.753847	+	2.81340i	0.209080	+	0.780295i	0.988167	+	0.153380i	\(0.0490160\pi\)
							−0.779088	+	0.626915i	\(0.784317\pi\)
\(17\)	−0.492674	−	0.853336i	−0.119491	−	0.206964i	0.800075	−	0.599900i	\(-0.204793\pi\)
							−0.919566	+	0.392935i	\(0.871460\pi\)
\(19\)	−3.40283	−	2.72410i	−0.780663	−	0.624952i
							\(23\)	−3.63798	+	6.30117i	−0.758571	+	1.31388i	0.185008	+	0.982737i	\(0.440769\pi\)
−0.943579	+	0.331147i	\(0.892564\pi\)
\(29\)	0.650803	+	2.42883i	0.120851	+	0.451022i	0.999658	−	0.0261553i	\(-0.00832646\pi\)
							−0.878807	+	0.477178i	\(0.841660\pi\)
\(31\)	3.24147			0.582186			0.291093	−	0.956695i	\(-0.405981\pi\)
							0.291093	+	0.956695i	\(0.405981\pi\)
\(37\)	1.25689	+	1.25689i	0.206631	+	0.206631i	0.802834	−	0.596203i	\(-0.203325\pi\)
							−0.596203	+	0.802834i	\(0.703325\pi\)
\(41\)	−0.960589	−	1.66379i	−0.150019	−	0.259840i	0.781215	−	0.624262i	\(-0.214600\pi\)
							−0.931234	+	0.364422i	\(0.881267\pi\)
\(43\)	0.117871	−	0.439902i	0.0179752	−	0.0670844i	−0.956356	−	0.292205i	\(-0.905611\pi\)
							0.974331	+	0.225121i	\(0.0722777\pi\)
\(47\)	8.26816	+	4.77363i	1.20604	+	0.696305i	0.961891	−	0.273434i	\(-0.0881594\pi\)
							0.244145	+	0.969739i	\(0.421493\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.bu.a.259.48	✓	320
16.11	odd	4	inner	912.2.bu.a.715.73	yes	320
19.8	odd	6	inner	912.2.bu.a.787.73	yes	320
304.27	even	12	inner	912.2.bu.a.331.48	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.bu.a.259.48	✓	320	1.1	even	1	trivial
912.2.bu.a.331.48	yes	320	304.27	even	12	inner
912.2.bu.a.715.73	yes	320	16.11	odd	4	inner
912.2.bu.a.787.73	yes	320	19.8	odd	6	inner