Embedded newform 912.2.bu.a.715.73

• Label: 912.2.bu.a.715.73
• Level: $912$
• Weight: $2$
• Character: 912.715
• 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			715.73
Character	\(\chi\)	\(=\)	912.715
Dual form			912.2.bu.a.787.73

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36839 + 0.357076i) q^{2} +(0.965926 + 0.258819i) q^{3} +(1.74499 + 0.977240i) q^{4} +(0.500613 - 1.86831i) q^{5} +(1.22935 + 0.699075i) q^{6} -1.82689 q^{7} +(2.03889 + 1.96034i) q^{8} +(0.866025 + 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36839	+	0.357076i	0.967599	+	0.252491i
							\(3\)	0.965926	+	0.258819i	0.557678	+	0.149429i
\(5\)	0.500613	−	1.86831i	0.223881	−	0.835536i								−0.758969	−	0.651127i	\(-0.774296\pi\)
							0.982850	−	0.184408i	\(-0.0590369\pi\)
\(7\)	−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	−0.464756	−	0.885439i	\(-0.653858\pi\)
							0.885439	+	0.464756i	\(0.153858\pi\)
\(13\)	2.81340	−	0.753847i	0.780295	−	0.209080i	0.153380	−	0.988167i	\(-0.450984\pi\)
							0.626915	+	0.779088i	\(0.284317\pi\)
\(17\)	−0.492674	−	0.853336i	−0.119491	−	0.206964i	0.800075	−	0.599900i	\(-0.204793\pi\)
							−0.919566	+	0.392935i	\(0.871460\pi\)
\(19\)	2.72410	−	3.40283i	0.624952	−	0.780663i
							\(23\)	−3.63798	+	6.30117i	−0.758571	+	1.31388i	0.185008	+	0.982737i	\(0.440769\pi\)
−0.943579	+	0.331147i	\(0.892564\pi\)
\(29\)	2.42883	−	0.650803i	0.451022	−	0.120851i	−0.0261553	−	0.999658i	\(-0.508326\pi\)
							0.477178	+	0.878807i	\(0.341660\pi\)
\(31\)	−3.24147			−0.582186			−0.291093	−	0.956695i	\(-0.594019\pi\)
							−0.291093	+	0.956695i	\(0.594019\pi\)
\(37\)	−1.25689	+	1.25689i	−0.206631	+	0.206631i	−0.802834	−	0.596203i	\(-0.796675\pi\)
							0.596203	+	0.802834i	\(0.296675\pi\)
\(41\)	0.960589	+	1.66379i	0.150019	+	0.259840i	0.931234	−	0.364422i	\(-0.118733\pi\)
							−0.781215	+	0.624262i	\(0.785400\pi\)
\(43\)	−0.439902	−	0.117871i	−0.0670844	−	0.0179752i	0.225121	−	0.974331i	\(-0.427722\pi\)
							−0.292205	+	0.956356i	\(0.594389\pi\)
\(47\)	−8.26816	−	4.77363i	−1.20604	−	0.696305i	−0.244145	−	0.969739i	\(-0.578507\pi\)
							−0.961891	+	0.273434i	\(0.911841\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.bu.a.259.48	✓	320	16.3	odd	4	inner
912.2.bu.a.331.48	yes	320	19.8	odd	6	inner
912.2.bu.a.715.73	yes	320	1.1	even	1	trivial
912.2.bu.a.787.73	yes	320	304.179	even	12	inner