Embedded newform 151.2.g.a.2.10

• Label: 151.2.g.a.2.10
• Level: $151$
• Weight: $2$
• Character: 151.2
• Analytic conductor: $1.206$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	151.g (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.20574107052\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(12\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			2.10
Character	\(\chi\)	\(=\)	151.2
Dual form			151.2.g.a.76.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.729613 - 1.26373i) q^{2} +(-1.15094 + 0.836205i) q^{3} +(-0.0646694 - 0.112011i) q^{4} +(2.41864 + 2.68617i) q^{5} +(0.216996 + 2.06458i) q^{6} +(-1.66941 - 1.85407i) q^{7} +2.72972 q^{8} +(-0.301633 + 0.928330i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 3 q^{2} - 4 q^{3} - 49 q^{4} - 9 q^{5} + 7 q^{6} - 7 q^{7} + 12 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/151\mathbb{Z}\right)^\times\).

\(n\)	\(6\)
\(\chi(n)\)	\(e\left(\frac{7}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.729613	−	1.26373i	0.515914	−	0.893589i	−0.483915	−	0.875115i	\(-0.660786\pi\)
							0.999829	−	0.0184745i	\(-0.00588096\pi\)
\(3\)	−1.15094	+	0.836205i	−0.664494	+	0.482783i	−0.868178	−	0.496254i	\(-0.834709\pi\)
							0.203684	+	0.979037i	\(0.434709\pi\)
\(5\)	2.41864	+	2.68617i	1.08165	+	1.20129i	0.978418	+	0.206633i	\(0.0662507\pi\)
							0.103229	+	0.994658i	\(0.467083\pi\)
\(7\)	−1.66941	−	1.85407i	−0.630979	−	0.700774i	0.339868	−	0.940473i	\(-0.389618\pi\)
							−0.970847	+	0.239700i	\(0.922951\pi\)
\(11\)	3.47434	−	1.54688i	1.04755	−	0.466401i	0.190532	−	0.981681i	\(-0.438979\pi\)
							0.857022	+	0.515280i	\(0.172312\pi\)
\(13\)	−3.94317	−	1.75561i	−1.09364	−	0.486920i	−0.220994	−	0.975275i	\(-0.570930\pi\)
							−0.872645	+	0.488356i	\(0.837597\pi\)
\(17\)	−5.79242	−	1.23122i	−1.40487	−	0.298614i	−0.557745	−	0.830012i	\(-0.688333\pi\)
							−0.847122	+	0.531398i	\(0.821667\pi\)
\(19\)	4.95124			1.13589			0.567946	−	0.823066i	\(-0.307738\pi\)
							0.567946	+	0.823066i	\(0.307738\pi\)
\(23\)	0.352224	−	0.610070i	0.0734438	−	0.127208i	−0.826965	−	0.562254i	\(-0.809934\pi\)
							0.900408	+	0.435045i	\(0.143268\pi\)
\(29\)	−1.52286	−	1.10643i	−0.282789	−	0.205458i	0.437344	−	0.899294i	\(-0.355919\pi\)
							−0.720133	+	0.693836i	\(0.755919\pi\)
\(31\)	−6.73145	−	1.43081i	−1.20900	−	0.256982i	−0.441048	−	0.897484i	\(-0.645393\pi\)
							−0.767957	+	0.640502i	\(0.778726\pi\)
\(37\)	0.970027	+	9.22920i	0.159472	+	1.51727i	0.722812	+	0.691045i	\(0.242849\pi\)
							−0.563340	+	0.826225i	\(0.690484\pi\)
\(41\)	5.08461	−	3.69418i	0.794083	−	0.576935i	−0.115090	−	0.993355i	\(-0.536716\pi\)
							0.909172	+	0.416420i	\(0.136716\pi\)
\(43\)	3.32758	−	3.69565i	0.507452	−	0.563582i	−0.433921	−	0.900951i	\(-0.642870\pi\)
							0.941373	+	0.337369i	\(0.109537\pi\)
\(47\)	−9.30900	−	4.14463i	−1.35786	−	0.604557i	−0.406784	−	0.913524i	\(-0.633350\pi\)
							−0.951073	+	0.308967i	\(0.900017\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	151.2.g.a.2.10	✓	96
151.76	even	15	inner	151.2.g.a.76.10	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
151.2.g.a.2.10	✓	96	1.1	even	1	trivial
151.2.g.a.76.10	yes	96	151.76	even	15	inner