Embedded newform 357.2.bh.a.151.19

• Label: 357.2.bh.a.151.19
• Level: $357$
• Weight: $2$
• Character: 357.151
• Analytic conductor: $2.851$
• Analytic rank: $0$
• Dimension: $192$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 357 = 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	357.bh (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.85065935216\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(24\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			151.19
Character	\(\chi\)	\(=\)	357.151
Dual form			357.2.bh.a.331.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.452018 - 1.68695i) q^{2} +(-0.991445 - 0.130526i) q^{3} +(-0.909440 - 0.525065i) q^{4} +(1.68494 + 1.29290i) q^{5} +(-0.668342 + 1.61352i) q^{6} +(0.416441 + 2.61277i) q^{7} +(1.17303 - 1.17303i) q^{8} +(0.965926 + 0.258819i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 8 q^{5}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/357\mathbb{Z}\right)^\times\).

\(n\)	\(52\)	\(190\)	\(239\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.452018	−	1.68695i	0.319625	−	1.19286i	−0.599981	−	0.800014i	\(-0.704825\pi\)
							0.919606	−	0.392842i	\(-0.128508\pi\)
\(3\)	−0.991445	−	0.130526i	−0.572411	−	0.0753593i
							\(5\)	1.68494	+	1.29290i	0.753528	+	0.578202i	0.912498	−	0.409081i	\(-0.134151\pi\)
−0.158970	+	0.987283i	\(0.550817\pi\)
\(7\)	0.416441	+	2.61277i	0.157400	+	0.987535i
							\(11\)	3.11222	+	4.05592i	0.938369	+	1.22291i	0.974689	+	0.223566i	\(0.0717698\pi\)
−0.0363199	+	0.999340i	\(0.511564\pi\)
\(13\)			0.853029i			0.236588i	0.992979	+	0.118294i	\(0.0377425\pi\)
							−0.992979	+	0.118294i	\(0.962258\pi\)
\(17\)	−4.07104	−	0.653203i	−0.987371	−	0.158425i
							\(19\)	0.799810	−	2.98493i	0.183489	−	0.684790i	−0.811460	−	0.584408i	\(-0.801327\pi\)
0.994949	−	0.100382i	\(-0.0320065\pi\)
\(23\)	6.00389	−	0.790427i	1.25190	−	0.164815i	0.524726	−	0.851271i	\(-0.324168\pi\)
							0.727172	+	0.686456i	\(0.240834\pi\)
\(29\)	−3.11130	+	1.28874i	−0.577754	+	0.239314i	−0.652373	−	0.757898i	\(-0.726226\pi\)
							0.0746181	+	0.997212i	\(0.476226\pi\)
\(31\)	−7.42289	−	0.977242i	−1.33319	−	0.175518i	−0.569975	−	0.821662i	\(-0.693047\pi\)
							−0.763216	+	0.646144i	\(0.776381\pi\)
\(37\)	2.45066	−	3.19376i	0.402885	−	0.525050i	−0.547216	−	0.836991i	\(-0.684312\pi\)
							0.950102	+	0.311941i	\(0.100979\pi\)
\(41\)	2.06379	+	0.854851i	0.322310	+	0.133505i	0.537972	−	0.842963i	\(-0.319191\pi\)
							−0.215661	+	0.976468i	\(0.569191\pi\)
\(43\)	4.32276	−	4.32276i	0.659215	−	0.659215i	−0.295979	−	0.955194i	\(-0.595646\pi\)
							0.955194	+	0.295979i	\(0.0956459\pi\)
\(47\)	1.22754	−	0.708719i	0.179055	−	0.103377i	−0.407794	−	0.913074i	\(-0.633702\pi\)
							0.586848	+	0.809697i	\(0.300368\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	357.2.bh.a.151.19	yes	192
7.2	even	3	inner	357.2.bh.a.100.6	yes	192
17.8	even	8	inner	357.2.bh.a.25.6	✓	192
119.93	even	24	inner	357.2.bh.a.331.19	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.bh.a.25.6	✓	192	17.8	even	8	inner
357.2.bh.a.100.6	yes	192	7.2	even	3	inner
357.2.bh.a.151.19	yes	192	1.1	even	1	trivial
357.2.bh.a.331.19	yes	192	119.93	even	24	inner