Embedded newform 357.2.bc.a.97.6

• Label: 357.2.bc.a.97.6
• Level: $357$
• Weight: $2$
• Character: 357.97
• 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.bc (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.85065935216\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(24\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			97.6
Character	\(\chi\)	\(=\)	357.97
Dual form			357.2.bc.a.265.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.678477 - 1.63799i) q^{2} +(0.555570 + 0.831470i) q^{3} +(-0.808463 + 0.808463i) q^{4} +(0.327699 - 1.64746i) q^{5} +(0.984996 - 1.47415i) q^{6} +(-0.339685 - 2.62385i) q^{7} +(-1.40320 - 0.581225i) q^{8} +(-0.382683 + 0.923880i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

Character values

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

\(n\)	\(52\)	\(190\)	\(239\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{13}{16}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.678477	−	1.63799i	−0.479756	−	1.15823i	−0.959723	−	0.280947i	\(-0.909352\pi\)
							0.479967	−	0.877286i	\(-0.340648\pi\)
\(3\)	0.555570	+	0.831470i	0.320759	+	0.480049i
							\(5\)	0.327699	−	1.64746i	0.146552	−	0.736765i	−0.835698	−	0.549189i	\(-0.814937\pi\)
0.982250	−	0.187576i	\(-0.0600631\pi\)
\(7\)	−0.339685	−	2.62385i	−0.128389	−	0.991724i
							\(11\)	−1.77376	+	2.65462i	−0.534808	+	0.800397i	−0.996227	−	0.0867853i	\(-0.972341\pi\)
0.461419	+	0.887182i	\(0.347341\pi\)
\(13\)	3.49202	−	3.49202i	0.968511	−	0.968511i	−0.0310082	−	0.999519i	\(-0.509872\pi\)
							0.999519	+	0.0310082i	\(0.00987181\pi\)
\(17\)	0.291609	−	4.11278i	0.0707255	−	0.997496i
							\(19\)	−1.17921	+	0.488444i	−0.270529	+	0.112057i	−0.513824	−	0.857896i	\(-0.671772\pi\)
0.243295	+	0.969952i	\(0.421772\pi\)
\(23\)	−7.08880	−	4.73659i	−1.47812	−	0.987647i	−0.993611	−	0.112859i	\(-0.963999\pi\)
							−0.484507	−	0.874788i	\(-0.661001\pi\)
\(29\)	0.614115	−	3.08737i	0.114038	−	0.573309i	−0.880941	−	0.473227i	\(-0.843089\pi\)
							0.994979	−	0.100083i	\(-0.0319108\pi\)
\(31\)	−0.853399	+	0.570223i	−0.153275	+	0.102415i	−0.629843	−	0.776723i	\(-0.716881\pi\)
							0.476568	+	0.879138i	\(0.341881\pi\)
\(37\)	3.36644	−	2.24938i	0.553439	−	0.369796i	−0.247180	−	0.968969i	\(-0.579504\pi\)
							0.800619	+	0.599173i	\(0.204504\pi\)
\(41\)	1.65293	+	8.30985i	0.258145	+	1.29778i	0.864519	+	0.502601i	\(0.167623\pi\)
							−0.606374	+	0.795180i	\(0.707377\pi\)
\(43\)	0.406866	−	0.982262i	0.0620465	−	0.149794i	−0.889815	−	0.456321i	\(-0.849167\pi\)
							0.951862	+	0.306527i	\(0.0991670\pi\)
\(47\)	4.56139	−	4.56139i	0.665347	−	0.665347i	−0.291288	−	0.956635i	\(-0.594084\pi\)
							0.956635	+	0.291288i	\(0.0940839\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	357.2.bc.a.97.6	yes	192
7.6	odd	2	inner	357.2.bc.a.97.5	✓	192
17.10	odd	16	inner	357.2.bc.a.265.5	yes	192
119.27	even	16	inner	357.2.bc.a.265.6	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.bc.a.97.5	✓	192	7.6	odd	2	inner
357.2.bc.a.97.6	yes	192	1.1	even	1	trivial
357.2.bc.a.265.5	yes	192	17.10	odd	16	inner
357.2.bc.a.265.6	yes	192	119.27	even	16	inner