Embedded newform 357.2.bc.a.97.16

• Label: 357.2.bc.a.97.16
• 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			97.16
Character	\(\chi\)	\(=\)	357.97
Dual form			357.2.bc.a.265.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.347777 + 0.839608i) q^{2} +(0.555570 + 0.831470i) q^{3} +(0.830220 - 0.830220i) q^{4} +(0.621082 - 3.12239i) q^{5} +(-0.504894 + 0.755628i) q^{6} +(-2.11985 - 1.58311i) q^{7} +(2.66501 + 1.10388i) q^{8} +(-0.382683 + 0.923880i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.347777	+	0.839608i	0.245916	+	0.593693i	0.997850	−	0.0655456i	\(-0.0208788\pi\)
							−0.751934	+	0.659238i	\(0.770879\pi\)
\(3\)	0.555570	+	0.831470i	0.320759	+	0.480049i
							\(5\)	0.621082	−	3.12239i	0.277756	−	1.39637i	−0.549943	−	0.835202i	\(-0.685351\pi\)
0.827699	−	0.561172i	\(-0.189649\pi\)
\(7\)	−2.11985	−	1.58311i	−0.801227	−	0.598360i
							\(11\)	0.0455245	−	0.0681322i	0.0137262	−	0.0205426i	−0.824543	−	0.565799i	\(-0.808568\pi\)
0.838269	+	0.545257i	\(0.183568\pi\)
\(13\)	1.47357	−	1.47357i	0.408696	−	0.408696i	−0.472588	−	0.881284i	\(-0.656680\pi\)
							0.881284	+	0.472588i	\(0.156680\pi\)
\(17\)	−4.09935	+	0.441955i	−0.994239	+	0.107190i
							\(19\)	3.94512	−	1.63412i	0.905073	−	0.374894i	0.118905	−	0.992906i	\(-0.462062\pi\)
0.786169	+	0.618012i	\(0.212062\pi\)
\(23\)	5.97005	+	3.98906i	1.24484	+	0.831777i	0.990789	−	0.135414i	\(-0.0432364\pi\)
							0.254053	+	0.967190i	\(0.418236\pi\)
\(29\)	−1.49458	+	7.51374i	−0.277536	+	1.39527i	0.550614	+	0.834760i	\(0.314393\pi\)
							−0.828150	+	0.560506i	\(0.810607\pi\)
\(31\)	−4.92595	+	3.29142i	−0.884728	+	0.591156i	−0.912781	−	0.408450i	\(-0.866069\pi\)
							0.0280530	+	0.999606i	\(0.491069\pi\)
\(37\)	−6.55734	+	4.38148i	−1.07802	+	0.720310i	−0.962031	−	0.272940i	\(-0.912004\pi\)
							−0.115989	+	0.993250i	\(0.537004\pi\)
\(41\)	2.34281	+	11.7781i	0.365885	+	1.83943i	0.523626	+	0.851948i	\(0.324579\pi\)
							−0.157741	+	0.987481i	\(0.550421\pi\)
\(43\)	4.42836	−	10.6910i	0.675319	−	1.63036i	−0.0971193	−	0.995273i	\(-0.530963\pi\)
							0.772438	−	0.635090i	\(-0.219037\pi\)
\(47\)	−3.53202	+	3.53202i	−0.515198	+	0.515198i	−0.916115	−	0.400917i	\(-0.868692\pi\)
							0.400917	+	0.916115i	\(0.368692\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	357.2.bc.a.97.16	yes	192
7.6	odd	2	inner	357.2.bc.a.97.15	✓	192
17.10	odd	16	inner	357.2.bc.a.265.15	yes	192
119.27	even	16	inner	357.2.bc.a.265.16	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.bc.a.97.15	✓	192	7.6	odd	2	inner
357.2.bc.a.97.16	yes	192	1.1	even	1	trivial
357.2.bc.a.265.15	yes	192	17.10	odd	16	inner
357.2.bc.a.265.16	yes	192	119.27	even	16	inner