Embedded newform 357.2.bl.a.10.22

• Label: 357.2.bl.a.10.22
• Level: $357$
• Weight: $2$
• Character: 357.10
• Analytic conductor: $2.851$
• Analytic rank: $0$
• Dimension: $384$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 357 = 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	357.bl (of order \(48\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.85065935216\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(24\) over \(\Q(\zeta_{48})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{48}]$

Embedding invariants

Embedding label			10.22
Character	\(\chi\)	\(=\)	357.10
Dual form			357.2.bl.a.250.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.19456 - 0.288919i) q^{2} +(-0.442289 - 0.896873i) q^{3} +(2.80077 - 0.750463i) q^{4} +(1.81750 - 1.59391i) q^{5} +(-1.22975 - 1.84045i) q^{6} +(-2.53809 + 0.747063i) q^{7} +(1.83962 - 0.761994i) q^{8} +(-0.608761 + 0.793353i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q+O(q^{10}) \)

Character values

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

\(n\)	\(52\)	\(190\)	\(239\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{16}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.19456	−	0.288919i	1.55179	−	0.204297i	0.694636	−	0.719361i	\(-0.255565\pi\)
							0.857152	+	0.515064i	\(0.172232\pi\)
\(3\)	−0.442289	−	0.896873i	−0.255355	−	0.517810i
							\(5\)	1.81750	−	1.59391i	0.812811	−	0.712817i	−0.148310	−	0.988941i	\(-0.547383\pi\)
0.961122	+	0.276124i	\(0.0890501\pi\)
\(7\)	−2.53809	+	0.747063i	−0.959308	+	0.282363i
							\(11\)	6.12220	−	0.401270i	1.84591	−	0.120987i	0.898095	−	0.439802i	\(-0.144952\pi\)
0.947817	+	0.318815i	\(0.103285\pi\)
\(13\)	−1.93919	−	1.93919i	−0.537834	−	0.537834i	0.385058	−	0.922892i	\(-0.374181\pi\)
							−0.922892	+	0.385058i	\(0.874181\pi\)
\(17\)	−1.67081	+	3.76940i	−0.405232	+	0.914214i
							\(19\)	−0.168515	−	1.28000i	−0.0386599	−	0.293651i	−0.999843	−	0.0177386i	\(-0.994353\pi\)
0.961183	−	0.275912i	\(-0.0889800\pi\)
\(23\)	−1.19810	−	0.590837i	−0.249821	−	0.123198i	0.313068	−	0.949731i	\(-0.398643\pi\)
							−0.562889	+	0.826533i	\(0.690310\pi\)
\(29\)	0.988225	+	4.96814i	0.183509	+	0.922561i	0.957295	+	0.289114i	\(0.0933606\pi\)
							−0.773786	+	0.633447i	\(0.781639\pi\)
\(31\)	0.796154	−	0.392620i	0.142994	−	0.0705166i	−0.369384	−	0.929277i	\(-0.620431\pi\)
							0.512378	+	0.858760i	\(0.328765\pi\)
\(37\)	−0.491054	+	7.49203i	−0.0807287	+	1.23168i	0.744787	+	0.667303i	\(0.232551\pi\)
							−0.825515	+	0.564380i	\(0.809115\pi\)
\(41\)	0.303942	−	1.52802i	0.0474678	−	0.238637i	−0.949766	−	0.312960i	\(-0.898680\pi\)
							0.997234	+	0.0743229i	\(0.0236795\pi\)
\(43\)	0.638340	+	1.54109i	0.0973460	+	0.235014i	0.965049	−	0.262068i	\(-0.0844046\pi\)
							−0.867703	+	0.497083i	\(0.834405\pi\)
\(47\)	1.49827	−	5.59161i	0.218545	−	0.815620i	−0.766344	−	0.642431i	\(-0.777926\pi\)
							0.984889	−	0.173189i	\(-0.0554072\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	357.2.bl.a.10.22	✓	384
7.5	odd	6	inner	357.2.bl.a.61.3	yes	384
17.12	odd	16	inner	357.2.bl.a.199.3	yes	384
119.12	even	48	inner	357.2.bl.a.250.22	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.bl.a.10.22	✓	384	1.1	even	1	trivial
357.2.bl.a.61.3	yes	384	7.5	odd	6	inner
357.2.bl.a.199.3	yes	384	17.12	odd	16	inner
357.2.bl.a.250.22	yes	384	119.12	even	48	inner