Embedded newform 357.2.bl.a.10.15

• Label: 357.2.bl.a.10.15
• 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			10.15
Character	\(\chi\)	\(=\)	357.10
Dual form			357.2.bl.a.250.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.559246 - 0.0736262i) q^{2} +(-0.442289 - 0.896873i) q^{3} +(-1.62452 + 0.435288i) q^{4} +(1.94651 - 1.70704i) q^{5} +(-0.313382 - 0.469009i) q^{6} +(2.53814 - 0.746878i) q^{7} +(-1.91873 + 0.794762i) q^{8} +(-0.608761 + 0.793353i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.559246	−	0.0736262i	0.395447	−	0.0520616i	0.0698192	−	0.997560i	\(-0.477758\pi\)
							0.325628	+	0.945498i	\(0.394424\pi\)
\(3\)	−0.442289	−	0.896873i	−0.255355	−	0.517810i
							\(5\)	1.94651	−	1.70704i	0.870505	−	0.763413i	−0.102369	−	0.994746i	\(-0.532642\pi\)
0.972874	+	0.231334i	\(0.0743090\pi\)
\(7\)	2.53814	−	0.746878i	0.959328	−	0.282293i
							\(11\)	0.958076	−	0.0627956i	0.288871	−	0.0189336i	0.0797210	−	0.996817i	\(-0.474597\pi\)
0.209150	+	0.977884i	\(0.432930\pi\)
\(13\)	−3.77608	−	3.77608i	−1.04730	−	1.04730i	−0.998825	−	0.0484722i	\(-0.984565\pi\)
							−0.0484722	−	0.998825i	\(-0.515435\pi\)
\(17\)	3.07312	−	2.74881i	0.745341	−	0.666684i
							\(19\)	−0.417008	−	3.16749i	−0.0956682	−	0.726672i	−0.970304	−	0.241891i	\(-0.922233\pi\)
0.874635	−	0.484781i	\(-0.161101\pi\)
\(23\)	−3.58688	−	1.76885i	−0.747916	−	0.368831i	0.0280499	−	0.999607i	\(-0.491070\pi\)
							−0.775966	+	0.630775i	\(0.782737\pi\)
\(29\)	0.353142	+	1.77536i	0.0655768	+	0.329677i	0.999622	−	0.0274769i	\(-0.00874727\pi\)
							−0.934046	+	0.357154i	\(0.883747\pi\)
\(31\)	9.23067	−	4.55206i	1.65788	−	0.817574i	0.660083	−	0.751193i	\(-0.270521\pi\)
							0.997794	−	0.0663814i	\(-0.0211454\pi\)
\(37\)	−0.606978	+	9.26070i	−0.0997866	+	1.52245i	0.595288	+	0.803512i	\(0.297038\pi\)
							−0.695075	+	0.718938i	\(0.744629\pi\)
\(41\)	−1.08348	+	5.44704i	−0.169212	+	0.850685i	0.799149	+	0.601133i	\(0.205284\pi\)
							−0.968361	+	0.249553i	\(0.919716\pi\)
\(43\)	−0.368499	−	0.889635i	−0.0561955	−	0.135668i	0.893288	−	0.449485i	\(-0.148392\pi\)
							−0.949484	+	0.313817i	\(0.898392\pi\)
\(47\)	−3.22686	+	12.0428i	−0.470685	+	1.75662i	0.166631	+	0.986019i	\(0.446711\pi\)
							−0.637316	+	0.770602i	\(0.719956\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	357.2.bl.a.10.15	✓	384
7.5	odd	6	inner	357.2.bl.a.61.10	yes	384
17.12	odd	16	inner	357.2.bl.a.199.10	yes	384
119.12	even	48	inner	357.2.bl.a.250.15	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.bl.a.10.15	✓	384	1.1	even	1	trivial
357.2.bl.a.61.10	yes	384	7.5	odd	6	inner
357.2.bl.a.199.10	yes	384	17.12	odd	16	inner
357.2.bl.a.250.15	yes	384	119.12	even	48	inner