Embedded newform 357.2.bl.a.10.16

• Label: 357.2.bl.a.10.16
• 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			10.16
Character	\(\chi\)	\(=\)	357.10
Dual form			357.2.bl.a.250.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.03243 - 0.135922i) q^{2} +(0.442289 + 0.896873i) q^{3} +(-0.884410 + 0.236977i) q^{4} +(-2.20366 + 1.93256i) q^{5} +(0.578538 + 0.865843i) q^{6} +(2.58525 + 0.562582i) q^{7} +(-2.80503 + 1.16188i) q^{8} +(-0.608761 + 0.793353i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.03243	−	0.135922i	0.730040	−	0.0961116i	0.243662	−	0.969860i	\(-0.421651\pi\)
							0.486378	+	0.873749i	\(0.338318\pi\)
\(3\)	0.442289	+	0.896873i	0.255355	+	0.517810i
							\(5\)	−2.20366	+	1.93256i	−0.985508	+	0.864267i	−0.990747	−	0.135719i	\(-0.956666\pi\)
0.00523945	+	0.999986i	\(0.498332\pi\)
\(7\)	2.58525	+	0.562582i	0.977131	+	0.212636i
							\(11\)	−5.38938	+	0.353239i	−1.62496	+	0.106506i	−0.850450	−	0.526056i	\(-0.823670\pi\)
−0.774510	+	0.632561i	\(0.782004\pi\)
\(13\)	1.52303	+	1.52303i	0.422413	+	0.422413i	0.886034	−	0.463621i	\(-0.153450\pi\)
							−0.463621	+	0.886034i	\(0.653450\pi\)
\(17\)	0.240895	−	4.11606i	0.0584255	−	0.998292i
							\(19\)	0.551372	+	4.18808i	0.126493	+	0.960812i	0.930526	+	0.366227i	\(0.119351\pi\)
−0.804032	+	0.594586i	\(0.797316\pi\)
\(23\)	5.63769	+	2.78020i	1.17554	+	0.579712i	0.921840	−	0.387571i	\(-0.126686\pi\)
							0.253699	+	0.967283i	\(0.418353\pi\)
\(29\)	1.80665	+	9.08263i	0.335486	+	1.68660i	0.668526	+	0.743689i	\(0.266926\pi\)
							−0.333040	+	0.942913i	\(0.608074\pi\)
\(31\)	5.24394	−	2.58602i	0.941839	−	0.464463i	0.0945051	−	0.995524i	\(-0.469873\pi\)
							0.847334	+	0.531061i	\(0.178206\pi\)
\(37\)	−0.257178	+	3.92377i	−0.0422797	+	0.645064i	0.923156	+	0.384424i	\(0.125600\pi\)
							−0.965436	+	0.260640i	\(0.916067\pi\)
\(41\)	0.761744	−	3.82955i	0.118964	−	0.598075i	−0.874604	−	0.484838i	\(-0.838878\pi\)
							0.993568	−	0.113236i	\(-0.0361216\pi\)
\(43\)	−1.20385	−	2.90635i	−0.183585	−	0.443214i	0.805115	−	0.593118i	\(-0.202103\pi\)
							−0.988700	+	0.149905i	\(0.952103\pi\)
\(47\)	2.29640	−	8.57027i	0.334964	−	1.25010i	−0.568945	−	0.822376i	\(-0.692648\pi\)
							0.903908	−	0.427726i	\(-0.140685\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	357.2.bl.a.10.16	✓	384
7.5	odd	6	inner	357.2.bl.a.61.9	yes	384
17.12	odd	16	inner	357.2.bl.a.199.9	yes	384
119.12	even	48	inner	357.2.bl.a.250.16	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.bl.a.10.16	✓	384	1.1	even	1	trivial
357.2.bl.a.61.9	yes	384	7.5	odd	6	inner
357.2.bl.a.199.9	yes	384	17.12	odd	16	inner
357.2.bl.a.250.16	yes	384	119.12	even	48	inner