Embedded newform 357.2.bl.a.10.5

• Label: 357.2.bl.a.10.5
• 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			10.5
Character	\(\chi\)	\(=\)	357.10
Dual form			357.2.bl.a.250.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.79328 + 0.236090i) q^{2} +(0.442289 + 0.896873i) q^{3} +(1.22827 - 0.329115i) q^{4} +(0.723589 - 0.634571i) q^{5} +(-1.00489 - 1.50393i) q^{6} +(-2.30150 - 1.30502i) q^{7} +(1.21721 - 0.504184i) q^{8} +(-0.608761 + 0.793353i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.79328	+	0.236090i	−1.26804	+	0.166941i	−0.734369	−	0.678750i	\(-0.762522\pi\)
							−0.533673	+	0.845691i	\(0.679189\pi\)
\(3\)	0.442289	+	0.896873i	0.255355	+	0.517810i
							\(5\)	0.723589	−	0.634571i	0.323599	−	0.283789i	−0.482097	−	0.876118i	\(-0.660125\pi\)
0.805696	+	0.592329i	\(0.201791\pi\)
\(7\)	−2.30150	−	1.30502i	−0.869887	−	0.493251i
							\(11\)	2.35430	−	0.154309i	0.709850	−	0.0465260i	0.293799	−	0.955867i	\(-0.405080\pi\)
0.416051	+	0.909341i	\(0.363414\pi\)
\(13\)	−4.23736	−	4.23736i	−1.17523	−	1.17523i	−0.980945	−	0.194287i	\(-0.937761\pi\)
							−0.194287	−	0.980945i	\(-0.562239\pi\)
\(17\)	−1.17306	−	3.95271i	−0.284508	−	0.958674i
							\(19\)	−0.426740	−	3.24141i	−0.0979008	−	0.743630i	−0.968035	−	0.250815i	\(-0.919301\pi\)
0.870134	−	0.492815i	\(-0.164032\pi\)
\(23\)	5.05716	+	2.49392i	1.05449	+	0.520018i	0.885037	−	0.465520i	\(-0.154133\pi\)
							0.169455	+	0.985538i	\(0.445799\pi\)
\(29\)	−0.871966	−	4.38367i	−0.161920	−	0.814027i	−0.973305	−	0.229515i	\(-0.926286\pi\)
							0.811385	−	0.584512i	\(-0.198714\pi\)
\(31\)	7.42551	−	3.66186i	1.33366	−	0.657689i	0.371060	−	0.928609i	\(-0.378995\pi\)
							0.962602	+	0.270920i	\(0.0873279\pi\)
\(37\)	0.411484	−	6.27804i	0.0676476	−	1.03210i	−0.820233	−	0.572029i	\(-0.806156\pi\)
							0.887881	−	0.460073i	\(-0.152177\pi\)
\(41\)	1.15543	−	5.80875i	0.180448	−	0.907174i	−0.779373	−	0.626561i	\(-0.784462\pi\)
							0.959821	−	0.280613i	\(-0.0905379\pi\)
\(43\)	−3.88360	−	9.37584i	−0.592243	−	1.42980i	−0.881331	−	0.472499i	\(-0.843352\pi\)
							0.289088	−	0.957303i	\(-0.406648\pi\)
\(47\)	1.23309	−	4.60194i	0.179864	−	0.671263i	−0.815808	−	0.578324i	\(-0.803707\pi\)
							0.995672	−	0.0929391i	\(-0.0296262\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	357.2.bl.a.10.5	✓	384
7.5	odd	6	inner	357.2.bl.a.61.20	yes	384
17.12	odd	16	inner	357.2.bl.a.199.20	yes	384
119.12	even	48	inner	357.2.bl.a.250.5	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.bl.a.10.5	✓	384	1.1	even	1	trivial
357.2.bl.a.61.20	yes	384	7.5	odd	6	inner
357.2.bl.a.199.20	yes	384	17.12	odd	16	inner
357.2.bl.a.250.5	yes	384	119.12	even	48	inner