Embedded newform 357.2.bc.a.97.18

• Label: 357.2.bc.a.97.18
• 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			97.18
Character	\(\chi\)	\(=\)	357.97
Dual form			357.2.bc.a.265.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.572289 + 1.38163i) q^{2} +(0.555570 + 0.831470i) q^{3} +(-0.167165 + 0.167165i) q^{4} +(-0.137019 + 0.688842i) q^{5} +(-0.830834 + 1.24343i) q^{6} +(-0.830945 + 2.51188i) q^{7} +(2.43663 + 1.00928i) q^{8} +(-0.382683 + 0.923880i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.572289	+	1.38163i	0.404669	+	0.976958i	0.986517	+	0.163660i	\(0.0523300\pi\)
							−0.581848	+	0.813298i	\(0.697670\pi\)
\(3\)	0.555570	+	0.831470i	0.320759	+	0.480049i
							\(5\)	−0.137019	+	0.688842i	−0.0612768	+	0.308059i	−0.999253	−	0.0386398i	\(-0.987698\pi\)
0.937976	+	0.346699i	\(0.112698\pi\)
\(7\)	−0.830945	+	2.51188i	−0.314068	+	0.949401i
							\(11\)	2.42374	−	3.62738i	0.730785	−	1.09370i	−0.260945	−	0.965354i	\(-0.584034\pi\)
0.991730	−	0.128343i	\(-0.0409659\pi\)
\(13\)	−0.391603	+	0.391603i	−0.108611	+	0.108611i	−0.759324	−	0.650713i	\(-0.774470\pi\)
							0.650713	+	0.759324i	\(0.274470\pi\)
\(17\)	−3.90663	+	1.31843i	−0.947496	+	0.319766i
							\(19\)	−1.21740	+	0.504264i	−0.279291	+	0.115686i	−0.517932	−	0.855422i	\(-0.673298\pi\)
0.238641	+	0.971108i	\(0.423298\pi\)
\(23\)	−3.99487	−	2.66929i	−0.832988	−	0.556585i	0.0643505	−	0.997927i	\(-0.479502\pi\)
							−0.897339	+	0.441342i	\(0.854502\pi\)
\(29\)	0.209210	−	1.05177i	0.0388492	−	0.195308i	−0.956487	−	0.291775i	\(-0.905754\pi\)
							0.995336	+	0.0964667i	\(0.0307541\pi\)
\(31\)	6.43653	−	4.30075i	1.15603	−	0.772438i	0.178651	−	0.983912i	\(-0.442827\pi\)
							0.977383	+	0.211475i	\(0.0678266\pi\)
\(37\)	2.55922	−	1.71002i	0.420733	−	0.281125i	−0.327128	−	0.944980i	\(-0.606081\pi\)
							0.747861	+	0.663855i	\(0.231081\pi\)
\(41\)	−1.15221	−	5.79257i	−0.179945	−	0.904647i	−0.960227	−	0.279219i	\(-0.909924\pi\)
							0.780282	−	0.625428i	\(-0.215076\pi\)
\(43\)	−3.54832	+	8.56641i	−0.541114	+	1.30637i	0.382823	+	0.923822i	\(0.374952\pi\)
							−0.923937	+	0.382544i	\(0.875048\pi\)
\(47\)	1.95463	−	1.95463i	0.285112	−	0.285112i	−0.550032	−	0.835144i	\(-0.685384\pi\)
							0.835144	+	0.550032i	\(0.185384\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	357.2.bc.a.97.18	yes	192
7.6	odd	2	inner	357.2.bc.a.97.17	✓	192
17.10	odd	16	inner	357.2.bc.a.265.17	yes	192
119.27	even	16	inner	357.2.bc.a.265.18	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.bc.a.97.17	✓	192	7.6	odd	2	inner
357.2.bc.a.97.18	yes	192	1.1	even	1	trivial
357.2.bc.a.265.17	yes	192	17.10	odd	16	inner
357.2.bc.a.265.18	yes	192	119.27	even	16	inner