Embedded newform 357.2.bc.a.97.4

• Label: 357.2.bc.a.97.4
• 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			97.4
Character	\(\chi\)	\(=\)	357.97
Dual form			357.2.bc.a.265.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.760462 - 1.83592i) q^{2} +(0.555570 + 0.831470i) q^{3} +(-1.37808 + 1.37808i) q^{4} +(-0.0595354 + 0.299305i) q^{5} +(1.10402 - 1.65228i) q^{6} +(0.257547 + 2.63319i) q^{7} +(-0.0938205 - 0.0388617i) q^{8} +(-0.382683 + 0.923880i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.760462	−	1.83592i	−0.537728	−	1.29819i	−0.926305	−	0.376774i	\(-0.877033\pi\)
							0.388577	−	0.921416i	\(-0.372967\pi\)
\(3\)	0.555570	+	0.831470i	0.320759	+	0.480049i
							\(5\)	−0.0595354	+	0.299305i	−0.0266251	+	0.133853i	−0.991812	−	0.127709i	\(-0.959238\pi\)
0.965187	+	0.261562i	\(0.0842376\pi\)
\(7\)	0.257547	+	2.63319i	0.0973435	+	0.995251i
							\(11\)	2.92087	−	4.37138i	0.880674	−	1.31802i	−0.0666601	−	0.997776i	\(-0.521234\pi\)
0.947334	−	0.320246i	\(-0.103766\pi\)
\(13\)	2.55384	−	2.55384i	0.708306	−	0.708306i	−0.257872	−	0.966179i	\(-0.583021\pi\)
							0.966179	+	0.257872i	\(0.0830214\pi\)
\(17\)	4.08898	+	0.529346i	0.991724	+	0.128385i
							\(19\)	4.57834	−	1.89641i	1.05034	−	0.435067i	0.210329	−	0.977631i	\(-0.432546\pi\)
0.840014	+	0.542564i	\(0.182546\pi\)
\(23\)	1.06986	+	0.714860i	0.223082	+	0.149059i	0.662090	−	0.749425i	\(-0.269670\pi\)
							−0.439008	+	0.898483i	\(0.644670\pi\)
\(29\)	0.888116	−	4.46486i	0.164919	−	0.829104i	−0.806409	−	0.591359i	\(-0.798592\pi\)
							0.971328	−	0.237745i	\(-0.0764083\pi\)
\(31\)	−2.58234	+	1.72546i	−0.463802	+	0.309902i	−0.765425	−	0.643525i	\(-0.777471\pi\)
							0.301624	+	0.953427i	\(0.402471\pi\)
\(37\)	−8.65962	+	5.78617i	−1.42363	+	0.951241i	−0.424687	+	0.905340i	\(0.639616\pi\)
							−0.998946	+	0.0459009i	\(0.985384\pi\)
\(41\)	0.522920	+	2.62889i	0.0816663	+	0.410564i	0.999895	+	0.0145129i	\(0.00461977\pi\)
							−0.918228	+	0.396051i	\(0.870380\pi\)
\(43\)	0.300847	−	0.726308i	0.0458787	−	0.110761i	−0.899279	−	0.437376i	\(-0.855908\pi\)
							0.945158	+	0.326615i	\(0.105908\pi\)
\(47\)	−4.88184	+	4.88184i	−0.712090	+	0.712090i	−0.966972	−	0.254882i	\(-0.917963\pi\)
							0.254882	+	0.966972i	\(0.417963\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	357.2.bc.a.97.4	yes	192
7.6	odd	2	inner	357.2.bc.a.97.3	✓	192
17.10	odd	16	inner	357.2.bc.a.265.3	yes	192
119.27	even	16	inner	357.2.bc.a.265.4	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.bc.a.97.3	✓	192	7.6	odd	2	inner
357.2.bc.a.97.4	yes	192	1.1	even	1	trivial
357.2.bc.a.265.3	yes	192	17.10	odd	16	inner
357.2.bc.a.265.4	yes	192	119.27	even	16	inner