Embedded newform 357.2.bc.a.97.12

• Label: 357.2.bc.a.97.12
• 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			97.12
Character	\(\chi\)	\(=\)	357.97
Dual form			357.2.bc.a.265.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0508407 - 0.122740i) q^{2} +(0.555570 + 0.831470i) q^{3} +(1.40173 - 1.40173i) q^{4} +(-0.498382 + 2.50553i) q^{5} +(0.0738092 - 0.110463i) q^{6} +(0.645550 + 2.56579i) q^{7} +(-0.488795 - 0.202465i) q^{8} +(-0.382683 + 0.923880i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0508407	−	0.122740i	−0.0359498	−	0.0867905i	0.904885	−	0.425656i	\(-0.139957\pi\)
							−0.940835	+	0.338866i	\(0.889957\pi\)
\(3\)	0.555570	+	0.831470i	0.320759	+	0.480049i
							\(5\)	−0.498382	+	2.50553i	−0.222883	+	1.12051i	0.693577	+	0.720382i	\(0.256034\pi\)
−0.916460	+	0.400126i	\(0.868966\pi\)
\(7\)	0.645550	+	2.56579i	0.243995	+	0.969777i
							\(11\)	−2.51133	+	3.75848i	−0.757195	+	1.13322i	0.229919	+	0.973210i	\(0.426154\pi\)
−0.987115	+	0.160013i	\(0.948846\pi\)
\(13\)	−0.662569	+	0.662569i	−0.183764	+	0.183764i	−0.792994	−	0.609230i	\(-0.791479\pi\)
							0.609230	+	0.792994i	\(0.291479\pi\)
\(17\)	−1.63099	−	3.78680i	−0.395573	−	0.918434i
							\(19\)	4.60192	−	1.90618i	1.05575	−	0.437307i	0.213810	−	0.976875i	\(-0.431413\pi\)
0.841942	+	0.539568i	\(0.181413\pi\)
\(23\)	4.07184	+	2.72071i	0.849037	+	0.567308i	0.902220	−	0.431277i	\(-0.141937\pi\)
							−0.0531831	+	0.998585i	\(0.516937\pi\)
\(29\)	1.22866	−	6.17690i	0.228157	−	1.14702i	−0.681550	−	0.731771i	\(-0.738694\pi\)
							0.909707	−	0.415250i	\(-0.136306\pi\)
\(31\)	−4.46125	+	2.98091i	−0.801264	+	0.535387i	−0.887457	−	0.460891i	\(-0.847530\pi\)
							0.0861928	+	0.996278i	\(0.472530\pi\)
\(37\)	7.14700	−	4.77547i	1.17496	−	0.785083i	0.194327	−	0.980937i	\(-0.437748\pi\)
							0.980633	+	0.195854i	\(0.0627479\pi\)
\(41\)	1.27225	+	6.39602i	0.198692	+	0.998891i	0.943439	+	0.331545i	\(0.107570\pi\)
							−0.744748	+	0.667346i	\(0.767430\pi\)
\(43\)	2.88257	−	6.95914i	0.439588	−	1.06126i	−0.536503	−	0.843898i	\(-0.680255\pi\)
							0.976091	−	0.217361i	\(-0.0697449\pi\)
\(47\)	5.34728	−	5.34728i	0.779981	−	0.779981i	−0.199847	−	0.979827i	\(-0.564044\pi\)
							0.979827	+	0.199847i	\(0.0640444\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	357.2.bc.a.97.12	yes	192
7.6	odd	2	inner	357.2.bc.a.97.11	✓	192
17.10	odd	16	inner	357.2.bc.a.265.11	yes	192
119.27	even	16	inner	357.2.bc.a.265.12	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.bc.a.97.11	✓	192	7.6	odd	2	inner
357.2.bc.a.97.12	yes	192	1.1	even	1	trivial
357.2.bc.a.265.11	yes	192	17.10	odd	16	inner
357.2.bc.a.265.12	yes	192	119.27	even	16	inner