Embedded newform 471.2.s.a.2.19

• Label: 471.2.s.a.2.19
• Level: $471$
• Weight: $2$
• Character: 471.2
• Analytic conductor: $3.761$
• Analytic rank: $0$
• Dimension: $1200$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	471.s (of order \(52\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.76095393520\)
Analytic rank:	\(0\)
Dimension:	\(1200\)
Relative dimension:	\(50\) over \(\Q(\zeta_{52})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{52}]$

Embedding invariants

Embedding label			2.19
Character	\(\chi\)	\(=\)	471.2
Dual form			471.2.s.a.236.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.932577 + 0.170901i) q^{2} +(1.58987 + 0.687239i) q^{3} +(-1.02954 + 0.390453i) q^{4} +(-0.457736 - 1.46893i) q^{5} +(-1.60013 - 0.369192i) q^{6} +(-2.14157 + 0.963841i) q^{7} +(2.51614 - 1.52106i) q^{8} +(2.05540 + 2.18525i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q - 26 q^{3} - 52 q^{4} - 20 q^{6} - 40 q^{7} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/471\mathbb{Z}\right)^\times\).

\(n\)	\(158\)	\(319\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{47}{52}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.932577	+	0.170901i	−0.659432	+	0.120845i	−0.499542	−	0.866290i	\(-0.666498\pi\)
							−0.159889	+	0.987135i	\(0.551114\pi\)
\(3\)	1.58987	+	0.687239i	0.917915	+	0.396778i
							\(5\)	−0.457736	−	1.46893i	−0.204706	−	0.656924i	−0.998793	−	0.0491201i	\(-0.984358\pi\)
0.794087	−	0.607804i	\(-0.207949\pi\)
\(7\)	−2.14157	+	0.963841i	−0.809437	+	0.364298i	−0.772496	−	0.635020i	\(-0.780992\pi\)
							−0.0369406	+	0.999317i	\(0.511761\pi\)
\(11\)	2.73094	−	1.88503i	0.823411	−	0.568359i	−0.0801681	−	0.996781i	\(-0.525546\pi\)
							0.903579	+	0.428422i	\(0.140930\pi\)
\(13\)			4.41374i			1.22415i	0.790799	+	0.612076i	\(0.209665\pi\)
							−0.790799	+	0.612076i	\(0.790335\pi\)
\(17\)	−2.45803	+	4.68339i	−0.596161	+	1.13589i	0.381237	+	0.924478i	\(0.375498\pi\)
							−0.977397	+	0.211412i	\(0.932194\pi\)
\(19\)	6.97923	+	1.72023i	1.60115	+	0.394647i	0.935981	−	0.352050i	\(-0.114515\pi\)
							0.665164	+	0.746697i	\(0.268362\pi\)
\(23\)	−0.543371	+	8.98299i	−0.113301	+	1.87308i	0.288668	+	0.957429i	\(0.406788\pi\)
							−0.401969	+	0.915653i	\(0.631674\pi\)
\(29\)	−1.62957	−	0.507794i	−0.302603	−	0.0942949i	0.142442	−	0.989803i	\(-0.454505\pi\)
							−0.445045	+	0.895508i	\(0.646812\pi\)
\(31\)	−5.68050	−	0.689737i	−1.02025	−	0.123880i	−0.406719	−	0.913553i	\(-0.633327\pi\)
							−0.613528	+	0.789673i	\(0.710250\pi\)
\(37\)	2.07502	−	1.08906i	0.341132	−	0.179040i	−0.285454	−	0.958392i	\(-0.592144\pi\)
							0.626585	+	0.779353i	\(0.284452\pi\)
\(41\)	−0.570397	−	9.42978i	−0.0890810	−	1.47268i	−0.718450	−	0.695578i	\(-0.755148\pi\)
							0.629369	−	0.777106i	\(-0.283313\pi\)
\(43\)	1.42751	+	0.642468i	0.217693	+	0.0979755i	0.516325	−	0.856393i	\(-0.327300\pi\)
							−0.298632	+	0.954368i	\(0.596530\pi\)
\(47\)	−0.506108	+	0.964308i	−0.0738235	+	0.140659i	−0.919570	−	0.392926i	\(-0.871463\pi\)
							0.845747	+	0.533584i	\(0.179155\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.s.a.2.19	✓	1200
3.2	odd	2	inner	471.2.s.a.2.32	yes	1200
157.79	odd	52	inner	471.2.s.a.236.32	yes	1200
471.236	even	52	inner	471.2.s.a.236.19	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.s.a.2.19	✓	1200	1.1	even	1	trivial
471.2.s.a.2.32	yes	1200	3.2	odd	2	inner
471.2.s.a.236.19	yes	1200	471.236	even	52	inner
471.2.s.a.236.32	yes	1200	157.79	odd	52	inner