Embedded newform 471.2.s.a.2.6

• Label: 471.2.s.a.2.6
• 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			2.6
Character	\(\chi\)	\(=\)	471.2
Dual form			471.2.s.a.236.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.25488 + 0.413222i) q^{2} +(-1.68169 + 0.414612i) q^{3} +(3.04369 - 1.15432i) q^{4} +(0.325864 + 1.04573i) q^{5} +(3.62069 - 1.62981i) q^{6} +(2.00200 - 0.901027i) q^{7} +(-2.46252 + 1.48865i) q^{8} +(2.65619 - 1.39450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q - 26 q^{3} - 52 q^{4} - 20 q^{6} - 40 q^{7} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.25488	+	0.413222i	−1.59444	+	0.292192i	−0.902803	−	0.430054i	\(-0.858494\pi\)
							−0.691636	+	0.722246i	\(0.743110\pi\)
\(3\)	−1.68169	+	0.414612i	−0.970927	+	0.239377i
							\(5\)	0.325864	+	1.04573i	0.145731	+	0.467667i	0.998508	−	0.0546084i	\(-0.0173910\pi\)
−0.852777	+	0.522275i	\(0.825083\pi\)
\(7\)	2.00200	−	0.901027i	0.756685	−	0.340556i	0.00492018	−	0.999988i	\(-0.498434\pi\)
							0.751764	+	0.659432i	\(0.229203\pi\)
\(11\)	−2.60663	+	1.79923i	−0.785930	+	0.542488i	−0.892088	−	0.451861i	\(-0.850760\pi\)
							0.106159	+	0.994349i	\(0.466145\pi\)
\(13\)		−	1.72680i		−	0.478929i	−0.970905	−	0.239464i	\(-0.923028\pi\)
							0.970905	−	0.239464i	\(-0.0769718\pi\)
\(17\)	2.21771	−	4.22549i	0.537874	−	1.02483i	−0.453186	−	0.891416i	\(-0.649713\pi\)
							0.991060	−	0.133417i	\(-0.0425949\pi\)
\(19\)	−6.69545	−	1.65028i	−1.53604	−	0.378600i	−0.621605	−	0.783331i	\(-0.713519\pi\)
							−0.914436	+	0.404730i	\(0.867365\pi\)
\(23\)	−0.0748603	+	1.23759i	−0.0156095	+	0.258055i	0.981996	+	0.188900i	\(0.0604920\pi\)
							−0.997606	+	0.0691556i	\(0.977969\pi\)
\(29\)	−4.21972	−	1.31492i	−0.783582	−	0.244174i	−0.120493	−	0.992714i	\(-0.538447\pi\)
							−0.663089	+	0.748540i	\(0.730755\pi\)
\(31\)	6.52788	+	0.792628i	1.17244	+	0.142360i	0.683496	−	0.729954i	\(-0.260459\pi\)
							0.488946	+	0.872314i	\(0.337382\pi\)
\(37\)	1.98717	−	1.04295i	0.326689	−	0.171460i	−0.293403	−	0.955989i	\(-0.594788\pi\)
							0.620092	+	0.784529i	\(0.287095\pi\)
\(41\)	−0.506170	−	8.36799i	−0.0790505	−	1.30686i	−0.794100	−	0.607787i	\(-0.792058\pi\)
							0.715050	−	0.699073i	\(-0.246404\pi\)
\(43\)	8.00906	+	3.60459i	1.22137	+	0.549694i	0.915477	−	0.402371i	\(-0.131814\pi\)
							0.305894	+	0.952066i	\(0.401045\pi\)
\(47\)	4.40524	−	8.39347i	0.642570	−	1.22431i	−0.318478	−	0.947930i	\(-0.603172\pi\)
							0.961047	−	0.276384i	\(-0.0891361\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.s.a.2.6	✓	1200
3.2	odd	2	inner	471.2.s.a.2.45	yes	1200
157.79	odd	52	inner	471.2.s.a.236.45	yes	1200
471.236	even	52	inner	471.2.s.a.236.6	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.s.a.2.6	✓	1200	1.1	even	1	trivial
471.2.s.a.2.45	yes	1200	3.2	odd	2	inner
471.2.s.a.236.6	yes	1200	471.236	even	52	inner
471.2.s.a.236.45	yes	1200	157.79	odd	52	inner