Embedded newform 471.2.s.a.2.20

• Label: 471.2.s.a.2.20
• 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			2.20
Character	\(\chi\)	\(=\)	471.2
Dual form			471.2.s.a.236.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.467978 + 0.0857602i) q^{2} +(-0.751208 + 1.56067i) q^{3} +(-1.65838 + 0.628942i) q^{4} +(-0.232282 - 0.745421i) q^{5} +(0.217706 - 0.794782i) q^{6} +(-3.20493 + 1.44242i) q^{7} +(1.53646 - 0.928823i) q^{8} +(-1.87137 - 2.34477i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q - 26 q^{3} - 52 q^{4} - 20 q^{6} - 40 q^{7} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.467978	+	0.0857602i	−0.330910	+	0.0606416i	−0.343142	−	0.939284i	\(-0.611491\pi\)
							0.0122314	+	0.999925i	\(0.496107\pi\)
\(3\)	−0.751208	+	1.56067i	−0.433710	+	0.901052i
							\(5\)	−0.232282	−	0.745421i	−0.103880	−	0.333362i	0.888118	−	0.459615i	\(-0.152013\pi\)
−0.991998	+	0.126253i	\(0.959705\pi\)
\(7\)	−3.20493	+	1.44242i	−1.21135	+	0.545184i	−0.912484	−	0.409111i	\(-0.865839\pi\)
							−0.298865	+	0.954295i	\(0.596608\pi\)
\(11\)	0.897960	−	0.619817i	0.270745	−	0.186882i	−0.424954	−	0.905215i	\(-0.639710\pi\)
							0.695700	+	0.718333i	\(0.255095\pi\)
\(13\)			2.73063i			0.757341i	0.925532	+	0.378671i	\(0.123619\pi\)
							−0.925532	+	0.378671i	\(0.876381\pi\)
\(17\)	2.81604	−	5.36551i	0.682989	−	1.30133i	−0.259063	−	0.965860i	\(-0.583414\pi\)
							0.942052	−	0.335466i	\(-0.108894\pi\)
\(19\)	−4.24766	−	1.04696i	−0.974481	−	0.240188i	−0.280239	−	0.959930i	\(-0.590414\pi\)
							−0.694241	+	0.719742i	\(0.744260\pi\)
\(23\)	−0.0940101	+	1.55417i	−0.0196025	+	0.324067i	0.975135	+	0.221611i	\(0.0711316\pi\)
							−0.994738	+	0.102456i	\(0.967330\pi\)
\(29\)	1.74138	+	0.542636i	0.323366	+	0.100765i	0.454889	−	0.890548i	\(-0.349679\pi\)
							−0.131523	+	0.991313i	\(0.541987\pi\)
\(31\)	−3.52167	−	0.427608i	−0.632510	−	0.0768007i	−0.201999	−	0.979386i	\(-0.564744\pi\)
							−0.430512	+	0.902585i	\(0.641667\pi\)
\(37\)	6.61632	−	3.47251i	1.08772	−	0.570877i	0.177126	−	0.984188i	\(-0.443320\pi\)
							0.910590	+	0.413311i	\(0.135628\pi\)
\(41\)	−0.214769	−	3.55055i	−0.0335412	−	0.554503i	−0.975669	−	0.219249i	\(-0.929639\pi\)
							0.942128	−	0.335254i	\(-0.108822\pi\)
\(43\)	−5.60993	−	2.52483i	−0.855507	−	0.385032i	−0.0652949	−	0.997866i	\(-0.520799\pi\)
							−0.790212	+	0.612834i	\(0.790030\pi\)
\(47\)	5.93764	−	11.3132i	0.866094	−	1.65020i	0.111740	−	0.993738i	\(-0.464358\pi\)
							0.754355	−	0.656467i	\(-0.227950\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.s.a.2.20	✓	1200
3.2	odd	2	inner	471.2.s.a.2.31	yes	1200
157.79	odd	52	inner	471.2.s.a.236.31	yes	1200
471.236	even	52	inner	471.2.s.a.236.20	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.s.a.2.20	✓	1200	1.1	even	1	trivial
471.2.s.a.2.31	yes	1200	3.2	odd	2	inner
471.2.s.a.236.20	yes	1200	471.236	even	52	inner
471.2.s.a.236.31	yes	1200	157.79	odd	52	inner