Embedded newform 471.2.s.a.236.20

• Label: 471.2.s.a.236.20
• Level: $471$
• Weight: $2$
• Character: 471.236
• 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			236.20
Character	\(\chi\)	\(=\)	471.236
Dual form			471.2.s.a.2.20

$q$-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 \(\left(\mathbb{Z}/471\mathbb{Z}\right)^\times\).

\(n\)	\(158\)	\(319\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{52}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.s.a.2.20	✓	1200	471.2	even	52	inner
471.2.s.a.2.31	yes	1200	157.2	odd	52	inner
471.2.s.a.236.20	yes	1200	1.1	even	1	trivial
471.2.s.a.236.31	yes	1200	3.2	odd	2	inner