Embedded newform 471.2.s.a.2.4

• Label: 471.2.s.a.2.4
• 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			2.4
Character	\(\chi\)	\(=\)	471.2
Dual form			471.2.s.a.236.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.31579 + 0.424384i) q^{2} +(1.59880 - 0.666207i) q^{3} +(3.31274 - 1.25636i) q^{4} +(-0.0565505 - 0.181477i) q^{5} +(-3.41976 + 2.22130i) q^{6} +(-2.19753 + 0.989028i) q^{7} +(-3.10880 + 1.87934i) q^{8} +(2.11234 - 2.13027i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q - 26 q^{3} - 52 q^{4} - 20 q^{6} - 40 q^{7} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.31579	+	0.424384i	−1.63751	+	0.300085i	−0.918650	−	0.395071i	\(-0.870720\pi\)
							−0.718859	+	0.695156i	\(0.755335\pi\)
\(3\)	1.59880	−	0.666207i	0.923069	−	0.384635i
							\(5\)	−0.0565505	−	0.181477i	−0.0252902	−	0.0811590i	0.941213	−	0.337814i	\(-0.109687\pi\)
−0.966503	+	0.256655i	\(0.917380\pi\)
\(7\)	−2.19753	+	0.989028i	−0.830588	+	0.373817i	−0.780676	−	0.624936i	\(-0.785125\pi\)
							−0.0499124	+	0.998754i	\(0.515894\pi\)
\(11\)	−4.34900	+	3.00190i	−1.31127	+	0.905106i	−0.998992	−	0.0448842i	\(-0.985708\pi\)
							−0.312280	+	0.949990i	\(0.601093\pi\)
\(13\)			3.69288i			1.02422i	0.858920	+	0.512110i	\(0.171136\pi\)
							−0.858920	+	0.512110i	\(0.828864\pi\)
\(17\)	−3.14601	+	5.99423i	−0.763020	+	1.45381i	0.124097	+	0.992270i	\(0.460396\pi\)
							−0.887118	+	0.461543i	\(0.847296\pi\)
\(19\)	−3.97771	−	0.980416i	−0.912548	−	0.224923i	−0.245029	−	0.969516i	\(-0.578798\pi\)
							−0.667519	+	0.744593i	\(0.732644\pi\)
\(23\)	−0.151601	+	2.50626i	−0.0316109	+	0.522591i	0.947563	+	0.319569i	\(0.103538\pi\)
							−0.979174	+	0.203023i	\(0.934924\pi\)
\(29\)	9.21396	+	2.87119i	1.71099	+	0.533166i	0.987561	−	0.157234i	\(-0.0502578\pi\)
							0.723428	+	0.690400i	\(0.242565\pi\)
\(31\)	3.35487	+	0.407354i	0.602552	+	0.0731630i	0.416127	−	0.909306i	\(-0.363387\pi\)
							0.186424	+	0.982469i	\(0.440310\pi\)
\(37\)	−6.01339	+	3.15607i	−0.988595	+	0.518855i	−0.879869	−	0.475217i	\(-0.842370\pi\)
							−0.108727	+	0.994072i	\(0.534677\pi\)
\(41\)	0.364104	+	6.01936i	0.0568636	+	0.940067i	0.909013	+	0.416768i	\(0.136837\pi\)
							−0.852149	+	0.523299i	\(0.824701\pi\)
\(43\)	−2.75110	−	1.23817i	−0.419539	−	0.188819i	0.189637	−	0.981854i	\(-0.439269\pi\)
							−0.609176	+	0.793035i	\(0.708500\pi\)
\(47\)	2.68419	−	5.11429i	0.391529	−	0.745996i	−0.607295	−	0.794476i	\(-0.707745\pi\)
							0.998824	+	0.0484799i	\(0.0154377\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.s.a.2.4	✓	1200
3.2	odd	2	inner	471.2.s.a.2.47	yes	1200
157.79	odd	52	inner	471.2.s.a.236.47	yes	1200
471.236	even	52	inner	471.2.s.a.236.4	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.s.a.2.4	✓	1200	1.1	even	1	trivial
471.2.s.a.2.47	yes	1200	3.2	odd	2	inner
471.2.s.a.236.4	yes	1200	471.236	even	52	inner
471.2.s.a.236.47	yes	1200	157.79	odd	52	inner