Embedded newform 471.2.s.a.2.17

• Label: 471.2.s.a.2.17
• 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			2.17
Character	\(\chi\)	\(=\)	471.2
Dual form			471.2.s.a.236.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16594 + 0.213666i) q^{2} +(-0.896153 + 1.48220i) q^{3} +(-0.556274 + 0.210967i) q^{4} +(0.649823 + 2.08535i) q^{5} +(0.728163 - 1.91963i) q^{6} +(3.50250 - 1.57635i) q^{7} +(2.63231 - 1.59129i) q^{8} +(-1.39382 - 2.65655i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q - 26 q^{3} - 52 q^{4} - 20 q^{6} - 40 q^{7} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16594	+	0.213666i	−0.824443	+	0.151085i	−0.575883	−	0.817532i	\(-0.695342\pi\)
							−0.248559	+	0.968617i	\(0.579957\pi\)
\(3\)	−0.896153	+	1.48220i	−0.517394	+	0.855747i
							\(5\)	0.649823	+	2.08535i	0.290610	+	0.932599i	0.978405	+	0.206695i	\(0.0662707\pi\)
−0.687796	+	0.725904i	\(0.741422\pi\)
\(7\)	3.50250	−	1.57635i	1.32382	−	0.595803i	0.379609	−	0.925147i	\(-0.376058\pi\)
							0.944210	+	0.329344i	\(0.106828\pi\)
\(11\)	3.22684	−	2.22732i	0.972928	−	0.671564i	0.0282629	−	0.999601i	\(-0.491002\pi\)
							0.944665	+	0.328037i	\(0.106387\pi\)
\(13\)		−	5.27919i		−	1.46418i	−0.681205	−	0.732092i	\(-0.738544\pi\)
							0.681205	−	0.732092i	\(-0.261456\pi\)
\(17\)	−0.389814	+	0.742729i	−0.0945438	+	0.180138i	−0.928135	−	0.372244i	\(-0.878589\pi\)
							0.833591	+	0.552382i	\(0.186281\pi\)
\(19\)	3.63599	+	0.896190i	0.834152	+	0.205600i	0.633188	−	0.773998i	\(-0.281746\pi\)
							0.200965	+	0.979598i	\(0.435592\pi\)
\(23\)	0.114216	−	1.88821i	0.0238156	−	0.393720i	−0.966663	−	0.256052i	\(-0.917578\pi\)
							0.990479	−	0.137667i	\(-0.0439604\pi\)
\(29\)	0.915725	+	0.285351i	0.170046	+	0.0529884i	0.381344	−	0.924433i	\(-0.375461\pi\)
							−0.211298	+	0.977422i	\(0.567769\pi\)
\(31\)	−0.228622	−	0.0277597i	−0.0410617	−	0.00498580i	0.0999800	−	0.994989i	\(-0.468122\pi\)
							−0.141042	+	0.990004i	\(0.545045\pi\)
\(37\)	7.23557	−	3.79752i	1.18952	−	0.624308i	0.250482	−	0.968121i	\(-0.419411\pi\)
							0.939038	+	0.343813i	\(0.111719\pi\)
\(41\)	−0.254912	−	4.21420i	−0.0398106	−	0.658147i	−0.962277	−	0.272071i	\(-0.912292\pi\)
							0.922467	−	0.386077i	\(-0.126170\pi\)
\(43\)	−7.27245	−	3.27307i	−1.10904	−	0.499138i	−0.228707	−	0.973495i	\(-0.573450\pi\)
							−0.880332	+	0.474358i	\(0.842680\pi\)
\(47\)	−2.61568	+	4.98375i	−0.381536	+	0.726955i	−0.998181	−	0.0602856i	\(-0.980799\pi\)
							0.616646	+	0.787241i	\(0.288491\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.s.a.2.17	✓	1200
3.2	odd	2	inner	471.2.s.a.2.34	yes	1200
157.79	odd	52	inner	471.2.s.a.236.34	yes	1200
471.236	even	52	inner	471.2.s.a.236.17	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.s.a.2.17	✓	1200	1.1	even	1	trivial
471.2.s.a.2.34	yes	1200	3.2	odd	2	inner
471.2.s.a.236.17	yes	1200	471.236	even	52	inner
471.2.s.a.236.34	yes	1200	157.79	odd	52	inner