Embedded newform 471.2.s.a.2.2

• Label: 471.2.s.a.2.2
• 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			2.2
Character	\(\chi\)	\(=\)	471.2
Dual form			471.2.s.a.236.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.63103 + 0.482154i) q^{2} +(-1.15550 - 1.29028i) q^{3} +(4.81980 - 1.82791i) q^{4} +(1.16001 + 3.72259i) q^{5} +(3.66226 + 2.83763i) q^{6} +(-2.99849 + 1.34951i) q^{7} +(-7.22155 + 4.36558i) q^{8} +(-0.329644 + 2.98183i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q - 26 q^{3} - 52 q^{4} - 20 q^{6} - 40 q^{7} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.63103	+	0.482154i	−1.86042	+	0.340934i	−0.988799	−	0.149251i	\(-0.952314\pi\)
							−0.871618	+	0.490185i	\(0.836929\pi\)
\(3\)	−1.15550	−	1.29028i	−0.667128	−	0.744943i
							\(5\)	1.16001	+	3.72259i	0.518770	+	1.66479i	0.726941	+	0.686700i	\(0.240941\pi\)
−0.208171	+	0.978092i	\(0.566751\pi\)
\(7\)	−2.99849	+	1.34951i	−1.13332	+	0.510067i	−0.888215	−	0.459428i	\(-0.848054\pi\)
							−0.245108	+	0.969496i	\(0.578824\pi\)
\(11\)	−0.541428	+	0.373720i	−0.163247	+	0.112681i	−0.646887	−	0.762586i	\(-0.723930\pi\)
							0.483641	+	0.875267i	\(0.339314\pi\)
\(13\)		−	1.25842i		−	0.349023i	−0.984655	−	0.174512i	\(-0.944165\pi\)
							0.984655	−	0.174512i	\(-0.0558347\pi\)
\(17\)	−2.67994	+	5.10620i	−0.649982	+	1.23844i	0.307940	+	0.951406i	\(0.400360\pi\)
							−0.957921	+	0.287031i	\(0.907332\pi\)
\(19\)	0.996162	+	0.245532i	0.228535	+	0.0563289i	0.351920	−	0.936030i	\(-0.385529\pi\)
							−0.123385	+	0.992359i	\(0.539375\pi\)
\(23\)	0.471286	−	7.79129i	0.0982700	−	1.62460i	−0.530973	−	0.847389i	\(-0.678174\pi\)
							0.629243	−	0.777208i	\(-0.283365\pi\)
\(29\)	−3.95855	−	1.23354i	−0.735085	−	0.229062i	−0.0929688	−	0.995669i	\(-0.529636\pi\)
							−0.642116	+	0.766607i	\(0.721943\pi\)
\(31\)	3.95707	+	0.480475i	0.710710	+	0.0862958i	0.467904	−	0.883779i	\(-0.345009\pi\)
							0.242806	+	0.970075i	\(0.421932\pi\)
\(37\)	3.87847	−	2.03558i	0.637617	−	0.334647i	−0.114747	−	0.993395i	\(-0.536606\pi\)
							0.752364	+	0.658747i	\(0.228913\pi\)
\(41\)	−0.745266	−	12.3207i	−0.116391	−	1.92417i	−0.324172	−	0.945998i	\(-0.605086\pi\)
							0.207781	−	0.978175i	\(-0.433376\pi\)
\(43\)	−3.67458	−	1.65379i	−0.560368	−	0.252201i	0.110377	−	0.993890i	\(-0.464794\pi\)
							−0.670744	+	0.741689i	\(0.734025\pi\)
\(47\)	−1.37521	+	2.62024i	−0.200595	+	0.382202i	−0.964886	−	0.262670i	\(-0.915397\pi\)
							0.764291	+	0.644872i	\(0.223089\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.s.a.2.2	✓	1200
3.2	odd	2	inner	471.2.s.a.2.49	yes	1200
157.79	odd	52	inner	471.2.s.a.236.49	yes	1200
471.236	even	52	inner	471.2.s.a.236.2	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.s.a.2.2	✓	1200	1.1	even	1	trivial
471.2.s.a.2.49	yes	1200	3.2	odd	2	inner
471.2.s.a.236.2	yes	1200	471.236	even	52	inner
471.2.s.a.236.49	yes	1200	157.79	odd	52	inner