Embedded newform 471.2.s.a.2.15

• Label: 471.2.s.a.2.15
• 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			2.15
Character	\(\chi\)	\(=\)	471.2
Dual form			471.2.s.a.236.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22356 + 0.224226i) q^{2} +(1.58808 + 0.691369i) q^{3} +(-0.423212 + 0.160503i) q^{4} +(-0.208958 - 0.670569i) q^{5} +(-2.09814 - 0.489842i) q^{6} +(2.87237 - 1.29275i) q^{7} +(2.61091 - 1.57835i) q^{8} +(2.04402 + 2.19590i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q - 26 q^{3} - 52 q^{4} - 20 q^{6} - 40 q^{7} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22356	+	0.224226i	−0.865187	+	0.158551i	−0.594480	−	0.804110i	\(-0.702642\pi\)
							−0.270707	+	0.962662i	\(0.587257\pi\)
\(3\)	1.58808	+	0.691369i	0.916880	+	0.399162i
							\(5\)	−0.208958	−	0.670569i	−0.0934488	−	0.299888i	0.896148	−	0.443755i	\(-0.146354\pi\)
−0.989597	+	0.143867i	\(0.954046\pi\)
\(7\)	2.87237	−	1.29275i	1.08566	−	0.488614i	0.213051	−	0.977041i	\(-0.431660\pi\)
							0.872604	+	0.488427i	\(0.162429\pi\)
\(11\)	−1.27832	+	0.882359i	−0.385427	+	0.266041i	−0.744984	−	0.667082i	\(-0.767543\pi\)
							0.359557	+	0.933123i	\(0.382928\pi\)
\(13\)		−	1.43479i		−	0.397940i	−0.980006	−	0.198970i	\(-0.936240\pi\)
							0.980006	−	0.198970i	\(-0.0637597\pi\)
\(17\)	2.01869	−	3.84629i	0.489605	−	0.932864i	−0.507842	−	0.861450i	\(-0.669557\pi\)
							0.997447	−	0.0714134i	\(-0.0227510\pi\)
\(19\)	−2.45353	−	0.604740i	−0.562877	−	0.138737i	−0.0523972	−	0.998626i	\(-0.516686\pi\)
							−0.510480	+	0.859890i	\(0.670532\pi\)
\(23\)	0.0428733	−	0.708780i	0.00893970	−	0.147791i	−0.990967	−	0.134108i	\(-0.957183\pi\)
							0.999906	−	0.0136826i	\(-0.00435545\pi\)
\(29\)	7.63445	+	2.37899i	1.41768	+	0.441767i	0.908136	−	0.418676i	\(-0.137506\pi\)
							0.509546	+	0.860444i	\(0.329813\pi\)
\(31\)	6.53115	+	0.793026i	1.17303	+	0.142432i	0.683764	−	0.729703i	\(-0.260342\pi\)
							0.489266	+	0.872135i	\(0.337265\pi\)
\(37\)	−3.01617	+	1.58301i	−0.495855	+	0.260245i	−0.694062	−	0.719916i	\(-0.744180\pi\)
							0.198207	+	0.980160i	\(0.436488\pi\)
\(41\)	0.667736	+	11.0390i	0.104283	+	1.72400i	0.555439	+	0.831557i	\(0.312550\pi\)
							−0.451156	+	0.892445i	\(0.648988\pi\)
\(43\)	−4.37874	−	1.97071i	−0.667752	−	0.300531i	0.0480392	−	0.998845i	\(-0.484703\pi\)
							−0.715791	+	0.698315i	\(0.753934\pi\)
\(47\)	0.367072	−	0.699398i	0.0535430	−	0.102018i	−0.857214	−	0.514960i	\(-0.827807\pi\)
							0.910757	+	0.412943i	\(0.135499\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.s.a.2.15	✓	1200
3.2	odd	2	inner	471.2.s.a.2.36	yes	1200
157.79	odd	52	inner	471.2.s.a.236.36	yes	1200
471.236	even	52	inner	471.2.s.a.236.15	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.s.a.2.15	✓	1200	1.1	even	1	trivial
471.2.s.a.2.36	yes	1200	3.2	odd	2	inner
471.2.s.a.236.15	yes	1200	471.236	even	52	inner
471.2.s.a.236.36	yes	1200	157.79	odd	52	inner