Embedded newform 560.2.cp.a.221.57

• Label: 560.2.cp.a.221.57
• Level: $560$
• Weight: $2$
• Character: 560.221
• Analytic conductor: $4.472$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	560.cp (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.47162251319\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(64\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			221.57
Character	\(\chi\)	\(=\)	560.221
Dual form			560.2.cp.a.261.57

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.29901 + 0.559084i) q^{2} +(-0.472296 + 0.126551i) q^{3} +(1.37485 + 1.45251i) q^{4} +(-0.965926 - 0.258819i) q^{5} +(-0.684270 - 0.0996620i) q^{6} +(2.60247 + 0.476612i) q^{7} +(0.973866 + 2.65548i) q^{8} +(-2.39103 + 1.38046i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 4 q^{4}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\)	\(241\)	\(337\)	\(351\)	\(421\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.29901	+	0.559084i	0.918538	+	0.395332i
							\(3\)	−0.472296	+	0.126551i	−0.272681	+	0.0730645i	−0.392568	−	0.919723i	\(-0.628413\pi\)
0.119888	+	0.992787i	\(0.461747\pi\)
\(5\)	−0.965926	−	0.258819i	−0.431975	−	0.115747i
							\(7\)	2.60247	+	0.476612i	0.983641	+	0.180143i
\(11\)	0.290336	+	1.08355i	0.0875395	+	0.326702i								0.995783	−	0.0917402i	\(-0.0292429\pi\)
							−0.908243	+	0.418442i	\(0.862576\pi\)
\(13\)	2.05265	+	2.05265i	0.569302	+	0.569302i	0.931933	−	0.362631i	\(-0.118122\pi\)
							−0.362631	+	0.931933i	\(0.618122\pi\)
\(17\)	1.64697	−	2.85264i	0.399449	−	0.691866i	−0.594209	−	0.804311i	\(-0.702535\pi\)
							0.993658	+	0.112445i	\(0.0358682\pi\)
\(19\)	−0.630383	+	2.35262i	−0.144620	+	0.539728i	0.855152	+	0.518377i	\(0.173464\pi\)
							−0.999772	+	0.0213515i	\(0.993203\pi\)
\(23\)	−1.24672	+	0.719795i	−0.259959	+	0.150088i	−0.624316	−	0.781172i	\(-0.714622\pi\)
							0.364357	+	0.931259i	\(0.381289\pi\)
\(29\)	4.19106	+	4.19106i	0.778261	+	0.778261i	0.979535	−	0.201274i	\(-0.0645081\pi\)
							−0.201274	+	0.979535i	\(0.564508\pi\)
\(31\)	1.56544	−	2.71142i	0.281161	−	0.486985i	−0.690510	−	0.723323i	\(-0.742614\pi\)
							0.971671	+	0.236338i	\(0.0759472\pi\)
\(37\)	−2.72095	−	0.729076i	−0.447321	−	0.119859i	0.0281244	−	0.999604i	\(-0.491047\pi\)
							−0.475446	+	0.879745i	\(0.657713\pi\)
\(41\)		−	6.02780i		−	0.941384i	−0.882298	−	0.470692i	\(-0.844004\pi\)
							0.882298	−	0.470692i	\(-0.155996\pi\)
\(43\)	−1.31670	+	1.31670i	−0.200795	+	0.200795i	−0.800341	−	0.599546i	\(-0.795348\pi\)
							0.599546	+	0.800341i	\(0.295348\pi\)
\(47\)	−5.45411	−	9.44679i	−0.795563	−	1.37796i	−0.922481	−	0.386043i	\(-0.873842\pi\)
							0.126918	−	0.991913i	\(-0.459492\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.cp.a.221.57	✓	256
7.2	even	3	inner	560.2.cp.a.541.30	yes	256
16.5	even	4	inner	560.2.cp.a.501.30	yes	256
112.37	even	12	inner	560.2.cp.a.261.57	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.cp.a.221.57	✓	256	1.1	even	1	trivial
560.2.cp.a.261.57	yes	256	112.37	even	12	inner
560.2.cp.a.501.30	yes	256	16.5	even	4	inner
560.2.cp.a.541.30	yes	256	7.2	even	3	inner