Embedded newform 560.2.cp.a.221.1

• Label: 560.2.cp.a.221.1
• 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			221.1
Character	\(\chi\)	\(=\)	560.221
Dual form			560.2.cp.a.261.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41403 - 0.0227292i) q^{2} +(2.02491 - 0.542574i) q^{3} +(1.99897 + 0.0642795i) q^{4} +(0.965926 + 0.258819i) q^{5} +(-2.87562 + 0.721191i) q^{6} +(2.64567 - 0.0210820i) q^{7} +(-2.82514 - 0.136328i) q^{8} +(1.20781 - 0.697328i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 4 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41403	−	0.0227292i	−0.999871	−	0.0160720i
							\(3\)	2.02491	−	0.542574i	1.16908	−	0.313255i	0.378495	−	0.925603i	\(-0.376442\pi\)
0.790588	+	0.612348i	\(0.209775\pi\)
\(5\)	0.965926	+	0.258819i	0.431975	+	0.115747i
							\(7\)	2.64567	−	0.0210820i	0.999968	−	0.00796823i
\(11\)	0.571074	+	2.13128i	0.172185	+	0.642604i								0.997014	+	0.0772215i	\(0.0246049\pi\)
							−0.824829	+	0.565383i	\(0.808728\pi\)
\(13\)	1.98113	+	1.98113i	0.549467	+	0.549467i	0.926287	−	0.376819i	\(-0.122982\pi\)
							−0.376819	+	0.926287i	\(0.622982\pi\)
\(17\)	−1.37575	+	2.38287i	−0.333668	+	0.577930i	−0.983228	−	0.182380i	\(-0.941620\pi\)
							0.649560	+	0.760310i	\(0.274953\pi\)
\(19\)	1.09761	−	4.09632i	0.251808	−	0.939761i	−0.718030	−	0.696012i	\(-0.754956\pi\)
							0.969838	−	0.243749i	\(-0.0783774\pi\)
\(23\)	−0.466181	+	0.269150i	−0.0972055	+	0.0561216i	−0.547815	−	0.836600i	\(-0.684540\pi\)
							0.450609	+	0.892721i	\(0.351207\pi\)
\(29\)	−0.196284	−	0.196284i	−0.0364490	−	0.0364490i	0.688647	−	0.725096i	\(-0.258205\pi\)
							−0.725096	+	0.688647i	\(0.758205\pi\)
\(31\)	−0.165376	+	0.286440i	−0.0297025	+	0.0514462i	−0.880495	−	0.474056i	\(-0.842789\pi\)
							0.850792	+	0.525502i	\(0.176123\pi\)
\(37\)	−10.2198	−	2.73839i	−1.68013	−	0.450189i	−0.712315	−	0.701860i	\(-0.752353\pi\)
							−0.967813	+	0.251671i	\(0.919020\pi\)
\(41\)		−	0.587931i		−	0.0918195i	−0.998946	−	0.0459097i	\(-0.985381\pi\)
							0.998946	−	0.0459097i	\(-0.0146187\pi\)
\(43\)	3.37025	−	3.37025i	0.513958	−	0.513958i	−0.401779	−	0.915737i	\(-0.631608\pi\)
							0.915737	+	0.401779i	\(0.131608\pi\)
\(47\)	−5.41133	−	9.37271i	−0.789324	−	1.36715i	−0.926381	−	0.376586i	\(-0.877098\pi\)
							0.137057	−	0.990563i	\(-0.456236\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.cp.a.221.1	✓	256
7.2	even	3	inner	560.2.cp.a.541.43	yes	256
16.5	even	4	inner	560.2.cp.a.501.43	yes	256
112.37	even	12	inner	560.2.cp.a.261.1	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.cp.a.221.1	✓	256	1.1	even	1	trivial
560.2.cp.a.261.1	yes	256	112.37	even	12	inner
560.2.cp.a.501.43	yes	256	16.5	even	4	inner
560.2.cp.a.541.43	yes	256	7.2	even	3	inner