Embedded newform 560.2.cp.a.221.21

• Label: 560.2.cp.a.221.21
• 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			221.21
Character	\(\chi\)	\(=\)	560.221
Dual form			560.2.cp.a.261.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.756293 - 1.19500i) q^{2} +(1.70799 - 0.457655i) q^{3} +(-0.856043 + 1.80754i) q^{4} +(0.965926 + 0.258819i) q^{5} +(-1.83864 - 1.69493i) q^{6} +(-2.09319 + 1.61819i) q^{7} +(2.80742 - 0.344057i) q^{8} +(0.109713 - 0.0633429i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 4 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.756293	−	1.19500i	−0.534780	−	0.844992i
							\(3\)	1.70799	−	0.457655i	0.986110	−	0.264227i	0.270494	−	0.962722i	\(-0.412813\pi\)
0.715616	+	0.698494i	\(0.246146\pi\)
\(5\)	0.965926	+	0.258819i	0.431975	+	0.115747i
							\(7\)	−2.09319	+	1.61819i	−0.791152	+	0.611619i
\(11\)	0.720715	+	2.68974i	0.217304	+	0.810988i								0.985343	+	0.170585i	\(0.0545659\pi\)
							−0.768039	+	0.640403i	\(0.778767\pi\)
\(13\)	4.34327	+	4.34327i	1.20461	+	1.20461i	0.972750	+	0.231856i	\(0.0744799\pi\)
							0.231856	+	0.972750i	\(0.425520\pi\)
\(17\)	1.21058	−	2.09678i	0.293608	−	0.508544i	−0.681052	−	0.732235i	\(-0.738477\pi\)
							0.974660	+	0.223691i	\(0.0718106\pi\)
\(19\)	0.556902	−	2.07839i	0.127762	−	0.476814i	−0.872161	−	0.489219i	\(-0.837282\pi\)
							0.999923	+	0.0124046i	\(0.00394859\pi\)
\(23\)	4.59296	−	2.65175i	0.957699	−	0.552928i	0.0622347	−	0.998062i	\(-0.480177\pi\)
							0.895464	+	0.445134i	\(0.146844\pi\)
\(29\)	4.73094	+	4.73094i	0.878514	+	0.878514i	0.993381	−	0.114867i	\(-0.0366443\pi\)
							−0.114867	+	0.993381i	\(0.536644\pi\)
\(31\)	−0.649182	+	1.12442i	−0.116596	+	0.201951i	−0.918417	−	0.395614i	\(-0.870532\pi\)
							0.801820	+	0.597565i	\(0.203865\pi\)
\(37\)	−2.17962	−	0.584027i	−0.358327	−	0.0960135i	0.0751642	−	0.997171i	\(-0.476052\pi\)
							−0.433491	+	0.901158i	\(0.642719\pi\)
\(41\)		−	4.85042i		−	0.757508i	−0.925497	−	0.378754i	\(-0.876353\pi\)
							0.925497	−	0.378754i	\(-0.123647\pi\)
\(43\)	−4.12156	+	4.12156i	−0.628532	+	0.628532i	−0.947699	−	0.319167i	\(-0.896597\pi\)
							0.319167	+	0.947699i	\(0.396597\pi\)
\(47\)	1.84154	+	3.18964i	0.268616	+	0.465257i	0.968505	−	0.248995i	\(-0.0801003\pi\)
							−0.699888	+	0.714252i	\(0.746767\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.cp.a.221.21	✓	256
7.2	even	3	inner	560.2.cp.a.541.22	yes	256
16.5	even	4	inner	560.2.cp.a.501.22	yes	256
112.37	even	12	inner	560.2.cp.a.261.21	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.cp.a.221.21	✓	256	1.1	even	1	trivial
560.2.cp.a.261.21	yes	256	112.37	even	12	inner
560.2.cp.a.501.22	yes	256	16.5	even	4	inner
560.2.cp.a.541.22	yes	256	7.2	even	3	inner