Embedded newform 560.2.cp.a.221.8

• Label: 560.2.cp.a.221.8
• 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			221.8
Character	\(\chi\)	\(=\)	560.221
Dual form			560.2.cp.a.261.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31828 + 0.511983i) q^{2} +(-0.0714125 + 0.0191349i) q^{3} +(1.47575 - 1.34988i) q^{4} +(-0.965926 - 0.258819i) q^{5} +(0.0843453 - 0.0617872i) q^{6} +(1.28741 - 2.31140i) q^{7} +(-1.25434 + 2.53508i) q^{8} +(-2.59334 + 1.49727i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 4 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31828	+	0.511983i	−0.932168	+	0.362026i
							\(3\)	−0.0714125	+	0.0191349i	−0.0412300	+	0.0110476i	−0.279375	−	0.960182i	\(-0.590127\pi\)
0.238145	+	0.971230i	\(0.423461\pi\)
\(5\)	−0.965926	−	0.258819i	−0.431975	−	0.115747i
							\(7\)	1.28741	−	2.31140i	0.486596	−	0.873627i
\(11\)	0.200747	+	0.749199i	0.0605276	+	0.225892i								0.989563	−	0.144098i	\(-0.0460279\pi\)
							−0.929036	+	0.369990i	\(0.879361\pi\)
\(13\)	3.55535	+	3.55535i	0.986076	+	0.986076i	0.999904	−	0.0138282i	\(-0.00440178\pi\)
							−0.0138282	+	0.999904i	\(0.504402\pi\)
\(17\)	2.94108	−	5.09409i	0.713316	−	1.23550i	−0.250290	−	0.968171i	\(-0.580526\pi\)
							0.963606	−	0.267328i	\(-0.0861408\pi\)
\(19\)	0.576876	−	2.15293i	0.132344	−	0.493916i	−0.867650	−	0.497175i	\(-0.834371\pi\)
							0.999995	+	0.00325877i	\(0.00103730\pi\)
\(23\)	0.800962	−	0.462436i	0.167012	−	0.0964245i	−0.414164	−	0.910202i	\(-0.635926\pi\)
							0.581176	+	0.813778i	\(0.302593\pi\)
\(29\)	−6.32528	−	6.32528i	−1.17458	−	1.17458i	−0.981106	−	0.193469i	\(-0.938026\pi\)
							−0.193469	−	0.981106i	\(-0.561974\pi\)
\(31\)	3.30314	−	5.72120i	0.593261	−	1.02756i	−0.400529	−	0.916284i	\(-0.631173\pi\)
							0.993790	−	0.111274i	\(-0.0354932\pi\)
\(37\)	7.46881	+	2.00126i	1.22786	+	0.329005i	0.813748	−	0.581218i	\(-0.197424\pi\)
							0.414117	+	0.910224i	\(0.364090\pi\)
\(41\)		−	4.60176i		−	0.718675i	−0.933208	−	0.359337i	\(-0.883003\pi\)
							0.933208	−	0.359337i	\(-0.116997\pi\)
\(43\)	7.39440	−	7.39440i	1.12764	−	1.12764i	0.137075	−	0.990561i	\(-0.456230\pi\)
							0.990561	−	0.137075i	\(-0.0437701\pi\)
\(47\)	3.78888	+	6.56253i	0.552665	+	0.957244i	0.998081	+	0.0619200i	\(0.0197224\pi\)
							−0.445416	+	0.895324i	\(0.646944\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.cp.a.221.8	✓	256
7.2	even	3	inner	560.2.cp.a.541.49	yes	256
16.5	even	4	inner	560.2.cp.a.501.49	yes	256
112.37	even	12	inner	560.2.cp.a.261.8	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.cp.a.221.8	✓	256	1.1	even	1	trivial
560.2.cp.a.261.8	yes	256	112.37	even	12	inner
560.2.cp.a.501.49	yes	256	16.5	even	4	inner
560.2.cp.a.541.49	yes	256	7.2	even	3	inner