Embedded newform 560.2.cp.a.221.48

• Label: 560.2.cp.a.221.48
• 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			221.48
Character	\(\chi\)	\(=\)	560.221
Dual form			560.2.cp.a.261.48

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.940904 + 1.05579i) q^{2} +(1.33880 - 0.358730i) q^{3} +(-0.229399 + 1.98680i) q^{4} +(0.965926 + 0.258819i) q^{5} +(1.63842 + 1.07596i) q^{6} +(0.771290 + 2.53083i) q^{7} +(-2.31349 + 1.62719i) q^{8} +(-0.934386 + 0.539468i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 4 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.940904	+	1.05579i	0.665320	+	0.746559i
							\(3\)	1.33880	−	0.358730i	0.772955	−	0.207113i	0.149278	−	0.988795i	\(-0.452305\pi\)
0.623676	+	0.781683i	\(0.285638\pi\)
\(5\)	0.965926	+	0.258819i	0.431975	+	0.115747i
							\(7\)	0.771290	+	2.53083i	0.291520	+	0.956565i
\(11\)	1.17752	+	4.39455i	0.355035	+	1.32501i								0.880441	+	0.474156i	\(0.157247\pi\)
							−0.525406	+	0.850851i	\(0.676087\pi\)
\(13\)	−4.64886	−	4.64886i	−1.28936	−	1.28936i	−0.935175	−	0.354187i	\(-0.884758\pi\)
							−0.354187	−	0.935175i	\(-0.615242\pi\)
\(17\)	1.12193	−	1.94324i	0.272108	−	0.471304i	−0.697294	−	0.716786i	\(-0.745613\pi\)
							0.969401	+	0.245481i	\(0.0789460\pi\)
\(19\)	1.19327	−	4.45335i	0.273755	−	1.02167i	−0.682916	−	0.730497i	\(-0.739288\pi\)
							0.956671	−	0.291171i	\(-0.0940450\pi\)
\(23\)	6.43197	−	3.71350i	1.34116	−	0.774318i	0.354181	−	0.935177i	\(-0.384760\pi\)
							0.986977	+	0.160858i	\(0.0514263\pi\)
\(29\)	4.81054	+	4.81054i	0.893295	+	0.893295i	0.994832	−	0.101537i	\(-0.0323759\pi\)
							−0.101537	+	0.994832i	\(0.532376\pi\)
\(31\)	2.77211	−	4.80143i	0.497885	−	0.862362i	−0.502112	−	0.864803i	\(-0.667443\pi\)
							0.999997	+	0.00244027i	\(0.000776761\pi\)
\(37\)	7.22610	+	1.93623i	1.18796	+	0.318314i	0.798081	−	0.602551i	\(-0.205849\pi\)
							0.389883	+	0.920865i	\(0.372516\pi\)
\(41\)			3.09833i			0.483878i	0.970291	+	0.241939i	\(0.0777834\pi\)
							−0.970291	+	0.241939i	\(0.922217\pi\)
\(43\)	0.340420	−	0.340420i	0.0519136	−	0.0519136i	−0.680673	−	0.732587i	\(-0.738313\pi\)
							0.732587	+	0.680673i	\(0.238313\pi\)
\(47\)	−3.39083	−	5.87310i	−0.494604	−	0.856679i	0.505377	−	0.862899i	\(-0.331354\pi\)
							−0.999981	+	0.00621949i	\(0.998020\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.cp.a.221.48	✓	256
7.2	even	3	inner	560.2.cp.a.541.37	yes	256
16.5	even	4	inner	560.2.cp.a.501.37	yes	256
112.37	even	12	inner	560.2.cp.a.261.48	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.cp.a.221.48	✓	256	1.1	even	1	trivial
560.2.cp.a.261.48	yes	256	112.37	even	12	inner
560.2.cp.a.501.37	yes	256	16.5	even	4	inner
560.2.cp.a.541.37	yes	256	7.2	even	3	inner