Embedded newform 560.2.cp.a.221.59

• Label: 560.2.cp.a.221.59
• 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			221.59
Character	\(\chi\)	\(=\)	560.221
Dual form			560.2.cp.a.261.59

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36056 - 0.385834i) q^{2} +(-0.623460 + 0.167056i) q^{3} +(1.70226 - 1.04990i) q^{4} +(0.965926 + 0.258819i) q^{5} +(-0.783801 + 0.467841i) q^{6} +(1.32042 + 2.29271i) q^{7} +(1.91095 - 2.08525i) q^{8} +(-2.23728 + 1.29170i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 4 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36056	−	0.385834i	0.962063	−	0.272826i
							\(3\)	−0.623460	+	0.167056i	−0.359955	+	0.0964496i	−0.434264	−	0.900786i	\(-0.642991\pi\)
0.0743092	+	0.997235i	\(0.476325\pi\)
\(5\)	0.965926	+	0.258819i	0.431975	+	0.115747i
							\(7\)	1.32042	+	2.29271i	0.499070	+	0.866561i
\(11\)	1.04117	+	3.88569i	0.313924	+	1.17158i								0.924987	+	0.379999i	\(0.124076\pi\)
							−0.611063	+	0.791582i	\(0.709258\pi\)
\(13\)	1.84641	+	1.84641i	0.512102	+	0.512102i	0.915170	−	0.403068i	\(-0.132056\pi\)
							−0.403068	+	0.915170i	\(0.632056\pi\)
\(17\)	2.84699	−	4.93112i	0.690496	−	1.19597i	−0.281180	−	0.959655i	\(-0.590726\pi\)
							0.971676	−	0.236318i	\(-0.0759408\pi\)
\(19\)	0.866312	−	3.23312i	0.198746	−	0.741729i	−0.792520	−	0.609846i	\(-0.791231\pi\)
							0.991265	−	0.131883i	\(-0.0421022\pi\)
\(23\)	−5.48061	+	3.16423i	−1.14279	+	0.659788i	−0.947119	−	0.320883i	\(-0.896020\pi\)
							−0.195667	+	0.980670i	\(0.562687\pi\)
\(29\)	−1.98290	−	1.98290i	−0.368216	−	0.368216i	0.498611	−	0.866826i	\(-0.333844\pi\)
							−0.866826	+	0.498611i	\(0.833844\pi\)
\(31\)	3.40370	−	5.89539i	0.611323	−	1.05884i	−0.379694	−	0.925112i	\(-0.623971\pi\)
							0.991018	−	0.133731i	\(-0.0426958\pi\)
\(37\)	−6.34275	−	1.69954i	−1.04274	−	0.279402i	−0.303493	−	0.952834i	\(-0.598153\pi\)
							−0.739249	+	0.673432i	\(0.764820\pi\)
\(41\)			0.0491454i			0.00767522i	0.999993	+	0.00383761i	\(0.00122155\pi\)
							−0.999993	+	0.00383761i	\(0.998778\pi\)
\(43\)	−1.96651	+	1.96651i	−0.299890	+	0.299890i	−0.840971	−	0.541081i	\(-0.818015\pi\)
							0.541081	+	0.840971i	\(0.318015\pi\)
\(47\)	5.79673	+	10.0402i	0.845540	+	1.46452i	0.885151	+	0.465303i	\(0.154055\pi\)
							−0.0396110	+	0.999215i	\(0.512612\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.cp.a.221.59	✓	256
7.2	even	3	inner	560.2.cp.a.541.15	yes	256
16.5	even	4	inner	560.2.cp.a.501.15	yes	256
112.37	even	12	inner	560.2.cp.a.261.59	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.cp.a.221.59	✓	256	1.1	even	1	trivial
560.2.cp.a.261.59	yes	256	112.37	even	12	inner
560.2.cp.a.501.15	yes	256	16.5	even	4	inner
560.2.cp.a.541.15	yes	256	7.2	even	3	inner