Embedded newform 560.2.cp.a.221.34

• Label: 560.2.cp.a.221.34
• 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			221.34
Character	\(\chi\)	\(=\)	560.221
Dual form			560.2.cp.a.261.34

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0439102 - 1.41353i) q^{2} +(-1.99697 + 0.535086i) q^{3} +(-1.99614 - 0.124137i) q^{4} +(0.965926 + 0.258819i) q^{5} +(0.668674 + 2.84628i) q^{6} +(-0.869562 + 2.49877i) q^{7} +(-0.263122 + 2.81616i) q^{8} +(1.10349 - 0.637103i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 4 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0439102	−	1.41353i	0.0310492	−	0.999518i
							\(3\)	−1.99697	+	0.535086i	−1.15295	+	0.308932i	−0.784147	−	0.620575i	\(-0.786899\pi\)
−0.368804	+	0.929507i	\(0.620233\pi\)
\(5\)	0.965926	+	0.258819i	0.431975	+	0.115747i
							\(7\)	−0.869562	+	2.49877i	−0.328664	+	0.944447i
\(11\)	−1.26999	−	4.73966i	−0.382916	−	1.42906i								−0.841426	−	0.540372i	\(-0.818283\pi\)
							0.458510	−	0.888689i	\(-0.348383\pi\)
\(13\)	1.84105	+	1.84105i	0.510614	+	0.510614i	0.914715	−	0.404100i	\(-0.132415\pi\)
							−0.404100	+	0.914715i	\(0.632415\pi\)
\(17\)	4.07288	−	7.05444i	0.987819	−	1.71095i	0.359152	−	0.933279i	\(-0.383066\pi\)
							0.628668	−	0.777674i	\(-0.283601\pi\)
\(19\)	0.782054	−	2.91867i	0.179416	−	0.669588i	−0.816342	−	0.577569i	\(-0.804001\pi\)
							0.995757	−	0.0920188i	\(-0.0293320\pi\)
\(23\)	−1.93102	+	1.11487i	−0.402645	+	0.232467i	−0.687625	−	0.726066i	\(-0.741346\pi\)
							0.284980	+	0.958534i	\(0.408013\pi\)
\(29\)	−3.57516	−	3.57516i	−0.663891	−	0.663891i	0.292404	−	0.956295i	\(-0.405545\pi\)
							−0.956295	+	0.292404i	\(0.905545\pi\)
\(31\)	−2.49559	+	4.32248i	−0.448220	+	0.776341i	−0.998270	−	0.0587912i	\(-0.981275\pi\)
							0.550050	+	0.835132i	\(0.314609\pi\)
\(37\)	6.58497	+	1.76444i	1.08256	+	0.290072i	0.755645	−	0.654981i	\(-0.227323\pi\)
							0.326918	+	0.945053i	\(0.393990\pi\)
\(41\)		−	2.78113i		−	0.434339i	−0.976134	−	0.217170i	\(-0.930317\pi\)
							0.976134	−	0.217170i	\(-0.0696825\pi\)
\(43\)	5.75005	−	5.75005i	0.876875	−	0.876875i	−0.116335	−	0.993210i	\(-0.537115\pi\)
							0.993210	+	0.116335i	\(0.0371147\pi\)
\(47\)	−4.34849	−	7.53181i	−0.634292	−	1.09863i	−0.986665	−	0.162767i	\(-0.947958\pi\)
							0.352372	−	0.935860i	\(-0.385375\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.cp.a.221.34	✓	256
7.2	even	3	inner	560.2.cp.a.541.11	yes	256
16.5	even	4	inner	560.2.cp.a.501.11	yes	256
112.37	even	12	inner	560.2.cp.a.261.34	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.cp.a.221.34	✓	256	1.1	even	1	trivial
560.2.cp.a.261.34	yes	256	112.37	even	12	inner
560.2.cp.a.501.11	yes	256	16.5	even	4	inner
560.2.cp.a.541.11	yes	256	7.2	even	3	inner