Embedded newform 560.2.cp.a.221.39

• Label: 560.2.cp.a.221.39
• 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			221.39
Character	\(\chi\)	\(=\)	560.221
Dual form			560.2.cp.a.261.39

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.355520 - 1.36880i) q^{2} +(-1.80630 + 0.483995i) q^{3} +(-1.74721 - 0.973269i) q^{4} +(0.965926 + 0.258819i) q^{5} +(0.0203180 + 2.64452i) q^{6} +(2.48022 - 0.921147i) q^{7} +(-1.95338 + 2.04556i) q^{8} +(0.430374 - 0.248477i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 4 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.355520	−	1.36880i	0.251390	−	0.967886i
							\(3\)	−1.80630	+	0.483995i	−1.04286	+	0.279435i	−0.739300	−	0.673376i	\(-0.764844\pi\)
−0.303565	+	0.952811i	\(0.598177\pi\)
\(5\)	0.965926	+	0.258819i	0.431975	+	0.115747i
							\(7\)	2.48022	−	0.921147i	0.937435	−	0.348161i
\(11\)	0.903429	+	3.37164i	0.272394	+	1.01659i								0.957568	+	0.288209i	\(0.0930597\pi\)
							−0.685173	+	0.728380i	\(0.740274\pi\)
\(13\)	1.97553	+	1.97553i	0.547913	+	0.547913i	0.925837	−	0.377923i	\(-0.123362\pi\)
							−0.377923	+	0.925837i	\(0.623362\pi\)
\(17\)	−2.71079	+	4.69523i	−0.657464	+	1.13876i	0.323807	+	0.946123i	\(0.395037\pi\)
							−0.981270	+	0.192637i	\(0.938296\pi\)
\(19\)	1.57532	−	5.87917i	0.361403	−	1.34878i	−0.510829	−	0.859683i	\(-0.670661\pi\)
							0.872232	−	0.489093i	\(-0.162672\pi\)
\(23\)	3.94573	−	2.27807i	0.822742	−	0.475010i	−0.0286191	−	0.999590i	\(-0.509111\pi\)
							0.851361	+	0.524580i	\(0.175778\pi\)
\(29\)	3.48472	+	3.48472i	0.647096	+	0.647096i	0.952290	−	0.305194i	\(-0.0987213\pi\)
							−0.305194	+	0.952290i	\(0.598721\pi\)
\(31\)	5.14703	−	8.91492i	0.924434	−	1.60117i	0.131966	−	0.991254i	\(-0.457871\pi\)
							0.792468	−	0.609913i	\(-0.208796\pi\)
\(37\)	9.37595	+	2.51228i	1.54140	+	0.413016i	0.926717	−	0.375761i	\(-0.122618\pi\)
							0.614680	+	0.788777i	\(0.289285\pi\)
\(41\)		−	2.40411i		−	0.375458i	−0.982221	−	0.187729i	\(-0.939887\pi\)
							0.982221	−	0.187729i	\(-0.0601127\pi\)
\(43\)	−9.00806	+	9.00806i	−1.37372	+	1.37372i	−0.518853	+	0.854864i	\(0.673641\pi\)
							−0.854864	+	0.518853i	\(0.826359\pi\)
\(47\)	−0.834366	−	1.44517i	−0.121705	−	0.210799i	0.798735	−	0.601683i	\(-0.205503\pi\)
							−0.920440	+	0.390884i	\(0.872169\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.cp.a.221.39	✓	256
7.2	even	3	inner	560.2.cp.a.541.5	yes	256
16.5	even	4	inner	560.2.cp.a.501.5	yes	256
112.37	even	12	inner	560.2.cp.a.261.39	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.cp.a.221.39	✓	256	1.1	even	1	trivial
560.2.cp.a.261.39	yes	256	112.37	even	12	inner
560.2.cp.a.501.5	yes	256	16.5	even	4	inner
560.2.cp.a.541.5	yes	256	7.2	even	3	inner