Embedded newform 560.2.cp.a.221.38

• Label: 560.2.cp.a.221.38
• 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			221.38
Character	\(\chi\)	\(=\)	560.221
Dual form			560.2.cp.a.261.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.313410 + 1.37905i) q^{2} +(0.215633 - 0.0577786i) q^{3} +(-1.80355 + 0.864415i) q^{4} +(0.965926 + 0.258819i) q^{5} +(0.147261 + 0.279259i) q^{6} +(0.618432 - 2.57246i) q^{7} +(-1.75732 - 2.21626i) q^{8} +(-2.55492 + 1.47508i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 4 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.313410	+	1.37905i	0.221614	+	0.975134i
							\(3\)	0.215633	−	0.0577786i	0.124496	−	0.0333585i	−0.196033	−	0.980597i	\(-0.562806\pi\)
0.320529	+	0.947239i	\(0.396139\pi\)
\(5\)	0.965926	+	0.258819i	0.431975	+	0.115747i
							\(7\)	0.618432	−	2.57246i	0.233745	−	0.972298i
\(11\)	1.49102	+	5.56455i	0.449559	+	1.67778i								0.703611	+	0.710586i	\(0.251570\pi\)
							−0.254052	+	0.967191i	\(0.581763\pi\)
\(13\)	2.05264	+	2.05264i	0.569300	+	0.569300i	0.931932	−	0.362632i	\(-0.118122\pi\)
							−0.362632	+	0.931932i	\(0.618122\pi\)
\(17\)	−2.61892	+	4.53611i	−0.635182	+	1.10017i	0.351295	+	0.936265i	\(0.385742\pi\)
							−0.986477	+	0.163902i	\(0.947592\pi\)
\(19\)	−1.69724	+	6.33419i	−0.389374	+	1.45316i	0.441782	+	0.897122i	\(0.354346\pi\)
							−0.831156	+	0.556040i	\(0.812320\pi\)
\(23\)	3.67258	−	2.12037i	0.765786	−	0.442127i	−0.0655831	−	0.997847i	\(-0.520891\pi\)
							0.831369	+	0.555720i	\(0.187557\pi\)
\(29\)	3.61371	+	3.61371i	0.671049	+	0.671049i	0.957958	−	0.286909i	\(-0.0926276\pi\)
							−0.286909	+	0.957958i	\(0.592628\pi\)
\(31\)	0.973069	−	1.68540i	0.174768	−	0.302708i	−0.765313	−	0.643659i	\(-0.777416\pi\)
							0.940081	+	0.340951i	\(0.110749\pi\)
\(37\)	−8.57181	−	2.29681i	−1.40920	−	0.377593i	−0.527559	−	0.849518i	\(-0.676893\pi\)
							−0.881639	+	0.471925i	\(0.843559\pi\)
\(41\)		−	8.93557i		−	1.39550i	−0.716341	−	0.697751i	\(-0.754184\pi\)
							0.716341	−	0.697751i	\(-0.245816\pi\)
\(43\)	8.54067	−	8.54067i	1.30244	−	1.30244i	0.375697	−	0.926742i	\(-0.377403\pi\)
							0.926742	−	0.375697i	\(-0.122597\pi\)
\(47\)	3.31009	+	5.73324i	0.482826	+	0.836279i	0.999806	−	0.0197184i	\(-0.00627696\pi\)
							−0.516979	+	0.855998i	\(0.672944\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.cp.a.221.38	✓	256
7.2	even	3	inner	560.2.cp.a.541.48	yes	256
16.5	even	4	inner	560.2.cp.a.501.48	yes	256
112.37	even	12	inner	560.2.cp.a.261.38	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.cp.a.221.38	✓	256	1.1	even	1	trivial
560.2.cp.a.261.38	yes	256	112.37	even	12	inner
560.2.cp.a.501.48	yes	256	16.5	even	4	inner
560.2.cp.a.541.48	yes	256	7.2	even	3	inner