Embedded newform 561.2.bn.a.7.8

• Label: 561.2.bn.a.7.8
• Level: $561$
• Weight: $2$
• Character: 561.7
• Analytic conductor: $4.480$
• Analytic rank: $0$
• Dimension: $1152$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 561 = 3 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	561.bn (of order \(80\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.47960755339\)
Analytic rank:	\(0\)
Dimension:	\(1152\)
Relative dimension:	\(36\) over \(\Q(\zeta_{80})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{80}]$

Embedding invariants

Embedding label			7.8
Character	\(\chi\)	\(=\)	561.7
Dual form			561.2.bn.a.481.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.856832 - 1.39822i) q^{2} +(-0.619094 - 0.785317i) q^{3} +(-0.312883 + 0.614068i) q^{4} +(1.08487 + 1.00284i) q^{5} +(-0.567588 + 1.53852i) q^{6} +(0.497789 + 4.20579i) q^{7} +(-2.14294 + 0.168653i) q^{8} +(-0.233445 + 0.972370i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/561\mathbb{Z}\right)^\times\).

\(n\)	\(188\)	\(409\)	\(496\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{11}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.856832	−	1.39822i	−0.605871	−	0.988692i	−0.997642	−	0.0686294i	\(-0.978137\pi\)
							0.391771	−	0.920063i	\(-0.371863\pi\)
\(3\)	−0.619094	−	0.785317i	−0.357434	−	0.453403i
							\(5\)	1.08487	+	1.00284i	0.485169	+	0.448485i	0.883276	−	0.468854i	\(-0.155333\pi\)
−0.398107	+	0.917339i	\(0.630333\pi\)
\(7\)	0.497789	+	4.20579i	0.188146	+	1.58964i	0.689426	+	0.724356i	\(0.257863\pi\)
							−0.501280	+	0.865285i	\(0.667137\pi\)
\(11\)	−0.587199	−	3.26423i	−0.177047	−	0.984202i
							\(13\)	−0.145453	−	0.918356i	−0.0403415	−	0.254706i	0.959273	−	0.282482i	\(-0.0911576\pi\)
−0.999614	+	0.0277755i	\(0.991158\pi\)
\(17\)	3.41135	−	2.31575i	0.827374	−	0.561651i
							\(19\)	5.18389	−	6.06955i	1.18927	−	1.39245i	0.288876	−	0.957366i	\(-0.406718\pi\)
0.900389	−	0.435085i	\(-0.143282\pi\)
\(23\)	−0.610021	−	0.912961i	−0.127198	−	0.190366i	0.762405	−	0.647101i	\(-0.224019\pi\)
							−0.889603	+	0.456735i	\(0.849019\pi\)
\(29\)	7.90355	−	4.42620i	1.46765	−	0.821925i	0.469515	−	0.882924i	\(-0.344429\pi\)
							0.998138	+	0.0609991i	\(0.0194287\pi\)
\(31\)	3.06035	+	8.29545i	0.549655	+	1.48991i	0.845133	+	0.534556i	\(0.179521\pi\)
							−0.295477	+	0.955350i	\(0.595479\pi\)
\(37\)	−0.463106	−	0.130609i	−0.0761341	−	0.0214720i	0.233161	−	0.972438i	\(-0.425093\pi\)
							−0.309295	+	0.950966i	\(0.600093\pi\)
\(41\)	2.44366	−	4.36347i	0.381636	−	0.681460i	−0.612396	−	0.790551i	\(-0.709794\pi\)
							0.994032	+	0.109092i	\(0.0347942\pi\)
\(43\)	2.39447	−	0.991823i	0.365154	−	0.151252i	−0.192560	−	0.981285i	\(-0.561679\pi\)
							0.557713	+	0.830034i	\(0.311679\pi\)
\(47\)	−0.789022	−	1.54854i	−0.115091	−	0.225878i	0.826274	−	0.563268i	\(-0.190456\pi\)
							−0.941365	+	0.337390i	\(0.890456\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	561.2.bn.a.7.8	✓	1152
11.8	odd	10	inner	561.2.bn.a.415.29	yes	1152
17.5	odd	16	inner	561.2.bn.a.73.29	yes	1152
187.107	even	80	inner	561.2.bn.a.481.8	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
561.2.bn.a.7.8	✓	1152	1.1	even	1	trivial
561.2.bn.a.73.29	yes	1152	17.5	odd	16	inner
561.2.bn.a.415.29	yes	1152	11.8	odd	10	inner
561.2.bn.a.481.8	yes	1152	187.107	even	80	inner