Embedded newform 539.2.r.b.23.20

• Label: 539.2.r.b.23.20
• Level: $539$
• Weight: $2$
• Character: 539.23
• Analytic conductor: $4.304$
• Analytic rank: $0$
• Dimension: $276$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.r (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.30393666895\)
Analytic rank:	\(0\)
Dimension:	\(276\)
Relative dimension:	\(23\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			23.20
Character	\(\chi\)	\(=\)	539.23
Dual form			539.2.r.b.375.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.167941 - 2.24102i) q^{2} +(1.19225 - 0.367760i) q^{3} +(-3.01630 - 0.454633i) q^{4} +(2.18464 + 2.02705i) q^{5} +(-0.623930 - 2.73361i) q^{6} +(1.44285 + 2.21770i) q^{7} +(-0.525258 + 2.30131i) q^{8} +(-1.19251 + 0.813036i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q + 3 q^{3} + 20 q^{4} + 6 q^{5} - 8 q^{6} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/539\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(442\)
\(\chi(n)\)	\(e\left(\frac{19}{21}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.167941	−	2.24102i	0.118752	−	1.58464i	−0.546136	−	0.837696i	\(-0.683902\pi\)
							0.664888	−	0.746943i	\(-0.268479\pi\)
\(3\)	1.19225	−	0.367760i	0.688346	−	0.212326i	0.0692001	−	0.997603i	\(-0.477955\pi\)
							0.619145	+	0.785276i	\(0.287479\pi\)
\(5\)	2.18464	+	2.02705i	0.977000	+	0.906523i	0.995638	−	0.0933030i	\(-0.0297425\pi\)
							−0.0186380	+	0.999826i	\(0.505933\pi\)
\(7\)	1.44285	+	2.21770i	0.545346	+	0.838211i
							\(11\)	0.826239	+	0.563320i	0.249120	+	0.169847i
\(13\)	1.43227	−	0.689746i	0.397241	−	0.191301i								−0.224589	−	0.974454i	\(-0.572104\pi\)
							0.621830	+	0.783152i	\(0.286390\pi\)
\(17\)	2.05798	−	5.24365i	0.499133	−	1.27177i	−0.430268	−	0.902701i	\(-0.641581\pi\)
							0.929402	−	0.369070i	\(-0.120324\pi\)
\(19\)	2.07748	−	3.59831i	0.476607	−	0.825508i	−0.523033	−	0.852312i	\(-0.675200\pi\)
							0.999641	+	0.0268042i	\(0.00853306\pi\)
\(23\)	−1.06933	−	2.72462i	−0.222971	−	0.568122i	0.774988	−	0.631976i	\(-0.217756\pi\)
							−0.997959	+	0.0638542i	\(0.979661\pi\)
\(29\)	0.610239	+	0.765216i	0.113319	+	0.142097i	0.835256	−	0.549861i	\(-0.185320\pi\)
							−0.721937	+	0.691959i	\(0.756748\pi\)
\(31\)	1.51134	+	2.61772i	0.271444	+	0.470156i	0.969232	−	0.246149i	\(-0.0791653\pi\)
							−0.697787	+	0.716305i	\(0.745832\pi\)
\(37\)	5.16751	−	0.778877i	0.849534	−	0.128047i	0.290176	−	0.956973i	\(-0.406286\pi\)
							0.559357	+	0.828927i	\(0.311048\pi\)
\(41\)	−1.03740	+	4.54512i	−0.162014	+	0.709829i	0.827024	+	0.562167i	\(0.190032\pi\)
							−0.989038	+	0.147663i	\(0.952825\pi\)
\(43\)	0.795086	+	3.48350i	0.121249	+	0.531228i	0.998672	+	0.0515110i	\(0.0164037\pi\)
							−0.877423	+	0.479718i	\(0.840739\pi\)
\(47\)	0.602471	−	8.03942i	0.0878795	−	1.17267i	−0.762675	−	0.646782i	\(-0.776114\pi\)
							0.850554	−	0.525888i	\(-0.176267\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.b.23.20	✓	276
49.32	even	21	inner	539.2.r.b.375.20	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.b.23.20	✓	276	1.1	even	1	trivial
539.2.r.b.375.20	yes	276	49.32	even	21	inner