Embedded newform 539.2.r.b.23.17

• Label: 539.2.r.b.23.17
• 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			23.17
Character	\(\chi\)	\(=\)	539.23
Dual form			539.2.r.b.375.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.110040 - 1.46838i) q^{2} +(-1.06483 + 0.328455i) q^{3} +(-0.166383 - 0.0250782i) q^{4} +(0.939178 + 0.871429i) q^{5} +(0.365125 + 1.59972i) q^{6} +(-0.817308 - 2.51635i) q^{7} +(0.600192 - 2.62961i) q^{8} +(-1.45275 + 0.990465i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q + 3 q^{3} + 20 q^{4} + 6 q^{5} - 8 q^{6} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.110040	−	1.46838i	0.0778102	−	1.03830i	−0.812503	−	0.582957i	\(-0.801896\pi\)
							0.890313	−	0.455348i	\(-0.150485\pi\)
\(3\)	−1.06483	+	0.328455i	−0.614777	+	0.189634i	−0.586477	−	0.809966i	\(-0.699486\pi\)
							−0.0283003	+	0.999599i	\(0.509009\pi\)
\(5\)	0.939178	+	0.871429i	0.420013	+	0.389715i	0.861670	−	0.507469i	\(-0.169419\pi\)
							−0.441657	+	0.897184i	\(0.645609\pi\)
\(7\)	−0.817308	−	2.51635i	−0.308913	−	0.951090i
							\(11\)	0.826239	+	0.563320i	0.249120	+	0.169847i
\(13\)	1.08595	−	0.522965i	0.301188	−	0.145044i								−0.277186	−	0.960816i	\(-0.589402\pi\)
							0.578374	+	0.815772i	\(0.303688\pi\)
\(17\)	1.84161	−	4.69235i	0.446657	−	1.13806i	−0.513373	−	0.858165i	\(-0.671604\pi\)
							0.960030	−	0.279897i	\(-0.0903004\pi\)
\(19\)	3.10110	−	5.37127i	0.711442	−	1.23225i	−0.252874	−	0.967499i	\(-0.581376\pi\)
							0.964316	−	0.264755i	\(-0.0852910\pi\)
\(23\)	2.25344	+	5.74168i	0.469876	+	1.19722i	0.947812	+	0.318829i	\(0.103290\pi\)
							−0.477937	+	0.878394i	\(0.658615\pi\)
\(29\)	−4.58890	−	5.75430i	−0.852138	−	1.06855i	−0.996869	−	0.0790763i	\(-0.974803\pi\)
							0.144730	−	0.989471i	\(-0.453768\pi\)
\(31\)	−0.375961	−	0.651184i	−0.0675246	−	0.116956i	0.830286	−	0.557337i	\(-0.188177\pi\)
							−0.897811	+	0.440381i	\(0.854843\pi\)
\(37\)	−5.07070	+	0.764284i	−0.833617	+	0.125648i	−0.551963	−	0.833868i	\(-0.686121\pi\)
							−0.281654	+	0.959516i	\(0.590883\pi\)
\(41\)	1.88691	−	8.26711i	0.294686	−	1.29111i	−0.583236	−	0.812303i	\(-0.698214\pi\)
							0.877922	−	0.478803i	\(-0.158929\pi\)
\(43\)	2.16146	+	9.46999i	0.329620	+	1.44416i	0.819856	+	0.572570i	\(0.194054\pi\)
							−0.490236	+	0.871590i	\(0.663089\pi\)
\(47\)	−0.562474	+	7.50569i	−0.0820452	+	1.09482i	0.792499	+	0.609873i	\(0.208779\pi\)
							−0.874545	+	0.484945i	\(0.838840\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.b.23.17	✓	276
49.32	even	21	inner	539.2.r.b.375.17	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.b.23.17	✓	276	1.1	even	1	trivial
539.2.r.b.375.17	yes	276	49.32	even	21	inner