Embedded newform 539.2.r.b.23.7

• Label: 539.2.r.b.23.7
• 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			23.7
Character	\(\chi\)	\(=\)	539.23
Dual form			539.2.r.b.375.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.113141 + 1.50976i) q^{2} +(2.71968 - 0.838911i) q^{3} +(-0.288924 - 0.0435483i) q^{4} +(-3.07265 - 2.85100i) q^{5} +(0.958850 + 4.20099i) q^{6} +(2.64504 + 0.0614254i) q^{7} +(-0.575355 + 2.52080i) q^{8} +(4.21419 - 2.87318i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q + 3 q^{3} + 20 q^{4} + 6 q^{5} - 8 q^{6} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.113141	+	1.50976i	−0.0800029	+	1.06756i	0.802312	+	0.596905i	\(0.203603\pi\)
							−0.882315	+	0.470659i	\(0.844016\pi\)
\(3\)	2.71968	−	0.838911i	1.57021	−	0.484346i	0.617078	−	0.786902i	\(-0.288316\pi\)
							0.953132	+	0.302556i	\(0.0978400\pi\)
\(5\)	−3.07265	−	2.85100i	−1.37413	−	1.27501i	−0.924183	−	0.381951i	\(-0.875252\pi\)
							−0.449947	−	0.893055i	\(-0.648557\pi\)
\(7\)	2.64504	+	0.0614254i	0.999730	+	0.0232166i
							\(11\)	0.826239	+	0.563320i	0.249120	+	0.169847i
\(13\)	1.64926	−	0.794241i	0.457422	−	0.220283i								−0.190960	−	0.981598i	\(-0.561160\pi\)
							0.648382	+	0.761315i	\(0.275446\pi\)
\(17\)	1.91925	−	4.89018i	0.465487	−	1.18604i	−0.484788	−	0.874632i	\(-0.661103\pi\)
							0.950275	−	0.311411i	\(-0.100801\pi\)
\(19\)	−0.890177	+	1.54183i	−0.204221	+	0.353721i	−0.949884	−	0.312602i	\(-0.898799\pi\)
							0.745663	+	0.666323i	\(0.232133\pi\)
\(23\)	−2.05968	−	5.24797i	−0.429472	−	1.09428i	−0.967840	−	0.251568i	\(-0.919054\pi\)
							0.538367	−	0.842710i	\(-0.319041\pi\)
\(29\)	2.28220	+	2.86179i	0.423794	+	0.531421i	0.947192	−	0.320668i	\(-0.103907\pi\)
							−0.523398	+	0.852089i	\(0.675336\pi\)
\(31\)	3.08828	+	5.34906i	0.554671	+	0.960719i	0.997929	+	0.0643246i	\(0.0204893\pi\)
							−0.443258	+	0.896394i	\(0.646177\pi\)
\(37\)	−2.50228	+	0.377158i	−0.411372	+	0.0620043i	−0.351468	−	0.936200i	\(-0.614317\pi\)
							−0.0599037	+	0.998204i	\(0.519079\pi\)
\(41\)	−0.344079	+	1.50751i	−0.0537361	+	0.235433i	−0.994663	−	0.103176i	\(-0.967099\pi\)
							0.940927	+	0.338610i	\(0.109957\pi\)
\(43\)	0.406601	+	1.78144i	0.0620061	+	0.271666i	0.996422	−	0.0845174i	\(-0.0269348\pi\)
							−0.934416	+	0.356184i	\(0.884078\pi\)
\(47\)	−0.378921	+	5.05635i	−0.0552713	+	0.737544i	0.898875	+	0.438204i	\(0.144385\pi\)
							−0.954147	+	0.299340i	\(0.903234\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.b.23.7	✓	276
49.32	even	21	inner	539.2.r.b.375.7	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.b.23.7	✓	276	1.1	even	1	trivial
539.2.r.b.375.7	yes	276	49.32	even	21	inner