Embedded newform 547.2.e.a.9.3

• Label: 547.2.e.a.9.3
• Level: $547$
• Weight: $2$
• Character: 547.9
• Analytic conductor: $4.368$
• Analytic rank: $0$
• Dimension: $264$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	547.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36781699056\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(44\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			9.3
Character	\(\chi\)	\(=\)	547.9
Dual form			547.2.e.a.304.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.569327 + 2.49438i) q^{2} +(2.72887 + 1.31416i) q^{3} +(-4.09588 - 1.97247i) q^{4} +(0.314234 + 0.394037i) q^{5} +(-4.83163 + 6.05868i) q^{6} +(-1.02655 + 0.494361i) q^{7} +(4.06156 - 5.09304i) q^{8} +(3.84928 + 4.82684i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 2 q^{2} - q^{3} - 44 q^{4} - 3 q^{5} - 10 q^{6} - 18 q^{7} - q^{8} - 35 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/547\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.569327	+	2.49438i	−0.402575	+	1.76380i	0.214333	+	0.976761i	\(0.431242\pi\)
							−0.616908	+	0.787035i	\(0.711615\pi\)
\(3\)	2.72887	+	1.31416i	1.57552	+	0.758729i	0.998324	−	0.0578790i	\(-0.0184338\pi\)
							0.577193	+	0.816608i	\(0.304148\pi\)
\(5\)	0.314234	+	0.394037i	0.140530	+	0.176219i	0.847116	−	0.531409i	\(-0.178337\pi\)
							−0.706586	+	0.707627i	\(0.749766\pi\)
\(7\)	−1.02655	+	0.494361i	−0.388000	+	0.186851i	−0.617706	−	0.786409i	\(-0.711938\pi\)
							0.229706	+	0.973260i	\(0.426224\pi\)
\(11\)	−4.96493			−1.49698			−0.748492	−	0.663144i	\(-0.769222\pi\)
							−0.748492	+	0.663144i	\(0.769222\pi\)
\(13\)	0.809473	+	3.54653i	0.224508	+	0.983632i	0.954039	+	0.299684i	\(0.0968813\pi\)
							−0.729531	+	0.683948i	\(0.760262\pi\)
\(17\)	1.71130	+	2.14591i	0.415052	+	0.520459i	0.944778	−	0.327710i	\(-0.106277\pi\)
							−0.529727	+	0.848168i	\(0.677705\pi\)
\(19\)	0.0141169	+	0.0618500i	0.00323863	+	0.0141894i	0.976521	−	0.215420i	\(-0.0691122\pi\)
							−0.973283	+	0.229610i	\(0.926255\pi\)
\(23\)	0.162767	+	0.713128i	0.0339392	+	0.148697i	0.989058	−	0.147524i	\(-0.0471304\pi\)
							−0.955119	+	0.296222i	\(0.904273\pi\)
\(29\)	−1.29090	+	1.61873i	−0.239713	+	0.300591i	−0.887106	−	0.461565i	\(-0.847288\pi\)
							0.647393	+	0.762157i	\(0.275859\pi\)
\(31\)	3.20341	+	1.54268i	0.575349	+	0.277074i	0.698847	−	0.715271i	\(-0.253697\pi\)
							−0.123498	+	0.992345i	\(0.539411\pi\)
\(37\)	−4.94949	−	6.20646i	−0.813690	−	1.02034i	−0.999289	−	0.0377098i	\(-0.987994\pi\)
							0.185598	−	0.982626i	\(-0.440578\pi\)
\(41\)	5.85506			0.914407			0.457203	−	0.889362i	\(-0.348851\pi\)
							0.457203	+	0.889362i	\(0.348851\pi\)
\(43\)	−3.58185	+	1.72493i	−0.546227	+	0.263049i	−0.686582	−	0.727052i	\(-0.740890\pi\)
							0.140355	+	0.990101i	\(0.455175\pi\)
\(47\)	13.5447			1.97570			0.987852	−	0.155396i	\(-0.0496655\pi\)
							0.987852	+	0.155396i	\(0.0496655\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.e.a.9.3	✓	264
547.304	even	7	inner	547.2.e.a.304.3	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.e.a.9.3	✓	264	1.1	even	1	trivial
547.2.e.a.304.3	yes	264	547.304	even	7	inner