Embedded newform 547.2.e.a.9.11

• Label: 547.2.e.a.9.11
• 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			9.11
Character	\(\chi\)	\(=\)	547.9
Dual form			547.2.e.a.304.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.379656 + 1.66338i) q^{2} +(2.89065 + 1.39206i) q^{3} +(-0.820767 - 0.395261i) q^{4} +(-2.52981 - 3.17229i) q^{5} +(-3.41299 + 4.27975i) q^{6} +(3.93562 - 1.89529i) q^{7} +(-1.15847 + 1.45267i) q^{8} +(4.54754 + 5.70244i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 2 q^{2} - q^{3} - 44 q^{4} - 3 q^{5} - 10 q^{6} - 18 q^{7} - q^{8} - 35 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.379656	+	1.66338i	−0.268458	+	1.17619i	0.643350	+	0.765572i	\(0.277544\pi\)
							−0.911808	+	0.410618i	\(0.865313\pi\)
\(3\)	2.89065	+	1.39206i	1.66892	+	0.803708i	0.998069	+	0.0621135i	\(0.0197841\pi\)
							0.670848	+	0.741595i	\(0.265930\pi\)
\(5\)	−2.52981	−	3.17229i	−1.13137	−	1.41869i	−0.894455	−	0.447158i	\(-0.852436\pi\)
							−0.236912	−	0.971531i	\(-0.576135\pi\)
\(7\)	3.93562	−	1.89529i	1.48752	−	0.716353i	0.498885	−	0.866668i	\(-0.333743\pi\)
							0.988638	+	0.150315i	\(0.0480288\pi\)
\(11\)	3.15167			0.950266			0.475133	−	0.879914i	\(-0.342400\pi\)
							0.475133	+	0.879914i	\(0.342400\pi\)
\(13\)	0.401614	+	1.75958i	0.111388	+	0.488021i	0.999592	+	0.0285719i	\(0.00909595\pi\)
							−0.888204	+	0.459449i	\(0.848047\pi\)
\(17\)	−1.69414	−	2.12439i	−0.410890	−	0.515239i	0.532723	−	0.846289i	\(-0.321169\pi\)
							−0.943613	+	0.331050i	\(0.892597\pi\)
\(19\)	0.770995	+	3.37795i	0.176878	+	0.774954i	0.983060	+	0.183286i	\(0.0586734\pi\)
							−0.806181	+	0.591669i	\(0.798469\pi\)
\(23\)	−1.46537	−	6.42021i	−0.305551	−	1.33871i	−0.861613	−	0.507566i	\(-0.830546\pi\)
							0.556062	−	0.831141i	\(-0.312312\pi\)
\(29\)	−4.52769	+	5.67755i	−0.840771	+	1.05429i	0.157002	+	0.987598i	\(0.449817\pi\)
							−0.997773	+	0.0666954i	\(0.978754\pi\)
\(31\)	−1.67408	−	0.806194i	−0.300674	−	0.144797i	0.277464	−	0.960736i	\(-0.410506\pi\)
							−0.578138	+	0.815939i	\(0.696220\pi\)
\(37\)	−1.04252	−	1.30728i	−0.171389	−	0.214915i	0.688717	−	0.725030i	\(-0.258174\pi\)
							−0.860106	+	0.510115i	\(0.829603\pi\)
\(41\)	3.10374			0.484722			0.242361	−	0.970186i	\(-0.422078\pi\)
							0.242361	+	0.970186i	\(0.422078\pi\)
\(43\)	−1.60005	+	0.770544i	−0.244006	+	0.117507i	−0.551890	−	0.833917i	\(-0.686093\pi\)
							0.307885	+	0.951424i	\(0.400379\pi\)
\(47\)	−11.4525			−1.67052			−0.835261	−	0.549854i	\(-0.814683\pi\)
							−0.835261	+	0.549854i	\(0.814683\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.e.a.9.11	✓	264
547.304	even	7	inner	547.2.e.a.304.11	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.e.a.9.11	✓	264	1.1	even	1	trivial
547.2.e.a.304.11	yes	264	547.304	even	7	inner