Embedded newform 547.2.e.a.9.19

• Label: 547.2.e.a.9.19
• 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			9.19
Character	\(\chi\)	\(=\)	547.9
Dual form			547.2.e.a.304.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.173579 + 0.760500i) q^{2} +(2.53318 + 1.21992i) q^{3} +(1.25371 + 0.603753i) q^{4} +(0.424209 + 0.531942i) q^{5} +(-1.36745 + 1.71473i) q^{6} +(-0.719571 + 0.346527i) q^{7} +(-1.64949 + 2.06839i) q^{8} +(3.05835 + 3.83505i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 2 q^{2} - q^{3} - 44 q^{4} - 3 q^{5} - 10 q^{6} - 18 q^{7} - q^{8} - 35 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.173579	+	0.760500i	−0.122739	+	0.537755i	0.875748	+	0.482768i	\(0.160369\pi\)
							−0.998487	+	0.0549865i	\(0.982488\pi\)
\(3\)	2.53318	+	1.21992i	1.46253	+	0.704319i	0.984720	−	0.174143i	\(-0.0557154\pi\)
							0.477813	+	0.878462i	\(0.341430\pi\)
\(5\)	0.424209	+	0.531942i	0.189712	+	0.237891i	0.867587	−	0.497286i	\(-0.165670\pi\)
							−0.677874	+	0.735178i	\(0.737099\pi\)
\(7\)	−0.719571	+	0.346527i	−0.271972	+	0.130975i	−0.564899	−	0.825160i	\(-0.691085\pi\)
							0.292927	+	0.956135i	\(0.405371\pi\)
\(11\)	−1.73806			−0.524046			−0.262023	−	0.965062i	\(-0.584390\pi\)
							−0.262023	+	0.965062i	\(0.584390\pi\)
\(13\)	−0.793933	−	3.47845i	−0.220197	−	0.964748i	−0.957329	−	0.288999i	\(-0.906678\pi\)
							0.737132	−	0.675749i	\(-0.236180\pi\)
\(17\)	−4.38844	−	5.50292i	−1.06435	−	1.33466i	−0.939526	−	0.342476i	\(-0.888734\pi\)
							−0.124825	−	0.992179i	\(-0.539837\pi\)
\(19\)	0.991867	+	4.34565i	0.227550	+	0.996961i	0.951630	+	0.307245i	\(0.0994072\pi\)
							−0.724081	+	0.689715i	\(0.757736\pi\)
\(23\)	0.633855	+	2.77710i	0.132168	+	0.579065i	0.997027	+	0.0770509i	\(0.0245504\pi\)
							−0.864859	+	0.502014i	\(0.832592\pi\)
\(29\)	6.00795	−	7.53374i	1.11565	−	1.39898i	0.208576	−	0.978006i	\(-0.433117\pi\)
							0.907073	−	0.420974i	\(-0.138312\pi\)
\(31\)	8.46906	+	4.07848i	1.52109	+	0.732517i	0.993159	−	0.116771i	\(-0.0372544\pi\)
							0.527929	+	0.849289i	\(0.322969\pi\)
\(37\)	−5.75055	−	7.21097i	−0.945385	−	1.18548i	−0.982518	−	0.186165i	\(-0.940394\pi\)
							0.0371330	−	0.999310i	\(-0.488177\pi\)
\(41\)	8.65515			1.35171			0.675854	−	0.737036i	\(-0.263775\pi\)
							0.675854	+	0.737036i	\(0.263775\pi\)
\(43\)	7.99847	−	3.85186i	1.21976	−	0.587403i	0.290512	−	0.956871i	\(-0.406174\pi\)
							0.929243	+	0.369468i	\(0.120460\pi\)
\(47\)	−6.26859			−0.914369			−0.457184	−	0.889372i	\(-0.651142\pi\)
							−0.457184	+	0.889372i	\(0.651142\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.e.a.9.19	✓	264
547.304	even	7	inner	547.2.e.a.304.19	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.e.a.9.19	✓	264	1.1	even	1	trivial
547.2.e.a.304.19	yes	264	547.304	even	7	inner