Embedded newform 547.2.h.a.13.7

• Label: 547.2.h.a.13.7
• Level: $547$
• Weight: $2$
• Character: 547.13
• Analytic conductor: $4.368$
• Analytic rank: $0$
• Dimension: $540$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	547.h (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36781699056\)
Analytic rank:	\(0\)
Dimension:	\(540\)
Relative dimension:	\(45\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			13.7
Character	\(\chi\)	\(=\)	547.13
Dual form			547.2.h.a.505.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.151474 + 2.02129i) q^{2} +(-0.549391 + 0.688914i) q^{3} +(-2.08499 - 0.314262i) q^{4} +(-0.809812 + 0.751396i) q^{5} +(-1.30927 - 1.21483i) q^{6} +(2.89077 - 0.435714i) q^{7} +(0.0489578 - 0.214498i) q^{8} +(0.494790 + 2.16782i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 540 q - 13 q^{2} - 10 q^{3} + 33 q^{4} - 15 q^{5} + 10 q^{6} - 2 q^{7} - 26 q^{8} - 96 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{11}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.151474	+	2.02129i	−0.107109	+	1.42927i	0.641942	+	0.766753i	\(0.278129\pi\)
							−0.749051	+	0.662512i	\(0.769490\pi\)
\(3\)	−0.549391	+	0.688914i	−0.317191	+	0.397745i	−0.914711	−	0.404109i	\(-0.867582\pi\)
							0.597520	+	0.801854i	\(0.296153\pi\)
\(5\)	−0.809812	+	0.751396i	−0.362159	+	0.336034i	−0.840174	−	0.542317i	\(-0.817547\pi\)
							0.478015	+	0.878352i	\(0.341357\pi\)
\(7\)	2.89077	−	0.435714i	1.09261	−	0.164684i	0.422074	−	0.906562i	\(-0.361302\pi\)
							0.670536	+	0.741877i	\(0.266064\pi\)
\(11\)	0.712738	−	1.23450i	0.214899	−	0.372215i	−0.738343	−	0.674426i	\(-0.764391\pi\)
							0.953241	+	0.302211i	\(0.0977246\pi\)
\(13\)	−5.41654	+	3.69293i	−1.50228	+	1.02424i	−0.516480	+	0.856299i	\(0.672758\pi\)
							−0.985798	+	0.167936i	\(0.946290\pi\)
\(17\)	−1.46268	−	0.451176i	−0.354752	−	0.109426i	0.112257	−	0.993679i	\(-0.464192\pi\)
							−0.467008	+	0.884253i	\(0.654668\pi\)
\(19\)	0.391405	+	5.22294i	0.0897946	+	1.19822i	0.842067	+	0.539373i	\(0.181339\pi\)
							−0.752272	+	0.658852i	\(0.771042\pi\)
\(23\)	−0.458794	−	6.12218i	−0.0956652	−	1.27656i	−0.814000	−	0.580865i	\(-0.802714\pi\)
							0.718335	−	0.695698i	\(-0.244905\pi\)
\(29\)	0.523375	−	2.29306i	0.0971883	−	0.425810i	−0.902803	−	0.430055i	\(-0.858494\pi\)
							0.999991	+	0.00424513i	\(0.00135127\pi\)
\(31\)	5.38436	−	6.75178i	0.967060	−	1.21265i	−0.0100564	−	0.999949i	\(-0.503201\pi\)
							0.977116	−	0.212705i	\(-0.0682275\pi\)
\(37\)	6.37060	+	1.96507i	1.04732	+	0.323055i	0.770215	−	0.637784i	\(-0.220149\pi\)
							0.277105	+	0.960840i	\(0.410625\pi\)
\(41\)	−4.52392	+	7.83566i	−0.706518	+	1.22373i	0.259623	+	0.965710i	\(0.416402\pi\)
							−0.966141	+	0.258015i	\(0.916932\pi\)
\(43\)	−2.15826	+	0.325306i	−0.329132	+	0.0496086i	−0.311530	−	0.950236i	\(-0.600841\pi\)
							−0.0176022	+	0.999845i	\(0.505603\pi\)
\(47\)	−0.841461	−	1.45745i	−0.122740	−	0.212591i	0.798107	−	0.602515i	\(-0.205835\pi\)
							−0.920847	+	0.389924i	\(0.872501\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.h.a.13.7	✓	540
547.505	even	21	inner	547.2.h.a.505.7	yes	540

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.h.a.13.7	✓	540	1.1	even	1	trivial
547.2.h.a.505.7	yes	540	547.505	even	21	inner