Embedded newform 531.2.m.a.130.46

• Label: 531.2.m.a.130.46
• Level: $531$
• Weight: $2$
• Character: 531.130
• Analytic conductor: $4.240$
• Analytic rank: $0$
• Dimension: $3248$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	531.m (of order \(87\), degree \(56\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.24005634733\)
Analytic rank:	\(0\)
Dimension:	\(3248\)
Relative dimension:	\(58\) over \(\Q(\zeta_{87})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{87}]$

Embedding invariants

Embedding label			130.46
Character	\(\chi\)	\(=\)	531.130
Dual form			531.2.m.a.241.46

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28205 + 1.12946i) q^{2} +(-0.120465 + 1.72786i) q^{3} +(0.115881 + 0.911998i) q^{4} +(3.02366 + 0.439803i) q^{5} +(-2.10598 + 2.07914i) q^{6} +(0.0341869 + 0.377666i) q^{7} +(1.03620 - 1.52828i) q^{8} +(-2.97098 - 0.416292i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3248 q - 29 q^{2} - 58 q^{3} + 29 q^{4} - 25 q^{5} - 66 q^{6} - 27 q^{7} - 116 q^{8} - 62 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/531\mathbb{Z}\right)^\times\).

\(n\)	\(119\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{26}{29}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.28205	+	1.12946i	0.906547	+	0.798647i	0.980071	−	0.198646i	\(-0.0636545\pi\)
							−0.0735241	+	0.997293i	\(0.523425\pi\)
\(3\)	−0.120465	+	1.72786i	−0.0695504	+	0.997578i
							\(5\)	3.02366	+	0.439803i	1.35222	+	0.196686i	0.781216	−	0.624261i	\(-0.214600\pi\)
0.571005	+	0.820947i	\(0.306554\pi\)
\(7\)	0.0341869	+	0.377666i	0.0129214	+	0.142744i	0.999828	−	0.0185492i	\(-0.00590473\pi\)
							−0.986907	+	0.161293i	\(0.948433\pi\)
\(11\)	−4.24602	+	4.32338i	−1.28022	+	1.30355i	−0.349908	+	0.936784i	\(0.613787\pi\)
							−0.930314	+	0.366763i	\(0.880466\pi\)
\(13\)	2.03614	+	0.0735575i	0.564723	+	0.0204012i	0.316993	−	0.948428i	\(-0.397327\pi\)
							0.247730	+	0.968829i	\(0.420315\pi\)
\(17\)	3.18643	−	1.47420i	0.772823	−	0.357546i	0.00648141	−	0.999979i	\(-0.497937\pi\)
							0.766342	+	0.642433i	\(0.222075\pi\)
\(19\)	−2.28422	−	0.502794i	−0.524036	−	0.115349i	−0.0549286	−	0.998490i	\(-0.517493\pi\)
							−0.469107	+	0.883141i	\(0.655424\pi\)
\(23\)	0.0436529	+	0.0164860i	0.00910226	+	0.00343758i	0.357853	−	0.933778i	\(-0.383509\pi\)
							−0.348751	+	0.937216i	\(0.613394\pi\)
\(29\)	1.46141	+	7.26130i	0.271376	+	1.34839i	0.847400	+	0.530956i	\(0.178167\pi\)
							−0.576024	+	0.817433i	\(0.695396\pi\)
\(31\)	2.37335	−	7.48801i	0.426267	−	1.34489i	−0.462538	−	0.886600i	\(-0.653061\pi\)
							0.888804	−	0.458287i	\(-0.151537\pi\)
\(37\)	−4.79080	−	7.06590i	−0.787603	−	1.16163i	−0.983483	−	0.180999i	\(-0.942067\pi\)
							0.195880	−	0.980628i	\(-0.437244\pi\)
\(41\)	−3.98441	−	3.26240i	−0.622260	−	0.509502i	0.268913	−	0.963164i	\(-0.413335\pi\)
							−0.891173	+	0.453663i	\(0.850117\pi\)
\(43\)	−11.0461	+	2.85316i	−1.68451	+	0.435103i	−0.965606	−	0.260009i	\(-0.916274\pi\)
							−0.718902	+	0.695112i	\(0.755355\pi\)
\(47\)	−5.78738	+	0.841798i	−0.844176	+	0.122789i	−0.552277	−	0.833660i	\(-0.686241\pi\)
							−0.291899	+	0.956449i	\(0.594287\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.m.a.130.46	✓	3248
9.7	even	3	inner	531.2.m.a.484.46	yes	3248
59.5	even	29	inner	531.2.m.a.418.46	yes	3248
531.241	even	87	inner	531.2.m.a.241.46	yes	3248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.2.m.a.130.46	✓	3248	1.1	even	1	trivial
531.2.m.a.241.46	yes	3248	531.241	even	87	inner
531.2.m.a.418.46	yes	3248	59.5	even	29	inner
531.2.m.a.484.46	yes	3248	9.7	even	3	inner