Embedded newform 531.2.p.a.128.34

• Label: 531.2.p.a.128.34
• Level: $531$
• Weight: $2$
• Character: 531.128
• 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.p (of order \(174\), degree \(56\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.24005634733\)
Analytic rank:	\(0\)
Dimension:	\(3248\)
Relative dimension:	\(58\) over \(\Q(\zeta_{174})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{174}]$

Embedding invariants

Embedding label			128.34
Character	\(\chi\)	\(=\)	531.128
Dual form			531.2.p.a.419.34

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.183378 + 0.232399i) q^{2} +(-0.0962418 - 1.72937i) q^{3} +(0.444753 - 1.85993i) q^{4} +(0.210423 - 1.15275i) q^{5} +(0.384256 - 0.339496i) q^{6} +(4.24856 - 2.35168i) q^{7} +(1.05115 - 0.486314i) q^{8} +(-2.98148 + 0.332876i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3248 q - 87 q^{2} - 54 q^{3} + 29 q^{4} - 87 q^{5} - 58 q^{6} - 27 q^{7} - 62 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(119\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{58}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.183378	+	0.232399i	0.129668	+	0.164331i	0.845560	−	0.533881i	\(-0.179267\pi\)
							−0.715892	+	0.698211i	\(0.753980\pi\)
\(3\)	−0.0962418	−	1.72937i	−0.0555652	−	0.998455i
							\(5\)	0.210423	−	1.15275i	0.0941039	−	0.515527i	−0.902323	−	0.431062i	\(-0.858139\pi\)
0.996426	−	0.0844656i	\(-0.0269183\pi\)
\(7\)	4.24856	−	2.35168i	1.60581	−	0.888850i	0.610506	−	0.792012i	\(-0.290966\pi\)
							0.995300	−	0.0968382i	\(-0.0308729\pi\)
\(11\)	0.891549	+	0.581270i	0.268812	+	0.175260i	0.673483	−	0.739203i	\(-0.264798\pi\)
							−0.404670	+	0.914463i	\(0.632614\pi\)
\(13\)	0.257966	−	0.235669i	0.0715469	−	0.0653629i	−0.637914	−	0.770107i	\(-0.720203\pi\)
							0.709461	+	0.704744i	\(0.248938\pi\)
\(17\)	3.10800	+	5.16553i	0.753800	+	1.25283i	0.961546	+	0.274643i	\(0.0885597\pi\)
							−0.207746	+	0.978183i	\(0.566613\pi\)
\(19\)	−0.805594	+	2.90149i	−0.184816	+	0.665647i	0.811836	+	0.583886i	\(0.198468\pi\)
							−0.996652	+	0.0817615i	\(0.973945\pi\)
\(23\)	−2.73896	+	5.65018i	−0.571112	+	1.17814i	0.394678	+	0.918820i	\(0.370856\pi\)
							−0.965790	+	0.259325i	\(0.916500\pi\)
\(29\)	−0.203729	+	1.40064i	−0.0378315	+	0.260092i	−0.999920	−	0.0126448i	\(-0.995975\pi\)
							0.962089	+	0.272737i	\(0.0879289\pi\)
\(31\)	−3.76520	−	3.69782i	−0.676250	−	0.664149i	0.278884	−	0.960325i	\(-0.410036\pi\)
							−0.955133	+	0.296176i	\(0.904289\pi\)
\(37\)	−0.0424636	+	0.0917836i	−0.00698097	+	0.0150891i	−0.911035	−	0.412330i	\(-0.864715\pi\)
							0.904054	+	0.427419i	\(0.140577\pi\)
\(41\)	0.328728	−	4.54381i	0.0513387	−	0.709624i	−0.906480	−	0.422249i	\(-0.861241\pi\)
							0.957818	−	0.287374i	\(-0.0927823\pi\)
\(43\)	−1.23447	+	2.43363i	−0.188255	+	0.371126i	−0.966433	−	0.256917i	\(-0.917293\pi\)
							0.778179	+	0.628043i	\(0.216144\pi\)
\(47\)	−6.51640	+	1.18950i	−0.950514	+	0.173506i	−0.632483	−	0.774574i	\(-0.717964\pi\)
							−0.318031	+	0.948080i	\(0.603022\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.p.a.128.34	yes	3248
9.5	odd	6	inner	531.2.p.a.482.34	yes	3248
59.6	odd	58	inner	531.2.p.a.65.34	✓	3248
531.419	even	174	inner	531.2.p.a.419.34	yes	3248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.2.p.a.65.34	✓	3248	59.6	odd	58	inner
531.2.p.a.128.34	yes	3248	1.1	even	1	trivial
531.2.p.a.419.34	yes	3248	531.419	even	174	inner
531.2.p.a.482.34	yes	3248	9.5	odd	6	inner