Embedded newform 531.2.p.a.128.35

• Label: 531.2.p.a.128.35
• 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			128.35
Character	\(\chi\)	\(=\)	531.128
Dual form			531.2.p.a.419.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.233609 + 0.296057i) q^{2} +(1.64141 + 0.552961i) q^{3} +(0.432058 - 1.80684i) q^{4} +(0.181210 - 0.992718i) q^{5} +(0.219741 + 0.615128i) q^{6} +(-0.0231777 + 0.0128294i) q^{7} +(1.32040 - 0.610881i) q^{8} +(2.38847 + 1.81527i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3248 q - 87 q^{2} - 54 q^{3} + 29 q^{4} - 87 q^{5} - 58 q^{6} - 27 q^{7} - 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.233609	+	0.296057i	0.165187	+	0.209344i	0.860621	−	0.509245i	\(-0.170075\pi\)
							−0.695435	+	0.718589i	\(0.744788\pi\)
\(3\)	1.64141	+	0.552961i	0.947670	+	0.319252i
							\(5\)	0.181210	−	0.992718i	0.0810395	−	0.443957i	−0.917854	−	0.396919i	\(-0.870079\pi\)
0.998893	−	0.0470377i	\(-0.0149781\pi\)
\(7\)	−0.0231777	+	0.0128294i	−0.00876036	+	0.00484906i	−0.488657	−	0.872476i	\(-0.662513\pi\)
							0.479897	+	0.877325i	\(0.340674\pi\)
\(11\)	2.07197	+	1.35088i	0.624724	+	0.407306i	0.819124	−	0.573616i	\(-0.194460\pi\)
							−0.194400	+	0.980922i	\(0.562276\pi\)
\(13\)	−2.16943	+	1.98192i	−0.601690	+	0.549684i	−0.916788	−	0.399375i	\(-0.869227\pi\)
							0.315097	+	0.949059i	\(0.397963\pi\)
\(17\)	−0.397663	−	0.660921i	−0.0964475	−	0.160297i	0.804877	−	0.593442i	\(-0.202231\pi\)
							−0.901324	+	0.433145i	\(0.857404\pi\)
\(19\)	0.000568369	−	0.00204708i	0.000130393	−	0.000469632i	−0.963485	−	0.267763i	\(-0.913716\pi\)
							0.963615	+	0.267294i	\(0.0861293\pi\)
\(23\)	3.30935	−	6.82684i	0.690047	−	1.42349i	−0.205558	−	0.978645i	\(-0.565901\pi\)
							0.895606	−	0.444849i	\(-0.146743\pi\)
\(29\)	0.741743	−	5.09950i	0.137738	−	0.946953i	−0.800097	−	0.599871i	\(-0.795219\pi\)
							0.937835	−	0.347082i	\(-0.112827\pi\)
\(31\)	−3.24300	−	3.18497i	−0.582461	−	0.572038i	0.348430	−	0.937335i	\(-0.386715\pi\)
							−0.930891	+	0.365296i	\(0.880968\pi\)
\(37\)	0.0733303	−	0.158501i	0.0120554	−	0.0260574i	−0.901454	−	0.432875i	\(-0.857499\pi\)
							0.913510	+	0.406817i	\(0.133361\pi\)
\(41\)	−0.288412	+	3.98655i	−0.0450425	+	0.622595i	0.925048	+	0.379851i	\(0.124025\pi\)
							−0.970090	+	0.242744i	\(0.921952\pi\)
\(43\)	−5.84941	+	11.5315i	−0.892027	+	1.75854i	−0.296905	+	0.954907i	\(0.595954\pi\)
							−0.595122	+	0.803636i	\(0.702896\pi\)
\(47\)	−11.7298	+	2.14115i	−1.71097	+	0.312319i	−0.944143	−	0.329537i	\(-0.893107\pi\)
							−0.766826	+	0.641855i	\(0.778165\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.p.a.128.35	yes	3248
9.5	odd	6	inner	531.2.p.a.482.35	yes	3248
59.6	odd	58	inner	531.2.p.a.65.35	✓	3248
531.419	even	174	inner	531.2.p.a.419.35	yes	3248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.2.p.a.65.35	✓	3248	59.6	odd	58	inner
531.2.p.a.128.35	yes	3248	1.1	even	1	trivial
531.2.p.a.419.35	yes	3248	531.419	even	174	inner
531.2.p.a.482.35	yes	3248	9.5	odd	6	inner