Embedded newform 531.2.p.a.128.38

• Label: 531.2.p.a.128.38
• 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			128.38
Character	\(\chi\)	\(=\)	531.128
Dual form			531.2.p.a.419.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.592454 + 0.750827i) q^{2} +(-1.23663 - 1.21274i) q^{3} +(0.252395 - 1.05550i) q^{4} +(0.656224 - 3.59498i) q^{5} +(0.177914 - 1.64699i) q^{6} +(-0.329974 + 0.182648i) q^{7} +(2.67808 - 1.23901i) q^{8} +(0.0585082 + 2.99943i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3248 q - 87 q^{2} - 54 q^{3} + 29 q^{4} - 87 q^{5} - 58 q^{6} - 27 q^{7} - 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.592454	+	0.750827i	0.418928	+	0.530915i	0.948258	−	0.317501i	\(-0.102844\pi\)
							−0.529329	+	0.848416i	\(0.677556\pi\)
\(3\)	−1.23663	−	1.21274i	−0.713969	−	0.700178i
							\(5\)	0.656224	−	3.59498i	0.293472	−	1.60772i	−0.420326	−	0.907373i	\(-0.638084\pi\)
0.713799	−	0.700351i	\(-0.246973\pi\)
\(7\)	−0.329974	+	0.182648i	−0.124718	+	0.0690344i	−0.545411	−	0.838169i	\(-0.683626\pi\)
							0.420692	+	0.907203i	\(0.361787\pi\)
\(11\)	1.78110	+	1.16124i	0.537023	+	0.350127i	0.785846	−	0.618422i	\(-0.212228\pi\)
							−0.248823	+	0.968549i	\(0.580044\pi\)
\(13\)	−1.64343	+	1.50139i	−0.455806	+	0.416410i	−0.869444	−	0.494032i	\(-0.835523\pi\)
							0.413637	+	0.910442i	\(0.364258\pi\)
\(17\)	−2.43288	−	4.04348i	−0.590061	−	0.980689i	−0.997716	−	0.0675528i	\(-0.978481\pi\)
							0.407655	−	0.913136i	\(-0.366347\pi\)
\(19\)	1.20289	−	4.33240i	0.275961	−	0.993921i	−0.687525	−	0.726161i	\(-0.741303\pi\)
							0.963486	−	0.267760i	\(-0.0862835\pi\)
\(23\)	−0.0542625	+	0.111938i	−0.0113145	+	0.0233406i	−0.905428	−	0.424499i	\(-0.860450\pi\)
							0.894114	+	0.447840i	\(0.147807\pi\)
\(29\)	−1.43346	+	9.85506i	−0.266186	+	1.83004i	0.243764	+	0.969835i	\(0.421618\pi\)
							−0.509950	+	0.860204i	\(0.670336\pi\)
\(31\)	−3.20022	−	3.14295i	−0.574776	−	0.564491i	0.353947	−	0.935265i	\(-0.384839\pi\)
							−0.928723	+	0.370775i	\(0.879092\pi\)
\(37\)	0.959245	−	2.07337i	0.157699	−	0.340861i	−0.812580	−	0.582850i	\(-0.801938\pi\)
							0.970279	+	0.241989i	\(0.0777998\pi\)
\(41\)	−0.101735	+	1.40623i	−0.0158884	+	0.219615i	0.983386	+	0.181527i	\(0.0581039\pi\)
							−0.999274	+	0.0380886i	\(0.987873\pi\)
\(43\)	−1.46475	+	2.88761i	−0.223372	+	0.440356i	−0.975904	−	0.218199i	\(-0.929982\pi\)
							0.752532	+	0.658556i	\(0.228832\pi\)
\(47\)	6.69063	−	1.22130i	0.975928	−	0.178145i	0.332040	−	0.943265i	\(-0.392263\pi\)
							0.643888	+	0.765120i	\(0.277320\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.p.a.128.38	yes	3248
9.5	odd	6	inner	531.2.p.a.482.38	yes	3248
59.6	odd	58	inner	531.2.p.a.65.38	✓	3248
531.419	even	174	inner	531.2.p.a.419.38	yes	3248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.2.p.a.65.38	✓	3248	59.6	odd	58	inner
531.2.p.a.128.38	yes	3248	1.1	even	1	trivial
531.2.p.a.419.38	yes	3248	531.419	even	174	inner
531.2.p.a.482.38	yes	3248	9.5	odd	6	inner