Embedded newform 731.2.bd.a.7.18

• Label: 731.2.bd.a.7.18
• Level: $731$
• Weight: $2$
• Character: 731.7
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $1024$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.bd (of order \(48\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(1024\)
Relative dimension:	\(64\) over \(\Q(\zeta_{48})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{48}]$

Embedding invariants

Embedding label			7.18
Character	\(\chi\)	\(=\)	731.7
Dual form			731.2.bd.a.209.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39138 - 0.576328i) q^{2} +(-0.0198737 + 0.303214i) q^{3} +(0.189566 + 0.189566i) q^{4} +(2.18482 + 2.49131i) q^{5} +(0.202402 - 0.410432i) q^{6} +(1.80655 - 2.05998i) q^{7} +(0.998149 + 2.40975i) q^{8} +(2.88279 + 0.379527i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q - 24 q^{3} - 32 q^{4} - 24 q^{5} - 8 q^{6} - 24 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{11}{16}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39138	−	0.576328i	−0.983853	−	0.407525i	−0.168002	−	0.985787i	\(-0.553731\pi\)
							−0.815851	+	0.578261i	\(0.803731\pi\)
\(3\)	−0.0198737	+	0.303214i	−0.0114741	+	0.175061i	0.988275	+	0.152681i	\(0.0487908\pi\)
							−0.999750	+	0.0223793i	\(0.992876\pi\)
\(5\)	2.18482	+	2.49131i	0.977080	+	1.11415i	0.993449	+	0.114280i	\(0.0364562\pi\)
							−0.0163685	+	0.999866i	\(0.505210\pi\)
\(7\)	1.80655	−	2.05998i	0.682812	−	0.778598i	−0.301796	−	0.953372i	\(-0.597586\pi\)
							0.984609	+	0.174774i	\(0.0559195\pi\)
\(11\)	−2.15004	+	1.43661i	−0.648262	+	0.433155i	−0.835748	−	0.549112i	\(-0.814966\pi\)
							0.187487	+	0.982267i	\(0.439966\pi\)
\(13\)	1.20122	+	4.48303i	0.333160	+	1.24337i	0.905850	+	0.423598i	\(0.139233\pi\)
							−0.572691	+	0.819772i	\(0.694100\pi\)
\(17\)	2.45523	+	3.31238i	0.595481	+	0.803370i
							\(19\)	−0.792346	−	6.01846i	−0.181777	−	1.38073i	−0.802477	−	0.596683i	\(-0.796485\pi\)
0.620701	−	0.784048i	\(-0.286848\pi\)
\(23\)	−2.15758	+	4.37514i	−0.449887	+	0.912280i	0.547272	+	0.836955i	\(0.315666\pi\)
							−0.997159	+	0.0753255i	\(0.976000\pi\)
\(29\)	−9.90000	−	3.36060i	−1.83838	−	0.624047i	−0.996423	−	0.0845073i	\(-0.973068\pi\)
							−0.841960	−	0.539540i	\(-0.818598\pi\)
\(31\)	0.113858	−	1.73713i	0.0204495	−	0.311999i	−0.975365	−	0.220598i	\(-0.929199\pi\)
							0.995814	−	0.0914003i	\(-0.0291343\pi\)
\(37\)	0.111614	−	1.70290i	0.0183492	−	0.279955i	−0.978818	−	0.204734i	\(-0.934367\pi\)
							0.997167	−	0.0752211i	\(-0.0239663\pi\)
\(41\)	−1.84897	+	9.29538i	−0.288760	+	1.45169i	0.515230	+	0.857052i	\(0.327706\pi\)
							−0.803990	+	0.594643i	\(0.797294\pi\)
\(43\)	6.53567	+	0.533883i	0.996680	+	0.0814163i
							\(47\)	−3.86724	+	3.86724i	−0.564095	+	0.564095i	−0.930468	−	0.366373i	\(-0.880599\pi\)
0.366373	+	0.930468i	\(0.380599\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bd.a.7.18	✓	1024
17.5	odd	16	inner	731.2.bd.a.566.47	yes	1024
43.37	odd	6	inner	731.2.bd.a.381.47	yes	1024
731.209	even	48	inner	731.2.bd.a.209.18	yes	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bd.a.7.18	✓	1024	1.1	even	1	trivial
731.2.bd.a.209.18	yes	1024	731.209	even	48	inner
731.2.bd.a.381.47	yes	1024	43.37	odd	6	inner
731.2.bd.a.566.47	yes	1024	17.5	odd	16	inner