Embedded newform 731.2.j.a.509.16

• Label: 731.2.j.a.509.16
• Level: $731$
• Weight: $2$
• Character: 731.509
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(64\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			509.16
Character	\(\chi\)	\(=\)	731.509
Dual form			731.2.j.a.135.16

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.73933 q^{2} +(1.10646 - 0.638817i) q^{3} +1.02526 q^{4} +(-2.28826 + 1.32113i) q^{5} +(-1.92450 + 1.11111i) q^{6} +(4.26158 + 2.46042i) q^{7} +1.69540 q^{8} +(-0.683825 + 1.18442i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 4 q^{2} + 116 q^{4} - 12 q^{8} + 70 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.73933			−1.22989			−0.614945	−	0.788570i	\(-0.710822\pi\)
							−0.614945	+	0.788570i	\(0.710822\pi\)
\(3\)	1.10646	−	0.638817i	0.638817	−	0.368821i	−0.145342	−	0.989382i	\(-0.546428\pi\)
							0.784159	+	0.620560i	\(0.213095\pi\)
\(5\)	−2.28826	+	1.32113i	−1.02334	+	0.590827i	−0.915070	−	0.403295i	\(-0.867865\pi\)
							−0.108272	+	0.994121i	\(0.534532\pi\)
\(7\)	4.26158	+	2.46042i	1.61072	+	0.929952i	0.989202	+	0.146556i	\(0.0468189\pi\)
							0.621522	+	0.783396i	\(0.286514\pi\)
\(11\)			0.464939i			0.140184i	0.997541	+	0.0700922i	\(0.0223293\pi\)
							−0.997541	+	0.0700922i	\(0.977671\pi\)
\(13\)	−2.87460	+	4.97895i	−0.797270	+	1.38091i	0.124117	+	0.992268i	\(0.460390\pi\)
							−0.921388	+	0.388645i	\(0.872943\pi\)
\(17\)	−0.621007	−	4.07607i	−0.150616	−	0.988592i
							\(19\)	−2.00439	−	3.47171i	−0.459839	−	0.796465i	0.539113	−	0.842234i	\(-0.318760\pi\)
−0.998952	+	0.0457685i	\(0.985426\pi\)
\(23\)	2.36314	−	1.36436i	0.492748	−	0.284488i	−0.232966	−	0.972485i	\(-0.574843\pi\)
							0.725714	+	0.687997i	\(0.241510\pi\)
\(29\)	−6.94958	−	4.01234i	−1.29050	−	0.745073i	−0.311761	−	0.950161i	\(-0.600919\pi\)
							−0.978744	+	0.205087i	\(0.934252\pi\)
\(31\)	−5.73772	+	3.31268i	−1.03053	+	0.594974i	−0.917135	−	0.398576i	\(-0.869504\pi\)
							−0.113391	+	0.993550i	\(0.536171\pi\)
\(37\)	0.115445	−	0.0666519i	0.0189790	−	0.0109575i	−0.490481	−	0.871452i	\(-0.663179\pi\)
							0.509459	+	0.860495i	\(0.329845\pi\)
\(41\)		−	0.696605i		−	0.108791i	−0.998519	−	0.0543957i	\(-0.982677\pi\)
							0.998519	−	0.0543957i	\(-0.0173232\pi\)
\(43\)	1.89825	+	6.27668i	0.289480	+	0.957184i
							\(47\)	−3.34439			−0.487830			−0.243915	−	0.969797i	\(-0.578432\pi\)
−0.243915	+	0.969797i	\(0.578432\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.j.a.509.16	yes	128
17.16	even	2	inner	731.2.j.a.509.15	yes	128
43.6	even	3	inner	731.2.j.a.135.15	✓	128
731.135	even	6	inner	731.2.j.a.135.16	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.j.a.135.15	✓	128	43.6	even	3	inner
731.2.j.a.135.16	yes	128	731.135	even	6	inner
731.2.j.a.509.15	yes	128	17.16	even	2	inner
731.2.j.a.509.16	yes	128	1.1	even	1	trivial