Embedded newform 731.2.p.a.594.6

• Label: 731.2.p.a.594.6
• Level: $731$
• Weight: $2$
• Character: 731.594
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.p (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(64\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			594.6
Character	\(\chi\)	\(=\)	731.594
Dual form			731.2.p.a.16.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.534495 + 2.34178i) q^{2} +(2.00623 - 0.457909i) q^{3} +(-3.39629 - 1.63557i) q^{4} +(-1.96143 - 1.56419i) q^{5} +4.94289i q^{6} -1.32734i q^{7} +(2.65019 - 3.32324i) q^{8} +(1.11237 - 0.535690i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q - 12 q^{2} - 72 q^{4} - 8 q^{8} + 62 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.534495	+	2.34178i	−0.377945	+	1.65589i	0.325804	+	0.945437i	\(0.394365\pi\)
							−0.703749	+	0.710448i	\(0.748492\pi\)
\(3\)	2.00623	−	0.457909i	1.15830	−	0.264374i	0.400145	−	0.916452i	\(-0.368960\pi\)
							0.758152	+	0.652078i	\(0.226102\pi\)
\(5\)	−1.96143	−	1.56419i	−0.877176	−	0.699525i	0.0775569	−	0.996988i	\(-0.475288\pi\)
							−0.954733	+	0.297463i	\(0.903859\pi\)
\(7\)		−	1.32734i		−	0.501689i	−0.968027	−	0.250845i	\(-0.919292\pi\)
							0.968027	−	0.250845i	\(-0.0807083\pi\)
\(11\)	−0.0593297	−	0.123199i	−0.0178886	−	0.0371460i	0.891833	−	0.452365i	\(-0.149419\pi\)
							−0.909722	+	0.415219i	\(0.863705\pi\)
\(13\)	−1.70718	+	2.14073i	−0.473485	+	0.593732i	−0.960021	−	0.279929i	\(-0.909689\pi\)
							0.486535	+	0.873661i	\(0.338260\pi\)
\(17\)	−2.08160	−	3.55907i	−0.504861	−	0.863201i
							\(19\)	−7.27116	−	3.50160i	−1.66812	−	0.803323i	−0.998141	−	0.0609409i	\(-0.980590\pi\)
−0.669976	−	0.742382i	\(-0.733696\pi\)
\(23\)	−2.13546	−	4.43433i	−0.445275	−	0.924623i	−0.995950	−	0.0899033i	\(-0.971344\pi\)
							0.550676	−	0.834719i	\(-0.314370\pi\)
\(29\)	−7.35127	−	1.67788i	−1.36510	−	0.311575i	−0.523659	−	0.851928i	\(-0.675433\pi\)
							−0.841438	+	0.540353i	\(0.818291\pi\)
\(31\)	10.5326	+	2.40399i	1.89170	+	0.431769i	0.999791	−	0.0204401i	\(-0.00650675\pi\)
							0.891912	+	0.452209i	\(0.149364\pi\)
\(37\)		−	2.07349i		−	0.340879i	−0.985368	−	0.170440i	\(-0.945481\pi\)
							0.985368	−	0.170440i	\(-0.0545187\pi\)
\(41\)	5.09880	+	1.16377i	0.796299	+	0.181750i	0.601269	−	0.799047i	\(-0.294662\pi\)
							0.195031	+	0.980797i	\(0.437519\pi\)
\(43\)	4.74464	+	4.52641i	0.723551	+	0.690271i
							\(47\)	−3.32640	−	1.60191i	−0.485205	−	0.233663i	0.175256	−	0.984523i	\(-0.443925\pi\)
−0.660461	+	0.750860i	\(0.729639\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.p.a.594.6	yes	384
17.16	even	2	inner	731.2.p.a.594.5	yes	384
43.16	even	7	inner	731.2.p.a.16.5	✓	384
731.16	even	14	inner	731.2.p.a.16.6	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.p.a.16.5	✓	384	43.16	even	7	inner
731.2.p.a.16.6	yes	384	731.16	even	14	inner
731.2.p.a.594.5	yes	384	17.16	even	2	inner
731.2.p.a.594.6	yes	384	1.1	even	1	trivial