Embedded newform 731.2.v.a.36.12

• Label: 731.2.v.a.36.12
• Level: $731$
• Weight: $2$
• Character: 731.36
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $512$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.v (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(512\)
Relative dimension:	\(64\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			36.12
Character	\(\chi\)	\(=\)	731.36
Dual form			731.2.v.a.264.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.48255 - 1.48255i) q^{2} +(-0.314627 - 0.241421i) q^{3} +2.39590i q^{4} +(-0.727962 + 0.948699i) q^{5} +(0.108530 + 0.824368i) q^{6} +(1.97310 + 2.57139i) q^{7} +(0.586944 - 0.586944i) q^{8} +(-0.735752 - 2.74586i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q - 16 q^{2} - 4 q^{3} - 12 q^{5} - 4 q^{6} - 4 q^{7} + 16 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.48255	−	1.48255i	−1.04832	−	1.04832i	−0.998772	−	0.0495486i	\(-0.984222\pi\)
							−0.0495486	−	0.998772i	\(-0.515778\pi\)
\(3\)	−0.314627	−	0.241421i	−0.181650	−	0.139385i	0.513934	−	0.857830i	\(-0.328188\pi\)
							−0.695584	+	0.718445i	\(0.744854\pi\)
\(5\)	−0.727962	+	0.948699i	−0.325555	+	0.424271i	−0.927243	−	0.374460i	\(-0.877828\pi\)
							0.601689	+	0.798731i	\(0.294495\pi\)
\(7\)	1.97310	+	2.57139i	0.745760	+	0.971893i	0.999989	+	0.00474919i	\(0.00151172\pi\)
							−0.254229	+	0.967144i	\(0.581822\pi\)
\(11\)	−4.11411	−	1.70412i	−1.24045	−	0.513811i	−0.336595	−	0.941650i	\(-0.609275\pi\)
							−0.903855	+	0.427838i	\(0.859275\pi\)
\(13\)	−5.22395	−	3.01605i	−1.44886	−	0.836502i	−0.450450	−	0.892802i	\(-0.648736\pi\)
							−0.998414	+	0.0562996i	\(0.982070\pi\)
\(17\)	3.74614	+	1.72233i	0.908573	+	0.417726i
							\(19\)	1.48754	−	5.55158i	0.341265	−	1.27362i	−0.555649	−	0.831417i	\(-0.687530\pi\)
0.896915	−	0.442203i	\(-0.145803\pi\)
\(23\)	1.14644	+	8.70811i	0.239050	+	1.81577i	0.517910	+	0.855435i	\(0.326710\pi\)
							−0.278860	+	0.960332i	\(0.589957\pi\)
\(29\)	1.96488	+	0.258681i	0.364868	+	0.0480358i	0.310732	−	0.950498i	\(-0.399426\pi\)
							0.0541367	+	0.998534i	\(0.482759\pi\)
\(31\)	7.11831	+	5.46207i	1.27849	+	0.981017i	0.999787	+	0.0206381i	\(0.00656977\pi\)
							0.278699	+	0.960379i	\(0.410097\pi\)
\(37\)	7.77763	+	5.96798i	1.27863	+	0.981130i	0.999784	+	0.0207955i	\(0.00661988\pi\)
							0.278850	+	0.960335i	\(0.410047\pi\)
\(41\)	0.0960526	−	0.231892i	0.0150009	−	0.0362154i	−0.916202	−	0.400718i	\(-0.868761\pi\)
							0.931203	+	0.364502i	\(0.118761\pi\)
\(43\)	5.34359	+	3.80079i	0.814891	+	0.579615i
							\(47\)		−	2.63325i		−	0.384099i	−0.981385	−	0.192049i	\(-0.938487\pi\)
0.981385	−	0.192049i	\(-0.0615133\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.v.a.36.12	✓	512
17.9	even	8	inner	731.2.v.a.638.12	yes	512
43.6	even	3	inner	731.2.v.a.393.12	yes	512
731.264	even	24	inner	731.2.v.a.264.12	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.v.a.36.12	✓	512	1.1	even	1	trivial
731.2.v.a.264.12	yes	512	731.264	even	24	inner
731.2.v.a.393.12	yes	512	43.6	even	3	inner
731.2.v.a.638.12	yes	512	17.9	even	8	inner