Embedded newform 731.2.bd.a.381.46

• Label: 731.2.bd.a.381.46
• Level: $731$
• Weight: $2$
• Character: 731.381
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $1024$
• 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			381.46
Character	\(\chi\)	\(=\)	731.381
Dual form			731.2.bd.a.566.46

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.29704 + 0.537254i) q^{2} +(1.63284 - 0.805230i) q^{3} +(-0.0205300 - 0.0205300i) q^{4} +(1.17834 - 3.47128i) q^{5} +(2.55048 - 0.167168i) q^{6} +(1.46865 + 4.32651i) q^{7} +(-1.09011 - 2.63175i) q^{8} +(0.191500 - 0.249568i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q - 24 q^{3} - 32 q^{4} - 24 q^{5} - 8 q^{6} - 24 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{11}{16}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.29704	+	0.537254i	0.917149	+	0.379896i	0.790789	−	0.612089i	\(-0.209671\pi\)
							0.126360	+	0.991984i	\(0.459671\pi\)
\(3\)	1.63284	−	0.805230i	0.942723	−	0.464899i	0.0950818	−	0.995469i	\(-0.469689\pi\)
							0.847641	+	0.530570i	\(0.178022\pi\)
\(5\)	1.17834	−	3.47128i	0.526970	−	1.55240i	−0.278907	−	0.960318i	\(-0.589972\pi\)
							0.805876	−	0.592084i	\(-0.201695\pi\)
\(7\)	1.46865	+	4.32651i	0.555098	+	1.63527i	0.755202	+	0.655492i	\(0.227539\pi\)
							−0.200104	+	0.979775i	\(0.564128\pi\)
\(11\)	1.77625	−	1.18685i	0.535560	−	0.357850i	−0.258189	−	0.966094i	\(-0.583126\pi\)
							0.793750	+	0.608244i	\(0.208126\pi\)
\(13\)	0.213025	+	0.0570798i	0.0590824	+	0.0158311i	0.288239	−	0.957558i	\(-0.406930\pi\)
							−0.229157	+	0.973390i	\(0.573597\pi\)
\(17\)	2.66214	−	3.14849i	0.645664	−	0.763622i
							\(19\)	−5.48781	+	4.21094i	−1.25899	+	0.966057i	−0.258990	+	0.965880i	\(0.583390\pi\)
−1.00000		0.000176708i	\(0.999944\pi\)
\(23\)	−1.51622	+	0.0993782i	−0.316153	+	0.0207218i	−0.222654	−	0.974898i	\(-0.571472\pi\)
							−0.0934994	+	0.995619i	\(0.529805\pi\)
\(29\)	−2.24976	+	1.97299i	−0.417770	+	0.366375i	−0.842272	−	0.539052i	\(-0.818783\pi\)
							0.424502	+	0.905427i	\(0.360449\pi\)
\(31\)	5.69715	−	2.80952i	1.02324	−	0.504605i	0.148352	−	0.988935i	\(-0.452603\pi\)
							0.874886	+	0.484330i	\(0.160936\pi\)
\(37\)	1.75642	−	0.866168i	0.288753	−	0.142397i	−0.292144	−	0.956374i	\(-0.594369\pi\)
							0.580897	+	0.813977i	\(0.302702\pi\)
\(41\)	−2.39453	+	12.0381i	−0.373962	+	1.88003i	0.0928537	+	0.995680i	\(0.470401\pi\)
							−0.466816	+	0.884355i	\(0.654599\pi\)
\(43\)	−3.95190	−	5.23282i	−0.602660	−	0.797998i
							\(47\)	5.57202	−	5.57202i	0.812763	−	0.812763i	−0.172284	−	0.985047i	\(-0.555115\pi\)
0.985047	+	0.172284i	\(0.0551148\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bd.a.381.46	yes	1024
17.5	odd	16	inner	731.2.bd.a.209.19	yes	1024
43.7	odd	6	inner	731.2.bd.a.7.19	✓	1024
731.566	even	48	inner	731.2.bd.a.566.46	yes	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bd.a.7.19	✓	1024	43.7	odd	6	inner
731.2.bd.a.209.19	yes	1024	17.5	odd	16	inner
731.2.bd.a.381.46	yes	1024	1.1	even	1	trivial
731.2.bd.a.566.46	yes	1024	731.566	even	48	inner