Embedded newform 731.2.n.a.208.19

• Label: 731.2.n.a.208.19
• Level: $731$
• Weight: $2$
• Character: 731.208
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.n (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(64\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			208.19
Character	\(\chi\)	\(=\)	731.208
Dual form			731.2.n.a.608.46

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.44677i q^{2} +(-2.41714 - 0.647670i) q^{3} -0.0931316 q^{4} +(0.138660 + 0.0371537i) q^{5} +(-0.937027 + 3.49703i) q^{6} +(0.626167 - 0.167781i) q^{7} -2.75879i q^{8} +(2.82501 + 1.63102i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.44677i		−	1.02302i	−0.859278	−	0.511509i	\(-0.829087\pi\)
							0.859278	−	0.511509i	\(-0.170913\pi\)
\(3\)	−2.41714	−	0.647670i	−1.39554	−	0.373933i	−0.518796	−	0.854898i	\(-0.673620\pi\)
							−0.876740	+	0.480965i	\(0.840286\pi\)
\(5\)	0.138660	+	0.0371537i	0.0620105	+	0.0166157i	0.289691	−	0.957120i	\(-0.406447\pi\)
							−0.227680	+	0.973736i	\(0.573114\pi\)
\(7\)	0.626167	−	0.167781i	0.236669	−	0.0634152i	−0.138535	−	0.990358i	\(-0.544239\pi\)
							0.375204	+	0.926942i	\(0.377573\pi\)
\(11\)	2.06781	+	2.06781i	0.623469	+	0.623469i	0.946417	−	0.322948i	\(-0.104674\pi\)
							−0.322948	+	0.946417i	\(0.604674\pi\)
\(13\)	2.14803	−	3.72050i	0.595757	−	1.03188i	−0.397682	−	0.917523i	\(-0.630185\pi\)
							0.993440	−	0.114358i	\(-0.0364812\pi\)
\(17\)	−1.99782	−	3.60676i	−0.484542	−	0.874768i
							\(19\)	0.179408	−	0.103581i	0.0411590	−	0.0237631i	−0.479279	−	0.877662i	\(-0.659102\pi\)
0.520438	+	0.853899i	\(0.325769\pi\)
\(23\)	1.70533	−	6.36436i	0.355585	−	1.32706i	−0.524162	−	0.851619i	\(-0.675621\pi\)
							0.879747	−	0.475443i	\(-0.157712\pi\)
\(29\)	1.54747	+	5.77523i	0.287357	+	1.07243i	0.947099	+	0.320940i	\(0.103999\pi\)
							−0.659742	+	0.751492i	\(0.729334\pi\)
\(31\)	3.69833	+	0.990964i	0.664239	+	0.177982i	0.575158	−	0.818042i	\(-0.304940\pi\)
							0.0890808	+	0.996024i	\(0.471607\pi\)
\(37\)	−10.5980	−	2.83972i	−1.74230	−	0.466847i	−0.759342	−	0.650692i	\(-0.774479\pi\)
							−0.982955	+	0.183845i	\(0.941146\pi\)
\(41\)	0.367490	+	0.367490i	0.0573923	+	0.0573923i	0.735220	−	0.677828i	\(-0.237079\pi\)
							−0.677828	+	0.735220i	\(0.737079\pi\)
\(43\)	−6.50342	+	0.839954i	−0.991762	+	0.128092i
							\(47\)	1.28573			0.187543			0.0937713	−	0.995594i	\(-0.470108\pi\)
0.0937713	+	0.995594i	\(0.470108\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.208.19	✓	256
17.13	even	4	inner	731.2.n.a.251.46	yes	256
43.6	even	3	inner	731.2.n.a.565.19	yes	256
731.608	even	12	inner	731.2.n.a.608.46	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.19	✓	256	1.1	even	1	trivial
731.2.n.a.251.46	yes	256	17.13	even	4	inner
731.2.n.a.565.19	yes	256	43.6	even	3	inner
731.2.n.a.608.46	yes	256	731.608	even	12	inner