Embedded newform 731.2.bh.a.13.5

• Label: 731.2.bh.a.13.5
• Level: $731$
• Weight: $2$
• Character: 731.13
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $1536$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.bh (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(1536\)
Relative dimension:	\(64\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			13.5
Character	\(\chi\)	\(=\)	731.13
Dual form			731.2.bh.a.225.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.40696 - 0.549373i) q^{2} +(2.99547 + 0.112083i) q^{3} +(3.68971 + 1.77687i) q^{4} +(0.0489901 + 0.0663792i) q^{5} +(-7.14841 - 1.91541i) q^{6} +(-0.797071 - 2.97471i) q^{7} +(-4.04437 - 3.22528i) q^{8} +(5.96869 + 0.447291i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 22 q^{3} + 208 q^{4} - 30 q^{5} - 12 q^{6} - 14 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{16}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.40696	−	0.549373i	−1.70198	−	0.388466i	−0.742411	−	0.669944i	\(-0.766318\pi\)
							−0.959567	+	0.281479i	\(0.909175\pi\)
\(3\)	2.99547	+	0.112083i	1.72944	+	0.0647110i	0.883460	−	0.468507i	\(-0.155208\pi\)
							0.845978	+	0.533218i	\(0.179018\pi\)
\(5\)	0.0489901	+	0.0663792i	0.0219090	+	0.0296857i	0.815415	−	0.578876i	\(-0.196509\pi\)
							−0.793506	+	0.608562i	\(0.791747\pi\)
\(7\)	−0.797071	−	2.97471i	−0.301265	−	1.12433i	−0.936113	−	0.351698i	\(-0.885604\pi\)
							0.634849	−	0.772636i	\(-0.281062\pi\)
\(11\)	−0.467988	+	1.33743i	−0.141104	+	0.403251i	−0.992649	−	0.121030i	\(-0.961380\pi\)
							0.851545	+	0.524281i	\(0.175666\pi\)
\(13\)	2.04457	+	5.20948i	0.567062	+	1.44485i	0.870581	+	0.492025i	\(0.163743\pi\)
							−0.303519	+	0.952825i	\(0.598162\pi\)
\(17\)	−1.91263	+	3.65265i	−0.463881	+	0.885898i
							\(19\)	4.47327	−	0.335225i	1.02624	−	0.0769059i	0.449041	−	0.893511i	\(-0.351766\pi\)
0.577197	+	0.816605i	\(0.304147\pi\)
\(23\)	5.09551	−	0.964123i	1.06249	−	0.201034i	0.374849	−	0.927086i	\(-0.377695\pi\)
							0.687639	+	0.726052i	\(0.258647\pi\)
\(29\)	0.129437	+	3.45928i	0.0240359	+	0.642372i	0.958296	+	0.285779i	\(0.0922523\pi\)
							−0.934260	+	0.356593i	\(0.883938\pi\)
\(31\)	3.16759	+	1.67412i	0.568916	+	0.300681i	0.726881	−	0.686764i	\(-0.240969\pi\)
							−0.157964	+	0.987445i	\(0.550493\pi\)
\(37\)	2.00702	−	7.49030i	0.329952	−	1.23140i	−0.579287	−	0.815123i	\(-0.696669\pi\)
							0.909239	−	0.416274i	\(-0.136664\pi\)
\(41\)	−3.29592	−	5.24543i	−0.514736	−	0.819198i	0.483692	−	0.875238i	\(-0.339295\pi\)
							−0.998429	+	0.0560402i	\(0.982153\pi\)
\(43\)	−5.46939	−	3.61742i	−0.834074	−	0.551652i
							\(47\)	−1.06650	−	0.513597i	−0.155564	−	0.0749158i	0.354482	−	0.935063i	\(-0.384657\pi\)
−0.510046	+	0.860147i	\(0.670372\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bh.a.13.5	✓	1536
17.4	even	4	inner	731.2.bh.a.701.60	yes	1536
43.10	even	21	inner	731.2.bh.a.268.60	yes	1536
731.225	even	84	inner	731.2.bh.a.225.5	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bh.a.13.5	✓	1536	1.1	even	1	trivial
731.2.bh.a.225.5	yes	1536	731.225	even	84	inner
731.2.bh.a.268.60	yes	1536	43.10	even	21	inner
731.2.bh.a.701.60	yes	1536	17.4	even	4	inner