Embedded newform 731.2.bh.a.13.20

• Label: 731.2.bh.a.13.20
• 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			13.20
Character	\(\chi\)	\(=\)	731.13
Dual form			731.2.bh.a.225.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.26218 - 0.288084i) q^{2} +(-2.65893 - 0.0994903i) q^{3} +(-0.291831 - 0.140538i) q^{4} +(2.58345 + 3.50045i) q^{5} +(3.32739 + 0.891572i) q^{6} +(-0.554513 - 2.06947i) q^{7} +(2.35224 + 1.87585i) q^{8} +(4.06842 + 0.304886i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 22 q^{3} + 208 q^{4} - 30 q^{5} - 12 q^{6} - 14 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.26218	−	0.288084i	−0.892496	−	0.203706i	−0.248406	−	0.968656i	\(-0.579907\pi\)
							−0.644090	+	0.764950i	\(0.722764\pi\)
\(3\)	−2.65893	−	0.0994903i	−1.53514	−	0.0574407i	−0.743626	−	0.668596i	\(-0.766895\pi\)
							−0.791511	+	0.611156i	\(0.790705\pi\)
\(5\)	2.58345	+	3.50045i	1.15535	+	1.56545i	0.764014	+	0.645200i	\(0.223226\pi\)
							0.391339	+	0.920247i	\(0.372012\pi\)
\(7\)	−0.554513	−	2.06947i	−0.209586	−	0.782187i	−0.988002	−	0.154438i	\(-0.950643\pi\)
							0.778416	−	0.627749i	\(-0.216023\pi\)
\(11\)	0.225899	−	0.645581i	0.0681110	−	0.194650i	−0.904797	−	0.425844i	\(-0.859977\pi\)
							0.972908	+	0.231194i	\(0.0742631\pi\)
\(13\)	0.356849	+	0.909238i	0.0989722	+	0.252177i	0.971781	−	0.235885i	\(-0.0757990\pi\)
							−0.872809	+	0.488063i	\(0.837704\pi\)
\(17\)	2.75691	−	3.06585i	0.668650	−	0.743578i
							\(19\)	−3.04257	+	0.228009i	−0.698014	+	0.0523089i	−0.419011	−	0.907981i	\(-0.637623\pi\)
−0.279003	+	0.960290i	\(0.590004\pi\)
\(23\)	−0.309494	+	0.0585594i	−0.0645339	+	0.0122105i	−0.218078	−	0.975931i	\(-0.569979\pi\)
							0.153544	+	0.988142i	\(0.450931\pi\)
\(29\)	−0.170031	−	4.54416i	−0.0315739	−	0.843830i	−0.921651	−	0.388020i	\(-0.873159\pi\)
							0.890077	−	0.455810i	\(-0.150650\pi\)
\(31\)	−1.50161	−	0.793622i	−0.269696	−	0.142539i	0.326953	−	0.945041i	\(-0.393978\pi\)
							−0.596650	+	0.802502i	\(0.703502\pi\)
\(37\)	0.643042	−	2.39986i	0.105715	−	0.394535i	−0.892710	−	0.450632i	\(-0.851199\pi\)
							0.998425	+	0.0560965i	\(0.0178654\pi\)
\(41\)	6.22564	+	9.90805i	0.972282	+	1.54738i	0.830158	+	0.557528i	\(0.188250\pi\)
							0.142124	+	0.989849i	\(0.454607\pi\)
\(43\)	4.59920	−	4.67412i	0.701371	−	0.712797i
							\(47\)	5.09981	+	2.45594i	0.743884	+	0.358236i	0.767128	−	0.641494i	\(-0.221685\pi\)
−0.0232439	+	0.999730i	\(0.507399\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bh.a.13.20	✓	1536
17.4	even	4	inner	731.2.bh.a.701.45	yes	1536
43.10	even	21	inner	731.2.bh.a.268.45	yes	1536
731.225	even	84	inner	731.2.bh.a.225.20	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bh.a.13.20	✓	1536	1.1	even	1	trivial
731.2.bh.a.225.20	yes	1536	731.225	even	84	inner
731.2.bh.a.268.45	yes	1536	43.10	even	21	inner
731.2.bh.a.701.45	yes	1536	17.4	even	4	inner