Embedded newform 731.2.bh.a.13.2

• Label: 731.2.bh.a.13.2
• 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			13.2
Character	\(\chi\)	\(=\)	731.13
Dual form			731.2.bh.a.225.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.66167 - 0.607509i) q^{2} +(2.00780 + 0.0751267i) q^{3} +(4.91349 + 2.36621i) q^{4} +(2.09544 + 2.83922i) q^{5} +(-5.29847 - 1.41972i) q^{6} +(1.25466 + 4.68245i) q^{7} +(-7.37161 - 5.87867i) q^{8} +(1.03402 + 0.0774891i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 22 q^{3} + 208 q^{4} - 30 q^{5} - 12 q^{6} - 14 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.66167	−	0.607509i	−1.88209	−	0.429574i	−0.882887	−	0.469586i	\(-0.844403\pi\)
							−0.999199	+	0.0400119i	\(0.987260\pi\)
\(3\)	2.00780	+	0.0751267i	1.15921	+	0.0433744i	0.610062	−	0.792353i	\(-0.291144\pi\)
							0.549144	+	0.835728i	\(0.314954\pi\)
\(5\)	2.09544	+	2.83922i	0.937110	+	1.26974i	0.962840	+	0.270071i	\(0.0870471\pi\)
							−0.0257300	+	0.999669i	\(0.508191\pi\)
\(7\)	1.25466	+	4.68245i	0.474216	+	1.76980i	0.624361	+	0.781136i	\(0.285359\pi\)
							−0.150145	+	0.988664i	\(0.547974\pi\)
\(11\)	−1.05668	+	3.01982i	−0.318602	+	0.910511i	0.667535	+	0.744578i	\(0.267349\pi\)
							−0.986137	+	0.165933i	\(0.946936\pi\)
\(13\)	−0.233958	−	0.596116i	−0.0648883	−	0.165333i	0.894752	−	0.446564i	\(-0.147352\pi\)
							−0.959640	+	0.281231i	\(0.909257\pi\)
\(17\)	−0.542766	−	4.08722i	−0.131640	−	0.991298i
							\(19\)	1.46128	−	0.109508i	0.335240	−	0.0251228i	0.0939533	−	0.995577i	\(-0.470050\pi\)
0.241287	+	0.970454i	\(0.422431\pi\)
\(23\)	6.64322	−	1.25697i	1.38521	−	0.262095i	0.560734	−	0.827996i	\(-0.310519\pi\)
							0.824474	+	0.565900i	\(0.191471\pi\)
\(29\)	−0.264712	−	7.07457i	−0.0491557	−	1.31371i	−0.777665	−	0.628679i	\(-0.783596\pi\)
							0.728509	−	0.685036i	\(-0.240214\pi\)
\(31\)	2.17654	+	1.15034i	0.390918	+	0.206606i	0.651166	−	0.758936i	\(-0.274280\pi\)
							−0.260247	+	0.965542i	\(0.583804\pi\)
\(37\)	0.745016	−	2.78044i	0.122480	−	0.457101i	−0.877257	−	0.480020i	\(-0.840629\pi\)
							0.999737	+	0.0229192i	\(0.00729603\pi\)
\(41\)	−3.80064	−	6.04868i	−0.593560	−	0.944646i	−0.999527	−	0.0307578i	\(-0.990208\pi\)
							0.405967	−	0.913888i	\(-0.366935\pi\)
\(43\)	5.82106	−	3.01915i	0.887703	−	0.460416i
							\(47\)	−2.68934	−	1.29512i	−0.392280	−	0.188912i	0.227337	−	0.973816i	\(-0.426998\pi\)
−0.619617	+	0.784904i	\(0.712712\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bh.a.13.2	✓	1536
17.4	even	4	inner	731.2.bh.a.701.63	yes	1536
43.10	even	21	inner	731.2.bh.a.268.63	yes	1536
731.225	even	84	inner	731.2.bh.a.225.2	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bh.a.13.2	✓	1536	1.1	even	1	trivial
731.2.bh.a.225.2	yes	1536	731.225	even	84	inner
731.2.bh.a.268.63	yes	1536	43.10	even	21	inner
731.2.bh.a.701.63	yes	1536	17.4	even	4	inner