Embedded newform 731.2.be.a.59.19

• Label: 731.2.be.a.59.19
• Level: $731$
• Weight: $2$
• Character: 731.59
• 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.be (of order \(56\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(1536\)
Relative dimension:	\(64\) over \(\Q(\zeta_{56})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{56}]$

Embedding invariants

Embedding label			59.19
Character	\(\chi\)	\(=\)	731.59
Dual form			731.2.be.a.508.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23009 - 0.772914i) q^{2} +(-1.98496 - 1.40841i) q^{3} +(0.0479473 + 0.0995635i) q^{4} +(-0.994433 - 3.45175i) q^{5} +(1.35310 + 3.26666i) q^{6} +(2.78266 - 1.15262i) q^{7} +(-0.307340 + 2.72772i) q^{8} +(0.965629 + 2.75961i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 20 q^{2} - 20 q^{3} - 28 q^{5} - 48 q^{6} - 48 q^{7} - 36 q^{8} - 44 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{4}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23009	−	0.772914i	−0.869802	−	0.546533i	0.0215895	−	0.999767i	\(-0.493127\pi\)
							−0.891392	+	0.453234i	\(0.850270\pi\)
\(3\)	−1.98496	−	1.40841i	−1.14602	−	0.813143i	−0.161232	−	0.986917i	\(-0.551547\pi\)
							−0.984786	+	0.173773i	\(0.944404\pi\)
\(5\)	−0.994433	−	3.45175i	−0.444724	−	1.54367i	−0.794754	−	0.606932i	\(-0.792400\pi\)
							0.350030	−	0.936739i	\(-0.386171\pi\)
\(7\)	2.78266	−	1.15262i	1.05175	−	0.435648i	0.211232	−	0.977436i	\(-0.432253\pi\)
							0.840515	+	0.541788i	\(0.182253\pi\)
\(11\)	−0.0828734	+	1.47570i	−0.0249873	+	0.444940i	0.960832	+	0.277131i	\(0.0893835\pi\)
							−0.985820	+	0.167809i	\(0.946331\pi\)
\(13\)	−1.01469	+	0.809187i	−0.281424	+	0.224428i	−0.754021	−	0.656851i	\(-0.771888\pi\)
							0.472597	+	0.881279i	\(0.343317\pi\)
\(17\)	−3.29000	−	2.48514i	−0.797942	−	0.602734i
							\(19\)	−2.74594	+	7.84745i	−0.629962	+	1.80033i	−0.0310453	+	0.999518i	\(0.509884\pi\)
−0.598917	+	0.800811i	\(0.704402\pi\)
\(23\)	−0.156703	+	2.79036i	−0.0326749	+	0.581830i	0.938767	+	0.344552i	\(0.111969\pi\)
							−0.971442	+	0.237277i	\(0.923745\pi\)
\(29\)	−0.0752499	+	0.442889i	−0.0139736	+	0.0822425i	−0.992000	−	0.126239i	\(-0.959709\pi\)
							0.978026	+	0.208482i	\(0.0668522\pi\)
\(31\)	4.58685	+	0.779338i	0.823823	+	0.139973i	0.564096	−	0.825709i	\(-0.309225\pi\)
							0.259727	+	0.965682i	\(0.416368\pi\)
\(37\)	−1.80264	+	4.35197i	−0.296353	+	0.715459i	0.703635	+	0.710561i	\(0.251559\pi\)
							−0.999988	+	0.00489766i	\(0.998441\pi\)
\(41\)	−0.540549	−	0.761833i	−0.0844196	−	0.118978i	0.771179	−	0.636619i	\(-0.219668\pi\)
							−0.855598	+	0.517641i	\(0.826810\pi\)
\(43\)	−1.48716	−	6.38658i	−0.226790	−	0.973944i
							\(47\)	1.94934	+	4.04784i	0.284340	+	0.590439i	0.993399	−	0.114711i	\(-0.0365942\pi\)
−0.709059	+	0.705149i	\(0.750880\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.be.a.59.19	✓	1536
17.15	even	8	inner	731.2.be.a.661.46	yes	1536
43.35	even	7	inner	731.2.be.a.637.46	yes	1536
731.508	even	56	inner	731.2.be.a.508.19	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.be.a.59.19	✓	1536	1.1	even	1	trivial
731.2.be.a.508.19	yes	1536	731.508	even	56	inner
731.2.be.a.637.46	yes	1536	43.35	even	7	inner
731.2.be.a.661.46	yes	1536	17.15	even	8	inner