Embedded newform 731.2.be.a.59.1

• Label: 731.2.be.a.59.1
• 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			59.1
Character	\(\chi\)	\(=\)	731.59
Dual form			731.2.be.a.508.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.34206 - 1.47161i) q^{2} +(0.145057 + 0.102923i) q^{3} +(2.45183 + 5.09127i) q^{4} +(0.480201 + 1.66682i) q^{5} +(-0.188268 - 0.454520i) q^{6} +(-3.24749 + 1.34515i) q^{7} +(1.13066 - 10.0349i) q^{8} +(-0.980389 - 2.80179i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 20 q^{2} - 20 q^{3} - 28 q^{5} - 48 q^{6} - 48 q^{7} - 36 q^{8} - 44 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.34206	−	1.47161i	−1.65609	−	1.04059i	−0.939185	−	0.343412i	\(-0.888417\pi\)
							−0.716900	−	0.697176i	\(-0.754440\pi\)
\(3\)	0.145057	+	0.102923i	0.0837486	+	0.0594228i	0.619782	−	0.784774i	\(-0.287221\pi\)
							−0.536034	+	0.844197i	\(0.680078\pi\)
\(5\)	0.480201	+	1.66682i	0.214753	+	0.745423i	0.993067	+	0.117548i	\(0.0375034\pi\)
							−0.778315	+	0.627874i	\(0.783925\pi\)
\(7\)	−3.24749	+	1.34515i	−1.22743	+	0.508420i	−0.899766	−	0.436373i	\(-0.856263\pi\)
							−0.327669	+	0.944793i	\(0.606263\pi\)
\(11\)	0.0856603	−	1.52532i	0.0258275	−	0.459902i	−0.958672	−	0.284514i	\(-0.908168\pi\)
							0.984499	−	0.175388i	\(-0.0561180\pi\)
\(13\)	−2.20829	+	1.76105i	−0.612470	+	0.488428i	−0.879906	−	0.475147i	\(-0.842395\pi\)
							0.267437	+	0.963575i	\(0.413823\pi\)
\(17\)	2.89626	+	2.93457i	0.702445	+	0.711738i
							\(19\)	0.766863	−	2.19157i	0.175930	−	0.502780i	−0.821825	−	0.569740i	\(-0.807044\pi\)
0.997756	+	0.0669593i	\(0.0213298\pi\)
\(23\)	−0.307691	+	5.47895i	−0.0641581	+	1.14244i	0.786800	+	0.617207i	\(0.211736\pi\)
							−0.850959	+	0.525233i	\(0.823978\pi\)
\(29\)	1.09726	−	6.45803i	0.203757	−	1.19923i	−0.680686	−	0.732576i	\(-0.738318\pi\)
							0.884442	−	0.466650i	\(-0.154539\pi\)
\(31\)	−7.32038	−	1.24378i	−1.31478	−	0.223390i	−0.532559	−	0.846393i	\(-0.678770\pi\)
							−0.782220	+	0.623003i	\(0.785913\pi\)
\(37\)	0.778174	−	1.87868i	0.127931	−	0.308853i	−0.846916	−	0.531726i	\(-0.821544\pi\)
							0.974847	+	0.222873i	\(0.0715436\pi\)
\(41\)	−2.46174	−	3.46949i	−0.384459	−	0.541844i	0.577018	−	0.816732i	\(-0.304216\pi\)
							−0.961476	+	0.274888i	\(0.911359\pi\)
\(43\)	−3.34187	−	5.64198i	−0.509631	−	0.860393i
							\(47\)	0.512795	+	1.06483i	0.0747989	+	0.155321i	0.935015	−	0.354608i	\(-0.115386\pi\)
−0.860216	+	0.509930i	\(0.829671\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.be.a.59.1	✓	1536
17.15	even	8	inner	731.2.be.a.661.64	yes	1536
43.35	even	7	inner	731.2.be.a.637.64	yes	1536
731.508	even	56	inner	731.2.be.a.508.1	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.be.a.59.1	✓	1536	1.1	even	1	trivial
731.2.be.a.508.1	yes	1536	731.508	even	56	inner
731.2.be.a.637.64	yes	1536	43.35	even	7	inner
731.2.be.a.661.64	yes	1536	17.15	even	8	inner