Embedded newform 731.2.j.a.509.17

• Label: 731.2.j.a.509.17
• Level: $731$
• Weight: $2$
• Character: 731.509
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(64\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			509.17
Character	\(\chi\)	\(=\)	731.509
Dual form			731.2.j.a.135.17

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.28957 q^{2} +(-0.316022 + 0.182455i) q^{3} -0.337012 q^{4} +(3.24573 - 1.87392i) q^{5} +(0.407532 - 0.235289i) q^{6} +(2.23124 + 1.28820i) q^{7} +3.01374 q^{8} +(-1.43342 + 2.48276i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 4 q^{2} + 116 q^{4} - 12 q^{8} + 70 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28957			−0.911863			−0.455932	−	0.890015i	\(-0.650694\pi\)
							−0.455932	+	0.890015i	\(0.650694\pi\)
\(3\)	−0.316022	+	0.182455i	−0.182455	+	0.105341i	−0.588446	−	0.808537i	\(-0.700260\pi\)
							0.405991	+	0.913877i	\(0.366927\pi\)
\(5\)	3.24573	−	1.87392i	1.45154	−	0.838045i	0.452967	−	0.891527i	\(-0.350365\pi\)
							0.998569	+	0.0534824i	\(0.0170321\pi\)
\(7\)	2.23124	+	1.28820i	0.843328	+	0.486896i	0.858394	−	0.512991i	\(-0.171463\pi\)
							−0.0150662	+	0.999886i	\(0.504796\pi\)
\(11\)			2.13325i			0.643198i	0.946876	+	0.321599i	\(0.104220\pi\)
							−0.946876	+	0.321599i	\(0.895780\pi\)
\(13\)	−0.772737	+	1.33842i	−0.214319	+	0.371211i	−0.953062	−	0.302776i	\(-0.902086\pi\)
							0.738743	+	0.673987i	\(0.235420\pi\)
\(17\)	1.07622	+	3.98017i	0.261021	+	0.965333i
							\(19\)	0.244239	+	0.423034i	0.0560322	+	0.0970507i	0.892681	−	0.450689i	\(-0.148822\pi\)
−0.836649	+	0.547740i	\(0.815488\pi\)
\(23\)	1.32968	−	0.767692i	0.277258	−	0.160075i	−0.354923	−	0.934895i	\(-0.615493\pi\)
							0.632181	+	0.774820i	\(0.282160\pi\)
\(29\)	−1.70110	−	0.982131i	−0.315886	−	0.182377i	0.333671	−	0.942690i	\(-0.391713\pi\)
							−0.649558	+	0.760312i	\(0.725046\pi\)
\(31\)	−4.62044	+	2.66761i	−0.829855	+	0.479117i	−0.853803	−	0.520596i	\(-0.825710\pi\)
							0.0239480	+	0.999713i	\(0.492376\pi\)
\(37\)	−7.15392	+	4.13032i	−1.17610	+	0.679020i	−0.955109	−	0.296256i	\(-0.904262\pi\)
							−0.220989	+	0.975276i	\(0.570929\pi\)
\(41\)			5.61023i			0.876171i	0.898933	+	0.438086i	\(0.144343\pi\)
							−0.898933	+	0.438086i	\(0.855657\pi\)
\(43\)	6.10384	−	2.39648i	0.930827	−	0.365460i
							\(47\)	8.49880			1.23968			0.619839	−	0.784729i	\(-0.287198\pi\)
0.619839	+	0.784729i	\(0.287198\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.j.a.509.17	yes	128
17.16	even	2	inner	731.2.j.a.509.18	yes	128
43.6	even	3	inner	731.2.j.a.135.18	yes	128
731.135	even	6	inner	731.2.j.a.135.17	✓	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.j.a.135.17	✓	128	731.135	even	6	inner
731.2.j.a.135.18	yes	128	43.6	even	3	inner
731.2.j.a.509.17	yes	128	1.1	even	1	trivial
731.2.j.a.509.18	yes	128	17.16	even	2	inner