Embedded newform 731.2.j.a.135.4

• Label: 731.2.j.a.135.4
• Level: $731$
• Weight: $2$
• Character: 731.135
• 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			135.4
Character	\(\chi\)	\(=\)	731.135
Dual form			731.2.j.a.509.4

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.53253 q^{2} +(1.51123 + 0.872510i) q^{3} +4.41371 q^{4} +(3.60596 + 2.08190i) q^{5} +(-3.82724 - 2.20966i) q^{6} +(-0.435566 + 0.251474i) q^{7} -6.11280 q^{8} +(0.0225487 + 0.0390554i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 4 q^{2} + 116 q^{4} - 12 q^{8} + 70 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.53253			−1.79077			−0.895385	−	0.445293i	\(-0.853100\pi\)
							−0.895385	+	0.445293i	\(0.853100\pi\)
\(3\)	1.51123	+	0.872510i	0.872510	+	0.503744i	0.868182	−	0.496247i	\(-0.165289\pi\)
							0.00432871	+	0.999991i	\(0.498622\pi\)
\(5\)	3.60596	+	2.08190i	1.61264	+	0.931055i	0.988757	+	0.149534i	\(0.0477774\pi\)
							0.623879	+	0.781521i	\(0.285556\pi\)
\(7\)	−0.435566	+	0.251474i	−0.164629	+	0.0950484i	−0.580051	−	0.814580i	\(-0.696967\pi\)
							0.415422	+	0.909629i	\(0.363634\pi\)
\(11\)		−	1.59267i		−	0.480208i	−0.970747	−	0.240104i	\(-0.922818\pi\)
							0.970747	−	0.240104i	\(-0.0771816\pi\)
\(13\)	0.640573	+	1.10950i	0.177663	+	0.307721i	0.941080	−	0.338185i	\(-0.109813\pi\)
							−0.763417	+	0.645906i	\(0.776480\pi\)
\(17\)	4.03682	+	0.839103i	0.979072	+	0.203512i
							\(19\)	−0.250258	+	0.433460i	−0.0574132	+	0.0994426i	−0.893304	−	0.449454i	\(-0.851619\pi\)
0.835890	+	0.548897i	\(0.184952\pi\)
\(23\)	3.28971	+	1.89932i	0.685953	+	0.396035i	0.802094	−	0.597198i	\(-0.203719\pi\)
							−0.116141	+	0.993233i	\(0.537053\pi\)
\(29\)	−1.95615	+	1.12938i	−0.363247	+	0.209721i	−0.670504	−	0.741906i	\(-0.733922\pi\)
							0.307257	+	0.951627i	\(0.400589\pi\)
\(31\)	−1.63893	−	0.946238i	−0.294361	−	0.169949i	0.345546	−	0.938402i	\(-0.387694\pi\)
							−0.639907	+	0.768452i	\(0.721027\pi\)
\(37\)	5.81239	+	3.35579i	0.955552	+	0.551688i	0.894801	−	0.446465i	\(-0.147317\pi\)
							0.0607506	+	0.998153i	\(0.480651\pi\)
\(41\)		−	6.19755i		−	0.967895i	−0.875097	−	0.483948i	\(-0.839202\pi\)
							0.875097	−	0.483948i	\(-0.160798\pi\)
\(43\)	−6.53940	−	0.485992i	−0.997250	−	0.0741130i
							\(47\)	−6.41023			−0.935028			−0.467514	−	0.883986i	\(-0.654850\pi\)
−0.467514	+	0.883986i	\(0.654850\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.j.a.135.4	yes	128
17.16	even	2	inner	731.2.j.a.135.3	✓	128
43.36	even	3	inner	731.2.j.a.509.3	yes	128
731.509	even	6	inner	731.2.j.a.509.4	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.j.a.135.3	✓	128	17.16	even	2	inner
731.2.j.a.135.4	yes	128	1.1	even	1	trivial
731.2.j.a.509.3	yes	128	43.36	even	3	inner
731.2.j.a.509.4	yes	128	731.509	even	6	inner