Embedded newform 731.2.n.a.208.11

• Label: 731.2.n.a.208.11
• Level: $731$
• Weight: $2$
• Character: 731.208
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.n (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(64\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			208.11
Character	\(\chi\)	\(=\)	731.208
Dual form			731.2.n.a.608.54

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.16713i q^{2} +(-3.18659 - 0.853843i) q^{3} -2.69647 q^{4} +(2.45506 + 0.657832i) q^{5} +(-1.85039 + 6.90576i) q^{6} +(2.85934 - 0.766159i) q^{7} +1.50935i q^{8} +(6.82721 + 3.94169i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	2.16713i		−	1.53240i	−0.642605	−	0.766198i	\(-0.722146\pi\)
							0.642605	−	0.766198i	\(-0.277854\pi\)
\(3\)	−3.18659	−	0.853843i	−1.83978	−	0.492967i	−0.840937	−	0.541133i	\(-0.817996\pi\)
							−0.998839	+	0.0481664i	\(0.984662\pi\)
\(5\)	2.45506	+	0.657832i	1.09794	+	0.294191i	0.761924	−	0.647666i	\(-0.224255\pi\)
							0.336012	+	0.941858i	\(0.390922\pi\)
\(7\)	2.85934	−	0.766159i	1.08073	−	0.289581i	0.325835	−	0.945426i	\(-0.394354\pi\)
							0.754895	+	0.655846i	\(0.227688\pi\)
\(11\)	−4.64082	−	4.64082i	−1.39926	−	1.39926i	−0.802184	−	0.597077i	\(-0.796329\pi\)
							−0.597077	−	0.802184i	\(-0.703671\pi\)
\(13\)	−0.964055	+	1.66979i	−0.267381	+	0.463117i	−0.968185	−	0.250237i	\(-0.919491\pi\)
							0.700804	+	0.713354i	\(0.252825\pi\)
\(17\)	−4.03817	−	0.832563i	−0.979401	−	0.201926i
							\(19\)	3.54102	−	2.04441i	0.812365	−	0.469019i	−0.0354112	−	0.999373i	\(-0.511274\pi\)
0.847777	+	0.530353i	\(0.177941\pi\)
\(23\)	0.0551197	−	0.205710i	0.0114933	−	0.0428934i	−0.959941	−	0.280202i	\(-0.909599\pi\)
							0.971434	+	0.237309i	\(0.0762652\pi\)
\(29\)	−1.74026	−	6.49473i	−0.323158	−	1.20604i	−0.916151	−	0.400833i	\(-0.868721\pi\)
							0.592993	−	0.805207i	\(-0.297946\pi\)
\(31\)	1.98698	+	0.532410i	0.356872	+	0.0956236i	0.432801	−	0.901490i	\(-0.357525\pi\)
							−0.0759285	+	0.997113i	\(0.524192\pi\)
\(37\)	3.31736	+	0.888885i	0.545371	+	0.146132i	0.520977	−	0.853571i	\(-0.325568\pi\)
							0.0243941	+	0.999702i	\(0.492234\pi\)
\(41\)	−0.466820	−	0.466820i	−0.0729050	−	0.0729050i	0.669714	−	0.742619i	\(-0.266417\pi\)
							−0.742619	+	0.669714i	\(0.766417\pi\)
\(43\)	1.44514	−	6.39622i	0.220381	−	0.975414i
							\(47\)	−12.3145			−1.79626			−0.898130	−	0.439730i	\(-0.855074\pi\)
−0.898130	+	0.439730i	\(0.855074\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.208.11	✓	256
17.13	even	4	inner	731.2.n.a.251.54	yes	256
43.6	even	3	inner	731.2.n.a.565.11	yes	256
731.608	even	12	inner	731.2.n.a.608.54	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.11	✓	256	1.1	even	1	trivial
731.2.n.a.251.54	yes	256	17.13	even	4	inner
731.2.n.a.565.11	yes	256	43.6	even	3	inner
731.2.n.a.608.54	yes	256	731.608	even	12	inner