Embedded newform 731.2.y.a.4.11

• Label: 731.2.y.a.4.11
• Level: $731$
• Weight: $2$
• Character: 731.4
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $768$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.y (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(768\)
Relative dimension:	\(64\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			4.11
Character	\(\chi\)	\(=\)	731.4
Dual form			731.2.y.a.183.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.59352 - 1.27079i) q^{2} +(-2.77311 - 0.312454i) q^{3} +(0.479356 + 2.10019i) q^{4} +(3.68453 - 1.28927i) q^{5} +(4.02194 + 4.02194i) q^{6} +(-1.17641 - 1.17641i) q^{7} +(0.136369 - 0.283173i) q^{8} +(4.66772 + 1.06538i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 14 q^{3} + 104 q^{4} - 6 q^{5} - 36 q^{6} - 28 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.59352	−	1.27079i	−1.12679	−	0.898583i	−0.131102	−	0.991369i	\(-0.541851\pi\)
							−0.995686	+	0.0927857i	\(0.970423\pi\)
\(3\)	−2.77311	−	0.312454i	−1.60105	−	0.180396i	−0.734192	−	0.678942i	\(-0.762439\pi\)
							−0.866863	+	0.498547i	\(0.833867\pi\)
\(5\)	3.68453	−	1.28927i	1.64777	−	0.576580i	0.662724	−	0.748864i	\(-0.269400\pi\)
							0.985047	+	0.172284i	\(0.0551146\pi\)
\(7\)	−1.17641	−	1.17641i	−0.444640	−	0.444640i	0.448928	−	0.893568i	\(-0.351806\pi\)
							−0.893568	+	0.448928i	\(0.851806\pi\)
\(11\)	−3.72410	+	2.34001i	−1.12286	+	0.705538i	−0.959690	−	0.281060i	\(-0.909314\pi\)
							−0.163168	+	0.986598i	\(0.552171\pi\)
\(13\)	0.823078	+	0.396373i	0.228281	+	0.109934i	0.544528	−	0.838743i	\(-0.316709\pi\)
							−0.316248	+	0.948677i	\(0.602423\pi\)
\(17\)	2.04965	−	3.57756i	0.497114	−	0.867685i
							\(19\)	1.99852	−	0.456148i	0.458491	−	0.104648i	0.0129639	−	0.999916i	\(-0.495873\pi\)
0.445527	+	0.895268i	\(0.353016\pi\)
\(23\)	6.39465	−	4.01802i	1.33338	−	0.837816i	0.338702	−	0.940894i	\(-0.390012\pi\)
							0.994673	+	0.103078i	\(0.0328691\pi\)
\(29\)	0.137516	+	1.22049i	0.0255362	+	0.226640i	0.999998	+	0.00208575i	\(0.000663916\pi\)
							−0.974462	+	0.224554i	\(0.927908\pi\)
\(31\)	−0.138980	−	1.23348i	−0.0249615	−	0.221539i	0.975038	−	0.222036i	\(-0.0712704\pi\)
							−1.00000		0.000497060i	\(0.999842\pi\)
\(37\)	−1.43810	+	1.43810i	−0.236422	+	0.236422i	−0.815367	−	0.578945i	\(-0.803465\pi\)
							0.578945	+	0.815367i	\(0.303465\pi\)
\(41\)	0.924947	−	0.104216i	0.144452	−	0.0162759i	−0.0394422	−	0.999222i	\(-0.512558\pi\)
							0.183895	+	0.982946i	\(0.441130\pi\)
\(43\)	−4.29901	+	4.95161i	−0.655594	+	0.755114i
							\(47\)	−2.79826	−	12.2600i	−0.408168	−	1.78830i	−0.592646	−	0.805463i	\(-0.701917\pi\)
0.184478	−	0.982837i	\(-0.440941\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.y.a.4.11	✓	768
17.13	even	4	inner	731.2.y.a.47.54	yes	768
43.11	even	7	inner	731.2.y.a.140.54	yes	768
731.183	even	28	inner	731.2.y.a.183.11	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.y.a.4.11	✓	768	1.1	even	1	trivial
731.2.y.a.47.54	yes	768	17.13	even	4	inner
731.2.y.a.140.54	yes	768	43.11	even	7	inner
731.2.y.a.183.11	yes	768	731.183	even	28	inner