Embedded newform 731.2.n.a.608.51

• Label: 731.2.n.a.608.51
• Level: $731$
• Weight: $2$
• Character: 731.608
• 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			608.51
Character	\(\chi\)	\(=\)	731.608
Dual form			731.2.n.a.208.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.81631i q^{2} +(-1.48008 + 0.396586i) q^{3} -1.29900 q^{4} +(-3.73299 + 1.00025i) q^{5} +(-0.720326 - 2.68829i) q^{6} +(4.01589 + 1.07605i) q^{7} +1.27324i q^{8} +(-0.564718 + 0.326040i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			1.81631i			1.28433i	0.766567	+	0.642164i	\(0.221963\pi\)
							−0.766567	+	0.642164i	\(0.778037\pi\)
\(3\)	−1.48008	+	0.396586i	−0.854525	+	0.228969i	−0.659384	−	0.751806i	\(-0.729183\pi\)
							−0.195141	+	0.980775i	\(0.562516\pi\)
\(5\)	−3.73299	+	1.00025i	−1.66944	+	0.447326i	−0.964960	−	0.262396i	\(-0.915487\pi\)
							−0.704482	+	0.709722i	\(0.748821\pi\)
\(7\)	4.01589	+	1.07605i	1.51786	+	0.406710i	0.919038	−	0.394170i	\(-0.128968\pi\)
							0.598825	+	0.800880i	\(0.295634\pi\)
\(11\)	−3.86791	+	3.86791i	−1.16622	+	1.16622i	−0.183131	+	0.983088i	\(0.558623\pi\)
							−0.983088	+	0.183131i	\(0.941377\pi\)
\(13\)	2.52598	+	4.37512i	0.700580	+	1.21344i	0.968263	+	0.249934i	\(0.0804088\pi\)
							−0.267683	+	0.963507i	\(0.586258\pi\)
\(17\)	0.158622	−	4.12005i	0.0384715	−	0.999260i
							\(19\)	−0.664650	−	0.383736i	−0.152481	−	0.0880350i	0.421818	−	0.906680i	\(-0.361392\pi\)
−0.574299	+	0.818645i	\(0.694725\pi\)
\(23\)	−0.631637	−	2.35730i	−0.131705	−	0.491532i	0.868284	−	0.496067i	\(-0.165223\pi\)
							−0.999990	+	0.00453550i	\(0.998556\pi\)
\(29\)	−0.382983	+	1.42931i	−0.0711182	+	0.265417i	−0.992325	−	0.123656i	\(-0.960538\pi\)
							0.921207	+	0.389073i	\(0.127205\pi\)
\(31\)	1.87430	−	0.502217i	0.336634	−	0.0902008i	−0.0865422	−	0.996248i	\(-0.527582\pi\)
							0.423176	+	0.906047i	\(0.360915\pi\)
\(37\)	−11.0117	+	2.95056i	−1.81030	+	0.485070i	−0.995508	−	0.0946769i	\(-0.969818\pi\)
							−0.814797	+	0.579747i	\(0.803152\pi\)
\(41\)	4.05589	−	4.05589i	0.633423	−	0.633423i	−0.315502	−	0.948925i	\(-0.602173\pi\)
							0.948925	+	0.315502i	\(0.102173\pi\)
\(43\)	3.35701	−	5.63298i	0.511939	−	0.859022i
							\(47\)	1.06126			0.154800			0.0774000	−	0.997000i	\(-0.475338\pi\)
0.0774000	+	0.997000i	\(0.475338\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.608.51	yes	256
17.4	even	4	inner	731.2.n.a.565.14	yes	256
43.36	even	3	inner	731.2.n.a.251.51	yes	256
731.208	even	12	inner	731.2.n.a.208.14	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.14	✓	256	731.208	even	12	inner
731.2.n.a.251.51	yes	256	43.36	even	3	inner
731.2.n.a.565.14	yes	256	17.4	even	4	inner
731.2.n.a.608.51	yes	256	1.1	even	1	trivial