Embedded newform 731.2.n.a.608.47

• Label: 731.2.n.a.608.47
• 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			608.47
Character	\(\chi\)	\(=\)	731.608
Dual form			731.2.n.a.208.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.62403i q^{2} +(-0.958554 + 0.256844i) q^{3} -0.637473 q^{4} +(3.71240 - 0.994733i) q^{5} +(-0.417122 - 1.55672i) q^{6} +(-0.920765 - 0.246718i) q^{7} +2.21278i q^{8} +(-1.74522 + 1.00760i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			1.62403i			1.14836i	0.818728	+	0.574181i	\(0.194680\pi\)
							−0.818728	+	0.574181i	\(0.805320\pi\)
\(3\)	−0.958554	+	0.256844i	−0.553421	+	0.148289i	−0.524682	−	0.851298i	\(-0.675816\pi\)
							−0.0287390	+	0.999587i	\(0.509149\pi\)
\(5\)	3.71240	−	0.994733i	1.66023	−	0.444858i	0.697782	−	0.716311i	\(-0.254171\pi\)
							0.962452	+	0.271452i	\(0.0875039\pi\)
\(7\)	−0.920765	−	0.246718i	−0.348016	−	0.0932507i	0.0805766	−	0.996748i	\(-0.474324\pi\)
							−0.428593	+	0.903498i	\(0.640991\pi\)
\(11\)	1.86389	−	1.86389i	0.561985	−	0.561985i	−0.367886	−	0.929871i	\(-0.619918\pi\)
							0.929871	+	0.367886i	\(0.119918\pi\)
\(13\)	−0.714120	−	1.23689i	−0.198061	−	0.343052i	0.749839	−	0.661621i	\(-0.230131\pi\)
							−0.947900	+	0.318569i	\(0.896798\pi\)
\(17\)	1.60840	+	3.79645i	0.390095	+	0.920775i
							\(19\)	3.40610	+	1.96651i	0.781413	+	0.451149i	0.836931	−	0.547309i	\(-0.184348\pi\)
−0.0555177	+	0.998458i	\(0.517681\pi\)
\(23\)	1.26739	+	4.72997i	0.264269	+	0.986266i	0.962696	+	0.270585i	\(0.0872171\pi\)
							−0.698427	+	0.715681i	\(0.746116\pi\)
\(29\)	−2.32060	+	8.66060i	−0.430925	+	1.60823i	0.319710	+	0.947515i	\(0.396414\pi\)
							−0.750635	+	0.660717i	\(0.770252\pi\)
\(31\)	5.90527	−	1.58231i	1.06062	−	0.284192i	0.313985	−	0.949428i	\(-0.398336\pi\)
							0.746633	+	0.665236i	\(0.231669\pi\)
\(37\)	4.18869	−	1.12236i	0.688616	−	0.184514i	0.102490	−	0.994734i	\(-0.467319\pi\)
							0.586126	+	0.810220i	\(0.300652\pi\)
\(41\)	−5.76895	+	5.76895i	−0.900958	+	0.900958i	−0.995519	−	0.0945606i	\(-0.969855\pi\)
							0.0945606	+	0.995519i	\(0.469855\pi\)
\(43\)	4.36535	−	4.89323i	0.665710	−	0.746210i
							\(47\)	4.05451			0.591411			0.295706	−	0.955279i	\(-0.404445\pi\)
0.295706	+	0.955279i	\(0.404445\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.608.47	yes	256
17.4	even	4	inner	731.2.n.a.565.18	yes	256
43.36	even	3	inner	731.2.n.a.251.47	yes	256
731.208	even	12	inner	731.2.n.a.208.18	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.18	✓	256	731.208	even	12	inner
731.2.n.a.251.47	yes	256	43.36	even	3	inner
731.2.n.a.565.18	yes	256	17.4	even	4	inner
731.2.n.a.608.47	yes	256	1.1	even	1	trivial