Embedded newform 731.2.n.a.608.55

• Label: 731.2.n.a.608.55
• 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			608.55
Character	\(\chi\)	\(=\)	731.608
Dual form			731.2.n.a.208.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+2.21617i q^{2} +(-0.0692057 + 0.0185436i) q^{3} -2.91143 q^{4} +(-1.25645 + 0.336665i) q^{5} +(-0.0410959 - 0.153372i) q^{6} +(0.0334492 + 0.00896268i) q^{7} -2.01988i q^{8} +(-2.59363 + 1.49743i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			2.21617i			1.56707i	0.621347	+	0.783536i	\(0.286586\pi\)
							−0.621347	+	0.783536i	\(0.713414\pi\)
\(3\)	−0.0692057	+	0.0185436i	−0.0399559	+	0.0107062i	−0.278742	−	0.960366i	\(-0.589917\pi\)
							0.238786	+	0.971072i	\(0.423251\pi\)
\(5\)	−1.25645	+	0.336665i	−0.561901	+	0.150561i	−0.528580	−	0.848884i	\(-0.677275\pi\)
							−0.0333215	+	0.999445i	\(0.510609\pi\)
\(7\)	0.0334492	+	0.00896268i	0.0126426	+	0.00338757i	0.265135	−	0.964211i	\(-0.414584\pi\)
							−0.252492	+	0.967599i	\(0.581250\pi\)
\(11\)	1.16878	−	1.16878i	0.352399	−	0.352399i	−0.508602	−	0.861001i	\(-0.669838\pi\)
							0.861001	+	0.508602i	\(0.169838\pi\)
\(13\)	0.160715	+	0.278367i	0.0445743	+	0.0772050i	0.887452	−	0.460901i	\(-0.152474\pi\)
							−0.842877	+	0.538106i	\(0.819140\pi\)
\(17\)	−4.09840	+	0.450672i	−0.994008	+	0.109304i
							\(19\)	−1.99124	−	1.14964i	−0.456821	−	0.263746i	0.253886	−	0.967234i	\(-0.418291\pi\)
−0.710707	+	0.703489i	\(0.751625\pi\)
\(23\)	−1.55573	−	5.80606i	−0.324392	−	1.21065i	−0.914921	−	0.403632i	\(-0.867748\pi\)
							0.590529	−	0.807016i	\(-0.298919\pi\)
\(29\)	1.16844	−	4.36068i	0.216974	−	0.809758i	−0.768488	−	0.639864i	\(-0.778991\pi\)
							0.985462	−	0.169894i	\(-0.0543425\pi\)
\(31\)	5.50658	−	1.47548i	0.989011	−	0.265005i	0.272176	−	0.962248i	\(-0.412257\pi\)
							0.716835	+	0.697243i	\(0.245590\pi\)
\(37\)	1.17164	−	0.313940i	0.192616	−	0.0516114i	−0.161221	−	0.986918i	\(-0.551543\pi\)
							0.353837	+	0.935307i	\(0.384877\pi\)
\(41\)	−8.15331	+	8.15331i	−1.27333	+	1.27333i	−0.329004	+	0.944329i	\(0.606713\pi\)
							−0.944329	+	0.329004i	\(0.893287\pi\)
\(43\)	5.85320	+	2.95635i	0.892605	+	0.450839i
							\(47\)	1.08884			0.158824			0.0794121	−	0.996842i	\(-0.474696\pi\)
0.0794121	+	0.996842i	\(0.474696\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.608.55	yes	256
17.4	even	4	inner	731.2.n.a.565.10	yes	256
43.36	even	3	inner	731.2.n.a.251.55	yes	256
731.208	even	12	inner	731.2.n.a.208.10	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.10	✓	256	731.208	even	12	inner
731.2.n.a.251.55	yes	256	43.36	even	3	inner
731.2.n.a.565.10	yes	256	17.4	even	4	inner
731.2.n.a.608.55	yes	256	1.1	even	1	trivial