Embedded newform 731.2.n.a.608.56

• Label: 731.2.n.a.608.56
• 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			608.56
Character	\(\chi\)	\(=\)	731.608
Dual form			731.2.n.a.208.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+2.24302i q^{2} +(-2.58532 + 0.692735i) q^{3} -3.03116 q^{4} +(-2.60177 + 0.697143i) q^{5} +(-1.55382 - 5.79894i) q^{6} +(2.01210 + 0.539141i) q^{7} -2.31291i q^{8} +(3.60594 - 2.08189i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			2.24302i			1.58606i	0.609184	+	0.793029i	\(0.291497\pi\)
							−0.609184	+	0.793029i	\(0.708503\pi\)
\(3\)	−2.58532	+	0.692735i	−1.49264	+	0.399951i	−0.910627	−	0.413230i	\(-0.864401\pi\)
							−0.582011	+	0.813181i	\(0.697734\pi\)
\(5\)	−2.60177	+	0.697143i	−1.16355	+	0.311772i	−0.788382	−	0.615185i	\(-0.789081\pi\)
							−0.375167	+	0.926957i	\(0.622414\pi\)
\(7\)	2.01210	+	0.539141i	0.760503	+	0.203776i	0.618172	−	0.786043i	\(-0.287874\pi\)
							0.142331	+	0.989819i	\(0.454540\pi\)
\(11\)	2.69807	−	2.69807i	0.813498	−	0.813498i	−0.171659	−	0.985156i	\(-0.554913\pi\)
							0.985156	+	0.171659i	\(0.0549127\pi\)
\(13\)	−3.41870	−	5.92137i	−0.948178	−	1.64229i	−0.749260	−	0.662276i	\(-0.769591\pi\)
							−0.198918	−	0.980016i	\(-0.563743\pi\)
\(17\)	3.91303	+	1.29930i	0.949050	+	0.315127i
							\(19\)	0.0486330	+	0.0280783i	0.0111572	+	0.00644160i	0.505568	−	0.862787i	\(-0.331283\pi\)
−0.494411	+	0.869228i	\(0.664616\pi\)
\(23\)	0.368975	+	1.37703i	0.0769367	+	0.287132i	0.993665	−	0.112378i	\(-0.0358469\pi\)
							−0.916729	+	0.399510i	\(0.869180\pi\)
\(29\)	1.14292	−	4.26543i	0.212235	−	0.792070i	−0.774887	−	0.632099i	\(-0.782193\pi\)
							0.987122	−	0.159971i	\(-0.0511399\pi\)
\(31\)	−8.96370	+	2.40182i	−1.60993	+	0.431379i	−0.948021	−	0.318207i	\(-0.896919\pi\)
							−0.661907	+	0.749586i	\(0.730253\pi\)
\(37\)	0.229473	−	0.0614872i	0.0377252	−	0.0101084i	−0.239907	−	0.970796i	\(-0.577117\pi\)
							0.277632	+	0.960687i	\(0.410450\pi\)
\(41\)	−2.14240	+	2.14240i	−0.334586	+	0.334586i	−0.854325	−	0.519739i	\(-0.826029\pi\)
							0.519739	+	0.854325i	\(0.326029\pi\)
\(43\)	4.95254	+	4.29795i	0.755255	+	0.655431i
							\(47\)	9.27352			1.35268			0.676341	−	0.736589i	\(-0.263565\pi\)
0.676341	+	0.736589i	\(0.263565\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.608.56	yes	256
17.4	even	4	inner	731.2.n.a.565.9	yes	256
43.36	even	3	inner	731.2.n.a.251.56	yes	256
731.208	even	12	inner	731.2.n.a.208.9	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.9	✓	256	731.208	even	12	inner
731.2.n.a.251.56	yes	256	43.36	even	3	inner
731.2.n.a.565.9	yes	256	17.4	even	4	inner
731.2.n.a.608.56	yes	256	1.1	even	1	trivial